Novel Lights Sources Beyond Free Electron Lasers (Particle Acceleration and Detection) [1st ed. 2022] 9783031042812, 9783031042829, 3031042816

Table of contents :
Preface
Contents
Acronyms
1 Introduction
References
2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium
2.1 Radiation from Relativistic Charges
2.1.1 Classical Description
2.1.2 Quantum Description
2.1.3 Quasi-classical Description of Radiation Emission
2.2 Bremsstrahlung
2.2.1 The Elementary Process of Bremsstrahlung
2.2.2 Coherent Bremsstrahlung
2.3 Synchrotron Radiation
2.4 Undulator Radiation
2.4.1 General Formalism
2.4.2 Spectral Distribution in the Forward Direction
References
3 Light Sources at High Photon Energies
3.1 Main Characteristics of Light Sources
3.2 Synchrotron Radiation Light Sources
3.3 Undulators and Wigglers
3.4 X-Ray Free-Electron Lasers
3.5 Alternative Schemes for Short-Wavelength Light Sources
3.5.1 Compton Scattering γ-Ray Light Source
3.5.2 Gamma Factory
3.5.3 Extremely Brilliant GeV γ-Rays From a Two-Stage Laser-Plasma Accelerator
References
4 Channeling Phenomenon and Channeling Radiation
4.1 Channeling of Ultra-Relativistic Particles
4.1.1 Crystallographic Axes and Planes
4.1.2 Continuous Potential Model
4.1.3 Positron Versus Electron Channeling
4.1.4 Classical Versus Quantum Description
4.2 Channeling Radiation: Basic Concepts
4.3 Radiation Damping
4.4 Overview of Numerical Approaches to Simulate Channeling Phenomenon
4.5 Atomistic Modeling of the Related Phenomena
4.5.1 Methodology
4.5.2 Statistical Analysis of Trajectories
4.5.3 Calculation of Spectral Distribution of Emitted Radiation
References
5 Radiation Emission in Bent Crystals
5.1 Channeling of Particles in Bent Crystals: Basic Concepts
5.2 Experimental Studies of the Phenomena
5.3 Results of Atomistic Simulations
5.3.1 Radiation Emission by 855 MeV Electrons in Bent Silicon (110) Crystals
5.3.2 Radiation Emission by 855 MeV Positrons in a Tungsten Crystal
5.3.3 Multi-GeV Electron and Positron Channeling in Bent Silicon Crystals
5.3.4 Electron Channeling and Radiation in Quasi-Mosaic Silicon Crystal
References
6 Crystalline Undulators
6.1 Crystalline Undulator: Basic Concepts, Feasibility
6.2 Positron and Electron-Based CUs: Illustrative Material
6.2.1 CU Radiation by Multi-GeV Electrons and Positrons
6.2.2 Specific Features of Radiation Emission by Electrons in Crystalline Undulators
6.2.3 Channeling and Radiation Emission in Diamond Hetero-Crystals
6.2.4 Channeling and Radiation Emission in SASP Periodically Bent Crystals
6.2.5 Experiments with SASP Periodically Bent Crystals
6.3 Stack of Periodically Bent Crystals
6.4 Brilliance of the CU Radiation
6.5 Atomistic Simulations of the CU Light Sources
References
7 Emission of Coherent CU Radiation
7.1 Introduction
7.2 ``Naive'' Approach to the Gamma-Laser Based on a Crystalline Undulator
7.2.1 Crude Estimate of the Gain
7.2.2 One-Crystal Amplifier
7.3 Beam Demodulation
7.4 Pre-bunching and Super-Radiance in CU
7.5 Brilliance of the CU-Based LSs
References
8 Conclusion
References
Appendix Index
Index

Citation preview

Particle Acceleration and Detection

Andrei Korol Andrey V. Solov’yov

Novel Lights Sources Beyond Free Electron Lasers

Particle Acceleration and Detection Series Editors Alexander Chao, SLAC, Stanford University, Menlo Park, CA, USA Katsunobu Oide, KEK, High Energy Accelerator Research Organization, Tsukuba, Japan Werner Riegler, Detector Group, CERN, Genèva, Switzerland Vladimir Shiltsev, Accelerator Physics Center, Fermi National Accelerator Lab, Batavia, IL, USA Frank Zimmermann, BE Department, ABP Group, CERN, Genèva, Switzerland

The series “Particle Acceleration and Detection” is devoted to monograph texts dealing with all aspects of particle acceleration and detection research and advanced teaching. The scope also includes topics such as beam physics and instrumentation as well as applications. Presentations should strongly emphasize the underlying physical and engineering sciences. Of particular interest are – contributions which relate fundamental research to new applications beyond the immediate realm of the original field of research – contributions which connect fundamental research in the aforementioned fields to fundamental research in related physical or engineering sciences – concise accounts of newly emerging important topics that are embedded in a broader framework in order to provide quick but readable access of very new material to a larger audience The books forming this collection will be of importance to graduate students and active researchers alike.

More information about this series at https://link.springer.com/bookseries/5267

Andrei Korol · Andrey V. Solov’yov

Novel Lights Sources Beyond Free Electron Lasers

Andrei Korol MBN Research Center Frankfurt am Main Hessen, Germany

Andrey V. Solov’yov MBN Research Center Frankfurt am Main Hessen, Germany

ISSN 1611-1052 ISSN 2365-0877 (electronic) Particle Acceleration and Detection ISBN 978-3-031-04281-2 ISBN 978-3-031-04282-9 (eBook) https://doi.org/10.1007/978-3-031-04282-9 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Development of coherent radiation sources for a wavelength λ below 1 angstrom (i.e., in the hard X-ray and gamma-ray ranges) is a challenging goal of modern physics. Sub-angstrom wavelength powerful spontaneous and, especially, coherent radiation light sources will have many applications in basic science, technology, and medicine. In particular, they may have a revolutionary impact on nuclear and solid-state physics as well as on life sciences. In this book, we discuss possibilities and perspectives for designing and practical realization of novel intensive gamma-ray Crystal-based Light Sources (CLS) operating at photon energies from 102 keV and above that can be constructed through exposure of oriented crystals—linear, bent, and periodically bent, to beams of ultrarelativistic positrons and electrons. CLSs can generate radiation in the photon energy range where the technologies based on the particle motion in the fields of permanent magnets become inefficient or incapable. One of the case studies presented in the book concerns a tunable CLS based on a periodically bent crystal. It is demonstrated that peak brilliance of radiation in the photon energy range 102 keV–102 MeV by currently available positron beams channeling in the crystal is comparable to or even higher than that achievable in conventional synchrotrons in the much lower photon energy range. The radiation intensity exceeds the values provided by LSs based on Compton scattering and can be made higher than the values predicted in the Gamma Factory proposal in CERN. By propagating a pre-bunched beam, the brilliance can be boosted by orders of magnitude reaching the values of spontaneous emission from the state-of-the-art magnetic undulators and being comparable with the values achievable at the X-Ray Free-Electron Laser (XFEL) facilities that operate in much lower photon energy range. Important is that by tuning the bending amplitude and period, one can maximize the brilliance for given parameters of the beam and/or chosen type of a crystalline medium. The size and the cost of CLSs are orders of magnitude less than that of modern LSs based on the permanent magnets. This opens many practical possibilities for the efficient generation of gamma-rays with various intensities and in various ranges of wavelength by means of the existing and to-be-constructed beam-lines. v

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Preface

We believe that the ideas, estimates, and conclusions presented in our book will be of interest to a broad audience of research. Construction of novel CLSs is an extremely challenging task which constitutes a highly interdisciplinary field. To accomplish this task, a broad collaboration is needed of research groups with different but complementary expertise, such as material science, nanotechnology, particle beam and accelerator physics, radiation physics, X-ray diffraction imaging, acoustics, solid-state physics, structure determination, advanced computational modeling, high-performance computing as well as industries specializing in manufacturing of crystalline structures and in design and construction of complete accelerator systems. In the past decade and a half, the studies of various phenomena relevant to the CLS field of research have been supported by several nationally funded projects (Germany, Denmark, Italy, South Africa). A unique experience has been gained within several EU-supported collaborative projects (FP6-PECU, FP7-CUTE, H2020PEARL, H2020-N-LIGHT, H2020-TECHNO-CLS) focused on the interdisciplinary and highly international R & D activities toward advancing the technologies for manufacturing of periodically bent crystals, their characterization, experimental investigation of the relevant channeling phenomena as well as further development of the theoretical methods and numerical algorithms. When developed, CLSs will have many applications in basic sciences, technology, and medicine. They could be used for disposing of nuclear waste, nuclear medicine, imaging techniques (positron tomography, nuclear chemistry), visualization of electron dynamics in atoms and molecules, production of rare isotopes, photo-induced nuclear reactions, medical applications (personalized medicine), and nondestructive imaging of complex molecular systems (proteins, viruses, nanodevices) with the resolution allowing detection of the positions of the nuclei and gamma-ray material diagnostics. We are grateful to Gennady Sushko who has contributed a lot to the developing of the computer codes used in the simulations and with whom many of the results presented here have been obtained, to Vadim Ivanov for his fruitful long-lasting collaboration, and to Roman Polozkov, Victor Bezchastnov, Alexander Pavlov, and Viktar Haurilovetz for their contribution to collaborative work within various projects supported by different agencies. We express our gratitude to Hartmut Backe, Werner Lauth, Ulrik Uggerhøj, Vincenzo Guidi, and Simon Connell for numerous stimulating and clarifying discussions as well as their support of our theoretical activity in the field and eagerness to carry out experimental investigations. Financial support from Deutsche Forschungsgemeinschaft (DFG), Alexander von Humboldt Foundation, and European Commission, granted to us at various stages of the research, is gratefully acknowledged. Frankfurt am Main, Germany

Andrei Korol Andrey V. Solov’yov

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 9

2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Radiation from Relativistic Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Classical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Quantum Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Quasi-classical Description of Radiation Emission . . . . . . . . 2.2 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Elementary Process of Bremsstrahlung . . . . . . . . . . . . . . 2.2.2 Coherent Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Undulator Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Spectral Distribution in the Forward Direction . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 17 18 20 21 21 23 25 27 27 32 38

3 Light Sources at High Photon Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Main Characteristics of Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Synchrotron Radiation Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Undulators and Wigglers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 X-Ray Free-Electron Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Alternative Schemes for Short-Wavelength Light Sources . . . . . . . . 3.5.1 Compton Scattering γ -Ray Light Source . . . . . . . . . . . . . . . . 3.5.2 Gamma Factory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Extremely Brilliant GeV γ -Rays From a Two-Stage Laser-Plasma Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 43 45 46 46 49 51 53

vii

viii

Contents

4 Channeling Phenomenon and Channeling Radiation . . . . . . . . . . . . . . . 4.1 Channeling of Ultra-Relativistic Particles . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Crystallographic Axes and Planes . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Continuous Potential Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Positron Versus Electron Channeling . . . . . . . . . . . . . . . . . . . . 4.1.4 Classical Versus Quantum Description . . . . . . . . . . . . . . . . . . 4.2 Channeling Radiation: Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Radiation Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Overview of Numerical Approaches to Simulate Channeling Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Atomistic Modeling of the Related Phenomena . . . . . . . . . . . . . . . . . 4.5.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Statistical Analysis of Trajectories . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Calculation of Spectral Distribution of Emitted Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 58 59 63 65 66 69

5 Radiation Emission in Bent Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Channeling of Particles in Bent Crystals: Basic Concepts . . . . . . . . . 5.2 Experimental Studies of the Phenomena . . . . . . . . . . . . . . . . . . . . . . . 5.3 Results of Atomistic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Radiation Emission by 855 MeV Electrons in Bent Silicon (110) Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Radiation Emission by 855 MeV Positrons in a Tungsten Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Multi-GeV Electron and Positron Channeling in Bent Silicon Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Electron Channeling and Radiation in Quasi-Mosaic Silicon Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 109 112

6 Crystalline Undulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Crystalline Undulator: Basic Concepts, Feasibility . . . . . . . . . . . . . . 6.2 Positron and Electron-Based CUs: Illustrative Material . . . . . . . . . . . 6.2.1 CU Radiation by Multi-GeV Electrons and Positrons . . . . . . 6.2.2 Specific Features of Radiation Emission by Electrons in Crystalline Undulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Channeling and Radiation Emission in Diamond Hetero-Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Channeling and Radiation Emission in SASP Periodically Bent Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Experiments with SASP Periodically Bent Crystals . . . . . . . 6.3 Stack of Periodically Bent Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Brilliance of the CU Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Atomistic Simulations of the CU Light Sources . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 137 139 139

71 73 74 79 87 97

112 118 122 128 132

142 146 149 152 154 157 164 176

Contents

7 Emission of Coherent CU Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 “Naive” Approach to the Gamma-Laser Based on a Crystalline Undulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Crude Estimate of the Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 One-Crystal Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Beam Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Pre-bunching and Super-Radiance in CU . . . . . . . . . . . . . . . . . . . . . . . 7.5 Brilliance of the CU-Based LSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

181 181 183 184 187 193 198 200 205

8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Acronyms

AW BC BrS BW ChR CLS CU CUL CUR FEL LALP LC LS MD PBCh PBCr SASE SASP UR

Acoustic Wave Bent Crystal Bremsstrahlung Band Width Channeling Radiation Crystal-based Light Source Crystalline Undulator Crystalline Undulator Laser Crystalline Undulator Radiation Free-Electron Laser Large-Amplitude Large-Period Linear Crystal Light Source Molecular Dynamics Periodically Bent Channel Periodically Bent Crystal Self-Amplified Spontaneous Emission Small-Amplitude Short-Period Undulator Radiation

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Chapter 1

Introduction

Development of light sources (LS) operating within the wavelengths λ well below 1 angstrom (the corresponding energies of the photons are E ph  10 keV) is a challenging goal of modern physics. Sub-angstrom wavelength, high brilliance, and tunable LSs will have a broad range of exciting potential cutting-edge applications. These applications include exploring elementary particles, probing nuclear structures and photonuclear physics, and examining quantum processes, which rely heavily on gamma-ray sources in the MeV to GeV range [1–3]. Modern X-ray Free-ElectronLaser (XFEL) can generate X-rays with wavelengths λ ∼ 1 Å [4–17]. However, no laser system has yet been commissioned for lower wavelengths due to the limitations of permanent magnet and accelerator technologies.1 Existing synchrotron facilities provide radiation of shorter wavelengths but of the intensity that is orders of magnitude less and which rapidly falls off as λ decreases [11, 19–21]. Therefore, to create a powerful LS in the range λ  1 Å, new approaches and technologies are needed. In this book, we discuss possibilities and perspectives for designing and practical realization of novel gamma-ray Crystal-based LSs (CLS) operating at photon energies E ph  102 keV and above that can be constructed through the exposure of oriented crystals (linear Crystals—LC, Bent Crystals—BC, Periodically Bent Crystals—PBC) to beams of ultra-relativistic charged particles. CLSs include Channeling Radiation (ChR) emitters, crystalline synchrotron radiation emitters, crystalline Bremsstrahlung (BrS) radiation emitters, Crystalline Undulators (CU), and stacks of CUs. This interdisciplinary research field combines theory, computational modeling, beam manipulation, design, manufacture, and experimental verification of high-quality crystalline samples and subsequent characterization of their emitted radiation as novel LSs. In an exemplary case study, we estimate the characteristics (brilliance, intensity) of radiation emitted in CU-LS by positron beams available at present. It is demonstrated that the peak brilliance of the CU Radiation (CUR) at E ph = 10−1 − 102 MeV is comparable to or even higher than that achievable in conventional synchrotrons but for much lower photon energies. The intensity of radiation from CU-LSs greatly exceeds that available in the laser-Compton scattering 1

In a recent study [18], a possibility of constructing an XFEL that operates at the wavelengths down to a picometer range is discussed. © Springer Nature Switzerland AG 2022 A. Korol and A. V. Solov’yov, Novel Lights Sources Beyond Free Electron Lasers, Particle Acceleration and Detection, https://doi.org/10.1007/978-3-031-04282-9_1

1

2

1 Introduction

Fig. 1.1 Selected examples of the novel CLSs: a bent crystal, b periodically bent crystal, and c a stack of periodically bent crystals. Black circles and lines mark atoms of crystallographic planes, wavy curves show trajectories of the channeling particles, and shadowed areas refer to the emitted radiation

LSs and can be made higher than predicted in the Gamma Factory proposal to CERN [22–24]. The brilliance can be boosted by orders of magnitude through the process of superradiance by a pre-bunched beam. We show that the brilliance of superradiant CUR can be comparable with the values achievable at the current XFEL facilities which operate in a much lower photon energy range. CLSs can generate radiation in the photon energy range where the technologies based on the charged particles’ motion in the fields of permanent magnets become inefficient or incapable. The limitations of conventional LS are overcome by exploiting very strong crystalline fields that can be as high as ∼1010 V/cm, which is equivalent to a magnetic field of 3000 Tesla while modern superconducting magnets provide 1–10 Tesla [25]. The orientation of a crystal along the beam enhances significantly the strength of the particle interaction with the crystal due to strongly correlated scattering from lattice atoms. This allows for the guided motion of particles through crystals of different geometry and for the enhancement of radiation. Examples of CLSs are shown in Fig. 1.1 [1]. The synchrotron radiation is emitted by ultra-relativistic projectiles propagating in the channeling regime through a bent crystal, panel (a). A CU, panel (b), contains a periodically bent crystal and a beam of channeling particles which emit CUR following the periodicity of the bending [26–28]. A CU-based LS can generate photons of E ph = 102 keV – 101 GeV range (corresponding to λ from 0.1 to 10−6 Å). Under certain conditions, CU can become a source of the coherent light within the range λ = 10−2 − 10−1 Å [27, 29, 30]. An LS based on a stack of CUs is shown in panel (c) [31]. Practical realization of CLSs often relies on the channeling effect. The basic phenomenon of channeling is in a large distance which a projectile particle penetrates moving along a crystallographic plane or axis and experiencing collective action

1 Introduction

3

Fig. 1.2 Left: Magnetic undulator for the European XFEL. Right: A Si1−x Gex superlattice CU (the upper layer) build atop the silicon substrate (the lower layer) with the face normal to the 100 crystallographic direction. In the superlattice, the (110) planes are bent periodically. The picture courtesy of J. L. Hansen, A. Nylandsted, and U. Uggerhøj (University of Aarhus). The whole figure is adapted from Ref. [37]

of the electrostatic fields of the lattice atoms [32] (see also reviews [33–35] and references therein). A typical distance covered by a particle before it stops moving in the channeling mode due to uncorrelated collisions is called the dechanneling length, L d . This quantity depends on the crystal type and its orientation, on the type of channeling motion, planar or axial, and on the energy and charge of an ultrarelativistic particle. In the planar regime, positrons channel in between two adjacent planes, whereas electrons propagate in the vicinity of a plane thus experiencing more frequent collisions. As a result, L d for electrons is much less than for positrons. To ensure enhancement of the emitted radiation due to the dechanneling effect, the crystal length L must be chosen as L ∼ L d [26–28]. The motion of a projectile and the radiation emission in bent and periodically bent crystals are similar to those in magnet-based synchrotrons and undulators. The main difference is that in the latter, the particles and photons move in vacuum, whereas in crystals they propagate in medium, thus leading to a number of limitations for the crystal length, bending curvature, and beam energy. However, the crystalline fields are so strong that they steer ultra-relativistic particles more effectively than the most advanced magnets. Strong fields bring bending radius in bent crystals down to the cm range and bending period λu in periodically bent crystals to the hundred or even ten microns range. These values are orders of magnitude smaller than those achievable with magnets [4]. As a result, the radiators can be miniaturized thus lowering dramatically the cost of CLSs as compared to that of conventional LSs. Figure 1.2 matches the magnetic undulator for the European XFEL with the CU manufactured in University of Aarhus and used in recent experiments [36]. Modern accelerator facilities make available intensive electron and positron beams of high energies, from the sub-GeV up to hundreds of GeV. These energies combined with large bending curvature achievable in crystals provide a possibility to consider novel CLSs of the synchrotron type (continuous spectrum radiation) and of the undulator type (monochromatic radiation) of the energy range up to tens of GeV. Manufacture of high-quality bent and periodically bent crystals is at the edge of current technologies.

4

1 Introduction

A number of theoretical and experimental studies of the channeling phenomenon in oriented linear crystals have been carried out (see, e.g., a review [38]). A channeling particle emits intensive channeling radiation (ChR), which was predicted theoretically [39] and shortly after observed experimentally [40]. Since then, there has been extensive theoretical and experimental investigation of ChR. The energy of emitted photons E ph scales with the beam energy as ε3/2 and thus can be varied by changing the latter. For example, by propagating electrons of moderate energies, ε = 10 − 40 MeV, through a linear crystal it is possible to generate ChR with photon energy E ph = 10 − 80 keV [41, 42]. This range can be achieved in magnetic undulators but with much higher beam energy. High-quality electron beams of (tunable) energies within the tens of MeV range are available at many facilities. Hence, it has become possible to consider ChR from linear crystals as a new powerful LS in the X-ray range [41]. In the gamma-range, ChR can be emitted by higher energy ε  102 MeV beams. However, modern accelerator facilities operate at a fixed value of ε (or, at several fixed values) [43–46]. This narrows the options for tuning the ChR parameters, in particular, the wavelength. Hence, the corresponding CLS lack the tunability option. From this viewpoint, the use of bent and, especially, periodically bent crystals can become an alternative as they provide tunable emission in the hard X- and gamma-ray range. Strong crystalline fields give rise to channeling in a bent crystal. Since its prediction [47] and experimental support [48], the idea to deflect high-energy beams of charged particles by means of bent crystals has attracted a lot of attention [34, 38]. The experiments have been carried with ultra-relativistic protons, ions, positrons, electrons, and π − -mesons [36, 49–64]. Steering of highly energetic electrons and positrons in bent crystals with small bending radius R gives rise to intensive synchrotron radiation with E ph  100 MeV [65, 66]. The parameters of radiation can be tuned by varying R within the range 100 − 102 cm [67, 67–70]. Even more tunable is a CU-LS. In this system, CUR and ChR are emitted in distinctly different photon energy ranges so that CUR is not affected by ChR. The intensity, photon energy, and line width of CUR can be varied and tuned by changing ε, bending amplitude a and period λu , type of crystal, its length, and detector aperture [30]. Since introducing the concept of CU, major theoretical studies have been devoted to the large-amplitude large-period bending λu  a > d [26–28]. In this regime, a projectile follows the shape of periodically bent planes. CUR is emitted at the frequencies ωu well below those of ChR, ωch . By varying a, λu , ε, and the crystal length, one can tune the CUR peak positions and intensities. Small-amplitude smallperiod regime implies a  d and λu less than period of channeling oscillations [71– 75]. This scheme allows the emission of photons of the higher energies, ωu > ωch , thus making feasible construction of a CLS which radiates in the GeV photon energy range [76]. Initially, the CU feasibility was justified for positrons [26, 28]. Positrons channel over larger distances passing a larger number of bending periods and, thus, increasing the CUR intensity. Experiments carried out so far to detect CUR from positrons

1 Introduction

5

have not been successful due to insufficient quality of periodic bending, large beam divergence, and high level of the background bremsstrahlung radiation [77–82]. The feasibility of CU for electrons was also proven [83, 84], but it was indicated that to obtain better CUR signal high-energy (GeV and above) electron beams are preferable. The CUR signal was detected in the experiments with electron beam of much lower energies at the Mainz Microtron [85, 86]. The radiation excess due to CUR was detected although it was not as intense as expected. In part, this discrepancy can be attributed to insufficient quality of the crystalline lattice although this issue has to be investigated in more detail. Also, the beam energy used was low (sub-GeV range) and as a consequence photon energies, as well as the choice of particles (electrons), were not optimal. A CU based on the heavy-projectile channeling is also feasible although in this case, the main restrictive factor is photon attenuation in a crystalline medium [30]. It has been demonstrated that the most feasible devices are the proton-based CU (for the projectile energies ε  1 TeV) and the muon-based CU (for ε  102 GeV). In both cases, the use of light crystals (diamond, silicon) is most promising. The first experimental evidence of proton channeling in periodically bent crystal was reported in Ref. [87]. The experiments were carried out with a 400 GeV proton beam at CERN and the evidence of planar channeling in the CU was firmly stated. Theoretical and experimental studies of the CU and CUR phenomena have ascertained the importance of the high quality of the undulator material needed to achieve strong effects in the emission spectra [88]. Up to now, several methods to create periodically bent crystalline structures have been proposed and/or realized. Figure 1.3 provides schematic illustration of the ranges of a and λu within which the emission of intensive CUR is feasible. Shadowed areas mark the ranges currently achievable by different technologies. Several approaches have been applied to produce static bending. The greenish area marks the area achievable by means of technologies based on surface deformations. These include mechanical scratching [81], laser ablation technique [89], grooving method [87, 90, 91], tensile/compressive strips deposition [90, 92–94], and ion implantation [95]. The most recent technique proposed is based on sandblasting, one of the major sides of a crystal to produce an amorphous layer capable of keeping the sample bent [96]. Another technique, which is under consideration for manufacturing periodically bent silicon and germanium crystals, is pulsed laser melting processing that produces localized and high-quality stressing alloys on the crystal surface. This technology is used in semiconductor processing to introduce foreign atoms in crystalline lattices [97]. Currently, by means of the surface deformation methods, the periodically bent crystals with a large period, λu  102 microns, can be produced. To decrease the period λu , one can rely on the production of graded composition strained layers in an epitaxially grown Si1−x Gex superlattice [98–101]. Both silicon and germanium crystals have a diamond structure with close lattice constants. Replacement of a fraction of Si atoms with Ge atoms leads to bending crystalline directions. By means of this method, sets of periodically bent crystals have been produced and used in channeling experiments [86]. A similar effect can be achieved by

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1 Introduction

Fig. 1.3 Shadowing indicates the ranges accessible by means of modern technologies: superlattices (gray), surface deformations (green), and acoustic waves (blue). Sloping dashed lines indicate the boundaries of the stable channeling for ε = 0.5 and 50 GeV projectiles. For each energy, the periodic bending corresponding to the CU regime (characterized by Large Period (LP), λu  a) lies to the right of the line. The horizontal line a/d = 1 (d is the interplanar spacing) separates the LargeAmplitude (LA) and Small-Amplitude (SA) bending. The boundaries of the most favorable CU regime, LALP, are marked by thick red lines. SASP area stands for Small-Amplitude Short-Period bending

graded doping during synthesis to produce diamond superlattice [102]. Both boron and nitrogen are soluble in diamond, however, higher concentrations of boron can be achieved before extended defects appear [102, 103]. The advantage of a diamond crystal is radiation hardness allowing it to maintain the lattice integrity in the environment of very intensive beams [38]. The gray area in Fig. 1.3 marks the ranges of parameters achievable by means of crystal growing. The bluish area indicates the range of parameters achievable by means of another method, realization of which is although still pending, based on the propagation of a transverse acoustic wave in a crystal [30]. In a Crystalline Undulator (CU), a projectile’s trajectory follows the profile of periodic bending. This is possible when the electrostatic crystalline field exceeds the centrifugal force acting on the projectile. This condition, which entangles bending amplitude and period, the projectile’s energy, and the crystal field strength, implies that the bending parameter C is less than one. The bending parameter is defined as the ratio of the interplanar force, which keeps a projectile in a channel, to the maximum centrifugal force acting on the projectile in a bent channel. Two sloping dashed lines in Fig. 1.3 show the dependences a = a(λu ) corresponding to the extreme value C = 1 for ε = 0.5 and 50 GeV projectiles. For each energy, the CU is feasible in the domain lying to the right from the line. In this domain, periodic bending is characterized by a Large Period (LP), which implies (i) λu  a, and (ii) λu greatly exceeds the period

1 Introduction

7

of channeling oscillations. The horizontal line a/d = 1 (d stands for the interplanar distance) divides the CU domain into two parts: the Large-Amplitude (LA), a > d, and the Small-Amplitude (SA), a < d, regions. Larger amplitudes are more favorable from the viewpoint of achieving higher intensities of CUR. The red lines delineate the domain where the LALP periodic bending can be considered. The necessary conditions, which must be met in order to treat a CU based on the LALP periodic bending a feasible scheme, are as follows [26, 28, 104, 105]: ⎧  C = 4π 2 εa/Umax λ2u < 1 – stable channeling, ⎪ ⎪ ⎪ ⎪ – large-amplitude regime, ⎪ ⎨ d < a  λu – N = L/λu  1 (1.1)  large number of periods, ⎪ ⎪ L ∼ min L (C), L (ω) – account for dechanneling and photon attenuation, ⎪ d a ⎪ ⎪ ⎩ ε/ε  1 – low energy losses. The formulated conditions are of a general nature since they are applicable to any type of a projectile undergoing channeling in PBCr. Their application to the case of a specific projectile and/or a crystal channel allows one to analyze the feasibility of the CU by establishing the ranges of ε, a, λu , L, N , and ω which can be achieved. • A stable channeling of a projectile in a periodically bent channel occurs if the  maximum centrifugal force Fcf is less than the maximal interplanar force Umax , i.e.,  C = Fcf /Umax < 1. Expressing Fcf through the projectile’s energy ε, the period λu and amplitude a of the bending one writes this condition in the form presented in Eq. (1.1). • The operation of a CU should be considered in the Large-amplitude regime. The limit a/d > 1 accompanied by the condition C  1 is mostly advantageous, since in this case the characteristic frequencies of UR and ChR are well separated: 2 ∼ Cd/a  1. As a result, the channeling motion does not affect the paramωu2 /ωch eters the UR, the intensity of which can be comparable to or higher than that of ChR. A strong inequality a  λu ensured elastic deformation of the crystal. • The term “undulator” implies that the number of periods, N , is large. Only then the emitted radiation bears the features of a UR (narrow, well-separated peaks in spectral-angular distribution). This is stressed by the third condition. • A CU essentially differs from a conventional undulator, in which the beams of particles and photons move in vacuum. In CU both the beams propagate in crystalline medium and, thus, are affected by the dechanneling and the photon attenuation. The dechanneling effect stands for a gradual increase in the transverse energy of a channeled particle due to inelastic collisions with the crystal constituents [32]. At some point, the particle gains a transverse energy higher than the planar potential barrier and leaves the channel. The average interval for a particle to penetrate into a crystal until it dechannels is called the dechanneling length, L d . In a straight channel, this quantity depends on the crystal, on the energy, and the type of a projectile. In a periodically bent channel, there appears an additional dependence on the parameter C. The intensity of the photon flux, which propagates through a crystal, decreases due to the processes of absorption and scattering. The inter-

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1 Introduction

val within which the intensity decreases by a factor of e is called the attenuation length, L a (ω). This quantity is tabulated for a number of elements and for a wide range of photon frequencies (see, e.g., Ref. [106, 107]). The fourth condition in (1.1) takes into account severe limitation of the allowed values of the length L of a CU due to the dechanneling and the attenuation. • Finally, let us comment on the last condition, which is of most importance for light projectiles, positrons, and electrons. For sufficiently large photon energies (ω  101 . . . 102 keV depending on the type of the crystal atom), the restriction due to the attenuation becomes less severe than due to the dechanneling effect. Then, the value of L d (C) effectively introduces an upper limit on the length of a CU. Since for an ultra-relativistic particle L d ∝ ε (see, e.g., [38]), it seems natural that to increase the effective length one can consider higher energies. However, at this point another limitation manifests itself [104]. The coherence of UR is only possible when the energy loss ε of the particle during its passage through the undulator is small, ε  ε. This statement, together with the fact that for ultrarelativistic electrons and positrons ε is mainly due to the photon emission, leads to the conclusion that L must be much smaller than the radiation length L r , the distance over which a particle converts its energy into radiation. For a positron-based CU, a thorough analysis of the system (1.1) was carried out for the first time in Refs. [26–28, 104, 105, 108]. Later on, the feasibility of the CU utilizing the planar channeling of electrons was demonstrated [83, 84]. Recently, a similar analysis was carried out for heavy ultra-relativistic projectiles (muon, proton, ion) [30]. Another regime of periodic bending, Small-Amplitude Short-Period (SASP), can be realized in the domain a < d and λu < 1 micron (these values of λu are much smaller that channeling periods of projectiles with ε  1 GeV). In the SASP regime, in contrast to the channeling in a CU, channeling particles do not follow the shortperiod bent planes but experience regular jitter-type modulations of their trajectories which lead to the emission of high-energy radiation. As mentioned, dynamic bending can be achieved by propagating a transverse acoustic wave along a particular crystallographic direction [26, 28, 109–113]. This can be achieved, for example, by placing a piezo sample atop the crystal and generating radio frequencies to excite the oscillations. The advantage of this method is its flexibility: the bending amplitude and period can be changed by varying the wave intensity and frequency [26, 28, 104]. Although the applicability of this method has not yet been checked experimentally, we note that a number of experiments have been carried out on the stimulation of ChR by acoustic waves excited in piezoelectric crystals [114]. Figure 1.4 allows one to estimate the acoustic wave frequencies ν needed to achieve the LALP periodic bending of the silicon(110) planes. The diagonal dashed lines correspond to the dependences a = a(λu ) obtained for several values (as indicated) of the bending parameter C. The CU cannot be realized in the domain lying to the left from the line C = 1. The solid curves present the dependences a = a(λu ) calculated for the fixed values (indicated in the legends) of the number of undulator peri-

References

9 AW frequency (MHz)

AW frequency (MHz) 4

ε=0.5 GeV

100

C=1

C=0.2

Nd=5 Nd=10 Nd=20 Nd=40

10 C=0.01

a/d=1 1 1

2

10

10

10

Bending period λu (micron)

C=0.1

Bending amplitude a (angstrom)

Bending amplitude a (angstrom)

3

3

10

1

10

ε=50 GeV

3

10

10 C=1

C=0.1

C=0.01

Nd=10 2

10

Nd=20 Nd=50 Nd=100 Nd=200

1

10

a/d=1 0 10 10

100

1000

Bending period λu (micron)

Fig. 1.4 Ranges of acoustic wave frequency ν (upper horizontal axis), of bending period equal to the wave wavelength λu = λAW (lower horizontal axis) and of amplitude a (vertical axis) to be probed to construct a silicon(110)-based CU. The data refer to ε = 0.5 GeV (left panel) and ε = 50 GeV positrons

ods Nd = L d (C)/λu within the dechanneling length L d (C). It is seen that the values λu ∼ 1 . . . 103 microns correspond to the frequencies ν = vs /λu ∼ 1 . . . 103 MHz, which are achievable experimentally (vs = 4.67 × 105 cm/s is the speed of sound) [114–119].

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58. Wienands, U., Markiewicz, T.W., Nelson, J., Noble, R.J., Turner, J.L., Uggerhøj, U.I., Wistisen, T.N., Bagli, E., Bandiera, L., Germogli, G., Guidi, V., Mazzolari, A., Holtzapple, R., Miller, M.: Observation of a remarkable deflection of multi-GeV electron beams by a thin crystal. SLAC Scientific Publications (2014) SLAC-PUB-15952 59. Wistisen, T.N., Uggerhøj, U.I., Wienands, U., Markiewicz, T.W., Noble, R.J., Benson, B.C., Smith, T., Bagli, E., Bandiera, L., Germogli, G., Guidi, V., Mazzolari, A., Holtzapple, R., Tucker, S.: Channeling, volume reflection, and volume capture study of electrons in a bent silicon crystal. Phys. Rev. Acc. Beams 19, 071001 (2016) 60. Sytov, A.I., Bandiera, L., De Salvador, D., Mazzolari, A., Bagli, E., Berra, A., Carturan, S., Durighello, C., Germogli, G., Guidi, V., Klag, P., Lauth, W., Maggioni, G., Prest, P., Romagnoni, M., Tikhomirov, V.V., Vallazza, E.: Steering of Sub-GeV electrons by ultrashort Si and Ge bent crystals. Eur. Phys. J. C 77, 901 (2017) 61. Wienands, U., Markiewicz, T.W., Nelson, J., Noble, R.J., Turner, J.L., Uggerhøj, U.I., Wistisen, T.N., Bagli, E., Bandiera, L., Germogli, G., Guidi, V., Mazzolari, A., Holtzapple, R., Miller, M.: Observation of deflection of a beam of multi-GeV electrons by a thin crystal. Phys. Rev. Let. 114, 074801 (2015) 62. Wienands, U., Gessner, S., Hogan, M.J., Markiewicz, T., Smith, T., Sheppard, J., Uggerhøj, U.I., Nielsen, C.F., Wistisen, T., Bagli, E., Bandiera, L., Germogli, G., Mazzolari, A., Guidi, V., Sytov, A., Holtzapple, R.L., McArdle, K., Tucker, S., Benson, B.: Channeling and radiation experiments at SLAC Int. J. Mod. Phys. A 34, 1943006 (2019) 63. Wienands, U., Gessner, S., Hogan, M.J., Markiewicz, T.W., Smith, T., Sheppard, J., Uggerhøj, U.I., Hansen, J.L., Wistisen, T.N., Bagli, E., Bandiera, L., Germogli, G., Mazzolari, A., Guidi, V., Sytov, A., Holtzapple, R.L., McArdle, K., Tucker, S., Benson, B.: Channeling and radiation experiments at SLAC. Nucl. Instrum Meth. B 402, 11 (2017) 64. Bandiera, L., Kyryllin, I.V., Brizzolari, C., Camattari, R., Charitonidis, N., De Salvador, D., et al.: Investigation on steering of ultrarelativistic e± beam through an axially oriented bent crystal. Europ. Phys. J C 81(3), 1–10 (2021) 65. Taratin, A.M., Vorobiev, S.A.: Radiation of high-energy positrons channeled in bent crystals. Nucl. Instrum. Meth. B 31, 551–557 (1988) 66. Taratin, A.M., Vorobiev, S.A.: Quasi-synchrotron radiation of high-energy positrons channeled in bent crystals. Nucl. Instrum. Meth. B 42, 41–45 (1989) 67. Bandiera, L., Sytov, A., De Salvador, D., Mazzolari, A., Bagli, E., Camattari, R., Carturan, S., Durighello, C., Germogli, G., Guidi, V., Klag, P., Lauth, W., Maggioni, G., Mascagna, V., Prest, M., Romagnoni, M., Soldani, M., Tikhomirov, V.V., Vallazza, E.: Investigation on radiation generated by sub-GeV electrons in ultrashort silicon and germanium bent crystals. Europ. Phys. J C 81(3), 1–9 (2021) 68. Polozkov, R.G., Ivanov, V.K., Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Radiation emission by electrons channeling in bent silicon crystals. Eur. Phys. J. D 68, 268 (2014) 69. Mazzolari, A., Bagli, E., Bandiera, L., Guidi, V., Backe, H., Lauth, W., Tikhomirov, V., Berra, A., Lietti, D., Prest, M., Vallazza, E., De Salvador, D.: Steering of a sub-GeV electron beam through planar channeling enhanced by rechanneling. Phys. Rev. Lett. 112, 135503 (2014) 70. Shen, H., Zha, Q., Zhang, F.S., Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Channeling and radiation of 855 MeV electrons and positrons in straight and bent tungsten (1 1 0)crystals. Nucl. Instrum. Meth. B 424, 26 (2018) 71. Kostyuk, A.: Crystalline undulator with a small amplitude and a short period. Phys. Rev. Lett. 110, 115503 (2013) 72. Wistisen, T.N., Andersen, K.K., Yilmaz, S., Mikkelsen, R., Hansen, J.L., Uggerhøj, U.I., Lauth, W., Backe, H.: Experimental realization of a new type of crystalline undulator. Phys. Rev. Lett. 112, 254801 (2014) 73. Uggerhoj, U.I., Wistisen, T.N., Hansen, J.L., Lauth, W., Klag, P.: Radiation collimation in a thick crystalline undulator. Eur. J. Phys. D 71, 124 (2017) 74. Korol, A.V., Bezchastnov, V.G., Sushko, G.B., Solov’yov, A.V.: Simulation of channeling and radiation of 855 MeV electrons and positrons in a small-amplitude short-period bent crystal. Nucl. Instrum. Meth. B 387, 41 (2016)

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75. Uggerhøj, U.I., Wistisen, T.N.: Intense and energetic radiation from crystalline undulators. Nucl. Instrum Meth. B 355, 35 (2015) 76. Bezchastnov, V.G., Korol, A.V., Solov’yov, A.V.: Radiation from multi-GeV electrons and positrons in periodically bent silicon crystal. J. Phys. B 47, 195401 (2014) 77. Baranov, V.T., Bellucci, S., Biryukov, V.M., et al.: First results of investigation of radiation from positrons in a crystalline undulator. JETP Lett. 82, 562–564 (2005) 78. Baranov, V.T., Bellucci, S., Biryukov, V.M., Britvich, G.I., Chepegin, V.N., et al.: Preliminary results on the study of radiation from positrons in a periodically deformed crystal. Nucl. Instrum. Meth. B 252, 32 (2006) 79. Backe, H., Krambrich, D., Lauth, W., Buonomo, B., Dabagov, S.B., Mazzitelli, G., Quintieri, L., Hansen, J.L., Uggerhøj, U.K.I., Azadegan, B., Dizdar, A., Wagner, W.: Future aspects of X-ray emission from crystal undulators at channeling of positrons. Nuovo Cimento C 34, 175 (2011) 80. Quintieri, L., Buonomo, B., Dabagov, S.B., Mazzitelli, G., Valente, P., Backe, H., Kunz, P., Lauth, W.: Positron channeling at the DA NE BTF facility: the CUP experiment. In: Dabagov, S.B., Palumbo, L., Zichichi, A. (eds.) Proceedings of the 51st Workshop Charged and Neutral Particles Channeling Phenomena Channeling 2008, Erice, Italy, Oct 2008, pp. 319–330. World Scientific, Singapore (2010). ISBN 9789814307017 81. Bellucci, S., Bini, S., Biryukov, V.M., Chesnokov, Yu.A., et al.: Experimental study for the feasibility of a crystalline undulator. Phys. Rev. Lett. 90, 034801 (2003) 82. Afonin, A.G., Baranov, V.T., Bellucci, S., Biryukov, V.M., Britvich, G.I., et al.: Crystal undulator experiment at IHEP. Nucl. Instrum. Methods Phys. Res. B 234, 122–127 (2005) 83. Tabrizi, M., Korol, A.V., Solov’yov, A.V., Greiner, W.: Feasibility of an electron-based crystalline undulator. Phys. Rev. Lett. 98, 164801 (2007) 84. Tabrizi, M., Korol, A.V., Solov’yov, A.V., Greiner, W.: Electron-based crystalline undulator. J. Phys. G: Nucl. Part. Phys. 34, 1581–1593 (2007) 85. Backe, H., Krambrich, D., Lauth, W., Hansen, J.L., Uggerhøj, U.K.I.: X-ray emission from a crystal undulator: experimental results at channeling of electrons. Nuovo Cimento C 34, 157–165 (2011) 86. Backe, H., Krambrich, D., Lauth, W., Andersen, K.K., Hansen, J.L., Uggerhøj, U.I.: Radiation emission at channeling of electrons in a strained layer Si1−x Gex undulator crystal. Nucl. Instum. Meth. B 309, 37 (2013) 87. Bagli, E., Bandiera, L., Bellucci, V., Berra, A., Camattari, R., De Salvador, D., Germogli, G., Guidi, V., Lanzoni, L., Lietti, D., Mazzolari, A., Prest, M., Tikhomirov, V.V., Valla, E.: Experimental evidence of planar channeling in a periodically bent crystal. Eur. Phys. J. C 74, 3114 (2014) 88. Kostyuk, A., Korol, A.V., Solov’yov, A.V., Greiner, W.: The influence of the structure imperfection of a crystalline undulator on the emission spectrum. Nucl. Instrum. Method B 266, 972–987 (2008) 89. Balling, P., Esberg, J., Kirsebom, K., Le, D.Q.S., Uggerhøj, U.I., Connell, S.H., Härtwig, J., Masiello, F., Rommeveaux, A.: Bending diamonds by femto-second laser ablation. Nucl. Instrum Meth. B 267, 2952 (2009) 90. Guidi, V., Antonioni, A., Baricordi, S., Logallo, F., Malagù, C., Milan, E., Ronzoni, A., Stefancich, M., Martinelli, G., Vomiero, A.: Tailoring of silicon crystals for relativistic-particle channeling. Nucl. Instrum. Meth. B 234, 40–46 (2005) 91. Lanzoni, L., Mazzolari, A., Guidi, V., Tralli, A., Martinelli, G.: On the mechanical behaviour of a crystalline undulator. Int. J. Eng. Sci. 46, 917–928 (2008) 92. Lanzoni, L., Radi, E.: Thermally induced deformations in a partially coated elastic layer. Int. J. Eng. Sci. 46, 1402–1412 (2009) 93. Guidi, V., Mazzolari, A., Martinelli, G., Tralli, A.: Design of a crystalline undulator based on patterning by tensile Si3 N4 strips on a Si crystal. Appl. Phys. Lett. 90, 114107 (2007) 94. Guidi, V., Lanzoni, L., Mazzolari, A.: Patterning and modeling of mechanically bent silicon plates deformed through coactive stresses. Thin Solid Films 520, 1074 (2011)

14

1 Introduction

95. Bellucci, V., Camattari, R., Guidi, V., Mazzolari, A., Paterno, G., Mattei, G., Scian, C., Lanzoni, L.: Ion implantation for manufacturing bent and periodically bent crystals. Appl. Phys. Lett. 107, 064102 (2015) 96. Camattari, R., Paternò, G., Romagnoni, M., Bellucci, V., Mazzolari, A., Guidi, V.: Homogeneous self-standing curved monocrystals, obtained using sandblasting, to be used as manipulators of hard X-rays and charged particle beams. J. Appl. Cryst. 50, 145–151 (2017) 97. Cristiano, F., Shayesteh, M., Duffy, R., Huet, K., Mazzamuto, F., Qiu, Y., Quillec, M., Henrichsen, H.H., Nielsen, P.F., Petersen, D.H., La Magna, A., Caruso, G., Boninelli, S.: Defect evolution and dopant activation in laser annealed Si and Ge. Mat. Scie. in Semicond. Process. 42, 188–195 (2016) 98. Bogacz, S.A., Ketterson, J.B.: Possibility of obtaining coherent radiation from a solid state undulator. J. Appl. Phys. 60, 177–188 (1986) 99. Breese, M.B.H.: Beam bending using graded composition strained layers. Nucl. Instrum. Method Phys. Res. B 132, 540–547 (1997) 100. Mikkelsen, U., Uggerhøj, E.: A crystalline undulator based on graded composition strained layers in a superlattice. Nucl. Instum. Meth. B 160, 435 (2000) 101. Avakian, R.O., Avetyan, K.N., Ispirian, K.A., Melikyan, E.G.: Bent crystallographic planes in gradient crystals and crystalline undulators. Nucl. Instrum. Method Phys. Res. A 508, 496–499 (2003) 102. Tran Thi, T.N., Morse, J., Caliste, D., Fernandez, B., Eon, D., Härtwig, J., Barbay, C., MerCalfati, C., Tranchant, N., Arnault, J.C., Lafford, T.A., Baruchel, J.: Synchrotron Bragg diffraction imaging characterization of synthetic diamond crystals for optical and electronic power device applications. J. Appl. Cryst. 50, 561 (2017) 103. de la Mata, B.G., Sanz-Hervás, A., Dowsett, M.G., Schwitters, M., Twitchen, D.: Calibration of boron concentration in CVD single crystal diamond combining ultralow energy secondary ions mass spectrometry and high resolution X-ray diffraction. Diam. Rel. Mat. 16, 809 (2007) 104. Korol, A.V., Solov’yov, A.V., Greiner, W.: Total energy losses due to the radiation in an acoustically based undulator: the undulator and the channeling radiation. Int. J. Mod. Phys. E 9, 77–105 (2000) 105. Krause, W., Korol, A.V., Solov’yov, A.V., Greiner, W.: Total spectrum of photon emission by an ultra-relativistic positron channeling in a periodically bent crystal. J. Phys. G Nucl. Part. Phys. 26, L87–L95 (2000) 106. Hubbel, J.H., Seltzer, S.M.: Tables of X-ray mass attenuation coefficients and mass energyabsorption coefficients from 1 keV to 20 MeV for elements Z = 1 to 92 and 48 additional substances of dosimetric interest. NISTIR 5632. Web version http://www.nist.gov/pml/data/ xraycoef/index.cfm 107. Nakamura, K., et al., (Particle Data Group): Review of particle physics. J. Phys. G: Nucl. Part. Phys. 37, 075021 (2010) 108. Korol, A.V., Solov’yov, A.V., Greiner, W.: Number of photons and brilliance of the radiation from a crystalline undulator. Proc. SPIE 5974, 597405 (2005) 109. Kaplin, V.V., Plotnikov, S.V., Vorobev, S.A.: Radiation by charged particles in deformed crystals. Sov. Phys. - Tech. Phys. 25, 650–651 (1980) 110. Baryshevsky, V.G., Dubovskaya, I.Ya., Grubich, A.O.: Generation of γ -quanta by channeled particles in the presence of a variable external field. Phys. Lett. 77A, 61–64 (1980) 111. Ikezi, H., Lin-Liu, Y.R., Ohkawa, T.: Channeling radiation in periodically distorted crystal. Phys. Rev. B 30, 1567–1569 (1984) 112. Mkrtchyan, A.R., Gasparyan, R.H., Gabrielyan, R.G., Mkrtchyan, A.G.: Channeled positron radiation in the longitudinal and transverse hypersonic wave field. Phys. Lett. 126A, 528–530 (1988) 113. Dedkov, G.V.: Channeling radiation in a crystal undergoing an action of ultrasonic or electromagnetic waves. Phys. Stat. Sol. (b) 184, 535–542 (1994) 114. Wagner, W., Azadegan, B., Büttig, H., Grigoryan, LSh., Mkrtchyan, A.R., Pawelke, J.: Channeling radiation on quartz stimulated by acoustic waves. Nuovo Cimento C 34, 157–165 (2011)

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115. Tzianaki, E., Bakarezos, M., Tsibidis, G.B., Orphanos, Y., Loukakos, P.A., Kosmidis, C., Patsalas, P., Tatarakis, M., Papadogiannis, N.A.: High acoustic strains in Si through ultrafast laser excitation of Ti thin-film transducers. Opt. Express 23, 17191–17204 (2015) 116. Bakarezos, M., Tzianaki, E., Petrakis, S., Tsibidis, G., Loukakos, P.A., Dimitriou, V., Kosmidis, C., Tatarakis, M., Papadogiannis, N.A.: Ultrafast laser pulse chirp effects on lasergenerated nanoacoustic strains in Silicon. Ultrasonics 86, 14–19 (2018) 117. Mkrtchyan, A.R., Gasparyan, R.A., Gabrielyan, R.G.: Channeled positron in the hypersonic wave field. Phys. Lett. 115A, 410–412 (1986) 118. Grigoryan, LSh., Mkrtchyan, A.R., Khachatryan, H.F., Wagner, W., Saharian, A.A., Baghdasaryan, K.S.: On the amplification of radiant energy during channeling in acoustically excited single crystal. Nucl. Instrum. Method B 212, 51–55 (2003) 119. Wagner, W., Azadegan, B., Büttig, H., Pawelke, J., Sobiella, M., Grigoryan, LSh.: Probing channeling radiation influenced by ultrasound. In: Dabagov, S.B., Palumbo, L., Zichichi, A. (eds.) Charged and Neutral Particles Channeling Phenomena - Channeling 2008, pp. 378–407. World Scientific, Singapore (2010)

Chapter 2

Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

2.1 Radiation from Relativistic Charges An important issue in the study of radiation formed in CLSa concerns the choice of the formalism used to describe the phenomenon. This point could have been seen as merely a technical one, but it is not so. Contrary to the case of conventional undulators, based on the action of magnetic fields, the physics of CUs is, essentially, a newly arisen research field. Therefore, any theoretical study of the effect, which pretends to go a bit farther than purely academic research, must contain a great part of numerical analysis and numerical data on the basis of which real experimental investigations can be planned. In turn, to obtain the reliable data it is necessary to choose a theoretical tool which allows one, on the one hand, to treat adequately all principal physical phenomena involved in the problem, and, on the other hand, to effectively carry out the corresponding numerical analysis. There are three basic phenomena which must be accurately described: (i) the motion of an ultra-relativistic particle in a strong external field (the electrostatic crystalline field), (ii) the process of photon emission by the particle, and (iii) the problem of the radiative recoil, which results in the radiative energy losses of the projectile.

2.1.1 Classical Description In many cases, the motion of an ultra-relativistic particle, moving in an external field, can be treated within the framework classical mechanics (see, e.g., [1, 2]). The general criterion of the applicability of the classical description is in the condition that the variation of the de Broglie wavelength λB = h/ p of the projectile must be negligible over the distances of the order of λB . This condition can be written in the form (see, e.g., [3]) mFmax / p 3  1, where m and p ≈ ε/c are the mass and momentum of a projectile, and Fmax is the maximum force exerted by the external field. Taking into account that Fmax ∼ 101 . . . 102 GeV/cm for an planar crystalline potential and by approximately an order of magnitude higher for an axial potential (see, e.g., [4]), it can be shown that the condition is well-fulfilled for positrons and © Springer Nature Switzerland AG 2022 A. Korol and A. V. Solov’yov, Novel Lights Sources Beyond Free Electron Lasers, Particle Acceleration and Detection, https://doi.org/10.1007/978-3-031-04282-9_2

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2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

electrons with ε ∼ 102 MeV and higher. This energy range is of prime interest in the CLS problem, as it will be demonstrated below in the book. The process of photon emission can be treated classically provided the photon energy is small compared to that of a projectile: ω/ε  1. Hence, if both of the mentioned conditions are met, one can calculate the spectralangular distribution the radiated energy E using the following formula of classical electrodynamics (see, e.g., [1]): e2 Z 2 ω2 d3 E = dω dΩ c 8π 2





τ

dt1 0



τ

dt2 e



iω ϕ(t1 )−ϕ(t2 )

0

 v1 · v2 −1 . c2

(2.1)

Here, dΩ is the solid angle in the direction n of the emission, Z is the projectile charge in units of the elementary charge e, and τ is the time of flight through a spatial domain within which the external field acts on the projectile. The quantities v1,2 ≡ v(t1,2 ) stand for the projectile velocities at the instants t1 and t2 . It is assumed that for an ultra-relativistic particle, v1,2 ≈ c. The phase function ϕ(t) is defined as follows: ϕ(t) = t −

n · r(t) . c

(2.2)

The dependence of the position vector on time, r = r(t), is found from the classical equations of motions. The classical description of the radiative process is illustrated by Fig. 2.1 left, where the solid curve represents the trajectory of the charged particle. The radiation formed in a segment of the trajectory is emitted predominantly within the cone θ ∼ 1/γ (γ = ε/mc2 is the Lorentz relativistic factor) along the vector of the instant velocity v(t). The classical approach based on Eq. (2.1) is commonly used to describe various types of electromagnetic radiation: bremsstrahlung, synchrotron radiation, undulator radiation, and channeling radiation. For more specific and detailed information, see Refs. [1, 2, 4–13]. The classical framework does not provide a self-consistent description of the decrease in the energy of a moving charge due to the radiation emission. Hence, this scheme implies that the particle moves along the trajectory having the constant value of the total energy, ε = const.

2.1.2 Quantum Description The most rigorous approach to the radiation process is based on the formalism of quantum electrodynamics (see, e.g., [14]), where the amplitude of the process is described in terms of a single free-free matrix element of the photon emission taken between the initial and final states of an ultra-relativistic particle in the interplanar

2.1 Radiation from Relativistic Charges

19

(ω,k) ω, k

v(t) r(t)

θ~γ

−1

ε1, p1

ε2, p2

Fig. 2.1 Classical (left) and quantum (right) approaches to the radiation process. Classical ultrarelativistic charged projectile (red dot on left panel), being accelerated (decelerated) by external field, moves along a well-defined trajectory r = r(t). The electromagnetic radiation of frequency ω and wave-vector k is essentially emitted within the cone θ ∼ γ −1 along the vector of the instant velocity. Within the quantum picture (the right panel represents the Feynman diagram), the radiative transition from the initial state of the projectile (initial energy and the asymptotic momentum are ε1 and p1 ) to the final state with ε2 , p2 is accompanied by the photon emission (dashed line). The circle denotes the vertex of the particle–photon interaction

field. The Feynman diagram of the process is presented in Fig. 2.1 right, where the solid line denotes the projectile in the initial (the subscript 1) and the final (the subscript 2) states, the dashed line stands for the emitted photon, and the dots marks the vertex of the particle–photon interaction. The energy conservation law implies ε1 − ε2 = ω. The corresponding analytical expression for the amplitude M21 is given by  M21 = Z e

dr Ψε†2 p2 ν2 (r) (e · α) exp (−ik · r) Ψε1 p1 ν1 (r).

(2.3)

Here, the bi-spinor wavefunction Ψεpν (r) stands for the solution of the Dirac equation with the external potential U (a so-called Furry approximation; see, e.g., [14]) corresponding to the total energy ε, the asymptotic momentum p. Other quantum numbers, including the polarization of the particle, are contained in the subscript ν. The symbol † denotes the Hermitian conjugation, α = γ 0 γ with γ 0 and γ standing for the Dirac matrices. The vectors k and e denote the photon wave-vector and polarization. The power of radiation P (the energy per unit time) emitted within the interval dω and the cone dΩ are related to the differential cross section d3 σ/dω dΩ of the photon emission process:  d3 σ ω3 p2 ε2  d3 P = jω = dΩp2 |M21 |2 dω dΩ dω dΩ (2π )5 c5 pol

(2.4)

(4π)

where j denotes the flux of the incoming particles. The sum is carried out over the particle polarizations in the initial and final states as well as over the photon polarizations; the integration is carried out over the scattering angles. Equations (2.3) and (2.4) are applicable in the whole range of the emitted photon energies, starting from the soft photons, ω  ε1 so that ε1 ≈ ε2 , up to the tip end of the spectrum, when nearly all the initial (kinetic) energy ε1 − mc2 is radiated.

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2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

In the ultra-relativistic domain, the quantum-electrodynamic approach has been used for theoretical and numerical studies of various radiative processes. These include BrS in electron–atom (or/and ion), electron–electron, etc., collisions (see, e.g., [14–16] and references therein), coherent BrS (e.g., [17]), synchrotron radiation [18], and channeling radiation (e.g., Refs. [11, 19–22]). In application to the channeling motion and ChR, as well as to the CUR, the main (technical) limitation of the quantum approach is due to the fact that in the ultra-relativistic limit, when γ  1, the number of the energy levels of the transverse motion in the effective interplanar (or, axial) potential increases significantly. Consequently, an accurate numerical calculation of the particle dynamics becomes a formidable task [22].

2.1.3 Quasi-classical Description of Radiation Emission An adequate approach to the radiation emission by ultra-relativistic projectiles was developed by Baier and Katkov in the late 1960s [23] and was called an “operator quasi-classical method”. The details of its formalism as well as applications to a variety of radiative processes can be found in Refs. [4, 14] and will not be reproduced here. A remarkable feature of this method is that it combines the classical description of the particle motion in an external field and the quantum effect of radiative recoil. The classical description of the motion is valid provided the characteristic energy of the projectile in an external field, ω˜ 0 , is much less than its total energy, ε = mγ c2 . The relation ω˜ 0 /ε ∝ γ −1  1 is fully applicable in the case of an ultra-relativistic projectile. The role of radiative recoil, i.e., the change of the projectile energy due to the photon emission, is governed by the ratio ω/ε. In the limit ω/ε  1, a purely classical description (2.1) of the radiative process can be used. For ω/ε ≤ 1, quantum corrections must be accounted for. The quasi-classical approach neglects the terms ∼ω˜ 0 /ε but explicitly takes into account the quantum corrections due to the radiative recoil. The method is applicable in the whole range of the emitted photon energies, except for the extreme high-energy tail of the spectrum (1 − ω) /ε  1. Within the framework of quasi-classical approach, the spectral-angular distribution of the energy radiated by an ultra-relativistic projectile reads as follows (see Refs. [4, 14]): Z 2 ω2 d3 E =α dω dΩ 4π 2





τ

τ

dt1 0

dt2 eiω







ϕ(t1 )−ϕ(t2 )

f (t1 , t2 ).

(2.5)

0

The notations used here, except for ω and f (t1 , t2 ), are as in the classical formula (2.1). The function f (t1 , t2 ) is defined as follows:

2.1 Radiation from Relativistic Charges

21

    v1 · v2 1  u2 2 1 + (1 + u) f (t1 , t2 ) = −1 + 2 . 2 c2 γ

(2.6)

The effect of the radiative recoil, i.e., the account for the terms ω/ε, is contained in the parameters ω and u, which are given by u=

ω , ε − ω

ω = (1 + u) ω.

(2.7)

In the classical limit u ≈ ω/ε → 0 and ω → ω, so that (2.5) and (2.6) reproduce Eq. (2.1). Application of the general quasi-classical formula (2.5) to a variety of radiative processes in ultra-relativistic collisions in linear crystals is discussed in Ref. [4]. It was also applied to the problem of synchrotron-type radiation emitted by an ultrarelativistic projectile channeling in a non-periodically bent crystal [24–26].

2.2 Bremsstrahlung 2.2.1 The Elementary Process of Bremsstrahlung In the elementary process of bremsstrahlung (BrS), a charged projectile emits a photon being accelerated by an electrostatic field of a target (nucleus, ion, atom, etc.). The basic description of BrS, both classical and quantum, can be found in textbooks (see, e.g., [1, 2, 14, 27]). The importance and the fundamental character of the BrS process were recognized long ago (for a review of the historical background, see, e.g., [28]). Since then, it has been intensively studied theoretically, numerically, and experimentally in a wide range of the projectile and the emitted photon energies, different geometries of the emission, and a variety of atomic and ionic targets (see the reviews [28, 29] and references therein). Diagrammatic representation of the BrS amplitude is given by Fig. 2.1 left. In the papers cited above (see also [30]), various approaches are described concerning the choice of the initial and final state wavefunctions of the projectile. For relativistic projectiles, the Bethe–Heitler (BH) approximation [31] is the simplest and the most widely used one, since all the related cross sections for radiation in a point-charge Coulomb field can be expressed in closed form in terms of the elementary functions (see Sect. 2.4.2 for a more detailed description). The mentioned cross sections include the spectral distribution dσ ≡ dσ/dω, which one obtains from the double cross section d2 σ ≡ d2 σ/dωdΩ by integrating over the emission angles cross Ω = (θ, φ); in turn, d2 σ is calculated by integrating the triply differential  section d3 σ ≡ d3 σ/dωdΩdΩp2 over the scattering angles Ωp2 = θp2 , φp2 . The BH formula results from the application of the Born approximation for the integration of the scattering relativistic electron with the potential, retaining only the first nonzero terms (for the details, see, e.g., Ref. [15]). To take into account the screening

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2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

of the atomic nucleus by the electrons, one multiplies the cross section d3 σ , calculated in the Coulomb field Z e/r of the nucleus, by the atomic form factor [31]

F(q) = 1 − Z −1 ρ(r) exp (iq · r) dr, where ρ(r) is the electron charge density in the atom (or ion), and q = p1 − p2 − k is the momentum transferred to the nucleus during the collision (see Fig. 2.1 (left) for the notations). Formally, the applicability of the BH formula is linked to that of the Born approximation, which implies that the initial (v1 ) and final (v2 ) projectile velocities satisfy the relation v1,2 /c  α Z (α is the fine structure constant). As a result, for heavy elements, α Z ∼ 1, the data on d2 σ and d3 σ within the BH approximation strongly deviate from the results of more accurate calculations (see Ref. [30] and references therein). However, the discrepancies become much less pronounced for the cross section dσ integrated over the emission and scattering angles [28]. The BH cross section dσ for the case of an ultra-relativistic electron (positron) scattering in the field of a neutral atom can be written as follows [15]: 16 Z 2 αr02 dσBH ≈ dω 3 ω

 1−x +

3x 2 4



  ln 183Z −1/3

(2.8)

where x = ω/ε1 and r0 is the classical electron radius. Equation (2.8) accounts for the photon emission in the process of “elastic” BrS, i.e., when the atom does not change its state in the collision. In addition to this, the processes of “inelastic” BrS, in which the photon emission is accompanied by excitation or ionization of the target, also contribute to the spectrum. To include this part, one recalls that for an ultra-relativistic particle the BrS cross section of a projectile electron does not depend on the mass of the target particle but is proportional to the square of its charge. Therefore, ignoring the differences in logarithmic factors, the cross section of elastic BrS in the field of a screened nucleus exceeds that of inelastic BrS in collision with an atomic electron by a factor Z 2 . Then, to account for the inelastic BrS in collision with an atom with Z electrons, one carries out the following substitution on the right-hand side of (2.8) Z 2 → Z (Z + 1) (see, e.g., [14]). Moving in a medium, an electron loses its energy, in particular, via photon emission. The efficiency of this process can be characterized by the radiation length L r , i.e., the mean distance over which a projectile loses all but 1/e of its energy. To calculate L r , one multiplies (2.8) by the photon energy ω and integrates over ω from 0 up to (ε1 − mc2 )/. Multiplying the result by the factor n a /ε1 (n a is the volume density of atoms in the medium) and accounting for the substitution mentioned above, one derives  −1   . L r = 4Z (Z + 1)αr02 n a ln 183Z −1/3

(2.9)

The radiation length is derived on the basis of (2.8), which defines the BrS spectrum in a single electron–atom collision. This approach is applicable provided L r exceeds greatly the characteristic length, L coh , termed the coherence length [32], over which the radiation is formed in a single collision. Omitting the details of evaluation (see, e.g., [14, 32]), we present the final result

2.2 Bremsstrahlung

23

L coh =

 qmin

≈

2ε1 ε2 , ωm 2 c3

(2.10)

where qmin = p1 − p2 − ω/c ≈ ωmc3 /2ε1 ε2 (the limit ε1,2 /mc2  1 is implied) is the minimum transferred momentum in the radiative collision. The formulae obtained for the BrS process on an isolated atom can be valid also for passage through a medium only in the absence of the secondary photon emission or/and electron scattering over the distance L coh . The absence of these processes is ensured by the condition L coh  L r . However, the condition of no electron scattering is violated much sooner, so that at the distances ∼L r a projectile experiences multiple scattering with the medium atoms. As a result, the distances, over which Eq. (2.8) is applicable for evaluation of the BrS spectrum in medium, become much smaller and result in the condition L coh  αL r . At larger distances, the yield of the BrS emission (as well of other electrodynamic processes, such as pair creation) is suppressed due to the multiple scattering of the projectile from the medium atoms, a so-called Landau– Pomeranchuk–Migdal effect [33, 34].

2.2.2 Coherent Bremsstrahlung In an amorphous medium, an ultra-relativistic projectile experiences uncorrelated collisions with the constituent atoms. As a result, the total intensity of BrS will be proportional to the number of atoms n a (per unit volume), i.e., it is equal to the incoherent sum of the intensities formed in collisions with individual atoms (a so-called incoherent BrS). Passing through a crystalline media along a line close to a crystallographic direction, a projectile feels the periodicity of the electrostatic potential of the crystal. The periodicity influences strongly the emission process provided the coherence length (2.10) noticeably exceeds the spatial period l of the potential [32, 35, 36]. In this case, the amplitudes of the waves, emitted in collisions of the projectile with each of the atoms of total number Ncoh ∼ L coh /l, add coherently, so that the resulting intensity of the emission is enhanced as compared to incoherent BrS by a factor ∼Ncoh . For each emission angle θ , the coherence effect is mostly pronounced when θ and ω (which enter via the coherence length) satisfy the condition l/(L coh θ ) = 2π n, with n being an integer [32, 35, 36]. This process, commonly termed as the coherent BrS, is associated with an overbarrier particle (see trajectory No. 3 in Fig. 4.5) which moves along a quasi-periodic trajectory, traversing the crystallographic planes (or axes) under the angle greater than Lindhard’s critical angle, Θ > ΘL . Within the framework of the Born approximation, it can be shown (see, e.g., [32]) that the triply differential cross section of BrS in a medium can be related to the BH cross section d3 σBH ≡ d3 σBH /dωdΩdΩp2 as follows:

24

2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

⎛ ⎞

2





d3 σ = d3 σBH eiq·ra / = d3 σBH ⎝ Na + eiq·Δaa / ⎠

a

a =a

= d σnc + d σcoh , 3

3

(2.11)

where q = p1 − p2 − k is the transferred momentum, Δaa = ra − ra , and the sums are carried out over the lattice atoms of total number Na . the contribution of the incoherent BrS, The term d3 σnc ≡ Na d3 σBH stands for  while the second one, d3 σcoh = Na d3 σBH a =a eiq·Δaa / , represents the coherent part of the BrS cross section. In an amorphousmedium, where the atomic position vectors ra and ra are not correlated, the sum a =a (. . . ) = 0, so that the coherent term vanishes.1 Then, integrating d3 σnc over the emission and scattering angles, one finds that the spectral distribution dσnc /dω of incoherent BrS is defined by formula (2.8) multiplied by Na . The BH spectrum is a smooth function of the photon energy, therefore, dσnc /dω represents a smooth background of the total BrS spectrum formed in a crystal. The coherent term d3 σcoh becomes dominant in a crystalline medium. Not pretending to overview all important theoretical results related to the phenomenon of coherent BrS (this one we can find in, for example, in Refs. [4, 11, 12, 17, 32, 37–41]), we just mention several qualitative features of d3 σcoh which originate from  the factor Σ(q) ≡ a =a eiq·Δaa / . The detailed evaluation of this factor for various types of cubic lattices is presented in [35]. As a function of the transferred momentum Σ(q) is strongly peaked in the vicinity of q = g, where g is a reciprocal lattice vector. Therefore, for particular geometries of radiation and scattered electron, d3 σcoh has very pronounced maxima. In the maximum d3 σcoh ∝ Na Ncoh with Ncoh standing for the number of atoms within the coherence length L coh related to the transferred momentum as L coh ∼ /q , where q = (p1 · q)/ p1 is the component of q along the incident momentum. These peaks will lead to the non-monotonous structure of the cross section dσcoh = dσcoh /dω integrated over the emission and scattering angles. This feature is illustrated by Fig. 2.2(left) which presents the experimental results on the enhancement factor, i.e., on the ratio dσcoh /dσnc for a 150 GeV electron traversing Si crystal as indicated in the capture [42]. Figure 2.2(right) stresses another feature of coherent BrS, namely its sensitivity to the sign of a projectile. It is seen that under the same conditions (incident energy, crystal, and the direction of traversing), the positron and electron are much alike (as well as the BH spectrum, which is just identical). This is in contrast with spectral dependences of ChR; see Fig. 4.8 in the subsequent section. To conclude this section, we mention that coherent BrS from high-energy electrons and positrons in a bent crystal was considered in Ref. [43], where analytical expressions for the spectrum of emitted radiation were derived. The process of coherent BrS by relativistic electrons in PBCr was considered in [44, 45]. The authors pre1

This term is also negligibly small when a projectile moves in a random direction through the crystal.

2.2 Bremsstrahlung

25

12 10

Enhancement

20

8 6 10 4 2

electron positron

0

0 0

0.2

0.4

0.6

0.8

1

0

Δε/ε

0.2

0.4

0.6

0.8

1

Δε/ε

Fig. 2.2 Left graph: Relative energy loss by 150 GeV electrons incident at the angle 0.08 mrad to the (110) planar direction in a 0.6 mm thick silicon crystal. The polar angle with respect to the 110 axis is 8.2 mrad. Right graph: Relative energy loss by electrons and positrons traversing 0.5 mm thick silicon crystal at 0.52 mrad from the axis and 27 μrad from the (100) plane. In both graphs, the enhancement is with respect to an amorphous target of the same material and thickness. The dependences were obtained by digitalizing Figs. 3b and 2a in Ref. [42]

dicted sharp jumps in the radiation spectrum in the high-frequency range of emitted photons.

2.3 Synchrotron Radiation The term synchrotron radiation (SR) is commonly used to indicate the bright emission of highly collimated electromagnetic waves from charged particles gyrating at ultra-relativistic energies in an external magnetic field. The basic developments and applications of SR are well-documented [4, 46–49]. Although SR can be considered within the quantum electrodynamics framework [18], in many cases the classical (see, e.g., [1]) or quasi-classical [4] description is applicable. Let us consider an ultra-relativistic charged projectile2 that moves along a circle with radius R; see Fig. 2.3. The radiation originates from different segments of the trajectory within different spatial cones. Therefore, when characterizing the spectralangular distribution of radiation, it is meaningful to consider the emission from a small arc the angular size of which is about the natural emission cone γ −1 . Let n be a unit vector in the direction of the emitted photon. Instead of commonly used polar 2

To be specific, below in this section we refer to an electron.

26

2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

z R x

v

y

ψ

Fig. 2.3 Coordinate system and emission angle used to characterize the radiation emission by an ultra-relativistic electron moving along the circular trajectory of radius R. The circle defines the x − y plane with the x-axis aligned with the instant velocity v. The radiation emission is calculated from a small arc (not shown) of the angular size ∼γ −1 . The quantity ψ denotes the angle between the direction of the radiation emitted and the x − y plane

angles θ and φ, one can introduce two other angles: (i) angle ψ between n and the plane of the orbit, and (ii) angle β (not shown in the figure) between the vector of instant velocity v and the projection of n on the plane of the orbit. Effectively, both angles are of the order of γ −1 . The integration over β can be carried out explicitly (see Ref. [4] for details). Within the quasi-classical framework, the spectral-angular distribution d2 E/dω dψ of the energy radiated can be written as follows [4]: 9 2 d2 E =α γ R(ω, ψ) (2.12) d(ω) dψ 4π 2  2  2 ω R(ω, ψ) = 1 + γ 2ψ 2 (2.13) ωc     u2 γ 2ψ 2 u2 2 2 × 1+u+ K 2/3 K (η) + (1 + u) + (η) 1/3 2 1 + γ 2ψ 2 2 where the quasi-classical parameter u, introduced in (2.7), accounts for the recoil, K 1/3 (η) and K 2/3 (η) are modified Bessel function of the second kind [50]. Their argument reads η = (ω/2ωc )(1 − δ)−1 (1 + γ 2 ψ 2 )3/2 with δ = ω/ε. The quantity ωc denotes the frequency beyond which the radiation intensity rapidly falls off any emission angle [1]. It is expressed in terms of the curvature radius and the energy of a projectile as follows: ωc =

3 3c γ . 2 R

(2.14)

In the limit ω/ε → 0, Eq. (2.12) reduces to the classical formula; see, e.g., [1].

2.3 Synchrotron Radiation

27 1.5

R(ω,ψ) (abs. u.)

R(ω,ψ) (abs. u.)

1.5

1

0.5

1

0.5 0.5

0.1

1 2

1

0.5

0.1

2

5

0

0 0

1

2

3

4

0

1

2

3

4

ω/ωc

γψ

Fig. 2.4 Factor R(ω, ψ), Eq. (2.13), as a function of (i) γ ψ for fixed values of ω/ωc as indicated, left panel, (ii) ω/ωc for fixed values of γ ψ as indicated, right panel

Dependences of the factor R(ω, ψ) on γ ψ for fixed values of scaled photon frequency, ω/ωc , and on ω/ωc for fixed values of γ ψ are shown in Fig. 2.4.

2.4 Undulator Radiation For the sake of completeness and for further referencing, let us present a collection of formulae describing the characteristics of radiation (spectral-angular and spectral distributions, position and width of the peaks of emitted harmonics, etc.) by an ultrarelativistic charged particle moving in vacuum with a constant velocity v ≈ c in the (y, z) plane along the trajectory y(z) = a sin ku z,

where ku = 2π zλu ,

(2.15)

consisting of N periods of the length λu , which is called an undulator period. We will term a device in which an ultra-relativistic projectile moves in vacuum along the sinusoidal line as an “ideal undulator”. Such a motion can be realized, in particular, in a planar magnetic undulator, where bending of the particle’s trajectory is achieved by applying a periodic magnetic field directed perpendicular to the (y, z) plane: B = (Bx , 0, 0) with Bx = B0 sin(ku z) (see, e.g., [8, 10, 51, 52]) (Fig. 2.5).

2.4.1 General Formalism In an undulator the particle moves quasi-periodically, i.e., during the time interval T it completes a full oscillation along the y direction and simultaneously advances by the interval λu along the z direction, which is called the undulator axis; see Fig. 2.6.

28

2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

Fig. 2.5 Operational principle of a magnetic undulator (proposed by Ginzburg [51] and verified experimentally by Motz [53, 54]). The beam of ultra-relativistic particles moves along the axis of a periodic lattice of alternating magnets. The magnetic field forces the particles to move periodically in the transverse direction with a spatial period λu . As a result, the particle experiences an undulator motion, i.e., moves along a periodic, sine-like trajectory. The periodicity of the motion gives rise to the electromagnetic radiation of a specific type, the undulator radiation (UR). Due to the interference effects, the UR is emitted only at particular wavelengths, λn = λ1 /n (where n = 1, 2, 3 . . . ). The fundamental wavelength λ1 is proportional to λu /γ , where γ is the relativistic Lorentz factor of the electron

Hence, the position vector and the velocity of the particle satisfy the conditions r(t + T ) = r(t) + v0 T,

v(t + T ) = v(t),

(2.16)

T where v0  = T −1 0 v(t)dt is the mean velocity which is directed along the undulator axis and v0  ≈ c. Assuming that the Lorentz relativistic factor satisfies a strong inequality γ  1, one expands the functions ϕ(t) and f (t1 , t2 ) in powers of γ −1 . Retaining the dominant non-vanishing terms, one represents the right-hand side of (2.5) as follows: d3 E ω2 (1 + u)(1 + w) = αq 2 d(ω)dΩ 4π 2



 γ −2 w|I0 |2 2 2 2 + |θ I0 − cos φ I1 | + sin φ|I1 | . 1+w

(2.17) Here, w = u 2 /2(1 + u) and (θ, φ) are the emission angles with respect to the undulator axis. The notations I0 and I1 denote the integrals  I0 =

∞

dt e −∞

iω Φ(t)

 ,

I1 =

∞

dt −∞

v y (t) iω Φ(t) e c

(2.18)

2.4 Undulator Radiation

29

y

θ

ψ

a

z λu Fig. 2.6 Schematic representation of the ideal planar undulator. An ultra-relativistic charged projectile (filled circle) moves along sinusoidal trajectory (2.15) (thick solid curve). The radiation (wavy lines) is emitted due to the charge acceleration. The maximum turning angle (with respect to  the undulator axis z) of the undulating particle is ψ ∼ a/λu ∼

v 2y /c, where v 2y is the mean square

γ −1 ,

then all radiation is emitted within the cone ∼γ −1 . This of the transverse velocity. If θ0  limit corresponds to small values of the undulator parameter, K < 1 (see (2.20)). In the opposite case, ψ  γ −1 (and, correspondingly, K 2  1), the emission occurs in the cone ∼ψ. Due to the interference of the waves emitted from spatially different but similar parts of the trajectory, for each θ the intensity of UR is proportional to the square of undulator periods

where      1 t v2y (t ) K2 K2 y(t) 2 1 + (γ θ ) + + . dt − − θ cos ϕ 2 2 2 2 c 2γ c (2.19) The quantity K —a so-called undulator parameter—is related to the mean-square 2 transverse velocity v⊥ (i.e., perpendicular to the undulator axis). For a planar undulator, this quantity is defined as follows: t Φ(t) = 2γ 2

K 2 = 2γ 2

v2y c2

=

2γ 2 c2



T 0

  dt 2 a 2 v y (t) = 2π γ . T λu

(2.20)

Hence, the undulator parameter can be defined as the ratio of the maximum turning 

angle, ψ ∼ v2y /c, of the undulating ultra-relativistic particle to the natural cone ∼γ −1 of the radiation emission from each part of the projectile trajectory: K ∼ γ ψ; see Fig. 2.6. The features of the spectral-angular distribution of radiation are somewhat different in the two limiting cases: (a) K < 1 (or ψ < γ −1 )—a so-called undulator mode— and (b) K  1 (ψ  γ −1 )—a wiggler regime. The differences will be discussed further in this section. Formulae (2.17)–(2.20) allow one to analyze, both analytically and numerically, the radiation emitted by an ultra-relativistic projectile moving along arbitrary periodic planar trajectory, y = S(z). In particular, they can be applied to the motion along the sinusoidal line described by (2.15). The spectral-angular distribution of the energy emitted by an ultra-relativistic particle in an ideal planar undulator can be written in the following form:

30

2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

d3 E = S(ω, θ, φ) D N (η). ˜ d(ω)dΩ

(2.21)

The function S(ω, θ, φ), which does not depend on the number of periods N , is given by ω2 (1 + u)(1 + w) S(ω, θ, φ) = αq 2 2 2 4π 2 γ ω0   w |F0 |2 + |γ θ F0 − K cos φ F1 |2 + K 2 sin2 φ |F1 |2 . (2.22) × 1+w The functions Fm ≡ Fm (θ, φ) (m = 0, 1) stand for the integrals 

2π

Fm = 0

   K 2 ω

K ω

dψ cosm ψ exp i ηψ + sin(2ψ) − θ cos ϕ sin ψ . 8γ 2 Ω0 γ Ω0 (2.23)

The parameter η is given by ω

η= 2γ 2 Ω0

  K2 2 2 1+γ θ + . 2

(2.24)

The factor D N (η) ˜ on the right-hand side of Eq. (2.21) is defined as follows:  D N (η) ˜ =

sin π ηN ˜ sin π η˜

2 ,

(2.25)

where η˜ = η − n and n is a positive integer such that n − 1/2 < η ≤ n + 1/2. For N  1, the function D N (η) ˜ has a sharp and powerful maximum in the point η˜ = 0 (which corresponds to the case η = n = 1, 2, . . . ), where D N (0) = N 2 . This ˜ results in a peculiar form of the spectral-angular distribution of UR behavior of D N (η) which clearly distinguishes it from other types of electromagnetic radiation formed by a charge moving in external fields [4, 8–10, 51]. Namely, for each value of the emission angle the spectral distribution consists of a set of narrow and equally spaced peaks (harmonics). In the soft-photon limit, when the emitted harmonic energy ωn is small compared to the projectile energy ε, the frequencies ωn are found from the relation ωn =

2γ 2 Ω0 n , n = 1, 2, 3, . . . , 1 + γ 2 θ 2 + K 2 /2

(2.26)

which coincides with the definition of the harmonic frequencies of UR within the framework of classical electrodynamics [8–10]. If the terms ∼ω/ε are not neglected, the right-hand side of (2.26) defines the values of ωn (see (2.7)) [4].

2.4 Undulator Radiation

31

The magnitude of the undulator parameter K determines the number n max ∼ K 3 /2 of the emitted harmonics [4, 10]. For K  1, the emission occurs mainly in the first harmonic, which is emitted within the cone θ  θ0 ∼ γ −1 , and the frequency of which, ω1 ≈ 2γ 2 Ω0 , does not vary noticeably for θ  θ0 . The peak intensity is proportional to N 2 . This factor reflects the constructive interference of radiation emitted from each of the undulator periods and is typical for any system which contains N coherent emitters. In the opposite limit, K 2  1, the number of emitted harmonics is large and they are emitted within a wider cone: θ  θ0 ∼ K /γ . The frequency ωn of each harmonic is the largest for the emission in the forward direction (i.e., at θ = 0). The emission cone ΔΩn and the natural bandwidth Δωn one derives from (2.26): ΔΩn = 2π

1 + K 2 /2 , 2n N γ 2

Δωn 1 = . ωn nN

(2.27)

Two 3D plots in Fig. 2.7 help one to visualize the spatial behavior of the angular distribution of UR (at fixed frequency) [55, 56]. The  data refer to  K = 3, N = 17, and γ = 105 . In these figures, the quantity log dE 3 /dω dΩ (measured along the z-axis) as a function of the azimuthal, φ, and the polar, θ , angles of the photon emission is plotted for two harmonics: n = 7 (left figure) and n = 6 (right figure). The undulator axis lies along the z direction, and yz is the undulator plane. The positive y direction corresponds to φ = 0◦ . In these figures, the dimensionless variable θ/θ0 (with θ0 = K /γ = 30 μrad) is used to characterize the distribution of the radiation with respect to the polar angle. The graphs illustrate general features intrinsic to the planar UR in the case K 2  1. We first note that the intensity in the odd harmonic is governed by a powerful maximum in the forward direction, whereas there is nearly no radiation in even harmonics for θ = 0◦ (see Eqs. (2.30) and (2.31) below). The latter reaches its maximal values in the off-axis direction. Apart from the main peak either in the forward (for odd n) or nearly forward (for even n) direction, the radiation in a particular harmonic is emitted in a wide range of polar angles. In both of the figures, there are several clearly distinct off-axis peaks in which the radiation intensity reaches the maxima, although the magnitudes of dE 3 /d(ω)dΩ in these secondary maxima rapidly decrease with the polar angle (recall the log scale along the z-axis). Another feature to be mentioned is the absence of the axial symmetry in the shape of the angular distribution. More specifically, the radiation emitted within the undulator plane is concentrated in the cone θ ∼ θ0 , whereas the emission cone in the x z plane is θ⊥ ∼ 1/γ . Thus, the ratio θ /θ⊥ ∼ K characterizes the asymmetry in the angular distribution with respect to the azimuthal angle φ. This peculiarity is more pronounced for odd harmonics.

32

2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

Fig. 2.7 Angular distribution of undulator radiation for the odd (left panel) and even (right panel) harmonics. The undulator axis is aligned with the z direction. The undulator plane is (yz). The xand y axes are scaled with respect to the dimensionless variable θ/θ0

2.4.2 Spectral Distribution in the Forward Direction For the sake of reference, let us outline the basic formulae which describe the spectrum of radiation emitted in the forward direction (i.e., θ = 0 with respect to the undulator axis). In this case, the integrals Fm from (2.23)  can be expressed in terms

π of Anger’s function Jν (ζ ) = π −1 0 cos νφ − ζ sin φ dφ (see, e.g., [57]): 

J (ζ ) F0 = 2π eiηπ cos ηπ 2 η/2

F1 = π e

iηπ

sin

ηπ 2

J η+1 (ζ ) − J η−1 (ζ ) 2



where ζ =

2

K 2 ω

. 8γ 2 Ω0

(2.28)

Using these relations in (2.22), one derives the following explicit formulae for the on-the-axis spectral distribution [58]:

⎧ d3 E

⎪ ⎪ = D N (η) ˜ S(ω, 0, φ) ⎪ ⎪ dωdΩ θ=0 ⎪ ⎪ ⎨ ω2 (1 + u)  ηπ 2 J η (ζ ) w cos2 S(ω, 0, φ) = αq 2 2 ω2 ⎪ 2 2 γ ⎪ 0  ⎪ ⎪ 2 ⎪ K 2 (1 + w) 2 ηπ  ⎪ ⎩ sin J η+1 (ζ ) − J η−1 (ζ ) + 2 2 4 2 We remind that w = u 2 /2(1 + u) with u = ω/(ε − ω).

(2.29)

2.4 Undulator Radiation

33

To obtain the on-axis intensity at the frequencies ω ≈ ωn , one notices that Anger’s function of an integer index ν = n reduces to the Bessel function Jn (ζ ) [57]. Then, taking into account the selection rules imposed by the factors sin2 ηπ/2 and cos2 ηπ/2, one derives for η = n:

d3 E

dωdΩ

θ =0 ω≈ωn

16αq 2 γ 2 n 2 (1 + u)(2 + K 2 )2 ⎧  ⎨ K 2 (1+w) J n+1 (ζn ) − J n−1 (ζn ) 4 2 2 × 2 ⎩ w J n (ζn ) = D N (η) ˜

2

2

n = 1, 3, 5, . . . (2.30) n = 2, 4, 6, . . .

where ζn = n K 2 /(4 + 2K 2 ). The argument η˜ in the factor D N (η) ˜ is kept to enable one to reproduce the profile of the in the vicinity of the resonance (i.e., for ω ≈ ωn where |η| ˜  1). At the resonance D N (0) = N 2 , so the peak intensity is proportional to the squared number of the periods. This factor reflects the constructive interference of radiation emitted from each of the undulator periods, and is typical for any system which contains N coherent emitters. In the limit ω/ε → 0, Eqs. (2.29) and (2.30) reproduce the formulae obtained by means of classical electrodynamics. In particular, setting u = w = 0 on the righthand side of (2.30), one arrives at the following well-known formula for the peak intensity of UR at ω = ωn [10, 13]:

d3 E cl

dωdΩ

 4αq 2 γ 2 n 2 K 2 J n+1 (ζn ) − J n−1 (ζn ) = D N (η) ˜ × 2 2 θ=0 (2 + K 2 )2 0 ω≈ωn

2

n = 1, 3, . . . n = 2, 4, . . .

(2.31)

The distinguishing feature of the classical description of UR is the absence of the on-axis emission into even harmonics. Comparing the right-hand side of (2.31) with that of (2.30), one notices that this restriction is lifted when the radiative recoil is taken into account. However, it can be shown, that if ω  ε (this case is of prime interest for the emission from the undulator), then the intensities of the on-axis emission into even harmonics is much smaller than into odd ones. Further simplification of the right-hand sides of (2.30) and (2.31) can be achieved in the limit of small undulator parameters, K 2  1. In this limit, the argument of the Bessel functions is also small, ζn = n K 2 /(4 + 2K 2 ) ≈ n K 2 /4  1, therefore, one can write (see, e.g., [50]) Jν (ζn ) ≈ (ζn /2)ν /Γ (ν + 1), where Γ (ν + 1) is the Gamma function. Hence, for the emission in odd harmonics (ν = (n ± 1)/2), one notes that the peaks with n > 1 are strongly suppressed compared with the fundamental peak n = 1. For even harmonics (see the quasi-classical formula (2.30)), only the term n = 2 can be kept. Therefore, for K 2  1 one derives

34

2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

K=0.8

10

0

_

K=3 20

3

6

d E/d(hω)dΩ|θ=0 (10 /sr)

20

10 0 1

2

3

4

5

6

7

ω/ω1 Fig. 2.8 Spectral distribution of undulator radiation emitted by a ε = 5 GeV projectile in the forward direction calculated for two values of the undulator parameter K as indicated. The undulator period λu = 150 microns and number of periods N = 10. The horizontal axis shows the photon frequency measured in the units of the first harmonic ω1 at θ = 0, Eq. (2.26). The values of ω1 are equal to 1.2 and 0.27 MeV for the top and bottom graphs, respectively. In both graphs, the pronounced peaks correspond to the emission in the odd harmonics, n = 1, 3, 5, . . . . See also explanation in the text

d3 E

dωdΩ

θ =0 ω≈ωn

≈ D N (η) ˜

! αq 2 γ 2 K 2 1 + w ω = ω1 × 4w ω = ω2 = 2ω1 . (1 + u)

(2.32)

In the classical limit u = w = 0, the emission in ω = ω2 is nullified, so the factor in front of the curly bracket describes the peak intensity of the fundamental harmonic. Two graphs in Fig. 2.8 illustrate the mentioned peculiarities of the UR emitted in the forward direction. The spectra are calculated for the two indicated values of the undulator parameter K , and for ε, λu , and N indicated in the caption. The pronounced peaks correspond to the emission in the odd harmonics: n = 1, 3, 5 are visible in the upper graph (although the intensities of the higher harmonics are much smaller than for the fundamental one, n = 1), whereas for a larger value of K — the lower graph, the total number of the emitted (odd) harmonic is ≈ 15; only the n = 1, 3, 5, 7 harmonics are plotted in the figure. The intensity of the emission into even harmonics is negligible due to the small value of the coefficient w in the first term in the brackets on the right-hand side of the second formula in (2.29). Indeed, for the indicated values of ε and ω1 (see the caption), this coefficient, which determines the even-harmonic peaks (see (2.30)), written as w ≈ u 2 /2 ≈ 0.5 (nω1 /ε)2 does not exceed 10−7 for n = 2 in the upper graph and for n = 10 in the lower graph. Hence, the presented spectra are, in fact, classical since they refer to the limit ω  ε.

Appendix: The Bethe–Heitler Approximation

35

Appendix: The Bethe–Heitler Approximation In the elementary process of bremsstrahlung (BrS), a charged projectile emits a photon being accelerated by the static field of a target (nucleus, ion, atom, etc.). For ultra-relativistic projectiles, the Bethe–Heitler (BH) approximation [31] (with various corrections due to Bethe et al. [59, 60] and Tsai et al. [61]) is the simplest and the most widely used one. For the sake of reference below in this section, we present the relevant formulae for the case of ultra-relativistic electrons/positrons scattering from a neutral atom treated within the Molière approximation [62]. Starting from Eq. (3.80) in Ref. [61], one can write the following formula for the cross section differential with respect to the photon energy ω and to the emission angle Ω = (θ, φ) (but integrated over the angles of the scattered electron): ! 4αr02 γ 2 4(1 − x) d2 σ = 2 − 2x + x 2 − d(ω)dΩ π ω 1+ξ  4(1 − x) F − 1 + ln(1 + ξ ) + (1 + ξ )2 (1 + ξ )2  "  1−x 6 6 . + −Z (Z + 1) 1 − 1+ξ (1 + ξ )2 (1 + ξ )2

(2.33)

Here, α ≈ 1/137 is the fine structure constant, r0 = e2 /mc2 ≈ 2.818 × 10−13 cm is the classical electron radius, x = ω/ε, and ξ = (γ θ )2 . The factor F is defined by Eqs. (3.5), (3.44), and (3.45) from Ref. [61]. In the ultra-relativistic limit (more exactly, for γ  103 ), it can be written as follows:  F =Z

2

    184 1194 2 ln 1/3 − 1 − f α Z ) + Z ln 2/3 − 1 , Z Z

(2.34)

     2 2 2 −1 where the function f (α Z )2 = (α Z )2 ∞ (with ζ = α Z ) n=1 n n + (α Z ) is the Coulomb correction to the first Born worked out in Refs. [59,   approximation 60]. In the limit (α Z )2  1, the term f (α Z )2 can be ignored. For example, for a   Si atom (Z = 14) f (α Z )2 ≈ 0.0126  1. The term proportional to Z 2 on the right-hand sides of (2.33) and (2.34) stands for the contribution of the elastic BrS process in which the target atom does not change its state during the collision. The terms ∝ Z are due to the inelastic BrS channels, when the atom becomes excited or ionized. To calculate the cross section of BrS radiated into the cone with the opening angle θ0 , one integrates Eq. (2.33) over the emission angles θ = [0, θ0 ] and φ = [0, 2π ]. The result reads

36

2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

 !  4αr02 4 dσ

dσ 26 1 − x + Z (Z + 1) 1 − = + d(ω) θ≤θ0 d(ω) ω D0 D0 9D02  "  2(1 − x) 4(1 − x) F + ln D0 2 , − 2 − 2x + x − + D0 D0 3D02

(2.35)

with D0 = 1 + (γ θ0 )2 . In the limit of large emission angles when θ0  1/γ , the second term on the right-hand side goes to zero. Therefore, the first term stands for the cross section differential in the photon energy but integrated over the whole range of the emission angles. Its explicit expression is as follows (cf. Eq. (3.83) in Ref. [61]): dσ = d(ω)

 4αr02 d2 σ ≈ (4 − 4x + 3x 2 ) F d(ω)dΩ 3ω 0 0  1−x . (2.36) +Z (Z + 1) 3





2π

dφ

∞

θ dθ

To calculate the cross section of the elastic BrS, one substitutes Z (Z + 1) → Z 2 on the right-hand sides of Eqs. (2.33), (2.35), and (2.36) as well as ignores the last term in Eq. (2.34). The latter approximation leads to the following reduction:    184 F → Fel = Z 2 ln 1/3 − 1 − f (α Z )2 . Z

(2.37)

Then, the single differential cross section of elastic BrS emitted within the cone 0 ≤ θ ≤ θ0 is given by

  2 ! 1−x dσel 26 dσel

4 2 Z + 4αr = + 1 − 0 d(ω) θ≤θ0 d(ω) ω D0 D0 9D02  "  2(1 − x) 4(1 − x) Fel + ln D0 , (2.38) − 2 − 2x + x 2 − + D0 Z 2 D0 3D02 where dσel 4αr02 Z 2  Fel 1−x = (4 − 4x + 3x 2 ) 2 + d(ω) 3 ω Z 3

(2.39)

is the Bethe–Heitler spectrum of elastic BrS. Within the framework of less accurate approximation, used frequently for quantitative estimates (see, e.g., [15, 63]), one ignores the non-logarithmic terms in (2.37) Fel ≈ ln

184 . Z 1/3

(2.40)

Appendix: The Bethe–Heitler Approximation

37 θ0=π rad

0.5

θ0=1.2 mrad

_

0

3

dEel/d(hω)⎥θ≤θ (×10 )

0.6

0.4 θ0=0.6 mrad

0.3 0.2

θ0=0.2 mrad

0.1 0 0

10

20

30

40

Photon energy (MeV) Fig. 2.9 Bethe–Heitler spectra of the energy dE el /d(ω) radiated via the elastic BrS channel by a ε = 855 MeV electron in amorphous silicon of the thickness 50 microns. Different curves correspond to different values of the emission cone angle θ0 as indicated. The curve θ0 = π rad stands for the spectral distribution integrated over the whole range of the emission angles. The elastic BrS cross section was calculated within the logarithmic approximation; see Eq. (2.40)

In order to calculate the spectral-angular distribution of the radiated energy d2 E/d(ω)dΩ in an amorphous target of the thickness L much less than the radiation length [64], one multiplies Eq. (2.33) by the photon energy ω, by the volume density n of the target atoms, and by L: d2 σ d2 E = n L ω . (2.41) d(ω)dΩ d(ω)dΩ

Spectral distribution dE/d(ω) of the energy radiated within the cone θ ≤ θ0 is θ≤θ0

obtained from

(2.41) by substituting the double differential cross section either with

(for the total emitted energy) or with dσel /d(ω) (if accounting dσel /d(ω) θ≤θ0

for elastic BrS only).

Figure 2.9 illustrates the spectral distributions dE el /d(ω)

θ≤θ0

θ≤θ0

of elastic BrS

formed during the passage of a ε = 855 MeV electron through a 50 μm thick amorphous silicon (n = 5 × 1022 cm−3 ). The curves were calculated for different values of the emission cone angle as indicated. The value θ0 = 0.21 mrad corresponds to the limit of small emission angles (γ θ0 )2  1 where γ −1 ≈ 6 × 10−3 for the indicated

steadily incident energy. For each photon energy, the magnitude of dE el /d(ω) θ≤θ0

increases with θ0 reaching its upper limit at θ0 = π which corresponds to the cross section integrated over the whole range of the emission angle; see Eq. (2.39).

38

2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

References 1. Jackson, J.D.: Classical Electrodynamics. Wiley, Hoboken (1999) 2. Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Course of Theoretical Physics, vol. 2. Pergamon Press, Oxford (1971) 3. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, vol. 3. Elsevier, Oxford (2004) 4. Baier, V.N., Katkov, V.M., Strakhovenko, V.M.: Electromagnetic Processes at High Energies in Oriented Single Crystals. World Scientific, Singapore (1998) 5. Kumakhov, M.A., Komarov, F.F.: Radiation from Charged Particles in Solids. AIP, New York (1989) 6. Kotredes, L., Talman, R.: An explicit formula for undulator radiation. Laboratory of Nuclear Studies, Cornell University, CBN 01-14 (2001) 7. Talman, R.: Exploiting undulator radiation for X-ray interferometry. Nucl. Instrum. Method A 489, 519–542 (2002) 8. Barbini, R., Ciocci, F., Datolli, G., Gianessi, L.: Spectral properties of the undulator magnets radiation: analytical and numerical treatment. Rivista del Nuovo Cimento 13, 1–65 (1990) 9. Talman, R.: Accelerator X-Ray Sources. Wiley-VCH, Weinheim (2006) 10. Alferov, D.F., Bashmakov, Yu.A., Cherenkov, P.A.: Radiation from relativistic electrons in a magnetic undulator. Sov. Phys. - Uspekhi 32, 200–227 (1989) 11. Andersen, J.U., Bonderup, E., Pantell, R.H.: Channeling radiation. Annu. Rev. Nucl. Part. Sci. 33, 453–504 (1983) 12. Bazylev, V.A., Zhevago, N.K.: Intense electromagnetic radiation from relativistic particles. Sov. Phys. Usp. 25, 565–595 (1982) 13. Kim, K.-J.: Characteristics of synchrotron radiation. In: X-ray Data Booklet, pp. 2.1–2.16. Lawrence Berkeley Laboratory, Berkley (2009). http://xdb.lbl.gov/xdb-new.pdf 14. Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P.: Course of Theoretical Physics. Quantum Electrodynamics, vol. 4. Pergamon Press, Oxford (1982) 15. Akhiezer, A.I., Berestetsky, V.B.: Quantum Electrodymanics. Interscience Publishers, New York (1965) 16. Gould, R.J.: Electromagnetic Processes. Princeton University Press, Princeton (2006) 17. Akhiezer, A.I., Shul’ga, N.S.: Radiation of relativistic particles in single crystals. Sov. Phys. Uspekhi 25, 541–564 (1982) 18. Klepikov, N.P.: Emission of Photons and Electron-Positron Pairs in a Magnetic Field. Zh. Eksp. Teor. Fiziki 26, 19 (1954). (in Russian) 19. Kumakhov, M.A., Wedell, R.: Theory of radiation of relativistic channeled particles. Phys. Stat. Sol. (b) 84, 581–593 (1977) 20. Saenz, A.W., Überall, H.: Calculation of electron channeling radiation with realistic potential. Nucl. Phys. A 372, 90–108 (1981) 21. Baryshevsky, V.G.: Channeling, Radiation and Reactions in Crystals at High Energies. Buelorussian State University Press, Minsk (1982). (in Russian) 22. Klenner, J., Augustin, J., Schäfer, A., Greiner, W.: Photon-photon interaction in axial channeling. Phys. Rev. A 50, 1019–1026 (1994) 23. Baier, V.N., Katkov, V.M.: Processes involved in the motion of high energy particles in magnetic field. Sov. Phys. - JETP 26, 854–860 (1968) 24. Solov’yov, A.V., Schäfer, A., Greiner, W.: Channeling process in a bent crystal. Phys. Rev. E 53, 1129–1137 (1996) 25. Arutyunov, V.A., Kudryashov, N.A., Samsonov, V.M., Strikhanov, M.N.: Radiation of ultrarelativistic charged particles in a bent crystal Nucl. Phys. B 363, 283–300 (1991) 26. Arutyunov, V.A., Kudryashov, N.A., Strikhanov, M.N., Samsonov, V.M.: Synchrotron and undulator radiation by fast particles in bent crystal. Sov. Phys. Tech. Phys. 36, 1–3 (1991) 27. Heitler, W.: The Quantum Theory of Radiation. Dover, Toronto (1984) 28. Pratt, R.H., Feng, I.J.: Electron-atom bremsstrahlung. In: Craseman, B. (ed.) Atomic InnerShell Physics, pp. 533–580. Plenum, New York (1985)

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29. Haug, E., Nakel, E.: The Elementary Process of Bremsstrahlung. World Scientific, Singapore (2004) 30. Shaffer, C.D., Pratt, R.H.: Comparison of relativistic partial-wave calculations of triply differential electron-atom bremsstrahlung with simpler theories. Phys. Rev. A 56, 3653–3658 (1997) 31. Bethe, H., Heitler, W.: On the stopping of fast particles and on the creation of positive electrons. Proc. Roy. Soc. Lond. Ser. A 146, 83–112 (1934) 32. Ter-Mikaelian, M.: High-Energy Electromagnetic Processes in Condensed Media. Wiley, New York (1972)) 33. Landau, L.D., Pomeranchuk, I.J.: The limits of applicability of the theory of bremsstrahlung and pair creation at high energies. Dokladi Akad. Nauk SSSR 92, 535–537 (1953) (English translation in: Landau, L.D.: The collected papers of L.D. Landau. Pergamon Press, Oxford (1965)) 34. Migdal, A.B.: Bremsstrahlung and pair production in condensed madia at high-energies. Phys. Rev. 103, 1811–1820 (1956) 35. Palazzi, G.D.: High-energy bremsstrahlung and electron pair production in thin crystals. Rev. Mod. Phys. 40, 611–631 (1968) 36. Überall, H.: High-energy interference effect of bremsstrahlung and pair production in crystals. Phys. Rev. 103, 1055–1067 (1965) 37. Andersen, J.U.: Channeling radiation and coherent bremsstrahlung. Nucl. Instrum. Method Phys. Res. 170, 1–5 (1980) 38. Ter-Mikaelian, M.: Electromagnetic radiative processes in periodic media at high energies. Phys. Uspekhi 44, 571–596 (2001) 39. Shul’ga, N.F.: Advances in coherent bremsstrahlung and LPM effect studies. In: Dabagov, S.B., Palumbo, L., Zichichi, A. (eds.) Charged and Neutral Particles Channeling Phenomena Channeling 2008, pp. 11–35. World Scientific, Singapore (2010) 40. Sørensen, A.H.: Channeling, bremsstrahlung and pair creation in single crystals. Nucl. Instrum. Meth. B 119, 2–29 (1996) 41. Uggerhøj, U.I.: The interaction of relativistic particles with strong crystalline fields. Rev. Mod. Phys. 77, 1131–1171 (2005) 42. Medenwaldt, R., Møller, S.P., Sørensen, A.H., Uggerhøj, E., Elsener, K., Hage-Ali, M., Siffert, P., Stoquert, J., Sona, P.: Coherent bremsstrahlung and channeling radiation from 40 and 150 GeV electrons and positrons traversing Si and diamond single crystals near planar directions. Phys. Lett. 260B, 235–239 (1991) 43. Bondarenco, M.V.: Coherent bremsstrahlung in a bent crystal. Phys. Rev. A 81, 052903 (2010) 44. Shul’ga, N.F., Boiko, V.V.: Coherent effect in the radiation of relativistic electrons in the field of bent atomic crystallographic planes. JETP Lett. 84, 305–307 (2006) 45. Shul’ga, N.F., Boyko, V.V., Esaulov, A.S.: New mechanism of jump formation in a spectrum of coherent radiation by relativistic electrons in the field of periodically deformed crystal planes of atoms. Phys. Lett. 372A, 2065–2068 (2008) 46. Sokolov, A.A., Ternov, I.M.: Synchrotron Radiation. Pergamon Press, Oxford (1968) 47. E.E. Koch: Handbook of Synchrotron Radiation, vol. 1–4. North-Holland Publishing Company, Amsterdam, New York, Oxford (1983–1996) 48. Jaeschke, E.J., Khan, Sh., Schneider, J.R., Hastings, J.B. (eds.): Synchrotron Light Sources and Free-Electron Lasers. Springer International Publishing Switzerland (2016) 49. Ternov, I.M.: Synchrotron radiation. Phys. Usp. 38, 409 (1995) 50. Abramowitz, M., Stegun, I.E.: Handbook of Mathematical Functions. Dover, New York (1964) 51. Ginzburg, V.L.: Radiation of microwaves and their absorption in air. Bull. Acad. Sci. USSR, Ser. Phys. 11, 165 (1947). (in Russian) 52. Rullhusen, P., Artru, X., Dhez, P.: Novel Radiation Sources Using Relativistic Electrons. World Scientific, Singapore (1998) 53. Motz, H.: Applications of the radiation from fast electron beams. J. Appl. Phys. 22, 527–534 (1951)

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2 Fundamental Mechanisms of the Radiation Emission in Vacuum and Medium

54. Motz, H., Thon, W., Whitehurst, R.N.: Experiments on radiation by fast electron beams. J. Appl. Phys. 24, 826–833 (1953) 55. Korol, A.V., Solov’yov, A.V., Greiner, W.: Photon emission by an ultra-relativistic particle channeling in a periodically bent crystal. Int. J. Mod. Phys. E 8, 49–100 (1999) 56. Korol, A.V., Solov’yov, A.V., Greiner, W.: Channeling and Radiation in Periodically Bent Crystals, 2nd edn. Springer, Berlin (2014) 57. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals. Series and Products. Academic, New York (1965) 58. Korol, A.V., Solov’yov, A.V., Greiner, W.: Parameters of the crystalline undulator and its radiation for particular experimental conditions. Proc. SPIE 6634, 66340P (2007) 59. Bethe, H.A., Maximon, L.C.: Theory of bremsstrahlung and pair production. I. Differential cross section. Phys. Rev. 93, 768–784 (1954) 60. Davis, H., Bethe, H.A., Maximon, L.C.: Theory of bremsstrahlung and pair production. II. Integral cross section for pair production. Phys. Rev. 93, 788–795 (1954) 61. Tsai, Y.-S.: Pair production and bremsstrahlung of charged leptons. Rev. Mod. Phys. 46, 815 (1974) 62. Molière, G.: Theorie der Streuung schneller geladener Teilchen I: Einzelstreuung am abgeschirmten Coulomb-Feld. Z. f. Naturforsch. A 2, 133–145 (1947) 63. Bak, J., Ellison, J.A., Marsh, B., Meyer, F.E., Pedersen, O., Petersen, J.B.B., Uggerhöj, E., Østergaard, K.: Channeling radiation from 2-55 Gev/c electrons and positrons: (I) planar case. Nucl. Phys. B. 254, 491–527 (1985) 64. Nakamura, K., et al. (Particle Data Group): Review of particle physics. J. Phys. G: Nucl. Part. Phys. 37, 075021 (2010)

Chapter 3

Light Sources at High Photon Energies

3.1 Main Characteristics of Light Sources One of the radiometric units, frequently used to compare different LS in the shortwavelength range, is brilliance, B. It is defined in terms of the number of photons ΔNω of frequency ω within the interval [ω − Δω/2, ω + Δω/2] emitted in the cone ΔΩ per unit time interval, unit source area, unit solid angle, and per bandwidth (BW) Δω/ω [1–3]. To calculate this quantity, it is necessary to know the beam electric current I , transverse sizes σx,y , and angular divergences φx,y as well as the divergence angle φ of the radiation and the “size” σ of the photon  beam. Explicit  2 2 expression for B measured in photons/s/mrad /mm /0.1% BW reads [4] B=

ΔNω I , 103 (Δω/ω) (2π )2 E x E y e

(3.1)

where e is the elementary charge. The quantities E x,y =

  2 2 σ 2 + σx,y φ 2 + φx,y

(3.2)

are the √ total emittance of the photon source in the transverse directions with φ = ΔΩ/2π and σ = λ/4π φ being the “apparent” source size calculated in the diffraction limit [5]. To ensure the aforementioned units for brilliance, the quantities σ, σx,y should be considered measured in millimeters, and φ, φx,y —in milliradians. Another quantity, frequently used to characterize a light source, is flux F. It stands for the number of photons per second emitted in the cone   ΔΩ and in a given bandwidth. Measured in the units of photons/s/0.1%BW , the flux is related to ΔNω as follows [4]: F=

ΔNω 3 10 (Δω/ω)

I . e

(3.3)

The product ΔNω I /e on the right-hand sides of Eqs. (3.1) and (3.3) represents the number of photons per second (intensity) emitted in the cone ΔΩ and frequency © Springer Nature Switzerland AG 2022 A. Korol and A. V. Solov’yov, Novel Lights Sources Beyond Free Electron Lasers, Particle Acceleration and Detection, https://doi.org/10.1007/978-3-031-04282-9_3

41

42

3 Light Sources at High Photon Energies

interval Δω. Using the peak value of the current, Imax , on the right-hand sides of Eqs. (3.1) and (3.3), one calculates the peak brilliance, Bpeak , and flux, Fpeak . The number of photons ΔNω emitted within BW Δω is related to the spectral distribution d3 E/dω dΩ of the radiated energy in the forward direction:   Δω ΔNω = d3 E/dω dΩ θ=0 ΔΩ . ω

(3.4)

The driving force behind the development of light sources is the optimization of their brilliance (or spectral brightness), which is the figure of merit of many experiments [3].

3.2 Synchrotron Radiation Light Sources Synchrotron radiation is the electromagnetic radiation emitted by charged particles when the particles’ trajectories are subjected to a magnetic field, which is, for example, generated in bending magnets in circular accelerators.1 In an ultra-relativistic limit, the radiation is collimated in a thin cone with an opening angle ∼ γ −1 ; see Fig. 2.4. Synchrotron radiation covers a wide spectral range and can be tuned from the infrared to the X-rays. The first description of the emission of relativistic particles following a circular orbit was given in the early 1910s [7]. Later on, in the 1940s, the problem attracted a lot of attention as it became clear that the rapid growth of the radiated power with electron energy (proportional to ε4 ) brings a problem in the construction of higher electron energy accelerators. The modern derivation of the basic expressions used in the description of synchrotron radiation was provided by Ivanenko and Pomeranchuk [8] and by Schwinger [9]. The first direct observation of the radiation was reported in 1947 [10] at the 70 MeV synchrotron in the General Electric Laboratories (USA). The radiation was seen as a spot of a brilliant white light by an observer looking into the vacuum tube tangent to the electron’s orbit. Initially, some electron storage rings designed and built for nuclear and subnuclear physics started to be for some fraction of the time used as sources of photons for experiments in atomic, molecular, and solid-state physics. These machines are nowadays referred to as “first-generation light sources” [3]. The experimental results obtained with the synchrotron radiation stimulated the construction of dedicated rings, designed and optimized to serve exclusively as light sources. Examples of these “second-generation” machines are the BESSYI ring in Berlin, the two National Synchrotron Light Source rings in Brookhaven, NY (USA), the SuperACO ring in Orsay, near Paris, and the Photon Factory in Tsulcuba (Japan).

1

In literature, one can find another term for this type of radiation—magnetic bremsstrahlung. This term is more frequently used in application to the astrophysical problems; see Ref. [6].

3.2 Synchrotron Radiation Light Sources

43

One of the most important challenges for synchrotron radiation sources has always been to reduce emittance (that is, the product of the spatial size and the angular spread) of the electron beam, because the brilliance of a source is inversely proportional to its emittance (3.1). The typical emittances of “third-generation” synchrotron radiation sources, which started operation in the 1990s (such as the European Synchrotron Radiation Facility (ESRF) in France, the Advanced Photon Source (APS) in the US, and SPring-8 in Japan, BESSY II), were initially several nanometer radian (nm rad), eventually decreasing to a few nm rad after the operation conditions were optimized. However, further improvements were not feasible [11]. This situation changed in the last decade with the ambitious proposal of the MAX IV ring in Sweden, which aims to significantly reduce emittance down for a 3 GeV storage ring [12]. The key innovation is to introduce a series of miniature dipole magnets to increase the total number of the magnets along the synchrotron ring and thus reduce the bend angle per dipole magnet. This brings important benefits in reducing emittance, which is proportional to the cube of the bend angle. The MAX IV facility was inaugurated in June 2016. Modifications aimed at decreasing emittance have been implemented (or are planned to be introduced) at many other synchrotron radiation facilities including the ESRF, APS, SPring-8, and PETRA III at DESY. These new sources are often called diffraction-limited synchrotron radiation (DLSR) sources, because their emittances approach the limitations set by the wavelength of light. The high beam quality of these facilities will stimulate increased use of coherence-related imaging technologies. Among them, coherent X-ray diffraction imaging is a method for enabling ultrahigh-resolution imaging of isolated objects without the use of any imaging optics. Another aspect of DLSR sources is that their reduced emittance decreases the horizontal size of the source from the submillimeter scale to a few tens of micrometers, allowing for a nearly circular-shaped beam. In contrast, the horizontally elongated beams seen in existing synchrotron radiation sources can only be turned into a circular shape by the use of a pinhole filter. Although the realization and the utilization of DLSR sources are technically challenging due to the requirement for ultimate stability down to a 10 nrad level, the resulting gain of photon flux for the nanobeam drastically increases by three orders of magnitude. Furthermore, a new optical scheme for harmonic separation, instead of a conventional X-ray monochromator with a narrower bandwidth, enables high-quality undulator radiation at a specific harmonic to be extracted, which will give an additional gain by two orders of magnitude for a broad range of applications [11].

3.3 Undulators and Wigglers The intensity of synchrotron radiation can be increased by reducing the curvature radius of particle’s trajectory; see Eq. (2.13). This is realized in the so-called insertion devices, undulators and wigglers, which create a periodic permanent magnetic field with a sinusoidal dependence along the electron trajectory; see Fig. 2.5. The resulting

44

3 Light Sources at High Photon Energies

Fig. 3.1 SOLEIL synchrotron radiation spectral range for the various insertion devices. Figure from Ref. [13]

Lorenz force on the drifting electrons modifies their straight trajectory into a zig-zag one, producing a large number of bends with intense radiation emission. In the case of a planar undulator, the synchrotron radiation on the axis is emitted at the resonance wavelength and its odd harmonics of order n: λn = λ0 (1 + K 2/2 + γ 2 θ 2 )/2nγ 2 where θ denotes the emission angle with respect to the axis. The undulator parameter K is related to the magnetic field period λ0 and the amplitude value B0 of the magnetic flux density as follows: K = 93.4λ0 (m)B0 (T) [2]. In the “undulator” regime K  1, the radiation emitted at each inversion interferes with the one produced in the previous inversions. These interferences are constructive for the resonance wavelength and the radiation is produced in a very intense spectral lines (harmonics) form. The sharpness of the harmonics can be affected by the observation angle, the energy spread, and the emittance of the electron beam. In the “wiggler” regime (K 2  10), the radiation of the different harmonics overlaps and the spectrum approaches the incoherent sum of the synchrotron radiation spectra formed in the fields of individual magnets. The wavelength of the emitted radiation can be varied by a modification of the undulator magnetic field (by changing the gap for permanent magnet insertion devices or the power supply current for electromagnetic insertion devices). Undulator radiation can be characterized as quasi-monochromatic and tunable. The particular choice of the undulator characteristics and technology enables to optimize the desired spectral range for a given beamline. Figure 3.1 presents the spectra calculated for different insertion devices of SOLEIL 2.75 GeV synchrotron LS [13]. The electric field of the radiation is in the plane of the electron trajectory. For a vertical magnetic field, the electron follows an undulator trajectory in the horizontal

3.4 X-Ray Free-Electron Lasers

45

plane. Combining magnetic fields in both planes with a possible phasing between them enable to provide various types of polarization from linear vertical, linear horizontal to circular one, or more generally elliptical one. Different technologies that provide the possibility of any type of polarization are discussed in Ref. [13]. Also, in the cited paper, one finds an extensive list of existing synchrotron radiation facilities around the world.

3.4 X-Ray Free-Electron Lasers The free-electron laser (FEL) concept was introduced by Madey in 1971 [14]. He has demonstrated that if an additional electromagnetic wave of appropriate wavelength and phase is propagating parallel to an electron in an undulator, the electron is coupled to this field exchanging energy. The energy exchange can result in deceleration of the electron and amplification of the radiation field. In this sense, the electron moving through the undulator operates as an amplifier. Placing the undulator inside an optical resonator with mirrors at both ends leads to the production of coherent light analogous to conventional lasers. Madey calculated the gain factor g that defines the increase in the number of emitted photons at a resonance frequency due to the emission stimulation of the beam particles. In a small-signal regime, the gain factor is proportional to the undulator length and the volume density of the beam particles and scales as γ −3 with the beam energy. In this operational mode (referred to as the oscillator mode), the laser field is stored in an optical cavity, enabling interaction with the electron beam on many passes. FEL oscillators cover a spectral range from the microwaves to vacuum ultra-violet, where mirrors are available. The first FEL oscillator reported [15] operated in the infrared wavelengths. One of the major advantages of the FEL LSs is the tunability of the wavelength of the emitted radiation. It is achieved by modifying the magnetic field of the undulator in a given spectral range set by the electron beam energy. Operation at short wavelengths requires high beam energies for reaching the resonant wavelength and thus long undulators and high beam density for ensuring a sufficient gain. As the wavelength of radiation decreases so does the effectiveness of storing the emitted light in an optical cavity due to the limited performance of the mirrors. To overcome this difficulty, a so-called “high-gain” operational regime has been proposed. It was shown that to achieve sufficient gain for a short-wavelength radiation, the FEL could operate as an amplifier in the so-called self-amplified spontaneous emission (SASE) mode, and the high-gain amplification of the initial spontaneous radiation is obtained with only one pass through a long undulator (100 m range for 1 angstrom) until saturation is reached [16–18]. One finds the details on the theory and physics behind the high-gain operational regime of FELs in [19–26] and references therein. Recent reviews on the achievements made in constructing and further advancing of the x-ray FEL facilities, both commissioned and planned, that generate coherent photon pulses with time duration of a few to 100 fs over a wavelength range

46

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extending from about 100 nm to less than 1 Å as well as on a number of new scientific results obtained in atomic and molecular sciences, in areas of physics, chemistry, biology, and applied science can be found in Refs. [11–13, 13, 24, 25, 27–30]. The main difference between spontaneous undulator radiation and FEL radiation is that in an undulator there is constructive interference between the electromagnetic waves emitted by one electron at different points of its trajectory. In addition to this, in the FEL process, the waves of different electrons also interfere constructively. This happens due to a positive feedback process in which electrons self-organize as it is illustrated by Fig. 3.2. Through interaction with the initially incoherent radiation emission, the intensity of which is proportional to the electron density, electrons form into microbunches separated by the radiation wavelength. The narrowbandwidth emission is then coherent, scaling as N 2 where N is the number of electrons emitting collectively (N  106 for x-ray FELs [25]). The amplification process results in almost full transverse coherence, and as linac-based accelerators for FELs deliver bunches with very high peak current, the output peak brightness can exceed that of storage ring sources by orders of magnitude. Nowadays, four FEL facilities, FERMI@Elletra, Italy [31], FLASH at DESY, Germany [32], LCLS at SLAC, USA [24, 27, 33], and SACLA at SPring-8, Japan [17], provide femtosecond short laserlike photon pulses to user experiments. Their wavelengths range from the EUV and soft x-rays (FERMI, FLASH) to hard x-rays (LCLS, SACLA). The peak brilliance usually exceeds 1030 photons s−1 mrad−2 mm−2 per 0.1% BW, orders of magnitude more than third-generation synchrotron-based light sources can provide; see Fig. 3.3 that compares peak brilliances of modern synchrotron facilities and undulators with that of several XFELs (TESLA Test Facility (TTF) [34], FLASH [32], and LCLS [33]).

3.5 Alternative Schemes for Short-Wavelength Light Sources 3.5.1 Compton Scattering γ -Ray Light Source Another type of modern LS, which does not utilize magnets, is based on the Compton scattering process [35, 36]. In this process, a low-energy (eV) laser photon backscatters from an ultra-relativistic electron, thus, acquiring an increase in the energy proportional to the squared Lorentz factor γ = ε/mc2 in Fig. 3.4. This method has been used for producing gamma-rays in a broad, 101 keV–101 MeV, energy range [37, 38]. Applying the four-momentum conservation law to a photon–electron collision (see, e.g., [40]), one finds the following relationship between the energies of the incoming, ω, and the scattered, ω , photons:

3.5 Alternative Schemes for Short-Wavelength Light Sources

47

Fig. 3.2 Illustrative representation of a SASE FEL, in which both the radiation power and the electron beam microbunching grow as a bunch of initially randomly phased electrons propagate through a magnetic undulator. The electric field of the emitted radiation couples to the electron transverse velocity allowing an energy exchange. This coupling enables a positive feedback process in which the electrons start to microbunch, coherently enhancing their emission, which acts back on the electrons further enhancing the microbunching, and so on, giving an exponential growth of the radiation intensity. (Figure’s layout from Ref. [25])

ω =

1 − β cos θ ω. 1 − β cos θ  + δ(1 − cos φ)

(3.5)

Here, β = v/c, δ = ω/ε, θ and θ  are the angles between the momenta of the incoming and scattered photons and that of the incident electron, and φ stands for the angle between the two photons. For a collision between an ultra-relativistic electron (γ = (1 − β 2 )−1/2  1, β ≈ 1) and a low-energy photon, ω  ε, the energy of the scattered photon is peaked along the direction of the incident electron. The backscattered photon has the maximum energy in a head-on collision with θ = π , θ  = 0, and φ = π : ω ≈

4γ 2 ω ≈ 4γ 2 ω . 1 + 4γ 2 δ

(3.6)

The latter equation is written in the limit of small recoil (i.e., when γ ω  mc2 ). The energy of a backscattered photon scales with the incoming electron energy as ε2 , so that 0.1 . . . 102 MeV photons can be obtained by scattering 1 eV laser photons from ∼ (0.1 . . . 1) GeV electrons.

48 Fig. 3.3 Peak brilliance of modern synchrotrons, undulators, and XFELs. (Figure from Ref. [34])

Fig. 3.4 Laser-Compton scattering (LCS) process (Figure’s layout Ref. [39])

3 Light Sources at High Photon Energies

3.5 Alternative Schemes for Short-Wavelength Light Sources

49

The first experimental demonstrations of gamma-ray production due to the Compton scattering were carried out by several groups over fifty years ago [41–43]. The first Compton gamma-ray source facility for nuclear physics research was brought to operation in Frascati [36]. This facility produced gamma-ray beams with energies up to 80 MeV and an on-target flux of up to 5 × 105 photons/s. Following the success of the facility at Frascati, several more Compton gamma-ray source facilities for nuclear physics research were brought to operation around the world starting in the 1980s. Reviews on Compton gamma-ray beams and some of the commissioned facilities are available in [44–49]. To be mentioned is the High-Intensity Gamma-ray Source (HIGS) at Duke University which is the first dedicated Compton gamma-ray facility employing as the photon driver a high-power FEL [50]. The HIGS facility is a high-flux, nearly monochromatic, and highly polarized gamma-ray source within the 1–100 MeV photon energy range. A maximum total flux of about 3 × 1010 photons/s at 10 MeV has been achieved at HIGS, which is two or three orders of magnitude more than produced by other existing facilities [49]. During the last decade or so, while a few Compton gamma-ray source facilities (e.g., LEGS and Graal) ceased operation after completing their research missions, other facilities continue to flourish with accelerator and laser system upgrades that improve beam performance and enable new capabilities. In the meantime, a few new facilities are under construction around the world. One finds a list of major operational laser Compton gamma-ray sources and new development projects in Ref. [49], Sect. 5.

3.5.2 Gamma Factory The Compton scattering also occurs if the scatterer is an atomic (ionic) electron which moves being bound to a nucleus. This phenomenon is behind the Gamma Factory (GF) proposal for CERN [51, 52] that implies using a beam of ultra-relativistic ions in the backscattering process. The GF project is aimed at creating, storing, and exploiting relativistic beams of partly stripped atomic ions that can be stored in the Super Proton Synchrotron (SPS) or the Large Hadron Collider (LHC) storage rings at very high energies (the corresponding relativistic factors within the range γ = 30 . . . 3000), at high bunch intensities (number of ions per bunch 108 . . . 109 ), and at a high bunch repetition rate (up to 20 MHz). The GF scheme is based on a resonant excitation of a partly stripped ions with the laser beam tuned to the atomic transitions frequencies, followed by the process of spontaneous emission of photons; see Fig. 3.5. Due to the relativistic Doppler effect, the energy of photons emitted in the direction of the beam is boosted by a factor of up to 4γ 2 as compared to the energy of the laser light. Due to huge excess (a factor up to 109 ) of resonant photon absorption cross section compared to that of photon scattering from a free electron, the intensity of an atomic-beam-driven LS is expected to be several orders of magnitude higher than what is possible with Compton gamma-ray sources driven by an electron beam.

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3 Light Sources at High Photon Energies

Fig. 3.5 Compton backscattering from partially stripped ions (Figure’s layout from Ref. [53])

The proposed LS could be realized at CERN by using the infrastructure of the existing accelerators. It could push the intensity limits of the presently operating light sources by at least 7 orders of magnitude, reaching the flux of the order of 1017 photons/s, in the particularly interesting gamma-ray energy domain of E ph = 1 − 400 MeV [52]. This domain is out of reach for the FEL-based light sources. The energy-tuned and quasi-monochromatic gamma beams together with the gammabeams-driven secondary beams of polarized positrons, polarized muons, neutrons, and radioactive ions would constitute the basic research tools of the proposed GF. To prove experimentally the concepts underlying the Gamma Factory proposal, feasibility tests have been and will continue to be performed at the SPS and at the LHC [49]. Since 2017, the experimental beam tests have started with various PSI beams. In 2018, for the first time, the 208 Pb81+ ions were injected into the LHC [54] aiming at demonstrating that bunches of hydrogen-like lead atoms can be efficiently produced and maintained at the LHC top energy with the lifetime and intensity fulfilling the GF requirements. Thus, the pivotal concept of the GF initiative that relativistic atomic beams can be produced, accelerated, and stored in the existing CERN SPS and LHC rings has been experimentally proven. It is planned that the SPS and LHC beam tests will be followed by the GF “proof-of-principle” SPS experiment [55] in which a beam of lithium-like lead ions, 208 Pb79+ , will be collided with the photon laser beam tuned to resonantly excite the 2s → 2 p1/2 atomic transition of the ions. It is expected that this experiment will provide a decisive proof and an experimental evaluation of the achievable intensities of the atomic-beam-based gamma-ray source.

3.5 Alternative Schemes for Short-Wavelength Light Sources

51

3.5.3 Extremely Brilliant GeV γ -Rays From a Two-Stage Laser-Plasma Accelerator Strong electric fields for the acceleration of particles can be produced by the separation of electrons and ions in a dense plasma. Powerful laser pulses propagating in plasma generate such charge separation through the excitation of wakefields due to the action of non-linear ponderomotive force [56, 57]. Wakes with electric fields orders of magnitude larger than in conventional accelerators are feasible allowing for reducing the size of accelerators. Compact laser-wakefield accelerators (LWFAs) have been developed [58] that offer a radically different approach: the acceleration length in plasmas is about three orders of magnitude smaller as compared to conventional accelerators, providing the ability to drive the acceleration and radiation of high-energy particles on a much smaller scale. Multi-GeV electron beams have been produced using LWFA, and X/γ -ray pulses in the keV to MeV range can be produced via LWFA-based betatron radiation and Compton backscattering (see Ref. [59] and references therein). The resulting radiation sources have typical peak brilliance of 1019 . . . 1023 photons/(s mm2 mrad2 0.1% BW), while the photon number per shot is limited to 107 . . . 108 photons due to the low level of the laser-to-photon energy conversion efficiency. Still, it remains a great challenge to significantly increase the energy conversion efficiency and to generate collimated γ -rays with high peak brilliance with energies in the MeV to GeV range. Although continuous development in ultrahigh-power laser technology provides possibilities for producing brilliant high-energy gamma-ray LSs, there are unavoidable physical limitations on the peak brilliance of gamma-rays produced by means of various methods based on laser pulses [59]. It has been noted in the cited paper that to produce GeV photons, an exceptionally high laser intensity of 1023 . . . 1025 W/cm2 (two to four orders of magnitude higher than the highest intensities available to date) is required. As soon as the laser intensity is reduced to the levels achievable in the current high-power laser systems, the methods mentioned become intrinsically inefficient for gamma-ray emission. To overcome the restrictions, a new scheme to produce extremely high-brilliance γ -rays with photon energies up to GeV has been proposed recently [59]. The scheme, which is illustrated by Fig. 3.6, is based on a two-stage LWFA driven by a single multi-petawatt laser pulse. In the first stage, the plasma electrons are self-injected and accelerated in the plasma bubble excited by the laser pulse propagating in an under-dense plasma, resulting in a low-divergence multi-GeV electron beam with a particle density close to the critical plasma density, 1021 cm−3 . The laser-to-electron energy conversion efficiency is quoted at the level of 40%. In the second stage, the laser pulse propagates into the relatively high-density plasma, resulting in a shrunken plasma bubble as the

52

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Fig. 3.6 Illustrative picture of the two-stage scheme. In the first acceleration stage, a plasma wake is driven by a multi-petawatt laser pulse propagating in an under-dense plasma channel, where the efficient electron injection and acceleration result in a multi-GeV, low-emittance, high-charge, and high-density electron beam. The laser pulse then enters a higher density plasma region that acts as a radiator, where collimated bright γ -rays are produced by the dense high-energy electrons in the enhanced electrostatic fields of the bubble in the denser plasma. (Adapted from Ref. [59])

density increases. Besides the accelerated GeV electrons from the previous stage, additional electrons are injected, which further increases the total charge of the accelerated electron beam with a peak density well above the critical density. The efficiency increases to above 50% for the total accelerated GeV electrons as well. This results in large quasi-static electromagnetic fields around the electron beam, which gives rise to the emission of a collimated beam of γ -rays with photon energies up to the GeV level. A distinct feature of this scheme is the high efficiency of both electron acceleration and radiation. Based on the results of fully relativistic three-dimensional numerical simulations carried out in Ref. [59], the authors predict that the photon number, peak brilliance, and power of the γ -rays emitted in the two-stage LWFA light source are several orders of magnitude higher than current LWFA betatron radiation and Compton sources. Numerical simulations, carried out in Ref. [59], demonstrate that more than 1012 γ -ray photons/shot are produced for photons above 1 MeV, and the peak brilliance is above 1026 photons s−1 mm−2 mrad−2 per 0.1% bandwidth at 1 MeV. The dependence of the calculated peak brilliance on the photon energy is shown in Fig. 3.7.

References

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26

10

Peak brilliance

25

10

24

10

23

10

22

10

21

10

0

10

1

10

2

10

3

10

Photon energy (MeV) Fig. 3.7 Peak brilliance (in photons/s mm2 mrad2 0.1% bandwidth ) of two-stage LWFA gammarays LS as a function of the photon energy E γ . (The dependence has been obtained by digitalizing the data presented in Figure 2(I) in [59])

References 1. Schmüser, P., Dohlus, M., Rossbach, J.: Ultraviolet and Soft X-Ray Free-Electron Lasers. Springer, Berlin (2008) 2. Rullhusen, P., Artru, X., Dhez, P.: Novel Radiation Sources Using Relativistic Electrons. World Scientific, Singapore (1998) 3. Altarelli, M., Salam, A.: The quest for brilliance: light sources from the third to the fourth generation. Europhysicsnews 35, 47–50 (2004) 4. Kim, K.-J.: Characteristics of synchrotron radiation. In: X-ray Data Booklet, pp. 2.1–2.16. Lawrence Berkeley Laboratory, Berkley (2009). http://xdb.lbl.gov/xdb-new.pdf 5. Kim, K.-J.: Brightness, coherence and propagation characteristics of synchrotron radiation. Nucl. Instrum. Meth. A 246, 71–76 (1986) 6. Ginzburg, V.L.: Theoretical Physics and Astrophysics (International seies in natural philosophy, vol. 99). Pergamon Press, Oxford (1979) 7. Schott, G.A.: Electromagnetic Radiation. Cambridge University Press, Cambridge (1912) 8. Ivanenko, D.D., Pomeranchuk, I.Ya.: On the maximum energy achievable in a betatron. Doklady Acad. Nauk 44, 343 (1944). (in Russian) 9. Schwinger, J.: On the classical radiation of accelerated electrons. Phys. Rev. 75, 1912 (1949) 10. Elder, F.R., Gurewitsch, A.M., Langmuir, R.V., Pollock, H.C.: Radiation from electrons in a synchrotron. Phys. Rev. 71, 829 (1947) 11. Yabashi, M., Tanaka, H.: The next ten years of X-ray science. Nat. Photonics 11, 12 (2017) 12. Tavares, P.F., Leemann, S.C., Sjöström, M., Andersson, Å: The MAX IV storage ring project. J. Synchrotron Rad. 21, 862 (2014) 13. Couprie, M.E.: New generation of light sources: present and future. J. Electr. Spectrosc. Rel. Phenomena 196, 3 (2014) 14. Madey, J.M.J.: Stimulated emission of bremsstrahlung in a periodic magnetic field. J. Appl. Phys. 42, 1906–1913 (1971) 15. Deacon, D.A.G., Elias, L.R., Madey, J.M.J., Ramian, G.J., Schwettman, H.A., Smith, T.I.: First operation of a free-electron laser. Phys. Rev. Lett. 38, 892 (1977)

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16. Kondratenko, A.M., E. Saldin, E.L.: Generating of coherent radiation by a relativistic electron beam in an ondulator. Part. Accel. 10, 207 (1980) 17. Kim, K.J.: Three-dimensional analysis of coherent amplification and self-amplified spontaneous emission in free-electron lasers. Phys. Rev. Lett. 57, 1871 (1986) 18. Bonifacio, R., Pellegrini, C., Narducci, L.M.: Collective instabilities and high-gain regime in a free electron laser. Opt. Commun. 50, 373–378 (1984) 19. Bonifacio, R., Casagrande, F., Cerchioni, G., de Salvo Souza, L., Pierini, P., Piovella, N.: Physics of the high-gain FEL and superradiance. Rivista del Nuovo Cimento 13, 1–69 (1990) 20. Luchini, P., Motz, H.: Undulators and Free-Electron Lasers. Oxford University Press, New York (1990) 21. Saldin, E.L., Schneidmiller, E.A., Yurkov, M.V.: The Physics of Free-Electron Lasers. Springer, Berlin (1999) 22. Huang, Zh., Kim, K-J.: Review of X-ray free-electron laser theory. Phys. Rev. ST Accel. Beams 10, 034801 (2007) 23. Pellegrini, C., Marinelli, A., Reiche, S.: The physics of x-ray free-electron lasers. Rev. Mod. Phys. 88, 015006 (2016) 24. Bostedt, Ch., Boutet, S., Fritz, D.M., Huang, Z., Lee, H.J., Lemke, H.T., Robert, A., Schlotter, W.F., Turner, J.J., Williams, G.J.: Linac coherent light source: the first five years. Rev. Mod. Phys. 88, 015007 (2016) 25. Seddon, E.A., Clarke, J.A., Dunning, D.J., Masciovecchio, C., Milne, C.J., Parmigiani, F., Rugg, D., Spence, J.C.H., Thompson, N.R., Ueda, K., Vinko, S.M., Wark, J.S., Wurth, E.: Short-wavelength free-electron laser sources and science: a review. Rep. Prog. Phys. 80, 115901 (2017) 26. Gover, A., Friedman, A., Emma, C., Sudar, N., Musumeci, P., Pellegrini, C.: Superradiant and stimulated-superradiant emission of bunched electorn beams. Rev. Mod. Phys. 91, 035003 (2019) 27. Emma, P., Akre, R., Arthur, J., Bionta, R., Bostedt, C., et al.: First lasing and operation of an Ångstrom-wavelength free-electron laser. Nat. Photonics 4, 641 (2010) 28. McNeil, B.W.J., Thompson, N.R.: X-ray free-electron lasers. Nat. Photonics 4, 814 (2010) 29. Milne, Ch.J., Schietinger, Th., Aiba, M., Alarcon, A., Alex, J., et al.: SwissFEL: the Swiss X-ray free electron laser. Appl. Sci. 7, 720 (2017) 30. Doerr, A.: The new XFELs. Nature Meth. 13, 33 (2018) 31. Di Mitri, S., Allaria, E.M., Cinquegrana, P., Craievich, P., Danailov, M., Demidovich, A., De Ninno, G., Diviacco, B., Fawley, W., Froehlich, L., Giannessi, L., Ivanov, R., Musardo, M., Nikolov, I., Penco, G., Sigalotti, P., Spampinati, S., Spezzani, C., Trovò, M., Veronese, M.: FERMI@Elettra, a seeded free electron laser source for a broad scientific user program. Proc. SPIE 8078, 807802 (2011) 32. http://flash.desy.de/ 33. LCLS Design Study Group (Arthur, J., et al.): Linac coherent light source (LCLS) design study report. SLAC-R-0521. see http://www.slac.stanford.edu/pubs/slacreports/slac-r-521. html (1998) 34. Materlik, G., Tschentscher, Th. (eds.): TESLA technical design report. Part V. The X-ray free electron laser (2001). http://tesla.desy.de/new_pages/TDR_CD/PartV/fel.html 35. Federici, L., Giordano, G., Matone, G., Pasquariello, G., Picozza, P., et al.: The LADON photon beam with the ESRF 5 GeV machine. Lett. Nuovo Cimento 27, 339 (1980) 36. Federici, L., Giordano, G., Matone, G., Pasquariello, G., Picozza, P.G., Caloi, R., Casano, L., de Pascale, M.P., Mattioli, M., Poldi, E.: Backward Compton scattering of laser light against high-energy electrons: the LADON photon beam at Frascati. Nuovo Cimento 59B, 247 (1980) 37. ur Rehman, H., Lee, J., Kim, Y.: Optimization of the laser-Compton scattering spectrum for the transmutation of high-toxicity and long-living nuclear waste. Ann. Nucl. Energy 105, 150 (2017) 38. Krämer, J.M., Jochmann, A., Budde, M., Bussmann, M., Couperus, J.P., et al.: Making spectral shape measurements in inverse Compton scattering a tool for advanced diagnostic applications. Sci. Reports 8, 139 (2018)

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39. ur Rehman, H., Lee, J., Kim, Y.: Comparison of the laser-Compton scattering and the conventional Bremsstrahlung X-rays for photonuclear transmutation. Int. J. Energy Res. 42, 236–244 (2018) 40. Berestetskii, V.B., Lifshitz, E.M. and Pitaevskii, L.P.: Course of Theoretical Physics. Quantum Electrodynamics, vol. 4. Pergamon Press, Oxford (1982) 41. Kulikov, O.F., Telnov, Y.Y., Filippov, E.I., Yakimenko, M.N.: Compton effect on moving electrons. Phys. Lett. 13, 344 (1964) 42. Bemporad, C., Milburn, R.H., Tanaka, N., Fotino, M.: High-energy photons from compton scattering of light on 6.0-GeV electrons. Phys. Rev. 138, B1546 (1965) 43. Ballam, J., Chadwick, G.B., Gearhart, R., Guiragossian, Z.G.T., Klein, P.R., Levy, A., Menke, M., Murray, J.J., Seyboth, P., Wolf, G., Sinclair, C.K., Bingham, H.H., Fretter, W.B., Moffeit, K.C., Podolsky, W.J., Rabin, M.S., Rosenfeld, A.H., Windmolders, R.: Total and partial photoproduction cross sections at 1.44, 2.8, and 4.7 GeV. Phys. Rev. Lett. 23, 498 (1969) (Erratum: Phys. Rev. Lett. 23, 817 (1969)) 44. D’Angelo, A., Bartalini, O., Bellini, V., Levi Sandri, P., Moricciani, D., Nicoletti, L., Zucchiatti, A.: Generation of compton backscattering γ -ray beams. Nucl. Instrum. Meth. A 455, 1 (2000) 45. Schaerf, C.: Polarized gamma-ray beams. Phys. Today 58, 44 (2005) 46. Weller, H.R., Ahmed, M.W., Gao, H., Tornow, W., Wu, Y.K., Gai, M., Miskimen, R.: Research opportunities at the upgraded HIgS facility. Prog. Part. Nucl. Phys. 62, 257 (2009) 47. Krafft, G.A., Priebe, G.: Compton sources of electromagnetic radiation. Rev. Accel. Sci. Technol. 3, 147 (2010) 48. Sei, N., Ogawa, H., Jia, Q.: Multiple-collision free-electron laser compton backscattering for a high-yield gamma-ray source. Appl. Sci. 10, 1418 (2020) 49. Howell, C.R., Ahmed, M.W., Afanasev, A., Alesini, D., Annand, J.R.M. et al.: International workshop on next generation gamma-ray source (2020). arXiv:2012.10843 International Workshop on Next Generation Gamma-Ray Source. arXiv preprint arXiv:2012.10843 (2020) 50. Wu, Y.K., Vinokurov, N.A., Mikhailov, S., Li, J., Popov, V.: High-gain lasing and polarization switch with a distributed Optical-Klystron free-electron laser. Phys. Rev. Lett. 96, 224801 (2006) 51. Xenon beams light path to gamma factory. CERN Courier (13 October 2017). https:// cerncourier.com/xenon-beams-light-path-to-gamma-factory/ 52. Krasny, M. W.: The gamma factory proposal for CERN (2015). arXiv:1511.07794 53. Krasny, M.W.: The gamma factory proposal for CERN. Photon-2017 Conference, May 22-29, 2017 8CERN, Geneva 54. Schaumann, M., Alemany-Fernández, R., Bartosik, H., Bohl, Th., Bruce, R. et al: First partially stripped ions in the LHC (208 Pb81+ ). In: Proceedings, 10th International Particle Accelerator Conference (IPAC2019), p. MOPRB055 (Melbourne, Australia, May 19–24, 2019 55. Krasny, M.W., Martens, A., Dutheil, Y.: Gamma factory proof-of-principle experiment: Letter of intent. CERN-SPSC-2019-031/SPSC-I-253, 25/09/2019. http://cds.cern.ch/record/ 2690736/files/SPSC-I-253.pdf 56. Tajima, T., Dawson, J.: Laser electron accelerator. Phys. Rev. Lett. 43, 267 (1979) 57. Pukhov, A., Meyer-ter-Vehn, J.: Laser wake field acceleration (LWFA). Appl. Phys. B 74, 355 (2002) 58. Esarey, E., Schroeder, C.B., Leemans, W.P.: Physics of laser-driven plasma-based electron accelerators. Rev. Mod. Phys. 81, 1229 (2009) 59. Zhu, X.-L., Chen, M., Weng, S.-M., Yu, T.-P., Wang, W.-M., He, F., Sheng, Z.-M., McKenna, P., Jaroszynski, D.A., Zhang, J.: Extremely brilliant GeV γ -rays from a two-stage laser-plasma accelerator. Sci. Adv. 6, eaaz7240 (2020)

Chapter 4

Channeling Phenomenon and Channeling Radiation

4.1 Channeling of Ultra-Relativistic Particles The basic effect of the channeling process in a straight single crystal is in an anomalously large distance which particle can penetrate moving along a crystallographic plane (the planar channeling) or an axis (the axial channeling) and experiencing the collective action of the electrostatic field of the lattice ions [1]. The field is repulsive at small distances for positively charged particles, and therefore, they are steered into the interatomic region, while negatively charged projectiles move in the close vicinity of the ion strings or planes.1 Figure 4.1 illustrates the propagation of positively and negatively charged particles through an oriented crystal. Channeling was discovered in the early 1960s by computer simulations of ion motion in crystals of various structures [22, 23]. Large penetration lengths were obtained for low-energy (up to 10 keV) CU ions incident along various crystallographic directions in several crystalline structures of the cubic symmetries. These calculations allowed researches to explain the results of earlier experiments [24, 25] on anomalously long propagation lengths for heavy ions in aluminum. Later, a comprehensive theoretical study by Lindhard [1] demonstrated that propagation of charged particles through crystals strongly depends on the relative orientation of beam and target. The important model of continuum potential for the interaction of energetic projectiles and lattice atoms was formulated. Based on this model, Lindhard demonstrated that under certain conditions the projectile can move through the crystal following a particular crystallographic direction. There are many reviews on the channeling phenomenon in straight crystals in which many of the topics described below are discussed in more detail. Let us indicate Refs. [1, 26–38].

1

Channeling effect can be discussed not only for crystals but, in principle, for any structured material which provides “passages”, moving along which a projectile has a much lower value of the mean square of the multiple scattering angles than when moving along any random direction. Examples of such materials are nanotubes and fullerites, for which the channeling effects have been also investigated; see, e.g., Refs. [2–21]. © Springer Nature Switzerland AG 2022 A. Korol and A. V. Solov’yov, Novel Lights Sources Beyond Free Electron Lasers, Particle Acceleration and Detection, https://doi.org/10.1007/978-3-031-04282-9_4

57

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4 Channeling Phenomenon and Channeling Radiation

Fig. 4.1 Artistic view of an oriented linear crystal: the crystal atoms (grayish spheres) are seen being arranged in arrays of well-separated planes. The electrostatic field of the plane is attractive for negatively charged ones (e.g., electrons) and repulsive for positively charged projectiles (e.g., positrons). Therefore, for positrons, the planes act as the “walls” of a “channel” in which the projectile bounces as it propagates (channels) through the crystal; see the red curve. In contrast, an electron channel experiences bouncing in the vicinity of a plane along which it channels; see the blue curve. For both projectiles, this type of motion is called “planar channeling”. The figure courtesy of G.B. Sushko.

4.1.1 Crystallographic Axes and Planes A specific feature of a crystalline structure (the case of a single crystal is assumed) is that the crystal atoms are arranged according to a regular pattern, which can be described in terms of three-dimensional translations of the unit cell, build with three primitive vectors a j ( j = 1, 2, 3) (see, e.g., [39]). The structure of a crystal as well as all its physical properties are invariant with respect to any translation characterized by the vector T = n 1 a1 + n 2 a2 + n 3 a3 ,

(4.1)

where n j are integers. Three integers define a crystallographic direction (axis) notated as n 1 , n 2 , n 3 . The plane perpendicular to this axis is notated as (n 1 , n 2 , n 3 ). Figure 4.2 (left) illustrates the main crystallographic axes for a primitive cubic cell. The dashed area marks the (110) plane which is perpendicular to the 110 axis. As a result of such a regular pattern of the crystal structure, the crystal ions can be viewed as been arranged in strings or/and planes. Seen along a crystallographic direction, a crystal can be represented as a set of regularly spaced strings, as it is schematically illustrated by Fig. 4.2 (right), where blue balls represent the atoms in a silicon crystal arranged along the 020 axis. Rotating the crystal, one can view its atoms aligned along the strings which, in turn, form a set of well-separated crystallographic planes; see Fig. 4.2 left. The right panel of the figure represents the crystal as a set of planes only. This can be achieved by looking at the crystal along any of its crystallographic planes from a random (i.e., not aligned with any crystallographic axis) direction (Fig. 4.3).

4.1 Channeling of Ultra-Relativistic Particles

59

Fig. 4.2 Left panel. A primitive cubic cell with the main crystallographic directions indicated. The dashed area marks the (110) plane. Right panel. Silicon single crystal seen along the 020 axis

Fig. 4.3 Si(100) planar channels seen from, approximately, 001 crystallographic direction (left panel) and from random (not aligned with any crystallographic axis) direction (right panel) Fig. 4.4 Trajectory and interaction of the charged particles with the string of atoms

Θ

v r⊥

d||

z

The atoms, arranged either in strings or in planes, create a periodic electrostatic potential which acts on the projectile. Under certain conditions, when the incident angle with respect to the string (or plane) is small enough, the collisions of the projectile with different atoms become strongly correlated, so that it moves further essentially following the direction along the string.

4.1.2 Continuous Potential Model A charged particle, moving along a string of atoms separated by the distance d , experiences the action of the electrostatic field created by the atoms; see Fig. 4.4.

60

4 Channeling Phenomenon and Channeling Radiation

Provided the incident angle θ is small enough, the motion of the particle will be governed not by the scattering from the field of individual atoms but rather by the correlated action of many centers of the force. To illustrate this statement, one can consider the following arguments. The characteristic time τ of the motion in the direction perpendicular to the string can be estimated as τ ∼ d⊥ /v⊥ , where d⊥ is the (average) distance between the strings and v⊥ is the mean velocity in the perpendicular direction. During this interval, the particle moves along the string a distance l = v τ = d⊥ v /v⊥ ≈ d⊥ /θ . Taking into account that in a crystalline structure d⊥ ∼ d , one derives l ∼ d /θ  d for θ 1. In other words, within l , the particle experiences a large number of correlated collisions. Therefore, to describe the transverse motion of the projectile, one can introduce the approximation [1] within which the potential V (r⊥ , z) of each atom is averaged along the string, resulting in the following continuous transverse potential Us (r⊥ ) of the string: ∞ Us (r⊥ ) =

V (r⊥ , z) −∞

dz , d

(4.2)

where r⊥ is the distance from the string; see Fig. 4.4. Similar to the case of the atomic string, the motion along a crystallographic plane at small incident angles is described by the continuous transverse potential Upl (y) of the atomic plane (which is aligned with the (x z) Cartesian plane) defined as follows: (see, e.g., [1, 27, 28, 33, 34]): ∞ ∞ Upl (y) = n a d

V (r ) dz dx,

(4.3)

−∞ −∞

crystal atoms, y—the where d is the interplanar distance, n a —the volume density of coordinate along the perpendicular direction, and V (r ) (r = x 2 + y 2 + z 2 ) is the potential of the projectile–atom interaction. Depending on the model applied to describe atomic potential V (r ), one can obtain different forms for Us (r⊥ ) and Upl (y). Many of these can be found in Refs. [1, 26– 29, 31, 33, 40, 41]. The comparison of atomic and interplanar potentials build using the parameterization due to Molière [42], Doyle and Turner [43], and Pacios [44] is presented in Appendix 1. Total continuous potential acting on the projectile in the crystal is the sum of the potentials created by individual planes (or axes). For the sake of clarity, below we consider the planar case. Taking into account that atomic potential V (r ) rapidly decreases at the distances exceeding the (mean) atomic radius,2 one can account for only the two nearest planes For a quantitative estimate, one can use the Thomas–Fermi radius aTF = Z −1/3 a0 as the measure of the mean atomic radius (see, e.g., [45]). Here Z is the atomic number of the crystal atom and a0 is the Bohr radius. 2

4.1 Channeling of Ultra-Relativistic Particles

61

Fig. 4.5 Schematic representation of the planar channeling motion (trajectories A and B) and the motion of the over-barrier particle (trajectory C). Trajectories A and B stand for a positively and a negatively charged projectile, respectively

C B

A

when calculating the sum. Then, the continuous interplanar potential can be presented in the form [27]:  U (ρ) = Upl

   d d − ρ + Upl + ρ + const. 2 2

(4.4)

Here, ρ = [−d/2, d/2] is the distance measured from the midplane. The additive constant can be chosen to ensure U = 0 at the midplane (the differences in choosing the midplane for positive and negative particles are discussed below). Illustrative examples of continuous interplanar potentials calculated within different approximations are presented in Appendix. 1.2. Within the framework of the continuous potential approximation, the transverse motion of an ultra-relativistic projectile of the mass m is described (in terms of classical mechanics) by the Newton equation but with a the “relativistic” mass equal to mγ (or, equivalently, to ε/c2 ). Written for the planar case, the equation of motion reads mγ y¨ = −

dU . dy

(4.5)

2 Within this picture, the energy of the transverse motion, ε⊥ = p⊥ /2mγ + U (y), is the integral of motion. Comparing ε⊥ with the depth U0 of the interplanar potential (4.4), one can distinguish between two modes of the transverse motion:

1. Channeling motion occurs when ε⊥ < U0 . In this case, the transverse motion is finite, having the form of oscillations in the vicinity of the midplane. 2. Over-barrier motion corresponds to ε⊥ > U0 when a particle moves unrestricted, sequentially crossing different crystal planes on its way. These modes are illustrated by Fig. 4.5. Taking into account that p⊥ = p, one re-writes the channeling condition in terms of the following restriction for the incident angle:

62

4 Channeling Phenomenon and Channeling Radiation

  < L =

2U0 ≈ pv



2U0 . ε

(4.6)

The right-hand side is written for an ultra-relativistic particle for which pv ≈ ε. The critical angle L was introduced by Lindhard [1]. √The values of critical angle in the case of planar channeling are lower by a factor ∼ 101 than those for the axial channeling due to the order of magnitude difference in the depths of the continuous potential well.

4.1.2.1

Dechanneling Effect

Lindhard [1] was the first who theoretically described the dechanneling process— the phenomenon of a gradual increase of the transverse energy of a channeled particle due to collisions with nuclei and electrons of the crystal. Within the continuous potential model, the transverse energy ε⊥ is conserved. However, the real potential, equal to the sum of potentials due to the point charges, differs from Us (r⊥ ) or/and Upl (y). Therefore, the projectile motion determined within the framework of the continuous potential model is perturbed by the action of the additional potential. This action can be considered in terms of uncorrelated scattering from crystal constituents. In each act of scattering, the magnitude of ε⊥ changes by some small value (except for, comparatively, rare events of close collisions with nuclei which may result in large-angle scattering and, thus, a noticeable change in ε⊥ ). As a consequence, initially channeled particle during its passage through the crystal may gain the transverse energy higher than the continuous potential barrier. At this point, the particle leaves the channel and is, basically, lost for the channeling process. The scale which defines the (average) interval for a particle to penetrate into a crystal until its dechannels is called a dechanneling length, L d . This quantity depends on the parameters of a crystal (these include the charge of nuclei, the interatomic spacing, the mean atomic radius, and the amplitude of thermal vibrations) and the parameters of a channeled particle—the energy and the charge. It is important to note that for negative and for positive projectiles, the dechanneling occurs in different regimes. As mentioned above, the interplanar (or axial) potential is repulsive at small distances for positively charged particles and is attractive for negatively charged ones. Therefore, negatively charged particles tend to channel in the regions around the nuclei whereas positive particles are pushed away. Consequently, the number of collisions with the crystal constituents is much larger for negatively charged particles and they dechannel faster. Typically, the dechanneling lengths of positive charges exceed those for negative ones (of the same energy and charge modulus) by the order of magnitude or more [32, 46, 47]. This statement is valid, both for axial and planar channeling, and for various pairs of positive/negative particles of the same charge modulus: for e+ /e− [35, 48], π + /π − [49], and p/ p¯ [50].

Longitudinal direction

24

3

n(y) (10 /cm ) Interplanar potential (eV)

4.1 Channeling of Ultra-Relativistic Particles

63

positron channels

electron channels ε⊥

20

ε⊥

10

0

nn(y)

8 6 4 2

nel(y)

0

-1

0

1

-1

0

1

Transverse coordinate, y (in units of d) Fig. 4.6 Positron versus electron planar channeling. Two top graphs show the continuous inteplanar potential (4.4) in Si(110) channel calculated within the Molière approximation. The middle graphs present distributions of the nucleus (solid line) and the electron (broken line) densities. The bottom graphs illustrate the channeling motion of the projectiles: while advancing along the crystallographic planes (filled circles stand for the atoms), the particle oscillates in the transverse direction, where its motion is restricted by the interplanar potential. The channeling oscillations occur with a fixed value of the transverse energy ε⊥ indicated in the top graphs. The figure illustrates that channeling positrons oscillate in the interplanar space whereas electrons channel in the vicinity of a plane

4.1.3 Positron Versus Electron Channeling At small distances from the atom, the field is repulsive for a positively charged particle but attractive for a negative one. The same feature characterizes the continuous potentials of planes and strings. Therefore, positively charged projectiles are steered into the interatomic region, while negatively charged projectiles move in the close vicinity of strings or planes (see, e.g., [34]). To be specific, we compare the motion of positrons and electrons, although the main features discussed below are applicable to any type of positively and negatively charged projectiles. For the planar case, the differences in the channeling regime for positrons and electrons are illustrated by Fig. 4.6 where the columns refer to the positron (left) and electron (right) channeling.

64

4 Channeling Phenomenon and Channeling Radiation

Fig. 4.7 Examples of the simulated trajectories of a positron (graph a) and an electron (graph b) in an oriented 110 Si crystal [52, 53]. Filled circles (red for positrons; blue for electrons) indicate the centers of the channels. Straight segments mark the channel boundaries (triangles for positrons; hexagons for electrons). The attractive character of the atomic string potential keeps electrons in the vicinity of a single string. Positrons, due to the repulsion from the string, move freely from channel to channel

The Si(110) interplanar potential U (y), calculated within the Molière approximation (see Appendix 1), versus the distance from the midplane is presented by the two upper graphs. The electron potential equals the inverted positron one but is shifted upward by the constant value of approximately U0 = 22.9 eV (the magnitude of the Si(110) interplanar potential well) to match the minimum value (which is set to zero for both particles). By definition, the midplanes are associated with the minima of U (y). In the case of positron channeling, the midplanes are located half-way between two neighboring crystal planes, whereas for an electron, they coincide with the crystal planes. As a result, a positron channels in between two planes—see left bottom graph, where the wavy line represents the positron trajectory and filled circles denote the crystal atoms. This domain is characterized by low content of crystal electrons and by, essentially, absence of the nuclei (see the middle graphs which represent the volume densities of crystal nuclei n n and electrons n el as functions of the distance from midplane). In contrast, an electron (as well as any other negatively charged projectile) channels the vicinity of the plane, i.e., in the domain with values of n n and n el . Hence, the probability to experience the uncorrelated collision is much higher for electrons rather than for positrons. Therefore, during the planar channeling, the electrons dechannel faster than positrons. In the case of axial channeling, the difference in channeling motion for positively and negatively charged projectiles is also well pronounced; see Fig. 4.7. The potential of atomic string (4.3), being repulsive for a positively charged projectile, pushes a positron away, so that the positron freely moves from axis to axis. In contrast, the trajectory of the channeling electron is mainly concentrated in the vicinity of a single string (see, e.g., [34]).

4.1 Channeling of Ultra-Relativistic Particles

65

4.1.4 Classical Versus Quantum Description In the case of a channeling particle, the notion of trajectory becomes meaningful provided the particle coordinate ρ and the transverse momentum p⊥ can be measured with accuracies (uncertainties) ρ and p y , which are much smaller than the channel width d and typical value p⊥, ch of the transverse momentum in the channeling regime. Thus, using Heisenberg’s uncertainty principle, one writes p⊥, ch d  p⊥ ρ > .

(4.7)

Kinetic energy, associated with the transverse motion of the channeling particle, cannot exceed the depth U0 of the interplanar potential well. Therefore, to estimate p⊥, ch , one can use the relation 2 p⊥, ch

2ε/c2

< U0 .

(4.8)

Combining (4.7) and (4.8), one finds the condition which validates the use of the classical description of the channeling motion [54]: ε

(c)2 . 2d 2 U0

(4.9)

Applying typical values d ∼ 2 Å and U0 ∼ 10 eV (the case of planar channels is considered), one obtains E  0.1 MeV. To demonstrate this, one can consider the interplanar potential within the harmonic approximation, which is adequate for a positively charged projectile: U (ρ) =  4ρ 2 U0 /d 2 . The frequency of channeling oscillations reads Ωch = c 8U0 /d 2 ε. The number of quantum levels in the potential well is N⊥ = U0 /Ωch = d 2 U0 ε/82 c2 . Hence, ε ≈ 16N⊥2

(c)2 . 2d 2 U0

(4.10)

The right-hand sides of (4.9) and (4.10) coincide up to the factor of 16N⊥2 . The classical description of the channeling oscillations is valid provided the number of levels is sufficiently large. Assuming N⊥ = 10, one obtains 16N⊥2 = 1, 600, thus specifying the sign in (4.9) [54]. Hence, the classical model can be always applied to ultra-relativistic heavy projectiles. It is also applicable to light positively charged projectiles (positrons) if their energy is in the hundred MeV range or higher. The applicability conditions for an electron are somewhat stricter than for a positron, because the interplanar potential well is narrower for negative projectiles; see Fig. 4.6. Numerical results presented in Appendix C in the book [51] indicate that

66

4 Channeling Phenomenon and Channeling Radiation

the number N⊥ of levels for a channeling electron is approximately two times smaller than for the positron of the same energy. For both types of projectiles, N⊥ ∝ ε1/2 . This functional dependence is well-known for the planar channeling [30]. For the axial case, N⊥ ∝ ε for both types of projectile [29, 34]. Therefore, the motion of positrons and electrons in the GeV range and higher can be described in classical terms for both planar and axial channeling.

4.2 Channeling Radiation: Basic Concepts The channeling of charged particles in crystals is accompanied by the channeling radiation (ChR) [55]. This specific type of electromagnetic radiation arises due to the transverse motion of the particle inside the channel under the action of the interplanar or axial field (the channeling oscillations). The phenomenon of ChR of a charged projectile in a linear crystal, see, e.g., [26, 29, 33, 35, 36, 47, 56–67], as well as in a “simple” (i.e., non-periodic, one-arc) bent channel [68–74], is known, although in the latter case the theoretical and experimental data are scarce. The study of the ChR, initially proposed for particles moving in crystals, was later extended to the case of nanotubes [2, 3, 6]. For our purposes, it is important to mention several well-established features of ChR. First, as well as in any other radiative process of a charge moving in an external potential, the intensity of ChR is inversely proportional to the squared mass of the projectile. Hence, a channeled electron/positron emits (m p /m e )2 ∼ 106 times more intensively than a proton with the same value of the relativistic factor γ . Second, for both electrons and positrons, the intensity of radiation in the channeling mode greatly exceeds (by more than an order of magnitude) that by the same projectile in an amorphous medium (see, e.g., [35, 64, 65]). This is a direct consequence of the motion of a channeling particle which, being quasi-periodic, bears close resemblance with the undulator motion [55]. As a result, constructive interference of the waves emitted from different but similar parts of the channeling trajectory increases the intensity. Similar to the UR, for each value of the emission angle θ , the coherence effect will be mostly pronounced for the radiation into harmonics. In the case of planar channeling, the harmonic frequencies can be calculated from (cf. Eq. (2.25)) ωch, n (ε⊥ ) =

2γ 2 Ωch (ε⊥ ) n, n = 1, 2, 3, . . . . 2 1 + γ 2 θ 2 + K ch (ε⊥ )/2

(4.11)

 2

Here, Ωch = 2π/Tch is the frequency of channeling oscillations and K ch = 2γ 2 v⊥ is the undulator parameter related to them. Within the framework of continuous potential approximation, these quantities are dependent on the magnitude of the transverse energy ε⊥ which, in turn, determines the amplitude of oscillations. The

4.2 Channeling Radiation: Basic Concepts

positron

Enhancement (abs. u.)

50

67

40

40

30

30

20

20

10

10

0

electron

50

0 0

20

40

60

80

100

Photon energy (MeV)

0

20

40

60

80

100

Photon energy (MeV)

Fig. 4.8 Spectral distribution of the enhancement factor for ChR by 6.7 GeV positrons and electrons channeling through an oriented L = 105 µ thick Si(110) crystal. Open circles show the experimental data, and solid lines stand for theoretical expectations. The dependences have been obtained by digitalizing the data from Fig. 1 in Ref. [36]

2 only exception is the case of harmonic continuous interplanar √

potential, U ∝ ρ (with ρ being the distance from the midplane), for Ωch = U (0)/mγ is independent of the amplitude. It was mentioned above that harmonic approximation is adequate for the positron channeling whereas for the electron, the interplanar potential is strongly anharmonic. Hence, the variation of Ωch (ε⊥ ) as well as of ωch, n (ε⊥ ) with ε⊥ will be more pronounced for an electron than for a positron. As a result, the spectrum of ChR, which one calculates by averaging over channeling trajectories with various ε⊥ from 0 up to U0 , will be more undulator-like for a positron but much less peaked for an electron. For any type of a projectile, the frequency Ωch is inversely proportional to (mγ )1/2 . Then, it follows from Eq. (4.11) that the ChR frequency scales as γ 3/2 ∝ ε3/2 . This qualitative statement is illustrated by Fig. 4.8, which was plotted by digitalizing the data from Fig. 1 in Ref. [36]. Two graphs in the figure represent the dependences of the spectral intensity of ChR scaled by that of incoherent bremsstrahlung (within the Bethe–Heitler approximation) on the emitted photon energy for 6.7 GeV positron (left graph) and electron (right graph) channeling in the planar channel Si(110). Open circles stand for the experimental data, and the curves represent the results of calculations. More details on the experimental setup and on the used theoretical framework can be found in [64]. It is clearly seen that in the positron case the emission spectrum resembles that of the undulator radiation with K 2 < 1. The first peak in Fig. 4.8 (left) corresponds to

68

4 Channeling Phenomenon and Channeling Radiation

Si(100)

1.5

Si(110)

1

ω(ε⊥)/ω(ε⊥=0)

1.4 0.9

1.3 1.2

0.8 ε=1 GeV ε=3 GeV ε=5 GeV ε=7 GeV

1.1 1 0

0.2

0.4

0.6

ε⊥/U0

0.7

0.8

1

0

0.2

0.4

0.6

0.8

1

ε⊥/U0

Fig. 4.9 Frequency ωch of the first harmonic of channeling radiation emitted in the forward direction as a function of energy ε⊥ of the transverse motion of a positron of different energies (as indicated) channeling in Si(100) (left) and Si(110) (right). The frequency is scaled by its value at ε⊥ = 0; the values of E ⊥ (horizontal axis) are scaled by the depth U0 of the interplanar potential. Reference [51]

the emission in the harmonic with n = 1, and the second, less accented one, is due to the emission into the second harmonic. In the case of electron channeling, the undulator effect is smeared completely due to strong anharmonicity of the interplanar potential. Hence, in contrast to coherent BrS, where the electron and positron radiative spectra are very close, the difference in the interplanar potentials for the two types of light projectiles results in a noticeable difference in the spectra of ChR. However, despite the close similarity of the positron ChR and UR spectra integrated over the emission angles, there are discrepancies in the spectra-angular distributions. Namely, for any value of the emission angle θ there will be sizable difference in the bandwidth ωn /ωn of the peaks. For the ideal undulator, ωn /ωn = 1/nN (see (2.27)) is the natural width determined by the finite size of the undulator ( ωn → 0 as N → ∞). Although this mechanism adds to broadening the lines of ChR, it is not dominant. Main contribution to the line width of ChR is due to the deviation of the positron interplanar potential from the harmonic shape [41]. To estimate ωch, n due to the anharmonic effects, in Fig. 4.9, we present the dependences of the first harmonic of ChR ωch, 1 (ε⊥ ) in the forward direction (see Eq. (4.11) with n = 1 and θ = 0) scaled by its value at ε⊥ → 0, ωch, 1 (0), i.e., in vicinity of the potential minimum where the parabolic approximation is fully applicable. The figure was plotted by digitalizing the data from Fig. 14a and b in Ref. [64]. The curves, calculated for different energies of the projectile positron (as indicated on

4.3 Radiation Damping

69

the left graph), correspond to the channeling in Si(100) (left graph) and Si(110) (right graph). The interplanar potential was treated in the Doyle–Turner approximation [43] (in Ref. [64], similar curves were plotted using the Molière potential). It is seen that the ratio ωch, 1 (ε⊥ )/ωch, 1 (0) varies from 10 to 50 % for ε⊥ within the interval of allowed values, from 0 up to U0 . This variation is the leading mechanism for broadening the lines of ChR.

4.3 Radiation Damping A charged particle moving in a medium or/and an external field loses its energy due to the emission of electromagnetic radiation. The radiation emission gives rise to a so-called radiative reaction force f acting on a projectile; see, e.g., [75]. For ultra-relativistic projectiles of very high energies (tens of GeV and above), this force becomes quite noticeable so that it must be accounted for to properly simulate the motion. Its action leads to the radiation damping, i.e., to a gradual decrease of the particle’s energy. In presence of the static electric field E only, the radiative reaction force can be written as follows [76]: f=

 e e 2 2e3  E − (β · E)2 β , (4.12) γ (v · ∇) E + (β · E) E − γ 2 3 3 mc mc mc

where β = v/c and the symbol ∇ stands for a vector differential operator. Equation (4.12) suggests that for an ultra-relativistic particle, the radiation damping is proportional to the square of its energy. Therefore, for sufficiently high energy of the particle, radiative reaction force can become much larger than the ordinary Lorentz force acting on the particle in the electromagnetic field. In this case, integrating the equations of motion, one derives the following expression for the total loss ε due to the radiation damping (see Sect. 76 in Ref. [76] for the details): ε =

ε2 . ε + εcrit

(4.13)

Here, ε is for the initial energy of the particle and the value εcrit is defined by the equation −1 = εcrit

2r0 3(mc2 )3



+∞

−∞

(eE⊥ )2 dz,

(4.14)

where r0 = 2.818 × 10−13 cm is the classical electron radius and E⊥ denotes the component of the electrostatic field intensity perpendicular to the the z-axis chosen along the projectile’s velocity v. Effectively, the integration is carried out over the spatial domain in which E = 0.

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4 Channeling Phenomenon and Channeling Radiation

Applying (4.14) to the channeling motion, one estimates the integral as F⊥2 L, where F⊥ stands for the electrostatic force acting on the projectile in the channel (planar or axial) and L denotes the crystal thickness along the beam direction. As a case study, consider an positron channeling in a L = 1 mm thick single silicon crystal along the (110) plane. In this case, setting mc2 = 0.511 MeV and F⊥ ≈ 5.7 GeV/cm, one obtains εcrit ≈ 200 GeV. Hence, the energy loss becomes noticeable in the energy range ε  10 GeV and is small in the sub-GeV range. The value of εcrit is inversely proportional to the crystal thickness L, which is limited, for the channeling particles, by the dechanneling length L d . Therefore, the value of εcrit at L ∼ L d calculated for positrons is an order of magnitude smaller than that for electrons of the same energy. Hence, the effect of the radiation damping in a crystal of thickness L ∼ L d is more pronounced for positrons than for electrons. Equation (4.12) can also be used to estimate the energy loss, $ ε, of an ultrarelativistic projectile due to the radiation emission in the course of its channeling motion in a crystal of thickness L. To be specific, the case of planar channeling is considered. Directing z-axis along the plane, one notices that for a channeling particle the longitudinal and transverse components of it satisfy the conditions: βz ≈ 1, βx,z 1. Then, retaining the dominant term on the right-hand side, one derives ε ε 2 e2 E 2 L = 1.4 × 10−3 εe2 E 2 L . ≈ − r0 ε 3 (mc2 )3

(4.15)

Here, the brackets ˙ indicate averaging over the channeling oscillations. The numeric factor on the right-hand side corresponds to ε measured in GeV, eE in GeV/cm, and L in cm. The averaging over channeling oscillations can be carried out analytically if one assumes the harmonic approximation for the positron interplanar potential and the Pöschl–Teller potential UPT (ρ) [77] to describe the channeling oscillations of an electron (see, e.g., [59]). The Pöschl–Teller potential reads UPT (ρ) = a tanh2 (ρ/b),

(4.16)

where the parameters a and b can be determined by matching the depth and the maximal gradient of the potential UPT to the values obtained within the continuous potential framework (for example, in Appendix C of the book [51], they were calculated within the Molière approximation). Carrying out averaging over the periods and amplitudes of the channeling oscillations, one finds [78]

e2 E 2  =

U02 d2

⎧ 8 ⎪ ⎨ for e+ 3   (4.17) × A     sinh A ⎪ ⎩ 7arccos cosh−1 A − 6 + cosh2 A for e− , 4 4 cosh A

4.4 Overview of Numerical Approaches to Simulate Channeling Phenomenon

71

where d is the interplanar distance, U0 is the depth of the potential, and A =

/8U0 with Umax standing for the maximum gradient. 33/2 dUmax

= 5.7 GeV/cm, Applying this result to the Si(110) channel (U0 = 23 eV, Umax d = 1.92 Å), one derives the following estimates for the radiative losses:  ε 0.005 for e+ =εL× 0.017 for e− ε

(4.18)

with ε in GeV and L in cm. The importance of taking into account the effect of radiation damping has been demonstrated recently in Refs. [79–81] where channeling of tens GeV electrons and positrons in has been considered for thick (1 mm and above) oriented diamond and silicon crystals. We also note that the influence of radiative energy losses on the operation of the crystalline undulator has been discussed in [82, 83].

4.4 Overview of Numerical Approaches to Simulate Channeling Phenomenon Various approximations have been used to simulate the channeling phenomenon in oriented crystals. While most rigorous description can be achieved within the framework of quantum mechanics (see recent paper [84] and references therein) quite often classical description in terms of particles trajectories is highly adequate and accurate. Indeed, the number of quantum states N of the transverse motion of √ a channeling electron and/or positron increases with its energy as N ∼ A γ where 2 A ∼ 1 and γ = ε/mc stands for the relativistic Lorentz factor of a projectile of energy ε and mass m [30, 54]. The classical description implies strong inequality N  1, which becomes well fulfilled for projectile energy in the hundred MeV range and above. Simulation of channeling and related phenomena has been implemented in several software packages within frameworks of different theoretical approaches. Below we briefly characterize the most recently developed ones.3 • The computer code Basic Channeling with Mathematica [91] uses continuous potential for analytic solution of the channeling-related problems. The code allows for the computation of classical trajectories of channeled electrons and positrons in continuous potential as well as for computation of wave functions and energy levels of the particles. Calculation of the spectral distribution of emitted radiation is also supported. • A toolkit for the simulation of coherent interactions between high-energy charged projectiles with complex crystalline structures called DYNECHARM++ has been 3

The list of earlier codes developed to simulate the channeling phenomenon includes, in particular, Refs. [85–90].

72

•

•

•

•

4 Channeling Phenomenon and Channeling Radiation

developed [92]. The code allows for the calculation of electrostatic characteristics (charge densities, electrostatic Potential, and field) in complex atomic structures and to simulate and track a particle’s trajectory. Calculation of the characteristics is based on their expansion in the Fourier series through the ECHARM (Electrical CHARacteristics of Monocrystals) method [93]. Two different approaches to simulate the interaction have been adopted, relying on (i) the full integration of particle trajectories within the continuum potential approximation and (ii) the definition of cross sections of coherent processes. Recently, this software package was supplemented with the RADCHARM+ module [94] which allows for the computation of the emission spectrum by direct integration of the quasi-classical formula of Baier and Katkov [26]. The CRYSTALRAD simulation code, presented in Ref. [95], is an unification of the CRYSTAL simulation code [96] and the RADCHARM++ routine [94]. The former code is designed for trajectory calculations taking into account various coherent effects of the interaction of relativistic and ultra-relativistic charged particles with straight or bent single crystals and different types of scattering. The program contains one- and two-dimensional models that allow for the modeling of classical trajectories of relativistic and ultra-relativistic charged particles in the field of atomic planes and strings, respectively. The algorithm for simulation of the motion of particles in presence of multiple Coulomb scattering is modeled accounting for the suppression of incoherent scattering [97]. In addition to this, nuclear elastic, diffractive, and inelastic scattering are also simulated. In Ref. [98], the algorithm based on the Fourier transform method for planar radiation has been presented and implemented to compute the emission spectra of ultra-relativistic electrons and positrons within the Baier–Katkov quasi-classical formalism. Special attention has been given to treat the radiation emission in the planar channeling regime in bent crystals with account for the contributions of both volume reflection and multiple volume reflection events. The simulation presented took into consideration both the nondipole nature and arbitrary multiplicity of radiation accompanying volume reflection. A large axial contribution to the hard part of the radiative energy loss spectrum as well as the strengthening of planar radiation, with respect to the single volume reflection case, in the soft part of the spectrum has been demonstrated. The codes described in Ref. [99] (see also [51]) allow for simulation of classical trajectories of ultra-relativistic projectiles in straight and periodically bent crystals as well as for computing spectra-angular distribution of the radiated energy within the quasi-classical formalism [26]. The trajectories are calculated by solving threedimensional equations of motion with account for (i) the continuous interplanar potential; (ii) the centrifugal potential due to the crystal bending; (iii) the radiative damping force; (iv) the stochastic force due to the random scattering of the projectile by lattice electrons and nuclei. Recently presented code [100] allows one to determine the trajectory of particles traversing oriented single crystals and to evaluate the radiation spectra within the quasi-classical approximation. To calculate the electrostatic field of the crystal lattice, the code uses thermally averaged Doyle–Turner continuous potential [43]

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73

(see also Appendix 1). Beyond this framework, included are multiple Coulomb scattering and energy loss due to radiation emission. It is shown that the use of Graphics Processing Units (GPU) instead of the CPU processors speeds up calculations by several orders of magnitude. • In Refs. [54, 101], a Monte Carlo code was described which allows one to simulate the electron and positron channeling. The code did not use the continuous potential concept but utilized the algorithm of binary collisions of the projectile with the crystal constituents. However, as it has been argued [51, 102, 103], the code was based on the peculiar model of the elastic scattering of the projectile from the crystal atoms. Namely, atomic electrons are treated as point-like charges placed at fixed positions around the nucleus. The model implies also that the interaction of an projectile with each atomic constituent, electrons included, is treated as the classical Rutherford scattering from a static, infinitely massive point charge. It was demonstrated in the cited papers that in practical simulations, non-zero statistical weight of hard collisions with spatially fixed electrons overestimates the increase of the root-mean-square scattering angle with increasing the propagation distance of the channeling particle. As a result, the model over-counts dechanneling–channeling events resulting from the hard collisions.

4.5 Atomistic Modeling of the Related Phenomena Numerical modeling of the channeling and related phenomena beyond the continuous potential framework can be carried out by means of the multi-purpose computer package MBN Explorer [104, 105]. The MBN Explorer package was originally developed as a universal computer program to allow the investigation of the structure and dynamics of molecular systems of different origins on spatial scales ranging from nanometers and beyond. In order to address the channeling phenomena, an additional module has been incorporated into MBN Explorer to compute the motion for relativistic projectiles along with dynamical simulations of the propagation environments, including the crystalline structures, in the course of the projectile’s motion [102]. The computation accounts for the interaction of projectiles with separate atoms of the environments, whereas a variety of interatomic potentials implemented in MBN Explorer supports rigorous simulations of various media. The software package can be regarded as a powerful numerical tool to reveal the dynamics of relativistic projectiles in crystals, amorphous bodies, as well as in biological environments. Its efficiency and reliability have been benchmarked for the channeling of ultra-relativistic projectiles (within the sub-GeV to tens of GeV energy range) in straight, bent, and periodically bent crystals [52, 53, 102, 106–118]. In these papers, the verification of the code against available experimental data and predictions of other theoretical models was carried out.

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4 Channeling Phenomenon and Channeling Radiation

4.5.1 Methodology The description of the simulation procedure is sketched below. Within the framework of classical relativistic dynamics, the propagation of an ultra-relativistic projectile of the charge q = Z e and mass m through a crystalline medium implies integration of the following two coupled equations of motion (EM): ⎧ ⎨

1 v˙ = mγ ⎩ r˙ = v



F·v F−v 2 c

 .

(4.19)

The force F is the sum of two terms: F = −q ∂U/∂r + f .

(4.20)

The first term is due to the electrostatic interaction of the particle with crystal atoms; U = U (r) stands for the electrostatic potential. The second term is the radiative reaction force (4.12). In MBN Explorer, the EM are integrated using the forth-order Runge–Kutta scheme with adaptive time step control. At each step, the potential U = U (r) is calculated as the sum of potentials Uat (r) of individual atoms U (r) =



  Uat r − R j  ,

(4.21)

j

where R j is the position vector of the jth atom. The code allows one to evaluate the atomic potential using the approximations due to Molière [42] and Pacios [44] (see also Appendix 1). A rapid decrease of these potentials with increasing the distances from the atoms allows the sum (4.21) to be truncated in practical calculations. Only atoms are located inside a sphere of the (specified) cut-off radius ρmax with the center at the instant location of the projectile. The value ρmax is chosen large enough to ensure negligible contribution to the sum from the atoms located at r > ρmax . The search for such atoms is facilitated by using the linked cell algorithm implemented in MBN Explorer [104, 119]. EM (4.19) describe the classical motion of a particle in the crystalline environment. They do not account for random events of inelastic scattering of a projectile from individual atoms leading to the atomic excitation or ionization. The impact of such events on the projectile motion is two-fold. First, they result in a gradual decrease in the projectile energy due to the ionization losses. Second, they lead to a chaotic change in the direction of the projectile motion. Rigorous treatment of the inelastic collision evens can only be achieved by means of quantum mechanics. However, taking into account that such events are random, fast, and local, they can be incorporated into the classical mechanics framework according to their probabilities. This approach, which is similar to the one developed

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75

in connection with the Irradiation-Driven Molecular Dynamics [120], is implemented in MBN Explorer following the scheme described in Ref. [99]. The differential probability (per path ds = cdt) of the relative energy transfer μ = (ε − ε )/ε by an ultra-relativistic projectile due to the the ionizing collisions with the quasi-free electrons is defined by the following expression [121, 122]: n e (r) d2 P 2 = 2πr02 Z 2 , dμ ds γ μ2

(4.22)

where n e (r) stands for the local concentration of electrons in the crystal. In the points away from the positions of the nuclei, it can be calculated from the Poisson equation ∇ · E = −4π en e (r) where E is the field strength in the point r. In a single collision with a quasi-free electron at rest, the relative energy transfer is sought within the interval [μmin , μmax ], where

μmin = I /ε,

μmax

⎧ ⎪ ⎪ ⎨

2γ ξ for ξ  1 1 + 2γ ξ + ξ 2 = for a positron (ξ = 1) 1 − γ −1 ⎪ ⎪ ⎩ (1 − γ −1 )/2 for an electron (ξ = 1).

(4.23)

Here, I is the (average) ionization potential of the crystal atom and ξ stands for the ratio of m to the electron mass m e . Formally, expression (4.22) is valid for μ 1. However, since the probability of collisions with μ ∼ 1 is negligibly small, it can be applied to the whole interval of μ. At each step s = c t of integration of the EM (4.19), the ionizing collisions are treated as probabilistic events. Once the event occurs and the value μ is determined, one calculates a round scattering angle θ measured with respect to the instant velocity v of the projectile:  cos θ =

1 + γ −1 1 − γ −1



1 − μ − γ −1 μ(ξ − 1)γ −1   − . (4.24) 1 − μ + γ −1 1 − γ −2 (1 − μ)2 − γ −2

The magnitude of the second (the azimuthal) scattering angle φ (also with respect to v) is not restricted by any kinematic relations and is obtained by random shooting (with a uniform distribution) into the interval [0, 2π ]. The two-fold probability d P 2 /dμ ds must satisfy the normalization condition  0

L ion



μmax

μmin

d2 P dμ ds = 1, dμ ds

(4.25)

where L ion is the spatial interval within which the probability of an ionizing collision accompanied by arbitrary energy transfer is equal to one. This quantity is expressed through μmin , μmax , and the local electron density:

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4 Channeling Phenomenon and Channeling Radiation

L −1 ion =

2πr02 Z 2 n e (r) 1 , γ μ0

μ0 =

μmax μmin ≈ μmin . μmax − μmin

(4.26)

Hence, probability of an ionizing collision with the energy transfer μ occurring within s can be written as dP =

s W (μ) dμ , L ion

W (μ) =

μ0 . μ2

(4.27)

Here, s/L ion defines the probability of the collision (with arbitrary μ) to happen on the scale s, whereas the factor W (μ)dμ represents the normalized probability of the energy transfer between μ and μ + dμ. To simulate the probability of the event to happen, one generates a uniform random deviate rs ∈ [0, 1] and matches it to s/L ion . If x ≤ s/L ion , then the event occurs and one generates the random deviate μ with the probability distribution μ0 /μ2 . The generated μ value is used in (4.24) to calculate the scattering angle θ . The second scattering angle φ is generated via φ = 2π rφ with rφ standing for a uniform random deviate rφ ∈ [0, 1]. The values of μ, θ , and φ are used to modify the velocity and energy of projectile at the start of the next integration step of the EM. The trajectory of a particle entering the initially constructed crystal at the instant t = 0 is calculated by integrating equations (4.19). Initial transverse coordinates, (x0 , y0 ), and velocities, (vx,0 , v y,0 ), are generated randomly accounting for the conditions at the crystal entrance (i.e., the crystal orientation and beam emittance). A particular feature of MBN Explorer is in simulating the crystalline environment “on the fly”, i.e., in the course of propagating the projectile. This is achieved by introducing a dynamic simulation box which moves following the particle (see Refs. [51, 102] for the details). As a first step in simulating the motion along a particular direction, a crystalline lattice is generated inside the rectangular simulation box of the size L x × L y × L z . The z-axis is oriented along the beam direction. To simulate the axial channeling, the z-axis is directed along a chosen crystallographic direction klm (here integers k, l, m stand for the Miller). In the case of planar channeling, the z-axis is parallel to the (klm) plane, and the y axis is perpendicular to the plane. The position vectors of the nodes R(0) j ( j = 1, 2, . . . , N ) within the simulation box are generated in accordance with the type of the Bravais cell of the crystal and using the pre-defined values of the lattice vectors. The simulation box can be cut along specified faces, thus allowing tailoring the generated crystalline sample to achieve the desired form of the sample. Several build-in options, characterized below, allow one to further modify the generated crystalline structure [53]. • The sample can be rotated around a specified axis, thus, allowing for the construction of the crystalline structure along any desired direction. In particular, this option allows one to choose the direction of the z-axis well away from major crystallographic axes, thus, avoiding the axial channeling (when not desired).

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77

Fig. 4.10 Construction of a crystalline medium is initiated by defining a unit cell of the crystal, graph (a), and is followed by a set of transformations: b translation, c rotation, and d bending (including periodic bending). This approach provides an efficient mapping of the coordinates into crystalline reference frame, i.e., cutting the periodic crystalline structure by a simulation box, graph (e). Reference [118]

• The nodes can be displaced in the transverse direction: y → y + R(1 − cos φ) where φ = arcsin(z/R). As a result, a crystal bent with a constant radius R is generated. • Periodic harmonic displacement of the nodes is achieved by means of the transformation r → r + a sin(k · r + ϕ). The vector a and its modulus, a, determine the direction and amplitude of the displacement, the wave-vector k specifies the axis along which the displacement is to be propagated, and λu = 2π/k defines the wavelength of the periodic bending. The parameter ϕ allows one to change the phase-shift of periodic bending. In a special case a ⊥ k, this option provides a simulation of linearly polarized periodically bent crystalline structure which is an important element of a crystalline undulator. These transformations, illustrated schematically in Fig. 4.10, are reversible and, therefore, allow for efficient construction of a crystalline structure in an arbitrary spatial area. Also, the simulation box can be cut along specified faces, thus, allowing tailoring of the generated structure to achieve the desired form of the sample. In addition to the aforementioned options, MBN Explorer allows one to model binary structures (for example, Si1−x Gex or diamond-boron superlattices;

78

4 Channeling Phenomenon and Channeling Radiation

Fig. 4.11 Illustrative Si1−x Gex binary crystal sample with concentration x = 0.05 of germanium atoms. The silicon and germanium atoms are shown in yellow and red, respectively. The Ge atoms occupy positions in the same grid structure as Si but the difference in the lattice constant leads to a deformation of the whole sample. Ref. [118]

see Fig. 4.11) by introducing random or regular substitution of atoms in the initial structure with the dopant atoms. Once the nodes are defined, the position vectors of the atomic nuclei are generated to account for random displacement from the nodes due to thermal vibrations corresponding to a given temperature T . For each atom, the displacement vector  is generated by means of the normal distribution w( ) =

  1 exp − 2 /2u 2T , (2π u 2T )3/2

(4.28)

where u T denotes the root-mean-square amplitude of the thermal vibrations. The values of u T for a number of crystals are summarized in [28]. By introducing an unrealistically large value of u T (for example, exceeding the lattice constants), it is possible to consider large random displacements. As a result, the amorphous medium can be generated.

Continuous interplanar potential (eV)

4.5 Atomistic Modeling of the Related Phenomena

79

ε⊥>ΔU ΔU 20

ε⊥ 0) Ge(110) channels. Reference [53]

Analytical fits for both channeling fractions can be constructed [118] using the parameters that enter the diffusion theory of the dechanneling process (see, e.g., [27]). Apart from the some initial interval [0, z 0 ], one approximates the fraction of accepted particles f ch,0 (z) by the decaying exponent: f˜ch,0 (z) = e−(z−z0 )/L d ,

(4.33)

where z 0 and L d are treated as the fitting parameters. The presence of the distance z 0 is due to the criterion formulated above to identify the accepted particles: the finite initial segment of the trajectory is used to classify the channeling mode of motion. The fraction f ch,0 (z) is used to calculate the penetration distance:  L p = z0 +

L z0

dz (z − z 0 ) f˜ch,0 (z) + (L − z 0 ) f˜ch,0 (L) . Ld

Here, the integral term accounts for the particles that dechannel within the interval [z 0 , L] whereas the non-integral term is due to the projectiles that pass through the whole crystal moving in the channeling regime. The integration produces   L p = z 0 + L d 1 − e−(L−z0 )/L d .

(4.34)

Primary channeling fraction, fch0(z)

4.5 Atomistic Modeling of the Related Phenomena

85

1 270 MeV diamond(110)

855 MeV diamond(110)

855 MeV Si(110)

0.1

simulation fit 0

5

10

15

0

10 20 30 40

0

10

20

30

Penetration distance, z (micron) Fig. 4.16 Primary channeling fraction f ch,0 (z) versus penetration distance for 270 and 855 MeV electrons in, correspondingly, 300 and 140 µ thick diamond(110) crystal (left and middle graphs) and for 855 MeV electrons in 500 µ thick Si(110) (right). Solid curves with open circles and error bars stand for the simulated data. Red dashed curves correspond to the fit (4.33). With account for statistical uncertainties, the intervals for the fitting parameter z 0 and λ are (from left to right) z 0 = (1.42 ± 0.05), (1.7 ± 0.1), (3.0 ± 0.1) µ and L d = (4.4 ± 0.1), (11.14 ± 0.14), (9.0 ± 0.1) µ

Figure 4.16 compares the approximate dependences f˜ch,0 (z) with those obtained by means of numerical simulations for 270 and 855 MeV electrons in diamond(110) and silicon(110) crystals. The values of the fitting parameters are indicated in the caption. The fitting formula for the channeling fraction f ch (z), which accounts for the rechanneling, can be derived within the basing on the framework of one-dimension diffusion approximation for the density of channeling particles, ρ(θ, z). The latter, being dependent on the penetration distance and the angle θ with respect to the channel midplane, satisfies the equation ∂ 2ρ ∂ρ = D 2, ∂z ∂θ

(4.35)

where one assumes the diffusion coefficient D to be independent of the variables. The solution that corresponds to the initial condition ρ(θ, 0) = δ(θ ) and the boundary condition ρ(±∞, z) = 0 reads as follows4 : 4

The initial condition and the solution are written for the case of ideally collimated beam. For a non-collimated beam, a substitution 2Dz → 2Dz + θ02 should be made, where θ0 stands for the initial beam divergence.

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4 Channeling Phenomenon and Channeling Radiation

  θ2 . ρ(θ, z) = √ exp − 4Dz 4π Dz 1

(4.36)

Then, the channeling fraction is expressed in terms of the error function  f ch (z) =

θL −θL

 ρ(θ, z)dθ = erf

√



θL 4Dz

,

(4.37)

where θL is Lindhard’s critical angle. The diffusion coefficient D is related to the root-mean-square angle of multiple scattering θ 2 (z): D=

θ 2  . 2z

(4.38)

For estimation purposes one can use the formula [133]:  θ 2  =

10.6(MeV) ε (MeV)

2

z

(4.39)

L rad

Channeling fraction, fch(z)

1 270 MeV diamond(110)

855 MeV diamond(110)

855 MeV Si(110)

0.5

simulation fit 0 0

100

200

0

50

100

0

200

400

Penetration distance, z (microns) Fig. 4.17 Channeling fraction f ch (z) = Nch (z)/Nacc versus penetration distance for 270 and 855 MeV electrons in, correspondingly, 300 and 140 µ thick diamond(110) crystal (left and middle graphs) and for 855 MeV electrons in 500 µ thick Si(110) (right). Solid curves with open circles and error bars stand for the simulated data. Red dashed curves correspond to the fit (4.40). With account for statistical uncertainties, the intervals for the fitting parameter A are (from left to right) A = (0.11 ± 0.07), (0.10 ± 0.5), and (0.092 ± 0.007) (µ/eV)1/2

4.5 Atomistic Modeling of the Related Phenomena

87

where L rad is the radiation length. It has been argued in the cited paper that the factor 10.6 (MeV) provides a more accurate approximation in the limit z L rad whereas a conventional value of 13.6 (MeV) (see, e.g., [122]) is applicable for z  L rad . From Eq. (33.20) in Ref. [122]), one calculates the following values of the radiation length in amorphous carbon, silicon, and germanium: 12.2, 9.47, and 2.36 cm, which by orders of magnitude exceed the thicknesses of the crystals used in channeling experiments with electrons and positrons. The fitting formula for the channeling fraction that follows from (4.38) and (4.39) reads    ε . (4.40) f˜ch (z) = erf A z The parameter A is written as follows A = (L rad U0 )1/2 /106 with L rad , U0 , ε, and ε measured in cm, eV, MeV, and microns, respectively. The calculated value can be used as the initial guess for the best fit to the f ch (z) obtained from numerical simulations. Figure 4.17 illustrates the applicability of the fitting formula (4.40). Solid curves with symbols stand for the dependences f ch (z) obtained from numerical simulations of electron channeling. The dashed curves  represent the fits. Note that for sufficiently  √ large penetration distances, erf A ε/z ∝ z −1/2 .

4.5.3 Calculation of Spectral Distribution of Emitted Radiation Spectral distribution of the energy emitted within the cone θ ≤ θ0 1 with respect to the incident beam is computed numerically using the following formula:  N0  1  dE(θ ≤ θ0 ) d3 E n = . dφ θ dθ dω N0 n=1 dω dΩ 2π

0

θ0

(4.41)

0

Here, ω is the radiation frequency and Ω is the solid angle corresponding to the emission angles θ and φ. The quantity d3 E n /dω dΩ stands for the spectral–angular distribution emitted by a particle that moves along the nth trajectory. The sum is carried out over all simulated trajectories, and thus, it takes into account the contribution of the channeling segments of the trajectories as well as of those corresponding to the non-channeling regime. The numerical procedures implemented in MBN Explorer to calculate the distributions d3 E n /dω dΩ [102] are based on the quasi-classical formalism [26]. In the limit ω/ε 1, the quasi-classical formula reduces to that known in classical electrodynamics (see, e.g., [75]). The classical description of the radiative process is

88

4 Channeling Phenomenon and Channeling Radiation

Fig. 4.18 Enhancement factor of the channeling radiation over the Bethe–Heitler spectrum for 6.7 GeV positrons (left) and electrons (right) in straight Si(110) crystal. Open circles stand for the experimental data [64]. The calculations performed with MBN Explorer [53, 102] are shown with black solid curves, which present the results obtained for fully collimated beams (zero emittance), and red dashed curves, which correspond to the emittance of 62 µrad as in the experiment. The symbols (closed circles and rectangles) mark a small fraction of the points and are drawn to illustrate typical statistical errors (due to a finite number of the trajectories simulated) in different parts of the spectrum. Green dashed curve, shown on the left figure, corresponds to the results presented in Ref. [135]. The data refer to the emission cone θ0 = 0.4 mrad. References [53, 118]

adequate to characterize the emission spectra by electrons and positrons of the subGeV and GeV energy range. The quantum corrections lead to strong modifications of the radiation spectra of multi-GeV projectiles channeling in bent and periodically bent crystals [107, 110, 111]. The calculated spectral intensity can be normalized to the Bethe–Heitler value (see, for example, Ref. [134] and also Appendix 2.4.2) and thus can be presented in the form of an enhancement factor over the bremsstrahlung spectrum in the corresponding amorphous medium. Figure 4.18 presents results of an exemplary case study of the emission spectra from 6.7 GeV positrons (left) and electrons (right) channeled in L = 105 μm thick oriented Si(110) crystal. The spectra were computed for the emission cone θ0 = 0.4 mrad [64] that exceeds the natural emission cone γ −1 by a factor of about five. Solid black and dashed red curves present the results of two sets of calculations. The first set corresponds to the case of zero beam emittance, when the velocities of all projectiles at the crystal entrance are tangent to Si(110) plane, i.e., the incident angle ψ is zero [102]. The second set of trajectories was simulated allowing for the distribution of the incident angle within the interval ψ = [−θL , θL ] with θL = 62 μrad being Lindhard’s critical angle. The calculated enhancement factors are compared with the

4.5 Atomistic Modeling of the Related Phenomena

89

Fig. 4.19 Simulated (lines) versus experimental (open circles, Ref. [64]) photon emission spectra for 6.7 GeV positrons (left) and electrons (right) in straight Si(110) crystal. The simulated spectra were obtained using (i) the Molière approximation (black curves) and (ii) the Pacios approximation (red curves) for atomic potentials. The results for zero emittance of the beam are shown with solid lines, and the results for beam emittance ψ = θ L = 62 μrad are shown with dashed lines. Refs. [53, 118]

experimental results presented in Ref. [64] and the results of numerical simulations for positrons from Ref. [135]. Figure 4.18 demonstrates that the simulated curves reproduce rather well the shape of the spectra and, in the case of the positron channeling, the positions of the main and the secondary peaks. With respect to the absolute values, both calculated spectra, ψ = 0 and |ψ| ≤ ψL , exhibit some deviations from the experimental results. For positrons, the curve with ψ = 0 perfectly matches the experimental data in the vicinity of the main peak but underestimates the measured yield of the higher (the second) harmonic. An increase in the incident angle results in some overestimation of the main maximum but improves the agreement above ω = 40 MeV. For electrons, the ψ = 0 curve exceeds the measured values; however, the increase in ψ leads to a very good agreement if one takes into account the statistical errors of the calculated dependence (indicated by symbols with error bars). The aforementioned deviations can be due to several reasons. First, the emission spectra can be sensitive to the choice of the approximation scheme used to describe the atomic potentials when constructing the crystalline field as a superposition of the atomic fields, Eq. (4.21). The results presented in Fig. 4.18 were obtained for the trajectories simulated within the Molière approximation framework. Though this approximation is a well-established and efficient approach, more realistic schemes for the crystalline fields, based, for example, on X-ray scattering factors [43, 136] or on accurate numerical approaches for calculation of the electron density in many-electron atoms [44], can also be employed for the channeling simulations. Figure 4.19 compares the experimentally measured spectra with those simulated numerically using the Molière and the Pacios approximations for atomic potentials [53]. For positrons, both approximations result in virtually the same dependences. In the case of electrons, the spectra obtained with the Pacios potential are

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4 Channeling Phenomenon and Channeling Radiation

5–10% less intensive. Within the statistical errors, both results are in a good agreement with the experimental measurements. Another source of the discrepancies can be attributed to some uncertainties in the experimental setup described in [36, 64]. In particular, it was indicated that the incident angles were in the interval [−ψL , ψL ] with the value ψL = 62 μrad for a 6.7 GeV projectile. However, no clear details were provided on the beam emittance which becomes an important factor for comparing theory vs experiment. In the calculations, a uniform distribution of the particles within the indicated interval of ψ was used, and this is also a source of the uncertainties. The spectra were also simulated for a larger cut-off angle equal to 2ψL (these curves are not presented in the figure). It resulted in a considerable (≈30%) decrease of the positron spectrum in the vicinity of the first harmonic peak. On the basis of the comparison with the experimental data, it can be concluded that the code produces reliable results and can be further used to simulate the propagation of ultra-relativistic projectiles along with the emitted radiation. In the subsequent chapter, we present several case studies of the channeling phenomena and radiation emission from ultra-relativistic projectiles traveling in various crystalline environments, incl. linear, bent, and periodically bent crystals as well as in crystals stacks. In most cases, the parameters used in the simulations, such as crystal orientation and thickness, the bending radii R, periods λu , and amplitudes a, as well as the energies of the projectiles, have been chosen to match those used in past and ongoing experiments. Wherever available, we compare the results of our simulations with available experimental data and/or those obtained by means of other numerical calculations.

Appendix 1: Atomic and Interplanar Potentials In this section, for the sake of reference, we compare atomic and interplanar potentials build using the parameterization due to Molière [42], Doyle and Turner [43], and Pacios [44]. The atomic system of units, e = m e =  = 1, is used in this section.

Appendix 1.1: Atomic Potential Parameterization Below we summarize the parameterizations for atomic potential, Uat (r ), and its at (q): Fourier image, U 4π Uat (r ) = (2π )3

 0

∞

sin(qr )  Uat (q)q 2 dq . qr

(4.42)

• The parameterization due to Molière [42] is based on the Thomas–Fermi atomic model:

Appendix 1: Atomic and Interplanar Potentials

91

⎧   3 ⎪ bjr Ze  ⎪ ⎪ U (r ) = a exp − j ⎪ ⎨ at,M r j=1 aTF 3 ⎪  ⎪ ⎪  ⎪ U (r ) = 4π Z e ⎩ at,M i=1

αj q 2 + γ j2

.

(4.43)

Here, Z is the nucleus charge and aTF = 0.8853Z −1/3 is the Thomas–Fermi radius. The dimensionless Molière coefficients are a1,2,3 = (0.1, 0.55, 0.35), b1,2,3 = (6.0, 1.2, 0.3). The short-hand notation γ j = β j /aTF is used. • Doyle and Turner [43] introduced parametric fits to kinematic scattering factors for X-rays and electrons with the use of relativistic Hartree–Fock approximation (see also Ref. [137]). From their formulae, one derives the following parameterization: ⎧   4 ⎪ (4π )3  a j 4π 2 r 2 ⎪ ⎪ ⎪ U (r ) = exp − √ ⎪ ⎨ at,DT bj 4 π j=1 b3/2 j .   4  ⎪ b j q2 ⎪ ⎪  ⎪ U (q) = 2π a j exp − ⎪ ⎩ DT (4π )2 j=1

(4.44)

For carbon, silicon, and germanium atoms, the Doyle–Turner parameters a j (in Å) and b j (in Å2 ) are listed in Table 4.2. It has been pointed out (see, e.g., Ref. [40]) that the D-T scheme does not provide correct behavior of the potential at small distances since limr →0 Uat,DT (r ) = ∞. • For atoms from H to Kr, Pacios [44] proposed the following parameterization based on the Hartree–Fock potentials: ⎧ M ⎪    4π  a j  ⎪ ⎪ ⎪ U 2 + b j r exp −b j r , (r ) = at,P ⎪ 3 ⎨ r j=1 b j   . M 2  ⎪ b a 1 ⎪ j j ⎪  (q) = 2(4π )2 ⎪ U + 2 ⎪ ⎩ at,P b3 q 2 + b2j (q + b2j )2 j=1 j

(4.45)

Sets of the coefficients a j and b j ( j = 1, . . . M) and values of the integer M for several selected atoms are presented in Table 4.3. Note that the nucleus charge Z is absent in Eqs. (4.45). Although it is not indicated either in Ref. [44] or in earlier papers  by the3 author, the nucleus charge and the coefficients are related as Z = 8π M j=1 a j /b j . Figure 4.20 compares the dependences rUa (r )/Z on r/aTF calculated for several atoms by means of different parameterizations; the Molière, Pacios, and Doyle– Turner approximations. We note that at large distances, the Pacios and Doyle–Turner curves practically coincide whereas the Molière approximation provides larger values for the potential. At small distances, where the Doyle–Turner parameterization fails, both Molière and Pacios schemes lead to the same result.

92

4 Channeling Phenomenon and Channeling Radiation

Table 4.2 Parameters a j (in Å) and b j (in Å2 ) for the Doyle–Turner fit for several neutral atoms as indicated Atom a1 b1 a2 b2 a3 b3 a4 b4 B C Si Ge

0.9446 0.7307 2.1293 2.4467

46.4438 36.9951 57.7748 55.8930

1.3120 1.1951 2.5333 2.7015

14.1778 11.2966 16.4756 14.3930

0.4188 0.4563 0.8349 1.6157

3.2228 2.8139 2.8796 2.4461

0.1159 0.1247 0.3216 0.6009

0.3767 0.3456 0.3860 0.3415

Table 4.3 Parameters a j and b j (in a.u.) of the Pacios potential Eq. (4.45) for several atoms as indicated M

a1

a2

a4

b1

b2

b3

b4

B

3

72.22775

–1.021225 0.778090

a3

–

9.828608

2.984085

1.689647

–

C

4

128.0489

–2.535155 2.041774

–

11.84981

3.508196

2.099930

–

Si

4

1713.363

158.9419

–107.9461 1.348130

29.95277

4.305803

3.906608

1.627379

Ge

4

20901.16

1399.193

169.1339

68.65812

22.95161

5.903443

1.541315

0.991756

B

C

Si

Ge

-1

Z rU(r) (angstrom-eV)

10

5

0

10 Moliere Pacios Doyle-Turner

5

0 0

2

4

r/aTF

6

8

0

2

4

6

8

r/aTF

Fig. 4.20 Dependences rU (r )/Z on r/aTF calculated for B, C, Si, and Ge atoms within the Molière, Pacios, and Doyle–Turner approximations. Reference [118]

Appendix 1: Atomic and Interplanar Potentials

93

These differences in the behavior of the atomic potentials reveal themselves in the scattering process of an ultra-relativistic projectile from an atom. Within the framework of classical small-angle scattering framework, the scattering angle θ is related to the change of the transverse momentum θ ≈ c| p⊥ |/ε. To calculate p⊥ , one assumes that the projectile moves along a straight line (the z direction) with a constant speed v ≈ c (see, e.g., Ref. [138]). As a result, the scattering angle as a function of the impact parameter ρ is written as follows:   ∞   2  ∂ Ua (r ) dz  √ . θ (ρ) ≈  ε ∂ρ 0 r = ρ 2 +z 2

(4.46)

Using Eqs. (4.43)–(4.45) in (4.46), one derives ⎧   3 ⎪ 2Z  ρ ⎪ ⎪ a j b j K1 b j Moliere approx. ⎪ ⎪ ⎪ aTF j=1 aTF ⎪ ⎪ ⎪   ⎪ 4 ρ2 1 ⎨ ρ  aj Doyle–Turner approx. exp − θ (ρ) = 4 j=1 B 2j 4B j ε⎪ ⎪ ⎪ ⎪ ⎪ M ⎪  ⎪ aj ⎪ ⎪ 8πρ K 2 (b j ρ) Pacios approx. ⎪ ⎩ b j=1 j

(4.47)

Here, K 1 (.) and K 2 (.) stand for the MacDonald functions of the first and second Order, respectively, and notation B j = b j /(4π )2 is introduced in the case of Doyle– Turner formula. For small arguments, K 1 (z) → z −1 and K 2 (z) → 2z −2 . Using these, one finds that in the limit of small impact parameters, ρ aTF , both the Molière and Pacios formulae reduce to a correct result 2Z /ερ which is the scattering angle in the point Coulomb field Z /r . The Doyle–Turner approximation produces an incorrect result, θ ∝ ρ, in this limit. Figure 4.21 shows the dependences θM,P,DT (ρ) calculated for a ε = 855 MeV electron/positron scattering by carbon and silicon atoms. For the sake of comparison, the dependence for the point Coulomb field is also shown.

Appendix 1.2: Continuous Planar Potential One obtains continuous potential Upl of a plane by summing up the potentials Uat of individual atoms assuming that the latter are distributed uniformly along the plane [1]. Directing the y-axis perpendicular to the plane, one writes

94

4 Channeling Phenomenon and Channeling Radiation

Scattering angle (mrad)

C

-2

-3

10

Si

10

point Coulomb Moliere Pacios Doyle-Turner

Scattering angle (mrad)

-2

10

-4

10

-5

10

-3

10

-4

10

-5

aTF=0.258 Å

point Coulomb Moliere Pacios Doyle-Turner

aTF=0.194 Å

10

-6

10 0

0.5

1

1.5

0

2

0.5

1

1.5

Impact parameter, ρ (angstrom)

Impact parameter, ρ (angstrom)

2

Fig. 4.21 Scattering angle θ versus impact parameter calculated for a 855 MeV electron (positron) scattered from a carbon (left) and silicon (right) atom. The dependencies obtained within the Moliére, Pacios, and Doyle–Turner approximations as well as for the point Coulomb field, Z /r , are presented. Reference [118]

 Upl (y) = N

∞ ∞ w( ) d 

dzdx Uat (|r − |) .

3

(4.48)

−∞ −∞

Here, N = nd denotes the mean surface density of the atoms expressed in terms of mean volume density n and interplanar distance d. Vector  stands for the displacement of an atom from its equilibrium position r due to thermal vibrations, which are accounted for via the distribution (4.28). To transform the r.h.s. of Eq. (4.48), one expresses Uat (|r − |) in terms of its Fourier transform U˜ a (q) and carries out the spatial integrals: N Upl (y) = π

∞

dq e−

q 2 u 2T 2

at (q) . cos(qy) U

(4.49)

0

Using here the Fourier transforms from Eqs. (4.43)–(4.45), one derives planar potentials within different parameterization schemes. Below we present the collection of formulae for Upl (y). • Moliére approximation.

Upl (y) = π Z N

3  αj i=1

γj

e

γ 2j u 2T 2



 F(y; γ j , u T ) + F(−y; γ j , u T ) ,

(4.50)

where F(±y; γ j , u T ) = e±γ j y erfc



γ j uT y √ ±√ 2 2u T

with erfc(x) being the complementary error function.

 (4.51)

Appendix 1: Atomic and Interplanar Potentials

95

These expressions coincide with those presented in Refs. [139, 140]. • Doyle–Turner approximation.

Upl (y) = 2π

1/2

N

4  j=1

aj

 4B j + 2u 2T

 exp −

y2 4B j + 2u 2T

 .

(4.52)

This expression coincides with the formulae presented in Refs. [40, 141]. The seeming deviations are due to different definitions of (i) the coefficients B j (in the cited papers, they are defined as B j = b j /4π 2 whereas here it is four times less) and (ii) the rms amplitudes of thermal vibrations: in [141], ρ 2 = 2u 2T stands for the two-dimensional rms amplitude. • Pacios approximation. M 2  √ − y2 a j b2j u2T 2π Z N u T e 2u T + 4π 2 N e 2 (4.53) b4 j=1 j   × (3 − b2j u 2T − b j y)F(y; b j , u T ) + (3 − b2j u 2T + b j y)F(−y; b j , u T )

Upl (y) =

with F(±y; b j , u T ) defined in (4.51).

Appendix 1.3: Continuous Interplanar Potentials The interplanar potential is obtained by summing the potentials Upl (y) of individual separate planes. For electrons, it can be presented in the form Upl (y) = Upl (y) +

Nmax  

 Upl (y + nd) + Upl (y − nd) + C ,

(4.54)

n=1

where y is the transverse coordinate with respect to an arbitrary selected reference plane, and the sum describes a balanced contribution from the neighboring planes. One chooses the constant term C to ensure Upl (y)(y = 0) = 0. The planar potentials (4.50), (4.52), and (4.53) fall off rapidly with increasing distance from the plane. Therefore, Eq. (4.54) provides a good approximation for the interplanar potential at already moderate numbers of the terms included in the sum. Numerical data presented below refer to Nmax = 2. For positrons, the interplanar potential can be obtained from Eq. (4.54) by reversing the signs of the Upl terms and selecting the constant C to adjust Upl (y = ±d/2) = 0.

Electron interplanar potential (eV)

Positron interplanar potential (eV)

96

4 Channeling Phenomenon and Channeling Radiation

20

diamond(110)

20

Si(110)

Ge(110) 30

Moliére Pacios Doyler-Turner

20 10

10

10

0

-0.4 -0.2 0

0.2 0.4

0

-0.4 -0.2 0

0.2 0.4

0

-0.4 -0.2 0

0.2 0.4

Distance from midplane (in units of d)

20

20 30

20 10

10 10

0

-0.4 -0.2 0

0.2 0.4

0

-0.4 -0.2 0

0.2 0.4

0

-0.4 -0.2 0

0.2 0.4

Distance from midplane (in units of d)

Fig. 4.22 Positron (upper row) and electron (lower row) planar (110) potentials in diamond, silicon, and germanium crystals calculated within frameworks of the Moliére, Pacios, and Doyle–Turner approximations as indicated in the common legend in the top right graph. Reference [118]

Three graphs in Fig. 4.22 compares the Moliére, Pacios, and Doyler–Turner electron and positron planar (110) potentials in diamond, silicon, and germanium crystals. For the sake of reference, we present Fig. 4.23 that compares the Moliére planar potentials in different oriented crystals as indicated.

References

97

Interplanar potential (eV)

140 120 100

graphite(002) diamond(110) Si(110) Ge(110) W(110)

80 60 40 20 0

-1.5

-1 -0.5

0

0.5

1

1.5

-1.5

-1 -0.5

0

0.5

1

1.5

Distance from midplane (angstrom) Fig. 4.23 Interplanar potentials for electrons (left) and positrons (right) in Graphite (002), C(110), Si(110), Ge(110), and W(110) calculated at T = 300◦ within the Moliére approximation

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66. Kumakhov, M.A., Wedell, R.: Theory of radiation of relativistic channeled particles. Phys. Stat. Sol. (b) 84, 581–593 (1977) 67. Saenz, A.W., Überall, H.: Calculation of electron channeling radiation with realistic potential. Nucl. Phys. A 372, 90–108 (1981) 68. Solov’yov, A.V., Schäfer, A., Greiner, W.: Channeling process in a bent crystal. Phys. Rev. E 53, 1129–1137 (1996) 69. Arutyunov, V.A., Kudryashov, N.A., Samsonov, V.M., Strikhanov, M.N.: Radiation of ultrarelativistic charged particles in a bent crystal Nucl. Phys. B 363, 283–300 (1991) 70. Arutyunov, V.A., Kudryashov, N.A., Strikhanov, M.N., Samsonov, V.M.: Synchrotron and undulator radiation by fast particles in bent crystal. Sov. Phys. Tech. Phys. 36, 1–3 (1991) 71. Bashmakov, Yu.A.: Radiation and spin separation of high energy positrons channeled in bent crystals. Radiat. Eff. 56, 55–60 (1981) 72. Taratin, A.M., Vorobiev, S.A.: Radiation of high-energy positrons channeled in bent crystals. Nucl. Instrum. Meth. B 31, 551–557 (1988) 73. Taratin, A.M., Vorobiev, S.A.: Quasi-synchrotron radiation of high-energy positrons channeled in bent crystals. Nucl. Instrum. Meth. B 42, 41–45 (1989) 74. Kaplin, V.V., Vorobev, S.A.: On the electromagnetic radiation of channeled particles in a curved crystal. Phys. Lett. 67A, 135–137 (1978) 75. Jackson, J.D.: Classical Electrodynamics Wiley, Hoboken (1999) 76. Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Course of Theoretical Physics, vol. 2. Pergamon Press, Oxford (1971) 77. Flügge, S.: Practical Quantum Mechanics I. Springer, Berlin/Heidelberg/New York (1971) 78. A.V. Korol, Solov’yov, A.V.: Unpublished (2020) 79. Di Piazza, A., Wistisen, T.N., Uggerhøj, U.I.: Investigation of classical radiation reaction with aligned crystals. Phys. Lett. 765, 1 (2017) 80. Wistisen, T.N., Di Piazza, A., Knudsen, H.V., Uggerhøj, U.I.: Experimental evidence of quantum radiation reaction in aligned crystals. Nat. Commun. 9, 795 (2018) 81. Nielsen, C.F., Justesen, J.B., Sørensen, A.H., Uggerhøj, U.I., Holtzapple, R.: Radiation reaction near the classical limit in aligned crystals. Phys. Rev. D 102, 052004 (2020) 82. Korol, A.V., Solov’yov, A.V., Greiner, W.: Total energy losses due to the radiation in an acoustically based undulator: the undulator and the channeling radiation. Int. J. Mod. Phys. E 9, 77–105 (2000) 83. Krause, W., Korol, A.V., Solov’yov, A.V., Greiner, W.: Photon emission by relativistic positrons in crystalline undulators: the high-energy regime. Nucl. Instrum. Method A 483, 455–460 (2002) 84. Wistisen, T.N., Di Piazza, A.: Complete treatment of single-photon emission in planar channeling. Phys. Rev. D 99, 116010 (2019) 85. Barrett, J.H.: Monte Carlo channeling oscillations. Phys. Rev. B 3, 1527–1547 (1971) 86. Smulders, P.J.M., Boerma, D.O.: Computer simulation of channeling in single crystals. Nucl. Instum. Methods B 29, 471–489 (1987) 87. Artru, X.: A simulation code for channeling radiation by ultrarelativistic electrons or positrons. Nucl. Instrum. Method Phys. Res. B 48, 278–282 (1990) 88. Biryukov, V.M.: Computer simulation of beam steering by crystal channeling. Phys. Rev. E 51, 3522–3528 (1995) 89. Fomin, S.P., Jejcic, A., Kasilov, V.I., Lapin, N.I., Maillard, J., Noga, V.I., Shcherbak, S.F., Shul’ga, N.F., Silva, J.: Investigation of the electron channeling by means of induced electronuclear reactions. Nucl. Instum. Methods B 129, 29–34 (1997) 90. Shul’ga, N.F., Syshchenko, V.V.: Simulation of incoherent bremsstrahlung of high energy electrons and positrons in a crystal. Nucl. Instum. Methods B 227, 125–131 (2005) 91. Bogdanov, O.V., Fiks, E.I., Korotchenko, K.B., Pivovarov, Yu.L., Tukhfatullin, T.A.: Basic channeling with mathematica: a new computer code. J. Phys. Conf. Ser. 236, 012029 (2010) 92. Bagli, E., Guidi, V.: Dynecharm++: a toolkit to simulate coherent interactions of high-energy charged particles in complex structures. Nucl. Instrum. Meth. B 309, 124 (2013)

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93. Bagli, E., Guidi, V., Maisheev, V.A.: Calculation of the potential for interaction of particles with complex atomic structures. Phys. Rev. E 81, 026708 (2010) 94. Bandiera, L., Bagli, E., Guidi, V., Tikhomirov, V.V.: RADCHARM++: A C++ routine to compute the electromagnetic radiation generated by relativistic charged particles in crystals and complex structures. Nucl. Instrum. Meth. B 355, 44 (2015) 95. Sytov, A.I., Tikhomirov, V.V., Bandiera, L.: Simulation code for modeling of coherent effects of radiation generation in oriented crystals. Phys. Rev. Accel. Beams 22, 064601 (2019) 96. Sytov, A., Tikhomirov, V.: CRYSTAL simulation code and modeling of coherent effects in a bent crystal at the LHC. Nucl. Instrum. Methods B 355, 383 (2015) 97. Tikhomirov, V.V.: Quantum features of relativistic particle scattering and radiation in crystals. Phys. Rev. Accel. Beams 22, 054501 (2019) 98. Guidi, V., Bandiera, L., Tikhomirov, V.: Radiation generated by single and multiple volume reflection of ultrarelativistic electrons and positrons in bent crystals. Phys. Rev. A 86, 042903 (2012) 99. Korol, A.V., Solov’yov, A.V., Greiner, W.: The influence of the dechannelling process on the photon emission by an ultra-relativistic positron channelling in a periodically bent crystal. J. Phys. G Nucl. Part. Phys. 27, 95–125 (2001) 100. Nielsen, C.F.: GPU accelerated simulation of channeling radiation of relativistic particles. Comp. Phys. Comm. 252, 107128 (2020) 101. Kostyuk, A.: Monte Carlo simulations of electron channeling a bent (110) channel in silicon. Eur. Phys. J. D 67, 108 (2013) 102. Sushko, G.B., Bezchastnov, V.G., Solov’yov, I.A., Korol, A.V., Greiner, W., Solov’yov, A.V.: Simulation of ultra-relativistic electrons and positrons channeling in crystals with MBN Explorer. J. Comp. Phys. 252, 404 (2013) 103. Bezchastnov, V.G., Korol, A.V., Solov’yov, A.V.: Reply to Comment on “Radiation from multi-GeV electrons and positrons in periodically bent silicon crystal.” J. Phys. B 51, 168002 (2018) 104. Solov’yov, I.A., Yakubovich, A.V., Nikolaev, P.V., Volkovets, I., Solov’yov, A.V.: MesoBioNano explorer - a universal program for multiscale computer simulations of complex molecular structure and dynamics. J. Comp. Chem. 33, 2412–2439 (2012) 105. http://mbnresearch.com/get-mbn-explorer-software 106. Polozkov, R.G., Ivanov, V.K., Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Radiation emission by electrons channeling in bent silicon crystals. Eur. Phys. J. D 68, 268 (2014) 107. Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Multi-GeV electron and positron channeling in bent silicon crystals. Nucl. Instrum. Methods B 355, 39 (2015) 108. Sushko, G.B., Bezchastnov, V.G., Korol, A.V., Greiner, W., Solov’yov, A.V., Polozkov, R.G., Ivanov, V.K.: Simulations of electron channeling in bent silicon crystal. J. Phys.: Conf. Ser. 438, 012019 (2013) 109. Sushko, G.B., Korol, A.V., Greiner, W., Solov’yov, A.V.: Sub-GeV electron and positron channeling in straight, bent and periodically bent silicon crystals. J. Phys.: Conf. Ser. 438, 012018 (2013) 110. Bezchastnov, V.G., Korol, A.V., Solov’yov, A.V.: Radiation from multi-GeV electrons and positrons in periodically bent silicon crystal. J. Phys. B 47, 195401 (2014) 111. Sushko, G.B., Korol, A.V., Solov’yov, A.V.: A small-amplitude crystalline undulator based on 20 GeV electrons and positrons: simulations. St. Petersburg Polytechnical Uni. J.: Phys. Math. 1, 341 (2015) 112. Korol, A.V., Bezchastnov, V.G., Sushko, G.B., Solov’yov, A.V.: Simulation of channeling and radiation of 855 MeV electrons and positrons in a small-amplitude short-period bent crystal. Nucl. Instrum. Meth. B 387, 41–53 (2016) 113. Korol, A.V., Bezchastnov, V.G., Solov’yov, A.V.: Channeling and radiation of the 855 MeV electrons enhanced by the re-channeling in a periodically bent diamond crystal. Eur. Phys. J. D 71, 174 (2017) 114. Shen, H., Zhao, Q., Zhang, F.S., Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Channeling and radiation of 855 MeV electrons and positrons in straight and bent tungsten (110) crystals. Nucl. Instrum. Meth. B 424, 26 (2018)

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115. Pavlov, A.V., Korol, A.V., Ivanov, V.K., Solov’yov, A.V.: Interplay and specific features of radiation mechanisms of electrons and positrons in crystalline undulators. J. Phys. B: At. Mol. Opt. Phys. 52, 11LT01 (2019) 116. Pavlov, A.V., Korol, A.V., Ivanov, V.K., Solov’yov, A.V.: Channeling of electrons and positrons in straight and periodically bent diamond(110) crystals. Eur. Phys. J. D 74, 21 (2020) 117. Haurylavets, V.V., Leukovich, A., Sytov, A., Mazzolari, A., Bandiera, L., Korol, A.V., Sushko, G.B., Solovyov, A.V.: MBN explorer atomistic simulations of electron propagation and radiation of 855 MeV electrons in oriented silicon bent crystal: theory versus experiment (2021). arXiv:2005.04138 118. Korol, A.V., Sushko, G.B., Solov’yov, A.V.: All-atom relativistic molecular dynamics simulations of channeling and radiation processes in oriented crystals. Eur. Phys. J. D 75, 107 (2021) 119. Solov’yov, I.A., Korol, A.V., Solov’yov, A.V.: Multiscale Modeling of Complex Molecular Structure and Dynamics with MBN Explorer. Springer International Publishing, Cham, Switzerland (2017) 120. Sushko, G.B., Solov’yov, I.A., Solov’yov, A.V.: Molecular dynamics for irradiation driven chemistry: application to the FEBID process. Europ. Phys. J. D 70, 217 (2016) 121. Rossi, B., Greisen, K.: Cosmic-ray theory. Rev. Mod. Phys. 13, 241 (1941). (New York: Prentice-Hall, Inc.) 122. Tanabashi, M., et al.: (Particle Data Group): review of particle physics. Phys. Rev. D 98, 030001 (2018) 123. Tsyganov, E.N.: Some aspects of the mechanism of a charge particle penetration through a monocrystal. Fermilab Preprint TM-682. Fermilab, Batavia (1976); Estimates of cooling and bending processes for charged particle penetration through a monocrystal. Fermilab Preprint TM-684. Fermilab, Batavia (1976) 124. Beloshitsky, V.V., Kumakhov, M.A., Muralev, V.A.: Multiple scattering of channeling ions in crystals - II. Planar channeling. Radiat. Eff. 20, 95 (1973) 125. Backe, H., Kunz, P., Lauth, W., Rueda, A.: Planar channeling experiments with electrons at the 855-MeV Mainz Microtron. Nucl. Instrum. Method B 266, 3835–3851 (2008) 126. Bogdanov, O.V., Dabagov, S.N.: Radiation spectra of channeled electrons in thick Si (111) crystals. J. Phys.: Conf. Ser. 357, 012029 (2012) 127. Scandale, W., Fiorini, M., Guidi, V., Mazzolari, A., Vincenzi, D., et al.: Measurement of the dechanneling length for high-energy negative pions. Phys. Lett. B 719, 70 (2013) 128. Mazzolari, A., Bagli, E., Bandiera, L., Guidi, V., Backe, H., Lauth, W., Tikhomirov, V., Berra, A., Lietti, D., Prest, M., Vallazza, E., De Salvador, D.: Steering of a sub-GeV electron beam through planar channeling enhanced by rechanneling. Phys. Rev. Lett. 112, 135503 (2014) 129. Wienands, U., Markiewicz, T.W., Nelson, J., Noble, R.J., Turner, J.L., Uggerhøj, U.I., Wistisen, T.N., Bagli, E., Bandiera, L., Germogli, G., Guidi, V., Mazzolari, A., Holtzapple, R., Miller, M.: Observation of deflection of a beam of multi-GeV electrons by a thin crystal. Phys. Rev. Let. 114, 074801 (2015) 130. Wistisen, T.N., Uggerhøj, U.I., Wienands, U., Markiewicz, T.W., Noble, R.J., Benson, B.C., Smith, T., Bagli, E., Bandiera, L., Germogli, G., Guidi, V., Mazzolari, A., Holtzapple, R., Tucker, S.: Channeling, volume reflection, and volume capture study of electrons in a bent silicon crystal. Phys. Rev. Acc. Beams 19, 071001 (2016) 131. Backe, H., Lauth, W.: Channeling experiments with Sub-GeV electrons in flat silicon single crystals. Nucl. Instrum. Meth. B 355, 24–29 (2015) 132. Sytov, A.I., Bandiera, L., De Salvador, D., Mazzolari, A., Bagli, E., Berra, A., Carturan, S., Durighello, C., Germogli, G., Guidi, V., Klag, P., Lauth, W., Maggioni, G., Prest, P., Romagnoni, M., Tikhomirov, V.V., Vallazza, E.: Steering of Sub-GeV electrons by ultrashort Si and Ge bent crystals. Eur. Phys. J. C 77, 901 (2017) 133. Backe, H.: Electron channeling experiments with bent silicon single crystals—a reanalysis based on a modified Fokker-Planck equation. JINST 13, C02046 (2018)

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Chapter 5

Radiation Emission in Bent Crystals

5.1 Channeling of Particles in Bent Crystals: Basic Concepts Let a crystallographic channel be bent with a constant curvature of the radius R, see illustrative Fig. 5.1. In a bent channel an ultra-relativistic particle with the energy ε ≈ pv experiences (in the co-moving frame) the action of a centrifugal force Fcf ≈

ε . R

(5.1)

 Channeling if Fcf does not exceed the maximum transverse force Umax due to the acting on the particle in the interplanar (or, axial) potential [1]. It is convenient to quantify this statement in terms of a dimensionless bending parameter C defined as the ratio of the centrifugal force to the interplanar one:

C=

ε Fcf ≈ ≤ 1.   Umax RUmax

(5.2)

The value C = 0 corresponds to a straight channel. The minimum radius consistent with the channeling motion (frequently called as Tsyganov’s radius) corresponds to  iC = 1 and is given by Rc = ε/Umax . A somewhat different formulation of the channeling condition presented in Ref. [2] is convenient in the case of an axial channeling in bent crystals. Namely, one can require the bending angle Θ over one period λch of the channeling oscillations, see Fig. 5.1, to be smaller than Lindhard’s critical angle ΘL . The effective potential in a bent channel one constructs adding the centrifugal  ρ (ρ is the distance from midplane) the continuous potential. Then, term −CUmax the potential U (C; ρ) is written as follows  ρ. U (C; ρ) = U (ρ) − CUmax

(5.3)

The centrifugal term affects the projectile’s motion in the transverse direction. The modified potential becomes asymmetric with respect to ρ = 0 and its depth ΔUC © Springer Nature Switzerland AG 2022 A. Korol and A. V. Solov’yov, Novel Lights Sources Beyond Free Electron Lasers, Particle Acceleration and Detection, https://doi.org/10.1007/978-3-031-04282-9_5

105

106 Fig. 5.1 In a bent channel a channeling particle experiences the action of the centrifugal force Fcf in addition to the interplanar force. The curvature radius R and the interplanar spacing d satisfy the condition R  d

5 Radiation Emission in Bent Crystals

Fcf

d R Θ

is a decreasing function of C. These features are illustrated by Fig. 5.2 where the effective potential in the Si(110) planar channel is presented for several values of bending parameter C. In accordance with (5.2), the potential well disappears at C = 1 making channeling channeling motion virtually nonexistent. For C < 1 the quantity ΔUC determines the maximum value of the transverse energy consistent with the channeling motion which, in turn, defines the critical angle Θc (C) in bent channel  Θc (C) =

2 ΔUC . ε

(5.4)

In a straight channel C = 0, and Θc (0) reduces to Lindhard’s critical angle ΘL . The condition to be met for a projectile to be accepted into the channeling mode at the crystal entrance reads Θ < Θc (C) where Θ is the angle between the particle velocity and the channel centerline [1]. Theoretical analysis of various phenomena related to the channeling in bent crystals one finds in [3–7]. The review of theoretical models and computational methods can be found in [8–11]. Since its theoretical prediction [1] and experimental support [12, 13], the idea to deflect high-energy beams of charged particles by means of bent crystals has attracted a lot of attention worldwide and still is of a great interest due to its growing practical application. Indeed, the crystals with bent crystallographic planes are used to steer high-energy charged particle beams replacing huge dipole magnets. We refer to the review papers which describe in detail initial stages of the development of the idea as well as the experiments carried out [8, 9, 14–16]. Bent crystal have been routinely used for beam extraction in the Institute for High Energy Physics, Russia [17]. A series of experiments on the bent crystal deflection and collimation of proton and heavy ion beams were performed at different accelerators [18–25] throughout the world. The bent crystal method has been proposed to extract

5.1 Channeling of Particles in Bent Crystals: Basic Concepts

107

Interplanar potential U(C;ρ) (eV)

40 C=0 C=0.2 C=0.4

30

20

U0

10

ΔUC

0

-10

-0.4

-0.2

0

0.2

Distance from centerline, ρ/d

0.4

Fig. 5.2 Effective interplanar potential U (C; ρ) in Si(110) as a function of the distance ρ from the midplane (the interplanar distance is d = 1.92 Å). The curve C = 0 stands for the potential U (ρ) in the straight channel calculated within the Molière approximation; its depth is U0 = 22.7 eV. ΔUC denotes the depth of the effective potential in a bent channel

particles from at CERN’s Large Hadron Collider [26]. The experiments have been carried with ultra-relativistic protons, ions, positrons, electrons, π − -mesons [20–23, 27–34]. Another possible application of bent crystal concerns a possibility to focus ultrarelativistic beams of heavy-ions [35] or protons [36]. To achieve this a crystal is needed in which the axes/planes are slanted more and more the farther away they are from the axis of the beam. Then the bending angle of the particles far away from the beam axis would be largest and a general focusing effect will result. Such a crystal can be produced by varying Ni to Cu (or Sb to Bi) ratio in a mixed crystal [35]. Similar approach has been also suggested and implemented for producing periodically bent crystals, see Chap. 1. Recently, it was suggested to use bent nanotubes to steer beams of ultra-relativistic charged particles [37–39]. The motion along the centerline of the bent channel gives rise to the synchrotrontype radiation, i.e. the one, which is emitted by a charge moving along a circle (or, its part) under the action of the magnetic field. Therefore, the total spectrum of radiation formed by an ultra-relativistic projectile in bent crystal contains the features of ChR and those of synchrotron radiation [40]. It was shown [41] that in the case of planar channeling the bending noticeably affects the spectrum only if the condition Θc γ ≤ 1 is met (Θc is the critical angle in the bent channel). In this case, the total spectrum preserves the features of ChR if L c  λ (here, L c = R/γ is the formation (coherence) length of the radiation emitted from an arc of the circle of the radius R), but becomes of the quasi-synchrotron type in the opposite limit [8, 41].

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Fig. 5.3 Selected trajectories of 855 MeV electrons in a L 1 = 25 µm thick oriented silicon (110) crystal bent with radius R = 1.3 cm. Reference [46]

The peculiarities which appear in spectral and spectra-angular distributions of the emitted radiation due to the interference of the two mechanisms of radiation were analyzed in Refs. [2, 42–45] by analytical and numerical means. The motion of a channeling particle in a bent crystal contains two components: the channeling motion itself and translation along the centerline of the bent channel. This feature is seen in the illustrative Fig. 5.1 as well in Fig. 5.3 [46]. The latter shows several selected trajectories of 855 MeV electrons in bent Si(110) crystal simulated by means of MBN Explorer package. The figure illustrates a variety of features which characterize the electron motion: the channeling mode, the dechanneling, the over-barrier motion, the rechanneling process, rare events of hard collisions etc. Recently, the method of relativistic molecular dynamics [47] has been applied for accurate computational modeling and numerical analysis of the channeling phenomena and the radiation emission various crystals irradiated by beams of ultra-relativistic electrons and positrons. The results of numerical simulations of the emission spectra by 855 MeV electrons were reported for straight and uniformly bent silicon (110) [46] and tungsten (110) crystals [48]. The influence of the detector aperture on the form of the spectral distribution of the emitted radiation was explored in [49]. The results of simulation of planar channeling as well as the calculated spectral distribution of the emitted radiation were reported in Ref. [50] for 3…20 GeV electrons and positrons in bent Si(111). In Ref. [51] the channeling phenomenon was analyzed for 855 MeV electrons in bent oriented silicon (111) crystal. Special attention was devoted to the transition from the axial channeling regime to the planar one in the course of the crystal rotation with respect to the incident beam. Distribution in the

5.1 Channeling of Particles in Bent Crystals: Basic Concepts

109

deflection angle of electrons and spectral distribution of the radiation emitted are analyzed in detail. The results of calculations are compared with the experimental data collected at the MAinzer MIctrotron (MAMI) facility.

5.2 Experimental Studies of the Phenomena A silicon crystal, due to its availability and high degree of purity of a crystalline structure, has been extensively used in the experiments carried out at various accelerator facilities starting from early days of the activity in the field up to nowadays, see e.g. [9, 14, 52–61]. Diamond crystals (natural or/and synthesized [62]) has also been used in channeling experiments (e.g. [57, 63–65]). The use of a diamond crystal is preferential in the experiments with high-intensity particle beams (such as the FACET beam at SLAC [66]) since it bears no visible influence from being irradiated [52]. Modern technologies available for preparing periodically bent crystals do not immediately allow for lowering the values of bending period down to tens of microns range or even smaller keeping, simultaneously, the bending amplitude in the range of several angstroms, see Chap. 1. These ranges of a and λu are most favourable to achieve high intensity of radiation [10]. One of the potential options to lower the bending period is related to using crystals heavier than diamond and silicon to propagate ultra-relativistic electrons and positrons. In recent experiments [67, 68], carried at the MAinzer MIkrotron, the steering of effect and the detection of radiation emitted by 855 electrons in bent silicon and germanium crystals has been reported. The germanium target was chosen to study the steering capability of planar channeling and to measure the intensity of channeling radiation in a higher-Z material. The first evidence of negative beam steering by planar channeling in a Ge crystal has been presented. Experimentally, it is more advantageous to study the interaction of charged particles with a bent crystal rather than with a straight one. The former case provides the opportunity to analyze the channeling efficiency by separating the channeling particles from the over-barrier particles, i.e. those which travel across the crystal planes (axes) at the angle larger than the Lindhard critical angle. Figure 5.4a shows a sketch of the experimental set-up which can be used to study the channeling efficiency by measuring the deflection angle of the incoming particles (electrons) due to the interaction with a bent crystal. The direction O Z of an incident electron beam is aligned with the tangent to the crystal planes at the crystal entrance. Placing a detector behind the crystal one can measure the distribution of electrons with respect to the deflection angle β. Different modes of the electrons motion contribute to different parts of the angular distribution (shown with a thick solid curve 1 − 2 − 3 on the top of the panel). Namely, the distribution of the electrons that move in the over-barrier mode from the entrance to the exit points (see schematic trajectory 1) is centered along the OZ direction, i.e. at β = 0. The particles that channel through the whole crystal of length L (trajectory 3) give rise to the maximum

110

5 Radiation Emission in Bent Crystals

Fig. 5.4 Schematic representation of the beam-crystal orientation for the study of a channeling efficiency, b volume reflection and volume capture effects in a bent crystal. In both panels, the incident electron beam enters the crystal along the direction OZ. Interaction with the crystal deflects the electrons. A detector, located behind the crystal, allows one to determine the distribution of electrons with respect to the deflection angle β (shown as thick solid curves 1 − 2 − 3 on the top of the panels). In the crystal, the electrons experience two basic types of motion: over-barrier motion (straight segments in the trajectories) and the channeling motion (wavy segments). In panel (a) the beam enters the crystal along the tangent to the crystal planes. The angular distribution of the initially over-barrier electrons at the entrance, label “1”, is centered along the OZ direction, β = 0. The electrons captured into the channeling mode at the entrance follow the crystal bending and thus are deflected by larger angles. Some of these electrons, which channel through a part of the crystal and then dechannel (label “2”), contribute to the central part 2 of the angular distribution. Those particles that channel through the whole crystal of length L (label “3”) contribute to the maximum 3 of the distribution at β = R/L. Panel (b) corresponds to the case when the angle between the incident beam and the tangent exceeds the Lindhard angle. As a result, most of the particles propagate in the over-barrier mode starting from the entrance. In this case, due to the crystal bending a particle can experience a volume reflection or a volume capture as marked by circles in the trajectories 1 and 2, respectively. The angular distribution of electrons provides the quantitative description of these events. Reference [51]

centered around β = R/L (marked as 3 ). Comparing the areas below these two maxima one can quantify the channeling efficiency, i.e. the relative number of the particles channelled through the whole crystal. Finally, the electrons captured into the channeling mode at the entrance but dechannelled somewhere inside the crystal (trajectory 2) experience deflection by the angle within the interval 0 < β < R/L. Their angular distribution (curve 2 ) allows one to deduce the dechanneling length of the particles. By means of bent crystals one can study other phenomena associated with the interaction of charged particles with oriented crystalline medium, such as volume

5.2 Experimental Studies of the Phenomena

111

Fig. 5.5 Schematic representation of the experimental setup at the MAMI facility. The collimated electron beam propagates through the oriented Bent Crystal (BC). On the exit, the electrons are deflected by a Bending Magnet (BM) towards the Detector (D) to measure the deflection angle distribution. The photon beam from the target is collimated and directed to the calorimeter where the photons energy is measured. The abbreviation “IC” stands for Ionization Chamber. For more details see, e.g., Refs. [53, 71, 72]. The figure’s layout is from Ref. [51]

reflection (VR) and volume capture (VC) [69, 70] . In this case, the incident beam is directed at the angle larger than the Lindhard angle, Fig. 5.4b. As a result, at the crystal entrance most of the particles start moving in the over-barrier mode across the bent channels. As a result of the bending of the crystal, at some point in the crystal volume the incident angle with respect to the local tangent direction can become less than the Lindhard angle so that the particle enters the channeling mode. This effect known as VC is illustrated by trajectory 2. In the process of VR the overbarrier particle is deflected to the side opposite the bend (see illustrative trajectory 1) due to the action of the centrifugal force. The efficiency of the VC and VR events in bent crystals can also be analyzed by measuring experimentally (or simulating numerically) the angular distribution of electrons behind the crystalline target. As an example, Fig. 5.5 shows experimental setup at the MAinz MIkrotron (MAMI) facility, which has been used to measure the radiation emission spectra and the distribution of electrons with respect to the deflection angle in bent crystals [34, 67, 68, 73] A 855 MeV electron beam, generated by the microtron, is aligned to the oriented bent crystal mounted on the goniometer. Beyond the target, the electrons, deflected by magnets, move towards a luminosity screen (a detector). Hitting the screen an electron causes an optical flash that is detected by a photo-camera. The angular distribution of electrons is built based on the processing of the detected flashes. An ionization chamber, indicated in the figure, allows one to measure the number of electrons reflected from the crystallographic axes and/or planes. This information

112

5 Radiation Emission in Bent Crystals

is used to determine the orientation of the crystalline target with respect to the incident beam. The radiation emitted in the target is registered by an electromagnetic calorimeter located behind the crystal along the incident beam direction. To decrease the background radiation the calorimeter is placed behind a lead protection which has a d = 40 mm hole to collimate the photon beam from the target. The hole size corresponds to the aperture of 4.63 mrad.

5.3 Results of Atomistic Simulations A quantitative analysis of the channeling motion and emission spectra based on the simulation with the MBN Explorer package has been carried out for sub-GeV electrons and positrons channeling in silicon [46], diamond [74] and tungsten [48] crystals. In Ref. [50] similar analysis has been presented for 4…20 GeV projectiles in oriented bent Si(111) crystal. In Ref. [51] the channeling phenomenon was analyzed for 855 MeV electrons in bent oriented silicon (111) crystal.

5.3.1 Radiation Emission by 855 MeV Electrons in Bent Silicon (110) Crystals MBN Explorer was used to simulate the trajectories 855 Mev electrons in bent silicon crystals incident along the (110) crystallographic planes [46]. The calculations were performed for two values of the crystal length, L = 25 and 75 µm, measured along the beam direction. The simulated trajectories were used to compute spectral distribution of the emitted radiation. The integration over the emission angle was performed for the aperture θ0 = 2.4 mrad which greatly exceeds the natural emission angle γ −1 ≈ 0.6 mrad.

5.3.1.1

Emission Spectra for L = 25 µm

Spectral enhancement factor, i.e. the spectral distribution dE/d(¯hω) of radiation emitted in the crystalline medium normalized to that in amorphous silicon, calculated for 855 MeV electrons is presented in Fig. 5.6. Solid curve represents the dependence obtained for straight Si(110), two other curves stand for the channel bent uniformly with the radii R = 2.5 and 1.3 cm. Several specific features of the emission spectra formed in the bent channels can be noted. First, the bending gives rise to the synchrotron radiation, since the channeled particle experiences the circular motion in addition to the channeling oscillations.

5.3 Results of Atomistic Simulations

113

This leads to the increase of the intensity in the photon energy range  102 keV. For these energies the radiation yield from the bent channel exceeds that from the straight channel. Second, there is a noticeable decrease in the intensity of the channeling radiation with the decrease of the bending radius. The ratio ξ of the maximum value of the spectrum in the bent channel to that in the straight one is 0.66 for R = 2.5 cm and 0.49 for R = 1.3 cm. The decrease of ξ can be explain as follows. In the vicinity of the maximum, the main contribution to the channeling radiation intensity comes from those segments of a trajectory where the particle stays in the same channel experiencing channeling oscillations. The intensity of radiation emitted from either of this segments and integrated over the emission angles is proportional to the segment length l and to the square of the undulator parameter K associated with the channeling oscillations. The undulator parameter K is related to the mean square of the transverse 2 /c2 . Then, to estimate the right-hand side of Eq. (4.41) velocity: K 2 = 2γ 2 v⊥ one can consider the contribution to the sum coming from the primary channeling segments of the accepted particles. Hence dE(θ ≤ θ0 ) ∝ A K 2 L p1 . h¯ dω

(5.5)

For bending radii much larger than the critical radius Rc = ε/Fmax = 0.15 the factor K 2 depends weakly on R. Hence, the factor ξ , defined above, can be calculated at the ratio of the products A L p1 calculated for the bent and the straight channels. Using

Enhancement

10

5

R=∞ R=2.5 cm R=1.3 cm 0 0

5

10

Photon energy (MeV) Fig. 5.6 Enhancement factor for 855 MeV electrons channeled in L = 25 µm straight (solid curve) and bent Si crystals along (110) planes. The curve with open circle corresponds to the bending radius R = 2.5 cm (the bending parameter C = 0.06), with open diamonsds – to R = 1.3 cm (C = 0.1). The data refer to the emission angle θ0 = 2.4 mrad. Reference [46]

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5 Radiation Emission in Bent Crystals

Table 5.1 Acceptance A , bending parameter C and penetration length L p (see Eqs. (4.29), (5.2) and (4.31), respectively) as functions of bending radius R for 855 MeV positrons in straight (R = ∞) and bent (R < ∞) planar channels Si(110) [46] R (cm) ∞ 13 7.0 2.5 1.3 0.75 0.45 C 0.00 0.01 0.02 0.06 0.10 0.20 0.30 A 0.66 0.65 0.64 0.55 0.44 0.34 0.22 L p (µm) 11.7 ± 0.3 11.4 ± 0.3 11.1 ± 0.3 10.2 ± 0.3 8.7 ± 0.3 7.0 ± 0.4 5.3 ± 0.3

the data from Table 5.1 one estimates: ξ = 0.72 ± 0.03 and 0.48 ± 0.02 for R = 2.5 and 1.3 cm, respectively. These values correlate with the ones quoted above. Thus, the decrease in the acceptance A and in the penetration length L p1 with R is the main reason for lowering the intensity of channeling radiation. Finally, comparing the curves in Fig. 5.6 one notices that the position ωmax of the maximum shifts to higher photon energies with the growth of the crystal curvature 1/R. This feature is specific for the electron channeling (more generally, for channeling of negatively charged projectiles) and is due to strong anharmonicity of the channeling oscillations. As a result, the frequency Ωch of the oscillations varies with the amplitude ach . Typically, smaller vales of ach correspond to larger frequencies (see, for example, the results of simulations presented in [49]). As the curvature increases, the allowed values of ach decrease due to the action of the centrifugal force. Hence, on average, the frequency of the channeling increases and so does the frequency of the emitted photons ωmax ∝ Ωch . The curves in Fig. 5.6 stand for the spectral distributions averaged over all simulated trajectories. The main contribution to the radiation enhancement comes from the electrons propagate in the channeling regime. In this context it is of interest to analyze the spectral distribution of radiation produced only by electrons propagating through the whole bent crystal staying in a single channel or in a few different channels (i.e. changing the channels due to the rechanneling effects). Such trajectories, although being quite rare, allow one to visualize the influence of the channeling oscillations as well as of the rechanneling effect on the synchrotron-like part of the spectrum. The radiation spectra, averaged over each group of electrons propagating in Si(110) crystal bent with R = 1.3 cm, are represented in Fig. 5.7 by solid curves labeled with “1” and “2”. At hω ¯ ≈ 5 MeV both curves have the maxima corresponding to the channeling radiation. It is not surprising that the enhancement factor averaged over the specially selected trajectories is much larger than the one averaged over all trajectories, see broken curve in Fig. 5.6. The increase of the enhancement factor at small photon energies is associated with the synchrotron radiation. To visualize this the spectral distribution of radiation due to the motion along a 25 µm arc of a circle of the radius 1.3 cm (i.e. along the bent channel centerline ignoring the channeling oscillations), is shown in Fig. 5.7 by the broken line. Within the statistical errors (not indicated) the curves 1 and 2 coincide with the synchrotron spectrum at hω ¯  10−1 MeV.

5.3 Results of Atomistic Simulations

115

Enhancement

20

10

2

0 -2 10

1

0

-1

10

10

1

10

Photon energy (MeV) Fig. 5.7 Spectral distribution of radiation by 855 MeV electrons in a L = 25 µm thick Si(110) crystal bend with the radius R = 1.3 cm. The data refer to the aperture θ0 = 2.4 mrad. Curves 1 and 2 correspond to the enhancement factor averaged over two specially selected sets of trajectories. The broken represents the spectrum of synchrotron radiation normalized to the Bethe-Heitler background. See also explanation in the text. Reference [46]

θ0=3.6 mrad

_

dE/d(hω) (×1000)

5 θ0=2.4 mrad 4 3 θ =1.2 mrad 0 2 1 θ0=0.6 mrad 0 1 10

2

10 Photon energy (keV)

3

10

Fig. 5.8 Low-energy part of radiation spectrum formed by 855 MeV electrons in a L = 25 µm thick silicon crystal with bending radius R = 1.3 cm. The curves correspond to different apertures, which are integer multiples of the natural emission angle γ −1 ≈ 0.6 mrad. Full curves stand for the spectra averaged over the electrons channeling through whole crystal in the same channel, Broken curves represents the spectra of synchrotron radiation. Reference [46]

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5 Radiation Emission in Bent Crystals

In Ref. [49] the influence of the detector aperture θ0 on the form of the spectral distribution of the channeling radiation in straight and bent Si(110) channels was explored. Figure 5.8 presents results of similar analysis but for the low-energy part of the spectrum. Full curves correspond to the spectra averaged over the simulated trajectories corresponding to the motion through the whole crystal staying in the same channel (“group 1”). Broken curves represent the distribution of the synchrotron radiation due to the motion along the circle arc. The calculations were performed for θ0 = 0.6, 1.2, 2.4 and 3.6 mrad which are integer multiples of the natural emission angle θγ = γ −1 ≈ 0.6 mrad. The latter characterizes the cone (along the instantaneous velocity) which accumulates most of the radiation emitted by an ultra-relativistic projectile. Two features of the presented dependences can be mentioned. First, it is seen that the intensity is quite sensitive to the detector aperture. This effect is more pronounced for the lower values of θ0 and within the 10 . . . 100 keV photon energy range where most of radiation is emitted via the synchrotron mechanism. For example, a two-fold change in the aperture from θγ to 2θγ results in a nearly four-fold increase of the intensity at in the lowest-energy part of the spectrum. Such a behaviour can be understood if one compares the quoted values of θ0 with the angle of rotation of the bent crystal centerline, θ L = L/R ≈ 1.9 mrad. For apertures smaller than θ L the radiation within the cone θ0 along the incident velocity will be effectively emitted only from the initial part of the crystal, the length l0 of which can be estimated as l0 ≈ Rθ0 = (θ0 /θ L )L < L. Thus, the raise of the aperture from 0.6 to 1.2 mrad results in the two-fold increase of l0 which, in turn, leads to additional augmentation of the emitted intensity. This effects becomes less pronounced for larger apertures, θ0 > θ L  θγ , which collects virtually all radiation emitted within L. Another feature to be mentioned is that the influence of the channeling motion on the emitted spectrum becomes more pronounced over wider range of photon energies with the increase of the aperture. Indeed, the deviation of the total emission spectrum (full curves) from the spectrum of synchrotron radiation (broken curves) ¯ ≈ 50 keV for θ0 = 3.6 mrad. starts at hω ¯ ≈ 250 keV for θ0 = 0.6 mrad but at hω To explain this one recalls that channeling motion bears close resemblance with the undulating motion. As a result, constructive interference of the waves emitted from different but similar parts of the trajectory increases the intensity. For each value of the emission angle θ the coherence effect is most pronounced for the radiation into harmonics. The  frequency of the  fundamental harmonic can be estimated as ω1 (θ ) = 2γ 2 Ωch / 1 + γ 2 θ 2 + K 2 /2 (see Eq. (2.26)). Anharmonicity of the electron channeling oscillations results in the variation of Ωch along the trajectory. Therefore, the radiation will be emitted within some frequency band which will form the main peak in the spectral distribution of the channeling radiation. For small apertures, when (γ θ0 )2  1, the emission of low-energy photons with ω  ω1 (0) is strongly suppressed. For larger apertures a big part of energy is radiated into the cone γ −1 < θ < θ0 . For (γ θ0 )2  1 the harmonic energy is strongly red-shifted ω1 (θ0 )  ω1 (0). As a result, the contribution of the channeling radiation to the lowenergy part of the spectrum increases with the aperture.

5.3 Results of Atomistic Simulations

5.3.1.2

117

Emission Spectra for L = 75 µm

For L = 75 µm thick Si(110) crystal the trajectories were simulated for a wider range of the bending radius. The results of calculations of the enhancement factor obtained for the aperture θ0 = 2.4 mrad are presented in Figs. 5.9 and 5.10. Figure 5.9 illustrates the modification of the spectral dependence of the enhancement factor in the vicinity of the maximum of channeling radiation. Comparing the presented dependences with those calculated for a shorter crystal, Fig. 5.6, one can state that in both cases the profiles of the enhancement factor are similar. and the maximum values of the enhancement factor are similar in both cases. In both figures there are two curves calculated for the same values of the bending radius: R = ∞ (straight crystal) and R = 1.3 cm. Comparing these one notices that the maximum values are larger (by factors ≈ 1.4 and 1.6, respectively) for a shorter crystal. To explain this one can use the following arguments. In a straight crystal, the intensity of channeling radiation is proportional to the total length of the channeling segments, L ch , whereas the intensity of the incoherent bremsstrahlung scales with the crystal length L. Hence, the enhancement factor is proportional to L ch /L. The numerically established values of L ch are 26 and 12.5 µm for a 75 and 25 µm thick crystals, respectively. Hence the ratio of the L ch /L calculated for L = 25 and 25 µm straight Si(110) crystals is 3/2, which correlates with the factor quoted above. Similar arguments can be applied to the case of a bent crystal. The only difference is that the intensity of the background bremsstrahlung radiation integrated over the aperture θ0 is proportional to the effective length l0 ∼ min {L , θ0 R}.

Enhancement

8 R=∞ R=3.0 cm R=1.3 cm R=0.75 cm R=0.45 cm R=0.38 cm

6

4

2

0 0

1

3 4 2 Photon energy (MeV)

5

6

Fig. 5.9 Enhancement factors for 855 MeV electrons channeled in L = 75 µm straight and bent Si(110) crystals. The numbers indicate the values of the bending radius in cm. The data refer to the aperture θ0 = 2.4 mrad. Reference [46]

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5 Radiation Emission in Bent Crystals

Enhancement

1.8

1.6

1.4

R=∞ R=3.0 cm R=1.3 cm R=0.75 cm R=0.45 cm R=0.38 cm

1.2 0

10

20

30

40

50

Photon energy (keV) Fig. 5.10 Enhancement factors for the low-energy part of radiation spectrum formed by 855 MeV electrons in a L = 75 µm straight and bent Si(110). The numbers indicate the values of the bending radius R in cm. The data refer to the aperture θ0 =2.4 mrad. Reference [46]

The synchrotron radiation influences the total spectrum in the photon energy range well below the maximum of channeling radiation. Figure 5.10 illustrates the behaviour the enhancement factor in the low-energy part of the spectrum. It is seen from the figure that the enhancement due to the synchrotron radiation is a non-monotonous function of the bending radius. At small curvatures the enhancement increases with 1/R. The maximum values are achieved at R ≈ 0.75 cm and the enhancement decreases with the curvature. This feature is due to the finite aperture which for sufficiently small values of R introduces the length l0 < L of the crystal where the radiation detected within the cone θ ≤ θ0 is effectively formed, and thus reduces the enhancement.

5.3.2 Radiation Emission by 855 MeV Positrons in a Tungsten Crystal In heavier crystals, both the depth, ΔU ∝ Z 2/3 , of the interplanar potential and its the  ∝ Z 2/3 , attain larger values, resulting in the enhancement maximum gradient, Umax  of the critical channeling angle and reduction of the critical radius Rc ∝ 1/Umax [1]. From this end, the tungsten crystal (Z = 74) is a good candidate for the study. Within the framework of the continuous interplanar potential model, one estimates  = 42.9 GeV/cm for the tungsten the maximum value of the interplanar field as Umax (110) planar channel, see Sect. 4.5.3 in Appendix 1. For a 855 MeV projectile, this results produces the critical bending radius Rc = ε/Fmax ≈ 0.02 cm is seven times

5.3 Results of Atomistic Simulations

119

1.5

θ0=8 mrad

θ0=0.24 mrad

15 straight R=0.86 cm R=0.43 cm R=0.29 cm R=0.17 cm R=0.12 cm R=0.09 cm R=0.04 cm

3

dE/h dω (×10 )

straight R=0.86 cm R=0.43 cm R=0.29 cm R=0.17 cm R=0.12 cm R=0.09 cm R=0.04 cm

10

_

1

_

3

dE/h dω (×10 )

up to 9.2

0.5

5

0 0

10

20

Photon energy (MeV)

30

0 0

10

20

30

Photon energy (MeV)

Fig. 5.11 Spectral distribution of radiation emitted by 855 MeV positrons in straight and bent W(110) crystals. The left and right graphs correspond to the emission angle θ0 = 0.24 and 8 mrad, respectively. All spectra refer to the crystal thickness L = 10 microns. Reference [48]

smaller than that in Si(110). This allows one, at least in theory, to consider periodic bending with λu  10 µm. This crystal was used in channeling experiments with both heavy [9, 75] and light [76, 77] ultra-relativistic projectiles. Quantitative analysis of both the channeling process and the emission of radiation of high-energy light projectiles in straight tungsten crystal was carried out within model approaches. In Refs. [78, 79] the continuous potential model was applied to construct the trajectories, whereas the dechanneling phenomenon was considered within the framework of the Fokker-Plank equation. The model of binary collisions was exploited in Ref. [80] to investigate the scattering angle of 0.5 GeV electrons and positrons in the process of axial channeling. No results have been presented for bent tungsten crystal. Planar channeling of 855 MeV electrons and positrons in L = 10 and 75 µm thick straight and bent tungsten (110) crystals has been simulated by means of the MBN EXPLORER software package in Ref. [48]. The results of simulations for a broad range of bending radii have been analyzed in terms of the channel acceptance, dechanneling length, and spectral distribution of the emitted radiation. Particular attention has been devoted to the formation of the synchrotron-type radiation in bent crystals. The modification of the emission spectrum with bending radius R > Rc is illustrated by Fig. 5.11. The dependences presented refer to 855 MeV positrons planar channeling in L = 10 µm thick crystal. Left graph refers to the spectral distribution of radiation emitted in the cone θ0 = 0.24 mrad (left graph), which is much smaller than the natural emission angle γ −1 = 0.6 mrad. Right graphs corresponds to a much larger cone θ0 = 8 mrad that collects virtually all radiation emitted by the ultra-relativistic projectiles. The channeling trajectories of positrons demonstrate nearly harmonic oscillations between the neighboring planes. As a result, for each value of the emission angle θ the spectral distribution of ChR in a straight crystal reveals a set of narrow and equally spaced peaks (harmonics). The harmonic frequencies, ωn , can be estimated from Eq. (2.26) where Ω0 and K stand the frequency and the undulator parameter

120

5 Radiation Emission in Bent Crystals

of the channeling oscillations. The maximum value of the latter can be estimated as 2π γ (d/2)/λch with d/2 and λch being, respectively, the maximum possible amplitude and the period of the oscillations. Within the framework of harmonic approximation for the interplanar potential, one derives Ωch = 2d/cΘL and K ch ≤ γ ΘL . For a 855 MeV positron channeling in straight W(110) crystal this estimate produces K ch  1. As a result, the emission spectrum contains few harmonics the intensities of which rapidly decrease with the number n. This feature is explicitly seen in the spectral distributions calculated from the simulated trajectories of positrons propagating in straight crystals, see the solid curves with open circles in Fig. 5.11. The well-defined peaks (more pronounced for the smaller aperture) correspond to the harmonics of the ChR. For both apertures, the most powerful first peak is located at ≈ 4 MeV This value corresponds to the energy of the first (or, fundamental) harmonic emitted in the forward direction which one obtains from Eq. (2.26) by setting n = 1, θ = 0 and using the aforementioned estimates for Ω0 and K . The intensities of the emission into higher harmonics (the peaks with n up to 5 are seen located at hω ¯ n ≈ 4n MeV) rapidly decrease with n. As the bending curvature increases, the spectral distributions of radiation emitted by positrons become modified following two different scenarios. The first one manifests itself as the decrease in the peak intensities with decrease in the bending radius R. This feature is much more pronounced for the smaller aperture. For example, the intensity of the first harmonic peak for R = 0.86 cm is six times less than in the straight crystal. To provide a qualitative explanation [48] we refer to Table 5.2 which contains the simulated data on the acceptance factor A (4.29) and the penetration length L p (4.31) of a 855 MeV positron in W(110) for different values of bending radius. The data on L p for positrons indicate that L p > L = 10 µm for all values of R presented. Hence, on average, all accepted particles propagate through the whole crystal moving in the channeling mode. In a straight crystal, the peak intensity dE is proportional to the product A (∞)L. In a bent channel, the emission in the cone θ0 aligned with the incident beam occurs over the whole crystal only if L < Rθ0 . In this case, the peak intensity dE ∝ A (R)L. In the opposite limit, the radiation within θ0 is emitted from the initial part of the crystal of the length Rθ0 , so that dE ∝ A (R)Rθ0 It is seen from the table, that even for the largest radius considered, R = 0.86 cm, the emission into the nearly forward cone θ0 = 0.24 mrad occurs from a short initial part of the length Rθ0 ≈ 2.1 µm. Hence, the estimate of the intensities ratio reads: dE| R 0) 10 µm thick W(110) crystals. The spectra correspond to the large emission cone θ0 = 8 mrad. The peaks at hω ¯ ≤ 1 MeV are due to the synchrotron radiation (dashed curves). The values of bending parameter C = ε/R = 0, 0.069, 0.174, 0.231, 0.463, indicated in the legend, correspond to the bending radii R = ∞, 0.29, 0.12, 0.09, 0.04 cm, see Table 5.2. Reference [48]

right show that for C > 0 the synchrotron radiation manifests itself as an additional structure in the low-energy part of the spectrum (¯hω  1 MeV). Its intensity increases with the bending parameter up to C ∼ L/θ0 , and then it decreases due to the reason discussed above: the larger curvature is, the smaller is the part of the trajectory which contributes to the radiation cone θ0 . To clearer visualize the relationship between the additional structure and the synchrotron radiation, we present Fig. 5.12 where the spectra calculated from the simulated trajectories for several representative values of C (solid curves) are plotted in a narrower range of photon energies. Also plotted are the spectral distributions of synchrotron radiation, Eq. (2.13), emitted by 855 MeV positrons moving along the circular arc of the length L = 10 µm and of the bending radius R corresponding to the indicated value of C. These dependencies were scaled to match the simulated distributions in the maxima.

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5 Radiation Emission in Bent Crystals

5.3.3 Multi-GeV Electron and Positron Channeling in Bent Silicon Crystals In Ref. [81] the dechanneling phenomenon of ε = 3 . . . 20 GeV electrons in Si(110) oriented crystal has been experimentally investigated at the SLAC facility. In the experiment, the beam of electrons was deflected due to channeling effect in the bent crystal, see illustrative Fig. 5.13. The crystal thickness in the beam direction was L = 60 µm, the bending radius R = 15 cm. These values correspond to the bending angle L/R = 400 µrad. In the experiment, the distribution of the beam particles with respect to the deflection angle θ was measured. Then, by means of the fitting procedure applied to the experimental data the values of dechanneling length L d and acceptance factor A have been estimated. In Ref. [50] the simulations of 3…20 GeV electrons and positrons channeling in bent Si(111) crystals have been carried out by means of MBN Explorer software package and the results were compared with the experimental data [81]. Apart from this, the emission spectra were calculated and analyzed. Figure 5.14 presents the calculated distributions of electrons (top graph) and positron (bottom graph) of different energies with respect to the deflection angle, i.e. the angle between the projectile velocities at the entrance and at the exit of the crystal. The vertical dashed lines mark the initial beam direction (zero angle) and the crystal bending angle. The maxima of the distributions are located in the vicinity of this direction. The first maximum is due to the particles which, being not captured at the entrance passed through the whole crystal in the non-channeling mode. Its position is displaced from the initial direction by the interval equal to the Lindhard critical angle ΘL . The width of the maximum is determined by the larger of the

θ

1 2 3

L R

Fig. 5.13 Schematic picture of particles deflection by a bent crystal. Thick (blue) solid curves illustrate trajectories of the particles which dechannel at the crystal entrance, a, dechannel somewhere inside the crystal, b, propagate through the whole crystal, (c). The distribution of particles with respect to the deflection angle θ can be measured in experiment

5.3 Results of Atomistic Simulations

123

Fig. 5.14 Angular distribution of electrons (top) and positrons (bottom) of several energies (as indicated in the top graph) at the exit from bent oriented Si(111) crystal. The crystal thickness is L = 60 µm, the bending radius R = 15 cm. Vertical lines correspond to the incident beam direction (0 µrad) and the bending angle L/R = 400 µrad. Reference [82]

two angles: ΘL and beam emittance. The distributions presented correspond to the zero beam emittance. The second maximum is formed by projectiles which leave the crystal moving in the channeling mode. It is positioned at exactly 400 µrad and has the width equal to ΘL . The distribution between the maxima corresponds to the particles which stayed in the channeling mode (including the segments due to the rechanneling events) throughout part of the crystal. Comparing the two graphs one notices that the second maximum is much more powerful for positrons since most of them channel through the whole crystal. All curves in Fig. 5.14 are normalized to the unit area. Approximating the maxima with the Gaussian distributions and the curve in between them by the exponential decay law ∝ exp(−Rθ/L d ), one can calculate the acceptance (subtracting the area under the first maximum from one), the channeling efficiency (the area under the second maximum), and estimate the dechanneling length. This methodology of estimating L d has been widely exploited recently for various ultra-relativistic projectiles in straight and bent crystals [30, 73, 81, 83]. We just notice that, intrinsically, this scheme has two drawbacks which influence the estimated value of L d . First, the rechanneling effect is neglected, and, second, the exponential decay law is assumed

124

5 Radiation Emission in Bent Crystals

Table 5.3 Calculated [50] values of A , L d and channeling efficiency for electrons vs. experimental data [81, 83] and results of previous simulations [84]. The term “acceptance” corresponds to “surface transmission” used in [83] ε (GeV) Method A (%) L d (µm) Efficiency (%) 3.35

6.3

sim. [50] exp. [83] calc. [83, calc. [81, sim. [50] exp. [83] calc. [83, calc. [81,

84] 84]

84] 84]

62 ± 2 64 ± 2 – 67 55 ± 2 57 ± 2 – 51

39 ± 2 43 ± 6 37 42 54 ± 6 33 + 5 − 2 42 31

22 ± 1 22 ± 1 23 21 26 ± 2 22 ± 1 23 20

to be valid for all penetration distances, whereas at z  L d it must be supplemented with additional terms [9].

5.3.3.1

Comparison with Experiment

Comparison of the results of simulations [50] with the experimental data [81, 83] is shown in Table 5.3 and Figs. 5.15, 5.16. Table 5.3 presents acceptance, channeling efficiency and dechanneling lengths for 3.35 and 6.3 GeV electrons. The experimental data as well as those calculated by means of the DYNECHARM++ code [84] are taken from Table I in [83] and Table I in [81]. We state the overall agreement for the lower energy. For 6.3 GeV electrons our results agree with experiment on the acceptance but slightly (on the level of 10–15%) overestimate the channeling efficiency. A larger discrepancy is seen in the L d values. The simulated lower value L d = 48 µm is 1.25 times larger than the experimentally measured upper boundary 38 µm. We note, though, that simulations performed with the code [83, 84] also lead to the overestimation. Figure 5.15 compares the simulated and experimentally detected angular distributions for 6.3 and 3.35 GeV electrons. The solid curve stands for the dependence obtained by means of the simulation procedure for the bending radius 15 cm as quoted in Ref. [83]. However, it is seen that the second maximum in the experimental distribution (broken curve) is shifted from the expected position of 400 µrad towards larger values. This may indicate that the bending radius used in the experiment was less than as quoted. To check this, we calculated the distribution for R = 14 cm (the chained line). The positions of its both of maxima coincide with those of the experimental curve. Comparing other features we state that the simulation overestimates slightly the channeling efficiency (the area under the second maximum) and underestimates the number of non-accepted particles (the area under the first maximum).

5.3 Results of Atomistic Simulations

125

Fig. 5.15 Angular distribution of 6.3 GeV (top) and 3.35 GeV (bottom) electrons in bent Si(111) planar channel. Broken curve represents the experimental data [83], dotted curve represent simulation results with DYNECHARM++ [81, 84], solid and chained curves represent our results obtained for two indicated values of the bending radius R. Vertical lines correspond to the incident beam direction (0 µrad) and the bending angle 400 µrad for R = 15 cm. Reference [82]

Dechanneling length as a function of electron energy is presented by Fig. 5.16. As mentioned, good correspondence is seen for the lower energy. The simulated data show steady increase of L d with ε which is, however, slower that in the straight crystal. The latter case, represented by diamonds, shows virtually a linear dependence (the straight line) in accordance with prediction of the diffusion theory of electron dechanneling [85] which is expected to be valid in the multi-GeV energy range. Open circles correspond to the model estimation of the dechanneling length in the straight crystal. These data are obtained by the division of the simulated L d values for bent Si(111) by the factor (1 − R/Rc )2 . The validity of this model, utilized in [83], can

5 Radiation Emission in Bent Crystals

Dechanneling length (micron)

126

100 80 60 40 Experiment Bent crystal Straight crystal Model

20 0 0

2

4

6

8

10

Beam energy (GeV) Fig. 5.16 Electron dechanneling length in L = 60 µm thick Si(111) crystals bent with R = 15 cm versus beam energy ε. The experimental data (squares) [83] are compared with the results of the simulation (filled circles and diamonds) Ref. [50]. Open circles and straight line stand for the model calculations, see explanations in the text

be proven for positively particles channeling in the harmonic interplanar potential [9] but is less obvious for negatively charged projectiles [86, 87]. Comparing the data marked in Fig. 5.16 by the open circles with the diamonds one concludes that the model can be used for quantitative estimations.

5.3.3.2

Radiation Spectrum

The simulated trajectories were used to compute spectral distribution of the radiation emitted within the cone θ < θmax = 1.2 mrad with respect to the incident beam. This cone exceeds the natural emission angle ∼ 1/γ by an order of magnitude for all ε considered and the bending angle by a factor of three. Hence, the dependencies correspond to virtually all emitted radiation. The calculated emission spectra calculated for θmax = 1.2 mrad are presented in Fig. 5.17. The resulting spectrum accounts for all mechanisms of the radiation formation: (a) channeling radiation due to the channeling segments, (b) bremsstrahlung (coherent and incoherent) due to the over-barrier motion. In addition to these, the motion along the arc in a bent crystal results in the synchrotron-type radiation. The distributions were calculated within the framework of the quasi-classical method [85] which accounts for the quantum corrections due to the radiative recoil quantified by the ratio hω/ε. ¯ In the limit hω/ε ¯  1 one can use the classical description of the radiative process which is adequate to describe the emission spectra by electrons and positrons of the sub-GeV energy range (see, for example, [10] and

5.3 Results of Atomistic Simulations

127

25

3

dE/d(h ω) (×10 )

3

dE/d(h ω) (×10 )

20

ε=4 GeV ε=6 GeV ε=8 GeV ε=10 GeV

15

ε=3.35 GeV (straight) ε=3.35 GeV (bent) ε=6.3 GeV (straight) ε=6.3 GeV (bent)

10

_

_

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10

5

5

0 0 10

1

10

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3

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Fig. 5.17 Left. Radiation spectra by 3.35 and 6.3 GeV electrons in straight and 60 micron thick bent Si(111). Dash-dotted line shows the spectrum in amorphous silicon for a 3.35 GeV projectile. Right. Radiation spectra of 4…10 GeV positrons in bent Si(111) crystal. Reference [50]

references therein). The quantum corrections must be accounted for if hω/ε ¯  1. It was demonstrated in [88] that the corrections lead to strong modifications of the radiation spectra of multi-GeV projectiles channeling in small-amplitude short-period crystalline undulators. In Fig. 5.17 left, comparing the solid curves that show the intensity of radiation in the straight crystal with the dashed ones, corresponding to the bent crystal, one notes lowering of the channeling radiation peaks due to the crystal bending. The synchrotron-type radiation, which is absent in the straight crystal, contributes to the low-energy part of the spectrum. It is clearly seen in the spectral dependences for a 6.3 GeV electron: in the photon energy range hω ¯ = 1 . . . 3 MeV the intensity in the bent crystal is higher than that in the straight one. The contribution of the synchrotron radiation to the emission spectrum becomes more pronounced for more energetic projectile, as it is illustrated by the right panel of the figure. Several features in the spectra can be noted. First, the planar direction (111) contains two types of channels, which results in a broad radiation spectrum with several maxima [49]. For positrons, the oscillations in these channels lead to distinguishable features in the spectrum. For electrons, the channeling oscillations are anharmonic, so that the radiation peaks broadened and merge into a single maximum. Second, crystal bending leads to the increase of the yield of low-energy photons due to the synchrotron radiation. The synchrotron-type radiation, which is absent in the straight crystal, contributes to the low-energy part of the spectrum. It is clearly seen in the spectral dependences for a 6.3 GeV electron: in the photon energy range hω ¯ = 1...3 MeV the intensity in the bent crystal is higher than that in the straight one. The contribution of the synchrotron radiation to the emission spectrum becomes more pronounced for more energetic projectile, as it is illustrated by the right panel of the figure. The third feature, illustrated by Fig. 5.17, is in lowering the peak of channeling radiation in a bent crystal. This is mainly due to the decrease in the channeling length due to the crystal bending. For the sake of comparison, the spectrum of incoherent bremsstrahlung in amorphous silicon is also presented in the left panel.

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5.3.4 Electron Channeling and Radiation in Quasi-Mosaic Silicon Crystal The method of relativistic molecular dynamics has been applied for accurate computational modeling and numerical analysis of the channeling phenomena for 855 MeV electrons in bent oriented silicon (111) crystal Ref. [51]. Special attention is devoted to the transition from the axial channeling regime to the planar one in the course of the crystal rotation with respect to the incident beam. Distribution in the deflection angle of electrons and spectral distribution of the radiation emitted are analyzed in detail. The results of calculations are compared with the experimental data collected at the MAinzer MIctrotron (MAMI) facility [34, 73]. In the experiments, a bent silicon single crystal of thickness 30.5 µm along the beam direction was used as a target. The crystal was bent using a mechanical holder that provided a uniform bending of the (111) planes with the bending radius R = 33.5 mm. The bending curvature was manifested as a result of the quasi-mosaic effect [89]. More details on the bending procedure and characterization can be found in Ref. [90]. The crystal was mounted on a high-precision goniometer with three degrees of freedom which allowed for a precise rotation of the target [71]. The orientation of the axes and planes in crystal was verified by means of high resolution X-ray diffraction [91]. In the experiment, the crystal was first oriented by aligning the crystalline axis 112 with the incident electron beam. Then, the crystal was rotated around the 111 axis to the position in which the beam is directed along the crystal plane (111). The choice of the (111) planes is due to the fact that they are most effective ones for reflecting negatively charged particles. The simulations [51] were carried out for a 855 MeV electron beam targeting the crystal at the directions corresponding to those probed experimentally as well as at the direction that were not explored. It has allowed one to compare the simulated dependences with the experimentally measured data and to study the evolution of the angular distributions of deflected electrons and the emission spectra with variation in the relative orientation of the crystal and the beam. The atomistic approach implemented in MBN Explorer allows one to simulate the trajectories of charged particles entering a crystal along an arbitrary direction. This feature was used to study the transition from electron capture by the crystalline axis 112 to the planar channeling mode in the bent (111) plane at different rotation angles. The crystal was oriented from the 112 axis along the beam direction to the channeling position in the (111) plane. Figure 5.18 shows the geometries of the beam-crystal orientation used two sets of the simulations (case studies). The first case study concerns the change in the channeling properties in the course of transition from the axial to the planar channeling regime, Fig. 5.18 left. The y-axis ¯ crystallographic axis which is normal to the (111) ¯ plane. is directed along the 111 The z-axis is aligned with the 112 crystallographic direction. At the entrance, the ¯ plane, is directed at the angle θ to the beam velocity v0 , being tangent to the (111)

5.3 Results of Atomistic Simulations

129

¯ plane is assumed with the y-axis. For given θ the uniform bending of the (111) curvature radius lying in the (v0 , y) plane. The case θ = 0 corresponds to the axial 112 channeling. Transition to the planar regime can be carried out by increasing the values of θ . In the simulations this was implemented by means of the following two routines. First, the values of θ were used that correspond to the axes with higher Miller indices 1n(n + 1) with n = 2, 3, . . . , 9. The limit n  1 corresponds to the planar channeling regime. The corresponding values of θ are within the hundreds of mrad range. Another option is to consider the values of θ within the range [0, θmax ] where θmax to be chosen much larger that Lindhard’s critical angle for the 112 axis but much smaller than 0.190 rad, which corresponds to the 123 axial direction. In the second case study, Fig. 5.18 right, the analysis of the volume reflection and volume capture phenomena was carried out. Here, the beam direction is characterized ¯ plane. Bending by angle θ defined as above by angle α between v0 and the (111) ¯ plane is assumed with the radius within the (y, z) plane. To match the of the (111) experimental conditions indicated in Refs. [34, 73], the value θ = 95 mrad was set in the simulations. The α angle was varied from −0.5 to 1.5 mrad. A detailed analysis of the modification of the particles distributions in the deflection angle one finds in Ref. [51]. Here we quote the comparison of the simulated results with the experimental data that refer to the second case study. Figure 5.19 compares the results of simulation with the experimental data for the angular distri¯ plane (α = 0) at the entrance bution of 855 MeV electrons incident along the (111) of bent silicon crystal. On the whole, there is good agreement between theory and experiment but there are also some differences. First, in the simulated curve, the

Fig. 5.18 Left. Beam-crystal orientation used to study the transient effects from axial to planar channeling. Right. Orientation of the crystal axes and the beam direction used in the second case ¯ study. The crystal bending occurs in the (y, z) plane, which is the (110) crystallographic plane. Reference [51]

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maximum centered at the zero deflection angle (corresponds to the distribution of electrons that become over-barrier at the entrance) is slightly shifted to the right as compared to that in the experimentally measured dependence. In the course of simulations it has been noticed that the closest agreement in the maxima positions is achieved at smaller incident angles 10 ≤ θ < 190 mrad. In this range the simulated angular distribution does not virtually depend on the incident angle. In contrast to this, for higher incident angles, corresponding to the axial directions 1(n + 1)(n + 2) with high indices n, the deviation from the experiment is more pronounced. This feature indicates that although the field intensities of the high-index axes are weak they, nevertheless, influence the angular distribution of particles. The second difference to be noted is that the peak, centered at the deflection angle about 0.8 mrad, in the simulated dependence is lower that its experimental counterpart. This peak is due to the electrons that channel through the whole length L of a crystal. To make this peak pronounced the value of L should be comparable with the dechanneling length L d . The simulations carried out in Ref. [49] by means of the MBN Explorer produced the result L d = 16.6 ± 0.8 µm for 855 MeV electrons channeled in oriented Si(111) crystal bent with radius R = 33 mm. The crystal thickness along the beam L = 30.5 mm which is very close to the value 33.5 mm used in the experiments [34, 73] was thin enough to observe the peak of channeled electrons. Figure 5.20 shows the modifications occurring in the spectral distributions of the emitted radiation due to the transition from the axial to planar channeling regimes. Figure 5.20 left presents the emission spectra calculated for the electrons entering ¯ bent plane along several axial directions 1 n (n + the crystal tangentially to (111) 1). These directions correspond to large values of the incident angles (indicated in the caption) measured with respect to the 112 axis. With n increasing, the averaged

Fig. 5.19 Simulated versus experimentally measured angular distributions of 855 MeV electrons exiting the oriented bent silicon crystal. The figure refer to the beam-crystal geometry with θ = 95 and α = 0 mrad (see Fig. 5.18 right). Reference [51]

5.3 Results of Atomistic Simulations

131

Fig. 5.20 Radiation spectra that correspond to the beam-crystal orientation indicated in Fig. 5.18 ¯ plane at different axial directions left. Left. Radiation spectra by the electrons incident along the (111) 1n(n + 1), as indicated. These directions correspond to large angles with the 112 axis: θ = 0.19, 0.28, 0.37, 0.41 mrad for n = 2, 3, 5, 7, respectively. Right. Radiation spectra by the electrons incident at small angles θ. Reference [51]

Fig. 5.21 Simulated vs. experimental radiation spectra by 855 MeV electrons entering the crystal as indicated by Fig. 5.18 right. Left panel corresponds to the incident angles θ = 95 and α = 0 mrad, right panel—to θ = 95 and α = 0.5 mrad. The latter geometry corresponds to the electron volume reflection from the bent plane. Reference [51]

axial electric field decreases leading to the decrease in the radiation intensity. At n = 7 the spectrum approaches that emitted in the planar channeling regime and becomes virtually insensitive to further increase in n. Similar modification of the spectra occurs when the incident angle varies within the interval θ  190 mrad, Fig. 5.20 right. In this case, small values of θ correspond to the motion close to the axial direction 112 where a projectile experiences action of the strong axial field and, thus, radiates intensively. As θ increases, a projectile moves across the axes experiencing smaller acceleration in the transverse direction. As a result, the intensity decreases (compare the curves corresponding to θ = 0.5, 1.0 and 1.5 mrad). For θ much larger than the critical angle the spectrum approaches its limit, which corresponds to the planar channeling regime. Comparison of the simulated spectra with the experimentally measured data is presented in Fig. 5.21. For angles θ = 95 mrad and α = 0 (the channeling regime), a comparison of the spectra obtained in the simulations with those measured experimentally is presented in Fig. 5.21 left. The dependence calculated by means of the RADCHARM code

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[92] is presented as well. The simulation is normalized for the experiment at a given orientation of the crystal. One can state that for this geometry, which corresponds to the planar channeling regime, there is a good agreement between the experiment and theory. For the geometry, which allows for the VR and VC regime (Fig. 5.21 right) the simulated and measured spectra follow the same shape but there is some discrepancy (on the level of 10%) in the absolute values differ by about 10 per cent in the region of the low- and mid-energy photons.

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60. Wienands, U., Gessner, S., Hogan, M.J., Markiewicz, T., Smith, T., Sheppard, J., Uggerhøj, U.I., Nielsen, C.F., Wistisen, T., Bagli, E., Bandiera, L., Germogli, G., Mazzolari, A., Guidi, V., Sytov, A., Holtzapple, R.L., McArdle, K., Tucker, S., Benson, B.: Channeling and radiation experiments at SLAC. Int. J. Mod. Phys. A 34, 1943006 (2019) 61. Mazzolari, A., Sytov, A., Bandiera, L., Germogli, G., Romagnoni, M., Bagli1, E., Guidi, V., Tikhomirov, V.V., De Salvador, D., Carturan, S., Durigello, C., Maggioni, G., Campostrini, M., Berra, A., Mascagna, V., Prest, M., Vallazza, E., Lauth, W., Klag, P., Tamisari, M.: Broad angular anisotropy of multiple scattering in a Si crystal. Eur. Phys. J. C 80, 63 (2020) 62. Tran Thi, T.N., Morse, J., Caliste, D., Fernandez, B., Eon, D., Härtwig, J., Barbay, C., MerCalfati, C., Tranchant, N., Arnault, J.C., Lafford, T.A., Baruchel, J.: Synchrotron Bragg diffraction imaging characterization of synthetic diamond crystals for optical and electronic power device applications. J. Appl. Cryst. 50, 561 (2017) 63. Nething, U., Galemann, M., Genz, H., Höfer, M., Hoffmann-Stascheck, P., Hormes, J., Richter, A., Sellschop, J.P.: Intensity of electron channeling radiation, and occupation lengths in diamond crystals. Phys. Rev. Lett. 72, 2411–2413 (1994) 64. Boshoff, D., Copeland, M., Haffejee, F., Kilbourn, Q., MacKenzie, B., Mercer, C., Osato, A., Williamson, C., Sihoyiya, P., Motsoai, M., Connell, M., Henning, C.A., Connell, S.H., Palmer, N.L., Brooks, T., Härtwig, J., Tran Thi, T.N., Uggerhøj, U.: The search for crystal undulator radiation. In: Peterson, S., Yacoob, S. (eds.) Proceedings of SAIP 2016, p. 112 (2017) 65. Brau, C.A., Choi, B.-K., Jarvis, J.D., Lewellen, J.W., Piot, P.: Channeling radiation as a source of hard X-rays with high spectral brilliance. Synchrotron Radiat. News 25, 20 (2012) 66. https://www6.slac.stanford.edu/facilities/facet.aspx 67. Sytov, A.I., Bandiera, L., De Salvador, D., Mazzolari, A., Bagli, E., Berra, A., Carturan, S., Durighello, C., Germogli, G., Guidi, V., Klag, P., Lauth, W., Maggioni, G., Prest, P., Romagnoni, M., Tikhomirov, V.V., Vallazza, E.: Steering of Sub-GeV electrons by ultrashort Si and Ge bent crystals. Eur. Phys. J. C 77, 901 (2017) 68. Bandiera, L., Sytov, A., De Salvador, D., Mazzolari, A., Bagli, E., Camattari, R., Carturan, S., Durighello, C., Germogli, G., Guidi, V., Klag, P., Lauth, W., Maggioni, G., Mascagna, V., Prest, M., Romagnoni, M., Soldani, M., Tikhomirov, V.V., Vallazza, E.: Investigation on radiation generated by sub-GeV electrons in ultrashort silicon and germanium bent crystals. Europ. Phys. J C 81(3), 1–9 (2021) 69. Taratin, A.M., Vorobiev, S.A.: Volume reflection of high-energy charged particles in quasichanneling states in bent crystals. Phys. Lett. 119, 425 (1987) 70. Taratin, A.M., Vorobiev, S.A.: Deflection of high-energy charged particles in quasi-channeling states in bent crystals. Nucl. Instrum. Meth. B 26, 512 (1987) 71. Backe, H., Kunz, P., Lauth, W., Rueda, A.: Planar channeling experiments with electrons at the 855-MeV Mainz Microtron. Nucl. Instrum. Method B 266, 3835–3851 (2008) 72. Backe, H., Krambrich, D., Lauth, W., Andersen, K.K., Hansen, J. Lundsgaard, and Uggerhøj, U. I.: Channeling and radiation of electrons in silicon single crystals and si1- xGex crystalline undulators. J. Phys. Conf. Ser. 438, 012017 (2013) 73. Mazzolari, A., Bagli, E., Bandiera, L., Guidi, V., Backe, H., Lauth, W., Tikhomirov, V., Berra, A., Lietti, D., Prest, M., Vallazza, E., De Salvador, D.: Steering of a sub-GeV electron beam through planar channeling enhanced by rechanneling. Phys. Rev. Lett. 112, 135503 (2014) 74. Polozkov, R.G., Ivanov, V. K., Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Channeling of ultra-pelativistic positrons in bent diamond crystals. St. Petersburg Polytechnical Uni. J: Phys. Math. 1, 212–218 (2015) 75. Kovalenko, A.D., Mikhailov, V.A., Taratin, A.M., Boiko, V.V., Kozlov, S.I., Tsyganov, E.N.: Bent tungsten crystal as deflector for high energy particle beams. JINR Rapid Comm. 72, 9–18 (1995) 76. Yoshida, K., Goto, K., Isshiki, T., Endo, I., Kondo, T., Matsukado, K., Takahashi, T., Takashima, Y., Potylitsin, A., Amosov, C. Yu., Kalinin, B., Naumenko, G., Verzilov, V., Vnukov, I., Okuno, H., and Nakayama, K.: Positron production in tungsten crystals by 1.2-GeV channeling electrons. Phys. Rev. Lett. 80, 1437–1440 (1998)

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77. Backe, H., Lauth, W., Scharafutdinov, A.F., Kunz, P., Gogolev, A.S., Potylitsyn, A.P.: Forward diffracted parametric x radiation from a thick tungsten single crystal at 855 MeV electron energy. Proc. SPIE 6634, 66340Z (2007) 78. Azadegan, B., Mahdipour, S.A., Wagner, W.: Simulation of positron energy spectra generated by channeling radiation of GeV electrons in a tungsten single crystal. J. Phys.: Conf. Ser. 517, 012039 (2014) 79. Azadegan, B., Wagner, W.: Simulation of planar channeling-radiation spectra of relativistic electrons and positrons channeled in a diamond-structure or tungsten single crystal (Classical approach). Nucl. Instrum. Meth. B 342, 144–149 (2015) 80. Efremov, V.I., Dolgikh, V.A., Pivovarov, Yu.L.: Multiple scattering of relativistic electrons and positrons in this tungsten crystals. Russian Phys. J. 50, 1237–1242 (2007) 81. Wienands, U., Markiewicz, T.W., Nelson, J., Noble, R.J., Turner, J.L., Uggerhøj, U.I., Wistisen, T.N., Bagli, E., Bandiera, L., Germogli, G., Guidi, V., Mazzolari, A., Holtzapple, R., Miller, M.: Observation of deflection of a beam of multi-GeV electrons by a thin crystal. Phys. Rev. Let. 114, 074801 (2015) 82. Sushko, G.B.: Atomistic Molecular Dynamics Approach for Channeling of Charged Particles in Oriented Crystals. (Doctoral dissertation), Goethe-Universität, Frankfurt am Main (2015) 83. Wienands, U., Markiewicz, T.W., Nelson, J., Noble, R.J., Turner, J.L., Uggerhøj, U.I., Wistisen, T.N., Bagli, E., Bandiera, L., Germogli, G., Guidi, V., Mazzolari, A., Holtzapple, R., Miller, M.: Observation of a Remarkable Deflection of Multi-GeV Electron Beams by a Thin Crystal. SLAC Scientific Publications (2014) SLAC-PUB-15952 84. Bagli, E., Guidi, V.: Dynecharm++: a toolkit to simulate coherent interactions of high-energy charged particles in complex structures. Nucl. Instrum. Meth. B 309, 124 (2013) 85. Baier, V.N., Katkov, V.M., Strakhovenko, V.M.: Electromagnetic Processes at High Energies in Oriented Single Crystals. World Scientific, Singapore (1998) 86. Tabrizi, M., Korol, A.V., Solov’yov, A.V., Greiner, W.: Feasibility of an electron-based crystalline undulator. Phys. Rev. Lett. 98, 164801 (2007) 87. Tabrizi, M., Korol, A.V., Solov’yov, A.V., Greiner, W.: Electron-based crystalline undulator. J. Phys. G: Nucl. Part. Phys. 34, 1581–1593 (2007) 88. Bezchastnov, V.G., Korol, A.V., Solov’yov, A.V.: Radiation from multi-GeV electrons and positrons in periodically bent silicon crystal. J. Phys. B 47, 195401 (2014) 89. Ivanov, Y., Petrunin, A., Skorobogatov, V.: Observation of the elastic quasi-mosaicity effect in bent silicon single crystals. JETP Lett. 81, 977 (2005) 90. Camattari, R., Guidi, V., Bellucci, V., Mazzolari, A.: The quasi-mosaic effect in crystals and its application in modern physics. J. Appl. Cryst. 48, 977 (2015) 91. Guidi, V., Mazzolari, A., De Dalvador, D., Carnera, A.: Silicon crystal for channelling of negatively charged particles. J. Phys. D: Appl. Phys. 42, 182005 (2009) 92. Bandiera, L., Bagli, E., Guidi, V., Tikhomirov, V.V.: RADCHARM++: A C++ routine to compute the electromagnetic radiation generated by relativistic charged particles in crystals and complex structures. Nucl. Instrum. Meth. B 355, 44 (2015)

Chapter 6

Crystalline Undulators

6.1 Crystalline Undulator: Basic Concepts, Feasibility A Crystalline Undulator (CU) device contains a periodically bent crystal (PBCr) and a beam of ultra-relativistic positrons or electrons undergoing planar channeling. In such a system, there appears, in addition to the channeling radiation (ChR) [1], the undulator radiation due to the periodic motion of the particles which follow the bending of the planes. A light source based on a CU can generate photons in the energy range from tens of keV up to the GeV region [2, 3] (the corresponding wavelengths range starts at 0.1 and goes down to 10−6 Å). The intensity and characteristic frequencies of the CU radiation (CUR) can be varied by changing the beam energy, the parameters of bending, and the type of a crystal. Under certain conditions a CU can become a source of the hard X- and gamma-ray laser light within the wavelength range 10−2 −10−1 Å [3–5], which cannot be reached in existing and planned FELs based on magnetic undulators. The mechanism of the photon emission by means of CU is illustrated by Fig. 6.1 which presents a cross section of a single crystal. The z axis is aligned with the midplane of two neighboring non-deformed crystallographic planes (not drawn in the figure) spaced by the interplanar distance d. The closed circles denote the nuclei of the planes which are periodically bent with the amplitude a and period λu . The harmonic (sine or cosine) shape of periodic bending, y(z) = a cos(2π z/λu ), is of a particular interest since it results in a specific pattern of the spectral-angular distribution of the radiation emitted by a beam of ultra-relativistic charged particles (the open circles in the figure) propagating in the crystal following the periodic bending. The operational principle of a CU does not depend on the type of a projectile. Provided certain conditions are met the particles will undergo channeling in PBCh [6, 8]. The trajectory of a particle contains two elements which are illustrated by Fig. 6.1. First, there are oscillations due to the action of the interplanar force—the   /dε (c is so-called channeling oscillations [9], whose frequency Ωch = c 2Umax the speed of light) depends on the projectile energy ε and on the parameters of the  and the interplanar channel: the maximal gradient of the interplanar potential Umax © Springer Nature Switzerland AG 2022 A. Korol and A. V. Solov’yov, Novel Lights Sources Beyond Free Electron Lasers, Particle Acceleration and Detection, https://doi.org/10.1007/978-3-031-04282-9_6

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y

spontaneous emission

stimulated emission

d

a

z channeling particles

λu

periodically bent channel

Fig. 6.1 Schematic representation of a CU [4, 6, 7]. Closed circles mark the atoms of crystallographic planes which are periodically bent with the amplitude a and period λu . Thin dotted line illustrates the trajectory of the particle (open circles), which propagates along the centerline (the undulator motion) and, simultaneously, undergoes so-called channeling oscillations. The periodic mode leads to the emission of the undulator-type radiation, and, under certain conditions, may result in the stimulated radiation

distance d. Second, there are oscillations due to the periodicity of the bending, the undulator oscillations, whose frequency is Ωu ≈ 2π c/λu . The spontaneous emission is associated with both of these oscillations. The typical frequency of the ChR is ωch ≈ 2γ 2 Ωch and [1, 10], where γ = ε/mc2 is the relativistic Lorentz factor of the projectile. The undulator oscillations give rise to photons with frequency ωu ≈ 4γ 2 Ωu /(2 + K 2 ), where K = 2π γ a/λu is the so-called undulator parameter. If Ωu  Ωch , then the frequencies of ChR and UR are well separated. In this case, the characteristics of undulator radiation are practically independent of channeling oscillations [4, 6, 8], and the operational principle of a crystalline undulator is the same as for a conventional one (see, e.g., [11–15]) in which the monochromaticity of radiation is the result of constructive interference of the waves emitted from similar parts of trajectory, see Fig. 2.5. Although the motion of a projectile and the process of photon emission in a CU are very similar to that in an conventional undulator based on the action of periodic magnetic (or, electric) field, there is an important distinguishing feature. Namely, the electrostatic fields inside a crystal are so strong that they are able to steer the particles much more effectively than even the most advanced superconductive magnets. The field strength is on the level of 1010 V/cm which is equivalent to the magnetic field of approximately 3000 Tesla. The present state-of-the-art superconductive magnets produce the magnetic flux density of the order of 100 −101 tesla [16, 17]. Strong crystalline fields allow one to bring the period λu of bending down to the hundred or even ten micron range, which is two to five orders of magnitude smaller than the period of a conventional undulator. As a result, the size of the undulator itself can be reduced by orders of magnitude as illustrated by Fig. 1.3, which matches the magnetic undulator for the X-ray laser XFEL [18] with a CU manufactured in University of Aarhus and used further in channeling experiments [19–21].

6.2 Positron and Electron-Based CUs: Illustrative Material

139

Table 6.1 Acceptances A and penetration lengths L p for 10 GeV electrons and positrons in linear (a = 0) and periodically bent (110) planar channels in silicon crystal. The parameter C =  ) denotes the ratio of the centrifugal force to the interplanar force a(2π/λu )2 (ε/Umax Projectile

a (Å)

C

A (%)

L p (microns)

Electron

0 4 0 2 4 6

0.0 0.16 0 0.08 0.16 0.24

65.8 ± 2.3 42.9 ± 3.3 97.1 ± 0.9 89.8 ± 2.1 81.6 ± 2.6 71.9 ± 5.8

82 ± 4 52 ± 4 302 ± 4 301 ± 5 287 ± 7 273 ± 15

Positron

6.2 Positron and Electron-Based CUs: Illustrative Material 6.2.1 CU Radiation by Multi-GeV Electrons and Positrons In Ref. [22], analysis of the channeling properties and the spectral intensities of radiation has been carried out for 195...855 MeV positrons and electrons in the CUs with the parameters as in the experiments at the Mainz Miktoron [23, 24]. The CUs used in the experiments were Si1−x Gex superlattices produced by means of ng the molecular beam epitaxy technology [25]. Later, similar calculations have been performed for multi-GeV projectile [26, 27] in connections with the channeling experiments planned to be carried out at the SLAC accelerator facility (USA) using highly intensive electron and positron beams with ε = 4−20 GeV [28]. The simulations have been performed to provide the benchmark data on the spectra of radiation emitted by the projectiles in silicon-based and diamond-based CUs with the parameters as in the experiments with sub-GeV electron beams at MAMI [23, 24]. The simulations have been carried out for the CUs with the following parameters: • Bending period λu = 40 microns. • Number of periods and crystal thickness Nu = 8 and L = Nu λu = 320 microns, respectively. • Bending amplitude a = 2 . . . 6 Å. The values of the channel acceptance A and penetration length L p calculated by means of statistical analysis of the trajectories simulated for 10 GeV electrons and positrons in linear (a = 0) and periodically bent Si(110) crystals are presented in Table 6.1 . The parameter C stands for the ratio of the maximum values of the centrifugal force, Fcf ≈ ε/Rmin with Rmin = a −1 (λu /2π )2 , to the interplanar force  Umax ≈ 5.7 GeV/c. The latter value one obtains using the Molière approximation for the (110) interplanar potential (at the temperature T = 300 K). The data on L p indicate that most of the positrons propagate travel through the whole crystal in the

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6 Crystalline Undulators 20

positron

20

electron

a=0 (straight) a=2 Å a=4 Å a=6 Å

15

Enhancement

Enhancement

25

15

10

a=0 (straight) a=4 Å

10

5 5

0 0 10

1

10

2

3

10

10

0 0 10

Photon energy (MeV)

1

10

2

10

3

10

Photon energy (MeV)

Fig. 6.2 Enhancement factor of the radiation over the background radiation calculated for 10 GeV positrons (left panel) and electrons (right panel) in straight and periodically bent Si(110) with different bending amplitudes indicated in the legends. The period of bending is 40 microns. All curves correspond to the emission angle equal to 7/γ ≈ 0.36 mrad

channeling mode. For electrons, both the acceptance and penetration length are much lower. These features reveal themselves in the radiation emission spectra. Figure 6.2 shows the enhancement factor (over the incoherent bremsstrahlung background) of the radiation emitted by positrons (left) and electrons (right) [27]. In the straight channel (red curves), the spectra are dominated by powerful peaks of the channeling radiation. The peak is more pronounced for positrons due to their harmonic-like channeling oscillations that lead to the emission within a comparatively narrow interval of photon energies centered at hω ¯ ch ≈ 70 MeV. Strong anharmonicity of the electron channeling oscillation leads to the noticeable broadening of the peak with the maximum located at hω ¯ ch ≈ 120 MeV. Periodical bending of the planes gives rise to the CU Radiation (CUR). The CUR peaks in the spectra are more pronounced for positrons since their dechanneling length is much larger than that of electrons. The energy of the first harmonic of CUR one estimates as follows (see, e.g., Eq. (6.14) in Ref. [4]): hω ¯ 1 [MeV ] =

ε2 9.5 1 + K 2 /2 λu

(6.1)

where ε is in GeV and λu in microns. The undulator parameter K accounts for the channeling oscillations as well as for those due to the periodicity of bending[29] K =



2 K u2 + K ch .

(6.2)

Here K u = 2π γ a/λu and K ch ∝ 2π γ ach /λch with λch and ach ≤ d/2 standing for the period and amplitude of the channeling oscillations. For positrons, assuming harmonicity of the channeling oscillations one derives the following expression for 2 averaged over the ach values(see [4], Eq. (B.5)): K ch

6.2 Positron and Electron-Based CUs: Illustrative Material θ0=1/γ

Enhancement

150

141 θ0=5/γ

positron

80

positron

60

100 40 50

20

0 Enhancement

60

0 electron

60

40

40

20

20

0 0

100 200 300 400 Photon energy (MeV)

0 0

electron

straight CU 100 200 300 400 Photon energy (MeV)

Fig. 6.3 Enhancement factor over the Bethe–Heitler spectrum for 10 GeV positrons (upper row) and electrons (lower row) in L = 320 microns thick straight diamond (thick black lines) and diamondbased CU with amplitude a = 4 Å and period λu = 40 microns (thick green lines). Thin blue lines stand for the dependences obtained for the ideal undulator with the same a and λu . Left column corresponds to the emission angle θ0 = 1/γ = 51.1 µrad; right column – to θ0 = 5/γ ≈ 256 µrad

2 K ch =

2γ U0 . 3mc2

(6.3)

where U0 is the depth of the interplanar potential well. For a ε = 10 GeV positron channeling in Si(110) (U0 ≈ 22 eV at T = 300 K, see, e.g., Fig. 4.22) this estimate 2  ≈ 0.56. produces K ch Using Eqs. (6.1)–(6.3) one estimates hω ¯ 1 for a = 2, 4, 6 Å as 16, 11.7 and 8 MeV, respectively. These values reproduce nicely the positions of the first peaks of CUR seen in the right panel in Fig. 6.2. Channeling of 4...20 GeV light projectiles in L = 320 microns thick straight and periodically bent oriented diamond(110) crystal has been considered in Ref. [26]. This study has been carried out aiming at producing benchmarks for the experiments planned to be carried out at the SLAC facility with intensive electron and positron beams. From this viewpoint, the use of diamond crystals looked preferential since diamond is highly sustainable with respect to the irradiation. The spectral distributions of the emitted radiation energy were computed for two values of the emission cone θ0 (see Eq. (4.41)): (i) for θ0 = 1/γ , which stands for

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6 Crystalline Undulators

the natural emission cone, and (ii) for a much wider one θ0 = 5/γ that collects virtually all radiation emitted by an ultra-relativistic projectile. Figure 6.3 shows the enhancement factor over the emission spectra in an amorphous medium calculated for 10 GeV positrons (upper row) and electrons (lower row). For the sake of comparison, the spectra formed by a positron moving in an ideal undulator (i.e., along the sine trajectory with the given values of a and λu ) are shown in the upper panels.

6.2.2 Specific Features of Radiation Emission by Electrons in Crystalline Undulators A numerical analysis of the evolution of the channeling properties and the radiation spectra has been performed in Refs. [30–32] for diamond(110)-based CUs. Special attention has been paid to the changes in the radiation spectra due to the increase in the bending amplitude a as well as to their sensitivity to the charge of the projectile. Some of the predictions formulated in the cited papers can be verified in channeling experiments with electrons at the MAMI facility. The simulations of the channeling process have been carried out for 270–855 MeV electrons and positrons propagating in 20 microns thick diamond crystal the (110) plane of which were periodically bent with a shape S(z) = a cos(2π z/λu ) with the z axis aligned with the direction of the incident beam. In the simulations, the bending amplitude has been varied within the interval a = 0−4 Å (a = 0 corresponds to the straight crystal), whereas the bending period was fixed at λu = 5 microns in accordance with the parameters of the crystalline samples used in the MAMI experiments [33]. Figure 6.4 shows the spectral distribution of radiation by ε = 855 MeV positrons and electrons emitted within the cone θ0 = 0.24 mrad, which is small compared with the natural emission angle γ −1 = 0.59 mrad. For both types of projectiles the spectra formed in the straight crystal, graph (a), are dominated by the peaks of ChR the intensity of which notably exceeds the intensity 2.5 × 10−5 of the incoherent bremsstrahlung background radiation. Harmonic-like character of the positron channeling oscillations results in the the narrow peak of the spectral distribution at hω ¯ ChR ≈ 3.6 MeV. In contrast, the ChR peaks in the electron spectra, indicated by the upward arrows, are significantly broadened and much less intensive (note the scaling factor ×5). In periodically bent crystals, Fig. 6.4b–d, the spectra exhibit additional features some of which evolve differently with increase in a as it is discussed below. • First to be noted are the CUR peaks that appear in the low-energy part of the spectrum. These peaks, most powerful of which correspond to the emission in the first harmonic at hω ¯ CUR ≈ 1 MeV, are seen in both electron and positron spectra. The peak intensity is a non-monotonous function of the amplitude a. This feature has been noted and discussed in detail in Refs. [31, 32].

6.2 Positron and Electron-Based CUs: Illustrative Material

_

3

dE/d(hω) (×10 )

6

(a) a=0

(b) a=1.2 Å

143

(c) a=2.5 Å

(d) a=4.0 Å

5 4

positron electron (×5)

3 2 1 0 0

10

20

0

10

20

0

10

20

0

10

20

Photon energy (MeV) Fig. 6.4 Spectral distributions of radiation emitted by 855 MeV positrons (red solid curves) and electrons (blue dashed curves; multiplied by a factor of 5) channeling in straight (a) and periodically bent (b)–(d) diamond (110) crystals. The upward arrows indicate the maxima of ChR for electrons, the downward arrows show the positions of the additional maxima appearing in the bent crystals. The error bars shown in graph a present the statistical errors due to the finite number of the simulated trajectories. The spectra correspond to the emission cone θ0 = 0.24 mrad. Reference [31]

• For positrons, the intensity of ChR decreases rapidly with the increase in the bending amplitude. It is seen that already for a = 1.2 Å the intensity is two times less than in the straight crystal. For larger amplitudes, the channeling radiation virtually disappears since the average amplitude of the channeling oscillations is a decreasing function of a [30, 31]. This happens due to the action of the centrifugal force the (mean) value of which increases with a. As a result, the centrifugal force, especially in the points of maximum curvature, drives the projectiles oscillating with large amplitudes away from the channel leading to strong quenching of the channeling oscillations. • The intensity of ChR emitted by electrons is less sensitive to the changes in the bending amplitude. As a increases, the ChR peak (marked with the upward arrow) is shifted to the region of higher photon energies and, additionally, there appears a structure (indicated by the downward arrow) on the right shoulder of the spectrum. The analysis has shown that these features are due to the radiation emitted by the dechanneled particles [32]. In a periodically bent crystal, these particles can undergo (i) the volume reflection (VR) [34, 35], occurring mainly in the vicinity of the maximum curvature points, and (ii) the over-barrier motion in the regions with small curvature. The emission that is due to these types of motion reveals itself in different parts of the radiation spectrum. The VR events contribute to the radiation emitted in same energy domain as ChR, whereas the over-barrier particles radiate at higher energies and this radiation reveals itself as an additional peak in the spectrum. The radiation emission by over-barrier particles propagating through a periodically bent crystal has been discussed in Ref. [36] within the continuous potential framework. More detailed quantitative analysis of the

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6 Crystalline Undulators 20

Enhancement

Electron channels

15

10

5

0

5

10

15

Penetration distance, z (microns)

20

0 0

10

20

30

Photon energy (MeV)

Fig. 6.5 Left panel presents an illustrative trajectory of an electron in a periodically bent diamond (110) crystal (a = 2.5 Å and λu = 5 microns). Highlighted are the segments corresponding to different types of motion: dashed maroon curves mark the channeling motion, dashed-dotted blue curves correspond to the segments of the over-barrier motion, and solid red curves indicate the segments where the VR events occur. The channels’ boundaries are drawn by thin wavy lines. Right. Dashed maroon, dashed-dotted blue, and solid red curves show the partial contributions to the spectrum coming from the segments of the channeling and over-barrier motions and due to the VR, respectively. Solid black curve with open circles shows the total spectrum (in the form of the enhancement factor) calculated as the incoherent sum of the three partial terms. Reference [32]

phenomena involved can be carried out on the basis of all-atom molecular dynamics. A brief overview of the results obtained and conclusions drawn in Ref. [32] is presented below. For a quantitative comparison of the contributions of the channeling and nonchanneling particles to the emission spectrum, the following approach has been applied. In each simulated trajectory, the segments corresponding to different types of motion were identified. These include (i) segments within which a particle moves in the channeling mode, (ii) segments corresponding to the over-barrier motion across the crystal planes, (iii) segments in the vicinity of maximum curvature points where a projectile experiences volume reflection. For a given trajectory, the total spectrum has been calculated as an incoherent sum of the partial terms corresponding to the radiation emitted from all segments identified. Thus, the interference of radiation emitted from different segments was no accounted for. The procedure described above is illustrated by Fig. 6.5. Its left panel shows a single trajectory of a 855 MeV electron in a periodically bent diamond (110) crystal with bending amplitude 2.5 Å. Segments of the trajectory corresponding to different types of motion are highlighted in different colors and line types. The emission spectra (calculated accounting for all trajectories simulated) are shown in the right panel. The dependences drawn allow one to relate the maxima in the total spectrum (black solid curve) to different types of the electron motion. The radiation emitted from the channeling segments (dashed maroon curves) contribute to the spectrum in the vicinity of the CUR peak (¯hωCUR ≈ 1 MeV) as well as in the region governed by the ChR, i.e., at hω ¯ ch ≈ 6 . . . 12 MeV. Numerical analysis of the trajectories has shown that the curvature of the segments in the points of VR is close to that of the

6.2 Positron and Electron-Based CUs: Illustrative Material

40

145

(a)

total channeling non-channeling

Enhancement

20 0 30 20 10 0 20

(b)

(c)

10 0 10 0

(d)

(e)

5 0 0

10

20

30

Photon energy (MeV) Fig. 6.6 Enhancement of radiation emitted by 855 MeV electrons in straight (a) and periodically bent (graphs (b)–(e) correspond to a = 1.2, 2.5, 4.0, 5.5 Å) diamond(110) crystals. Solid black curves show the total spectra, dashed maroon curves ones correspond to the radiation emitted from the channeling segments only, and dashed-dotted blue curves stand the spectra emitted from all non-channeling parts of the trajectories. Reference [32]

channeling trajectories. Hence, the peak centered at ≈ 9 MeV is due both to the channeling motion and to the VR events. Trajectories of the over-barrier particles acquire quasi-periodical modulation in the electrostatic field of periodically bent channels. The mean period of these modulations is less than that of the channeling motion and it decreases with the increase of the bending amplitude. For a = 2.5 Å, this period is approximately two times smaller than the period of channeling oscillations. As a result, the radiation emitted from the over-barrier segments is most intensive in the range hω ¯ ch ≈ 15 . . . 20 MeV where it results in the additional structure in the total spectrum. Figure 6.6 compares the contributions of the channeling and non-channeling particles to the emission spectrum calculated for different bending amplitudes. In each graph the solid (black) curve shows the total spectrum, the contributions of the channeling segments and the non-channeling segments (both over-barrier and VR) are shown by the dashed (maroon) and dash-dotted (blue) curves, respectively. In the straight crystal as well as in the periodically bent one with small bending amplitude (a = 1.2 Å) the spectrum above 1 MeV is mainly due to the channeling particles the emission of which forms the peak of ChR. With a increasing, the contribution from the non-channeling segments to the total emission spectrum becomes more pronounced, whereas the channeling particles radiate less intensively. The

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increase in a causes (i) the increase of the curvature of the segments containing the VR events, (ii) the decrease in the period of the quasi-periodic modulation of the trajectories for over-barrier particles. As a result, the following maxima seen in graphs (b)–(e) become shifted towards higher photon energies: the maxima marked with upward arrows that are associated with the channeling motion and the VR events, and those marked with downward arrows, which are due to the over-barrier particles. For large bending amplitudes, graphs (d)–(e), these maxima are virtually due to the emission by the non-channeling particles only. For a > 0 the low-energy part of the spectrum is dominated by the peak located at hω ¯ ≈ 1 MeV. For moderate amplitudes, a ≤ 2.5 Å, when the bending parameter C (see (4.30)) is small enough, this peak is associated with CUR emitted by the accepted particles which cover a distance of at least one bending period λu . For larger amplitude, a = 4.0 Å (C = 0.77), the penetration length L p becomes less than λu /2 leading to noticeable broadening of the CUR peak. For larger amplitudes, the peak undergoes further modifications due to the phenomenon other than channeling. Figure 6.6e shows the dependences obtained for a = 5.5 Å, which corresponds to C = 1.15. The bending parameter exceeds one, hence, only a small fraction of the particles is accepted at the entrance. The accepted electrons channel has very small amplitude of the channeling oscillations, ach  d/2. As a result, the ChR is suppressed for these particles but they do contribute to the CUR part of the spectrum (see the dashed curve in the graph). However, even in the low-energy part of the spectrum, this contribution is not a dominant one. The main part of the CUR peak intensity in the total spectrum comes from the non-channeling particles, see the dash-dotted curve. The explanation of this feature is as follows. A trajectory of a non-channeling particle consists of short segments corresponding to VR separated by longer segments Δz ≈ λu /2 where it moves in the over-barrier mode. In the course of two sequential VR events the particle experiences “kicks” in the opposite directions, see the trajectory in Fig. 6.5a. Therefore, the whole trajectory acquires periodic modulation with the period 2Δz ≈ λu , which results in the emission of radiation at the CUR frequency. These effects, which are due to the interplay of different radiation mechanisms in periodically bent crystals, can be probed experimentally. In this connection, one can mention successful experiments on detecting the excess of radiation emission due to VR in oriented bent Si(111) crystal by 855 MeV electrons [37] and 12.6 GeV electrons [38].

6.2.3 Channeling and Radiation Emission in Diamond Hetero-Crystals As mentioned in Chap. 1, periodic bending can be achieved by graded doping during the synthesis process of a diamond superlattice [39]. Both boron and nitrogen can be used as the dopant, however, boron is preferable as it allows for achieving higher

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Fig. 6.7 a Sketch of the crystal geometry. The diamond single crystal is cut with its surface perpendicular to the [100] direction. The crystal is tilted by 45◦ to orient the (110) planes along the incident beam, panel (a). The hetero-crystal consists of two segments: a straight (S) L S = 141 microns thick single crystal substrate and a boron-doped L PB = 20 microns thick periodically bent (PB) segment which accommodates four bending periods [42]. Gradient shading shows the boron concentration which results in the PB of the (110) planes. Panels b and c show two possible orientations of the hetero-crystal with respect to the incident beam. In panel b the beam enters the PB segment, in panel c—the S segment. These two orientations are called “PB-S crystal” and “S-PB crystal”, respectively. An exemplary trajectory of a positron channeled through the whole PB-S crystal is shown in panel b. An exemplary trajectory of an electron in the S-PB crystal is presented in panel c. Note that several channeling and over-barrier parts of the electron’s trajectory that are outside the drawing are not shown. Reference [43]

dopant concentrations before extended defects appear [40]. The advantage of using a diamond crystal in channeling experiments is due to its radiation hardness that maintains the lattice integrity under the exposure to intensive beams of charged particles [41]. Boron-doped diamond layer cannot be separated from a straight/unstrained substrate (SC) on which the superlattice is synthesized. Therefore, unlike Si1−x Gex superlattice, a diamond-based superlattice has essentially a hetero-crystal structure, i.e., it consists of two segments, a straight single diamond crystal substrate and a periodically bent (PB) layer [42]. Reference [43] presents the results of computational analysis of channeling and radiation properties in experimentally realized diamond-based CU, Fig. 6.7. Special attention has been paid to the analysis of the new effects which appear due to the presence of the interface between the straight and PB segments in the hetero-crystal. The experiment has been carried out with the 270–855 MeV electron beams [33, 44, 45]. For the sake of comparison, the simulations have been carried out for both electron and positron beams. The positron beam of the quoted energy range is available at the DA NE acceleration facility [23, 46]. Panel (a) in the figure shows the geometry of the system. The incident beam can enter the crystal at either PB or straight (S) part, panels (b) and (c), respectively. To distinguish the crystal orientation with respect to the incident beam, in the text below the crystal shown in panel (b) is labeled as the PB-S crystal and the one in panel (c) as the S-PB crystal. To illustrate the particle’s propagation through the crystal, the

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PB-S crystal S-PB crystal

40

PB-S crystal S-PB crystal ChR

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Fig. 6.8 Spectra of radiation emitted within the cone θ0 = 0.24 mrad by ε = 855 MeV positrons (left) and electrons (right) in propagating in the oriented PB-S and S-PB hetero-crystals. The intensity of the background incoherent bremsstrahlung estimated within the Bethe–Heitler approximation is 2.5 × 10−5 (not indicated in the figure). Reference [43]

selected trajectories of a positron (red curve, panel (b)) and an electron (blue curve, panel (c)) are shown. The parameters of the hetero-crystal used in the simulations of channeling along the (110) plane matched those used in the experiment [42]: total thickness in the beam direction is L = 161 microns out of which 141 microns correspond to the straight segment and 20 microns—to the PB segment; the bending amplitude and period are a = 2.5 Å and λu = 5 microns, respectively. Figure 6.8 compares the calculated spectral dependences of the radiation emitted within the cone θ0 = 0.24 mrad by ε = 855 MeV positrons (left graph) and electrons (right graph) channeling in the PB-S and S-PB hetero-crystals. In the straight segment of the crystal, a projectile can experience channeling oscillations. In addition to these, in the PB segment a projectile is involved in the undulator motion due to the periodic bending of the channels. Spectral distributions of the radiation bear features of both types of the oscillatory motion. The features are more pronounced for positrons. The positron spectra clearly exhibit two main peaks: the one of ChR centered at about hω ¯ ≈ 3.6 MeV and the CUR peak athω ¯ ≈ 1.1 MeV. These peaks correspond to the fundamental harmonics of two types of radiation. The second harmonic manifests themselves as a small bump around hω ¯ ≈ 2.2 MeV (CUR) and hω ¯ ≈ 7.2 MeV (ChR). It is seen that the intensity of CUR is virtually insensitive to the hetero-crystal orientation (PB-S or S-PB), whereas the intensity of ChR for the S-PB crystal is ca 2 times higher than for the PB-S one. To provide qualitative explanation of these features, one can consider the following arguments. The intensity of ChR is proportional to the (average) amplitude of channeling oscillations ach squared and to the (average) length L ch of the channeling segment: 2 L ch . When a projectile enters the S-PB crystal it moves initially in the dE ∝ ach straight part of the system so that the values of ach consistent with the channeling condition are within interval [0, d/2], where d = 1.26 Å is the interplanar distance. If entering the PB segment, the centrifugal force acting on a projectile decreases

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the allowed amplitude values by the factor equal approximately to (1 − C), where  is the bending parameter. For the bending amplitude and period C = 4π 2 aε/λu Umax  ≈ 8 GeV/cm in diamond(110) one finds indicated above and for the value Umax C ≈ 0.4. Further, let us note that dechanneling length L d of a 855 MeV positron in a straight diamond(110) channel is about 500 microns [4], and in the PB one is it ca (1 − C) less, i.e., equals to ≈ 300 microns. Therefore, in the case of the S-PB orientation, most of the accepted particles move in the channeling mode over the straight segment (i.e., the largest part of the hetero-crystal, L S = 141 microns out of total L = 161 microns) with the values of ach distributed within the interval [0, d/2]. At the interface, S-to-PB, the particles oscillate with ach  (1 − C)d/2 dechannel. The remaining positrons channel in the short L PB = 20 microns PB segment with the amplitudes distributed within the narrower interval. For the PB-S orientation, the amplitude of the accepted particles are distributed over the interval 0 ≤ ach ≤ (1 − C)d/2, and most part of these particles channel through the whole crystal. Taking into account these arguments one estimates the ratio of the peak intensities of ChR for the two orientations of the hetero-crystal as follows: L S + (1 − C)2 L PB dE S−PB ≈ ≈ 2.5 . dE PB−S (1 − C)2 (L S + L PB ) The ratio obtained corresponds to the ratio of the peak intensities of ChR seen in Fig. 6.8 left. Weak dependence of the CUR intensity on the hetero-crystal orientation is also clear. In this case, the intensity is mostly determined by the number of particles which undergo undulator motion in the PB segment. Due to the strong inequality L d L, for either orientation the number of particles accepted at the PB segment entrance can be estimated as (1 − C)N , where N is the total number of particles incident on the crystal. Thus, the CUR intensities are (approximately) the same in both cases. Compared to positrons, electrons have significantly shorter dechanneling lengths. As a result, the primary fraction of channeled electrons virtually dies out for crystal thicknesses greater than 50 microns for both PB-S and S-PB crystals. This explains why CUR radiation is seen in the spectral distribution only in the case of PB-S crystal, Fig. 6.8 right. More discussion on the differences in the electron spectra as well as comparison of the spectral dependences obtained for different energies of the beam one finds in Ref. [43].

6.2.4 Channeling and Radiation Emission in SASP Periodically Bent Crystals The crystalline undulator concept implies that the projectiles move in a crystal following the profile of its periodically bent planes (or axes). Such undulator motion is modulated with the frequencies Ωu that are smaller than the frequencies Ωch

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of the channeling oscillations. Additionally, this regime assumes periodic bending characterized by large amplitude, a > d, and long period, λu a. In the spectral distribution of the emitted radiation, the CUR peaks appear at the energies lower that those typical for ChR [6–8]. Another type of periodic bending, called Small-Amplitude Short-Period (SASP) bending, was suggested in Ref. [47]. It implies the bending amplitude to be much less than the interplanar distance, a  d, and, simultaneously, the bending period much shorter than the period of channeling oscillations. In contrast to the motion in a CU, a particle moving in a SASP crystal does not follow the profile of the bent planes but acquires a short-period jitter-like modulations due to the bending. These fast modulations lead to the emission of electromagnetic radiation in the energy domain beyond the channeling peaks [2, 47–52]. It is noted that a similar radiative mechanism has been investigated in connection with the radiation produced by ultrarelativistic charged particles in interstellar environments with turbulent fluctuations of the magnetic field [53, 54]. In Ref. [48], results of numerical simulations of the channeling and radiation processes have been presented for ε = 855 MeV positrons and electrons passing through an oriented SASP silicon crystal. The numerical modeling has been carried out by means of all-atom relativistic molecular dynamics. Specific features that are due to the SASP bending have been established and elucidated further within an analytically developed model approach (see Sect. 6.5). The parameters of the SASP bending were chosen to match those used in the electron channeling experiment at MAMI [49]. The crystal used in the experiment was a Si1−x Gex superlattice with the content x of germanium atoms varied from 0.3 % to 1.3 %. As reported, the periodic bending of the (110) planes was achieved with the amplitude a = 0.12 ± 0.03 Å and period λu = 0.43 ± 0.004 microns. The number of SASP periods was 10. No further details on the bending profile have been provided although in a more recent paper [52] it was noted that “…the shape is roughly sinusoidal”. A thicker crystalline sample L = 12 microns has been considered in the simulations [48]. A perfect cosine SASP profile with the period 400 nm has been assumed. The bending amplitude was varied from a = 0 up to a = 0.9 Å. The latter value is close to the half of the (110) interplanar distance in a silicon crystal (d = 1.92 Å). The emission spectra have been calculated in the photon energy range up to hω ¯ = 40 MeV. Two emission cones considered were θ0 = 0.21 and 4 mrad. The natural emission cone for a 855 MeV projectile is γ −1 ≈ 0.6 mrad, Therefore, the smallest value of θ0 accounts mainly for a nearly forward emission, whereas the largest value, being much larger than γ −1 , collects almost all the radiation emitted. Figure 6.9 illustrates the variety of features seen in the emission spectra. The pronounced peaks of ChR are clearly seen in the spectra for the straight (a = 0) crystal, see black solid-line curves. Harmonic-like channeling oscillations of positrons result in the undulator-type spectral dependence with small values of the undulator parameter, K 2  1. Namely, the positron spectra display the fundamental peaks of ChR at hω ¯ 1 ≈ 2.5 MeV whereas the higher harmonics are suppressed. In particular, for ¯ 1 is an order of magnitude larger the cone θ0 = 0.21 mrad the peak intensity at hω than that for the second harmonics at hω ¯ 2 ≈ 5 MeV, and only a bulge of the third

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Fig. 6.9 Spectral distribution of radiation emitted by 855 MeV positrons (upper row) and electrons (lower row) in a L = 12 µm thick straight (a = 0) and periodically bent (a > 0) oriented silicon (110) crystal. The period of the SASP bending is 400 nm and various bending amplitudes are indicated in the common legend given in the right bottom graph. The left and right columns refer to the emission cones θ0 = 0.21 and 4 mrad. The intensity of the incoherent bremsstrahlung radiation in amorphous silicon (not shown) is 0.016 × 10−3 and 0.15 × 10−3 for the smaller and larger cones, respectively. Reference [55]

harmonics can be identified at about 7.5 MeV (see the top left graph). For electrons, the ChR peaks are less intensive and much broader than the positrons due to strong anharmonicity of the channeling oscillations. In the SASP bent crystals, there are additional peaks in the spectra that are due to the short-period modulations of the trajectories of channeling particles (see discussion in Sect. 6.5). The peaks, being more pronounced for the smaller emission cone, lie in the photon energy domain well beyond the peaks of ChR. For both positrons and electrons, the fundamental peaks of the SASP radiation are seen at the energies hω ¯ ≈ 16 MeV. For positrons, the peaks of the SASP radiation virtually disappear for smaller values of a. It happens because positrons experience mainly “regular” channeling staying away from the crystalline atoms and being therefore less affected by the modification of the crystalline field due to the SASP bending (see Sect. 6.5 for details). In contrast, the electrons experience stronger impact of the SASP bending at lower values of a. As seen in the left lower graph, the peak for a = 0.1 Å is only two times lower than the highest peak displayed for a = 0.4 Å.

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To be noted are the spectral properties for the smaller aperture value (left column graphs in the figure). The electron spectra display peaks at the energies around 32 MeV for a ≥ 0.2 Å which correspond to the second harmonic of the SASP radiation. In addition, the peaks of channeling radiation decrease in heights and shift towards the lower emission energies. In contrast, the positron spectra exhibit less peculiarities and gradually converge to the Bethe–Heitler background with increasing radiation energies. For the larger emission cone, a sizable part of the energy is radiated at the angles θ > γ −1 . The harmonics energies decrease with θ approximately as (1 + K 2 /2 + (γ θ )2 )−1 . As a result, the peaks of ChR and those of the radiation due to the SASP bending broaden and shift towards softer radiation energies.

6.2.5 Experiments with SASP Periodically Bent Crystals The radiation from SASP crystals has been investigated in the experiments with 855 MeV electrons at the MAMI facility [51]. The crystal used in the experiment was fabricated by adding a linearly increasing fraction 0.5% < x < 1.5% of germanium atoms to a silicon substrate. By alternating successively the linear increase with the decrease a sawtooth bending pattern was achieved with 120 periods of λu = 0.44 microns and with “the expected oscillation amplitude” [51] of the (110) planes equal to a ≈ 0.12 Å. In the experiment, the enhancement factor over the radiation intensity in amorphous silicon was measured. Two sets of the measurements were performed: (i) with collimation to an emission angle θ0 ≈ 0.24 mrad, and (ii) with no collimation that correspond to a large (θ0 γ −1 ) emission cone. It was noted that the latter case corresponded to the emission In Fig. 6.10, the experimental data on the radiation spectra enhancement are compared to the results of numerical simulations carried out with MBN Explorer [55]. Left graph corresponds to the collimated case, θ0 = 0.24 mrad, whereas the right graph presents the data obtained without collimation. The simulations were performed for several values of the bending amplitude as indicated in the common legend. The same silicon crystal with the SASP bending containing 120 periods each of λu = 0.44 microns was used in the experiment a highly intensive 16 GeV electron beam carried out at the SLAC facility [52]. In the experiment, the SASP signal can only be expected to appear when the crystal is properly aligned and should reveal itself as a peak in the spectrum detected in a narrow emission angle. Therefore, in Ref. [52] the enhancement was looked for as the crystal was rotated in the beam, passing through the aligned condition, and a narrow radiation cone when scanning the angular distribution with the detector. However, as mentioned in the cited paper, the measurements of the spectrum have not been successful due to difficulties with the experimental setup and variations in beam energy that had not been expected. Initially, the experiments mentioned above had been planned to be carried out with both electron and positron beams. In this connection, aiming to provide theoretical benchmarks the simulations have been carried out for 15–35 GeV projectiles by

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Fig. 6.10 Radiation spectra enhancements for 855 MeV electrons in straight and SASP bent Si(110) with respect to the amorphous silicon. Left and right graphs refer to the emission within the opening cones θ0 = 0.24 and 4 mrad, respectively. Experimental data from Ref. [51] is shown by symbols. The curves stand for the results of simulations [55]. Common legend is presented in the right graph

electron, straight electron, SASP positron, straight positron, SASP

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Fig. 6.11 Spectral distribution of radiation emitted by 20 GeV electrons and positrons in straight and SASP bent Si(110). The data refer to the crystal thickness L = 52.3 µm, bending period λu = 436 nm and bending amplitude a = 0.12 Å. Left and right graphs refer to the emission cones with opening angle θ0 = 13 and 130 µrad, correspondingly. Common legend is presented in the right graph. Reference [55]

means of the MBN Explorer package [55, 56] . The simulations of trajectories were supplemented with computation of the spectra of the emitted radiation for various detector apertures. The conclusion drawn from the results of calculations was to carry out experiments with electrons and with the smallest aperture available. In this case, the SASP signal was predicted to be the highest. Figure 6.11 illustrates theoretical predictions by presenting the spectral distribution of radiation emitted by 20 GeV projectiles in the narrow 13 µrad ≈ 1/2γ (left graph) and wide 13 µrad ≈ 5/γ (right graph) cones along the beam direction. The peaks centered around 7 GeV are due to the SASP bending. The ChR peaks are located at much lower energy, hω ¯ = 0.2 − 0.5 GeV.

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Fig. 6.12 Spectral distribution of radiation emitted by 20 GeV electrons and positrons in straight and SASP periodically bent 4 microns thick diamond(110) (left) and silicon(110) (right) oriented crystals. Bending amplitude and period are 0.4 Å and 0.4 µm, respectively. References [27, 57]

6.3 Stack of Periodically Bent Crystals In recent series of experiments at MAMI [49] with 600 and 855 MeV electrons, the effect of the radiation enhancement due to the SASP periodic bending has been observed (see discussion in Sect. 6.2.4). Another set of experiments with thin SASP diamond crystals was planned within the E-212 collaboration at the SLAC facility (USA) with 10–20 GeV electron beams [50]. As a case study aimed at producing theoretical benchmarks for the SLAC experiments, a series of numerical simulations have been performed of the planar channeling of 10–20 GeV electrons and positrons in straight and SASP periodically bent thin crystals of silicon and diamond [27, 57]. The crystal thickness L was set to 4 microns, the period of bending λu = 0.4 microns and the bending amplitude a = 0.4 Å, which is lower than half of the (110) interplanar distance in both cases. In Fig. 6.12, the results of the simulation of radiation of 20 GeV projectiles are compared for the cases of straight and periodically bent diamond(110) crystals. The beam emittance was taken equal to ψ = 5 µrad. The spectra presented refer to the emission cone θ0 = 150 µrad, which is 5.8 times higher than natural emission angle 1/γ = 25.6 µrad and thus collects virtually all radiation emitted. In both figures, the peaks located below 1 GeV correspond to the channeling radiation. For periodically bent targets, the peaks at hω ¯ ≈ 6 GeV and above are due to the SASP bending. Note that bending of a crystal leads to significant suppression of the channeling peak. This effect can be explained qualitatively in terms of the continuous potential modification in a SASP channel (see Sect. 6.5). With increase of bending amplitude, the depth of the potential well decreases and the width of the potential well grows resulting in decrease of the number of channeling projectiles and in lowering frequencies of channeling oscillations. Another factor that leads to the suppression of the channeling radiation is that for the 20 GeV projectiles the characteristic period of channeling oscillations, deduced from the simulated trajectories, is about λch  ≈ 10 microns for both diamond and silicon crystals, so that the crystal is too thin to allow for even a single channeling

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Photon energy (GeV) Fig. 6.13 Emission spectra for 20 GeV electrons in SASP bent diamond(110) crystal calculated for two different thicknesses, as indicated. The value L = 24 µm exceeds characteristic channeling oscillations period while L = 4 µm is lower than that. Note the absence of channeling radiation peak around 150 MeV in the latter case. For the sake of comparison, the curve for L = 4 µm is multiplied by six. References [27, 57]

oscillation. Therefore, the peak of channeling radiation is not that pronounced as in the case of thicker, L > λch . The emission spectra formed in thick (L = 24 microns) and thin (L = 4 microns) crystals are compared in Fig. 6.13 where the latter spectrum is multiplied by a factor of 6 for the sake of convenience. A sharp peak of the channeling radiation at hω ¯ ≈ 150 MeV is present for the thick crystal whereas for the thin one it reduces to a small hump, which is due to the synchrotron-type radiation emitted by projectiles moving along the one-ark trajectory. Remarkable feature, seen in the figure, is that the peaks due to the SASP bending in both curves virtually coincide. The phenomenon of suppression of ChR in thin crystals accompanied by maintaining the level of radiation due to the SASP bending can be used to produce intensive radiation emitted in the high photon energy domain. One possibility to increase the intensity of this radiation is to use thicker crystals. However, this extensive approach may not be optimal because of technological complications (increase in the duration of the crystal growth as well as in the costs associated, accumulation of the defects in the crystalline structure, etc.) Alternatively, one can use a stack of several thin aligned crystals [27, 57], see illustrative Fig. 6.14. In this scheme, a projectile passes sequentially several layers of SASP crystals and the radiation emitted in each element of the stack adds to the total emission spectrum. Thickness L of each layer in the stack can be chosen to be smaller than characteristic period of the channeling oscillations of a projectile thus leading to the suppression of channeling radiation.

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1

2

N

..... λu

L

l

Fig. 6.14 Illustrative representation of a stack of N crystal layers each of thickness L separated by the intervals l. In each layer, the SASP periodic bending of the crystalline structure is shown by thin wavy curves. Thick red line illustrates a projectile’s trajectory consisting of the undulating parts inside the layers and the straight segments between the layers. Reference [57]

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Fig. 6.15 Radiation spectra for the small θ0 = 0.3/γ (left panel) and large θ0 = 5/γ (right) apertures calculated for different number n of diamond(110) periodically bent crystals in stack (as indicated in the common legend). The data refer to 20 GeV positrons, the bending amplitude and period are 0.4 Å and 400 nm, respectively. Reference [57]

In contrast, the emission via the SASP radiation mechanism remains and becomes enhanced with the increase in the number of layers. Modeling of particles propagation through a stack of crystals and computation of the spectral distribution of the radiation emitted was carried out in Refs. [27, 57]. A set of several L = 4 microns thick layers of bent periodically with the λu = 0.4 microns and separated with l = 4 microns gaps were generated in the simulation box. A 20 GeV projectile (a positron) enters the first layer of the stack tangent to its (110) crystallographic plane. Because of the multiple scattering in the crystalline medium, the projectile that leaves the layer at some non-zero angle is with respect to the initial direction and this angle is considered as the incident angle at the entrance

6.4 Brilliance of the CU Radiation

157

to the second layer, etc. The process of multiple scattering in the layers leads to a gradual increase in the angular divergence of the particle so that a probability to be accepted into the channeling mode decreases as the particle propagates through the stack. Nevertheless, for a moderate number N of layers this destructive effect is not too pronounced, so that the radiation intensity increases being proportional to N . For sufficiently large N values, the intensity saturates and becomes independent on N . Figure 6.15 compares the radiation spectra calculated for the stack built of different number N of identical Si(110) layers. Two panels in the figure correspond to the radiation emitted in a small θ0 = 0.3/γ = 15.3 µrad (left) and large θ0 = 5/γ = 256 µrad (right) emission cones. For the smaller cone, the radiation intensity scales linearly with the number of layers up to N = 6. For larger values of N , the spread in the projectiles’ transverse velocities broadens leading to the saturation of the intensity of radiation emitted within a narrow cone. For the larger cone, the nearly the intensity grows linearly up to N = 24. Upper panel in Fig. 6.16 compares the intensities of radiation emitted by 20 GeV positrons traveling in a single SASP diamond(110) crystal of thickness 24 microns and in a stack of six layers each 4 microns thick. It is seen that the peak of ChR is suppressed for the stack, whereas the higher energy undulator peaks are of the same intensity for both targets. Lower panel in the figure compares the emission spectra for 20 GeV positrons and electrons of the same energy passing through the stacks with different numbers of layers, N = 1, 6 and 24. The figure illustrates weak sensitivity of the spectra formed in the SASP periodically bent crystals to the sign of a projectile’s charge. Therefore, this regime is favorable for the construction of CLSs based on the propagation of intensive electron beams which, at present, are more available than positron beams.

6.4 Brilliance of the CU Radiation In this chapter, quantitative estimates are presented for the CUR brilliance using the parameters of high-energy positron beams either available at present or planned to be commissioned in near future (see Table 7.2 in Sect. 7.5). It is demonstrated that by means of the LALP CU-based LS is it feasible to achieve the photon yield that is much higher than the values provided by modern LS facilities operating in the gamma-ray range, E ph  102 keV [3]. The relevant modern facilities are synchrotrons and undulators based on the action of magnetic field.1 Another type of a short-wavelength LS, which does not utilize magnets, is based on the Compton scattering process [58]. In this case, a low-energy laser photon backscatters from an ultra-relativistic electron thus acquiring increase in the energy proportional to the squared Lorentz factor γ = ε/mc2 of the electron.

1

Fore the sake of comparison the data presented below is compared to the brilliance available at the XFEL facilities which operate at much lower energies of the emitted radiation.

158

6 Crystalline Undulators

L=24 μm, θ0=15 μrad 6xL=4 μm, θ0=15 μrad L=24 μm, θ0=256 μrad 6xL=4 μm, θ0=256 μrad

3

dE/d(hω) (×10 )

3

_

2

1

0 0

5

10

Photon energy (GeV)

_

3

dE/d(hω) (×10 )

8

6

4

2

N=24

N=6 N=1

0 0

5

10

Photon energy (GeV) Fig. 6.16 Upper graph. Spectral distributions of radiation emitted by 20 GeV positrons in a single 24 microns thick crystal and in a stack of six L = 4 microns thick layers. Dashed curves correspond to the smaller emission cone, θ0 = 15 µrad, solid curves—to the larger cone, θ0 = 256 µrad. Refs. [27, 57]. Lower graph. Comparison of the spectral distributions of radiation emitted by 20 GeV electrons (dashed curves) and positrons (dashed curves) in the stacks of SASP diamond(110) crystals with different number N of layers. The data refer to the emission cone θ0 = 5/γ = 256 µrad

This method has been used for producing gamma rays in a broad, 10−2 −101 MeV, energy range [59, 60]. The Compton scattering can also occur from an atomic (or ionic) electron, which moves being bound to a nucleus. This phenomenon is the basis for the Gamma Factory proposal for CERN [61–63] within which a beam of ultra-relativistic partly stripped ions is proposed to be used in the backscattering process. In this scheme, an ionic electron is resonantly excited by absorbing a laser photon. The subsequent radiative de-excitation produces a gamma-photon. Due to the relativistic Doppler effect, the energy of photons emitted in the direction of the beam is boosted by a factor of up to 4γ 2 as compared to the energy of the laser light. Because of a huge excess of the

6.4 Brilliance of the CU Radiation

159

resonant photon absorption cross section compared to that of the photon scattering from a free electron, the intensity of an atomic-beam-driven LS is expected to be several orders of magnitude higher than what is possible with Compton gamma-ray sources driven by an electron beam. Let us overview the model approach which can be used to estimate the main characteristics of a CU-based LS (number of photons, brilliance) accounting for the two main parasitic effects that are the dechanneling of the beam particles and the photon attenuation. The number of photons ΔNωn of frequency within the interval   ωn − Δωn /2, ωn + Δωn /2 emitted within the cone ΔΩn along the beam direction is given by the following expression (see Refs. [4, 64] for the details):  2 Δωn n+1 (nζ ) (nζ ) − J Neff , ΔNωn = A (C) 4π α nζ J n−1 2 2 ωn

(6.4)

Here, A stands for the channel acceptance, i.e., the fraction of the beam particles captured into the channeling mode at the crystal entrance (another term used is surface transmission). Other notations used include: Jν (nζ ) - the Bessel function, ζ = K 2 /(4 + 2K 2 ), and K = 2π γ a/λu is the undulator parameter of the CU. The integer n enumerates harmonics, ωn = nω1 , of CUR. The fundamental harmonic is given by ω1 =

2γ 2 2π c . 1 + K 2 /2 λu

(6.5)

Equation (6.4) differs from the corresponding formula for an ideal undulator (see, e.g., [65]). The main difference (apart from the factor A ) is that the number of undulator periods, which enters the latter, is substituted with the effective number of periods, Neff , which depends on the number of periods within the dechanneling length, Nd = L d /λu , and on the ratios x = L d /L a and κ = L/L d , where L d and L a stand for the dechanneling length and the photon attenuation length, respectively. Within the model approach, the effective number of periods can be calculated as follows [4, 64]: 4Nd Neff = xκ



xe−xκ e−κ 2e−(2+x)κ/2 − + (1 − x)(2 − x) 1−x 2−x

 1 + κ2

(x − 1)2 + 1 . 4π 2

(6.6)

If the dechanneling attenuation are neglected, i.e., in the limit L d , L a → ∞, then Neff → Nu = L/λu , as in the case of an ideal undulator. In an ideal undulator, it is possible, in principle, to increase infinitely the number of periods by considering larger values of the undulator length. This will lead also to the increase in the number of photons and the brilliance since these quantities are proportional to Nu . The limitations on L and Nu are mainly due to technological reasons. In a CU, however, the situation is different since the number of channeling particles and the number of photons, which can emerge from the crystal, decrease as L grows. From Eq. (6.6) follows that if L → ∞ the values of κ and xκ = L/L a also become

160

6 Crystalline Undulators

infinitely large resulting in Neff → 0. This result is quite clear, since in this limit L L a so that all emitted photons are absorbed inside the crystal. Another formal (and physically trivial) fact is that Neff = 0 also for L = 0. Vanishing of a positivelydefined function Neff (Nd , x, κ) at two extreme boundaries suggests that at some length L(x) the function has a maximum. To define the value of L(x) or, what is equivalent, of the quantity κ(x) = L(x)/L d , one carries out the derivative of f (x, κd ) with respect to κ and equalizes it to zero. The analysis of the resulting equation shows that for each value of x = L d /L a ≥ 0 there is only one root κ. Hence, the equation defines, in an inexplicit form, a singlevalued function κ d (x) = L(x)/L d which ensures the maximum of Neff (x, κd ) for given dechanneling and attenuation lengths as well as for the bending period. It was shown [4, 64] that the quantity L(x) ensures the highest values of the number of photons and the brilliance of CUR. Therefore, L(x) can be called the optimal length that corresponds to a given value of the ratio x = L d /L a . The following multi-step procedure can be used to calculate the highest brilliance of CUR. • Fix crystal and crystallographic direction. In the current paper, we have focused on the (110) planar channels in diamond and silicon crystals, which are commonly used in channeling experiments. We note that other crystals/channels, available or/and studied experimentally, can also be considered [6, 66, 67]. • Fix parameters of the positron beam: energy ε, sizes σx,y and divergence φx,y , peak beam current Imax . • Scan over photon energy ω. For each ω value: 1. Determine the attenuation length L a (ω) (for the photon energies above 1 keV the data are compiled in Ref. [68]). 2. Scan over a and λu consistent with the stable channeling condition (5.2), where one uses R = λ2u /4π 2 a for the curvature radius in the points of the maximum curvature. Then, the condition (5.2) C = 4π 2

a ε < 1.  λ2u Umax

(6.7)

3. Determine dechanneling length L d (C). The data on the dechanneling length can be extracted (when available) from the experiments [45, 69] or obtained by means of numerical simulation of the channeling process [4, 70, 71]. For positrons, the following formulae provide good estimation for the dechanneling lengths in the straight channel, L d (0), and in the periodically bent one L d (C) [4, 72]: L d (C) = (1 − C)2 L d (0), 256 aTF d ε L d (0) = 9π 2 r0 m e c2 Λ

(6.8)

6.4 Brilliance of the CU Radiation

161

where r0 cm is the classical electron radius, Z and aTF are the atomic number and the Thomas–Fermi radius of the crystal atom, Λ = 13.55 + 0.5 ln(ε[GeV]) − 0.9 ln(Z ). 4. Determine the maximum value of Neff and the optimal length L. 5. Estimate the acceptance A (C) of the bent channel from A (C) = (1 − C) A0

(6.9)

where A0 = 1 − 2u T /d stands for the acceptance of the straight channel (u T is the amplitude of thermal vibrations atoms in the crystal) [72]. 6. Substituting the quantities obtained into Eqs. (6.4) and (3.1) one calculates the highest value of peak brilliance Bpeak (ω) achievable for the channel/crystal chosen and for the fixed parameters of the beam. As formulated, the items (3)–(6) are applicable for a positron beam with zero divergence, φ = 0. In reality, φ > 0, see Table 7.2, so that only a fraction ξ of the beam particles, which are incident with the angle θ less than Lindhard’s critical angle ΘL , become accepted at the entrance. Assuming the normal distribution of the beam particles with respect to θ one calculates ξ as follows: ξ = (2π φ 2 )−1/2





θ2 exp − 2 dθ. 2φ −ΘL ΘL

(6.10)

The values of ξ calculated for the beams from Table 7.2 √ are listed in Table 6.2. For each beam indicated, Lindhard’s critical angles ΘL = 2U0 /ε have been estimated for U0 = 20 eV, which corresponds, approximately, to the interplanar potential well in Si(110) and diamond(110). The values of beam divergence indicated in the table have been calculated as φ = min[φx , φ y ]. To account for the non-zero divergence one multiplies the value Bpeak (ω), calculated as described above, by the factor ξ . Figures 6.17 and 6.18 show the results of calculations performed for silicon(110)and diamond(110)-based CU using the parameters of the beams from Table 7.2. The dependences presented were obtained by maximizing the CUR brilliance following the multi-step procedure described above. It is seen, that within the range of moderate values of the bending amplitude it is possible to construct a CU with sufficiently large number of periods, Neff ≈ 10 . . . 100, graphs (c). The corresponding ranges of undulator periods λu ≈ 101 . . . 102 µm (graphs (d)) can be achieved by means of the existing technologies for preparation of periodically bent crystalline materials, see Chap. 1. Figures 6.17 and 6.18 show the peak brilliance Bpeak (ω), graphs (f), is highly sensitive to the parameters of a positron beam: The magnitude of Bpeak (ω) varies over six orders of magnitude (compare the DAΦNE and CEPC curves). Let us compare the brilliance of CUR with that available at modern synchrotron facilities. Figure 6.19 presents the peak brilliance calculated for positron-based diamond(110) and Si(110) CUs and that for several synchrotrons. The CUR curves refer to the optimal parameters of CU, i.e., those which ensure the highest values of

162

6 Crystalline Undulators

(a)

(d)

0.3

2

C

λu (μm)

10

0.2

1

10

0.1

(b)

(e)

a/d

L/Ld

2

SuperKEKB FACET-II CEPC

(c)

Neff

DAΦNE BEPC-II SuperB

50

0 -1 10

0

10

1

10

Photon energy (MeV)

0

Peak Brilliance

1 100

1

10

10 26 10

(f)

24

10

22

10

20

10

18

10

-1

10

0

10

1

10

Photon energy (MeV)

Fig. 6.17 Graphs a–e show parameters of the silicon(110)-based CU: bending parameter C, bending period λu and amplitude a (measured in the interplanar distance d = 1.92 Å), effective number of period Neff , optimal length L (measured in the units of L d (C)), that ensure the highest peak brilliance Bpeak (ω), graph (f), for several positron beams as indicated in the legend (see also Table 7.2). Reference [3]

Bpeak (ω) of CUR for each positron beam indicated. To calculate Bpeak (ω) we used Eq. (3.1) where the peak currents Imax , listed in Table 7.2, have been substituted. To be noted is that for the well-collimated intensive beams with small transverse sizes (SuperB, FACET, SuperKEK, CEPC) the peak brilliance of CUR in the photon energy range from 102 keV to 102 MeV (the corresponding wavelengths vary from 10−1 down to 10−4 Å) is comparable to (the case of SuperB, FACET, and SuperKEK beams) or even higher (CEPC beam) than that achievable in conventional LS for much lower photon energies. We stress that the values of bending amplitude and periods, which maximize the CUR brilliance over broad range of photon energies, are accessible by means of modern technologies (compare Figs. 6.17 and 6.18 with Figs. 1.3 and 1.4 in Sect. 7.5). The Gamma Factory proposal for CERN discusses a concept of the LS based on the resonant absorption of laser photons by the ultra-relativistic ions [61, 62, 75]. It is expected that the intensity (photon rate) of the LS will be orders of magnitude higher than that of the presently operating LS aiming at the values of IGF = 1017 photons/s in the gamma-ray domain 1 MeV ≤ E ph ≤ 400 MeV. The quoted value of

6.4 Brilliance of the CU Radiation

163

(a)

(d)

λu (μm)

0.3 2

C

10

0.2

1

10 0.1

(b)

(e)

a/d

L/Ld

2

SuperKEKB FACET-II CEPC

(c)

Neff

DAΦNE BEPC-II SuperB

50

0 -1 10

0

10

1

10

Photon energy (MeV)

0

Peak Brilliance

1 100

1

10

10 26 10

(f)

24

10

22

10

20

10

18

10

-1

10

0

10

1

10

Photon energy (MeV)

Fig. 6.18 Same as in Fig. 6.17 but for the diamond(110)-based CU (the interplanar distance d = 1.26 Å). Reference [3]

the intensity refers to the average beam current. To calculate the corresponding peak value one multiplies IGF by a factor ≈ 100/0.64 ≈ 150, which is the ratio of the bunch spacing (≈100 ns) to the bunch length (≈0.64 ns). To this end, it is instructive to compare the intensity of CUR with the quoted value as well as with the intensities currently achievable by means of the LS based on laser-Compton scattering from electron beam [59]. Figure 6.20 presents the peak intensities, ΔNω Imax /e, of the first (solid lines) and third (dashed lines) harmonics of CUR from diamond(110)-based CU with the optimized parameters (see Fig. 6.18). Different curves correspond to different positron beams as specified in the caption. The curves presented show orders of magnitude higher intensities in the photon energy range one to tens of MeV than that from the laser-Compton scattering LS (1013 −1014 photon/s, not shown in the figure). Within the same photon energy interval, the CUR intensity can be comparable with or even much higher (see the curves for the FACET-II beam) than the value predicted in the Gamma Factory proposal (marked with the horizontal dash-dotted line). Figures 6.19 and 6.20 demonstrate also the tunability of a CU-LS. For any positron beam with specified parameters, the photon yield can be maximized (more generally, varied) over broad range of photon energies by properly choosing parameters of the CU (bending amplitude and period, crystal, plane).

6 Crystalline Undulators

27

10

CEPC

2

26

10

2

Peak brilliance (photons/s mrad mm 0.1%BW)

164

10

25

SuperB

SPring8

FACET

24

10

SuperKEKB

ESRF

23

10

PETRA

22

10

APS

C(110) Si(110)

21

10

-2

10

-1

0

10

10

1

10

2

10

Photon energy (MeV) Fig. 6.19 Comparison of the peak brilliance of CUR from diamond(110)- and Si(110)-based CUs for several positron beams listed in Table 7.2 with the brilliance available at several modern synchrotron radiation facilities (APS, ESRF, PETRA, SPring8). The data on APS (USA), ESRF (France), PETRA (DESY, Germany), SPring8 (Japan) are from [73, 74]. Reference [3] Table 6.2 Fraction ξ of the beam particles with incident angle less than Lindhard’s critical angle ΘL (in mrad). For each beam indicated the parameter φ (in mrad) stands for the minimum of two divergences φx and φ y , see Table 7.2. Facility

VEPP4M

BEPCII

DAΦNE

SuperKEKB

SuperB

FACETII

CEPC

φ ΘL ξ

0.2 0.08 0.31

0.35 0.14 0.31

0.54 0.28 0.40

0.18 0.1 0.42

0.125 0.08 0.48

0.044 0.063 0.85

0.03 0.03 0.67

6.5 Atomistic Simulations of the CU Light Sources To verify the predictions made in Ref. [3], accurate all-atom molecular dynamics simulations have been performed [76] aiming at at providing reliable quantitative data on the brilliance of CUR. The exemplary case study presented in this section shows that using the positron beam available at present it is realistic to achieve brilliances that exceed those of the laser-Compton scattering LSs and predicted in the GF proposal. The model approach developed in Ref. [3] and overviewed in Sect. 6.4 has allowed one to establish optimal parameters of a periodically bent crystal (these include crystal

6.5 Atomistic Simulations of the CU Light Sources

165

21

10

5

Intensity (photons/s)

20

10

GF 19

6

10

3

4 2 18

10

1

17

10

-1

10

0

10

1

10

2

10

photon energy (MeV) Fig. 6.20 Peak intensity (number of photons per second, ΔNω Imax /e) of diamond(110)-based CUs calculated for positron beams at different facilities: 1 - DAΦNE, 2 - BEPC-II, 3 - SuperB, 4 SuperKEK, 5 - FACET-II, 6 - CEPC. Solid lines and a dashed line (for FACET-II only) correspond to the emission in the first and third harmonics, respectively. The horizontal dash-dotted line marks the intensity 1017 photon/s, indicated in the Gamma Factory (GF) proposal [62], multiplied by a factor 150, which is the ratio of the bunch spacing to the bunch length. The intensities of the laserCompton backscattering being on the level 1013 −1014 photon/s [59] are not shown in the figure

thickness L, bending amplitude a and period λu ) that ensure the highest values of brilliance of the CU-LS for a positron beam of given energy ε, transverse beam sizes σx,y and angular divergence σφx,y . The molecular dynamics simulations have been performed for a 10 GeV (γ = 1.96 × 104 ) positron beam propagating through an oriented periodically bent diamond (110) single crystal. The following values of the beam sizes and divergence were used: σx,y = 32, 10 microns and σφx,y = 10, 30 µrad, respectively. These values correspond to normalized emittance γ x,y = γ σφx,y σx,y = 6.3, 5.9 m-µrad and are within the ranges indicated for the FACET-II beam available at the SLAC facility, which can be found in literature (Table 4.6 in Ref. [??]). These ranges are listed in Table 6.3. The peak current of the beam is Ipeak = 3.1 kA. In Ref. [76], the simulations have been performed for the positron beam with the sizes σx,y = 32, 10 microns and divergence σφx,y = 10, 30 µrad. The values of bending amplitude and period as well as the number of periods and the crystal thickness used in the simulations are listed in Table 6.4. The energies hω ¯ 1 of the fundamental harmonic of CUR in the forward direction, also shown in the table, are calculated from Eq. (2.26). Within the framework of the model approach [??], it has

166

6 Crystalline Undulators

Table 6.3 Parameters of a 10 GeV (γ = 1.96 × 104 ) positron beam (before longitudinal compression) at FACET-II: the normalized emittance γ x,y = γ σφx,y σx,y and the transverse sizes σx,y (the nominal value as well as the accessible range) [77]. The values of beam divergence σφx,y have been calculated using the data on x,y and σx,y . The peak current of the beam is Ipeak = 3.1 kA γ x (m-µrad)

σx (µm) Nominal

Range

σφx (µrad) γ  y Range (m-µrad)

σ y (µm) Nominal

Range

σφ y (µrad) Range

6.3

10

10–35

32–9

7

7–10

43–30

5.9

Table 6.4 Two sets of parameters of diamond-based CUs used in the simulations [76]: bending amplitude a, period λu , number of periods Nu , crystal thickness L = Nu λu . The last column presents the energies hω ¯ 1 of the fundamental harmonic of CUR emitted in the forward direction Set a (Å) λu (µm) Nu L (mm) hω ¯ 1 (MeV) (I) (II)

20.9 5.3

85 38

85 180

7.06 6.84

2.0 10.0

been established that these values provide the maximum brilliance of the CU-LS radiation emitted in the forward direction at the fundamental harmonic energy hω ¯ 1, Eq. (6.5). To simulate the trajectories of the beam particles, the y-axis was chosen along the 110 axial direction. This choice ensures that a sufficiently big fraction ξ of the incident beam is accepted in the channeling mode at the crystal entrance. For a Gaussian  Θ beam, this fraction can be estimated as ξ = (2π σφ2y )−1/2 −ΘL L exp −φ 2 /2σφ2y dφ where ΘL = (2U0 /ε)1/2 stands for Lindhard’s critical angle. Using U0 ≈ 20 eV for the interplanar potential depth in diamond(110) (see Fig. 4.22) one calculates ΘL ≈ 63 µrad that is large enough in comparison with the divergence φ y = 30 µrad. The beam divergence φx = 10 µrad along the x transverse direction is much smaller than the natural emission angle θγ = γ −1 ≈ 50 µrad. As a result, a big fraction of radiation is emitted by the channeling particles within the cone θ0 ≤ θγ centered along the incident beam. In the simulations, the spectral distribution dE j (θ ≤ θ0 )/d(¯hω) has been calculated for each trajectory simulated ( j = 1, . . . , N0 where N0 ∼ 5 × 103 is the total number of trajectories). The resulting spectral distribution used to calculate the brilliance (3.1) has  been obtained by averaging the individual spectra: dE(θ ≤ θ0 )/d(¯hω) = N0−1 j dE j (θ ≤ θ0 )/d(¯hω). Figure 6.21 presents the spectral distribution of radiation emitted in the CU with the parameters labeled as “Set (I)” in Table 6.4. Left graph corresponds to the narrow emission cone θ0 = (5γ )−1 = 10 µrad; the right one—to the wider cone θ0 = γ −1 = 50 µrad. The calculations were performed for different number of undulator periods, Nu , as indicated in the common legend shown in the left graph. For the sake of comparison, the dashed lines in the figure show the emission spectra in the ideal planar undulator where a projectile moves along perfect cosine trajectory y(z) = a cos(2π z/λu ).

6.5 Atomistic Simulations of the CU Light Sources

0.6

_

dE/d(hω)

_

dE/d(hω)

0.15

Nu=20 Nu=30 Nu=40 Nu=80

0.1

0.05

0 1.7

up to 1.5

θ0=50 μrad

up to 0.75

θ0=1/5γ=10 μrad

0.2

167

0.4

0.2

1.8

1.9

2

2.1

0 1.5

1.6

1.7

1.8

1.9

2

2.1

Photon energy (MeV)

Photon energy (MeV)

Fig. 6.21 Spectral distribution dE(θ ≤ θ0 )/d(¯hω) emitted within the cones θ0 = (5γ )−1 = 10 µrad (left panel) and γ −1 = 50 µrad (right panel) corresponding to different number of undulator periods Nu as indicated in the left graph. The data refer to the bending amplitude and period indicated for Set (I) in Table 6.4. Solid lines stand for results of numerical simulation, dashed lines correspond to ideal planar undulators of the same a and λu . Reference [76]

_

dE/d(hω)

0.3

θ0=50 μrad

Nu=20 Nu=30 Nu=40 Nu=50 Nu=80

0.2

0.1

0

1.5

1.6

1.7

1.8

1.9

2

2.1

Photon energy (MeV) Fig. 6.22 Spectral distribution emitted within the cone θ0 = γ −1 = 50 µrad computed for different number of undulator periods as indicated. Solid lines show the emission spectra averaged over all simulated trajectories. Dashed lines represent the spectra averaged over those particles that move in the channeling mode through the whole crystal. The data refer to the bending amplitude and period indicated for Set (I) in Table 6.4. Reference [76]

Several features of the computed spectra can be noted. First, all peaks are redshifted from the value hω ¯ 1 = 2 MeV estimated from Eq. (6.5). The shift increases with the emission cone. Second, for a large number of periods the spectrum becomes virtually independent on the crystal length: in both panels all spectra become very close for Nu  50.

6 Crystalline Undulators Bpeak (ph/s/mrad /mm /0.1%BW)

168

0.2

0.1

0 1.4

24

10

2

0.3

2

_

dE/d(hω)

0.4

θ0=0.2/γ θ0=0.5/γ θ0=0.7/γ θ0=1/γ θ0=2/γ

1.5

1.6

1.7

1.8

Photon energy (MeV)

1.9

2.0

2.1

23

10

22

θ0=0.1/γ θ0=0.2/γ θ0=0.5/γ θ0=0.7/γ θ0=1/γ θ0=2/γ

10

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

Photon energy (MeV)

Fig. 6.23 Left graph: Spectral distribution dE(θ ≤ θ0 )/d(¯hω) of radiation emitted within different cones θ0 indicated in the units of the natural emission cone γ −1 ≈ 50 µm. Right graph: Peak brilliance of CU-LS calculated for different emission cones. Both panels refer to the bending amplitude and period as Set (I) in Table 6.4; the crystal thickness is L = 5.1 mm (Nu = 60). Reference [76]

To provide an explanation to both of these features can notice that in the vicinity of maximum the emission spectrum is mainly due to the particles that move in the channeling mode through the whole crystal, see Fig. 6.22. Therefore, the evolution of the spectrum is related, to a great extent, to the dynamics of channeling particles in the course of their propagation through the crystalline medium. Relation (6.5) is valid for an ideal planar undulator. In a CU, a channeling particle experiences (i) channeling oscillations while moving along a periodically bent centerline, and (ii) stochastic motion along the x axis due to the multiple scattering from crystal constituents. The channeling oscillations lead to the following modification of the 2 [29]. Here K ch ∝ 2π γ ach /λch with ach ≤ undulator parameter: K 2 → K 2 + K ch d/2 and λch being the amplitude and period of the channeling oscillations. In the case of positron channeling, assuming harmonicity of the oscillations, one carries 2  = 2γ U0 /3mc2 [4]. out averaging over the allowed values of ach and finds K ch 2  ≈ 0.56. The For ε = 10 GeV in diamond(110) (U0 ≈ 20 eV) this results in K ch stochastic motion along the x axis results in a gradual increase in the rms scattering angle θx2  with the penetration distance z. Hence, the motion of the particle is not restricted to the (yz) plane and its emission within the cone centered along the incident beam direction becomes off-axis. As a result, the denominator of the fraction in (6.5) 2 /2 + γ 2 θx2 (z) + γ 2 σφ2x , leading to the increases, 1 + K 2 /2 → 1 + K 2 /2 + K ch decrease in the fundamental harmonic frequency. The increase in the rms scattering angle leads to the saturation of the emission spectrum with the crystal thickness. Indeed, taking into account that an ultra-relativistic particle radiates, predominantly, within the cone 1/γ along its instant velocity, the emission within the cone θ0 along the z axis becomes negligibly small at the penetration distances z˜ when the relation 1/2  2 − γ −1  θ0 becomes valid. Here, the term σφ2x accounts for the θx (˜z ) + σφ2x beam divergence at the crystal entrance. The spectrum saturates at L ∼ z˜ . The increase in the multiple scattering angle is also the main reason for a suppression of the simulated intensity as compared to the ideal undulator, Fig. 6.21.

Bpeak (photons/s mrad mm 0.1%BW)

6.5 Atomistic Simulations of the CU Light Sources

169

25

10

SPring8

24

(II)

ESRF

(I)

2

2

10

Model

Simulation

23

10

PETRA 22

10

APS 21

10

-2

10

-1

10

0

10

1

10

Photon energy (MeV)

Fig. 6.24 Comparison of the peak brilliance available at several synchrotron radiation facilities (APS, ESRF, PETRA, SPring8) with that for the diamond(110)-based CU-LS. The filled diamond symbols show the brilliances that correspond to the sets (I) and (II) of the bending parameters indicated in Table 6.4. The dashed line stands for the model estimation [??]; the filled circles marked the estimated brilliances for the sets (I) and (II). Reference [76]

The CU radiation spectra calculated for different emission cones are presented in the left graph of Fig. 6.23. All curves correspond to the crystal length L = 5.1 mm (N u = 60) with bending amplitude and period indicated for Set (I) in Table 6.4. These data supplemented with the aforementioned values for the beam size, divergence and peak current allow one to calculate the peak brilliance of the CU-LS. The results of calculations are shown in Fig. 6.23 right together with the estimate (marked with ¯ = 2 MeV filled green circle) of 1.50 × 1024 photons/s/mrad2 mm2 /0.1 % BW at hω obtained in Ref. [??] within the framework of the model approach. It is seen that in contrast to dE(θ ≤ θ0 )/d(¯hω), which is an increasing function of the emission cone, the peak brilliance is a non-monotonous function due to the presence of the terms proportional to θ0 in the total emittance E x,y that enter the denominator in max ≈ 3.5 × 1023 Eq. (3.1). In the case study considered here the maximum value Bpeak 2 2 photons/s/mrad mm /0.1 % BW is achieved athω ¯ = 1.85 MeV for the emission cone θ0 = 0.7/γ = 35 µrad. Due to the reasons discussed above the maximum is redshifted and less intensive (approx. 4 times less) as compared to the model prediction. Figure 6.24 compares peak brilliance Bpeak (ω) of several operational synchrotron radiation facilities with that achievable by means of the diamond-based CU exposed to the FACET-II positron beam. The dashed line shows the results of the model calculations [??] that have provided, for each photon energy hω, ¯ the highest value of Bpeak (ω) by scanning through the ranges of bending amplitude a and period λu . Symbols show the values of Bpeak (ω) obtained for the two sets, (I) and (II), of bending parameters indicated in Table 6.4: the results of model calculations estimations

170

6 Crystalline Undulators

(circles) are to be compared with the results of the accurate numerical simulations (diamonds). By tuning the bending amplitude and period, one can maximize brilliance for given parameters of a positron beam and/or chosen type of a crystalline medium. As a result, extremely high values of brilliance can be achieved in the photon energy range 101 . . . 102 MeV by currently available (or planned to be available in near future) positron beams [??]. For each set of the input parameters (beam energy and emittance, bending amplitude and period, crystal type and thickness, detector aperture, etc.) to provide accurate data on the brilliance from a CLS rigorous numerical simulations, similar to those presented above, must be carried out. When doing this, the results of the model approach can be used as the initial estimates of the the ranges of the parameters to be used in the simulations. It is worth mentioning that the size and the cost of CLSs are orders of magnitude less than those of modern LSs based on the permanent magnets. This opens many practical possibilities for the efficient generation of gamma-rays with various intensities and in various ranges of wavelength by means of the CLSs on the existing and newly constructed beam-lines.

Appendix: Continuous Potential and Transverse Motion in a SASP Crystal In this supplementary section, explicit formulae are derived that describe nonperiodic and periodic parts of the continuous planar potential in a SASP bent crystal. The analytical and numerical analysis of the results obtained allow us to qualitatively explain the peculiar features in the motion of ultra-relativistic projectiles as well as in the radiative spectra.

Continuous Potential Consider a crystallographic plane which coincides with the (x z) Cartesian plane. For the sake of clarity, let us introduce a cosine periodic bending, a cos(ku z) with ku = 2π/λu , of the plane in the transverse y direction. The bending amplitude a and period λu satisfy the SASP bending condition a < d  λu ,

(6.11)

where d stands for the interplanar distance. Similar to the procedure used for a straight plane (see Sect. 4.5.3), the continuous potential of a periodically bent plane one obtains summing up the potentials of individual atoms assuming that the latter is distributed uniformly along the plane:

Appendix: Continuous Potential and Transverse Motion in a SASP Crystal

(x,y,z) r

y x

y’ z

λu

Δ

A

171

a

z’

Fig. 6.25 Supplementary figure illustrating the derivation of the continuous potential of a periodically bent crystallographic plane (the thick curve represents the bending profile). The atoms are displaced randomly from their equilibrium positions (x  , y  , z  ) due to to thermal vibrations (the vector Δ shows the position of a displaced atom “A”)

Upl (y, z) = N

∞ w(Δ) dΔ

dz

−∞



∞

dx  Uat (|r − Δ|) .

(6.12)

−∞

Vector Δ stands for the displacement of an atom from its equilibrium position, characterized by the coordinates (x  , y  , z  ) with with y  = a cos ku z  (see illustrative Fig. 6.25) due to thermal vibrations. Expressing Uat in terms of its Fourier transform and using (4.28) one integrates over x  , z  , Δ and presents the planar potential in the form of a series: Upl (y, z) = V0 (y; a) +

∞

cos(nku z) Vn (y; a)

(6.13)

at (q) cos(qy) J0 (qa)U

(6.14)

n=1

with V0 (y; a) =

2N Vn (y; a) = π

N π

∞ dq e 0

∞

dq e−

q 2 u 2T 2

0

−

2 Q2 n uT 2

 at (Q n ) × Jn (qa) U

n

(−1) 2 cos(qy) n−1 (−1) 2 sin(qy)

(6.15)

where the short-hand notation Q 2n = q 2 + (nku )2 is used. The upper line stands for even n values, the lower line—for the odd ones. In the limit of a straight channel, a = 0, the right-hand side of Eq. (6.13) reduces to that in (4.49). Indeed, taking into account that J0 (0) = 0 and Jn (0) = 0 for n > 0, one notices that this leads to Vn (y; 0) ≡ 0 for all terms defined by (6.15) whereas the term V0 (y; 0), Eq. (6.14), reduces to (6.13). To determine the interplanar potential U (y, z), one uses Eq. (4.54) where the potentials Upl (y, z) of individual planes are to be inserted. The integrals on the right-hand sides of Eqs. (6.14) and (6.15) can be evaluated explicitly for a number of analytic approximations for Uat which can be found in

172

6 Crystalline Undulators

literature [9, 78–83]. For reference purposes, we present the explicit formulae derived within the framework of the Molière approximation (4.43): Vn (y; a) = (1 + δn0 )N Z e

3

αj i=1 Γn j

e

γ 2j u 2T 2

Tn (y; a, Γn j )

(6.16)

1/2

, Tn stands for the inteHere δn0 is the Kronecker symbol, Γn j = γ j2 + (nku )2 gral: Tn (y; a, Γ ) =

π/2

 dθ cos(nθ ) F (y − a cos θ ; Γ ) + (−1)n F (y + a cos θ ; Γ )

(6.17)

0

where F (Y ; Γ, u T ) = F(Y ; Γ, u T ) + F(−Y ; Γ, u T )

(6.18)

with F(±Y ; Γ, u T ) defined as in (4.51). For n = 0, Eq. (6.16) reproduces the expression derived in Ref. [48]. Similar to the case of a straight crystal, the interplanar potential U (y, z) in a SASP bent crystal is obtained by summing up the potentials (6.13) of individual planes. For the electron channel, the result can be written in the form U (y, z) =

∞ 

cos(nku z) Un (y)

(6.19)

n=0

where Un (y) = Vn (y) +

K max 

 Vn (y + kd) + Vn (y − kd) + Cn .

(6.20)

k=1

Here, y is the transverse coordinate with respect to an arbitrary selected reference plane, and the sum describes a balanced contribution from the neighboring planes. The constants Cn can be chosen to satisfy the condition Un (0) = 0. For positrons, the interplanar potential can be obtained from Eq. (6.20) by reversing the signs of the planar potentials and selecting the constants Cn to ensure Vn (±d/2) = 0. Similar summation schemes allow one to calculate the charge densities, nuclear and electronic, across the periodically bent channels.

Transverse Motion in a SASP Channel The function y(t) describes the transverse motion of a particle with respect to the centerline of the channel. The equation of motion (EM) reads

Appendix: Continuous Potential and Transverse Motion in a SASP Crystal

y¨ = −

173

1 ∂U (y, z) . mγ ∂y

(6.21)

In what follows we outline a perturbative solution of the EM. Assuming the longitudinal coordinate z changes linearly with time, z ≈ ct, one re-writes the potential (6.19) substituting z with ct. Then, the EM is written as follows:   ∞  1 cos(nΩu t) f n (y) (6.22) y¨ = f 0 (y) + mγ n=1 with Ωu = ku c =

2π c , λu

f 0 (y) = −

dU0 , dy

f n (y) = −

dUn . dy

(6.23)

The action of the time-independent force f 0 results in the channeling oscillations. At the same time, the projectile experiences local small-amplitude oscillations (the jitter-like motion) due to the driving forces f n cos(nΩu t) (n = 1, 2, . . . ). In a SASP channel, the frequency Ωu (and, respectively, its higher harmonics nΩu ) exceeds greatly the frequency of the channeling oscillations: Ωu Ωch . As a result, the EM (6.22) can be integrated following the perturbative procedure outlined in Ref. [84], Sect. 30, for the motion in a rapidly oscillating field. Namely, y(t) is represented as a sum y(t) = Y (t) + Ξ (t)

(6.24)

where Ξ (t) stands for a small (but rapidly oscillating) correction to the smooth dependence Y (t) which describes the channeling oscillations. The function Ξ (t) satisfies the equation in which the coordinate Y is treated as a parameter: Ξ¨ =

∞ 

cos(nΩu t)

n=1

f n (Y ) . mγ

(6.25)

Its solution reads Ξ (t, Y ) =

∞ 

ξn (Y ) cos(nΩu t) .

(6.26)

f n (Y ) 1 2 mγ Ωu n 2

(6.27)

n=1

where ξn (Y ) = −

174

6 Crystalline Undulators

The presence of the term Ξ (t, Y ) modifies the EM for Y (t). In addition to the force f 0 = −dU0 (Y )/dY due to the static potential, a ponderomotive force f pond appears. It can be calculated as follows [84] (below, the overline denotes averaging over the period 2π/Ωu which is much smaller than the characteristic time of the channeling motion and, thus, does not affect the value of Y (t)): f pond (Y ) = Ξ (t, Y )

∞ n=1

n (Y ) cos(nΩu t) d fdY =−

dUpond dY

(6.28)

The ponderomotive potential, Upond , introduced here is defined as follows: Upond (Y ) =

∞ λ2u  f n2 (Y ) 16π 2 ε n=1 n 2

(6.29)

Note that the ponderomotive correction to the potential becomes smaller as the energy increases since Upond ∝ 1/ε. Therefore, the channeling oscillations Y = Y (t) are described by the EM 1 dUeff Y¨ = − mγ dY

(6.30)

where the total effective potential reads Ueff (Y ) = U0 (Y ) + Upond (Y ) .

(6.31)

The non-periodic potential U0 and the ponderomotive term Upond calculated within the Molière approximation for positron SASP Si(110) channel are presented in Fig. 6.26. The curves correspond to different bending amplitude as indicated. The right panel shows the dependence of the product εUpond (with ε measured in GeV) which is independent on the projectile energy. In both panels, the vertical lines mark the (110)-planes in the straight crystal. The modification of U0 with increase of the bending amplitude is clearly seen on the left panel in Fig. 6.26. In detail, this issue was discussed in Ref. [48]. Here, for the sake of consistency, we mention several features relevant to the topic of the current paper. For small and moderate amplitude values, a ≤ 0.4 Å, the major change in the potential is the decrease of the interplanar potential barrier. As the a values approach the 0.4 . . . 0.6 Å range, the volume density of atoms becomes more friable leading to flattening of the potential maximum. For larger amplitudes, the potential changes in a more dramatic way as additional potential well appears. In the figure, this feature is clearly seen in the behavior of the the U0 curve for a = 0.8 Å: in addition to the “regular” channels centered at the midplanes, i.e., at y/d = . . . , −1, 0, 1, . . . , “complementary” channels appear centered at y/d = . . . , −0.5, 0.5, . . . . As a result, a positron can experience channeling oscillations moving in the channels of the two different types. Similar feature characterizes the electron channeling phenomenon at sufficiently large values of bending amplitude [48].

Non-periodic potential, U0 (eV)

0

20

0.1 0.2

0.4

10

0.6

0

0.8

-0.5

0

0.5

1

1.5

Ponderomotive potential, εUpond (GeV×eV)

Appendix: Continuous Potential and Transverse Motion in a SASP Crystal

0.3

175

0.15

0.2

0.1

0.2

0.4

0.1

Transverse coordinate (in units of d)

0.6 0.8

0

-0.5

0

0.5

1

1.5

Transverse coordinate (in units of d)

Fig. 6.26 Left. The non-periodic part U0 (y) of the continuous interplanar potential for positrons in Si(110) calculated at different values of bending amplitude indicated in Å near the curves (a = 0 stands for the straight crystal). Right. The ponderomotive correction Upond multiplied by ε in GeV calculated for various a and for fixed bending period λu = 308 microns. The potentials shown are evaluated for temperature 300 K by using the Molière atomic potentials. The vertical dashed lines mark the adjacent (110) planes in straight crystal (the interplanar spacing is d = 1.92 Å)

Comparing the absolute values of the ponderomotive term Upond (curves in the right panel of Fig. 6.26 correspond to the terms at ε = 1 GeV) and those of U0 one can state that for all bending amplitudes Upond is a negligibly small correction to U0 for the projectile energies above 1 GeV. For much lower energies, say for a few

0

Potentials U0, Ueff (eV)

20 0.2

0.4

10 0.6

0 -0.5

0.8

0

0.5

Transverse coordinate (in units of d) Fig. 6.27 The effective interplanar potential Ueff , Eq. (6.31), (solid lines) for a 250 MeV positron in Si(110) at different values of bending amplitude indicated in Å near the curves (a = 0 stands for the straight crystal). The dashed curves show the continuous potential U0 without the ponderomotive corrections. The bending period is λu = 308 microns

176

6 Crystalline Undulators

hundreds-MeV, the contribution of Upond becomes more noticeable, reaching the eV range in the regions 2|y|/d ≈ . . . , −1, 0, 1, . . . . Thus, it should be accounted if an accurate integration of the EM (6.30) is desired. Figure 6.27 illustrates the change in the continuous interplanar Si(110) potential due to the ponderomotive correction. Solid curves who the corrected potentials, the dashed ones—the term U0 . The data refer to 250 MeV positron channeling in Si(110) bent periodically with the period λu = 308 microns; the values of bending amplitude (in Å) are indicated in the figure.

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39. Tran Thi, T.N., Morse, J., Caliste, D., Fernandez, B., Eon, D., Härtwig, J., Barbay, C., MerCalfati, C., Tranchant, N., Arnault, J.C., Lafford, T.A., Baruchel, J.: Synchrotron Bragg diffraction imaging characterization of synthetic diamond crystals for optical and electronic power device applications. J. Appl. Cryst. 50, 561 (2017) 40. de la Mata, B.G., Sanz-Hervás, A., Dowsett, M.G., Schwitters, M., Twitchen, D.: Calibration of boron concentration in CVD single crystal diamond combining ultralow energy secondary ions mass spectrometry and high resolution X-ray diffraction. Diam. Rel. Mat. 16, 809 (2007) 41. Uggerhøj, U.I.: The interaction of relativistic particles with strong crystalline fields. Rev. Mod. Phys. 77, 1131–1171 (2005) 42. Boshoff, D., Copeland, M., Haffejee, F., Kilbourn, Q., Mercer, C., Osatov, A., Williamson, C., Sihoyiya, P., Motsoai, M., Henning, C.A., Connell, S.H., Brooks, T., Härtwig, J., Tran Thi, T.N., Palmer, N., Uggerhøj, U.: The search for diamond crystal undulator radiation. In: 4th International Conference “Dynamics of Systems on the Nanoscale” (Bad Ems, Germany, Oct. 3-7 2016) Book of Abstracts, p. 38 (2016) 43. Pavlov, A., Korol, A., Ivanov, V, Solov’yov, A.: Channeling and radiation of electrons and positrons in diamond hetero-crystals (2021). arXiv:2004.07043 44. Backe, H., Krambrich, D., Lauth, W., Andersen, K.K., Hansen, J., Lundsgaard, and Uggerhøj, U.I.: Channeling and radiation of electrons in silicon single crystals and si1- xGex crystalline undulators. J. Phys. Conf. Ser. 438, 012017 (2013) 45. Backe, H., Lauth, W., Tran Thi, T.N.: Channeling experiments at planar diamond and silicon single crystals with electrons from the Mainz Microtron MAMI. J. Instrum. (JINST) 13, C04022 (2018) 46. Si111-qm-general2-60.sh The DA NE Beam-Test Facility. http://www.lnf.infn.it/ acceleratori/btf/ 47. Kostyuk, A.: Crystalline undulator with a small amplitude and a short period. Phys. Rev. Lett. 110, 115503 (2013) 48. Korol, A.V., Bezchastnov, V.G., Sushko, G.B., Solov’yov, A.V.: Simulation of channeling and radiation of 855 MeV electrons and positrons in a small-amplitude short-period bent crystal. Nucl. Instrum. Meth. B 387, 41–53 (2016) 49. Wistisen, T.N., Andersen, K.K., Yilmaz, S., Mikkelsen, R., Hansen, J.L., Uggerhøj, U.I., Lauth, W., Backe, H.: Experimental realization of a new type of crystalline undulator. Phys. Rev. Lett. 112, 254801 (2014) 50. Uggerhøj, U.I., Wistisen, T.N.: Intense and energetic radiation from crystalline undulators. Nucl. Instrum Meth. B 355, 35 (2015) 51. Uggerhoj, U.I., Wistisen, T.N., Hansen, J.L., Lauth, W., Klag, P.: Radiation collimation in a thick crystalline undulator. Eur. J. Phys. D 71, 124 (2017) 52. Wienands, U., Gessner, S., Hogan, M.J., Markiewicz, T.W., Smith, T., Sheppard, J., Uggerhøj, U.I., Hansen, J.L., Wistisen, T.N., Bagli, E., Bandiera, L., Germogli, G., Mazzolari, A., Guidi, V., Sytov, A., Holtzapple, R.L., McArdle, K., Tucker, S., Benson, B.: Channeling and radiation experiments at SLAC. Nucl. Instrum Meth. B 402, 11 (2017) 53. Medvedev, M.V.: Theory of “jitter” radiation from small-scale random magnetic fields and prompt emission from gamma-ray burst shocks. Astrophys. J. 540, 704–714 (2000) 54. Kellner, S.R., Aharonian, F.A., Khangulyan, D.: On the jitter radiation. Astrophys. J. 774, 61 (2013) 55. Korol, A.V., Sushko, G.B., Solov’yov, A.V.: All-atom relativistic molecular dynamics simulations of channeling and radiation processes in oriented crystals. Eur. Phys. J. D 75, 107 (2021) 56. Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Unpublished (2016) 57. Sushko, G.B., Korol, A.V., Solov’yov, A.V.: A small-amplitude crystalline undulator based on 20 GeV electrons and positrons: Simulations. St. Petersburg Polytechnical Uni. J.: Phys. Math. 1, 341 (2015) 58. Federici, L., Giordano, G., Matone, G., Pasquariello, G., Picozza, P., et al.: The LADON photon beam with the ESRF 5 GeV machine. Lett. Nuovo Cimento 27, 339 (1980)

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59. ur Rehman, H., Lee, J., Kim, Y.: Optimization of the laser-Compton scattering spectrum for the transmutation of high-toxicity and long-living nuclear waste. Ann. Nucl. Energy 105, 150 (2017) 60. Krämer, J.M., Jochmann, A., Budde, M., Bussmann, M., Couperus, J.P., et al.: Making spectral shape measurements in inverse Compton scattering a tool for advanced diagnostic applications. Sci. Rep. 8, 139 (2018) 61. Xenon beams light path to gamma factory. CERN Courier (13 October 2017). https:// cerncourier.com/xenon-beams-light-path-to-gamma-factory/ 62. Krasny, M.W.: The Gamma Factory proposal for CERN (2015). arXiv:1511.07794 63. Krasny, M.W., Martens, A., Dutheil, Y.: Gamma factory proof-of-principle experiment: Letter of intent. CERN-SPSC-2019-031/SPSC-I-253, 25/09/2019 http://cds.cern.ch/record/2690736/ files/SPSC-I-253.pdf 64. Korol, A.V., Solov’yov, A.V., Greiner, W.: Number of photons and brilliance of the radiation from a crystalline undulator. Proc. SPIE 5974, 597405 (2005) 65. Kim, K.-J.: Characteristics of synchrotron radiation. In: X-ray Data Booklet, pp. 2.1–2.16. Lawrence Berkeley Laboratory, Berkley (2009). http://xdb.lbl.gov/xdb-new.pdf 66. Kostyuk, A., Korol, A.V., Solov’yov, A.V., Greiner, W.: Demodulation of a positron beam in a bent crystal channel. Nucl. Instrum. Method B 269, 1482–1492 (2011) 67. Sytov, A.I., Bandiera, L., De Salvador, D., Mazzolari, A., Bagli, E., Berra, A., Carturan, S., Durighello, C., Germogli, G., Guidi, V., Klag, P., Lauth, W., Maggioni, G., Prest, P., Romagnoni, M., Tikhomirov, V.V., Vallazza, E.: Steering of Sub-GeV electrons by ultrashort Si and Ge bent crystals. Eur. Phys. J. C 77, 901 (2017) 68. Hubbel, J.H., Seltzer, S.M.: Tables of X-ray mass attenuation coefficients and mass energyabsorption coefficients from 1 keV to 20 MeV for elements Z = 1 to 92 and 48 additional substances of dosimetric interest. NISTIR 5632. Web version http://www.nist.gov/pml/data/ xraycoef/index.cfm 69. Wienands, U., Markiewicz, T.W., Nelson, J., Noble, R.J., Turner, J.L., Uggerhøj, U.I., Wistisen, T.N., Bagli, E., Bandiera, L., Germogli, G., Guidi, V., Mazzolari, A., Holtzapple, R., Miller, M.: Observation of deflection of a beam of multi-GeV electrons by a thin crystal. Phys. Rev. Let. 114, 074801 (2015) 70. Sushko, G.B., Bezchastnov, V.G., Solov’yov, I.A., Korol, A.V., Greiner, W., Solov’yov, A.V.: Simulation of ultra-relativistic electrons and positrons channeling in crystals with MBN Explorer. J. Comp. Phys. 252, 404 (2013) 71. Solov’yov, I.A., Korol, A.V., Solov’yov, A.V.: Multiscale Modeling of Complex Molecular Structure and Dynamics with MBN Explorer. Springer International Publishing, Cham, Switzerland (2017) 72. Biryukov, V.M., Chesnokov, Yu.A., Kotov, V.I.: Crystal Channeling and Its Application at High-Energy Accelerators. Springer, Berlin/Heidelberg (1996) 73. Ayvazyan, V., Baboi, N., Bohnet, I., Brinkmann, R., Castellano, M., et al.: A new powerful source for coherent VUV radiation: demonstration of exponential growth and saturation at the TTF free-electron laser. Europ. Phys. J. D 20, 149 (2002) 74. Schmüser, P., Dohlus, M., Rossbach, J.: Ultraviolet and Soft X-Ray Free-Electron Lasers. Springer, Berlin/Heidelberg (2008) 75. Krasny, M.W.: The Gamma Factory proposal for CERN. Photon-2017 Conference, May 22-29, 2017 8CERN, Geneva 76. Sushko, G.B., Korol, A.V., Solov’yov, A.V.: Extremely brilliant crystal-based light sources. In preparation (2021) 77. SLAC Site Office: Preliminary Conceptual Design Report for the FACET-II Project at SLAC National Accelerator Laboratory. Report SLAC-R-1067, SLAC (2015) 78. Gemmell, D.S.: Channeling and related effects in the motion of charged particles through crystals. Rev. Mod. Phys. 46, 129–227 (1974) 79. Baier, V.N., Katkov, V.M., Strakhovenko, V.M.: Electromagnetic Processes at High Energies in Oriented Single Crystals. World Scientific, Singapore (1998)

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80. Doyle, P.A., Turner, P.S.: Relativistic Hartree-Fock X-ray and electron scattering factor. Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. A 24, 390–397 (1968) 81. Pacios, L.F.: Analytical density-dependent representation of Hartree-Fock atomic potentials. J. Comp. Chem. 14, 410–421 (1993) 82. Dedkov, G.V.: Interatomic potentials of interactions in radiation physics Sov. Phys. - Uspekhi 38, 877–910 (1995) 83. Chouffani, K., Überall, H.: Theory of low energy channeling radiation: application to a Germanium crystal. Phys. Status Sol. (b) 213, 107–151 (1999) 84. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, vol. 1. Mechanics. Elsevier, Oxford (2003)

Chapter 7

Emission of Coherent CU Radiation

7.1 Introduction The radiation emitted in an undulator is not coherent with respect to the emitters, i.e., the undulating particles of total number Np . Indeed, the intensity of the emitted  Np radiation, proportional to the square of the total electric field Etot = j=1 E j , where E j stands for the electric field of the electromagnetic wave emitted by the jth particle. In an undulator, the positions of the particles (in particular, in the longitudinal direction) are not correlated,1 —see graph (a) in the illustrative Fig. 7.1. As a result, the phasefactors exp(iψ j ),contained in E j , are not correlated as well. Therefore, the sum over the cross terms j=k E j · E∗k ∝ exp(i(ψ j − ψk )), which appear in |Etot |2 , cancels out and the intensity is proportional to the number of emitters:

Iinc

Np   2 E j  = Np |E1 |2 ∝ Np N 2 . ∝ |Etot | → 2

(7.1)

j=1

This relation points out the two important features of the incoherent spontaneous UR (the subscript “inc” on the left-hand side of stands for “incoherent”). First, Iinc ∝ |E1 |2 ∝ N 2 , i.e., the radiation is coherent (at the harmonics frequencies) with respect to the number of undulator periods, N . The proportionality to N 2 makes the UR a powerful source of spontaneous electromagnetic radiation. In modern undulators, based on the action of magnetic field, the number of undulator periods is on the level of 103 . . . 104 [1, 2]. The second feature is that the UR is incoherent with respect to the number of the radiating particles, Iinc ∝ Np . Hence, the increase in the beam density will cause a moderate (linear) increase in the radiated energy. More powerful coherent radiation can be emitted if the beam particles are modulated in the longitudinal direction with the period equal to integer multiple of the 1

To be specific, we assume the emission in the forward direction. This is why the longitudinal coordinate, i.e., the one along the undulator axis, plays the key role. © Springer Nature Switzerland AG 2022 A. Korol and A. V. Solov’yov, Novel Lights Sources Beyond Free Electron Lasers, Particle Acceleration and Detection, https://doi.org/10.1007/978-3-031-04282-9_7

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7 Emission of Coherent CU Radiation

(a)

i

travsverse direction

travsverse direction

182

j longitudinal direction ϕi-ϕj=random

nλ

(b)

longitudinal direction ϕi-ϕj=2πn

2

Iinc∝⎢Etot⎢ ∝Np

2

2

Icoh∝⎢Etot⎢ ∝Np

Fig. 7.1 In an unmodulated beam, graph a, positions of the beam particles are random in the longitudinal direction, hence, the difference in phases ψ j − ψk of the waves emitted by particle j and k acquires random value. As a result, the total intensity is incoherent sum of the intensities of radiation emitted by each particle. In the modulated beam, graph b, the longitudinal distance between any two particles is an integer multiple of the wavelength λ. Therefore, the phase difference is ψ j − ψk = 2π n (n = 0, ±1, ±2, . . .), and the total is proportional to the number of particles squared

radiation wavelength λ, see Fig. 7.1b. In this case, the electromagnetic waves emitted in the forward direction by different particles have approximately the same phase (more exactly, φ j − φk ≈ nλ where n is an integer) [3]. Therefore, the total amplitude of the emitted radiation is a coherent sum of individual electromagnetic waves,  Np i.e., Etot = j=1 E j ∝ Np E1 , so that the intensity Icoh becomes proportional to the square of the radiating particles: Icoh ∝ |Etot |2 ∝ Np2 N 2 .

(7.2)

Comparing (7.2) and (7.1) one sees, that Icoh /Iinc ∝ Np . Thus, the increase in the photon flux due to the beam modulation (other terms used are “bunching” [4–7] or “microbunching” [8]) can reach orders of magnitudes relative to the UR of an unmodulated beam of the same density.

7.2 “Naive” Approach to the Gamma-Laser Based on a Crystalline Undulator

183

7.2 “Naive” Approach to the Gamma-Laser Based on a Crystalline Undulator A possibility to obtain stimulated emission of high-energy photons by means of a bunch of ultra-relativistic positrons channeling in PBC was considered in [9, 10], The photons, emitted in the forward direction at the points of the maximum curvature of the bent channel by a group of particles, travel parallel to the beam and stimulate the photon generation by particles of the same bunch in the vicinity of the successive maxima and minima (see illustrative Fig. 7.2). This scheme implies that the stimulation is due to the motion of the same bunch along the PBCh centerline. In the theory FEL, this principle is called Self-Amplified Spontaneous Emission (SASE) when the emission amplification starts from a shot noise in the particle beam [11–13]. The advantage, which justifies the use of a CU to create a FEL-type radiation in comparison with conventional FEL devices, is that in the former case, despite a number of restrictive factors, it is feasibly to generate the stimulated emission in the photon energy range which is not achievable in conventional FELs. In Ref. [10] it was shown, that it is possible to separate the stimulated photon emission in the CU from the ChR in the regime of large bending amplitudes a  d. Then, the scheme illustrated by Fig. 7.2 is applicable for the stimulated emission of high-energy photons up to the MeV range. Following [14] we call this scheme a Gamma-laser. Below in this section, we carry out estimation of the gain factor for a CU-based Gamma-laser assuming that the characteristic frequencies of CUR are well-separated from those of ChR. spontaneous CUR

channeling particles

stimulated CUR

periodically bent channel

Fig. 7.2 Illustrative figure of the radiation stimulation in CU. Photons emitted in the forward direction by channeling positrons stimulate the radiation emission by the particles of the same bunch. The length of the bunch L b , its velocity v ≈ c, and the undulator period λu satisfy the condition (1 − v/c)λu  L b , which ensures that the photon slippage against the positron bunch during one undulator period is much less than λu [10]

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7 Emission of Coherent CU Radiation

7.2.1 Crude Estimate of the Gain To carry out the estimation of the gain factor for a CU-based Gamma-laser we recall that in FEL-like devices the emission stimulation occurs at the frequencies of the harmonics of the spontaneous UR ωn = nω1 , n = 1, 2, . . . [15]. In what follows we consider the stimulation only for the fundamental harmonic emitted along the undulator axis (see Eq. (2.26) with n = 1 and θ = 0). In the formulae below ω is used instead of ω1 . The dechanneling effect and the photon attenuation are ignored. The gain factor, g, defines the increase in the total number Nph of the emitted photons at a frequency ω due to the emission stimulation the beam particles: d Nph = g Nph dz. By definition, the gain factor can be written as a difference between the numbers of emitted and absorbed photons: g = n ch,0 [σe (ε, ε − ω) − σa (ε, ε + ω)] ,

(7.3)

where σe (ε, ε − ω) and σa (ε, ε + ω) are the cross sections of, correspondingly, the spontaneous emission and absorption of the photon ω by a particle of the beam, n ch,0 stands for the volume density (measured in cm−3 ) of the beam particles. Using the known relationship between the cross sections σe,a and the spectral-angular intensity of the emitted radiation (see, e.g., [16]), and taking into account the relation ω  ε, one derives the following expression for the gain:  3  d E c2 d Δω ΔΩ. g = −(2π ) 2 n ch,0 ω dε dω dΩ ϑ=0 3

(7.4)

Here Δω is the width of the first harmonic peak, and ΔΩ is the effective cone (with respect to the undulator axis) into which the emission of the ω-photon occurs. Neglecting the dechanneling and the photon attenuation one calculates these quantities from Eq. (2.27). For a CU of the length L, the total increase in the number of photons is Nph = Nph,0 eG(L) ,

(7.5)

where G(L) = gL is the total gain on the scale L. Evaluating the right-hand side of (7.4) (see [10, 17] for the details), one derives the following expression for G(L) [18]: L3 K2 J 2 (η) , (7.6) G(L) = n ch,0 (2π )3r0 3 γ λu (1 + K 2 /2)2 where r0 is classical electron radius and J (η) = J0 (η) − J1 (η),

η=

K2 , 4 + 2K 2

(7.7)

7.2 “Naive” Approach to the Gamma-Laser Based on a Crystalline Undulator

185

with J0,1 (η) standing for the Bessel functions. Note the strong inverse dependence of G(L) on γ , which is due to the radiative recoil, and the proportionality to L 3 . The main difference, of a principal character, between a conventional FEL and a CU-based FEL is that in the former the bunch of particles and the photon flux both travel in vacuum, whereas in the latter they propagate in a crystalline medium. Consequently, in a conventional FEL one can, in principle, increase infinitely the length of the undulator L. This will result in the increase of the total gain. The situation is different for a CU, where the dechanneling effect and the photon attenuation lead to the decrease of the channeling particles density and of the photon flux density with the penetration length and, therefore, result in the limitation on the L-values, see the discussion in Refs. [10, 14, 19]. In the cited papers it has been established that ω  10 keV is a reasonable range of photon energies, for which the operation of CU and CU-based Gamma-laser will not be affected by the photon attenuation. The analysis carried out in [10, 17, 18], has demonstrated that to optimize the parameters of the stimulated emission in the photon energy range ω > 10 keV in the case of a positron channeling in one should consider the following ranges of parameters: ε = 0.5 . . . 2.5 GeV, a/d = 10 . . . 20, C = 0.1 . . . 0.3 which are common for all the crystals which were investigated. For these ranges the energy of the first harmonic lies within the interval 50 . . . 150 keV and the length of CU can be taken to be equal to the dechanneling length L = L d (C). Illustrative results of calculations are presented in Fig. 7.3 where the dependences of the undulator period λu , the number of undulator periods within the dechanneling length Nd = L d (C)/λu , the first harmonic energy and the ratio G(L)/n ch,0 versus the bending parameter C are presented for 0.5 GeV positrons channeling in several channels as indicated. The data correspond to the ratio a/d = 20 except for the case of Si for which a/d = 10. The dechanneling length L d (C) was calculated within the model described by Eqs. (6.7) and (6.8). For each crystal the curves ω and G(L)/n ch,0 were truncated at those C values for which the number of undulator periods becomes less than 10 (see graph (a)). For high-energy photons the emission stimulation must occur during a single pass of the bunch of particles through the crystal—the SASE regime [11–13]. Indeed, for such photon energies there are no mirrors, and, therefore, the photon flux must develop simultaneously with the bunch propagation. In the theory of conventional FEL the SASE regime usually implies that the FEL operates with high gain G(L) > 1, which ensures that the exponential factor in (7.5) is large. From graph (c) it is seen that for a CU G(L)/n ch,0 is a rapidly varying function of C, which attains the maximum value ≈10−21 cm3 at C ≈ 0.1. The maximum value allows one to estimate the magnitude of the volume density n ch,0 of a positron bunch needed to achieve the total gain G(L) = 1 over the length L = L d (C).2 Then it follows from the graph that 2

Equation (7.6) was derived within the framework of the low-gain approximation (see, e.g., [20]), i.e. it implies that G(L)  1. Therefore, the mentioned estimate of n ch,0 is a crude one since it extends (7.6) beyond its range of applicability.

186

7 Emission of Coherent CU Radiation -21

10

(a)

(c)

3

G(ω)/n (cm )

25

Nd

20 15 10

-22

10 5

(b)

(d)

120 100

hω1 (keV)

20

80

10

0 0

_

λu (microns)

30

60 0.1

0.2

0.3

0.4

Parameter C

40 0

0.1

0.2

0.3

0.4

Parameter C

Fig. 7.3 Number of undulator periods within the dechanneling length Nd = L d (C)/λu , (a); undulator period λu , (b); the ratio G(L)/n ch,0 , (c); the first harmonic energy ω1 , (d) as functions of the bending parameter C for ε = 0.5 GeV positron channeling in Si (111) (curves with no symbols), Ge (111) (curves with open circles), W (110) (curves with filled circles) [14, 18]

to achieve the emission stimulation within the range ω = 50 . . . 150 keV on the basis of the SASE mechanism it is necessary to operate with very dense, n ch,0  1021 cm−3 positron bunches of the energy within the GeV range. However, it looks like such high values of positron beam densities can be achieved. To support this hope of ours we present the quotation from recent review by U. Uggerhøj of the achievements and prospects of experiments with relativistic particles interacting with strong crystalline fields [21]. On p. 1160 (left column) of the review, where the author comments on the perspective of investigation of the stimulated emission by means of CU, one finds the following commentary: This would require positron densities of the order 1021 cm−3 …, only about two orders of magnitude less than the electron density in a typical metal. Although at first sight this might seem far-fetched, (electron) densities of the order of 1021 cm−3 are actually available at the final focus test beam at SLAC and theoretical schemes to increase this by a factor of 30 have been devised (Emma et al., 2001). Furthermore, it is known from experimental tests that a diamond crystal bears no visible influence from being irradiated by the final focus test beam, whereas amorphous aluminum simply evaporates …

The paper cited in this quotation is Ref. [22].

7.2 “Naive” Approach to the Gamma-Laser Based on a Crystalline Undulator

187

7.2.2 One-Crystal Amplifier In Ref. [23] the estimation of the gain factor G(L) and, on its basis, of the volume densities n ch,0 of a positron beam propagating through a CU has been carried out within the framework of classical electrodynamics. The latter is applicable in the limit ω  ε. The classical approach [24] has been widely used for the description of conventional FEL lasers (see, e.g., [4, 20]) along with the quantum theory (see, e.g., [25]). The lasing effect in a CU takes place if the positions of the channeling particles are correlated ensuring the coherent emission of electromagnetic waves, see Fig. 7.2. In a conventional FEL, this is accomplished by a spatial modulation (termed usually as “bunching” [4, 5] or “microbunching” [8]) of the particle density along the beam with the period equal to the wavelength of the emitted radiation. To obtain such a modulation, initial (or, seed) radiation from an external source is needed. For this purpose, the spontaneous emission of charged particles either in a CU or in the field of an infrared laser wave can be used. In both cases, the initial radiation has to be well collimated to ensure sufficient monochromaticity and coherence. Under certain conditions, revealed in [23], the seed radiation modulates the density of the particles channeling in a PBCr. The resulting bunched beam produces additional radiation of the same wavelength, thus amplifying the initial radiation. The amplifier, based on a single CU, “a one-crystal gamma ray amplifier”, is shown in Fig. 7.4. A beam of positrons, aligned with the initial radiation, enters a PBCr of the length L along the direction tangential to the bent crystallographic planes. Being captured into the channeling mode, the particles propagate through the crystal following the shape of PBCh. Let us discuss the conditions which ensure that due to the channeling particles the radiation becomes amplified when exiting the crystal [23]. Assuming the beam propagates along the z axis, we consider the following harmonic shape of the periodic bending in the (yz) plane: (7.8) y = a cos(ku z), where ku = 2π/λu .

beam

amplified radiation

initial radiation

Fig. 7.4 A scheme of the one-crystal amplifier [23]. A positron beam and the initial radiation enter a periodically bent crystal. The particles follow the shape of the crystallographic planes and move along nearly sinusoidal trajectories. The radiation is amplified due to its interaction with the beam in the crystal. [14, 23]

188

7 Emission of Coherent CU Radiation

The condition a  d allows one to neglect the channeling oscillations and to assume that the particle follows the trajectory (7.8) under the action of the interplanar field. Then, to characterize the position of the particle one can use the length s measured along the trajectory. The conjugate momentum ps = 

m s˙ 1 − s˙ 2 /c2

1/2 ,

(7.9)

is tangential to the trajectory in the point s. The evolution of ps , which is due to the radiation field of intensity E, is described by the following EM p˙s = e E · n ,

(7.10)

where n is the unit tangent vector. The amplified radiation is sought in the form of a plane wave linearly polarized along the y direction, so that E = (0, E, 0) with E = E 0 cos(kz − ωt + φ) ,

(7.11)

where k = 2π/λ = ω/c with λ being the wavelength of radiation, and φ is an arbitrary phaseshift. For gamma- or/and hard X-rays, the attenuation length exceeds the dechanneling lengths of GeV positrons by at least an order of magnitude (see Figs. 4.3 and 4.4 in [14]). Therefore, for a crystal length L ∼ L d the attenuation of radiation can be neglected. Using (7.11) in (7.10) and accounting for a  λu , one derives 

a 2 ku (k + ku ) eE 0 aku sin ψ + sin(2κku s) p˙s = − 2 8

 a 2 ku (k − ku ) sin(2κku s) − 2κku s , − sin ψ + 8 where ψ = (k + ku )κs − ωt + φ,

κ =1−

(aku )2 . 4

(7.12)

(7.13)

A gradual energy transfer from the particle to the electromagnetic wave occurs when the ponderomotive phase ψ stays nearly constant [1, 23]. If otherwise, the first sine on the right-hand side of (7.12) oscillates, thus averaging out the energy exchange. The ponderomotive phase is constant provided the following resonant condition is fulfilled: (7.14) (k + ku )κ s˙ − ω = 0. Using the second equation from (7.13) and writing s˙ in terms of the Lorentz factor of the ultra-relativistic projectile, s˙ = c(1 − γ −2 )1/2 ≈ c(1 − γ −2 /2), one reduces (7.14) to the following relation between γ and k:

7.2 “Naive” Approach to the Gamma-Laser Based on a Crystalline Undulator

189

 γ =

2k/ku ≡ γr . 4 − a 2 kku

(7.15)

Resolving this equation for k one finds3 In what follows, for the sake of simplicity, we consider the emission in the first harmonic only. k=

4γr2 ku , 2 + K2

(7.16)

where K = 2π γr a/λu is the undulator parameter. Equation (7.16) suggests that k  ku in the vicinity of the resonance (7.14). Therefore, setting k ± ku ≈ k, one expands the right-hand side of (7.12) in the Fourier series and omits the oscillating terms. The result reads p˙s = −

eE 0 aku J (η) sin ψ . 2

(7.17)

The function J (η) and the argument η are defined in (7.7). Differentiating (7.9) and carrying out the second derivative in time of ψ from (7.13), one compares the results and, accounting for s ≈ ct, derives the pendulum equation which describes the evolution of the phase ψ: d2 ψ = −Ω 2 sin(ψ) ds 2

(7.18)

where the oscillation frequency of the corresponding simple pendulum is given by Ω=

eE 0 k K J (η) 2mc2 γr4

1/2 .

(7.19)

The derivative dψ/ds is related to the deviation of the particle energy ε from its resonance value εr = γr mc2 : dψ 4ku ε − εr ≡ ζ (s) = ds 2 + K 2 εr

(7.20)

Following Ref. [23], we apply Eq. (7.18) to the emission stimulation in CU in the limit of small gain and small signal. The former implies that the change in the electromagnetic wave intensity is much less than its value E 0 at the entrance. As a result, the intensity and the frequency Ω are approximately constant along

3

In accordance with the general theory of FEL (e.g., [4]) the emission stimulation in an undulator occurs only at the frequencies corresponding to the harmonics of the spontaneous undulator radiation. As it is written, Eq. (7.16) defines the wavenumber of the fundamental harmonic k = ω1 /c.

190

7 Emission of Coherent CU Radiation

the undulator. Within the small signal limit, the intensity of E 0 is considered to be low enough to ensure the condition Ω L  1. In this case the iterative solution of Eq. (7.18) yields ⎧

sin ψ0 cos ψ0 ⎪ 2 sin(ψ0 + ζ0 s) ⎪ − − ⎨ ψ(s) ≈ ψ0 + ζ0 s + (Ωs) (ζ0 s)2 (ζ0 s)2 ζ0 s  2 Ω ⎪ ⎪ ⎩ ζ (s) ≈ ζ0 + cos(ψ0 + ζ0 s) − cos ψ0 ζ0

(7.21)

where ψ0 = ψ(0) and ζ0 = dψ(0)/ds denote the quantities at the undulator entrance. In terms of classical electrodynamics, the gain factor G(L) characterizes the relative increase in the energy of the electromagnetic wave over the undulator length: G(L) =

ΔE , E0

(7.22)

where E0 = E 02 /8π is the radiation energy density at the entrance, and ΔE = E (L) − E0 is the increase in the energy density (E (L) is the density at the exit from CU). The energy conservation implies that the radiated energy equals the decrease in the energy of the channeling particles due to the radiative losses. Therefore, to calculate G(L) one can analyze the radiative energy losses. The energy density of the channeling particles, Ech (s), in the point s can be written as [23] (7.23) Ech (s) = ε(s) n ch (s). Here n ch (s) stands for the volume density of channeling particles and ε(s) denotes the average energy of a particle at the distance s. The averaging procedure assumes that there is no correlation in the instants of entry into the crystal for different particles of the beam. Thus, it is implied that the particles are randomly distributed with respect to the phase ψ0 . According to Eq. (7.12), the interaction of a particle with the radiation field depends on ψ0 . Therefore, to obtain ε(s) one averages the energy  2π ε(s) = mc2 γ (s) with respect to ψ0 : ε(s) = (2π )−1 0 ε(s)dψ0 . The dechanneling effect leads to a decrease in the volume density of channeling particles n ch (s) with the penetration distance s. For the estimation purposes, this dependence can be modeled by the exponential decay law n ch (s) = A(C) n ch, 0 exp(−s/L d (C)), where n ch, 0 is the beam density at the entrance, L d (C) is the dechanneling length, A(C) stands for the channel acceptance and C =

is the bending parameter. aku2 ε/Umax From (7.23), it follows that the change dEch over the interval ds contains two terms. One of these, proportional to dn ch (s)/ds, is due to the dechanneling process. The second term, proportional to dε(s) /ds, describes the radiative losses, and, thus, it is responsible for the change dE of the electromagnetic field energy. Therefore, E satisfies the equation dE /ds = −mc2 n ch (s)(dγ /ds), where the derivative of γ is calculated using (7.18) and (7.20). Then, integrating the equation, one obtains

7.2 “Naive” Approach to the Gamma-Laser Based on a Crystalline Undulator



mc2 γr3 2 Ω ΔE = k

L

  n ch (s) sin ψ(s) ds .

191

(7.24)

0

Using (7.21) and taking into account Ωs ≤ Ω L  1, one   carries out the averaging over ψ0 : sin(ψ) = Ω 2 (2ζ02 )−1 sin(ζ0 s) − ζ0 s cos(ζ0 s) . Substituting this relation into (7.24) and recalling (7.22), one derives [23]: G(L , ζ0 ) = 8π

r0 ku ηJ 2 (η) γr3 ζ02



L

  n ch (s) sin(ζ0 s) − ζ0 s cos(ζ0 s) ds.

(7.25)

0

The second argument in G(L , ζ0 ) indicates the dependence of the gain factor on ζ0 . In the limit of a short crystal, L  L d , the dechanneling can be neglected, n ch (s) ≈ An ch, 0 . Then, the right-hand side of (7.25) reduces to a well-known formula for the gain factor of the conventional FEL obtained within the small signal and small gain approximation (see, e.g., [4]). In reality, the dechanneling cannot be neglected. Indeed, the dechanneling length of ultra-relativistic positrons does not exceed several millimeters for ε within the GeV range [14]. Therefore, the limit of a long crystal, L  L d , is of a great interest for an amplifier based on a CU. Assuming L  L d = L d (C) in (7.25), one extends the integration over the infinite interval. The analytical result reads G(L , ζ0 )| LL d = 16πr0 ku

L 3d (C) w A(C) n ch, 0 ηJ 2 (η) , γr3 (1 + w2 )2

(7.26)

√ where w√= ζ0 L d (C). The factor w/(1 + w2 )2 attains its maximum of 3 3/16 at w√ = 1/ 3. Therefore, the maximum of G with respect to ζ0 is reached for ζ0 = 1/ 3L d (C), and it does not depend on the crystal length [23]: √ L 3 (C) G ≡ G(L , ζ0 )| LL d , ζ0 =(√3L d )−1 = 3 3πr0 ku A(C) n ch, 0 d 3 ηJ 2 (η) . γr (7.27) Up to a numeric factor ∼1 and the factor A, this formula coincides with expression (7.6) obtained in [10, 17, 18] within the quantum approach. The difference is due to additional approximations in the course of evaluation of (7.27) and (7.6). Expressing ku via C and undulator parameter K ku =

C Umax , K mc2

(7.28)

and using explicit dependences of L d (C) and A(C) on the bending parameter C, one re-writes (7.27) as follows [23]

192

7 Emission of Coherent CU Radiation

√ U G = 3 3πr0 n ch, 0 max mc2



L d (0) γr

3

  C(1 − C)7 K −1 ηJ 2 (η)

η=K 2 /(4+2K 2 )

.

(7.29) The ratio L d (0)/γr weakly (logarithmically) depends on the beam energy. Therefore, it can be considered constant for a given crystal. Then, G allows further optimization in terms of two independent variables, C and K . The factor C(1 − C)7 reaches the maximum value of ≈ 0.05 at C = 1/8. The maximum of K −1 ηJ 2 (η) with respect to the undulator parameter is ≈ 0.15 at K ≈ 1.2. These optimum values of C and K ensure the maximum gain [23]: G max = r0

Umax L 3d (0) n ch, 0 . mc2 γr3

(7.30)

This gain corresponds to the optimal value λopt u ≈ 60

mc2

Umax

(7.31)

of the undulator period, which is found from (7.28) for C = 1/8 and K ≈ 1.2. The optimal relativistic factor γr is found from K = γr aku and (7.28), the corresponding energy of emitted photons ω one deduces from (7.16):

, γr ≈ 11.5mc2 aUmax

ω ≈ 16

mc3 .

a 2 Umax

(7.32)

It is seen that the radiation energy and the optimal relativistic factor are not fixed by the choice of the optimum values of the parameters K and C, but also depend on the bending amplitude a. Thus, the variations if a under the constraints K = const and C = const do not destroy the optimum regime. opt The values of λu and the related parameters for different crystals and positron beam are shown in Table 7.1. The optimum values of the beam energy ε and the photon energy ω depend on the ratio χ = d/a, which is of the order of 0.1. For each channel the beam density, which correspond to G max = 1, is estimated from (7.30) and presented in the last column in the table. It is seen that extremely high positron densities in the beam are needed to obtain a lasing effect in a simple one-crystal amplifier even in optimized regime. This is consistent with the results obtained within the quantum formalism outlined in Sect. 7.2.1, although in the latter case the values n ch, 0 (G max = 1) are lower (see graph (c) in Fig. 7.3). This discrepancy is due to (a) difference in numerical factors in (7.27) and (7.6), (b) the absence of the

of the interplanar acceptance A(C) in (7.6). Additionally, the maximal gradient Umax potential was differently defined in Sect. 7.2.1 and in the present section. In Table 7.1,

are calculated as the derivative dU/dρ in the point ρ = d/2 − aTF , the values of Umax

= dU/dρ at ρ = d/2 which are larger whereas in Sect. 7.2.1 it is implied that Umax by (approximately) a factor 1.5.

7.2 “Naive” Approach to the Gamma-Laser Based on a Crystalline Undulator

193

Table 7.1 Parameters of the one-crystal amplifier in the optimum regime for different crystals and planes at the temperature T = 4 K [23]. The notation χ stands for the ratio d/a  1. The last column presents the values of the positron beam density needed to achieve G max = 1 Crystal

Plane

C(diamond) (111) C(graphite) (100) Si Si Ge Ge W

(110) (111) (110) (111) (110)

d (Å)

Umax (GeV/cm)

λu (μm)

opt

ε (GeV)

ω (MeV)

n ch, 0 (cm−3 )

1.54

5.16

59.4

37.7χ

132χ 2

1.4 × 1023

10.2χ

165χ 2

5.3 × 1021

31.4χ

89χ 2

1.2 × 1023

20.4χ

47χ 2

4.5 × 1022

13.7χ

37χ 2

7.3 × 1022

9.1χ

20χ 2

2.9 × 1022

3.3χ

8χ 2

2.0 × 1022

3.35 1.92 2.35 2.00 2.45 2.24

8.77 4.98 6.28 10.94 13.55 40.74

35.0 61.6 48.8 28.0 22.6 7.5

The main reason why an appreciable gain cannot be reached at lower densities is that both the beam evolution and the emission radiation take place in one crystal whose length is limited by a few dechanneling lengths. Increasing further the length of a CU will not increase the gain due to is the exponential decay of the density of the channeling particles. However, it has been shown in Ref. [23] that it is possible to achieve reasonable gain at much less dense beams by means of a two-crystal gamma ray amplifier—the gamma klystron.

7.3 Beam Demodulation The radiation emitted in an undulator at the harmonics frequencies is coherent with respect to the number of periods, Nu , but not with respect to the radiating particles since their positions are not correlated. As a result, the intensity of radiation emitted in a certain direction is proportional to Nu2 and to the number of particles, Iinc ∝ Np Nu2 (the subscript “inc” stands for “incoherent”). In conventional undulators, Nu is on the level of 103 . . . 104 [1], therefore, the enhancement due to the squared number of periods is large. This makes undulators a powerful source of spontaneous radiation. However, the incoherence with respect to the number of the radiating particles causes a moderate (linear) increase in the radiated energy with the beam density. More powerful and coherent radiation will be emitted by a beam in which position of the particles is modulated in the longitudinal direction with the period equal to integer multiple to the radiation wavelength λ. In this case, the electromagnetic waves emitted by different particles have approximately the same phase. Therefore, the amplitude of the emitted radiation is a coherent sum of the individual waves, so that the intensity becomes proportional to the number of particles squared, Icoh ∝ Np2 Nu2 [26]. Thus, the increase in the photon yield due to the beam pre-bunching (other terms used are “bunching” [6] or “microbunching” [27]) can reach orders of magnitudes relative to radiation by a non-modulated beam of the same density (see the data on Np in Table 7.2 in 7.5). Following Ref. [28] we use the term “superradiant” to designate the coherent emission by a pre-bunched beam of particles.

194

7 Emission of Coherent CU Radiation

In what follows we assume that the beam is fully modulated at the crystal entrance. The description of the methods of preparation of a pre-bunched beam with the parameters needed to amplify CUR is presented in Appendix (see also Ref. [29] and Sect. 8.5 in Ref. [14]). In a CU, a channeling particle, while moving along the channel centerline, undergoes two other types of motion in the transverse directions with respect to the CU axis z. First, there are channeling oscillations along the y direction perpendicular to the crystallographic planes. Second, the particle moves along the planes (the x direction). To be noted is that different particles have different (i) amplitudes ach of the channeling oscillations, and (ii) momenta px in the (x z) plane due to the distribution in the transverse energy of the beam particles as well as the result of multiple scattering from crystal atoms. Therefore, even if the speed of all particles along their trajectories is the same, the difference in ach or/and in px leads to different values of the velocities with which the particles move along the undulator axis. As a result, the beam loses its modulation while propagating through the crystal. For an unmodulated beam, the CU length L is limited mainly by the dechanneling process. A dechanneled particle does not follow the periodic shape of the channel, and, thus, does not contribute to the CUR spectrum. Hence, it is reasonable to estimate L on the level of several dechanneling lengths L d (see panels (b) in Figs. 6.17 and 6.18). Longer crystals would attenuate rather than produce the radiation. Since the intensity of CUR is proportional to the undulator length squared, the dechanneling length and the attenuation length are the main restricting factors which must be accounted for. For a modulated beam, the intensity is sensitive not only to the shape of the trajectory but also to the relative positions of the particles along the undulator axis. If these positions become random because of the beam demodulation, the coherence of CUR is lost even for the channeled particles. Hence, the demodulation becomes the phenomenon which imposes most restrictions on the parameters of a CU. In Ref. [30] an important quantity—the demodulation length, was introduced. It represents the characteristic scale of the penetration depth at which a modulated beam of channeling particles becomes demodulated. Within the framework of the approach developed in the cited papers the demodulation length L dm is related to the dechanneling length L d (C) in a bent channel: L dm =

L d (C) . √ α(ξ ) + ξ /j0,1

(7.33)

Here j0,1 = 2.4048 . . . is the first root of the Bessel function J0 (x). The dimensionless parameter ξ is expressed in terms of the emitted radiation frequency ω, the dechanneling length L d (C) and Lindhard’s critical angle ΘL (C) in the bent channel: ξ = ωL d (C)ΘL2 (C)/2c (see [31] for the details). The function α(ξ ) is related to the real and imaginary parts of the first root (with respect to ν) of the equation [31]   F(−ν, 1, z)

= √ z=(1+ı) j0,1 ξ /2

0

(7.34)

7.3 Beam Demodulation

195 diamond(110)

0.8

0.6

Ldm/Ld(C)

Ldm/Ld(C)

0.6

0.4

0.2

0 -1 10

Si(110)

0.8

C=0 C=0.1 C=0.2 C=0.3 C=0.4 C=0.5

0.4

0.2

0

1

10

10

0 -1 10

Photon energy (MeV)

0

10

1

10

2

10

Photon energy (MeV)

Fig. 7.5 The ratio L dm /L d (C) as a function of the photon energy for diamond(110) and silicon(110) channels calculated for at different values of bending parameter C. [33]

where F(.) stands for Kummer’s confluent hypergeometric function (see, e.g., [32]). Equations (7.33) and (7.34) can be analyzed numerically to derive the dependence of the demodulation length on the radiation energy ω for a particular crystal channel. The result of such analysis is illustrated by Fig. 7.5 where the dependences of the ratio L dm /L d (C) on the photon energy are presented for the (110) planar channels in diamond and silicon and for several values of the bending parameter C as indicated. To be noted is that for all values of the bending parameter C and over broad energy range of the emitted radiation, the demodulation length is noticeably less than the dechanneling one. To preserve the beam modulation during its channeling in a crystal and, as a result, to maintain the coherence of the radiation the crystal length L must be less than the demodulation length: L  L dm < L d (C) .

(7.35)

It follows from (7.33) that in a periodically bent crystal L dm depends on the crystal type, on the parameters of the channel (its width, strength of the interplanar field), on the bending amplitude and period, on the projectile energy and its type (these are “hidden” in L d (C), C, and ξ ) as well as on the emitted photon energy (enters the parameter ξ ). Therefore, Eq. (7.35) imposes addition restriction on the CU length as compared to the case of the CUR emission by the unmodulated beam. The phase velocity of the modulated beam along the CU channel is also modified as compared to the unmodulated one [31]. This modification changes the resonance condition that links the parameters of the undulator and the wavelength (energy) of the radiation. The expression for the fundamental harmonic frequency ω ≡ ω1 acquires the following form (compare with Eq. (6.5)): ω=

2π c 2γ 2 2 2 1 + K /2 + Δβ /2 λu

where the additional term in the denominator is given by

(7.36)

196

7 Emission of Coherent CU Radiation

Δ2β = 4γ 2 Θ L2 (C) β(ξ ) +

1

√

(7.37)

2 j0,1 ξ

with β(ξ ) being another function related to the real and imaginary parts of the first root of Eq. (7.34) (details can be found in Refs. [14, 31]). The quantity ξ = ωL d (C)ΘL2 (C)/2c depends on ω. Therefore, Eq. (7.36) represents a transcendent equation which relates ω to the projectile energy and to the bending amplitude and period. Analysis of the formulae written above shows that for given values of ω and ε all other quantities which characterize the CU and the demodulation process can be expressed in terms of a single independent variable, for example, the bending amplitude a. Then, scanning over the a values it is possible to determine the whole set of the parameters (these include a, λu , C, L dm (C)) which maximize the peak brilliance of the superradiant emission (see Sect. 7.4). Figure 7.6 shows the results of calculations of the parameters of the diamond(110)based CU which maximize the peak brilliance of the radiation of energy ω emitted by the FACET-II positron beam, see Table 7.2. The dependences presented correspond

λu (microns)

200 2

a/d

10

100

1

10

0

0

10 0.6

20

C

Ndm

0.4 0.2

10

Ldm(C) (cm)

0.0 10 -1

K

10

5

-2

10

-1

10

0

10

1

10

Photon energy (MeV)

0 -1 10

0

10

1

10

Photon energy (MeV)

Fig. 7.6 Parameters of the diamond(110)-based CU,—λu , a (measured in units of the interplanar distance d = 1.26 Å), C, the undulator parameter K = 2π γ a/λu , the demodulation length L dm (C) and the number of periods within L dm , Ndm = L dm /λu that ensure the highest peak brilliance for fully modulated FACET-II positron beam

7.3 Beam Demodulation

197

Table 7.2 Parameters of positron (“p”) and electron (“e”) beams: beam energy, ε, bunch length, L b , number of particles per bunch, N , beam size, σx,y , beam divergence φx,y , volume density n = N /(π σx σ y L b ) of particles in the bunch, peak current Imax = eN c/L b . In the cells with no explicit reference to either “e” or “p” the data refer to both modalities Facility

VEPP4M

BEPCII

DANE

SuperKEKB SuperB

FACET-II

CEPC

References

[35]

[35]

[35]

[35]

[36, 37]

[38]

[39]

ε (GeV)

6

1.9–2.3

0.51

10

45.5 8

N (units 1010 )

15

3.8

L b (cm)

5

1.2

p: 4

p: 6.7

e: 7

e: 4.2

p: 2.1

p: 9.04

p: 6.5

p: 0.375

e: 3.2

e: 6.53

e: 5.1

e: 0.438

1–2

p: 0.6

0.5

e: 0.5 σx (μm) σ y (μm)

1000 30

347 4.5

260 4.8

p: 10

8

p: 10.1

e: 11

8

e: 5.5

p: 0.048

0.04

e: 0.062 φx (mrad) φ y (mrad) Imax (A)

0.2 0.67 144

n 3.2 (1013 cm−3 )

0.35 0.35 152

65

1 0.54

p: 0.00076

0.85

e: 0.00011

p: 7.3

6 0.04

e: 5.9

p: 0.32

p: 0.250

p: 0.178

e: 0.42

e: 0.313

e: 0.073

p: 0.18

p: 0.125

p: 0.044

e: 0.21

e: 0.150

e: 0.044

p: 50-100

p: 723

p: 624

p: 12.1×103

e: 77-154

e: 627

e: 490

e: 75.5×103

p: 54

p:1.0 × 106 p:1.3 × 106 p: 2 × 105

e: 82

e:0.6 × 106 e:1.0 × 106 e: 3.9 × 106

0.03 0.04 452

1.25 × 106

to the emission in the fundamental harmonic. The crystal thickness was set to the demodulation length L = L dm (C), graph (e). The quantity Ndm stands for the number of undulator periods within the demodulation length, Ndm = L dm (C)/λu . Only the data corresponding to Ndm ≥ 10 are shown in the panels. The dependences presented refer to the Large-Amplitude regime of the periodic bending, which implies that the amplitude a exceeds the interplanar distance d. Noticing that the factor 2π/λu can be written in terms of the undulator parameter K = 2π γ a/λu , one writes Eq. (7.36) as a quadratic equation with respect to K . Resolving it one finds that K is a two-valued function of ω, which is reflected in graph (f). As a result, all dependences presented contain two branches related to the smaller (black curves) and larger (blue curves) allowed values of K .

198

7 Emission of Coherent CU Radiation

7.4 Pre-bunching and Super-Radiance in CU Powerful superradiant emission by ultra-relativistic particles channeled can be achieved if the probability density of the particles in the beam is (uniformly) modulated in the longitudinal direction with the period equal to integer multiple to the wavelength λ of the emitted radiation [28]. To prevent the demodulation of the beam as it propagates through the crystal, the crystal length L must satisfy condition (7.35). In a wide range of photon energies, starting with ω ∼ 102 keV, the demodulation length is noticeably less than the dechanneling length L d . In addition to this, in this energy range the photon attenuation length L a in silicon and diamond greatly exceeds the dechanneling length of positrons with energies up to several tens of GeV [14]. Therefore, on the spatial scale of L dm the dechanneling and the photon attenuation effects can be disregarded. In what follows, we carry out quantitative estimates of the characteristics of the superradiant CU radiation (CUR) emitted by a fully modulated positron beam channeled in periodically bent diamond and silicon (110) oriented crystals in the absence of the dechanneling and the photon attenuation. The beam represents a train of bunches each of the length L b containing N particles. The crystal length (along the beam direction) is set to the demodulation length, L = L dm . The transverse sizes of a crystal are assumed to be larger than those of the beam, i.e., than σx,y . For the sake of clarity, below we consider the emission in the first harmonics of CUR, see Eq. (7.36). Final width Δω of the CUR peak introduces a time interval τcoh = 1/Δω within which two particles separated in space can emit coherent waves. Hence, one can introduce a coherence length [1] L coh = cτcoh =

λ ω 2π Δω

(7.38)

where λ is the radiation wavelength, and the band-width (BW) Δω/ω ≈ 1/Ndm with Ndm = L dm /λu standing for the number of periods within the demodulation length. The number of the particles from the bunch which emit coherently is calculated as Ncoh =

L coh N . Lb

(7.39)

2 . The number of such sub-bunches Their radiated energy is proportional to Ncoh is L b /L coh , therefore, the energy emitted by the whole bunch contains the factor 2 = N Ncoh . (L b /L coh )Ncoh Another important quantity to be estimated is the solid angle ΔΩcoh within which the waves emitted by the particles of the sub-bunch are coherent. This angle can be chosen as the minimum value from the natural emission cone of the first harmonics ΔΩ = 2π λu /L dm and the angle ΔΩ⊥ which ensures transverse coherence of the emission due to the finite sizes σx,y of the bunch. Assuming the elliptic form for

7.4 Pre-bunching and Super-Radiance in CU

199

the bunch cross section one derives ΔΩ⊥ ≤ λ2 /4π σx σ y . Therefore, the solid angle ΔΩcoh is found from   ΔΩcoh = min ΔΩ⊥ , ΔΩ . (7.40) The number of photons ΔNω emitted by the bunch particles one obtains multiplying the spectral-angular distribution of the energy emitted by a single particle by the factor N Ncoh ΔΩcoh (Δω/ω). The result reads: ΔNω = 4π α N Ncoh ζ [J0 (ζ ) − J1 (ζ )]2 Ndm

ΔΩcoh Δω . ΔΩ ω

(7.41)

where ζ = (K 2 + Δ2β )/2(2 + K 2 + Δ2β ) with Δ2β defined in (7.37). The number of photons emitted by the particles of the unmodulated beam in a CU of the same length and number of periods one calculates from Eq. (6.4) written for n = 1 by setting Neff = Ndm , substituting K 2 → K 2 + Δ2β and multiplying the right-hand side by N . Comparing the result with Eq. (7.41) one notices that the enhancement factor due to the coherence effect is Ncoh ΔΩcoh /ΔΩ.  Another quantity of interest is the flux Fω of photons. Measured in the units of photons/s/0.1%BW , it is related to ΔNω as follows: Fω =

ΔNω 1 103 (Δω/ω) Δtb

(7.42)

where Δtb = L b /c = eN /Imax is the time flight of the bunch and Imax stands for the peak current. The peak brilliance, Bpeak , of the superradiant CUR one obtains substituting ΔNω from (7.41) into Eq. (1) in the main text and using there peak current Imax instead of I . Figure 7.7 shows peak brilliance of radiation formed in the diamond(110)-based CU as functions of the first harmonic energy. Four graphs correspond to the positron beams (as indicated) the parameters of which are listed in Table 7.2. In each graph, the dashed line refers to the emission of the spontaneous CUR formed in the undulator with optimal parameters, see Fig. 6.18. The thick curves present the peak brilliance of the superradiant CUR maximized by the proper choice of the bending amplitude and period (as described in Sect. 7.3). Two branches of this dependence, seen in graphs (a)–(c), are due to the two-valued character of the dependence of undulator parameter K on the radiation frequency ω. For the CEPC beam, graph (d), this peculiarity manifests itself in the frequency domain beyond 40 MeV, therefore it is not seen in the graph.

200

7 Emission of Coherent CU Radiation

(a) FACET-II

27

2

2

Peak brilliance (ph/s/mm /mrad /0.1%BW)

10

(b) SuperKEKB

29

10

28

10 26

10

27

10

26

10

25

10

25

10 24

10

24

10

30

(c) SuperB

10

33

10

(d) CEPC

32

29

10

28

10

10

31

10

30

10

27

10

29

10

26

10

28

10

25

27

10

10

24

26

10

0.1

10 1

10

Photon energy (MeV)

0.1

1

10

Photon energy (MeV)

Fig. 7.7 Peak brilliance of superradiant CUR (thick curves) and spontaneous CUR (dashed curves) emitted in periodically bent diamond(110) crystal. The graphs refer to four positron beams

7.5 Brilliance of the CU-Based LSs Quantitative analysis and numerical data on the parameters of a CU which maximize the brilliance of CUR in presence of the demodulation process is presented in Sect. 7.4. These data have been used to calculate the peak brilliance of the superradiant CUR. Figure 7.8 illustrates a boost in peak brilliance due to the beam modulation. Thick curves correspond to superradiant CUR calculated for fully modulated positron beams (as indicated) propagating in the channeling mode through diamond(110)based CU. In the photon energy range 10−1 . . . 101 MeV the brilliance of superradiant CUR by orders of magnitudes (up to 8 orders in the case of CEPC) exceeds that of the spontaneous CUR (dash-dotted curves) emitted by the random beams. Remarkable feature is that the superradiant CUR brilliance can not only be much higher that the spontaneous emission from the state-of-the-art magnetic undulator (see the curves for the TESLA undulator) but also be comparable with the values achievable at the XFEL facilities (LCLC (Stanford) and TESLA SASE FEL) which operate in much lower photon energy range.

Appendix 1: Beam Parameters

201

TESLA SASE FEL

32

10

2

Peak brilliance (ph/s mrad mm 0.1%BW)

34

10

LCLS

30

10

2

TESLA undulator

28

10

CEPC

26

10

SuperB SPring8

24

10

FACET

ESRF SuperKEKB

PETRA 22

10

-2

10

-1

10

0

10

1

10

2

10

Photon energy (MeV) Fig. 7.8 Peak brilliance of superradiant CUR (thick solid curves) and spontaneous CUR (dashed lines) from diamond(110) CUs calculated for the SuperKEKB, SuperB, FACET-II, and CEPC positron beams versus modern synchrotrons, undulators, and XFELs. The data on the latter are taken from Ref. [34]

Appendix 1: Beam Parameters The data on positron and electron beams energy ε, bunch length L b , number of particles per bunch N , beam sizes σx,y and divergences φx,y (the subscripts x, y refer to the horizontal and vertical dimensions, respectively), volume density n, and peak current Imax are summarized in Table 7.2. The table compiles the data for the following facilities: VEPP4M (Russia), BEPCII (China), DANE (Italy), SuperKEKB (Japan) [35], SLAC (the FACET-II beams, Ref. [38]), SuperB (Italy) [36], and CEPC (China) [39]. Note that the SuperB data are absent in the latest review by Particle Data Group [35] since its construction was canceled [37].

Appendix 2: Beam Pre-bunching for a Super-Radiant CUR In this section, we describe a scheme which may provide a possibility to prepare a pre-bunched beam with the parameters needed to amplify CUR of a wavelength in the range of hard X- and gamma-rays, λ ∼ 10−2 . . . 10−1 Å (see also Ref. [29] and Sect. 8.5 in Ref. [14]).

202

7 Emission of Coherent CU Radiation

Positron beam

Crystalline Undulator

λ

XFEL undulator

XFEL radiation, λXFEL

Super-radiant radiation

Fig. 7.9 A scheme for pre-bunching a beam to achieve superradiant CUR. The initial positron (or electron) beam with layered energy distribution (see Fig. 7.11) propagates in the conventional XFEL emitting electromagnetic radiation of the wavelength λXFEL . Under the joint action of the XFEL magnetic field and that of the radiation, the beam becomes spatially modulated at the entrance to CU, i.e. it becomes a train of microbunches each of which is modulated further with shorter period, see Fig. 7.10. The output coherent radiation of wavelength λ  λXFEL is due to the channeling of the modulated beam in the CU [29]. The figure’s layout is from Refs. [14, 29]

The scheme is illustrated by Fig. 7.9 [29]. It combines two basic elements, a CU and an XFEL. The electromagnetic radiation from the CU is tuned to the desired wavelength λ. The average energy ε of the beam particles as well as the parameters of the CU, which include the period λu , the amplitude a of the periodic bending and the crystal length, satisfy the general conditions but with additional restriction imposed on the crystal length L which must be equal or less than the demodulation length L dm (C) corresponding to the bending parameter C (see Sect. 7.3). The conventional XFEL, indicated in the scheme, is supposed to operate in the high-gain regime (see, e.g., Refs. [1, 41]). Its radiation wavelength λXFEL must satisfy the following conditions: λXFEL /λ ≈ M  1

(7.43)

with M standing for an integer. The operational principle of the device is as follows [29]. In the scheme Fig. 7.9, the conventional XFEL is used to produce a spatially modulated particle beam. In accordance with the general FEL theory (see, e.g., [1, 20, 41]) the initially unmodulated beam, while traveling through the undulator, becomes modulated (micro-bunched) with the period λXFEL in the longitudinal direction. This occurs under the joint action of the undulator field and that of the emitted radiation. In the high-gain regime, the longitudinal modulation at sufficiently large penetration distances still has the periodicity of the wavelength λXFEL but it is no longer a simple harmonic distribution. Thus, its Fourier decomposition contains higher harmonics λXFEL /m where m > 1 are integers (see, e.g., [1, 41–43]). The amplitudes of these harmonics oscillate along the FEL undulator axis. For any harmonic one

Denstity of beam particles (arb.u.)

Appendix 2: Beam Pre-bunching for a Super-Radiant CUR

203

1/M

0

Distance (in units of λXFEL)

1

Fig. 7.10 Sequence of microbunches additionally modulated with a shorter period λ ≈ λFEL /M (see Eq. (7.43)) at the entrance to the CU in the scheme shown in Fig. 7.9 [29]. The figure’s layout is from Refs. [14, 29]

can determine a spatial point (or, points) where its amplitude reaches the maximum value. This is valid, in particular, for the harmonic in which number M is defined by Eq. (7.43). In this point each micro-bunch of the beam is modulated further with a shorter period λ ≈ λFEL /M, see Fig. 7.10. Hence, placing the entrance of the CU at this point one achieves the coherent emission of radiation by the particles of the micro-bunch as they propagate through the crystal. It is noted in Ref. [29] that it is feasible to further improve the coherence of the emission from CUL. To this end, the beam entering the XFEL must be arranged in layers with respect to the energy of its particles. An example of such a layered energy distribution dn(ε)/dε is represented in Fig. 7.11 by a solid curve. In contrast to the Gaussian energy distribution (the dashed curve), the layered distribution, being centered around ε = ε0 , contains sets of decreasing maxima at |ε − ε0 | > 0 separated with the minima where dn(ε)/dε = 0. While propagating through the XFEL undulator, the bunch with N energy layers is converted into longitudinal spatial modulation with a short period l < λXFEL /N [40]. Thus, the parameters of the initial layered energy distribution can be adjusted so that the period l becomes approximately equal to λXFEL /M. In this case the shortperiod modulation of the micro-bunch corresponding to the Mth harmonic, Fig. 7.10, will become more pronounced leading to the enhancement of the coherent emission. To create the layered energy distribution, the scheme presented in Fig. 7.12 can be implemented [29]. Due to the difference in energies ε, the particles of the initial Gaussian beam are spatially separated after passing through the first bending magnet. The second magnet directs the particles to a lattice with N slits which absorbs the particles having energies in particular intervals Δε and passes through other particles. The last two magnets collect the passed particles into a single beam distributed in N energy layers.

7 Emission of Coherent CU Radiation

Distribution in energy, dn(ε)/dε (arb. u.)

204

ε0 Energy (arb. u.)

Fig. 7.11 The Gaussian energy distribution (dashed curve) and the layered energy distribution (solid curve) of the beam particles. Both distributions are centered around the mean value ε0 of the beam energy [29]. The figure’s layout is from Refs. [14, 29]

BM

BM

BM

BM

1

5 2

4

3 Fig. 7.12 The particles of the beam (1) with Gaussian energy distribution are (i) spatially separated (2) by bending magnets (BM), and (ii) directed to the lattice with N slits (3). The particles (4) which pass through the slits are collected into the output beam (5) with the layered energy distribution [29, 40]. The figure’s layout is from Refs. [14, 29]

References

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26. Bessonov, E.G.: Theory of parametric free-electron lasers. Sov. J. Quantum Electron. 16, 1056 (1986) 27. Bostedt, Ch., Boutet, S., Fritz, D.M., Huang, Z., Lee, H.J., Lemke, H.T., Robert, A., Schlotter, W.F., Turner, J.J., Williams, G.J.: Linac coherent light source: the first five years. Rev. Mod. Phys. 88, 015007 (2016) 28. Gover, A., Friedman, A., Emma, C., Sudar, N., Musumeci, P., Pellegrini, C.: Superradiant and stimulated-superradiant emission of bunched electorn beams. Rev. Mod. Phys. 91, 035003 (2019) 29. Greiner, W., Korol, A.V., Kostyuk, A., Solov’yov, A.V.: Vorrichtung und Verfahren zur Erzeugung electromagnetischer Strahlung. Application for German Patent, June 14, Ref.: 10 2010 023 632.2 (2010) 30. Kostyuk, A., Korol, A.V., Solov’yov, A.V., Greiner, W.: Stable propagation of a modulated particle beam in a bent crystal channel. J. Phys. B At. Mol. Opt. Phys. 43, 151001 (2010) 31. Kostyuk, A., Korol, A.V., Solov’yov, A.V., Greiner, W.: Demodulation of a positron beam in a bent crystal channel. Nucl. Instrum. Method B 269, 1482–1492 (2011) 32. Abramowitz, M., Stegun, I.E.: Handbook of Mathematical Functions. Dover, New York (1964) 33. Korol, A.V., Solov’yov, A.V.: Crystal-based intensive gamma-ray light sources. Europ. Phys. J. D 74, 201 (2020) 34. Ayvazyan, V., Baboi, N., Bohnet, I., Brinkmann, R., Castellano, M., et al.: A new powerful source for coherent VUV radiation: demonstration of exponential growth and saturation at the TTF free-electron laser. Europ. Phys. J. D 20, 149 (2002) 35. Tanabashi, M., et al.: (Particle Data Group): Review of particle physics. Phys. Rev. D 98, 030001 (2018) 36. Nakamura, K., et al.: (Particle Data Group): Review of particle physics. J. Phys. G: Nucl. Part. Phys. 37, 075021 (2010) 37. Banks, M.: Italy cancelse1bn SuperB collider. Physics World (2012) https://physicsworld. com/a/italy-cancels-1bn-superb-collider/ 38. SLAC Site Office: Preliminary Conceptual Design Report for the FACET-II Project at SLAC National Accelerator Laboratory. Report SLAC-R-1067, SLAC (2015) 39. The CEPC Study Group: CEPC Conceptual Design Report (2018). Preprint at arXiv:1809.00285 40. Bessonov, E.G.: Methods of electron beam bunching. Nucl. Instrum. Method A 528, 511–515 (2004) 41. Huang, Zh., Kim, K-J.: Review of X-ray free-electron laser theory. Phys. Rev. ST Accel. Beams 10, 034801 (2007) 42. Tanikawa, T., Lambert, G., Hara, T., Labat, M., Tanaka, Y., Yabashi, M., Chubar, O., Courpie, M.E.: Nonlinear harmonic generation in a free-electron laser with high harmonic radiation. Europ. Phys. Lett. 94, 34001 (2011) 43. Tremaine, A., Wang, X.J., Babzien, M., Ben-Zvi, I., Cornacchia, M., Murokh, A., Nuhn, H.-D., Malone, R., Pellegrini, C., Reiche, S., Rosenzweig, J., Skaritka, J., Yakimenko, V.: Fundamental and harmonic microbunching in a high-gain self-amplified spontaneous-emission free-electron laser. Phys. Rev. E 66, 036503 (2002)

Chapter 8

Conclusion

Construction of novel Crystal-Based LSs is a challenging technological task, which requires a highly interdisciplinary approach combining theoretical analysis and computational modeling, development of technologies for crystalline sample preparation (engaging material science, nanotechnology, acoustics, solid-state physics) together with a detailed experimental program (for CLS characterization, design of incident particle beams, experimental characterization of the emitted radiation). To accomplish this task, one has to assemble a consortium that has the necessary broad range of competences to realize the science-toward-technology breakthrough that will enable the practical realization of the CLS. The practical realization of CLS will include elaboration of the key theoretical, experimental, and technological aspects, demonstration of the device functionality, designing the novel technology for the construction of CLS that will allow for their mass production in the future and establishing the standards required for the adoption of CLS by different user communities. Realization of this program implies a broad range of correlated and entangled activities including 1. Fabrication of linear, bent, and periodically bent crystalline structures with lattice quality necessary for delivering pre-defined bending parameters within the ranges indicated in Fig. 1.3; 2. Advanced control of the lattice quality by means of the highest quality nondestructive X-ray diffraction techniques. The same techniques are to be applied to detect possible structural modification following particle irradiation; 3. Validation of functionality of the manufactured structures through experiments with high-quality (low energy spread, low emittance, high particle density, and current) beams of ultra-relativistic electrons and positrons with ε = 10−1 − 101 GeV, including an authoritative study of the structure sustainability with respect to beam intensity, as well as explicit experimental characterization of the emission spectra; 4. Advance in computational and numerical methods for multiscale modeling of nanostructured materials with extremely high, reliable levels of prediction (from atomistic to mesoscopic scale), of particle propagation, of irradiation-induced solid-state effects, and for calculation of spectral-angular distribution of emitted © Springer Nature Switzerland AG 2022 A. Korol and A. V. Solov’yov, Novel Lights Sources Beyond Free Electron Lasers, Particle Acceleration and Detection, https://doi.org/10.1007/978-3-031-04282-9_8

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208

8 Conclusion

radiation and for modeling [1]. Ultimately, this will enable better experimental planning and minimization of experimental costs. The knowledge gained from the studies (1)–(4) will provide CLSs prototypes and a roadmap for practical implementation by CLS system manufacturers and accelerator laboratories/users worldwide. Sub-angstrom wavelength ultrahigh brilliance, tunable CLSs will have a broad range of exciting potential cutting-edge applications [2]. These applications include exploring elementary particles, probing nuclear structures and photonuclear physics, and examining quantum processes, which rely heavily on gamma ray sources in the MeV to GeV range [3]. Gamma rays induce nuclear reactions by phototransmutation. For example, in the experiment [4], a long-lived isotope can be converted into a short-lived one by irradiation with a gamma ray bremsstrahlung pulse. However, the intensity of bremsstrahlung is orders of magnitudes less than CUR. Moreover, to increase the effectiveness of the photo-transmutation process, it is desirable to use photons whose energy is in resonance with the transition energies in the irradiated nucleus [5, 6]. By varying the CU parameters, one can tune the energy of CUR to values needed to induce the transmutation process in various isotopes. This opens the possibility for a novel technology for disposing of nuclear waste. Photo-transmutation can also be used to produce medical isotopes. Another possible application of the CU-LSs concerns photo-induced nuclear fission, where a heavy nucleus is split into two or more fragments due to the irradiation with gamma-quanta whose energy is tuned to match the transition energy between the nuclear states. This process can be used in a new type of nuclear reactor—the photonuclear reactor [6]. A nondestructive assay system for radioactive waste management by means of nuclear resonance fluorescence triggered by gamma rays generated from the Compton scattering of laser photons by relativistic electrons has been discussed [7]. This problem can also be attacked by means of the CLSs radiation. Powerful monochromatic radiation within the MeV range can be used as an alternative source for producing beams of MeV protons by focusing a photon pulse on to a solid target [4, 8]. Such protons can induce nuclear reactions in materials producing, in particular, light isotopes which serve as positron emitters to be used in Positron Emission Tomography (PET). The production of PET isotopes using CUR exploiting the (γ ; n) reaction in the region of the giant dipole resonance (typically 20–40 MeV) is an important application of CLS since PET isotopes are used directly for medial PET and for Positron Emission Particle Tracking (PERP) experiments. Irradiation by hard X-ray strongly decreases the effects of natural surface tension of water [9]. The possibility to tune the surface tension by CUR can be exploited to study the many phenomena affected by this parameter in physics, chemistry, and biology such as, for example, the tendency of oil and water to segregate. The last but not least, a micron-sized narrow CLS photon beam may be used in cancer therapy [10] to improve the precision and effectiveness of the therapy for the destruction of tumors by collimated radiation allowing delicate operations to be performed in close vicinity of vital organs.

8 Conclusion

209

The exemplary case study of a tunable CU-based LS, considered in Chap. 6, demonstrates that peak brilliance of CUR emitted in the photon energy range 102 keV up to 102 MeV by currently available (or planned to be available in near future) positron beams channeling in periodically bent crystals is comparable to or even higher than that achievable in conventional synchrotrons in the much lower photon energy range. Intensity of CUR greatly exceeds the values provided by LSs based on Compton scattering and can be made higher than the values predicted in the Gamma Factory proposal in CERN. By propagating a pre-bunched beam, the brilliance in the energy range 102 keV up to 101 MeV can be boosted by orders of magnitude reaching the values of spontaneous emission from the state-of-the-art magnetic undulators and being comparable with the values achievable at the XFEL facilities which operate in a much lower photon energy range. Important is that, by tuning the bending amplitude and period, one can maximize brilliance for given parameters of a positron beam and/or chosen type of a crystalline medium. Last but not least, it is worth mentioning that the size and the cost of CLSs are orders of magnitude less than that of modern LSs based on the permanent magnets. This opens many practical possibilities for the efficient generation of gamma rays with various intensities and in various ranges of wavelength by means of the CLSs on the existing and newly constructed beamlines. Though we expect that, as a rule, the highest values of brilliance can be reached in CU-based LSs (or, in those based on stacks of CUs), the analysis similar to the one presented can be carried out for other types of CLSs based on linear and bent crystals. This will allow one to make an optimal choice of the crystalline target and the CLS type to be used in a particular experimental environment or/and to tune the parameters of the emitted radiation matching them to the needs of a particular application. The case study presented has been focused on the positron beams, which have a clear advantage since the dechanneling length of positrons is the order of magnitude larger than that of electrons of the same energy. This allows one to use thicker crystals in channeling experiments with positrons, thus enhancing the photon yield. Nevertheless, experimental studies of CLSs with electron beams are worth to be carried out. Indeed, high-quality electron beams of energies starting from sub-GeV range and onward are more available than their positron counterparts. Therefore, these laboratories provide more options for the design, assembly, and practical implementation of a full suite of correlated experimental facilities needed for operational realization and exploitation of the novel CLSs. In this connection, we note that in the course of channeling experiments at the Mainz Microtron facility with ε = 190−855 MeV electrons propagating in various CUs, which have been carried out over the last decade within the frameworks of several EU-supported collaborative projects (FP6PECU, FP7-CUTE, H2020-PEARL), a unique experience has been gained. This experience has ascertained that the fundamental importance of the quality of periodically bent crystals, which, in turn, is based on the cutting-edge technologies used to manufacture the crystalline structures, of modern techniques for nondestructive characterization of the samples, of the necessity of using advanced computational methods for numerical modeling of a variety of phenomena involved. On the basis

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8 Conclusion

of this experience, the bottlenecks on the way to the practical realization of the CLSs concept have been established. To quantify the scale of the impact within Europe and worldwide, which the development of radically novel CLSs might have, we can draw historical parallels with synchrotrons, optical lasers, and XFELs. In each of these technologies, there was a time lag between the formulation of a pioneering idea, its practical realization, and follow-up industrial exploitation. However, each of these inventions has subsequently launched multi-billion dollar industries. The implementation of CLS, operating in the photon energy range up to hundreds of MeV, is expected to lead to a similar advance and CLSs have the potential to become the new synchrotrons and lasers of the mid to late twenty-first century, stimulating many applications in basic sciences, technology, and medicine. The development of CLS will, therefore, herald a new age in physics, chemistry, and biology.

References 1. Solov’yov, I.A., Yakubovich, A.V., Nikolaev, P.V., Volkovets, I., Solov’yov, A.V.: MesoBioNano explorer—a universal program for multiscale computer simulations of complex molecular structure and dynamics. J. Comp. Chem. 33, 2412–2439 (2012) 2. Zhu, X.L., Chen, M., Weng, S.M., Yu, T.P., Wang, W.M., He, F., Sheng, Z.M., McKenna, P., Jaroszynski, D.A., Zhang, J.: Extremely brilliant GeV γ -rays from a two-stage laser-plasma accelerator. Sci. Adv. 6, eaaz7240 (2020) 3. Howell, C.R., Ahmed, M.W., Afanasev, A., Alesini, D., Annand, J.R.M. et al.: International workshop on next generation gamma-ray source (2020). arXiv:2012.10843 International Workshop on Next Generation Gamma-Ray Source. arXiv preprint arXiv:2012.10843 (2020) 4. Ledingham, K.W.D., McKenna, P., Singhal, R.P.: Applications for nuclear phenomena generated by ultra-intense lasers. Science 300, 1107 (2003) 5. ur Rehman, H., Lee, J., Kim, Y.: Optimization of the laser-Compton scattering spectrum for the transmutation of high-toxicity and long-living nuclear waste. Ann. Nucl. Energy 105, 150 (2017) 6. ur Rehman, H., Lee, J., Kim, Y.: Comparison of the laser-compton scattering and the conventional bremsstrahlung X-rays for photonuclear transmutation. Int. J. Energy Res. 42, 236–244 (2018) 7. Hajima, R., Hakayama, T., Kikuzawa, N., Minehara, E.: Proposal of nondestructive radionuclide assay using a high-flux gamma-ray source and nuclear resonance fluorescence. J. Nucl. Sci. Technol. 45, 441–451 (2008) 8. Ledingham, K.W.D., Singhal, R.P., McKenna, P., Spencer, I.: Laser-induced nuclear physics and applications. Europhys. News 33, 120 (2002) 9. Weon, B.M., Je, J.H., Hwu, Y., Margaritondo, G.: Decreased surface tension of water by hardX-ray irradiation. Phys. Rev. Lett. 100, 217403 (2008) 10. Solov’yov, A.V. (ed.): Nanoscale Insights into Ion-Beam Cancer Therapy. Springer International Publishing, Cham, Switzerland (2017)

Index

A Acceptance, 79, 81, 114, 120, 124, 139, 161 Atomic potential Doyle–Turner approximation, 90 Molière approximation, 74, 89, 90 Pacios approximation, 74, 89, 90 Atomistic modelling, 73, 112, 120, 124, 130, 142, 165 trajectories simulation, 79

B Bandwidth, 41 Beam divergence, 41, 161, 165 size, 41, 161, 165 Bending parameter, 7, 81, 105, 114, 120, 124 Bethe–Heitler approximation, 22, 35, 67 Bremsstrahlung Bether–Heitler formula, 22 coherent, 23, 24 elastic and inelastic, 22 elementary process, 21, 22, 35 Brilliance, 41, 157

C Channeling in bent crystals, 106 condition, 81, 105 fraction, 83, 85, 86 in bent crystal, 4, 107, 112 experiments, 4, 107, 124, 130 radiation, 4, 108, 112, 114, 117, 119, 126, 130 motion, 61, 65, 80, 114 of electrons, 58, 62, 63, 122, 128, 139

of positrons, 58, 62, 63, 122, 139 oscillations, 65 phenomenon, 2, 57 radiation, 4, 66, 68, 137, 150 stable, 7 Channeling radiation, 66 spectral distribution, 88, 114, 117–119, 130, 150, 152, 154 Coherence length, 23 Compton scattering, 46, 47, 49, 50 light source, 46, 47, 159, 163, 169 Continuous potential model, 60, 62, 71, 79, 170 Critical angle, 62 Critical bending radius, 81, 119 Crystal-based light source, 1 Crystalline structure modelling, 76 Crystalline undulator, 137 amplitude, 7, 139 bending parameter, 138, 160 brilliance of radiation, 159 feasibility, 4, 7, 8, 137 light source, 4 number of periods, 7 optimal length, 160 period, 139 radiation, 1, 137, 139, 154, 157 stack of, 1, 2, 154 undulator parameter, 138 Crystalline undulator radiation spectral distribution, 141, 142, 148, 150, 152, 154, 166, 169 Crystallographic directions, 58

D Dechanneling, 7, 8, 62, 80, 114, 125

© Springer Nature Switzerland AG 2022 A. Korol and A. V. Solov’yov, Novel Lights Sources Beyond Free Electron Lasers, Particle Acceleration and Detection, https://doi.org/10.1007/978-3-031-04282-9

211

212 Dechanneling length, 3, 7, 62, 82, 85, 123, 125, 161 Diamond hetero-crystal, 147 Diffusion approximation, 85 Distribution in deflection angle, 122, 130 Dynamic simulation box, 76

E Energy losses, 7, 8, 69, 71 Equations of motion, 61, 74

F Flux of photons, 41, 49, 50 Free electron laser, 3, 45

G Gamma-factory, 49, 50, 159, 163, 169

H Harmonics, 66, 68, 90, 116, 120, 138, 142, 159

I Interplanar potential, 60, 107 Doyle–Turner approximation, 60, 93, 95 harmonic approximation, 67, 70 in bent channel, 105 Molière approximation, 60, 64, 93, 95, 140, 174 Pacios approximation, 60, 93, 95 Pöschl-Teller approximation, 70 Ionizing collisions, 72, 75 Irradiation-Driven Molecular Dynamics, 75

L Landau-Pomeranchuk-Migdal effect, 23 Large-amplitude regime, 7 Laser-wakefield accelerator, 51 light source, 51, 52 Linked-cell algorithm, 74

M MBN Explorer computer package, 73, 112

N Natural emission cone, 18, 25

Index Number of photons, 42, 51, 52

O Over-barrier motion, 61, 80

P Peak brilliance, 42, 46, 51, 52, 162, 169 Penetration length, 81, 85, 114, 120, 124, 139 Periodically bent crystal, 2 Periodic bending acoustic wave propagation, 6, 8 Large-Amplitude Large Period, 7 Small-Amplitude Short Period, 7, 8, 150, 152, 154 superlattice, 5, 147 surface deformations, 5 Photon attenuation, 7, 8

Q Quasi-mosaic crystal, 128

R Radiation damping, 69 Radiation emission classical description, 17, 19, 30, 32 quantum description, 19 quasi-classical description, 20, 72 Radiation length, 22, 23, 87 Rechanneling, 80, 83

S Superlattice, 78, 147 Synchrotron radiation, 25, 42, 116, 118, 126 light source, 42, 163, 169 quasi-classical formula, 26

T Technologies for crystal bending, 5 Thermal vibrations, 78

U Undulator ideal, 27, 28 motion, 27 number of periods, 7, 31 parameter, 28, 29, 44, 66

Index

213

Undulator radiation, 27, 32, 33, 43, 44, 68 constructive interference, 33 harmonics, 27, 30

Volume reflection, 72, 110, 111, 144

V Volume capture, 110, 111

X X-ray free-electron lazer, 46, 158