[1st Edition] 9780128121931, 9780128120903

Table of contents :
Content:
Series PagePage ii
CopyrightPage iv
ContributorsPage vii
PrefacePage ixPeter W. Hawkes
Future ContributionsPages xi-xiii
Chapter One - Past and Present Attempts to Attain the Resolution Limit of the Transmission Electron MicroscopePages 1-59Ernst Ruska
Chapter Two - Phase Plates for Transmission Electron MicroscopyPages 61-102Christopher J. Edgcombe
Chapter Three - X-Ray Lasers in Biology: Structure and DynamicsPages 103-152John C.H. Spence
IndexPages 153-157

Citation preview

EDITOR-IN-CHIEF

Peter W. Hawkes CEMES-CNRS Toulouse, France

Cover photo credit: Courtesy SLAC National Accelerator Laboratory Academic Press is an imprint of Elsevier 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 125 London Wall, London, EC2Y 5AS, United Kingdom First edition 2017 Copyright © 2017 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-12-812090-3 ISSN: 1076-5670 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Zoe Kruze Acquisition Editor: Poppy Garraway Editorial Project Manager: Shellie Bryant Production Project Manager: Magesh Kumar Mahalingam Cover Designer: Mark Rogers Typeset by SPi Global, India

CONTRIBUTORS Christopher J. Edgcombe University of Cambridge, Cambridge, United Kingdom Ernst Ruska (1906–1988) Institut f€ ur Elektronenmikroskopie am Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin-Dahlem, Germany John C.H. Spence Arizona State University, Tempe, AZ, United States

vii

PREFACE For this 200th volume of these Advances, Volume 1 of which appeared in 1948, we have contributions on subjects of great current interest by J.C.H. Spence and C.J. Edgcombe. The volume opens, however, with a chapter that bears the name of the most distinguished name of all in electron microscopy, Ernst Ruska. This article was first published in Advances in Optical and Electron Microscopy, a serial that is not available online. It contains many timeless articles by such authors as Albert Septier, Alan Metherell, Karl-Joseph Hanszen, Gaston Dupouy, Pieter Kruit, Gottfried M€ ollenstedt, Fred Bok, Eric Plies, John Rouse, Hannes Lichte, and many others just as well known. We plan to reprint some of these in future volumes. In the first of the regular articles, C.J. Edgcombe surveys electron phase plates. These have been studied intermittently since 1947, when Hans Boersch suggested that the phase plates employed in light microscopy could be beneficial in electron microscopy after suitable adaptation. Since then, many attempts to modify the image-forming properties of the electron microscope by introducing such a plate have been made, with varying degrees of success. The subject has received a new impetus in the past few years and a new use for such plates has emerged: prespecimen plates designed to generate vortex beams. The theory and practice of the various types of plate are presented in detail in this chapter, which concludes with predictions for future developments. The third chapter, by J.C.H. Spence, describes a very exciting and rapidly developing subject, the use of the X-ray laser to explore the structure of biological material at the atomic level and its changes over very short times. “It is the free-electron laser which now, for the first time, has allowed us to image hydrated molecules in motion during a chemical reaction at atomic resolution with femtosecond time resolution and negligible radiation damage,” writes John Spence and it is this remarkable development that he describes in detail. This is a new field of science, still in its infancy, and we can look forward to a rich future. I am very grateful to C.J. Edgcombe and J.C.H. Spence for presenting these far-reaching subjects so clearly and for making this 200th volume a milestone in the series. PETER W. HAWKES

ix

FUTURE CONTRIBUTIONS S. Ando Gradient operators and edge and corner detection J. Angulo, S. Velaso-Forero Convolution in (max, min)-algebra and its role in mathematical morphology J. Angulo, S. Velaso-Forero Non-negative sparse mathematical morphology A. Ashrafi, (Vol. 201) Walsh functions and their applications D. Batchelor Soft x-ray microscopy E. Bayro Corrochano Quaternion wavelet transforms C. Beeli Structure and microscopy of quasicrystals C. Bobisch, R. M€ oller Ballistic electron microscopy F. Bociort Saddle-point methods in lens design K. Bredies Diffusion tensor imaging A. Broers A retrospective A. Cornejo Rodriguez, F. Granados Agustin Ronchigram quantification J. Elorza Fuzzy operators R.G. Forbes Liquid metal ion sources P.L. Gai, E.D. Boyes Aberration-corrected environmental microscopy R. Herring, B. McMorran Electron vortex beams M.S. Isaacson Early STEM development

xi

xii K. Ishizuka Contrast transfer and crystal images K. Jensen, D. Shiffler, J. Luginsland Physics of field emission cold cathodes U. Kaiser The sub-A˚ngstr€ om low-voltage electron microcope project (SALVE) S.A. Khan, (Vol. 201) Quantum methodologies in Maxwell optics O.L. Krivanek Aberration-corrected STEM M. Kroupa The Timepix detector and its applications B. Lencova´ Modern developments in electron optical calculations H. Lichte Developments in electron holography M. Matsuya Calculation of aberration coefficients using Lie algebra J.A. Monsoriu Fractal zone plates L. Muray Miniature electron optics and applications M.A. O’Keefe Electron image simulation V. Ortalan Ultrafast electron microscopy D. Paganin, T. Gureyev, K. Pavlov Intensity-linear methods in inverse imaging N. Papamarkos, A. Kesidis The inverse Hough transform H. Qin Swarm optimization and lens design Q. Ramasse, R. Brydson, (Vol. 202) The SuperSTEM laboratory B. Rieger, A.J. Koster Image formation in cryo-electron microscopy P. Rocca, M. Donelli Imaging of dielectric objects

Future Contributions

Future Contributions

J. Rodenburg Lensless imaging J. Rouse, H.-n. Liu, E. Munro The role of differential algebra in electron optics J. Sa´nchez Fisher vector encoding for the classification of natural images P. Santi Light sheet fluorescence microscopy R. Shimizu, T. Ikuta, Y. Takai Defocus image modulation processing in real time T. Soma Focus-deflection systems and their applications I.J. Taneja, (Vol. 201) Inequalities and information measures J. Valdes Recent developments concerning the Syste`me International (SI) J. van de Gronde, J.B.T.M. Roerdink Modern non-scalar morphology

xiii

CHAPTER ONE

Past and Present Attempts to Attain the Resolution Limit of the Transmission Electron Microscope Ernst Ruska✠ Institut f€ ur Elektronenmikroskopie am Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin-Dahlem, Germany

Contents 1. The Limits of Resolving Power and the Sources of Aberrations 1.1 The First Estimate of the Resolution Limit and the Resolution Achieved at the Present Time 1.2 Sources of Errors Which Have to Be Diminished in Order to Attain the Resolution Limit 1.3 Resolution Limit due to the Combination of Diffraction and Spherical Aberration in the Objective for Two Image Points Radiating Incoherently 1.4 Nature and Magnitude of Permissible Disturbances If the Resolution Limit for Two Points Radiating Incoherently Is to Be Attained in the Image 1.5 Resolution Limit of Linear Lattices, in Partially Coherent Illumination, due to Spherical and Chromatic Aberrations of the Objective 1.6 Zone Plates for Improving Resolution and Contrast in Bright- and Dark-Field Images 2. The Single-Field Condenser Objective 2.1 Principle, Ray Paths, and Construction 2.2 Electron Beam Energy and Resolution Limit for Two Incoherently Radiating Points 2.3 Nature and Magnitude of Permissible Disturbances If the Resolution Limit for Two Points Radiating Incoherently Is to Be Attained in the Image 2.4 Resolution Limit for Linear Lattices, in Partially Coherent Illumination, When the Disturbances Are so Small That They Do Not Prevent the Attainment of the Resolution Limit for Two Points Radiating Incoherently 3. Movement of the Image During Photographic Exposure 3.1 Movement and Heating of the Specimen 3.2 Movement of the Electron Image due to Electric and Magnetic Fields 4. Lack of Sharpness of the Image 4.1 Instability in the Lenses ✠

2 2 5 7 10 12 23 23 23 26 32

33 35 35 37 38 38

Deceased. “Reprinted from E. Ruska (1966). Advances in Optical and Electron Microscopy, vol. 1, pp. 115–179 (Academic Press, London & New York)”.

Advances in Imaging and Electron Physics, Volume 200 ISSN 1076-5670 http://dx.doi.org/10.1016/bs.aiep.2017.03.001

#

2017 Elsevier Inc. All rights reserved.

1

2

Ernst Ruska

4.2 Noncircularity of Lenses 5. Changes in the Specimen 5.1 Interaction Between Specimen, Electron Beam, and Residual Gases 5.2 Specimen-Space Cooling as a Means of Preventing Changes in the Specimen due to Residual Gas During Electron Irradiation 5.3 Cooling of the Specimen as Well as Its Surroundings 6. Self-Structure of Supporting Films References

39 39 39 44 49 53 58

1. THE LIMITS OF RESOLVING POWER AND THE SOURCES OF ABERRATIONS 1.1 The First Estimate of the Resolution Limit and the Resolution Achieved at the Present Time Although the wave nature of the electron had already been theoretically postulated by de Broglie in 1925 and proved experimentally by Davisson and Germer in 1927, it was not invoked in the first publications on the electron microscope for evaluating its resolving power (Br€ uche & Johannson, 1932; Knoll & Ruska, 1932a). But already in the next publication on the subject (Knoll & Ruska, 1932b), an estimate was made of the resolution limit of the transmission electron microscope with magnetic lenses. In these early experiments—as at the present time—the aperture of illumination was much smaller than the imaging aperture. Further, it was usual at that time only to irradiate the objective aperture with illumination coaxial with respect to the objective lens axis. Correspondingly, the attainable resolution was evaluated according to the light microscopical theory of the imaging of linear lattices illuminated with a beam of aperture much smaller than that of the objective and coaxial to it. Under these conditions the light microscope will resolve linear lattices with a lattice constant equal to or greater than δLL, given by δLL
2δL ¼

λL , n sin α

(1)

where δL is the ultimate resolution limit of the light microscope. This limit can only be attained by irradiating the specimen either with illumination coaxial with the objective and with an angular aperture of the same size as the objective aperture α, or with a beam of narrow aperture in the direction of maximum inclination α to the objective lens axis. In their original experiments, Knoll and Ruska used a gas discharge tube, which worked

Past and Present Attempts to Attain the Resolution Limit

3

at a voltage of 75 kV. In Eq. (1), therefore, they replaced the wavelength of light λL by the wavelength λE of 75 keV electrons, and on the basis of their initial experimental experience, they estimated the maximum permissible imaging aperture α of magnetic lenses to be 0.02. Thus, they obtained at this early date the right order of magnitude for the resolution limit of the mag˚ . They concluded their netic transmission electron microscope, 2δL ¼ 2.2 A discussion as follows: “Compared with the normal microscope the theoretical resolution limit of the electron microscope, as set by the wave nature of the electron, is thus higher by at least two to three orders of magnitude and is already of atomic dimensions. Whether this high resolving power can be used to make visible structures of this order of magnitude cannot be decided in the present state of knowledge. Further investigations are required, involving detailed studies of the aberrations of the image and an increase in the intensity of the electron source. Investigations in these directions are in progress.” After 30 years’ development the resolution limit originally estimated for images of nonperiodic specimens has been almost reached by the imaging of two points separated ˚ , as shown in Fig. 1 (Engel, Koppen, & Wolff, 1962), and the by 4 A ˚ for linear lattices has recently been achieved by Komoda limit of 2 A and Otsuki (1964) (Fig. 2).a Thus the transmission electron microscope can now resolve particles 500 times more closely adjacent, and lattices with a spacing 1000 times smaller, than is possible in the normal microscope using visible light with either coaxial illumination of wide aperture ˚ , from Eq. (1) or inclined illumination of narrow aperture (δL ¼ 2000 A with λL ¼ 0.56 μm and n sin α ¼ 1.4). When we look back over the development of the electron microscope so far, we see a striking difference in comparison with that of the light microscope. In the decisive stages of the history of the light microscope, extending over more than 300 years, its performance was increased by improvements to its individual lenses and to the lens system as a whole. In particular the spherical and chromatic aberrations of the objective were reduced, the magnification increased by adopting the two-stage design, and the resolution improved by introducing the immersion lens; later the contrast was also increased by means of the phase-contrast method. In comparison with this, even the best modern electron microscopes still have magnetic lenses with spherical and chromatic aberration constants as great as those of the a

The author is grateful to Dr. Komoda for making this micrograph available before it was published.

4

Ernst Ruska

Fig. 1 Pair of micrographs of the same area of a platinum–iridium evaporated film, with a measured point resolution δPP ¼ 4 Å. Siemens-Elmiskop I with hair-pin filament U ¼ 80 kV, narrow coaxial illumination of angular aperture σ  103 (A), and “wide” coaxial objective aperture α  102 (B), ME ¼ 200,000 , Guilleminot-Electroguil-plate, t ¼ 5 s. Image data: Taken by Weichan (1962).

pole-piece lenses first used for electron microscopical imaging 30 years ago. That we have been satisfied with such imperfect electron lenses up to now is due to the fact that the wavelength of 100 kV electrons, the voltage used nowadays in transmission electron microscopy, is 100,000 times shorter than the wavelength of light. Thus, the resolution of nonperiodic specimen structures, as limited only by the spherical and diffraction aberration

Past and Present Attempts to Attain the Resolution Limit

5

Fig. 2 Set of lattice planes (200) of a gold single crystal, d ¼ 2.04 Å. Electron microscope Hitachi HU-11A, with hair-pin filament, U ¼ 100 kV, inclined illumination of angular aperture σ ¼ 0.3  103 at α ¼ 15  103, ME ¼ 240,000 , Fuji-H.S. plate, t ¼ 7 s. Image data: Taken by Komoda, T., & Otsuki, M. (1964). Japan Journal of Applied Physics, 3, 666–667.

of the objective, has the excellent value of about 3 A˚, in spite of the relatively large spherical aberration constant of the electron objectives in use today. Even though the resolution of periodic structures (lattice planes of single crystals) is also limited ultimately by the spherical and diffraction aberrations, it is several times better than this, so long as other defects are sufficiently small.

1.2 Sources of Errors Which Have to Be Diminished in Order to Attain the Resolution Limit The attainable resolution of the imaging process is limited by the wavelength of the imaging beam and by the aperture of the objective. For an objective with no spherical aberration the objective aperture is defined by the distance between the specimen and the objective and by the diameter of the objective. When spherical aberration is present, a limit is set on the permissible size of the objective aperture by the magnitude of the spherical aberration constant. As the objectives of present-day electron microscopes have a high spherical aberration constant, the attainable electron microscopical resolution thus depends on the electron wavelength and on the imaging aperture, as limited by the spherical aberration of the objective. In addition, the resolution limit will also be influenced by the nature of the resolvable structure

6

Ernst Ruska

and other properties of the specimen. The resulting resolution limit of an optical instrument for a particular type of specimen can only be realized, however, if all its other errors are so far reduced that their effect on the image is smaller than this limit. In the light microscope with its comparatively poor resolving power, it was relatively easy to keep all aberrations sufficiently low. But up to now, it has been, and still is, far more difficult to keep all the aberrations of the transmission electron microscope low enough for its theoretical resolution—1000 times better than that of the light microscope—to be attained while at the same time paying due regard to the special properties of electron beams, which are so very different from those of light beams. In particular the following sources of error have to be taken into account: (i) insufficient magnification of the image of the specimen on the photographic plate; (ii) movement of the specimen relative to the objective; (iii) movement of the image relative to the photographic plate (independent of 2); (iv) decentering of the electron gun and lenses with respect to the objective lens axis; (v) spread of wavelengths in the electron beam; (vi) variation with time of the focal length of the objective; (vii) noncircularity of the objective; (viii) inaccurate focusing of electron image (defocusing error); and (ix) changes in the specimen due to the illuminating beam. All these sources of error cannot be kept sufficiently small by the construction of the electron microscope alone. Some of them must be kept under observation by the operator and reduced as far as possible by adjustments to the instrument. The art of microscopy, of which one already speaks in connection with light microscopy, has to meet higher requirements in the case of the electron microscope. During the development of the electron microscope so far, our efforts have been concerned almost exclusively with the reduction of errors (i) to (viii) in the above list. By introducing objectspace cooling, we have recently succeeded in decreasing error (ix) also, i.e., an effective reduction of the changes in the specimens which are the most disturbing factors in this range of resolution. Thus, there is now a greater interest in efforts to improve the resolving power for nonperiodic specimen ˚ , by using a magnetic objective with structures to a value smaller than 3 A a reduced spherical aberration constant, the “single-field condenser objective,” and by an electron optical and constructional design of the whole microscope adequate for this purpose.

Past and Present Attempts to Attain the Resolution Limit

7

1.3 Resolution Limit due to the Combination of Diffraction and Spherical Aberration in the Objective for Two Image Points Radiating Incoherently Even in the case of solids, the most frequently investigated microscopic objects, nonperiodic specimens are more usual than periodic specimens in the size ranges of both the light and the electron microscopes. Hence, the resolution attainable with nonperiodic specimens is the primary consideration in both instruments. For about 100 years now, it has been possible to produce light microscope objectives free from spherical aberration and having an aperture of almost π/2. For a long time, therefore, structure elements (e.g., two discrete points or two lines) of nonperiodic specimens have been optically resolvable with spacings as small as the lattice constant of the smallest resolvable linear lattice. The best resolution is attained by using a narrow beam aperture and inclined illumination or a wide aperture and coaxial illumination. In both cases, beams scattered by nearly 2α can pass through the objective as well as the unscattered beam. Also, no optical path differences can occur, between beams that have the same phase but different directions with respect to the optical axis, in an objective which is free of spherical aberration. The resolution limit of the light microscope δL thus has the same value for linear lattice structures as for nonperiodic structures: δL ¼

λL λL  : 2n sin α 2n

(2)

As has already been mentioned, the rotationally symmetric objectives, which have so far been used in electron microscopes, have relatively large spherical aberration constants. With such an objective, linear lattice structures may be better resolved than nonperiodic structures, which will scatter electrons into angles distributed over the whole objective aperture. This will be discussed in fuller detail in Section 1.5. In passing through the objective, optical path differences arise between beams scattered in different directions with respect to the optical axis, owing to the spherical aberration of the objective. These path differences must be restricted because they reduce the image contrast. For this reason the permissible aperture, and consequently also the resolving power for nonperiodic structures, is subject to lower limits than would be the case for an objective free from spherical aberration. Almost no calculations exist for the resolution limit of the transmission electron microscope in the case of nonperiodic specimen structures of

8

Ernst Ruska

defined properties. But some time ago, Glaser and Scherzer made estimates of the resolution limit of the transmission electron microscope for two closely adjacent specimen points radiating incoherently in surroundings which are not radiating. Assuming that the radiation from the points has a Lambert distribution, the optimum imaging aperture is given by (Glaser, 1943) pffiffiffiffiffiffiffiffiffiffiffiffi (3) αPP ¼ 1:13 4 λE =Cs : The resolution limit for two points radiating incoherently in the Gaussian image plane (Glaser, 1943) is then qffiffiffiffiffiffiffiffiffiffi 0:633λE 4 δPP ¼ 0:56 λ3E Cs ¼ : (4) αPP In the plane of the circle of least confusion, i.e., at maximum underfocusing, which is more favorable for the resolution, the optimum image aperture is pffiffiffiffiffiffiffiffiffiffiffiffi (5) αPP ¼ √2  4 λE =Cs and the resolution limit of two points radiating incoherently is: δPP ¼ 0:43

qffiffiffiffiffiffiffiffiffiffi 0:61λΕ 4 λ3E Cs ¼ αPP

(6)

again assuming a Lambert distribution of the radiation from the specimen (Scherzer, 1949). Even with a magnetic objective, which has a spherical aberration constant smaller by about an order of magnitude than that of an equivalent electrostatic objective, the optimum imaging aperture is limited to αPP  102

(7)

and the corresponding resolution limit of two points radiating incoherently to δPP  60λE :

(8)

The point resolution given by Eqs. (4), (6), and (8) is analogous to the resolution limit of the light microscope given by Eq. (2), insofar as it is assumed in both cases that beams which are scattered or diffracted at the object through angles up to 2α contribute to image formation, as well as the unscattered beam. The conditions (radiating points in nonradiating

Past and Present Attempts to Attain the Resolution Limit

9

surroundings), which are the basis for these equations, do not correspond to the actual circumstances in transmission electron microscopy. Here the main interest is in the separation of closely adjacent particles, mostly located on supporting films, or of thin layers, as in microtome sections—i.e., particles or layers which are of greater or smaller density than their surroundings. The separation of particles in an irradiated specimen, which simply scatter more or less strongly than their surroundings, will naturally not be as good as the separation of two radiating points in nonradiating surroundings. The contrast between the bright images of radiating points and their dark background is optimal. On the other hand, very small scattering or phase-shifting particles in the specimen plane of the electron microscope will be imaged with a contrast which is the weaker the smaller the particles are. By increasing the beam voltage, and thus reducing the electron wavelength, the point resolution can only be improved until the reduced intensity in the electron image just suffices to distinguish the details of the specimen structure from their surroundings, depending on their power of scattering and phase-shifting the electron waves. If we simply increase the beam voltage, when imaging nonperiodic specimens, the image contrast will be reduced. When the beam voltage is kept constant, the contrast will increase with the weight of the particles imaged, i.e., with the number and the atomic weight of the atoms or molecules composing the particles. If we want to image single atoms, for instance, the limit of weight below which atoms are invisible, owing to inadequate image contrast, will become higher and higher as the beam tension is increased, and so the number of kinds of atoms which can be visibly imaged will become smaller and smaller. Fortunately, in the range of very high beam voltages, the relativistic increase in the electron mass has the effect of preventing contrast from becoming indefinitely small. It may be that in these rather complicated conditions an optimum beam voltage exists for the resolution of atoms lying at the closest possible separation from each other, and this might have a higher value for the more easily resolved heavy atoms. More exact calculations of the contrast and resolution to be expected when imaging single atoms in transmission have recently been carried out (Niehrs, 1962) for the single-field condenser objective, which has a particularly low spherical aberration constant (cf. Section 2). These calculations showed, for instance, that carbon atoms with a separation ˚ produce a phase contrast of 10% at 300 kV, which is just sufficient for of 3 A observation. If these calculations are confirmed by experiment, the objections often expressed to the use of higher beam voltages, on the ground of poor contrast, will lose much of their weight.

10

Ernst Ruska

1.4 Nature and Magnitude of Permissible Disturbances If the Resolution Limit for Two Points Radiating Incoherently Is to Be Attained in the Image In the transmission electron microscope the best resolution is attained if the velocity (and so the wavelength) of the electrons has negligible spread and is constant with time, and if the spatial distribution of field strength in the lenses is constant with time and also uniform with respect to symmetry around the optical axis. Deviations from this ideal case will lead to the circle of confusion becoming so large that the ultimate resolution, as limited by the combination of spherical aberration and diffraction, cannot be attained. The relative deviations from the exact values required for optimum imaging are called disturbances or errors. When using the transmission electron microscope, special attention has to be paid to the following factors: 2Δca U U

Spread of emission energy of electrons with respect to accelerating voltage, including the possible broadening due to interaction of electrons in the beam (Boersch effect)

2Δsp U U

Spread of energy loss in the specimen with respect to incident electron energy

2Δinc U U

Relative inconstancy of accelerating voltage due to superposition of ripple, fluctuations, and drift during photographic exposure

2Δinc J J

Relative inconstancy of objective current due to superposition of ripple, fluctuations, and drift during photographic exposure

2Δast J J

Relative difference of current in an astigmatic objective between the settings for the first and second astigmatic focal planes in the electron image—a measure of the astigmatism of the lens

Δdef J J

Smallest relative change of objective current between focused and nonfocused electron image—a measure of amount of defocusing

The radii of the circles of confusion arising from the individual disturbing factors for a point on the lens axis are given by Δca U Cc α, U Δsp U Cc α, ρsp U ¼ U Δinc U ρinc U ¼ Cc α, U ρca

U

¼

(9a) (9b) (9c)

Past and Present Attempts to Attain the Resolution Limit

2Δinc J Cc α, J Δast J Cc α, ρast J ¼ J Δdef J Cc α: ρdef J ¼ J

ρinc J ¼

11

(9d) (9e) (9f )

The strength of a magnetic lens, in which the field distribution along the z-axis is BðzÞ ¼

Bmax , 1 + ðz=hÞ2

(10)

is measured by the parameter k2 (Glaser, 1941): k2 ¼

e B2max h2 J 2 ðkAÞ2 ¼ 3:52 Ur ðkVÞ 8m0 Ur

(11)

e being the charge and m0 the rest mass of the electron, h the half width of the lens field on the z-axis, J the ampere-turns of the lens, and   Ur ¼ U 1 + 0:98  103 U (12) the relativistically corrected value of the accelerating voltage U (in kV). The chromatic aberration constant Cc is   πk2 1 + z2G =h2 Cc ¼ h (13) 2ð1 + k2 Þ3=2 when using such a lens with the object at the position zG. The radius of the resultant circle of confusion due to all the disturbing factors, and assuming we use the optimum aperture αPP, can be allowed to be no more than a small fraction of δPP: ρca U + ρsp

U

+ ρinc

U

+ ρinc J + ρast J + ρdef J ¼ pδPP :

(14a)

With αPP and δPP given by Eqs. (5) and (6), the maximum permissible sum of all the disturbances is then Δca U + Δsp U + Δinc U 2Δinc J + Δast J + 2Δdef J pδPP + ¼ U J Cc αPP pffiffiffiffiffiffiffiffiffiffi p ¼ 0:3 λE Cs : Cc

(14b)

12

Ernst Ruska

1.5 Resolution Limit of Linear Lattices, in Partially Coherent Illumination, due to Spherical and Chromatic Aberrations of the Objective For about 10 years now the lattice planes of single crystals have been increasingly the subject of electron microscopy. So we must also estimate the resolving power of the transmission electron microscope for linear lattices in partially coherent illumination. For this purpose we start from a study of contrast in the electron microscopical image of a set of lattice planes (Dowell, 1963). A linear lattice of lattice constant d can only be resolved by an objective of aperture α so long as at least one diffracted beam as well as the directly transmitted beam passes through the objective to form the image, i.e., so long as we have θ1 λ ¼ arc sin  α: 2 2d

(15)

The interference fringe system formed by the transmitted and diffracted beams is the image of the linear lattice. To image the smallest resolvable spacing, the crystal must be irradiated at the angle α to the axis (exact inclined illumination) so that the angle θ1 between the directly transmitted and diffracted beams may reach the maximum value 2α, as limited by the objective aperture. To obtain a symmetrical position of the transmitted and diffracted beams to the optical axis, the lattice planes to be imaged by the electron microscope have to be parallel to it. Because of the high spherical aberration of an electron objective, the irradiation must be effected with an electron beam having the smallest possible angular aperture, σ. The apertures of the undiffracted beam and that diffracted at the angle θ1 ¼ 2α will be of the same angular width as σ. Thus when using precise inclined illumination, both beams producing the interference image only fill a zone of the objective aperture which is of radius fα and breadth fσ. So only those optical path differences occurring within this zone, and the consequent phase differences, can be effective in reducing the contrast and resolution of the interference image. They are, of course, smaller than the phase differences occurring within a beam which fills the whole aperture, α. In order that both beams entering the objective at angles of α to the axis shall cross in the Gaussian image plane, the focal length must be longer by Δf ¼ Csα2 than the Gaussian focal length. This defocusing will prevent nonperiodic specimen structures, such as the outlines of particles in the specimen, from being sharply imaged.

Past and Present Attempts to Attain the Resolution Limit

13

If the structure resolved in the electron image is to be visible, it must be sufficiently large in comparison with the resolving power of the viewing screen or at least that of the photographic emulsion. So, a higher resolution will naturally require a higher electron magnification. As the highly magnified electron image must be bright enough for observation and focusing, it is necessary to irradiate the specimen more intensely, and this can only be achieved by using a wider angle of illumination, when working with a given electron gun of limited brightness. Therefore, the resolution limit for linear lattices is also ultimately fixed by the spherical aberration, although a linear lattice will, in the limit, be better resolved than two closely adjacent image points, because inclined illumination of narrow angular aperture can be used. The resolution of linear lattices can thus be improved still further by employing an electron gun of higher brightness, such as a point filament. The specimen can then be illuminated with sufficient brightness by a beam of smaller angular aperture σ, so that the angle (θ1/2 ¼ α) of the beam to the axis can be increased until the limiting phase difference is reached which is set by the desired resolution. With a sufficiently great but still finite brightness of the electron gun, and a correspondingly small angle of illumination, linear lattices with a spacing of only λE/2 should in principle be resolved by an objective uncorrected for spherical aberration and with an aperture of α  π/2, provided that all other errors do not exceed a certain limit. Even when using parallel coaxial illumination, an uncorrected objective will resolve a smaller lattice spacing than its point–point resolution. For this purpose an underfocusing of Δf ¼ + Csα2 will again be necessary, so that the Bragg reflections of first order, diffracted to the left or to the right respectively, shall form images which lie on the optical axis and coincide with the image formed by the undiffracted beam. When focusing on the Gaussian image plane, both these dark-field images are displaced with respect to the optical axis by a small amount. In the plane perpendicular to the imaging axis where the two diffracted beams intersect each other and also the axis, the sharpness of nonperiodic specimen structures is again below the optimum value due to the fact that their dark-field images are already overfocused, while the bright-field image formed by the undiffracted beam is still underfocused. However, the improvement in the resolution of linear lattices in partially coherent illumination over that of two points radiating incoherently is less in parallel coaxial illumination than with inclined illumination of narrow aperture. The objective is now illuminated not only by the two first-order diffracted beams over a zone of width fσ at the radius f θ1/2 ¼ fα but also by the undiffracted beam over an aperture fσ around the optical axis.

14

Ernst Ruska

As a test specimen we assume a thin single-crystal platelet appropriately orientated with respect to the beam. The fringe system of spacing Md, which is superimposed on the image of the crystal, is formed only by interference of the primary beam with a beam diffracted in first order at the selected set of planes. We treat the aperture diaphragm of the condenser lens as an extended incoherent electron source. The inhomogeneity of the incident beam with respect to direction and energy, as well as any fluctuations within the imaging system during the photographic exposure, entails an incoherent superposition of elementary images. On the other hand, each individual elementary image is a coherent superposition of the undiffracted and diffracted partial waves of all electrons having the same direction and wavelength in the imaging beam, when working with a constant imaging system. Between the two waves forming the elementary image, there exists a phase difference ϕ at an arbitrarily fixed origin of a coordinate system in the plane of observation. The distribution of intensity (i.e., of electron current density) in the elementary image depends on ϕ and has the form (Dowell, 1963):  w  I ðw Þ ¼ ðIU ÞE + ðIS ÞE  cos 2π ϕ , (16) Md where λE is the wavelength of the electron beam; (IU)E ¼ 1/2(Imax + Imin)E is 1 2

the background intensity of the elementary image; ðIS ÞE ¼ ðImax  Imin ÞE is the amplitude of the intensity fluctuations due to interference of the partial waves, which appear as a fringe system in the elementary image; w is the coordinate orthogonal to the direction of the fringes in the plane of observation; M is the magnification of the elementary image produced by the objective; d is the lattice plane spacing; and Md is the fringe period in the image, produced by interference of the two partial waves. (IU)E and (IS)E are in the first place dependent on the properties of the specimen. If jθ0 is the current density in the undiffracted beam and jθ1 the current density in the beam diffracted by θ1, the maximum and minimum intensities in the elementary image will be:  2 pffiffiffiffiffiffiffiffiffi ðImax ÞE ¼ ðIU ÞE + ðIS ÞE ¼ √jθ0 + √jθ1 ¼ jθ0 + jθ1 + 2 jθ0 jθ1 , (17a)  2 pffiffiffiffiffiffiffiffiffi ðImin ÞE ¼ ðIU ÞE  ðIS ÞE ¼ √jθ0  √jθ1 ¼ jθ0 + jθ1  2 jθ0 jθ1 (17b) and thus ðIU ÞE ¼ jθ0 + jθ1 ,

(18a)

15

Past and Present Attempts to Attain the Resolution Limit

pffiffiffiffiffiffiffiffiffi ðIS ÞE ¼ 2 jθ0 jθ1 :

(18b)

The intensity contrast in the elementary image is defined as pffiffiffiffiffiffiffiffiffiffiffiffi   pffiffiffiffiffiffiffiffiffi ðImax  Imin ÞE IS 2 jθ0 jθ1 2 jθ1 =jθ0 ¼ ¼ ¼ : KE ¼ ðImax + Imin ÞE IU E jθ0 + jθ1 1 + jθ1 =jθ0

(19)

The positions of the fringe maxima in the elementary image are determined by the phase difference ϕ, which depends on the direction and wavelength of the electrons forming the image. The composite image is a superposition of a large number of intensity distributions given by Eq. (16). So long as the angle of illumination is not too large, jθ0 and jθ1 , and consequently (IU)E and (IS)E, may be taken as being the same for all the elementary images. But the phase differences ϕ between undiffracted and diffracted partial waves in the elementary images may be strongly dependent on small variations in the illumination (direction and wavelength) or on fluctuations in the electron optical system. In the composite image, this variation of ϕ over the elementary images causes a decrease in the observable fringe contrast. If the ϕ-values of all elementary images are homogeneously distributed over a range 2Δϕ, superposition of all the elementary images gives for the composite image (Dowell, 1963): Imax ¼ ðIU ÞE + ðIS ÞE

sinΔϕ , Δϕ

(20a)

Imin ¼ ðIU ÞE  ðIS ÞE

sin Δϕ , Δϕ

(20b)

  ðImax  Imin Þ IS sinΔϕ sinΔϕ K¼ ¼ KE : ¼ ðImax + Imin Þ IU E Δϕ Δϕ

(21)

For the interference fringes in the superposition image of a single-crystal platelet to be resolved, the image contrast K must reach the level KR given by the particular definition of resolution adopted. The maximum permissible   phase range 2ΔR ϕ in the superposition image is thus also dependent on the elementary contrast KE in the test specimen. Therefore, one has to come to some agreement about the test specimen to be used for defining the resolution. When its elementary contrast is known, ΔR ϕ is given by the image contrast transfer constant KR/KE necessary for resolution, from Eq. (21)

16

Ernst Ruska

sin ΔR ϕ KR KR ¼ ¼ : ðIS =IU ÞE KE ΔR ϕ

(22)

A test specimen with jθ0 ¼ jθ1 would yield the maximum conceivable elementary contrast KE max ¼ (IS/IU)E ¼ 1 and thus maximum values for the   phase range ΔR ϕ max . Experiments in the electron microscope (Dowell, 1963) on the elementary contrast of thin single-crystal platelets show that the value KE ¼ (IS/IU)E ¼ 1/4 can be attained if the crystals are kept free from contamination in the electron microscope before the image is photographed. For two different test specimens, having elementary contrast KE max ¼ 1 (column II) and KE ¼ 1/4 (column III), Table 1 gives the values ΔR ϕ (lines 4 and 7) corresponding to three assumed values of the contrast KR, connected with different definitions of resolution. For the test specimen with KE ¼ 1/4, the contrast KR ¼ 1/2π (column V) leads to a permissible Table 1 Quantities Determining the Contrast of Interference Images for Two Values of the Elementary Contrast KE and Three Values of the Contrast KR Required by Differing Definitions of Resolution Limit Imin 1  IS =IU IS 1  Imin =Imax sin ΔR ϕ KR KR ¼ ; K¼ ¼ ; ¼ ¼ Imax 1 + IS =IU IU 1 + Imin =Imax KE ðIS =IU ÞE ΔR ϕ I





Imin Imax E   IS IU E

KE max

II

III

IV

0

3/5

1

1/4

1

—

V

VI

VII

Imin Imax R   IS IU R

29/40

3/4

19/21

1

1/(2π)

1/7

1/20

2

sin ΔR ϕ ΔR ϕ

1/(2π)

1/7

1/20

3

ΔR ϕ

155 π 180 155 ¼ 0:86 180

157 π 180 157 ¼ 0:87 180

172 π 180 172 ¼ 0:95 180

4

sin ΔR ϕ ΔR ϕ

2/π

4/7

1/5

6

ΔR ϕ

90 π 180 90 ¼ 0:5 180

99 π 180 90 ¼ 0:55 180

149 π 180 149 ¼ 0:83 180

7





ΔR ϕ=π

KE

—

1/4

ΔR ϕ=π

5

8

Past and Present Attempts to Attain the Resolution Limit

17

range 2ΔR ϕ ¼ π in the image. The contrast KR ¼ 1/7 (column VI) corresponds to Imin/Imax ¼ 3/4, a value often used in defining the resolution limit. The lowest of these contrasts KR min ¼ 1/20 (column VII) is just sufficient for perceiving the fringes on a photographic plate. For agreed values of the elementary contrast KE of the test specimen, and of the contrast KR in the superposition image required for defining the resolution, the range 2ΔR ϕ that can be tolerated for imaging is fixed by Eq. (22). The resolution limit for linear lattices δLL in partially coherent illumination can then be estimated in the following way (Ruska, 1964, 1965). We consider the situation in which an electron microscope is used to image increasingly smaller linear lattice spacings of given elementary contrast KE with inclined illumination, of θ1 =2 ¼ α. The superposition image of the linear lattice will be resolved until the limit at which the distribution of ϕvalues of the elementary images, over a range ϕmax–ϕmin, has decreased the contrast of the superposition image down to the value KR set by the agreed definition of resolvability. The actual distribution of ϕ-values, of course, is determined by the aberrations in the image. In electron microscopical imaging, differing ϕ-values in the elementary images are produced not only by spherical aberration, which increases with increasing angle of illumination. The specimen is illuminated not by an electron beam of constant wavelength, but by electrons of a range of wavelengths, the width of which depends on the distribution of the thermal velocities of emission of the electrons, their Boersch effect, and the loss of energy occurring within the specimen, as well as on the inconstancy of the accelerating voltage U. Defocusing of the electron image of the linear lattice also leads to an increase of the range ϕmax–ϕmin over which the ϕ-values for the elementary images are distributed, thus decreasing the image contrast. This defocusing may be due to the spread of electron velocities, or due to inconstancy of the objective lens field, as well as due to incorrect focusing. The resolution of the image, i.e., the interference fringe system, is thus limited on the one hand by the angle of illumination σ, through the spherical aberration constant Cs, and on the other hand by the chromatic aberration constant Cc, through the disturbances considered in Section 1.2. These disturbances are partly unavoidable and partly, in principle, avoidable. They comprise on the one hand fluctuations in the accelerating voltage and objective lens current, and on the other hand the focusing error, i.e., the imaging defects (v), (vi), and (viii) of the list above. The imaging defects (i) to (iv), as well as (vii) and (ix), of this list are not considered in the following discussion. The electron magnification of the specimen at the photographic plate

18

Ernst Ruska

(i) can quite easily be made sufficiently large. The movement of the specimen (ii), and the deflection of the image (iii) caused by electric and magnetic stray fields, can only with great difficulty be calculated quantitatively, but at the cost of some trouble, they may be kept sufficiently small. It is also possible to keep the decentering of the microscope (iv) below a fixed limit, so that it can be neglected here, as also can be the astigmatism (vii) since it does not influence the resolution of linear lattices. Last but not least, even changes in the specimen (ix) can now be prevented. Apart from the aperture of illumination σ, the angular error ζ in the correct angle θ1/2 ¼ α of inclined illumination, and the constants of spherical and chromatic aberration, Cs and Cc, only the following disturbances are thus to be considered as essential parameters of the optical system when imaging linear lattices in partially coherent illumination: (a) Relative energy spread of beam electrons   2ΣΔU 2 Δca U + Δsp U + Δinc U ¼ U U 2Δinc J ; from (a), (b), and J equation (II) the consequent variation in the lens strength gives a relative error

(b) Relative inconstancy of objective current

(c) 2Δk2 ΣΔU Δinc J ¼2 +4 2 k U J Δdef J . J In the assumed case of inclined illumination the dependence of the ϕ-values on all these imaging parameters is especially simple (Dowell, 1963): (d) Focusing error of the objective current

ϕ  ϕ0 F ¼ , 2π d where F is only a function of the parameters of the optical system, and so (an essential result) is no longer dependent on the lattice period d. In the following, the aperture of inclined illumination is assumed to be limited to sin α  α  1=10

(23)

Past and Present Attempts to Attain the Resolution Limit

19

and so only the spherical aberration of third order (that of fifth order being negligible) needs to be considered. The resulting limitation to δLL ¼

λE
5λE 2 sin α

(24)

is of no practical importance in present circumstances. The properties of existing fluorescent screens and photographic emulsions are such that specimens cannot be illuminated by present-day electron guns with the intensity needed for achieving a better resolution than this. Depending on the relative magnitudes of the imaging parameters, the maximum differences ϕmax–ϕmin may occur between two elementary images for which the rays, both diffracted and undiffracted, behave in any one of the following ways: (a) both rays pass through the limiting aperture at different parts of the edge; (b) one ray passes through the limiting aperture at the edge, and the other in the center; and (c) both rays pass through the limiting aperture at the same place on the edge, but they possess the two extreme values of the variation range of voltage and current. Thus, it is not possible to give an equation for ϕmax–ϕmin which will comprise all orders of magnitude of the imaging parameters. But ϕmax–ϕmin lies below a certain limit, the derivation of which can only be indicated here. From eq. (20) of the paper by Dowell (1963), with the substitutions E+

θ1 ¼ζ 2

(25)

and  2    2Δdef J θ1 ΣΔU 2Δinc J 2 Δf0 ¼ Cs  Cc + 3ζ  Cc + , J 4 U J

(26)

it follows that  

 ϕmax  ϕmin ΣΔU 2Δinc J 2Δdef J 2 Cc < σ σ Cs + + + d J 4π U J   3 2 ΣΔU 2Δinc J +ζ σ Cs + + Cc : 2 U J

(27)

20

Ernst Ruska

It is important to note that the quantity on the right-hand side does not depend on d. We now define an equivalent phase range 2Δϕ in which the ϕ-values are homogeneously distributed and such that the decrease in contrast of the superposition image is the same as it is with the ϕ-values of the elementary images which occur in practice, inhomogeneously distributed over the whole range ϕmin … ϕmax. It is not possible to take account of the probability distribution of ϕ-values owing to lack of knowledge of the probability distribution of the values of the accelerating voltage and objective current within their range of variation. Even in the simplest case, it would lead to an integration which could only be carried out numerically. So the equivalent phase range will be  

 2π ΣΔU 2Δinc J 2Δdef J 2 Δϕ  Cc σ σ Cs + + + J d U J   3 ΣΔU 2Δinc J +ζ σ 2 Cs + + Cc : 2 U J

(28)

For increasingly smaller linear lattices the range ϕmax–ϕmin, which is inversely proportional to the lattice constant, will increase until the equivalent phase range 2Δϕ reaches the value 2ΔR ϕ for which the electron image just has the contrast KR assumed in defining the resolution. The lattice constant d is then, by Eq. (28), the resolution limit δLL for linear lattices in partially coherent illumination:  

 2π ΣΔU 2Δinc J 2Δdef J 2 Cc σ σ Cs + δLL  + + J U J ΔR ϕ   3 ΣΔU 2Δinc J +ζ σ 2 Cs + + Cc : 2 U J

(29)

The corresponding aperture αLL of the objective is αLL ¼

λE : 2δLL

(30)

So from Eq. (29), we may estimate the resolution limit for linear lattices from the given elementary contrast KE of the specimen and from the assumed value of the required image contrast KR, i.e., from the permissible phase range 2ΔR ϕ and the aberration constants of the electron microscope.

21

Past and Present Attempts to Attain the Resolution Limit

As the estimated resolution is naturally strongly dependent on the value assumed for σ, it is necessary to investigate whether the specimen can be illuminated with a sufficiently high current density when using an angular aperture of this size. The suitability of an electron gun for such a purpose is measured by the value of the brightness of the beam in the immediate neighborhood of the axis, which will depend on the operating parameters of the electron gun. In magnetic imaging systems, this axial brightness RA is constant along the beam and so, in principle, can be measured in any cross section of the beam. It is given by the current density jA at the axis divided by the solid angle σ 2π within which electrons are incident on the center of the measuring area, for the limiting case σ ! 0. Thus RA ¼ lim ð jA =σ 2 πÞ: σ!0

(31)

An electron gun illuminating the specimen with brightness RA and aperture σ produces in the immediate neighborhood of the beam axis the current density jsp ¼ σ 2 πRA :

(32)

The current density jL on the fluorescent screen follows from the current density at the specimen and the specimen magnification ME in the electron image on the screen or the photographic plate, after taking into account the loss of the scattered beams. If jθ0 + jθ1 jsp

(33)

qjsp jθ0 + jθ1 ¼ ME2 ME2

(34)

q¼ then jL ¼

where qjsp is the fraction of the current density in the specimen, which passes through the objective aperture, after being scattered by a small amount. For interference fringes of separation δLL  ME to be visible in the photographic image, they must be larger than the resolving power of the photographic emulsion for linear lattices (δLL)ph, which will depend on the contrast. Therefore, it follows that ME


ðδLL Þph δLL

:

(35)

22

Ernst Ruska

In order to attain the best resolution the specimen must be illuminated through the smallest angular aperture compatible with an electron image bright enough for focusing. The minimum current density on the luminescent screen jL min which can be tolerated naturally depends also on the observer. The optimum aperture of illumination follows from Eqs. (32) and (35) as: sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jL min 1 ðδLL Þph jL min jsp : (36) σ opt ¼ ME ¼ πRA q δLL πRA jθ0 + jθ1 The quantities in Eq. (36) relate either to the electron microscope or to the specimen, and all depend on the beam voltage used. It follows that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jL min jsp σ opt  δLL ¼ ðδLL Þph : (37) πRA jθ0 + jθ1 The left-hand side of Eq. (37) can be obtained from experimental results. For instance, using the Elmiskop I electron microscope at 100 kV and σ ¼ 0.5  103, micrographs were taken of tremolite lattice planes with a ˚ (Dowell, 1963). Here we have resolution of δLL ¼ 3.2 A ˚ ¼ 1:6  103 A ˚: ðσ  δLL Þexp ¼ 0:5  103  3:2 A So the value of σ used in estimating δLL must be adjusted until the product of the assumed σ-value and the value δLL calculated from it agrees with that obtained by experiment, (σδLL)exp. In agreement with experimental experience, Eq. (29), with the parameters of the best present-day electron microscopes, predicts a better resolution for linear lattices in transmission than do Eqs. (4) and (6), which were derived by Glaser and Scherzer for the resolution limit when imaging two points radiating incoherently. When judging different electron microscopes, only resolutions of the same kind should be compared, of course. Eq. (29) may be used in the following three ways to estimate particular quantities, thus allowing a comparison of different transmission electron microscopes to be made in well-defined conditions: (1) the linear lattice resolution δLL can be estimated if the sum of all the errors is known and values for KE and KR are agreed upon; (2) the elementary contrast KE of the test specimen can be estimated if the sum of all the errors is known, and also, from a test micrograph, the lattice resolution δLL at a contrast K; and

Past and Present Attempts to Attain the Resolution Limit

23

(3) the sum of all the errors may be estimated if KE for the test specimen is known, and also, from a test micrograph, δLL at a contrast K.

1.6 Zone Plates for Improving Resolution and Contrast in Bright- and Dark-Field Images Recently a new idea has been put forward, which should enable us to improve the resolution and contrast of the electron microscopic image still further—and perhaps even by a factor of 2. In the electron microscope, as is well known, a narrow aperture is situated in the back focal plane of the objective, which cuts off electrons diffracted or scattered at large angles to the optical axis. Hence a structural element of the specimen, which is strongly scattering, will contribute fewer electrons to the image than its surroundings, and so the corresponding image element will be darker. It has been suggested by Hoppe (1961), Lenz (1963), and Riecke (1964) (see also Hanszen, Morgenstern, & Rosenbruch, 1964) that this aperture could be made in the form of a zone plate, the opaque areas of which would prevent the passage of those electrons, which at the image plane have unfavorable phase differences with respect to the paraxial beam. The realization of this idea will certainly be beset with difficulties. But when we consider how many difficulties, which at one time seemed insurmountable, have already been overcome during the development of the electron microscope, we have some reason to be confident for the future.

2. THE SINGLE-FIELD CONDENSER OBJECTIVE 2.1 Principle, Ray Paths, and Construction The stage of development now attained justifies an attempt to improve the resolving power still further by reducing the spherical aberration of the objective and by using a shorter electron wavelength. The attack on the problem, now being made in several laboratories, aims at achieving a limiting res˚ between two points radiating incoherently and it is probable olution of 1 A that this can be realized in the not-too-distant future. To attain the optimum resolution at a given beam voltage, it is necessary to find the objective lens having the smallest spherical aberration constant. This must be a magnetic objective, since, under similar conditions, the spherical aberration constant of a magnetic lens is smaller than that of an electrostatic lens. Another reason why only magnetic lenses need to be considered is the intention to use electrons of as short a wavelength as possible and therefore of high voltage.

24

Ernst Ruska

Already 20 years ago such an “optimum electron objective” was designed and its focal length, as well as the spherical and chromatic aberrations, calculated by Glaser (1941). It is a strong magnetic lens, the first half of its field being used as condenser and the second half as objective. In this lens, the specimen must lie in the position at which the axial field strength is a maximum. Such a magnetic field, which simply decreases from a maximum, has a spherical aberration constant about 10 times smaller than the magnetic objectives hitherto used, where the specimen is generally situated in front of the whole field; the chromatic aberration constant is half as small, and the resulting resolution limit nearly twice as good. Though their low aberration has been known for more than 20 years, such single-field condenser objectives have not so far been used in electron microscopes, for two reasons. On the one hand, complications associated with the electron optics and the construction of such an unusual lens system were feared, not unreasonably. On the other hand, the smaller spherical aberration of the new type of objective could not bring any improvement in the resolving power of the electron microscope, so long as other defects were limiting its performance. Now that a sufficiently effective reduction of these limiting factors has been achieved, the lens design indicated by Glaser has recently been explored (Riecke, 1962a; Ruska, 1962). Fig. 3 shows the principle of the single-field condenser objective. The first half of the field, which acts as a low-aberration condenser of short focal

Condenser field

Selected area aperture

Objective field

Condenser aperture

Objective aperture

Intermediate image

Specimen

Fig. 3 Illumination and imaging of the specimen by means of the single-field condenser objective, the “optimum magnetic lens” due to Glaser (1943).

Past and Present Attempts to Attain the Resolution Limit

25

length, will produce at the specimen a highly reduced image of a “field aperture” situated within the convergent beam and before the field. The illuminated area may be varied as desired, for example, reduced to a cross section of less than 0.1 μm, by inserting apertures of different sizes. Control of the intensity of the beam passing through the specimen is achieved by varying the angular aperture of the beam illuminating the field aperture. For this purpose, as shown in Fig. 4, two additional strong condenser lenses

Electron source

Aperture Reducing lens One-stage reduced source

Long focal-length condenser Selected area aperture Two-stage image of source Plane of entrance aperture Prefield condenser One-stage reduced image of the selected area aperture Specimen Objective lens Three-stage image of source and diffraction pattern Plane of exit aperture Objective aperture Diffraction lens

One-stage magnified image of the specimen Projective lens One-stage image of the diffraction pattern

Fluorescent screen and photosensitive layer

Two-stage magnified Two-stage magnified image of the specimen image of the diffraction pattern

Fig. 4 Beam paths for imaging a selected specimen area and its diffraction pattern with the single-field condenser objective.

26

Ernst Ruska

Soft iron case

Vacuum seals

Upper lens coil

Ring-shaped chambers for cooling water

Pole-piece system Insert for pole-piece system

Crosswise control for aperture adjustment

Specimen insertion Specimen airlock

Lower lens coil

0

100 mm

Fig. 5 Cross section of the single-field condenser objective.

are situated between the source of illumination and the aperture. The first of them, of short focal length, produces a reduced image of the illuminating source. The second, of long focal length, projects the reduced image of the illuminating source at approximately unit magnification into the plane of the aperture of the forefield condenser. The size of this second image of the illuminating source defines the beam intensity with which the specimen is irradiated. To prevent any loss of intensity in the second image of the source, the long focal length condenser is provided with two stigmators of second and third order (Riecke, 1962b). Figs. 5 and 6 are simplified cross sections of the condenser objective and of its pole-piece system.b Fig. 7 shows how a very small area of the specimen may be illuminated with this lens.

2.2 Electron Beam Energy and Resolution Limit for Two Incoherently Radiating Points The combined condenser objective has been designed with sufficient iron in the magnetic circuit to ensure that at excitations up to 12,000 ampere-turns, b

The author is grateful to Mr. F. St€ ocklein for the careful execution of this difficult construction.

27

Past and Present Attempts to Attain the Resolution Limit

Protection tube Protection cap

Pole piece

Cross-stage for specimen

Adjusting control for cooled diaphragms

Adjusting control for aperture diaphragm

Stigmator poles

0

Stigmator poles

mm

10

Fig. 6 Pole-piece system of the single-field condenser objective.

no stray fields, separate from the main lens field, are present either before the forefield condenser or after the objective lens field proper. It was fitted with a pole-piece system of short focal length in which the bore diameter and the gap width together amount to 4.5 mm. Measurements on this single-field condenser objective (Hildebrandt, 1964) show an axial field distribution (Fig. 8), which corresponds only approximately to the form of Eq. (10). Owing to saturation of the iron circuit, the magnetic field spreads out somewhat along the axis with increasing excitation. The increase in the half-width h of the axial field distribution with ampere-turns I is represented by h ðmmÞ ¼ 1:27 + 0:057 J ðkAÞ:

(38)

The properties of a single-field condenser objective can be calculated on the assumption that the field distribution corresponds to Eq. (10) (Glaser, 1943), and this is taken as basis for the following discussion, together with a variable half-width of the lens field (h) according to Eq. (38). The strength of the condenser-objective lens is k2 ¼ 3 and the lens flux is given by Eq. (11) as J ðkAÞ ¼ 0:924√Ur ðkVÞ:

(39)

28

Ernst Ruska

Fig. 7 Investigation of a small region of the specimen by means of a selected area aperture in an electron microscope with single-field condenser objective: (A) image. (B) Electron diffraction pattern corresponding to (A).

Spreading of the field will thus occur when the beam voltage is increased, resulting in a corresponding increase in the focal length, in the spherical aberration constant, and in the diffraction error, so that the improvement of resolving power with increasing voltage is somewhat less than expected. Table 2 shows values for the following parameters of the single-field condenser objective for 10–1000 kV electrons: 

ΣΔU ΣΔJ Ur , λE ,J,h, f , Cs , Cc , αPP , δPP , + U J

 : max

29

Past and Present Attempts to Attain the Resolution Limit

1.0

8 kA

12 kA

B(z) / Bmax

0.8

0.6

0.4 Measured Glaser

Measured

Glaser 0.2

−8 −6 −4 −2

0

2

4

6

8 0 −8 −6 −4 −2 z (mm)

0

2

4

6

8

Fig. 8 Comparison of the measured axial field strength distribution of the single-field Bmax condenser objective with the field strength distribution Bðz Þ ¼ , on which the 1 + ðz=hÞ2 focal lengths and aberration constants—calculated by Glaser (1941)—are based (Hildebrandt, 1964).

The optimum apertures αPP and resolutions δPP for two points, radiating incoherently with a Lambert distribution, have been calculated for the Gaussian image plane (Glaser, 1943) as well as for the plane of least confusion (Scherzer, 1949). Though the resolutions given relate to points radiating incoherently, they approximately indicate the variation of resolution with beam voltage for two points in the specimen which are more strongly scattering than their surroundings. From these data, we reach the following conclusions: due to the low spherical aberration of the single-field condenser objective, it will be possible to achieve a high resolution even at very low beam voltages, and consequently under especially favorable conditions of contrast. For this purpose, it is necessary to find means of overcoming the severe difficulties occurring when the electron microscope is operated at low voltage, difficulties resulting from specimen heating, charging phenomena, and high sensitivity to stray electrical and magnetic alternating fields.

Table 2 Parameters of Single-Field Condenser Objective (k2 ¼ 3) With Pole-Piece System Having d + s ¼ 4.5 mm for Voltages From 10 to 1000 kV

U

kV

10

20

50

100

200

500

1000

kV ˚ A

10.098

20.39

52.45

109.8

239.8

745.0

1980

0.122

0.086

0.054

0.037

0.025

0.014

0.009

J ¼ 0.924 √Ur

kA

2.94

4.17

6.57

9.68

14.29

25.22

41.10

Values for h

Verified by measurement

h ¼ 1.27 + 0.057, J ¼ f

mm

1.44

1.51

1.64

1.82

2.08

2.71

3.61

Cs ¼ 0.3, h  Cc/2

mm

0.43

0.45

0.49

0.55

0.62

0.81

1.08

4J h

kG

8.2

11.0

16.0

21.5

27.5

37.5

45.5

102

1.46

1.33

1.16

1.03

0.90

0.73

0.62

˚ A

5.27

4.00

2.94

2.29

1.77

1.23

0.98

102

1.83

1.66

1.45

1.28

1.12

0.91

0.77

˚ A

4.06

3.16

2.27

1.76

1.36

0.95

0.75

p  106

25.7

21.3

16.0

12.6

9.7

6.4

4.4

3

Ur ¼ U(1 + 0.98  10 U) λE ¼

0  388 √Ur

Bmax ¼

Glaser (1943) pffiffiffiffiffiffiffiffiffiffiffiffi αPP ¼ 1:13 4 λE =Cs qffiffiffiffiffiffiffiffiffiffi δPP ¼ 0:56 4 λ3E Cs Scherzer (1949) pffiffiffiffiffiffiffiffiffiffiffiffi αPP ¼ 1:41 4 λE =Cs qffiffiffiffiffiffiffiffiffiffi δPP ¼ 0:43 4 λ3E Cs   pffiffiffiffiffiffiffiffiffiffi ΣΔU ΣΔJ + ¼ 0:28p λE =h U J max

Extrapolated

Beam voltage (U), electron wavelength (λE), ampere-turns in the objective (J), half-width of the lens field (h), spherical and chromatic aberration constants (Cs, Cc), maximum induction (Bmax), optimum objective aperture and resolution limit for two points radiating incoherently (αPP and δPP), and maximum tolerance for instabilities (ΣΔU/U + ΣΔJ/J)max.

Past and Present Attempts to Attain the Resolution Limit

31

Already at the beam voltage of 100 kV, as usually used at present, the new lens will yield the same resolution as could be attained with a conventional objective at 400 kV. Thus the new condenser objective will achieve a further improvement in the resolving power of the existing electron microscope with less complication and cost than a corresponding increase in the high voltage would do—a point of great interest for the ordinary user. Though they require a considerable expenditure, which restricts the number of users, very high beam voltages are especially valuable for metal-physical investigations. Several high tension electron microscopes, with conventional objective lenses, have already been built with maximum voltages of 300–400 kV and in one case (Dupouy, Perrier, & Fabre, 1961) even greater than 1000 kV. The new type of lens would be advantageous for such microscopes since its focal length is relatively short even at high voltages and consequently the magnification remains relatively high. The single-field condenser objective has the further advantage that its chromatic aberration constant Cc is only half that of conventional objectives of strength k2  0.6. Also the chromatic aberration is proportional to the sum of all the errors due to current and voltage inconstancy, of which the relative spread 2ΔcaU/U of the energy of the electrons emitted from the cathode is the most serious, because it cannot be reduced effectively. As this quantity is inversely proportional to the beam voltage, the radius of the corresponding circle of confusion, ρca U, will, at high voltage, be still smaller on this account (as well as because the chromatic constant Cc is itself decreased in value) than it is in electron microscopes with low-powered lenses at low voltages. However, as the required lens flux increases appreciably with beam voltage, it will become more and more difficult to achieve an adequate dimensioning of the iron circuit and to dissipate the power from the lens windings. It is of some interest to compare the single-field condenser objective with objectives of conventional dimensioning and considerably lower k2-values. The measurements of the pole-piece system and the consequent possibilities for specimen manipulation differ considerably. Table 3 compares the bore (d) and gap width (s) of the pole-piece systems of three 400 kV-transmission electron microscopes, two of them equipped with strong objectives of k2 ¼ 3 and one with a weak objective of k2 ¼ 0.4. In the first place the strong and weak lenses have the same maximum field strength Bmax, and in the second place, they have the same resolution limit for two points radiating incoherently δPP. The single-field condenser objective has the advantage not only that the specimen lies in the gap between the pole pieces, but also that appreciably higher values may be chosen for the pole-piece measurements

32

Ernst Ruska

Table 3 Comparison of Objectives of Different Strengths for a 400-kV Transmission Electron Microscope Ur 5 557 kV U 5 400 kV

(a)

(b)

k2

1

3.0

0.4

3.0

J

kA

21.8

7.9

21.8

Bmax

kG

20

20

7

f

mm

4.3

3.4

12

h

mm

4.3

1.6

12

d+s

mm A˚

12.1

4.4

30

1.6

2.2

2.2

δPP

(a) At the same maximum field Bmax. (b) At the same resolution limit δPP.

(bore and gap) at given beam voltage and resolution limit. It is thus easier to design devices for introducing the specimen into the objective and manipulating it into the position needed for microscopic operation. In addition, it is possible to produce pole pieces with wider bores and consequently lower astigmatism. The possibility of access to the specimen from all sides through the wide gap permits a better combination of the devices now frequently demanded for orientating, tempering, deforming, or acting chemically on the specimen, and also for providing specimen space cooling. Only with lenses having values of k2 greater than about 1.2 is it possible to make use of these advantages (von Ardenne, 1944).

2.3 Nature and Magnitude of Permissible Disturbances If the Resolution Limit for Two Points Radiating Incoherently Is to Be Attained in the Image In the single-field condenser objective the spherical aberration constant is particularly small and thus the optimum aperture is particularly large. To realize the very high resolution which is consequently made possible, the circle of confusion due to the sum of all disturbances has to be sufficiently small with respect to this resolution. For the single-field condenser objective with k2 ¼ 3 and zG ¼ 0, it follows from Eq. (14b), with Cc ¼ 0.59 h from Eq. (13) and with Cs ¼ 0.3 h (Glaser, 1941), that

Past and Present Attempts to Attain the Resolution Limit

33

pffiffiffiffiffiffiffiffiffiffi Δca U + Δsp U + Δinc U 2Δinc J + Δast J + 2Δdef J + ¼ p  0:28 λE =h: (40) U J For U ¼ 100 kV the highest permissible sum of disturbances, with ˚ and h ¼ 1.82 mm ¼ 18.2  106 A ˚ , will be λE ¼ 0.037 A Δca U + Δsp U + Δinc U 2Δinc J + Δast J + 2Δdef J + ¼ 12:6  106 p: U J

(41)

Even allowing a relatively large radius for the resulting circle of confusion, having the value pδPP ¼ (1/3)δPP, the sum of all disturbances must not exceed the value 4.2  106. Relative to U ¼ 100 kV the spread 2ΔcaU of the emission energy of electrons from the cathode by itself almost reaches this fraction so that all the other relative fluctuations of J and U must be smaller by at least an order of magnitude. However, Δdef J can hardly be made smaller than Δinc J, and the smallest exactly adjustable current interval 2Δdef J hardly smaller than 2Δinc J. On the other hand, we can set ΔspU/U ¼ 0 as the specimen must be very thin if its structure is to be so highly resolved. Thus we can only allow the following very small disturbances: Δca U 0:4V ¼ ¼ 4  106 U 100kV and Δinc U 2Δinc J Δast J 2Δdef J ¼ 2  107 : + + + J U J J This result shows that more precise limitation of the spread of emission energies is the first necessity if we are to achieve a further increase in resolving power.

2.4 Resolution Limit for Linear Lattices, in Partially Coherent Illumination, When the Disturbances Are so Small That They Do Not Prevent the Attainment of the Resolution Limit for Two Points Radiating Incoherently For the single-field condenser objective, with Cs ¼ 0.3 h and Cc  0.6 h (Glaser, 1941), the resolution limit δLL for linear lattices in partially coherent illumination is given by Eq. (29) as:

34

Ernst Ruska

   2 

1:2h σ ΣΔU 2Δinc J 2Δdef J 3 2 ΣΔU 2Δinc J + +ζ σ + + + + : δLL  σ 2 J U J 4 U J ΔR ϕ=π (42)

The permissible magnitudes of the various disturbances have been evaluated in the previous section in respect of the realization of the resolution limit for two points radiating incoherently. We now proceed to estimate the resolution limit for linear lattices, in partially coherent illumination, which can be obtained with the single-field condenser objective in the presence of these disturbances. For this estimation, we suppose the test specimen to be a thin single crystal with KE ¼ 1/4 and the contrast for defining the resolution limit to be KR ¼ 1/20. From Eq. (22), or from Table 1, the equivalent half-width of the phase-range permissible in the image is ΔR ϕ=π ¼ 149=180 ¼ 0:83. As we want to achieve the best possible resolution, we assume precise inclined illumination: i.e., ζ ¼ 0. We use an aperture of illumination σ ¼ 2  103. ˚ , we have For U ¼ 100 kV, and with h ¼ 1.82 mm ¼ 18.2  106 A δLL 

˚   1:2  18:2  106 A ˚ ¼ 9λE : 2  103 2  106 + 4:2  106 ¼ 0:33A 0:83

From Eq. (30) the corresponding aperture of the objective will be αLL ¼

0:037 ¼ 0:056: 2  0:33

In order to check whether the aperture of illumination σ is adequate for irradiating the linear lattice intensely enough for the image to be sufficiently bright for focusing, we also calculate the left-hand side of Eq. (37): ˚ ¼ 0:66  103 A ˚: σ  δLL ¼ 2  103  0:33A This calculated value for 100 kV in fact exceeds the experimental value actually achieved when using 100 kV: ˚ ¼ 0:6  103 A ˚ ðσ  δLL Þexp ¼ 0:3  103  2A (Komoda & Otsuki, 1964, Hitachi HU-11A, 100 kV, hair-pin filament, ˚ for linear lattices in cf. Fig. 2). The estimated resolution limit of δLL ¼ 0.33 A partially coherent illumination, for the single-field condenser objective in the presence of the assumed disturbances, can thus be realized even with the use of a hair-pin filament.

Past and Present Attempts to Attain the Resolution Limit

35

3. MOVEMENT OF THE IMAGE DURING PHOTOGRAPHIC EXPOSURE 3.1 Movement and Heating of the Specimen To obtain a sharp micrograph of an image, it is an elementary requirement that the image must not move during the time of exposure. When recording electron microscopical images, it has above all to be ensured that the specimen does not move unduly, while the photographic emulsion is being exposed to the imaging beam. The component of the specimen movement perpendicular to the objective axis and the electron beam is the most critical. It is essential that the specimen does not move in this direction by more than ˚ at the most during an exposure of about one minute, i.e., by not more 1A than the closest separation of two neighboring atoms. In the course of time, it has been found possible largely to fulfill these extreme requirements of mechanical stability of the specimen by the following two simple measures. First the occurrence of an elastic displacement between the specimen and the objective was restricted. To achieve this, the mechanical path from the specimen support to the specimen stage, and onto the pole-piece system of the objective, was made as short and as free of back lash as possible. The stage was prevented from moving spontaneously relatively to the pole-piece system by means of sliding friction. Fig. 9 shows an example of such a design (Ruska, 1950). Second, the strong interaction of fast electrons with matter leads to rapid heating even in the case of very thin specimens, and this will be intensified when the electron beam happens to fall on the bars of the specimen grid. As they are much thicker than the specimen, they will absorb the entire beam energy. Thus the thermal drift caused by uneven local heating ˚ /min during the exposure must also be kept down to the order of a few A time. For this purpose the electron source is imaged on the specimen with high demagnification (Ruska, 1956), so that only that area of the specimen is illuminated, which appears in the final image on the fluorescent screen. This area can be the smaller, the higher the electron magnification is raised. For instance, a specimen area just 1 μm in diameter corresponds to an image 10 cm in diameter on the fluorescent screen at an electron microscopical magnification of 100,000 times. Furthermore, spectacular progress in specimen preparation techniques has made it possible on the one hand to produce support films which, ˚ thick, are capable of adequately withstanding although only about 150 A mechanical stress and chemical action, and on the other hand to cut sections

36

Ernst Ruska

Pushrod

Iron flange

Specimen stage Wheel

Iron core

Hold down spring Specimen cartridge

Return spring

Coil Pole vpieces

Fig. 9 Specimen stage for an electromagnetic objective. This design corresponds to the € 100 (1949) and the Elmiskop I electron microscopes (1954) specimen stages of the UM produced by Siemens and Halske AG.

˚ . Thus the specimen volume of biological material as thin as about 300 A which is irradiated, and consequently heated by the electron beam, may today be restricted to the order of a cube having an edge length of 1/2000 mm. The beam energy absorbed in this extremely minute specimen volume is so small that it will be dissipated to its surroundings with only a very small increase in the specimen temperature. Accordingly, the specimen may be kept practically at room temperature, even when it is being irradiated ˚ /cm2, as is required for an electron with a beam current density of about 1 A image magnification of 200,000 times. Fig. 10 compares the older and the present-day conditions of specimen irradiation. It is also now possible to fix films and thin sections on specimen supports of relatively large free bore, up to 1 mm in diameter (Dowell, 1957). The much narrower electron beam now used need not strike the specimen support itself, even when the specimen is searched, and so should not give rise to any intermittent thermal drift. Apart from other objections which have since been proved wrong, the original doubt whether it would be possible to eliminate the strong heating of the specimen in the electron beam, and its consequent movement and destruction, hindered the development of electron microscopy for many years.

37

Past and Present Attempts to Attain the Resolution Limit

Holding screw for specimen carrier Electron beam

Electron beam

≈1000 mm

1 mm

≈0.5 mm

70 mmf 1933

f

Specimen carrier

1000 mmf

Specimen

1963

≈0.05 mm

Fig. 10 Comparison of the early and present methods of illuminating the specimen in the electron microscope: by decreasing the irradiated area and increasing the free area of the specimen, the heating effect is reduced.

3.2 Movement of the Electron Image due to Electric and Magnetic Fields Although the deflection of electron beams by electric and magnetic fields is the first condition for the design of electron lenses and electron microscopes, it may unfortunately also lead to rather inconvenient effects during operation. Analogous difficulties of this kind had not to be overcome during the development of the light microscope. If the deflection of the electron beam between the specimen and the final image varies with time, the electron image may move on the fluorescent screen or the photographic emulsion even when the specimen is completely steady with respect to the objective. In the case of a fast deflection, for instance by magnetic alternating fields of 50 c/s, the resolution on the fluorescent screen and photographic image will suffer. In the case of a slower deflection, which allows the movement of the image on the fluorescent screen to be followed with the eye, the resolution of the image will still suffer. A deflection of the electron beam which is constant with time leads to a misalignment with respect to the lenses and thus also decreases the image resolution. By striking the inner surfaces of the microscope, which are often covered with thin insulating films, the electron beam may produce undesired electric fields within the electron microscope. The locally varying potential differences, which will usually also vary with time, produce correspondingly varying electric fields which deflect the electron beam. Such an electric stray

38

Ernst Ruska

field most frequently occurs at the objective lens aperture, which is bombarded by electrons scattered in the specimen. Owing to its lack of rotational symmetry, this field—even if it is steady with time—will disturb the completely axially symmetric focusing of the electron beam, which is necessary for stigmatic imaging, i.e., imaging free of astigmatic aberration. Such detrimental electric fields can at present only be eliminated by regularly cleaning all metal surfaces exposed to the beam, so that their conductivity is maintained. Magnetic alternating fields of 50 c/s occur from time to time in nearly all rooms with electric installations. If the electron microscope is not properly shielded, these fields will periodically deflect the electron beam and the electron image. Also any steady stray magnetic fields from the iron casings of the magnetic electron lenses may penetrate into the microscope column at other points. As the shielding around the microscope column is usually not rotationally symmetrically distributed, any perturbing fields which occur there are also not rotationally symmetric and thus will have a disturbing effect even when steady with time. Depending on the strength of the lens current, these fields will cause a deflection and misalignment, of variable strength, of the electron beam with respect to the whole lens system, which will also result in a worsening of the resolution. From this two requirements follow: on the one hand the electron beam must be shielded as completely as possible by the wall of the microscope column against magnetic alternating fields, generated either by the electrical supplies of the microscope itself or by electrical instruments and circuits in its immediate neighborhood. On the other hand, it is necessary to install the microscope as far away as possible from such perturbing magnetic fields.

4. LACK OF SHARPNESS OF THE IMAGE 4.1 Instability in the Lenses The fact that electron beams are deflected by magnetic fields gave rise to a second difficult problem, i.e., how to keep constant the focal length of a lens, which depends on the velocity of the electrons and on the strength of the lens field. For this purpose, it was necessary to produce direct voltages of the order of 100 kV with a constancy of better than 1 part in 105 for accelerating the electron beam, and also to supply the lens coils with currents of even higher stability. For two reasons, these stabilization requirements will become still more stringent in the future. In order to penetrate thicker specimens, and also to get a still better resolution with thinner ones, the

Past and Present Attempts to Attain the Resolution Limit

39

microscope will be operated at higher voltages and with stronger lens fields, which will require more highly stabilized beam voltages and lens currents. Again, analogous problems did not arise in the development of the light microscope, since the focal length of optical lenses is dependent on factors which are effectively invariable with time, such as the curvature of the lens surfaces and the refractive index of the glasses composing them.

4.2 Noncircularity of Lenses Owing to the shortness of electron wavelengths, much higher precision is required in the rotational symmetry of electron lenses than in light optical lenses. In magnetic lenses the degree of symmetry of the field depends on the degree of circularity of the pole-piece bore and on the degree of homogeneity of the magnetic material. The deviation from true rotational symmetry which can be tolerated lies below 1 part in 105. So, up to the present time, it has not been possible to produce adequately symmetrical fields in lenses of very short focal length by machining pole-piece bores which are true enough, in magnetic material which is sufficiently homogeneous. But a method was found of restoring the symmetry of a noncircular lens field by means of an electron optical cylinder lens situated in the immediate neighborhood of the lens to be corrected. The power of refraction and the azimuth of this stigmator may be adjusted in such a way that the noncircularity of the latter lens is compensated. As the imaging beam has its largest angular aperture in the first image stage of the microscope, the spherical aberration and astigmatism of the objective lens are the main factors limiting the quality of the recorded image. The noncircularity of the objective field varies with time, particularly owing to the charging of the aperture, so that the operator must be able to change the stigmator adjustment while he is working. The introduction of a stigmator which can be adjusted during operation of the electron microscope was a decisive contribution to the attainment of the present high resolving power.

5. CHANGES IN THE SPECIMEN 5.1 Interaction Between Specimen, Electron Beam, and Residual Gases The utilization of the high resolving power of the electron microscope has, however, been hindered by yet another circumstance which gives rise to no difficulty in the case of the light microscope. The electron beam may change the composition and structure of the specimen within the illuminated region

40

Ernst Ruska

by ionization. Such radiation effects may occur either between the beam electrons and the specimen alone, or with the participation of gas and vapor molecules as well, since these strike the surfaces of the specimen in great numbers, even though the electron microscope is evacuated down to a pressure of 105 torr. This last named phenomenon includes the so-called contamination of the specimen by carbon layers built up on the surface, which has long been known. These result when the molecules of various hydrocarbons land on the specimen, molecules evaporating permanently into the evacuated space, either from the oil required for the vacuum pumps or from the rubber rings and greases used for making the microscope vacuum-tight. The lower the temperature of the specimen, the longer will they stay on its surface before they are evaporated back into the evacuated space. During this time, they may be struck by beam electrons. The first result is the production of polymerizates which will not reevaporate, and further irradiation will release hydrogen, so that solid carbon remains. Thus a carbon layer builds up on both surfaces of the specimen, growing thicker with time. The growth of carbon layers may be observed best when using a supporting film with very small holes; these holes get smaller during irradiation and finally fill up altogether. At low magnifications the structure of this carbon layer, which is imaged at the same time as the specimen, will not be resolved in the image, and so only the image contrast of the coarse structure of the specimen will decrease. At high magnification and sufficient resolution, however, the structure of the two carbon layers as well as the fine structure of the specimen will be resolved. Their superposition in the image makes it difficult to observe and interpret the fine structure of the specimen. Similarly, nitrogen and oxygen, as well as hydrogen, may be released by the ionizing action of beam electrons on organic specimens and supporting films, while the carbon atoms remain in position. It must be considered a fortunate circumstance that the morphology of biological and medical specimens is preserved by this carbon skeleton. In the usual conditions of electron microscopy the decomposition of organic compounds does not occur very rapidly, so that it is still possible to stain organic specimens, and in particular tissue sections, after they have been observed with the electron beam for a short time. But this is only possible so long as the organic nature of the compounds is preserved to a certain degree. The basic carbon skeleton will no longer react with chemical stains or contrast media. As has also been known for a long time, another radiation-chemical process can occur in the case of carbon-containing specimens: carbon may be

Past and Present Attempts to Attain the Resolution Limit

41

removed from the surface. Nearly all organic specimens in biology and medicine will undergo this change. The main cause of the process, counteracting the formation of a deposited layer, has recently been found in the watervapor molecules which still exist in the residual gas and thus continuously bombard the specimen (Heide, 1963; he also gives extensive references to the development of our knowledge of specimen contamination). The beam electrons ionize the water molecules absorbed on the surface, the carbon in the organic compounds is then oxidized, and the resulting carbon monoxide or carbon dioxide is pumped off. By this process, carbon-containing structures in the specimen will be changed or may even disappear. This effect is nearly always undesirable, of course, although in certain cases it may actually be useful, when a controlled degradation of the specimen is sought. The speed with which the carbon layer is built up increases with the partial pressure of hydrocarbons, whereas the speed at which carbon is removed increases with the partial pressure of water vapor. Both processes increase with the electron current density, until a limiting rate is attained, and decrease with increasing specimen temperature. The temperature also increases with the electron current density and with the size of the illuminated region, as well as with the mass thickness of the specimen, whereas it decreases with increasing heat conductivity of the latter. Let us now consider what happens to the specimen during observation in the transmission electron microscope. For simplicity, we shall discuss only the case where the specimen temperature is not appreciably increased, i.e., when working with beams of small diameter and very thin specimens. The specimen is being bombarded on the one hand by the electrons needed for imaging it, and on the other hand by the various molecules, atoms, and ions which still remain in the evacuated space of the electron microscope. Electrons may ionize some atoms in the specimen, so that the structure appearing in the electron image may be changed, and against ionization very little can be done. Of course one can reduce the beam intensity and the exposure time to a minimum, but this can only be done within certain limits. Even if an ideal photosensitive surface or some other method of recording the final image were available, which utilized every individual electron, a minimum number of incident electrons are needed on unit area of the specimen in order to obtain a given resolution. This is shown by information theory: for example, at least 1013 electrons are needed per square centimeter ˚ . But this already corof specimen for a resolution of no better than 150 A responds to a dose of 1,000,000 rad in biological matter when using electrons accelerated by 100 kV. Living matter is the faster killed by ionization, the

42

Ernst Ruska

more it is organized. At present, therefore, the electron microscopy of living matter is still in a very early stage. Other effects of ionization may be the breaking down of crystal lattices, as well as various chemical changes which occur without heating of the specimen; furthermore, some organic substances undergo a loss of material. Fortunately, it is not ionization itself which is disturbing, but the subsequent chemical and structural changes which ionization initiates. The ionization does not depend on temperature, whereas the resulting chemical or structural changes may be retarded by reducing the specimen temperature. From observations made independently of electron microscopy, it is known that many types of radiation damage are reduced at lower temperatures. We shall revert to this subject below. Consider now what particles strike the specimen in the microscope apart from the beam electrons, i.e., the various molecules which are still flying about in the vacuum in the form of residual gas. Of these, the hydrocarbon molecules are particularly harmful since they form the polymerized layers of contamination. Methods for partially preventing contamination by hydrocarbons have been known for some time, and they are certainly worth using. But we must first of all consider what happens if hydrocarbons are completely kept away from the specimen. Suppose that the object is surrounded with a cooling chamber, which prevents any hydrocarbon molecules from striking the specimen. Then, for the first time, the specimen surface will be freely exposed to the vacuum. It is no longer shielded by a layer of hydrocarbons, which, under irradiation, becomes a steadily growing carbon layer, but it is directly exposed to the residual gas molecules incident on it. It would indeed be strange if no new radiation-chemical processes should then occur during electron bombardment. In order to be clear about possible processes of this kind, one first has to know which molecules, and in what numbers, are likely to bombard the two surfaces of the specimen. To find an answer to this question, we have experimentally studied the composition of the residual gas in the microscope by means of a Topatron mass spectrometer (Heide, 1964a). Without going into the details of the measurements, the results are briefly set out in Fig. 11. The individual measurements show considerable, and to some extent characteristic, fluctuations. The reason for this lies in the slightly varying vacuum conditions in the microscope, to which one normally pays no attention. Nevertheless, the experimental values remain within one order of magnitude. In addition, we found that water molecules (2 + 16 ¼ 18) form the largest component of the molecules incident on the specimen. The OH radicals

43

Past and Present Attempts to Attain the Resolution Limit

OH 10

12 14 16 18

32

39

A

C4H...

C3H3 C3H5 C3H7 CO2

2

C3H7

CH 10−13

C 2H

CH3

CH4

10−12

C 2H 2 C2H3

C

10−11

O2

H2

N,CH2 CH4

10−10

I (A)

N2,CO

H2O

10−9

28

20

30

41 43 44

40

56

50

(A) 10−9

C12H...

135

148

165

C13H...

C11H...

97

C10H...

84

C9H...

C7H...

71

C8H...

C6H...

C4H...

C5H...

I (A)

C3H...

10−10

10−11

10−12

10−1339/41/43 ~56 50

100

109

m/e

122

177

150

(B) Fig. 11 (A) Ion currents measured by means of a Topatron in the Elmiskop I. Mean values from numerous measurements plotted on a logarithmic scale ¼ black solid lines; measured extreme values ¼ horizontal lines. (B) As in (A) but for greater mass/charge ratios. In this mass region the individual mass numbers are no longer separated by the Topatron and the lines plotted may in reality consist of a whole group of values due to a particular hydrocarbon.

44

Ernst Ruska

detected also originate from water molecules. Nitrogen (2  14 ¼ 28) and carbon monoxide (12 + 16 ¼ 28), which unfortunately could not be separated in these measurements, form the next largest component and then follow decreasing contributions of hydrogen, methane, oxygen, and carbon dioxide. In the mass region above 56 all the hydrocarbons are represented; the groups with 5, 6, 7 carbon atoms, and so on, follow each other regularly with a separation of about 14 mass units. This knowledge of the composition of the residual gas provides the basis for an understanding of the radiation-chemical reactions which may be expected. According to the chemical composition of the specimen, one has to decide in each case which processes are possible or probable. It must, however, be taken into account that under the influence of the ionizing electron beam, short-lived compounds may be created which otherwise do not occur. Even reactions between noble gas and carbon atoms have been observed. The greatest role in radiation-chemical reactions is normally played by water vapor, however; every water molecule absorbed on the surface and ionized by electron impact has a high probability of reacting with any oxidizable substance. It is, of course, always undesirable for carbon atoms to be removed from the specimen, both when it is biological in nature and in many other instances. Therefore a method of preventing contamination will only be of value if water molecules as well as hydrocarbons are, as far as possible, kept away from the specimen.

5.2 Specimen-Space Cooling as a Means of Preventing Changes in the Specimen due to Residual Gas During Electron Irradiation Several decades ago, it was recognized that hydrocarbons in the residual gas in the electron microscope were the cause of the growth of a carbonaceous layer on the specimen in the region irradiated by the electron beam. In order to prevent this “contamination,” attempts were made in several laboratories to reduce the partial pressure of hydrocarbons in the specimen region by cooling with liquid air, but with only partial success. A first experiment on these lines (Hillier, 1948) was not successful, probably because the relatively large specimen holder (at room temperature) was not completely surrounded by a chamber cooled down to 180°C. The first success was achieved by an arrangement (Leisegang, 1956) which allowed the specimen and the chamber surrounding it to be cooled to any given low temperature, down to a maximum of 150°C. When the specimen and the cartridge reached temperatures of about 80°C contamination ceased. At still lower

Past and Present Attempts to Attain the Resolution Limit

45

temperatures, carbon was removed at an increasing rate from carboncontaining specimens. In order to prevent carbon deposition or removal the temperature had to be maintained at 80°C. But it is difficult and sometimes impossible to fulfill this requirement because of the influence of the irradiation on the specimen temperature. A method of specimen-space cooling for preventing contamination, in which the specimen is kept at room temperature, was later developed by Philips for the 100-kV electron microscope EM 200 (van Dorsten & van den Broek, 1960). But they gave no data regarding the temperature of the cooling chamber, the degree to which it surrounds the specimen or the rate of carbon removal. Shortly afterward a device was developed for the Hitachi microscope (Komoda & Morito, 1960) with which contamination could also be reduced: a comparatively large space around the specimen is isolated from the rest of the microscope by throttling apertures and is pumped separately. In addition to the specimen, there is a copper block in this space, cooled to low temperature. Here, of course, one has to take into account that the gas desorbed by the numerous warm surfaces near the specimen will have a considerable influence on the effectiveness of such a device. This influence, which is difficult to control in an electron microscope because of the methods of cleaning and evacuating the relevant parts, has not yet been investigated. More recent investigations, particularly by Heide (1962, 1963, 1964a), have proved that water vapor in the residual gas of the electron microscope is mainly responsible for the removal of carbon from the area of the specimen irradiated by electrons, and that this can, therefore, be eliminated by adequate means for condensing water vapor. The method of specimen-space cooling developed in the course of these investigations (Heide, 1964b) for the Siemens Elmiskop I electron microscope will now be described in detail. As shown in Fig. 12, the specimen is maintained at approximately room temperature through good thermal contact with the object stage and is surrounded by a chamber which is as completely closed as possible. By means of a copper rod, one end of which dips into liquid air, this chamber is cooled down to 190°C, so that hydrocarbons as well as water vapor molecules will condense on the walls. The partial pressure in the cooled specimen space is thus decreased to a value insufficient for the two types of modification of the specimen. Molecules from the evacuated space surrounding the chamber can now only get into the chamber through the two small apertures which allow the electron beam to pass through the chamber and the specimen accommodated in it. The effectiveness of the cooling chamber mainly

46

Ernst Ruska

Spring Specimen stage for specimen cartridge

Entrance aperture of the upper part of the cooling chamber

Centering and thermally insulating ring

Cooling rod for Cooling cone

Lower part of the cooling chamber with exit apertures

Pole-piece system Specimen support

0

10 mm

Fig. 12 Cartridge developed for the Elmiskop I for inserting the specimen and the thermally insulated cooling chamber (for description, see text).

depends on the fact that the solid angle—seen from the specimen—which is not shielded by the cooling chamber, is made as small as possible. Consequently, the cooling chamber surrounding the specimen must be provided with two apertures of comparatively small bore, one in front of the specimen and the other between the specimen and the objective aperture. When these two apertures are cooled, a layer of ice quickly condenses on them (up to ˚ /min), and when the beam electrons strike them, hydrocarbons will 1000 A also be embedded in the deposited layer. It is, therefore, not surprising that after a certain time, this second aperture produces astigmatism, since it is situated in such a critical position in the field of the objective lens. According to our experiments, this disturbance can be of great importance when operating at high resolution, so special attention was given to preventing it in the design of the object-space cooling device. Fig. 13 shows the cold trap attached to the objective; it is evacuated together with the microscope column. When starting work, the container, which is surrounded by the vacuum of the electron microscope, is filled with liquid nitrogen. Its content only amounts to 40 mL, and one filling will last for more than 4 h. Fig. 12 shows the construction of the cartridge by which the specimen and the object-space cooling chamber are introduced through

47

Past and Present Attempts to Attain the Resolution Limit

Opening for filling 100 90

Thin-walled steel tube

80 70 60

Cooling rod

Copper foils

Container of cooling agent

Evacuated casing

mm 50 40 30 20 10 0

Fig. 13 Cooling reservoir for the specimen-space cooling chamber in the Elmiskop I electron microscope. A casing flanged to the side of the microscope objective, and evacuated together with the microscope column, accommodates a container for liquid air, which is thermally insulated from it and cools the copper rod leading to the cooling ring.

the air lock. A heat-insulated ring of synthetic material is fixed to the pole piece of the objective, well centered to it, and carries a copper ring, the bore of which is conical. This copper ring is cooled by the cooling rod and is the only part that always remains in the microscope. The whole cooling chamber is loosely connected to the specimen cartridge and is put in and out of the air lock along with the specimen. Cooling chamber and specimen cartridge interlock like two links of a chain. When the specimen cartridge, conical on the outside, is introduced into the inner cone of the object stage by the air lock mechanism, the cooling chamber, which is also conical outside, makes contact with the inner cone of the cooling ring. In this way the cooling chamber sits in the microscope cooled and centered but without being connected with the specimen cartridge, which remains at room temperature. The specimen can be moved to and fro within the cooled chamber by means of the object stage. Thus, the critical aperture between the specimen and the

48

Ernst Ruska

objective remains in the microscope for no longer than the specimen itself and is cooled only for this period of time. If, nevertheless, the aperture becomes contaminated, it is only a matter of a few seconds to change it by means of the air lock, i.e., without letting air into the microscope. Normal aperture diaphragms are used as chamber apertures. When astigmatism does occur, it is important to determine quickly whether it is due to the lower cooling chamber aperture. For this purpose, it is best to use a specimen cartridge without cooling chamber. One advantage of this design is that it is always possible to operate with a normal specimen holder without cooling chamber, no matter whether the device is cooled or not. Testing for astigmatism may, of course, also be carried out by using the cooling cartridge or simply by unscrewing the lower part of the cooling chamber. The quickest test of the cleanliness of the lower cooling chamber aperture is the following: if any charging of the edge of this aperture is occurring, a characteristic displacement of the final image will be seen when the objective aperture, situated just below it, is moved in or out. For this test, one does not even need a special type of specimen. So far we have had very good results with this improved object-space cooling chamber. Normal specimen support disks or grids of 2.4 mm diameter are employed. The only additional procedure when changing the specimen is to screw the lower part of the cooling chamber off and on. In the course of microscopic operation with this cooling device, phenomena have sometimes occurred which cannot as yet be explained. Normally when using the cooling chamber, a given specimen area can be irradiated with an intense beam of extremely small diameter for at least 1 h before any significant change is observed, and this is nearly always carbon removal. In some few instances, however, considerable contamination occurred after an exposure of 10 min, although the cooling chamber was in use. The causes of this phenomenon have still to be solved. But as there are a number of suspected factors, and as it is not easy to reproduce experiments such as this in which many parameters have to be kept constant, one must be prepared for lengthy investigations. One source of suspicion is the degree of cleanliness of those surfaces of the device which remain warm in the cooling chamber. Working with specimen-space cooling is particularly advantageous when observing extremely fine structures of poor contrast in thin sections of cells and tissues at the highest possible resolution and at direct magnifications of 100,000 times or more. In order to prevent the resolution in the

Past and Present Attempts to Attain the Resolution Limit

49

recorded image from being impaired by thermal drift of the specimen, only a very small specimen area should be illuminated. If specimen-space cooling were not available, the specimen would contaminate in a few seconds, since even then it does not heat up, but remains at approximately room temperature. So it has been very difficult previously to record images with good resolution and sufficient contrast, during the short time between the beginning of irradiation and the appearance of contamination. By means of specimen space cooling, together with selected-area irradiation, it has now become possible to observe specimens for hours at such high magnifications and sufficient brightness, and to take during that time many photographs of the unchanged specimen at very high resolution. In other words, it is only now possible to make full use of the high resolving power of the electron microscope, particularly in the case of organic specimens. An example of such a biological specimen is shown in Fig. 14, a thin section of the microorganism, Listeria. It is a remarkable fact that the cell was irradiated for 1 h with a beam of about 4 μm diameter before micrograph (B) was taken. Hardly any difference can be detected compared with micrograph (A) which was taken 5 min after the selected-area irradiation began. For comparison, Figs. 15 and 16 show micrographs of the same tremolite crystal after differing periods of irradiation, without and with object-space cooling. Fig. 16 demonstrates the breaking down of the tremolite lattice under the action of the incident electron dosage, without causing contamination, when the specimen was at low temperature. The micrographs have ˚ , proving that no disturbing thermal drift was a lattice resolution of 8–9 A produced by cooling the object-space chamber.

5.3 Cooling of the Specimen as Well as Its Surroundings A subject which is certain to become more important in the future is the observation of cooled specimens. It has already been mentioned that several types of radiation damage occur less frequently at lower temperatures. But it goes without saying that there are many other reasons for electron microscopical observations of specimens maintained at low temperatures. According to our present knowledge of contamination and partial pressure effects (Heide, 1964b), it seems that the following may be stated: specimen cooling will be successful and structure will not be disturbed, if the specimen is brought down to a temperature intermediate between that of the cooling chamber and the microscope column. In other words, the cooling chamber

50

Ernst Ruska

Fig. 14 Listeria monocytogenes in the final stage of cell division, recorded by means of selected-area irradiation with specimen space cooling to 190°C, after (A) irradiation for 5 min; (B) irradiation for 60 min. Initial magnification ME ¼ 40,000, U ¼ 60 kV, finebeam diameter ¼ 5 μm. Preparation and photography by S. Grund.

must always be colder than the specimen. As shown in Fig. 17, this requirement can be realized by adding a few extra parts inside the cooling chamber of the specimen cartridge already described. For this purpose a kind of thermal potentiometer arrangement is used: well-defined thermal resistances are put on the one hand between the specimen and the warm object stage, and on the other hand between the specimen and the cold chamber. The resistance on the warm side of the specimen should be as high as possible and of such a nature that—even in the case of heat flow—the specimen remains

Past and Present Attempts to Attain the Resolution Limit

51

Fig. 15 Rapid contamination of tremolite crystals when imaged with selected-area illumination and without specimen-space cooling: fine-beam diameter 2 μm, current density 1 Å/cm2, beam voltage 100 kV, and electron magnification 130,000 . The electron micrographs were recorded: (A) immediately after insertion of the selected area in the fine beam; (B) after irradiation of the selected area for 4 min; (C) for 11 min; (D) for 15 min. Taken by H. G. Heide.

stable, so that no drift can be observed at the highest magnification. For this purpose, two thin-walled, concentric tubes were used, made from a suitable steel. The thermal resistance between the specimen and the cooled chamber is a spring made of copper wire, which can readily be changed. It may be dimensioned in such a way that any desired specimen temperature between 20°C and 70°C can be obtained. The first experiments with this device were made with specimen temperatures between about 30°C and 50°C. No disturbing specimen shift was observed and no contamination, even after hours of exposure to fine-beam irradiation. No water or ice condensation occurred on the specimen, only a slow removal of carbon which did not ˚ per hour. Thus these experiments seem to be quite promising, exceed 30 A although the design of this latest device is certainly still capable of improvement.

Fig. 16 Behavior of tremolite crystals, under selected area illumination, with specimenspace cooling to 190°C: fine-beam diameter 2 μm, current density 1 Å/cm2, beam voltage 100 kV, electron magnification 130,000 . (A) Survey picture recorded after irradiation for 5 min. (B) Selected area from (A); the resolved set of lattice planes has a lattice constant of 8.9 Å. (C) The same area with the survey picture irradiated for 10 min. (D) The same area with the survey picture irradiated for 15 min. Taken by H. G. Heide.

Specimen cartridge

Thermal resistance (steel) Cooling chamber

Thermal resistance (Cu-spring)

Specimen support

Cooled apertures 0

mm

10

Fig. 17 Modified device for cooling the specimen as well as its surroundings. With the specimen chamber at a temperature of 190°C the specimen may be cooled to temperatures between 20°C and 70°C as desired.

Past and Present Attempts to Attain the Resolution Limit

53

6. SELF-STRUCTURE OF SUPPORTING FILMS Specimen space cooling now allows us to study the structure of uncontaminated supporting films. We should naturally prefer to have structureless films available for high-resolution electron microscopy, so that only the structure of the specimen itself appears in the micrograph. But, since we hope that the resolution of the electron microscope can be improved to the level of the separation between atoms, this desire may seem almost utopian. Up to now, all available supporting films have shown the structure when imaged in a high-resolution electron microscope, although this might have been the structure of a contamination layer. The use of specimen-space cooling now makes it possible to image the structure of a supporting film uncontaminated and unaffected by carbon removal, and perhaps even to find structureless films. We have made through-focal series of micrographs of such films in the Siemens Elmiskop I electron microscope, with very small steps of lens current.c By means of an additional “superfine” focusing knob in the objective circuit, the plane of focus of the object could be changed by ˚ ), i.e., approximately the thickness of a very increments of 17 nm (¼170 A thin specimen. The smallest adjustable change in current is then of the same order of magnitude as the current change due to instability. Another object of the focal series was to find out to what extent, if at all, the astigmatism of the objective might influence the structure seen in the images of the supporting film. In order to eliminate photographic factors when comparing micrographs, a single plate was used for a through-focus series of 32 individual exposures. Fig. 18 shows the results for a circular hole in a carbon film. The objective was still somewhat astigmatic, as is evident at the edges of the hole. Fig. 19 shows the middle micrographs of a similar through-focus series at a higher light optical magnification. Carbon films that have circular holes are relatively thick, and even the uncontaminated film shows a structure which—in best focus—reminds one of the appearances of crossed parquetry. The two mutually perpendicular directions correspond with the directions of the under- and overfocused partial fringes, which are visible at the holes owing to the astigmatic imaging. In the case of astigmatic imaging and optimum focusing, all image points and thus also the images as a whole are focused in two directions at right angles. In one of the directions c

Unpublished work in collaboration with A. M. D’Ans and G. Tochtermann.

54

Ernst Ruska

Fig. 18 Film with circular holes (20 nm thick) imaged with specimen-space cooling to 190°C. Through-focus series of 32 micrographs on one plate 6.5  9.0 cm. ME ¼ 80,000; U ¼ 60 kV; Δf ¼ 121 nm; Δfast ¼ 30 nm (1 nm ¼ 10 Å).

Fig. 19 Film with circular holes (20 nm thick) with astigmatic imaging. Through-focus series with specimen-space cooling to 190°C. ME ¼ 80,000; U ¼ 60 kV; Δfast ¼ 242 nm.

Past and Present Attempts to Attain the Resolution Limit

55

Fig. 20 Tempered collodion film (5 nm thick) coated with carbon by evaporation. Through-focus series with specimen-space cooling to 190°C. ME ¼ 80,000; U ¼ 60 kV; Δfast ¼ 0.

bisecting these two directions the image is underfocused, and in the other bisecting direction it is overfocused. Accordingly, since all the micrographs in the through-focus series are defocused—although the middle ones only in certain directions—they all show the phase-contrast effects which are characteristic of defocusing. Fig. 20 shows the middle micrographs of a throughfocus series of an evaporated carbon film, prepared by the method developed by Bradley. The film was extremely thin (about 5 nm) so that 97% of the incident electrons could contribute to the image, and the objective in this case had very little astigmatism. Those micrographs that are most accurately focused show no structure at an electron optical magnification of 80,000 times. Owing to the absence of astigmatic defocusing, the phase contrast has also disappeared here. Fig. 21 shows a through-focus series of a thicker carbon film (about 140 nm thick) having a circular hole covered with a thinner collodion film (about 10 nm thick) at an electron microscopic magnification of 157,000 , taken at 80 kV and with negligible astigmatism. At optimum focus the weak phase contrast of the thin covering film again disappears, whereas the stronger phase contrast of the thicker carbon film does not completely

56

Ernst Ruska

Fig. 21 Film with circular holes (140 nm thick) with covering film of collodion (100 Å thick). Through-focus series with specimen-space cooling to 190°C. ME ¼ 157,000 ; U ¼ 80 kV; Δfast ¼ 0.

disappear: it reaches its minimum at exact focus and thus its optimum resolution. In Fig. 22, the thin covering film was lightly coated with platinum–iridium by evaporation. Single groups of platinum–iridium parti˚ can naturally be seen only within the cles having a size of about 10–20 A holes in the thicker film. Although their atomic weight exceeds that of carbon by nearly an order of magnitude, their optimum contrast is still relatively low, due to the fact that the covering film is thicker by an order of magnitude. In the case of optimum focusing on the covering film, their contrast is good because the phase contrast of the film is then a minimum. It becomes still better when slightly underfocused (+ in Fig. 22) and then slowly decreases with increasing underfocus. With slight overfocus on the other hand, it soon disappears. In order to observe heavier particles on a light carbon film undisturbed by phase contrast, it is thus preferable to arrange the supporting film in the usual way, i.e., the electrons first striking the supporting film and then the specimen, rather than in the reverse position. From our present experience of these fine through-focus series, the following conclusion may be drawn: so long as it is not certain that the image is

Past and Present Attempts to Attain the Resolution Limit

57

Fig. 22 Film with circular holes (140 nm thick) with covering film of collodion (100 Å thick) and light evaporated coating of platinum–iridium. Through-focus series with specimen-space cooling to 190°C. ME ¼ 157,000 ; U ¼ 80 kV; Δfast ¼ 0.

sufficiently free of astigmatism and accurately focused, one should be most reserved in deducing the true specimen structure from the finest image details that can be seen at high electron microscopical magnification. Even the best electron microscopes of today are not good enough to obtain such images with certainty. As optimum images can only be recognized and selected from the finest through-focus series, electron microscopes of high performance should in the future be provided with appropriately fine controls. Even so, when working near the resolution limit, one should be careful about drawing conclusions as to the specimen structure, until we have a better theoretical understanding of the imaging of such small groups of atoms or molecules in the electron microscope. A critical investigation (Albert, Schneider, & Fischer, 1964) recently published shows, for instance, that in the electron microscope, particles of a size in the same order of magnitude as the resolving power, and of mass density not sufficiently different from their environment for them to be visible in the focused image by scattering contrast, can still be made visible in an under- or overfocused image by the phase-contrast effects.

58

Ernst Ruska

To sum up the present situation, we can say with certainty that improvement of the resolving power of the electron microscope will not continue to advance as rapidly as in the early years of its development. It will become a hard fight for even the smallest progress in the various aspects of the problem. But since the high resolution already realized can in principle be improved still further, the molecular structure of matter both animate and inanimate will become ever more accessible to direct imaging during the next few years.

REFERENCES Albert, L., Schneider, R., & Fischer, H. (1964). Elektronenmikroskopische Sichtbarmachung von  10 A großen Fremdstoffeinschl€ ussen in elektrolytisdi abgesdiiedenen Nickelschichten mittels Phasenkontrast durdi Defokussieren. Zeitschrift f€ ur Naturforschung, 19a, 1120–1124. Br€ uche, E., & Johannson, H. (1932). Elektronenoptik und Elektronenmikroskop (electron optics and the electron microscope). Naturwissenschaften, 20, 353–358. Dowell, W. C. T. (1957). Carbon-stabilized collodion substrates for electron microscopy. Journal of Applied Physics, 28, 634–635. Dowell, W. C. T. (1963). Das elektronenmikroskopische Bild von Netzebenenscharen und sein Kontrast. Optik, 20, 535–568. Dupouy, G., Perrier, F., & Fabre, R. (1961). Microscope electronique fonctionnant sous tre`s haute tension. Comptes Rendus Academie des Sciences, 252, 627–632. Engel, A., Koppen, G., & Wolff, O. (1962). Versuche zur Verbesserung der Hochspannungsmesstechnik f€ ur die Elektronenmikroskopie. In Proceedings of the fifth international congress for electron microscopy, Philadelphia: Vol. 1 (p. E-13). New York: Academic Press. Glaser, W. (1941). Strenge Berechnung magnetischer Linsen der Feldform H ¼ Ho/[1 + (z/a)2]. Zeitschrift f€ ur Physik, 117, 285–315. Glaser, W. (1943). Bildentstehung und Aufl€ osungsverm€ ogen des elektronenmikroskops vom standpunkt der Wellenmechanik. Zeitschrift f€ ur Physik, 121, 647–666. Hanszen, K.-J., Morgenstern, B., & Rosenbruch, K.-J. (1964). Aussagen der optischen € Ubertragungstheorie u €ber Auflo¨sung und Kontrast im elektronenmikroskopischen Bild. Zeitschrift f€ ur Angewandte Physik, 16, 477–486. Heide, H. G. (1962). Electron microscopic observation of specimens under controlled gas pressure. Journal of Cell Biology, 13, 147–152. Heide, H. G. (1963). Die Objektverschmutzung im Elektronenmikroskop. Zeitschrift f€ ur Angewandte Physik, 15, 116–128. Heide, H. G. (1964a). Die Restgaszusammensetzung im Elektronenmikroskop. I. Zeitschrift f€ ur Angewandte Physik, 17, 70–72. Heide, H. G. (1964b). Die Restgaszusammensetzung im Elektronenmikroskop. II. Zeitschrift f€ ur Angewandte Physik, 17, 73–75. Hildebrandt, H.-J. (1964). Diplomarbeit Freie Universit€at Berlin. Hillier, J. (1948). On the investigation of specimen contamination in the electron microscope. Journal of Applied Physics, 19, 226–230. Hoppe, W. (1961). Ein neuer Weg zur Erh€ ohung des Aufl€ osungsvermogens des Electronenmikroskops. Naturwissenschaften, 48, 736–737. Knoll, M., & Ruska, E. (1932a). Beitrag zur geometrischen Electronenoptik. Annalen der Physik, 12(5), 607–661.

Past and Present Attempts to Attain the Resolution Limit

59

Knoll, M., & Ruska, E. (1932b). Das Elektronenmikroskop. Zeitschrift f€ ur Physik, 78, 318–339. Komoda, T., & Morito, N. (1960). Experimental study on the specimen contamination in electron microscopy. Journal of Electronmicroscopy (Japan), 9, 77–80. Komoda, T., & Otsuki, M. (1964). Resolution of 2.0 A˚ in lattice images with an electron microscope. Japan Journal of Applied Physics, 3, 666–667. ˚ in lattice images with an electron microscope. Leisegang, S. (1956). Resolution of 2.0 A In Proceedings of the third international conference on electron microscopy, London 1954 (pp. 176–184). London: Royal Microscopical Society. € Lenz, F. (1963). Zonenplatten zur Offnungsfehlerkorrektur und zur Kontrasterho¨hung. Zeitschrift f€ ur Physik, 172, 498–502. Niehrs, H. (1962). Das Problem des Kontrasts und der Auflo¨sung bei einer Elektronenmikroskopie von Atomen bzw. Atomgruppen. In Proceedings of the fifth international congress for electron microscopy, Philadelphia: Vol. 1 (p. AA-2). New York: Academic Press. Riecke, W. D. (1962a). Ein Kondensorsystem f€ ur eine starke Objektivlinse. In Proceedings of the fifth international congress for electron microscopy, Philadelphia: Vol. 1 (p. KK-5). New York: Academic Press. Riecke, W. D. (1962b). Die Objektraumk€ uhlung im Elektronenmikroskop. Optik, 19, 81–116. Riecke, W. D. (1964). Kann man die “atomare Aufl€ osung” im Elektronenmikroskop nur mit dem Dunkelfeld erreichen? Zeitschrift f€ ur Naturforschung, 19a, 1228–1230. € Ruska, E. (1950). Uber neue magnetische Durchstrahlungs-Elektronenmikroskope im Strahlungsbereich von 40 … 220 kV. Teil I. Kolloid-Zeitschrift, 116, 102–120. Ruska, E. (1956). Ein Hochauflo¨sendes 100-kV-Elektronenmikroskop mit Kleinfelddurchstrahlung. In Proceedings of the third international conference on electron microscopy, London 1954 (pp. 673–693). London: Royal Microscopical Society. Ruska, E. (1962). What is the theoretical resolution limit of the electron microscope and when will it be reached? In Proceedings of the fifth international congress for electron microscopy, Philadelphia: Vol. 1 (p. A-1). New York: Academic Press. Ruska, E. (1964). Das Aufl€ osungsproblem in der Elektronenmikroskopie. Naturwissenschaftliche Rundschau, 17, 125–135. € Ruska, E. (1965). Uber die Aufl€ osungsgrenzen des Durchstrahlungs-Elektronenmikroskops. Optik, 22, 319–349. Scherzer, O. (1949). The theoretical resolution limit of the electron microscope. Journal of Applied Physics, 20, 20–29. van Dorsten, A. C., & van den Broek, S. L. (1960). Some design features of a new Philips electron microscope. “VierterInt. Kongr. f€ ur Elektronenmikroskopie Berlin” 1958: Verh. 1 (pp. 171–174). Berlin: Springer. € von Ardenne, M. (1944). Uber ein neues Universal-Elektronenmi kroskop mit Hochleistungsmagnet-Objektiv und herabgesetzter thermischer Objektbelastung. Kolloid-Zeitschrift, 108, 195–208.

CHAPTER TWO

Phase Plates for Transmission Electron Microscopy Christopher J. Edgcombe1 University of Cambridge, Cambridge, United Kingdom 1 Corresponding author: e-mail address: [email protected]

Contents 1. 2.

Introductory Survey Postspecimen Plates 2.1 Geometry 2.2 Methods of Controlling Phase Shift 3. Theory for Postspecimen Plates 3.1 Illumination Function 3.2 Object Function 3.3 Plate Function 4. Images With Zernike Plates 4.1 Size Parameter 4.2 A Rotationally Symmetric Object and a Zernike Plate 4.3 A Disc Object and Zernike Plate 4.4 Images for Spherical Object and Zernike Plate 5. Images With Straight-Edged Plates on the Cylindrical Axis 5.1 A Disc Object With an Opaque (Foucault) Plate 5.2 A Disc Object With a Half-Plane Phase-Changing (Hilbert) Plate 6. Discussion 7. Conclusions Acknowledgments Appendix A. The 2D Transform in Cylindrical Coordinates A.1. Fourier Series A.2. Hankel Transform A.3. Conversion From Cartesian Form A.4. Inverse Transform References

Advances in Imaging and Electron Physics, Volume 200 ISSN 1076-5670 http://dx.doi.org/10.1016/bs.aiep.2017.01.007

#

2017 Elsevier Inc. All rights reserved.

62 68 68 70 74 76 77 80 81 81 82 83 86 89 89 90 93 94 95 95 95 95 96 98 98

61

62

Christopher J. Edgcombe

1. INTRODUCTORY SURVEY A phase plate is a device placed in the column of a microscope, either before or after the illumination reaches the specimen, for the purpose of producing a specific effect on the image. One of the earliest devices used in this way was the straight edge of an opaque screen or knife edge, used to observe an optical mirror’s surface by interception in the focal plane (Foucault, 1858). Slightly later, knife edges were also used to produce “schlieren” imaging, showing gradients in the density of gases (section 13.2.5 of Hecht, 2002). The knife edge thus produced optical images of the phase gradients in transparent objects (section 12.4.3 of Lipson, Lipson, & Tannhauser, 1995). Further developments of this technique are described in Section 2. Until recently, phase plates have been used mainly as “postspecimen” plates, inserted between the specimen and the image. Suitably designed plates can produce contrast at the image from “phase objects” that absorb little of the incident beam but change its phase on transmission. The principle of the postspecimen type most widely known for this purpose was published by Zernike (1942; reviewed in 1955). When a phase object scatters a wave weakly, the scattered part of the wave is normally a quarter cycle out of phase with the unscattered or direct wave, and so makes only a very small change to the intensity at the detector. Zernike investigated ways of changing the phase of a scattered (optical) wave relative to the direct wave by a further quarter cycle so that the scattered wave creates a change in intensity which the detector can record. In 1947, Boersch proposed the translation of this method from optics to transmission electron microscopy and suggested some structures that might provide the desired phase difference between the direct and scattered electrons (Fig. 1). One of these, now known as the Zernike plate, was a thin A

Aperturblende

Folie

B

Aperturblende

ΔU

C

Aperturblende

ΔU

Fig. 1 Structures proposed by Boersch to act as postspecimen phase plates for electron microscopy. (A) Phase shift by means of a foil with an inner potential. (B and C) Phase shift € die kontraste by electrostatic fields. Reprinted by permission from Boersch, H. (1947). Uber von atomen im elektronenmikroskop. Zeitschrift Fur Naturforschung—Section A Journal of Physical Sciences, 2(11–12), 615–633. https://doi.org/10.1515/zna-1947-11-1204.

Phase Plates for TEM

63

sheet of material, perforated with a central hole large enough to allow the direct beam to pass without the change in phase that is experienced by the scattered electrons. The second structure was a small electrode assembly which caused the potential distributions and electrical path lengths to differ for the direct and scattered electrons, so inducing a suitable phase difference. Following the development of methods of fabrication at the nanoscale, both these techniques have been realized in a variety of ways. The theory and practice of postspecimen plates are described in the following surveys and discussions. Starting from Schr€ odinger’s equation, a detailed derivation of bright-field imaging in the TEM, the effect of imperfections in the instrument and relevant transfer-function theory, with an extensive bibliography of earlier work, are provided by chapters 64–66 of Hawkes and Kasper (1994). Section 66.5.3 of the same work lists earlier work on postspecimen plates, including other forms of aperture function besides those considered here. The major current interest in phase plates for use in cryomicroscopy of biological molecules is described in the review by Henderson (1995). A survey of methods for cryomicroscopy, with practical details and comparison of many types of phase plate, has been given by Glaeser (2013). Marko has described experience with Zernike plates (2013) and with hole-free plates fabricated in his lab (2016). Some recent results from Marko, Meng, Hsieh, Roussie, and Striemer (2013) are shown in Fig. 2. This and other developments of geometries and methods of producing phase difference by postspecimen plates are described in Section 2. Independently of these techniques for modifying the scattered wave after the specimen, there has been much interest recently in plates located before the specimen to produce electron beams with orbital angular momentum (OAM), known as vortex beams. The analysis of these electron beams was developed from their optical equivalents by Bliokh, Bliokh, Savel’Ev, and Nori (2007). Uchida and Tonomura (2010) imaged holographically a small area of a graphite sample containing a phase singularity, around which the phase varied continuously with angle (Fig. 3). They found that the interference with a plane wave produced the pattern of stripes with a central dislocation that would be expected from a phase change of 2π at the singularity, and suggested that electron beams with OAM might be useful for energy loss spectroscopy (EELS). Soon after, Verbeeck, Tian, and Schattschneider (2010) reported on the inverse behavior. They created a plate from the interference pattern between a wave with OAM and a plane wave and demonstrated that it produced separated beams in the far field with magnitudes of

64

Christopher J. Edgcombe

Fig. 2 Comparison of images of Spirochete bacteria, without and with Zernike phase plate. (A) In-focus, without phase plate; (B) underfocused by 15 μm, without phase plate; (C) near focus, with phase plate. In (C) the arrows show clear imaging of filamentary structures. Reprinted from Marko, M., Meng, X., Hsieh, C., Roussie, J., & Striemer, C. (2013). Methods for testing Zernike phase plates and a report on silicon-based phase plates with reduced charging and improved ageing characteristics. Journal of Structural Biology, 184(2), 237–244. https://doi.org/10.1016/j.jsb.2013.08.008J. Copyright (2013), with permission from Elsevier.

Fig. 3 Phase singularity in graphite thin film observed by Uchida and Tonomura: (A) holographic image; (B) phase distribution reconstructed from hologram. Reprinted by permission from Macmillan Publishers Ltd: Uchida, M., & Tonomura, A. (2010). Generation of electron beams carrying orbital angular momentum. Nature, 464(7289), 737–739. https://doi.org/10.1038/nature08904. Copyright (2010).

OAM corresponding to those of the interfering waves (Fig. 4). The separated parts of this vortex beam were used in an EELS system to measure magnetic circular dichroism in a sample of magnetized iron. Work at NIST by McMorran (2012) and McMorran et al. (2011) showed the generation of vortex beams with a topological charge as large as 100 (angular momentum of 100ℏ). Analyses of the production of vortex wave functions by a “grid”

Phase Plates for TEM

65

Fig. 4 (A) Structure used by Verbeeck, Tian, and Schattschneider to generate OAM in beam; (B) diffraction pattern from (A), showing separation of beam into parts with l ¼ 1, 0, and +1. Reprinted by permission from Macmillan Publishers Ltd: Verbeeck, J., Tian, H., & Schattschneider, P. (2010). Production and application of electron vortex beams. Nature, 467(7313), 301–304. https://doi.org/10.1038/nature09366. Copyright (2010).

(or grating) with central dislocations have been presented by Idrobo and Pennycook (2011) and Schattschneider and Verbeeck (2011). Further applications are suggested by Verbeeck et al. (2014). The spiral zone plate, in which the central dislocations are spaced uniformly in azimuth, was investigated by Verbeeck, Tian, and Beche (2012) (Fig. 5) and by Saitoh, Hasegawa, Tanaka, and Uchida (2012) (Fig. 6). These plates have the advantage that they can produce many beams of different OAMs that are in focus at different axial positions, allowing selection of a specific one at the specimen. The undesirable loss of beam intensity by interception at a simple grating has stimulated the development of transparent gratings, intended to modulate the phase of the beam with minimum interception (Grillo et al., 2014; Harvey et al., 2014; Yuan, 2014). Recently, this method of modulating the phase of the entire beam has been used in conjunction with a matched detector in STEM imaging (Ophus et al., 2016; Yang, Ercius, Nellist, & Ophus, 2016). The combined technique appears to achieve a desirable increase in contrast at low

66

Christopher J. Edgcombe

Fig. 5 Spiral phase plate of diameter 50 μm. Reprinted from Verbeeck, J., Tian, H., & Beche, A. (2012). A new way of producing electron vortex probes for STEM. Ultramicroscopy, 113, 83–87. https://doi.org/10.1016/j.ultramic.2011.10.008. Copyright (2012), with permission from Elsevier.

Fig. 6 Spiral phase plate of diameter 20 μm producing topological charge m ¼ 10, fabricated by Saitoh et al. (2012). Reprinted with permission from Saitoh, K., Hasegawa, Y., Hirakawa, K., Tanaka, N., & Uchida, M. (2013). Measuring the orbital angular momentum of electron vortex beams using a forked grating. Physical Review Letters, 111, 074801. Copyright (2013) by the American Physical Society.

Phase Plates for TEM

67

spatial frequencies. The possibility of providing plates both before and after the lens, with properties matched to provide a specific type of filtering, has been suggested by Verbeeck (2016). Another technique for generating OAM is the use of an approximation to a magnetic monopole. This is obtained by placing a magnetized needle transverse to the microscope axis, with one tip close to the axis (Beche, Van Boxem, Van Tendeloo, & Verbeeck, 2013; Blackburn & Loudon, 2014) (Fig. 7). While the returning flux in the needle introduces some asymmetry, the mainly radial flux distribution outside the body of the needle resembles that expected from a monopole. In these experiments, electron holography was particularly effective for determining details of the behavior near the needle tip. Verbeeck et al. (2015) have reported the use of an energizing coil with a ferromagnetic needle, by which the chirality of vortices can be varied.

Fig. 7 A nickel rod fastened over an aperture of diameter about 20 μm to simulate a magnetic monopole. Reprinted by permission from Macmillan Publishers Ltd.: Beche, A., Van Boxem, R., Van Tendeloo, G., & Verbeeck, J. (2013). Magnetic monopole field exposed by electrons. Nature Physics, 10(1), 26–29. https://doi.org/10.1038/nphys2816. Copyright (2013).

The next section describes some of the geometric forms and the many physical effects that have been used for postspecimen plates. Sections 3–5 give some theory for the imaging of simple objects by rotationally symmetric lenses, described here as “round” lenses, when used with the two bestknown types of postspecimen plate. This work uses the spectral transfer theory defined for optics by Abbe, as outlined by Born and Wolf (1993), and as also derived from Schr€ odinger’s equation by Hawkes and Kasper (1994).

68

Christopher J. Edgcombe

It uses cylindrical coordinates so that in the simplest case where the object, lens, and plate are all rotationally symmetric, the system can be specified by a single parameter. This analysis describes first the behavior of plates with a central hole, known as Zernike plates. The method is then extended to apply to systems with azimuthal variation, by restating the known twodimensional Cartesian transform as a pair in cylindrical coordinates. This enables analysis of a knife edge or a dielectric plate with a straight edge, in the special case that the edge is located on the cylindrical axis. The results of this analysis show why the Hilbert plate as commonly implemented has not shown ideal performance, and suggest a step that may help to improve the behavior.

2. POSTSPECIMEN PLATES 2.1 Geometry The type of plate recorded earliest (Foucault, 1858) uses a straight edge placed near or on the unscattered (optical) beam. Consequently, opaque plates with straight edges are known as Foucault plates. In a later development, this plate was made thin enough to transmit some of the incident wave, and the newer form is known as a Hilbert plate (Danev, Okawara, Usuda, Kametani, & Nagayama, 2002). The ability to choose the phase change introduced by the plate by varying its thickness gives an additional degree of freedom relative to the Foucault plate, with effects that are described in Section 5. The type of postspecimen plate that has been most widely used since its demonstration by Danev and Nagayama (2001) consists of a film of selected material with controlled thickness, containing a small circular hole which is located so that the unscattered part of the electron beam passes through it. The rotational symmetry of this system corresponds to that of Zernike’s proposal, and consequently any construction that is rotationally symmetric is liable to be called a Zernike plate. Further types of geometry have been proposed, many of which offer the benefit of avoiding interception of the direct beam. The Zach plate (Hettler et al., 2012, 2015; Schultheiss et al, 2010; Zach, 2007; Fig. 8) includes a conducting probe placed at the back focal plane (BFP), thereby varying the phase changes along the paths of scattered electrons as a function of azimuthal angle. The “anamorphotic” plate described by Schr€ oder, Barton, Rose, and Benner (2007) (Fig. 9) also provides a variation in the electrical path length with azimuth and can be incorporated in an aberration corrector.

69

Phase Plates for TEM

Fig. 8 Zach phase plate, consisting of conductor and coaxial shielding tube. Reprinted from Hettler, S., Wagner, J., Dries, M., Oster, M., Wacker, C., Schro€der, R. R., & Gerthsen, D. (2015). On the role of inelastic scattering in phase-plate transmission electron microscopy. Ultramicroscopy, 155, 27–41. https://doi.org/10.1016/j.ultramic.2015.04.001. Copyright (2015), with permission from Elsevier.

Cross-section A–B y Insulator z Conductor

A y u

x

B

Fig. 9 Anamorphotic plate, suitable for incorporation (in pairs) into an aberration cor€der et al. (2007). Reprinted from Rose, H. (2010). Theoretical rector, proposed by Schro aspects of image formation in the aberration-corrected electron microscope. Ultramicroscopy, 110, 488–499. Copyright (2010), with permission from Elsevier.

70

Christopher J. Edgcombe

Perhaps the least obvious form of “plate” yet proposed consists of a very high alternating electric field, to be provided by a laser-excited resonator (M€ uller et al. 2010). In contrast, a recent development, the “hole-free” phase plate (Buijsse, van Duinen, Sader, & Danev, 2014; Danev, Buijsse, Khoshouei, Plitzko, & Baumeister, 2014; Malac, Kawasaki, Beleggia, Peng, & Egerton, 2010; Marko, Hsieh, Leith, Mastronarde, & Motoki, 2016) relies on interception by the plate at a site which can easily be varied, which clearly offers an advantage in practice.

2.2 Methods of Controlling Phase Shift The original Foucault plate consisting of a straight knife edge has recently been refined with the aim of providing phase correction over a limited range of low spatial frequencies (Buijsse et al., 2011) as described by Glaeser (2013) (Fig. 10). The resulting “tulip” plate, in the form of a short half-plane supported in an otherwise open aperture, can be made by ion beam machining. Other opaque probes such as the Zach design can also now be made by this versatile form of machining or by controlled etching techniques.

Fig. 10 The “Tulip” plate. Reprinted from Buijsse, B., van Laarhoven, F. M. H. M., Schmid, A. K., Cambie, R., Cabrini, S., Jin, J., & Glaeser, R. M. (2011). Design of a hybrid double-sideband/single-sideband (schlieren) objective aperture suitable for electron microscopy. Ultramicroscopy, 111(12), 1688–1695. https://doi.org/10.1016/j.ultramic.2011.09. 015. Copyright (2011), with permission from Elsevier.

Phase Plates for TEM

71

Plates that provide a phase change by transmission through material of controlled thickness are conveniently made by thin-film techniques, possibly in conjunction with ion beam or other etching. These include films of carbon for Zernike and Hilbert plates. Other materials have also been tested (Dries et al., 2016; Marko et al., 2013). The logical inversion of the Zernike plate, into a central phase-changing spot supported on a very thin conductive layer, offers reduced attenuation of the scattered wave as discussed by Rhinow (2016), but clearly this spot should be able to withstand the focused direct beam, without charging. Another method of obtaining a relative phase difference between electron paths is the use of electrodes to vary the electrostatic potential distribution along different paths. Examples are the einzel lens (Majorovits et al., 2007; Schultheiß, Perez-Willard, Barton, Gerthsen, & Schr€ oder, 2006) (Fig. 11);

Fig. 11 (A) Einzel lens with three support struts, whose effect on the image can be removed by image processing; (B) detail of electrode structure on axis of (A). Reprinted from Majorovits, E., Barton, B., Schultheiß, K., Perez-Willard, F., Gerthsen, D., & Schro€der, R. R. (2007). Optimizing phase contrast in transmission electron microscopy with an electrostatic (Boersch) phase plate. Ultramicroscopy, 107(2–3), 213–226. https://doi.org/10.1016/j. ultramic.2006.07.006. Copyright (2007), with permission from Elsevier.

72

Christopher J. Edgcombe

use of materials of different work functions (Tamaki, 2013); a drift tube constructed by deep-etched microfabrication (Cambie, Downing, Typke, Glaeser, & Jin, 2007) (Fig. 12); the anamorphotic structure compatible with an aberration corrector (Schr€ oder et al., 2007) (Fig. 9); and the Zach electrode (Schultheiss et al., 2010) (Fig. 8). Most of these have the benefit that the added phase is controllable by adjustment of the electrode potentials. The more recently developed forms of the hole-free phase plate (Danev et al., 2014; Malac, Beleggia, Kawasaki, Li, & Egerton, 2012; Malac et al., 2010) also appear to work by an electrostatic phase change. Since the phase change along an electron’s path is an integral not only of the electric field but also of the magnetic vector potential, it is possible to produce a phase difference by a suitably arranged magnetic field distribution. Tonomura’s group noted (Osakabe, Nomura, Matsuda, Endo, & Tonomura, 1983) that a magnetized ring of permalloy carrying a complete vortex of flux could produce a phase difference between electrons that travel through the ring and those that travel outside it, as an application of the Aharonov–Bohm effect (Aharonov & Bohm, 1959; Ehrenberg & Siday, 1949). Another way of using the effect was suggested by Valdre` (1979), who proposed to form a toroidal coil of dimensions small enough to fit in a diffraction plane of a TEM. More recently, the idea of using a ring of flux was reinvented (Edgcombe, 2010) and a thin-film ring of cobalt was tested (Edgcombe et al., 2012) (Fig. 13). Cobalt is expected to withstand a greater axial (out-of-plane) field than the softer material permalloy can, but not as much as the typical field at an objective aperture under conditions of

Fig. 12 Electrode system consisting of concentric cylinders. Reprinted from Cambie, R., Downing, K. H., Typke, D., Glaeser, R. M., & Jin, J. (2007). Design of a microfabricated, twoelectrode phase-contrast element suitable for electron microscopy. Ultramicroscopy, 107(4–5), 329–339. https://doi.org/10.1016/j.ultramic.2006.09.001. Copyright (2007), with permission from Elsevier.

Phase Plates for TEM

73

Fig. 13 Effect of a ring of magnetic flux on the phase integrated along the beam path (normal to plane of ring). (A, B) Ring is in “onion” state with two opposed semicircles of flux; (C, D) ring is in vortex state with continuous flux around ring; (A, C) images of cos (2*phase) obtained from holograms; (B, D) brightness increases with phase. Reprinted from Edgcombe, C. J., Ionescu, A., Loudon, J. C., Blackburn, A. M., Kurebayashi, H., & Barnes, C. H. W. (2012). Characterisation of ferromagnetic rings for Zernike phase plates using the Aharonov-Bohm effect. Ultramicroscopy, 120, 78–85. https://doi.org/10.1016/j. ultramic.2012.06.011. Copyright (2012), with permission from Elsevier.

maximum magnification. As shown in Section 4, the size of the phase object that can be imaged without distortion is proportional to the focal length of the lens used as objective, so if there were no other considerations, one would want to maximize the focal length. For biological applications, a minilens may be usable, placing the BFP at the position of the selected-area aperture. Alternatively, a transfer lens system that lengthens the effective focal length may be suitable. In addition to the rotationally symmetric magnetic devices, a magnetic equivalent of the Hilbert plate can be obtained by using a straight transverse wire carrying flux (Nagayama, 2008). While such a wire appears to block

74

Christopher J. Edgcombe

part of the diffraction plane, the device offers more possibility of phase control by external excitation than is available with simple rings. In many of the reports of phase plates mentioned in this section, a recurring problem is the need to minimize charging at the plate, thought to occur at insulating material deposited on the surface even when the material of the plate is a good conductor. One way of reducing the problem by depositing a layer of carbon is described by Nagayama (2008) and Glaeser (2013). The papers noted in this and the previous sections include only a sample, collected to illustrate general lines of development. The aim has been to cite the first publication of each development. This inevitably has led to the omission from this review of many excellent papers, notably from the recent abundance of publications on vortex beams. Those who wish to read further are recommended to browse among recent volumes of Ultramicroscopy, including the pages of enjoyable reviews emanating from Peter Hawkes.

3. THEORY FOR POSTSPECIMEN PLATES Analysis of bright-field imaging in the TEM often considers the phasecontrast transfer function (PCTF), which defines the relative phase introduced by the system as a function of spatial frequency. The PCTF is typically specified as (sin χ(q)), where χ (denoted by some writers as λ) is a function of defocus and aberrations. It is well known that if the magnitude of the PCTF can be changed from (sin χ) toward (cos χ), then a substantial increase in contrast at low frequencies can be obtained. For this to happen, the phase added by the plate needs to be an odd multiple (preferably small) of π/2 and ideally independent of q. In practice, this is difficult to achieve because the direct beam is present at q ¼ 0, where the added phase is taken as zero. These transfer functions are well established, but by themselves, they do not show the image produced by an extended object that contains many spatial frequencies and many sources of scattering. However, the spectral transfer theory defined for optical systems by Abbe (Born & Wolf, 1993), and for TEM, from Schr€ odinger’s equation, by Hawkes and Kasper (1994), shows how the image can be obtained from the spectrum of the object exit wave and the frequency response of other parts of the system. Here we give some results of applying this spectral theory to simple models. This analysis for postspecimen plates is intended for biological objects and is entirely classical; for production of vortex beams by prespecimen plates, more detailed analysis (for example, Babiker, Yuan, & Lembessis, 2015; Bliokh et al., 2007) should be considered.

Phase Plates for TEM

75

According to diffraction theory, imaging can be defined by two successive processes. First, the object wave can be specified by its spectrum in the radial spatial frequency q. The lens (assumed here to be round) redistributes the spectrum so that all rays that have the same magnitude of q before reaching the lens converge after leaving the lens to one physical radius that is proportional to q, on a specific plane transverse to the cylindrical axis. If the illumination is parallel, this plane is the BFP. It is this relation between q and physical radius that allows a phase plate at the BFP to modify particular ranges of q (eqs. 65.15–17 of Hawkes & Kasper, 1994). Second, the modified spectrum then propagates further, and its inversion shows the focused image that may be obtained. The complex amplitude of the image then shows the phases of the various terms relative to the incident wave and hence their contributions to modulation of the image intensity. Most lenses used in electron optics for imaging are rotationally symmetric, and analysis for them is made clearer by using cylindrical coordinates (r, ϕ). The spectrum of the object wave can in some simple cases be found analytically, allowing interpretation of how the spectrum depends on parameters of the object. Then much of the second transform, which yields the image amplitude and intensity distribution, can also be specified analytically, requiring only a relatively brief numerical calculation. When also both the object and plate are rotationally symmetric, the calculation is independent of ϕ and so can be specified entirely by use of the radial coordinate and radial spatial frequency q. Results for the image intensity with some Zernike plates as a function of object “size” (to be defined) are shown in Section 4. The transmission of plates that have some azimuthal structure can in general be specified as a function of both q, the magnitude of radial frequency, and an angle θ. The transmission of the simpler types can be specified as a Fourier series in θ, with azimuthal periodicity ‘ and coefficients that are functions of q. The postplate spectrum for each ‘ can be converted to the image function for the same ‘ by using an inverse Hankel transform of order ‘. The complete 2D transform and its inversion are defined in Appendix A. The radial parts of this transform (Eqs. (A.3) and (A.4)) have no dependence on ϕ, so these equations seem more suitably described as a Hankel transform pair (of the nth order) than with the name of Fourier, as the author did earlier. The 2D transform is used in Section 5 to show the behavior of Foucault and Hilbert plates located on the cylindrical axis. The aim of these theoretical sections is to show the effects produced by some combinations of model object and model phase plate, independently of any other possible instrumental effect or signal-to-noise ratio. Thus the

76

Christopher J. Edgcombe

theory given here will ignore the usual contributions such as defocus, aberrations, energy spread, and any other defects. The illumination is assumed to be fully coherent and parallel at the object so that the radial ordering in q is located at the BFP. The objects to be described are assumed to produce zero absorption and so are “pure phase objects.” A condition for a phase object to be described as “weak” appears in Section 4.3.2. The coordinates (ri, ϕi) of the image are assumed to be scaled by reduction from the actual coordinates by the magnification factor and by rotation to remove any rotational effect of the lens system, as in Hawkes and Kasper (1994; section 65.2); or, if you prefer, the image coordinates are referred to object space. The inversion of the modified spectrum then gives the image functions directly in these scaled coordinates. The general plan for notation is that lower-case names (such as f(r, ϕ), g(ri, ϕi)) specify functions of physical coordinates, while upper-case ones (such as F(q, θ), T(q, θ)) define spectra that are functions in q-space. In the 2D transform, fn(r) is both a coefficient of a Fourier series and a source for a Hankel transform. Some of the relations for Zernike plates have been published previously (Edgcombe, 2016).

3.1 Illumination Function In classical optics, it is often assumed that the illumination has infinite transverse extent and varies simply as exp i(kz  ωt), but this raises a question about the transform of the incident wave. When Cartesian coordinates are used, the Fourier transform in each coordinate direction is represented by a Dirac delta function and justified by distribution theory (Lighthill, 1959). Here we wish to use cylindrical coordinates, and there appears to be a question of how to find the Hankel transform of a radially uniform wave of infinite extent. The excursion into distribution theory can be avoided by noting that in practice the illumination, instead of being transversely unbounded, will always be limited radially. We therefore assume that at the plane of the object exit wave the illumination is nonzero only within a radius a that is much greater than the specimen radius. Here we omit the factor exp i(kz  ωt) and represent the illumination at the specimen by ψ ¼ uðr=aÞ where u is a unit step function: uðρÞ ¼




1, 0 < ρ < 1 0, 1 < ρ

77

Phase Plates for TEM

The Hankel transform H0 of this rotationally symmetric wave is (using Eq. (A.4)) Z ∞ H0 ðq, aÞ ¼ uðr=aÞJ0 ðqr Þr dr 0

¼ a J1 ðqaÞ=qa 2

Then provided a is not infinite, H0 is well defined even when q ! 0.

3.2 Object Function Here we assume that the object advances the phase of the wave (relative to free-space propagation) by γw(r, ϕ), where γ is a constant angle, w is a specified function of approximately unit magnitude and zero for r greater than some value b < a, and r and ϕ are the cylindrical coordinates transverse to the axis of the system. Any absorption by the object will be ignored here. Then, with the incident wave as in Section 3.1, the exit wave is described by ψ ¼ uðr=aÞ exp iγw ðr, ϕÞ (With this definition, the phase change γw is defined only relative to an integer multiple of 2π.) For convenience in discussing weak-phase behavior, ψ is now written as the sum of the incident wave and a scattered wave f: ψ ¼ uðr=aÞ + f ðr, ϕÞ from which

    f ¼ uðr=aÞ eiγw  1 ¼ eiγw  1 ¼ iγw ðr, ϕÞ + O γ 2

(1)

The second equality follows because u is unity over the range smaller than a where w is nonzero. If some part of the direct beam is present at the image, then linear modulation of the image intensity by the object phase may be possible. We shall assume that the plates defined here introduce zero additional phase shift on the axis. Then the amplitude of the image of the incident wave is some fraction of u(r/a). This quantity is real, so at the image the phase of the scattered wave, relative to the incident one, can be determined easily from the (complex) amplitude of the image of the scattered wave. If the phases of the images of the incident and scattered waves happen to differ by π/2, so that their amplitudes at the image can be represented as u

78

Christopher J. Edgcombe

and (iη), then the total image intensity is ju + iηj2 ¼ u2 + η2 and the intensity is modulated only as η2. However, if some part of the image of the scattered wave is in phase with the incident wave and has the real amplitude η, then the total intensity is ju + ηj2 ¼ u2 + 2ηu + O(η2), and the intensity is linearly modulated by η. Thus for weak-phase objects with η < 1, the variation of image intensity with object phase will be linear for any part of the image amplitude that is in phase with the image of the incident wave. Three types of objects will be defined here. The disc object, though not typical of most biological objects, provides the simplest analysis for a range of plates. A more realistic object might be the spherical form, for which a weakphase result has been obtained with a Zernike plate. An object with a phase shift proportional to azimuthal angle has also been defined, for possible future use. 3.2.1 Disc Object This object is assumed to produce an uniform-phase change γ (relative to freespace propagation) over its diameter, 2b, while outside this diameter the phase change is zero (Fig. 14A). The scattered wave at this object can be expressed as   fd ðρÞ ¼ eiγ  1 uðρÞ (2) which is identical with fd ðρÞ ¼ exp iγuðρÞ  1

(3)

where u(ρ) is a unit step as defined in Section 3.1, and ρ ¼ r=b The Hankel transform of (2) according to (A.4) with n ¼ 0 can be written as   (4) Fd ðq, bÞ ¼ eiγ  1 b2 J1 ðQÞ=Q A

B

u

v 1

1

r

r 2b

2b

Fig. 14 Form of phase profile used for two simple objects: (A) disc object; (B) spherical object.

79

Phase Plates for TEM

where Q ¼ qb and J1( ) is a Bessel function of the first kind. 3.2.2 Spherical Object For an exit wave that might be produced by a spherical object of radius b, centered on the axis, the phase advance at radius r from the axis is modeled as proportional to the axial extent of the object at the same radius (Fig. 14B). Then the scattered wave is fs ðρÞ ¼ exp iγvðρÞ  1 where γ is constant and vðρÞ ¼




ð1  ρ2 Þ 0,

1=2

, 0 1. For a weak-phase object, the choice 1.0

1.0

Ui – ld B=1

0.5 5

10

15

20 –0.5

–1.0

–1.0

5

10

B=4

0.5

q0ri

–0.5

1.0

1.0 B=2

0.5

15

20 –0.5 –1.0

5

10

15

B=8

0.5 20 –0.5

5

10

15

20

–1.0

Fig. 16 Plots of ui (proportional to disc object phase) (dashed) and (ui – Id) (proportional to image intensity in WPA, solid line) as functions of q0ri for objects of the size parameter B ¼ 1, 2, 4, and 8 with a Zernike plate of step profile. The polarity of intensity modulation at the detector depends on the sign of (sin α). These radial functions do not depend on α.

85

Phase Plates for TEM

A1

B 4

0

3 2

–1 –2

0.5p a =p

h

–3 –4

0.5p a =p 1.5p

h

1 0

1.5p 0

2

4

6 Object phase

–1 8

0

2

4

6

8

Object phase

Fig. 17 Variation of h(γ, α, Id) with object phase, for (A) Id ¼ 0 and α ¼ π/2, π, and 3π/2; (B) Id ¼ 1 and α ¼ π/2, π, and 3π/2. By definition, h(γ, α, 0) ¼ h(γ, 2π  α, 1).

of α does not affect the relative radial variation of contrast. When γ is larger, the relation between intensity and γ depends on α, as shown by the plots of h(γ, α, Id) in Fig. 17 for Id ¼ 0 and 1 and for three values of α. The value Id ¼ 0 (minimizing the loss integral) is obtained approximately in imaging objects with B < 1, and so is of major interest for the range of object sizes over which the plate is effective; it also occurs outside larger objects. Then when α ¼ 3π/2, the intensity decreases continuously as γ increases from zero to about 2.4 radians, while if α ¼ π/2, the range of monotonic increase of intensity is reduced to about 0.8 radian. This result agrees qualitatively with that found for a Cartesian system by Beleggia (2008). The value Id ¼ 1 approximates the imaging of the interior of objects for which B > 1. For both values of Id, the plots for different α differ widely in their monotonic ranges, showing that the second term in (15) has a substantial influence on these ranges. In particular, when α ¼ π, the WPA would predict no variation of intensity, but inclusion of the second term in (15) causes the range of response to exceed that for α ¼ π/2 or 3π/2. Thus the WPA is not a good guide to the monotonic range of a plate. 4.3.3 A Disc Object With a Continuous Phase Transition The plots of intensity in Fig. 16 for step transitions in both γ and θ show noticeable radial fluctuation or “ringing,” for values of B  1, that is absent from the object phase distribution. Similar behavior in electrical filter circuits can sometimes be reduced by arranging that the response changes more smoothly with frequency. It thus seemed worth investigating the behavior of a Zernike plate in which the change of phase is made to occur continuously over a range of radius, as in Fig. 15B. By using the general definition for the phase function of the plate, varying for frequencies below q0, the image of the disc object can be written in the form

86

Christopher J. Edgcombe

  gdn ðri Þ ¼ 1 + eiϕ  1 ζ n

(16)

where the transform of F(q, b) has been replaced by the scattered wave as defined by (2), ζn is defined by Z B  iα  iα e  eiθpn J1 ðQÞJ0 ðρi QÞdQ (17) ζn ¼ u e  0

and the function θp(q) must be inserted for each specific plate. The intensity for a disc object can be found from (16) as

(18) jgdn j2 ¼ 1  2Imðζ Þ sinγ + 2 jζj2  ReðζÞ ð1  cos γ Þ 4.3.3.1 A Weak-Phase Disc Object With a Continuous Phase Transition

The approximation to (18) to the first order in γ is jgdw j2  1  2ηdp γ sin α where ηdp ¼ Im(ζ)/sin α can be obtained from (17) as Z B

1  sinθp = sin α J1 ðQÞJ0 ðρi QÞdQ ηdp ðρi , BÞ ¼ uðρi Þ  0

In Fig. 18, ηd2 for the ramped profile of Fig. 15B is shown for the two plate phases α ¼ π/2 and α ¼ 3π/2 with objects of sizes B ¼ 1, 2, 4, and 8. The difference between plots for the two values of α is substantial: with α ¼ π/2 the central response remains above the mean value for the largest object considered here, while with α ¼ 3π/2 the contrast reverses for objects with B values 4 and 8. The form of the transition in phase near the central hole in the plate thus has a strong influence on the low-frequency components of the images of disc objects sufficiently large that B > 1. Where the phase transition has the form of a ramp, the choice α ¼ π/2 gives more realistic images than α ¼ 3π/2 for weak-phase large disc objects.

4.4 Images for Spherical Object and Zernike Plate 4.4.1 A Weak-Phase Spherical Object With a Stepped Plate The image for the spherical object can be obtained from (10) and (11) with w(ρi) ¼ v(ρi). At the date of writing, an analytic transform for a spherical object has been found only by using the WPA described in Section 3.2.2. Thus with this object the loss integral can be found only in WPA. After

87

Phase Plates for TEM

1.0

1.0

1.0

1.0

0.5

0.5

0.5

0.5

q0ri 5

-0.5

10

15

B=1

-1.0

20 -0.5

5

10

15

B=2

-1.0

20 -0.5

5

10

15

B=4

-1.0

20 -0.5

1.0

1.0

1.0

0.5

0.5

0.5

0.5

5 -0.5 -1.0

10

15

B=1

20 0.5

5

10

15

B=2

1.0

20 -0.5 -1.0

5

10

15

B=4

10

20

15

20

B=8

-1.0

1.0

q0ri

5

5

10

-0.5 -1.0

15

20

B=8

Fig. 18 Responses of full-ramp profile to large disc objects of weak phase. Upper row: phase added by plate is π/2; lower row, phase added by plate is 3π/2.

combining the transform (7) with TZ1 for the stepped Zernike plate, and using (10) and (11), the weak-phase image function and intensity can be written as r    i + eiα ð exp iγvðρi Þ  1Þ  iγ eiα  1 Isw1 ðρi , BÞ gsw1 ðri Þ ¼ u a   jgsw1 j2 ¼ uðri =aÞ½1  2γ ðv  Isw1 Þ sinα + O γ 2 where v(ρ) is the radial distribution of object phase as defined by (6) and a new loss function Isw1 is defined by Z

B

Isw1 ðρi , BÞ ¼

 sin Q=Q2  cosQ=Q J0 ðρi QÞdQ

0

These results for gsw contain approximations to first order in γ. Plots of Isw and (v – Isw) for B ¼ 1, 2, 4, and 8 are shown in Figs. 19 and 20, respectively. Just as for the disc object, the modulation of the image intensity in Fig. 20 resembles the object distribution closely when B < 1. As B increases, the intensity modulation reduces in magnitude and shows some ringing, but less than the disc object and without the discontinuity at the object’s edge. At ri ¼ 0, Isw ¼ (1  sin B/B). 4.4.2 A Weak-Phase Spherical Object With a Ramped Zernike Plate The effect of distributing the phase change at the plate can be found as for the disc object. The image of the spherical object in the WPA can be written as gsw2 ðri Þ ¼ uðri =aÞ + eiα ð exp iγvðρi Þ  1Þ  iγ eiα Isw2 ðρi , BÞ

88

Christopher J. Edgcombe

1.

1. B=1

0.5

1. B=2

0.5

1. B=4

0.5

B=8

0.5

q0ri 5

10

15

20

–0.5

5

10

15

20

–0.5

5

10

15

20

–0.5

5

10

15

20

–0.5

Fig. 19 Function Isw(ρ, B) plotted as a function of ρB ¼ q0r for B ¼ 1, 2, 4, and 8. The dashed lines show the corresponding object functions v(ρ) (as defined by (6)) for the same values of B. 1.

1. B=1

0.5

1.

1. B=2

0.5

B=4

0.5

B=8

0.5

q0ri 5

10

15

5

20

10

15

20

–0.5

–0.5

5

10

15

5

20

10

15

20

–0.5

–0.5

Fig. 20 Plots of (v – Isw) representing radial variation of intensity from spherical objects of relative radii B ¼ 1, 2, 4, and 8 with a stepped Zernike plate (as Fig. 15A). At ri ¼ 0, (v – Isw) ¼ (sin B/B).

1.

1. B=1

0.5

1.

1. B=2

0.5

B=4

0.5

B=8

0.5

q0ri 5

10

15

20

–0.5

10

5

15

20

–0.5

1.

1.

B=1

0.5 5 –0.5

10

15

15

10

–0.5

15

20

1.

B=4

10

15

20

15

20

B=8

0.5 5

–0.5

5

20

0.5 5

20

10

–0.5

1.

B=2

0.5

q0ri

5

–0.5

10

15

5

20

10

–0.5

Fig. 21 Plots of v (dashed lines) and (v – Isw2) (solid) representing radial variation of intensity from weak-phase spherical objects of relative radii B ¼ 1, 2, 4, and 8 with a fully ramped Zernike plate (as Fig. 15B). Upper row: phase added by plate is π/2; lower row, phase added by plate is 3π/2.

and   jgj2 ¼ uðri =aÞ  2γ ðv  Isw2 Þ sinα + O γ 2 where Z Isw2 ðρi , BÞ ¼

B

ð1  sinθ2 ðQ=BÞ=sin αÞ

 sinQ=Q2  cos Q=Q J0 ðρi QÞ dQ 0

Plots of v and (v – Isw2) for B ¼ 1, 2, 4, and 8 and a fully ramped plate are shown in Fig. 21. It can be seen that for the fully ramped plate with α ¼ π/2,

89

Phase Plates for TEM

the images reduce in amplitude progressively while maintaining the same sense, but when α ¼ 3π/2 then contrast reversal appears for B  4. The effect of the plate is thus similar to that for the disc object.

5. IMAGES WITH STRAIGHT-EDGED PLATES ON THE CYLINDRICAL AXIS Here we consider a straight-edged plate, located so that the plate edge just touches the axis of the cylindrical system. The 2D transform in cylindrical coordinates that will be used here is described in Appendix A.

5.1 A Disc Object With an Opaque (Foucault) Plate The transmission function TF of the opaque plate was found in Section 3.3.2. The spectrum from a disc object at exit from the plate is the product TF(θ) Fd(qb) and contains a mean value together with a Fourier series with odd periodicities whose coefficients are functions of q. It can be written in the form of (A.8) as TF Fd ¼ H0 ðqÞ +

∞ X

ið2m1Þ exp ið2m  1ÞθH2m1 ðqÞ

(19)

m¼∞

where H0 ¼ Fd ðQÞ=2, H2m1 ðqÞ ¼ iFd ðQÞ=π ð2m  1Þ   Fd ðQÞ ¼ eiγ  1 b2 J1 ðQÞ=Q, Q ¼ qb Eq. (19) is already resolved into angular harmonics and the Hn are defined. We can therefore use (A.11) to find the image function at (ri, ϕi) as   Z ∞ ∞ X ri gðri , ϕi Þ ¼ u H0 ðqÞ J0 ðqri Þqdq + exp ið2m  1Þϕi + a 0 m¼∞ Z ∞  H2m1 ðqÞ J2m1 ðqri Þqdq 0

" #   ∞  iγ  1 ri 2X cos ð2m  1Þϕi uðρ Þ + i + e 1 L2m1 ðρi Þ ¼u 2 i π m¼1 ð2m  1Þ a

ð20Þ

90

Christopher J. Edgcombe

where ρi ¼ ri =b and

Z

L2m1 ðρi Þ ¼

∞

J1 ðQÞ J2m1 ðρi QÞ dQ , m > 0

0

The integral L0 has been replaced by its known value which is u(ρi), the unit step function (eq. 10.22.63 of DLMF with μ ¼ 1, and Fig. 21A). It thus provides a copy of the object, reduced by the factor of 1/2. We seek to know whether the image amplitude g has a real part that is in phase with the direct beam and can contribute linearly to the image intensity. The second term on the right of Eq. (20) shows that, for weak-phase objects with (eiγ  1)  iγ, a real part can occur only if the square bracket contains an imaginary part that is nonzero. The copy of the object provided by u/2 in (20) is real, so its phase remains as that of the scattered wave and it does not produce a linear response in the intensity. The other radial components in (20) are all changed in phase by π/2, as is desired. However, Fig. 22 shows that the amplitudes of the image components for (2m  1 ¼ 3, 5, 7) are very small at radii less than that of the disc. The component for 2m  1 ¼ 1 provides some contribution for periodicity 1 at image radii near that of the object, but all components show some form of singularity at the object radius. This type of plate thus may emphasize gradients, outlines and discontinuities, but seems unlikely to produce an image of a phase object in the way that is desired.

5.2 A Disc Object With a Half-Plane Phase-Changing (Hilbert) Plate This plate has the same geometry as the Foucault plate and is assumed to be similarly located, but we now assume that the solid half-plane, instead of intercepting half the wave, is nonabsorbing and provides a phase change of α. The contribution to the image from the half-plane of free space remains as for the Foucault plate, but is augmented by the phase-changed contribution from the other half plane. The transmission function now has the form
1, π=2 < θ < π=2 TH ðθÞ ¼ iα e , π=2 < jθj < π Since the transforms are linear, we can find the image that is added to that of the Foucault plate (Eq. (20)) by adding the spectrum of another Foucault plate, as in Section 3.3.3 but with θ replaced by θ + π and advanced in phase by α. The additional transmission factor ΔT and the total TH are

91

Phase Plates for TEM

1.0

1.0 0.5

0.5

n=0

n=1

r 0.5

1.0

1.5

0.5

2.0

–0.5

–0.5

–1.0

–1.0

1.0

1.0

0.5

0.5

n=2

0.5

1.0

1.5

2.0 –0.5

–1.0

–1.0

1.0

1.0

0.5

0.5

n=4

0.5

1.0

1.5

2.0 –0.5

–1.0

–1.0

1.0

1.0

0.5

n=6

0.5

0.5

1.0

1.5

2.0 –0.5

–1.0

–1.0

1.0

1.5

2.0

1.0

1.5

2.0

1.0

1.5

2.0

n=7 0.5

–0.5

2.0

n=5 0.5

–0.5

1.5

n=3

0.5

–0.5

1.0

Fig. 22 Plots of function Ln(ρ) for n ¼ 0–7. It is known that L0 is the unit step function, and that L2m ¼ 0 for ρ < 1.

"

#  X m1 ∞ 1 1 ð 1 Þ ΔT ðθÞ ¼ eiα exp ið2m  1Þðθ + π Þ + 2 π m¼∞ ð2m  1Þ " #  X ∞ 1 ð1Þm1 iα 1 exp ið2m  1Þθ  ¼e 2 π m¼∞ ð2m  1Þ

92

Christopher J. Edgcombe

  ∞ 1 + eiα 1  eiα X ð1Þm1 TH ðθÞ ¼ + exp ið2m  1Þθ 2 π ð2m  1Þ m¼∞ By comparison with (8) and (20), the image of a disc object obtainable with a Hilbert plate is then r    i + eiγ  1 a " # ∞     X cos ð2m  1Þϕi iα uðρi Þ iα 2  1+e L2m1 ðρi Þ +i 1e 2 π m¼1 ð2m  1Þ

gðρi , ϕi Þ ¼ u

(21) Eq. (21) shows that if α is chosen to be π, as often happens for Hilbert plates, then the part of the image that is the copy of the (uniform-phase) object u(ri/b) vanishes. Also the ϕ-periodic components are imaginary, as desired, but the plots of the radial functions L2m1 shown in Fig. 22 make clear that for 2m  1 ¼ 3, 5, and 7, where there should be images of the body of the object (r < b), they also vanish almost completely. Only the component varying as cos ϕi contributes appreciably, and even that vanishes linearly toward the center of the image. The ϕi-varying components all have some spiky response at the boundary of the disc object analyzed. It seems worthwhile to consider other possible values for α. The imaginary part of the coefficient of u in (21) is maximized when α is π/2 or 3π/2. Then the amplitude g of the complete image and its intensity jgj2 can be expected to contain linear copies of the disc object function, even for strong-phase changes. A recent study by Koeck (2015), which takes account of defocus and spherical aberration, shows that a phase shift of π/2 has the further advantage of extending the first passband, when operated at the Scherzer defocus. This form of plate is relatively simple to make but, as usual, some method will be needed to minimize possible charging from the direct beam near the edge of the plate. Many other objects and forms of the plate can be imagined. However, in most cases the integrals required are likely to take longer to evaluate than those reported here.

Phase Plates for TEM

93

6. DISCUSSION The spectral transfer method outlined in Sections 4 and 5 gives details of the images to be expected from round lenses with some simple forms for the object and the postspecimen phase plate. By including the transverse phase distribution of objects, it makes analytic statements about topics not normally accessible by use of single-frequency transfer functions alone. These statements include: (1) A quantitative measure has been defined for the “size” of an object in a system with a Zernike plate, beyond which appreciable distortion of the image can be expected. From the calculations shown here, it seems that imaging may be acceptably accurate when B as defined in Section 4.1 is less than about unity. This condition agrees with that found by Danev, Glaeser, and Nagayama (2009), who state that “the cut-on periodicity should be at least twice the particle size.” The angular frequency q0 used here is (2π/cut-on periodicity), so in terms of our B ¼ q0b, the condition given by Danev, Glaeser, and Nagayama is that B  π/2. Also, Hall, Nogales, and Glaeser (2011) find in detailed simulations that maximum contrast is achieved when the period at cut-on approaches three times the particle size, that is when B  1.05. The condition obtained here for simple objects, that B should be less than about 1 for accurate imaging, thus agrees well with that found experimentally and by simulation for more realistic objects. (2) Calculations of image intensity show the forms of distortion to be expected for some combinations of object and plate. Comparison of Fig. 16 with Fig. 20, and Fig. 18 with Fig. 21, shows that the artifacts introduced by the imaging system are very similar for the two different objects considered, in WPA. (3) The artifacts introduced when the phase response of the plate cuts on with a sharp step can be reduced by making the transition in phase more gradual, if the phase added by the plate is sufficiently small. When this added phase is 3π/2, severe contrast reversal can occur for objects of size parameter B ¼ 4 and B ¼ 8, but the contrast falls smoothly without reversing, as B increases, when the added phase is π/2. (4) The calculated image amplitudes show the phase relation between the direct beam and components of the image of the scattered wave. This

94

Christopher J. Edgcombe

enables identification of terms that will provide a linear response of intensity to phase variation in the object. (5) In the systems simulated, it is found that plates that merely obstruct part of the wavefront without altering phase do not produce linear modulation of image intensity. Zernike and Hilbert plates that provide a phase shift relative to the direct wave do give the possibility of linear intensity modulation. (6) This theory predicts that the traditional choice of π for the phase added by a Hilbert plate cancels the low-frequency components in the image of a uniform-phase disc, when the plate edge touches the cylindrical axis. The added phase needed to maximize the low-frequency contrast is π/2. This deduction clearly needs further testing and comparison with the earlier theory. (7) The maximum diameter of objects that can be imaged with little distortion is proportional to the focal length of the imaging lens. This maximum can therefore be raised by changing from the standard objective to a minilens, as reported by Minoda, Okabe, and Iijima (2011).

7. CONCLUSIONS Prespecimen phase plates are used to generate beams with quantized OAM, so they need to be designed or analyzed with attention to quantization. Postspecimen plates are used at present to improve the contrast in the microscopy of biological molecules that are large enough to be modeled by classical analysis. In analysis of imaging, evaluation of the spatial frequency transform of an object, together with the response of the plate, can give more information about possible image distortion than is available from contrast transfer functions. Applied to some simple model systems, this method shows that a Zernike plate can provide accurate imaging up to a definable size of object; this size depends not only on object diameter but also on the system parameters. Analysis for a Hilbert plate located exactly on the cylindrical axis suggests that the usual choice of π for the added phase destroys the linear response of intensity to the phase of a weak-phase object, but that a linear response can be obtained if the added phase is reduced to about π/2. While this method of analysis can be applied at present only to limited types of object and plate, it has given useful information about the two types of plate that are most widely applied.

95

Phase Plates for TEM

ACKNOWLEDGMENTS The author thanks the editorial staff of the following journals for permission to reprint figures, as acknowledged in the individual captions: Journal of Structural Biology Nature Nature Physics Physical Review Letters Ultramicroscopy Zeitschrift f€ ur Naturforschung

APPENDIX A. THE 2D TRANSFORM IN CYLINDRICAL COORDINATES Before defining the full 2D transform, we state first the Fourier series and Hankel (1D) transform to be used here.

A.1. FOURIER SERIES A given function f(r, ϕ) that is periodic in the angle ϕ with period 2π can be represented by a Fourier series in exp iϕ as f ðr, ϕÞ ¼

∞ X

ei‘ϕ f‘ ðr Þ

(A.1)

‘¼∞

where ‘ is integral and the (complex) f‘ are defined by Z 1 π i‘ϕ f‘ ðr Þ ¼ e f ðr, ϕÞ dϕ 2π π

(A.2)

A.2. HANKEL TRANSFORM A given function of r, f‘(r), can be represented by a function H‘(q) in a variable q as Z ∞ f‘ ðr Þ ¼ H‘ ðqÞJ‘ ðrqÞ q dq (A.3) 0

96

Christopher J. Edgcombe

where H‘ is defined by

Z H‘ ðqÞ ¼

∞

f‘ ðr ÞJ‘ ðqr Þ rdr

(A.4)

0

and J‘ (qr) is a Bessel function of the first kind and integral order ‘. The definitions (A.3) and (A.4) are one way of writing a Hankel pair. Section A.3 shows that relations in this form are useful when the 2D Cartesian transform is converted to cylindrical coordinates. An alternative pair, defined earlier by Hankel with different weighting functions, is used in DLMF (2016) and some other tables. In applications, it may be useful to think of q as a radial phase constant in the angular measure, while bearing in mind that the local wavelength of a Bessel function J‘ (qr) differs from 2π/q. The transform denoted here by H‘ has no relation to the Hankel function H‘(n) (¼J‘ iY‘). If f‘ is a constant independent of r, the integral in (A.4) is divergent, so some constraint on f‘ is needed. In practice, f0 will be limited since the wave will have zero amplitude outside some maximum radius, but also we should avoid assuming that f(r, ϕ) can be specified simply as cos ‘ϕ , for instance. It may seem that, when (A.2) has provided the Fourier coefficient functions f‘(r), suitable spectra could be obtained from transforms other than the Hankel transform, by using any convenient basis. However, the following derivation shows that if we restate the Cartesian transform by defining the same operation in cylindrical variables, the Hankel transform is necessarily obtained.

A.3. CONVERSION FROM CARTESIAN FORM The Cartesian 2D transform of f can be specified by (for example, section 2.1.1 of Goodman (1996) with 2πfX replaced by qx and similarly for fy) Z Z   1 ∞ ∞ F¼ f exp  i qx x + qy y dxdy (A.5) 2π ∞ ∞ Here the initial constant has been chosen for later convenience. Relation (A.5) can be expressed in cylindrical coordinates r, ϕ, q, θ by converting the Cartesian components to their cylindrical equivalents: qx ¼ q cos θ, qy ¼ q sin θ, x ¼ r cos ϕ, y ¼ r sin ϕ

97

Phase Plates for TEM

In applications, θ is the azimuthal angle of q ¼ (q, θ), on the same axis and with the same origin as ϕ. After substituting into (A.5) with suitable changes to the ranges of integration, and specifying f and F as functions of cylindrical coordinates at the appropriate transverse planes, Z Z 1 π ∞ F ðq, θÞ ¼ f ðr, ϕÞexp  i½qr cos ðϕ  θÞ rdrdϕ (A.6) 2π π 0 An object function specified as f(r, ϕ) is necessarily periodic in ϕ with period 2π and so can be expressed as a Fourier series in exp iϕ. To obtain the harmonic dependence on θ, we first substitute in (A.6) from (A.1). We then define a new angle ψ by ψ ¼ϕθ Then, with θ held constant as ϕ varies, F can be expressed as Z ∞ Z π ∞ 1 X i‘θ F ðq, θÞ ¼ e f‘ ðr Þ cos‘ψ exp  i½qr cos ψ  dψ r dr π ‘¼∞ 0 0 and after using Bessel’s integral (10.9.2 of DLMF, 2016) for ψ with θ constant, F can be written as F ðq, θÞ ¼

∞ X

‘ ilθ

i

Z

∞

e

f‘ ðr Þ J‘ ðqr Þ r dr

(A.7)

0

‘¼∞

or, by using (A.4), F ðq, θÞ ¼

X

i‘ ei‘θ H‘ ðqÞ

(A.8)

‘

The series (A.7) has been obtained in harmonics of (exp iθ) and so the range of ‘ remains from ∞ to +∞. On substituting (A.2) in (A.7), the first relation of the 2D transform then follows as a function of the original object function: Z Z 1 X ‘ i‘θ ∞ π i‘ϕ F ðq, θÞ ¼ i e e f ðr, ϕÞ dϕJ‘ ðqr Þrdr (A.9) 2π ‘ 0 π

98

Christopher J. Edgcombe

A.4. INVERSE TRANSFORM An application such as Section 5 may provide the spectrum in the form of (A.8), from which the H‘ can be identified. If not, the H‘ can be found from F(q, θ) by the Fourier inversion of (A.8): Z 1 π ‘ i‘θ H‘ ðqÞ ¼ i e F ðq, θÞ dθ (A.10) 2π π When the H‘ are known, f can be found by combining (A.1) and (A.3) as Z ∞ ∞ X i‘ϕ e H‘ ðqÞJ‘ ðqr Þ qdq (A.11) f ðr, ϕÞ ¼ ‘¼∞

0

The relation (A.11) is used in Section 5 to find the image function from a known Fourier series for the spectrum. The second relation of the 2D transform, for f as a function of F, can be found by inserting (A.10) in (A.11): Z ∞Z π ∞ 1 X ‘ i‘ϕ ie ei‘θ F ðq, θÞdθ J‘ ðqr Þ q dq (A.12) f ðr, ϕÞ ¼ 2π ‘¼∞ 0 π Alternatively, on inserting (A.10) in (A.3) and substituting the resulting expression for f‘ into (A.1), the complete inversion agrees with (A.12).

REFERENCES Aharonov, Y., & Bohm, D. (1959). Significance of electromagnetic potentials in the quantum theory. Physical Review, 115, 485–491. Babiker, M., Yuan, J., & Lembessis, V. E. (2015). Electron vortex beams subject to static magnetic fields. Physical Review A: Atomic, Molecular, and Optical Physics, 91(1), 1–5. https://doi.org/10.1103/PhysRevA.91.013806. Beche, A., Van Boxem, R., Van Tendeloo, G., & Verbeeck, J. (2013). Magnetic monopole field exposed by electrons. Nature Physics, 10(1), 26–29. https://doi.org/10.1038/ nphys2816. Beleggia, M. (2008). A formula for the image intensity of phase objects in Zernike mode. Ultramicroscopy, 108(9), 953–958. https://doi.org/10.1016/j.ultramic.2008.03.003. Blackburn, A. M., & Loudon, J. C. (2014). Vortex beam production and contrast enhancement from a magnetic spiral phase plate. Ultramicroscopy, 136, 127–143. https://doi.org/ 10.1016/j.ultramic.2013.08.009. Bliokh, K. Y., Bliokh, Y. P., Savel’Ev, S., & Nori, F. (2007). Semiclassical dynamics of electron wave packet states with phase vortices. Physical Review Letters, 99, 190404. https:// doi.org/10.1103/PhysRevLett.99.190404. € die kontraste von atomen im elektronenmikroskop. Zeitschrift Fur Boersch, H. (1947). Uber Naturforschung—Section A Journal of Physical Sciences, 2(11–12), 615–633. https://doi.org/ 10.1515/zna-1947-11-1204. Born, M., & Wolf, E. (1993). Principles of optics (6th ed.). Oxford: Pergamon Press.

Phase Plates for TEM

99

Buijsse, B., van Duinen, G., Sader, K., & Danev, R. (2014). Challenges in phase plate product development. Microscopy and Microanalysis, 20(Suppl. 3), 218–219. https://doi.org/ 10.1017/S1431927614002815. Buijsse, B., van Laarhoven, F. M. H. M., Schmid, A. K., Cambie, R., Cabrini, S., Jin, J., & Glaeser, R. M. (2011). Design of a hybrid double-sideband/single-sideband (schlieren) objective aperture suitable for electron microscopy. Ultramicroscopy, 111(12), 1688–1695. https://doi.org/10.1016/j.ultramic.2011.09.015. Cambie, R., Downing, K. H., Typke, D., Glaeser, R. M., & Jin, J. (2007). Design of a microfabricated, two-electrode phase-contrast element suitable for electron microscopy. Ultramicroscopy, 107(4–5), 329–339. https://doi.org/10.1016/j.ultramic.2006. 09.001. Danev, R., Buijsse, B., Khoshouei, M., Plitzko, J., & Baumeister, W. (2014). Volta potential phase plate for in-focus phase contrast transmission electron microscopy. Proceedings of the National Academy of Sciences of the United States of America, 111, 15635–15640. Danev, R., Glaeser, R. M., & Nagayama, K. (2009). Practical factors affecting the performance of a thin-film phase plate for transmission electron microscopy. Ultramicroscopy, 109(4), 312–325. https://doi.org/10.1016/j.ultramic.2008.12.006. Danev, R., & Nagayama, K. (2001). Transmission electron microscopy with Zernike phase plate. Ultramicroscopy, 88(4), 243–252. https://doi.org/10.1016/S0304-3991(01)00088-2. Danev, R., Okawara, H., Usuda, N., Kametani, K., & Nagayama, K. (2002). A novel phasecontrast transmission electron microscopy producing high-contrast topographic images of weak objects. Journal of Biological Physics, 28(4), 627–635. https://doi.org/10.1023/ A:1021234621466. DLMF. (2016). NIST Digital Library of Mathematical Functions. In F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller & B. V. Saunders (Eds.), http://dlmf.nist.gov/, Release 1.0.14 of 2016-12-21. Dries, M., Hettler, S., Schulze, T., Send, W., M€ uller, E., Schneider, R., … Samwer, K. (2016). Thin-film phase plates for transmission electron microscopy fabricated from metallic glasses. Microscopy and Microanalysis, 22(5), 955–963. https://doi.org/10.1017/ S143192761601165X. Edgcombe, C. J. (2010). A phase plate for transmission electron microscopy using the Aharonov-Bohm effect. Journal of Physics Conference Series, 241, 12005. https://doi. org/10.1088/1742-6596/241/1/012005. Edgcombe, C. J. (2014). Imaging of weak phase objects by a Zernike phase plate. Ultramicroscopy, 136, 154–159. Edgcombe, C. J. (2016). Imaging by Zernike phase plates in the TEM. Ultramicroscopy, 167, 57–63. https://doi.org/10.1016/j.ultramic.2016.05.003. Edgcombe, C. J., Ionescu, A., Loudon, J. C., Blackburn, A. M., Kurebayashi, H., & Barnes, C. H. W. (2012). Characterisation of ferromagnetic rings for Zernike phase plates using the Aharonov-Bohm effect. Ultramicroscopy, 120, 78–85. https://doi.org/ 10.1016/j.ultramic.2012.06.011. Ehrenberg, W., & Siday, R. E. (1949). The refractive index in electron optics and the principles of dynamics. Proceedings of the Physical Society Section B, 62(1), 8. https://doi.org/10. 1088/0370-1301/62/1/303. Foucault, L. (1858). Description des procedees employes pour reconnaitre la configuration des surfaces optiques. Comptes Rendus de l’Academie des Sciences, 47, 958–959. Retrieved from http://gallica.bnf.fr/ark:/12148/bpt6k3004t/f960.item.r¼Foucault. Glaeser, R. M. (2013). Methods for imaging weak-phase objects in electron microscopy. The Review of Scientific Instruments, 84, 111101. Goodman, J. W. (1996). Introduction to Fourier optics (2nd ed.). London: McGraw-Hill. Gradsteyn, I. S., & Ryzhik, I. M. (2000). Table of integrals, series and products (6th ed.). London: Academic Press.

100

Christopher J. Edgcombe

Grillo, V., Gazzadi, G. C., Karimi, E., Mafakheri, E., Boyd, R. W., & Frabboni, S. (2014). Highly efficient electron vortex beams generated by nanofabricated phase holograms. Applied Physics Letters, 104(4), 043109. https://doi.org/10.1063/1.4863564. Hall, R. J., Nogales, E., & Glaeser, R. M. (2011). Accurate modeling of single-particle cryoEM images quantitates the benefits expected from using Zernike phase contrast. Journal of Structural Biology, 174(3), 468–475. https://doi.org/10.1016/j.jsb.2011.03.020. Harvey, T. R., Pierce, J., Agrawal, A., Ercius, P., Linck, M., & McMorran, B. J. (2014). Efficient diffractive phase optics for electrons. New Journal of Physics, 16, 93039. https://doi.org/10.1088/1367-2630/16/9/093039. Hawkes, P. W., & Kasper, E. (1994). Wave Optics (Principles of Electron Optics. Vol. 3). London: Academic Press. Hecht, E. (2002). Optics (4th ed.). London: Addison-Wesley. Henderson, R. (1995). The potential and limitations of neutrons, electrons and X-rays for atomic resolution microscopy of unstained biological molecules. Quarterly Reviews of Biophysics, 28(2), 171. https://doi.org/10.1017/S003358350000305X. Hettler, S., Gamm, B., Dries, M., Frindt, N., Schr€ oder, R. R., & Gerthsen, D. (2012). Improving fabrication and application of Zach phase plates for phase-contrast transmission electron microscopy. Microscopy and Microanalysis, 18(5), 1010–1015. https://doi. org/10.1017/S1431927612001560. oder, R. R., & Hettler, S., Wagner, J., Dries, M., Oster, M., Wacker, C., Schr€ Gerthsen, D. (2015). On the role of inelastic scattering in phase-plate transmission electron microscopy. Ultramicroscopy, 155, 27–41. https://doi.org/10.1016/j.ultramic.2015. 04.001. Idrobo, J. C., & Pennycook, S. J. (2011). Vortex beams for atomic resolution dichroism. Journal of Electron Microscopy, 60(5), 295–300. https://doi.org/10.1093/jmicro/dfr069. Koeck, P. J. B. (2015). Improved Hilbert phase contrast for transmission electron microscopy. Ultramicroscopy, 154, 37–41. https://doi.org/10.1016/j.ultramic.2015.03.002. Lighthill, M. J. (1959). Introduction to Fourier analysis and generalised functions. Cambridge: Cambridge University Press. Lipson, S. G., Lipson, H., & Tannhauser, D. (1995). Optical physics (3rd ed.). Cambridge: Cambridge University Press. Majorovits, E., Barton, B., Schultheiß, K., Perez-Willard, F., Gerthsen, D., & Schr€ oder, R. R. (2007). Optimizing phase contrast in transmission electron microscopy with an electrostatic (Boersch) phase plate. Ultramicroscopy, 107(2–3), 213–226. https:// doi.org/10.1016/j.ultramic.2006.07.006. Malac, M., Beleggia, M., Kawasaki, M., Li, P., & Egerton, R. F. (2012). Convenient contrast enhancement by a hole-free phase plate. Ultramicroscopy, 118, 77–89. https://doi.org/10. 1016/j.ultramic.2012.02.004. Malac, M., Kawasaki, M., Beleggia, M., Peng, L., & Egerton, R. F. (2010). Convenient contrast enhancement by hole free phase plate in a TEM. Microscopy and Microanalysis, 16(Suppl. 2), 526–527. https://doi.org/10.1017/S1431927610055807. Marko, M., Hsieh, C., Leith, E., Mastronarde, D., & Motoki, S. (2016). Practical experience with hole-free phase plates for cryo electron microscopy. Microscopy and Microanalysis, 22, 1316–1328. https://doi.org/10.1017/S143192761601196X. Marko, M., Meng, X., Hsieh, C., Roussie, J., & Striemer, C. (2013). Methods for testing Zernike phase plates and a report on silicon-based phase plates with reduced charging and improved ageing characteristics. Journal of Structural Biology, 184(2), 237–244. https://doi.org/10.1016/j.jsb.2013.08.008. McMorran, B. J. (2012). System and method for producing and using multiple electron beams with quantized orbital angular momentum in an electron microscope. US Patent 8680488. McMorran, B. J., Agrawal, A., Anderson, I. M., Herzing, A. A., Lezec, H. J., McClelland, J. J., & Unguris, J. (2011). Electron vortex beams with high quanta of orbital angular momentum. Science (New York, NY), 331(6014), 192–195. https://doi.org/10.1126/science.1198804.

Phase Plates for TEM

101

Minoda, H., Okabe, T., & Iijima, H. (2011). Contrast enhancement in the phase plate transmission electron microscopy using an objective lens with a long focal length. Journal of Electron Microscopy, 60(5), 337–343. https://doi.org/10.1093/jmicro/dfr067. M€ uller, H., Jin, J., Danev, R., Spence, J., Padmore, H., & Glaeser, R. M. (2010). Design of an electron microscope phase plate using a focused continuous-wave laser. New Journal of Physics, 12, 073011. https://doi.org/10.1088/1367-2630/12/7/073011. Nagayama, K. (2008). Development of phase plates for electron microscopes and their biological application. European Biophysics Journal, 37(4), 345–358. https://doi.org/10.1007/ s00249-008-0264-5. Ophus, C., Ciston, J., Pierce, J., Harvey, T. R., Chess, J., McMorran, B. J., … Ercius, P. (2016). Efficient linear phase contrast in scanning transmission electron microscopy with matched illumination and detector interferometry. Nature Communications, 7, 10719. https://doi.org/10.1038/ncomms10719. Osakabe, N., Nomura, S., Matsuda, T., Endo, J., & Tonomura, A. (1983). Phase-contrast electron microscope. Japan. Japanese Patent Application JP 60-007048. Rhinow, D. (2016). Towards an optimum design for thin film phase plates. Ultramicroscopy, 160, 1–6. https://doi.org/10.1016/j.ultramic.2015.09.003. Saitoh, K., Hasegawa, Y., Tanaka, N., & Uchida, M. (2012). Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates. Journal of Electron Microscopy, 61(3), 171–177. https://doi.org/10.1093/jmicro/dfs036. Schattschneider, P., & Verbeeck, J. (2011). Theory of free electron vortices. Ultramicroscopy, 111(9–10), 1461–1468. https://doi.org/10.1016/j.ultramic.2011.07.004. Schr€ oder, R. R., Barton, B., Rose, H. H., & Benner, G. (2007). Contrast enhancement by anamorphotic phase plates in an aberration corrected TEM. Microscopy and Microanalysis, 13(S03), 136–137. https://doi.org/10.1017/S143192760708004X. oder, R. R. (2006). Schultheiß, K., Perez-Willard, F., Barton, B., Gerthsen, D., & Schr€ Fabrication of a Boersch phase plate for phase contrast imaging in a transmission electron microscope. Review of Scientific Instruments, 77(3), 1–4. https://doi.org/10. 1063/1.2179411. Schultheiss, K., Zach, J., Gamm, B., Dries, M., Frindt, N., Schr€ oder, R. R., & Gerthsen, D. (2010). New electrostatic phase plate for phase-contrast. Microscopy and Microanalysis, 16(6), 785–794. https://doi.org/10.1017/S1431927610093803. Tamaki, H. (2013). Development of a contact-potential-type phase plate. Microscopy and Microanalysis, 19(Suppl. 2), 1148–1149. https://doi.org/10.1017/S1431927613007733. Uchida, M., & Tonomura, A. (2010). Generation of electron beams carrying orbital angular momentum. Nature, 464(7289), 737–739. https://doi.org/10.1038/nature08904. Valdre`, U. (1979). Electron microscope stage design and applications. Journal of Microscopy, 117, 55–75. Verbeeck, J. (2016). Can phase manipulation turn TEM into an even more versatile instrument? In 16th European microscopy congress (EMC2016) (Plenary talk). Lyon. Verbeeck, J., Beche, A., Guzzinati, G., Clark, L., Juchtmans, R., van Boxem, R., & van Tendeloo, G. (2015). Construction of a programmable electron vortex phase plate. MC2015 Proceedings (pp. 582–583). Regensburg: DGE (German Society for electron Microscopy). Verbeeck, J., Guzzinati, G., Clark, L., Juchtmans, R., Van Boxem, R., Tian, H., … Van Tendeloo, G. (2014). Shaping electron beams for the generation of innovative measurements in the (S)TEM. Comptes Rendus Physique, 15(2–3), 190–199. https://doi.org/10. 1016/j.crhy.2013.09.014. Verbeeck, J., Tian, H., & Beche, A. (2012). A new way of producing electron vortex probes for STEM. Ultramicroscopy, 113, 83–87. https://doi.org/10.1016/j.ultramic. 2011.10.008. Verbeeck, J., Tian, H., & Schattschneider, P. (2010). Production and application of electron vortex beams. Nature, 467(7313), 301–304. https://doi.org/10.1038/nature09366.

102

Christopher J. Edgcombe

Yang, H., Ercius, P., Nellist, P. D., & Ophus, C. (2016). Enhanced phase contrast transfer using ptychography combined with a pre-specimen phase plate in a scanning transmission electron microscope. Ultramicroscopy, 171, 117–125. https://doi.org/10.1016/j. ultramic.2016.09.002. Yuan, J. (2014). Bright electron twisters. Nature, 509, 37–38. Zach, J. (2007). Phase plate, image producing method and electron microscope. Germany. European Patent PCT/EP2007/009289. Zernike, F. (1942). Phase contrast, a new method for the microscopic observation of transparent objects. Physica, 9(10), 974–986. https://doi.org/10.1016/S0031-8914(42) 80079-8. Zernike, F. (1955). How I discovered phase contrast. Science, 121(11), 345–349.

CHAPTER THREE

X-Ray Lasers in Biology: Structure and Dynamics John C.H. Spence1 Arizona State University, Tempe, AZ, United States 1 Corresponding author: e-mail address: [email protected]

Contents 1. 2. 3. 4. 5. 6. 7.

Introduction The X-Ray Free-Electron Laser Data Acquisition Modes and Sample Delivery for XFEL Structural Biology Radiation Damage Limits Resolution Serial Crystallography at XFELs for Structural Biology Molecular Machines and Single-Particle Imaging Time-Resolved Serial Crystallography, Optical Pump-Probe Methods, and Photosynthesis 8. Time-Resolved SFX for Slower Processes: Mixing Jets and Other Excitations 9. Fast Solution Scattering and Angular Correlation Function Methods 10. Data Analysis 10.1 Serial Crystallography 10.2 Single Particles 11. Summary Acknowledgments References

103 107 111 116 120 122 126 129 130 133 133 138 143 144 144

1. INTRODUCTION To place EXFEL research in context, we begin by summarizing complementary techniques for atomic resolution imaging. Richard Feynman once commented that “Everything that living things do can be understood in terms of the jigglings and wigglings of atoms.” He believed that the atomic hypothesis, that matter consists of atoms, was mankind’s most important discovery. The first person to see an atom clearly was a student of Erwin Mueller’s at Penn State around 1951, using his field-ion microscope with a cooled tip (Muller, 1951). (Earlier work at room temperature had been less Advances in Imaging and Electron Physics, Volume 200 ISSN 1076-5670 http://dx.doi.org/10.1016/bs.aiep.2017.01.008

#

2017 Elsevier Inc. All rights reserved.

103

104

John C.H. Spence

successful.) From the ancient Greeks to Robert Hooke’s “Micrographica” in 1665 (where the angles between the facets he saw on microcrystals in his new microscope suggested the packing of tiny spheres), the idea that atoms are the basic building blocks of matter has motivated efforts to see them in order to explain everything from biochemical reactions to the performance of turbine blades in jet engines. The equally important challenge of understanding why atoms stick together in the particular arrangements they do, responsible for the properties of matter, from rusting ships to strong, light materials, explosives, and protein–ligand interactions, has proven far more difficult (Pauling, 1960; Spence, Huang, & Sankey, 1993; Zuo, Kim, O’Keeffe, & Spence, 1999). But it is the X-ray Free-electron Laser (XFEL) which now, for the first time, has allowed us to image hydrated molecules in motion during a chemical reaction, at atomic resolution, with femtosecond time resolution and negligible radiation damage (Pande et al., 2016). It was Albert Crewe in 1970 at Argonne National Laboratory who first saw an isolated atom in an electron microscope (STEM) (Crewe, Wall, & Langmore, 1970). Crewe dedicated years of work to the imaging of dehydrated DNA on carbon films in vacuum. Just prior to that work, the first three-dimensional reconstructions of stained viruses had been achieved by TEM (Derosier & Klug, 1968), and in 1975 molecular images were obtained using two-dimensional unstained crystals of bacteriorhodopsin at 0.7 nm resolution (Henderson & Unwin, 1975), leading eventually to the modern method of cryoelectron microscopy (cryo-EM). This field has made admirable steady incremental progress since, through the development of fast computers, field-emission guns, frozen-hydrated samples in vitreous ice, automated sample preparation methods, tomography, powerful new computer algorithms capable of sorting single-particle images by orientation and conformation, and most recently single-electron detectors, which bring the single-particle 3D reconstruction resolution to about 0.2 nm in favorable cases. The field is reviewed elsewhere (Glaeser, 2007; Spence, 2013). In biology, unlike materials science, DNA provides the ability to make unlimited numbers of “identical” copies of molecules, whose images (following orientation determination) may be merged to minimize radiation damage, and provide a three-dimensional reconstruction. Each molecule receives less radiation dose (about 10 beam electrons per A˚2) than would destroy it, resulting in an image projection that is far too noisy to form a useful image. After orientation determination, only the merged results of thousands of projections then produce a high-resolution three-dimensional image.

X-Ray Lasers in Biology: Structure and Dynamics

105

For smaller proteins, the NMR method has also proven very powerful and can also provide dynamical information (Lewandowski, Halse, Blackledge, & Emsley, 2015). Neutron diffraction avoids damage altogether, but requires the use of large crystals. The question of resolution is crucial to progress in molecular imaging, as we have recently seen from the huge impact which an improvement of only about 0.15 nm has had on the impact of the field of cryoEM, due to the introduction of better detectors. Imaging must provide sufficient resolution to faithfully reveal the three-dimensional building blocks of molecular biology. Very roughly, the important milestones for imaging the secondary structure of proteins are as follows: at 0.7 nm we first resolve the alpha-helices in proteins, at 0.47 the beta sheets, at 0.35 nm the individual amino acid residues, at 0.28 nm the water molecules, at 0.2 nm the protein side chains, and at 0.15 nm, atoms themselves become individually resolved. (At this resolution the “direct methods” ab initio numerical approach to solving the phase problem becomes effective for small molecules.) But chemical reactions such as enzyme catalysis involve time-dependent conformational changes in proteins, not the static molecular shapes, constrained by crystal formation, which crystallography provides. Molecular movies of a kind can, however, be obtained from cryo-EM, since samples rapidly quenched from an equilibrium ensemble provide images which may be sorted by conformational similarity, and so displayed as a movie. And the field of time-resolved crystallography (Moffat, 2014; Schlichting et al., 1989) allows the small atomic motions which do not destroy a crystal to be studied. A reacting species can be diffused into the crystal; however, this may take much longer than in solution. Here, it is important to note that certain proteins have been shown to remain chemically active as enzymes in crystalline form (Hajdu, Acharya, Stuart, Barford, & Johnson, 1988). In all this work, radiation damage imposes the most severe limit on data quality and resolution. Rather than spreading the dose over many molecules to reduce damage, and merging the resulting real-space images (essentially synthesizing an artificial crystal in the computer), the XFEL approach to molecular movies uses the ability of very brief pulses to outrun radiation damage, and deals with diffraction patterns (which require phasing) rather than the real-space images formed by an electron microscope. It has thus far required the use of small protein crystals in order to achieve atomic resolution, because even when the 1012 X-ray photons in a single XFEL pulse are focused onto a single virus, the scattering (which falls off as the inverse

106

John C.H. Spence

fourth power of the scattering angle for a spherical virus) is negligible at the high angles needed to reveal fine detail in the sample. For crystals, however, the “Bragg boost” is a powerful effect, since the intensity at the Bragg peak (not the angle-integrated intensity) which brings the peak above noise level is proportional to the square of the number of molecules in the crystal, so that even a nanocrystal consisting of 10
10
10 molecules will therefore provide a million times more peak intensity than one molecule, if instrumental and background effects are ignored. In the method of time-resolved crystallography (often based on the Laue method at synchrotrons; Moffat, 2014), a chemical reaction cycle can be triggered, involving a few atoms only at the center of the large identical protein molecules in each unit cell, which may contain mostly water. In that case the molecular motions may not destroy the crystal. If a femtosecond laser pulse is used to trigger the reaction (the “pump-probe” method), its progress should be synchronized in all the unit cells. However, there may be several reaction paths, and these experiments detect only the Bragg beams from the entire crystal, providing a periodic spatial average of charge density as a function of time. The method therefore depends on accurate knowledge of the unexcited structure (usually from previous crystallography) and modeling to extract the several simultaneously excited structures within the crystal from the Bragg intensities. Repeated exposures are needed to collect the many projections needed for phasing and three-dimensional reconstruction. This is an entirely different situation from the first-order phase transitions common in materials science, which generally nucleate at special sites, spreading throughout the crystal. These crystals are often best described as continuously bonded network of atoms, rather than the many individual molecules, each within a unit cell which may consist mainly of water, in protein crystallography. This difference has made it impossible to record atomic resolution time-resolved movies in three dimensions in materials science, except perhaps at surfaces. The field of XFEL applications to structural biology has been reviewed in Spence, Weierstall, and Chapman (2012), Bostedt et al. (2016), Schlichting (2015), and, with emphasis on new approaches to time-resolved diffraction, in Spence (2014) and special issues of journals have been devoted to it (see, for example, Spence and Chapman (2014)). A comparison of pulsed MeV electron diffraction with XFELs for the purpose of outrunning radiation damage is given in Spence (2017). We will see that the unique capability of this new form of pulsed radiation is its ability to outrun radiation damage, and so provide damage-free movies of molecular machines at work under physiological conditions.

107

X-Ray Lasers in Biology: Structure and Dynamics

2. THE X-RAY FREE-ELECTRON LASER The invention of the free-electron laser has a fascinating history, dating mainly from the development of traveling wave vacuum tubes for high-power microwave generation in the early 1950s. Klystrons, developed during World War II, were the first sources of coherent radiation from a free-electron beam. A simple explanation of the operation of the XFEL can be found in Ribic and Margaritondo (2012), the history of the XFEL is given in Pellegrini (2012), and a review of the principles of XFEL operation and its applications can be found in Seddon et al. (2017). The world’s first hard X-ray FEL, the Linac coherent light source (LCLS), commenced operation in 2009 at the US Department of Energy laboratory SLAC, near Stanford University in California. For the XFEL, a pulsed electron beam is generated by the photoelectric effect, using a powerful pulsed laser to illuminate a photocathode, perhaps of pure copper, as the electron beam source. The development of the high-brightness RF cavity photocathode source, a spin-off from the Star Wars program, was an important breakthrough which made the development of the XFEL possible. The laser (and resulting electron) pulse duration may be a few femtoseconds (fs), 1 fs (1015 s) being the time for a singleelectron orbit around an atom. The electron beam is accelerated through a series of RF cavities (a Linac) to an energy of a few GeV, and then passes between a row of alternating small magnets, as shown in Fig. 1, causing Electron beam

Undulator Photon beam

(1) (2) Spontaneous emission Energy modulation / bunching radiated

log( power )

l

Beam dump

(3)

Coherent emission

(4) Saturation

106–109

z

Fig. 1 Schematic version of XFEL resonant bunching (Bonifacio, Pellegrini, & Narducci, 1984).

108

John C.H. Spence

lateral undulations. As all physics students know, accelerated charges radiate, and the undulator radiation from such a linear electron beam subject to small lateral oscillations was analyzed by Motz (1951). Later, the interaction between the radiation so generated and the undulating electrons themselves was studied, leading to the idea that such an effect could amplify the radiation. This occurs because of a microbunching effect on the electrons due to the radiation they produce, with the period of the radiation. Normally, we think of lasers as quantum devices, in which amplification and oscillation occur in a cavity due to the induced emission process during a population inversion among atoms. This process is hardly possible for hard X-rays because of the prohibitively large amount of power needed to sustain a population inversion among inner-shell atomic electrons. For the XFEL there is no cavity or oscillation, and the amplification occurs by a classical process (if a small recoil term is neglected) akin to spontaneous emission in a free particle beam. Is it therefore a true laser? Laser stands for Light Amplification (not oscillation) by Stimulated Emission of Radiation (not spontaneous emission). But we might agree to a more general definition of a laser, as having the crucial characteristic of feedback, which results in gain. In that case the XFEL qualifies as a laser, since the radiation generated feeds back onto the electrons which created it, to make more radiation, resulting in an exponential increase in radiation with distance, until saturation occurs (Kondratenko & Saldin, 1980). Furthermore, whereas the intensity of undulator radiation is proportional to the number N of electrons per bunch, in the XFEL the magnetic field associated with the emitted radiation and the transverse electron velocity create a Lorentz force which causes the electrons to collect in microbunches whose period is equal to the wavelength of the radiation. This collective oscillation of the microbunches produces correlated photon emission, so that we must sum the electric fields of radiation from each electron, instead of their intensity, and the intensity of radiation is then proportional to the square of N. As a result, the radiation is amplified exponentially with distance x along the undulator as I ¼ I0 exp ðx=LG Þ where LG is the gain length. The photons which are amplified originate from noise due to spontaneous emission, so this process is known as self-amplified spontaneous emission, or SASE. It is also possible to inject an external weak monochromatic beam into the device, which will be amplified, a method

X-Ray Lasers in Biology: Structure and Dynamics

109

known as “seeding.” FEL oscillators using a cavity have also been demonstrated at longer wavelengths. The X-ray wavelength can be found by considering the field due to the undulator magnets (with period L along the beam in the lab frame) on either side of the electron beam, in the rest frame of the fast electron. This field appears as an electromagnetic wave, whose wavelength is reduced by the relativistic Lorentz length contraction, to become L/γ, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ ¼ 1= 1  v2 =c 2 for an electron velocity v. The electron energy is E ¼ γmoc2. Finally, the radiation must be corrected for the relativistic Doppler effect, so that the emitted X-ray wavelength from an XFEL is approximately   λ ¼ L= 2γ 2 Under these conditions the time delay between X-rays emitted at two successive electron oscillation maxima (multiplied by the speed of light) is equal to the wavelength. There is one further correction involving the strength K of the undulator magnetic field, which multiplies this expression by (1 + K2/2). The radiation is peaked in the forward direction around an opening angle 1/γ. For modern XFELs the electron beam energy is a few GeV, so that, for the LCLS, where L  3 cm, γ ¼ 30,000, and K ¼ 3, we can obtain the 0.1 nm wavelength X-rays needed for crystallography. The X-ray pulse duration can be selected from 10 to 200 fs, and these occur at a repetition rate of 120 Hz. Much higher repetition rates are planned at the European XFEL (commencing operation in 2018 in Hamburg, Germany), but for most applications the data acquisition speed will be limited by the detector readout speed to perhaps 20 kHz in the near future. At this point it is worth noting that the cost and the size of an XFEL depend mainly on the energy of the electron beam and its linear accelerator. The above equation, however, shows that we can obtain the same X-ray wavelength in a much smaller and less expensive machine if both numerator and denominator could be divided by, say, 10,000, resulting in a room-sized machine operating at a few MeV, if we can use a value of L of micron dimensions. This is the approach used in the compact XFEL (CXFEL) under construction at Arizona State University, where the transverse electric field of a powerful optical laser will be used, running antiparallel to the electron beam, to provide undulation of the electron beam on the scale of the wavelength of light (Graves, K€artner, Moncton, & Piot, 2012).

110

John C.H. Spence

The bandwidth of the X-ray beam is Δλ 1 ¼ λ M where M is the number of undulator periods. A derivation of the expression for the gain length LG is complex, but the result is proportional to γ and to j1/3 where j is the electron current density. This argues for the smallest possible beam area and highest current per pulse for a short gain length. The required currents (perhaps 3 kA at LCLS) can only be achieved very briefly by using femtosecond pulses of electrons generated by the photocathode electron source. Thus the time-average power of the X-ray beam may be not much more than a handheld laser pointer at low repetition rates. Additional requirements for achieving gain are that the energy spread in the electron beam be smaller than the gain bandwidth, the size and beam divergence of the electron and X-ray beams should be matched (emittance is the product of these), and diffraction losses should be small. For a typical XFEL operating with a charge per pulse of 1 nC, a typical pulse duration of 50 fs, and beam energy 10 GeV, the pulse energy is about 1 mJ, corresponding to 6
1012 photons per pulse for 1 kV X-rays. This is sufficient to immediately drill a hole in sheet steel if focused to micron dimensions. The modern free-electron laser concept, together with a quantum mechanical theory, was introduced by Madey (1971), who constructed the first such machine at Stanford, operating at 24 MeV, producing infrared radiation (10 μm wavelength) with 7% single-pass gain. Additional machines, operating in the visible and infrared region, and important new theoretical work on the high-gain SASE mode led eventually to the construction of the V-UV FEL “FLASH” at the DESY laboratory in Hamburg, Germany, which commenced operation in 2005, and the first hard X-ray XFEL, the LCLS at SLAC (proposed by Pellegrini in 1992) which started in 2009, as shown in Fig. 2. The Japanese SACLA XFEL commenced operation in 2011, and new hard X-ray machines will come online around 2018 in Germany at DESY (the European XFEL, or EXFEL, with a superconducting Linac), in South Korea (PAL), and Switzerland (SwissFEL). A superconducting upgrade (LCLS II) is under construction at SLAC which will provide a second independent X-ray laser. The development of these new user facilities is reviewed in Bostedt et al. (2016), and more details can be found on their respective web pages, including the procedure for submission of a research proposal for use of the machines. Unlike synchrotrons which use a storage ring to circulate electrons, allowing many experimental X-ray facilities to be arranged around the ring and used simultaneously, the

X-Ray Lasers in Biology: Structure and Dynamics

111

Fig. 2 The Linac coherent light source (LCLS) at SLAC within its 3-km tunnel.

XFEL is a linear device, which, initially, allowed only one experiment to be performed at a time. More recently, productivity has been improved by placing several experiments in series, using the direct unscattered X-ray beam which has passed through one sample to be used further downstream (since absorption by a thin transmission sample is small) for a different experiment, which shares the same time structure. Unlike fast electrons, an X-ray photon which generates a photoelectron (the dominant inelastic process) is annihilated, and does not continue to the detector to generate background, as is the case in electron microscopy.

3. DATA ACQUISITION MODES AND SAMPLE DELIVERY FOR XFEL STRUCTURAL BIOLOGY Several modes of data collection are possible for structural biology at an XFEL, from solution scattering to the use of protein microcrystals and single particles (with one particle per shot). The history and invention of the most popular mode at present, serial crystallography (SFX), can be traced to early proposals for the delivery of samples across a beam by liquid jet (Spence & Doak, 2004), to the first application and development of that method at a synchrotron (Shapiro et al., 2008) in preparation for its use in the first crystallography experiments at the first XFEL (Chapman et al., 2011) using a gas-dynamic virtual nozzle (GDVN). In this approach, femtosecond X-ray pulses diffract from successive hydrated nanocrystals, running in single file across the focused XFEL beam, in random orientations. Each nanocrystal is destroyed by the beam following diffraction. Diffraction patterns are recorded and read out at 120 Hz at the LCLS. Since the intense

112

John C.H. Spence

XFEL pulse destroys the sample (after providing a diffraction pattern from undamaged crystal), methods had to be developed to provide a continuously refreshed supply of identical samples for this “destructive readout” method. For biological significance, samples need to be hydrated and so somehow either sprayed in single file across the beam (hopefully synchronized) or, if mounted on a silicon wafer, scanned across it. The method has been described as “diffract-and-destroy” or “diffract-then-destroy.” Several other modes of diffraction analysis are possible. The four most common are shown in Fig. 3. They are serial femtosecond crystallography (SFX), with one protein nanocrystal per shot, fast solution scattering (FSS) (or “snapshot WAXS”), single-particle diffraction (SP) with one particle, such as a virus, per shot, and mix-and-inject studies for snapshot imaging of chemical reactions (this may use either solution scattering or nanocrystals). The GDVN, which uses gas focusing of a liquid stream to avoid clogging, is shown in Fig. 4. A coaxial jacket of helium gas under high pressure flows around the emerging liquid, causing it to speed up, and hence shrinks in diameter, since area times velocity is constant for incompressible A

B Single-crystal diffraction

C Solution scattering

D Single-particle diffraction

Mixing jet injector

t

I ∝MN 2

I ∝MN

I∝ M

For M patterns and N molecules

Vary reaction time by sliding inner tube (blue)

And viscous variants for SFX One virus per droplet ? X Micron-sized droplet beam

X-ray beam into page

Fig. 3 Left to right. (A) Serial femtosecond X-ray diffraction (SFX) with one protein nanocrystal per shot. (B) Fast solution scattering (FSS) with many similar molecules per shot. (C) The single-particle (SP) mode with one particle per shot, more commonly driven from an electrospray. (D) A mixing jet for snapshot imaging of slow dynamics. Below is shown a simple Rayleigh jet, in which the water stream breaks up into perfectly spherical droplets a few microns in diameter (Weierstall et al., 2008). Fixed sample systems are also used (see text) and all of these may be combined with X-ray and emission spectroscopy.

X-Ray Lasers in Biology: Structure and Dynamics

113

Fig. 4 Environmental SEM image of an operating gas dynamic virtual nozzle (GDVN) system. The liquid can be seen to narrow as the outer jacket of high-pressure gas speeds it up as it enters vacuum at about 10 m/s, where it breaks up into droplets which freeze at about 106°C/s (Weierstall, Spence, & Doak, 2012). A Bragg beam is seen scattered from a microcrystal in the stream to top left. Image courtesy of D. DePonte.

flow under these conditions. In this way the liquid from a nozzle large enough to avoid clogging is focused down to a much smaller stream, a little larger in diameter than the X-ray beam of a few microns. Full details of this GDVN design are given in Weierstall et al. (2008, 2012). A major advance on this method has been the development of the double-focusing liquid jet, based on the earlier mixing jet designs discussed in Section 8. By using alcohol as an intermediate fluid which focuses but does not mix, one has probably the most generally applicable sample delivery system of all at present, which conserves protein (the innermost stream may be reduced to zero), accepts both membrane and soluble proteins, and is compatible with the high repetition rate of future XFEL sources for time-resolved diffraction (Oberthuer et al. (2017)). Other delivery modes, such as viscous media “toothpaste” jets, conveyor belts supplied with droplets, on-demand acoustic droplet methods (Roessler et al., 2016), the electrokinetic injector (Sierra et al., 2016), and particle trap arrays on chips (Lyubimov et al., 2015), have also been developed, as shown in Fig. 5, together with scanning “fixed” sample mounts for both 2D and 3D crystals. Sample delivery methods are compared in Weierstall (2014). Because of the high speed (about 10 m/s) of the GDVN liquid jet, most protein nanocrystals run to waste between shots with a 120 Hz repetition rate. In order to conserve precious protein, the lipid cubic phase (LCP) jet was developed, using a medium with the high viscosity of automobile grease which emerges slowly from the nozzle. This LCP also has the great advantage that it forms a growth medium for growth

114

John C.H. Spence

Fig. 5 Environmental SEM image of protein crystals “wicked” down to fill holes in a thin silicon crystal for the road-runner fixed-target system of XFEL sample delivery (Roedig et al., 2016).

Hydraulic stage

Sample reservoir

LCP nozzle

Gas line

Water line

Asymmetric pistion

PTFE spheres

LCP

Fig. 6 Lipid cubic phase injector. The rigidly coupled asymmetric pistons act as a pressure amplifier to drive the viscous LCP into vacuum within an outer gas sheath which keeps it straight. From Weierstall, U., James, D., Wang, C., White, T. A., Wang, D. J., Liu, W., . . . Cherezov, V. (2014). Lipidic cubic phase injector facilitates membrane protein serial femtosecond crystallography. Nature Communications, 5, 3309. http://dx.doi.org/10.1038/ncomms4309.

of membrane proteins nanocrystals, such as GPCRs (Landau & Rosenbusch, 1996), so that crystals can be grown and delivered from the same system. Details of construction of the LCP jet are shown in Fig. 6 and are given in Weierstall et al. (2014). The use of the LCP jet for time-resolved diffraction at LCLS has been reported (Nogly et al., 2016)—the use of this approach at higher repetition rates remains to be determined, since the shock wave generated during destruction of one sample in the liquid tube should not interfere

X-Ray Lasers in Biology: Structure and Dynamics

115

with the next nanocrystal. This constrains the distance between crystals, the flow velocity, and the intensity of the X-ray pulse (Stan et al., 2016). While the liquid jets are well suited to the high repetition rates expected in future XFELs with their fast cameras, and can be adapted for timeresolved pump-probe diffraction at high repetition rate, a more efficient method for static structure determination for soluble proteins which may not grow in LCP has been developed using a silicon membrane containing windows with small holes slightly larger than the protein nanocrystals. The crystals are “wicked” down into the holes from solution above by a filter paper below, and become jammed in the holes. The first-order Braggdiffracted beams from the single-crystal wafer diffract to a high angle beyond the edge of the detector, so that background is very low. The crystals are kept hydrated with flowing wet helium gas, and the data are collected in air or a helium environment at atmospheric pressure. This promising “road-runner” arrangement (Roedig et al., 2016) is under further development for time-resolved diffraction. A second scanned fixed sample arrangement, which uses spectroscopy to locate the microcrystalline samples, with high hit rate, is described in Oghbaey et al (2016). The GDVN was originally constructed by precision grinding methods (after much experimentation with semiconductor lithography and other methods unsuited to this cylindrical tube-in-tube geometry); however, more recently these nozzles have been made using two-photon 3D printers (Nelson et al., 2016), which have a submicron resolution. A 3D CAD drawing is needed, and printing may take several hours per nozzle. This method opens up many new possibilities for testing prototypes for all sample delivery modes, including mixing jets for SFX and sheet jets for FSS. For single particles, most sample delivery has used the aerosol gasfocusing injector system (Hantke et al., 2014), driven by a GDVN or electrospray. This system has been extensively developed by the Hajdu group in Uppsala. Whereas hit rates for LCP injectors may run as high as 40% for nanocrystals, it has proven very difficult to obtain a hit rate above about 1% for single particles in these systems. This hit rate is H ¼ Tf συ=ðcd Þ where T is a transmission coefficient, f the particle injection frequency, σ is the sum of the X-ray beam diameter and the particle diameter ν the XFEL repetition rate, c the particle speed, and d the particle-beam diameter, which is assumed to be larger than the X-ray beam diameter. We see that hit rate can be increased most readily by increasing the injection frequency,

116

John C.H. Spence

increasing the repetition rate, or reducing the particle speed. The quality of data collected in single-particle experiments also depends on accurate detector characterization, lateral jitter in beam position (perhaps equal to the beam diameter, which at LCLS may be either about 3 μm or about 0.2 μm, with extended tails), the impact parameter for the hits, the amount of salts which may “plate out” onto the surface of the particle (giving, for example, a false impression of the size of an icosahedral virus and reducing the resolution of merged data), and the X-ray background from stray scattering. This background has been greatly reduced using a system of shadowing apertures, in particular a small aperture placed slightly downstream of the sample, which blocks upstream background sources, such as X-ray scattering from aperture edges and asperities. Since the scattering from a dielectric sphere falls of as the inverse fourth power of the scattering angle, the limited dynamic range of current X-ray detectors is a serious problem. The use of viruses lying on a hydrated graphene substrate may have advantages. A simple convergent nozzle has given a 2 μm focus of 200 nm particles (Kirian et al., 2015), while the possibility of running viruses along a hollow tube of light (a Bessel beam) has also been explored (Eckerskorn et al., 2013). New optical imaging methods now allow, for the first time, bioparticles to be directly observed during injection at XFELs, a most important advance to assist alignment (Awel et al., 2016). Since all these modes have time-resolved variants, the full taxonomy of data collection modes might be labeled SFX, FSS, SP, TR-SFX, TR-FSS, and TR-SP. The time-resolved modes may use a variety of means to initiate reactions, including optical pulses (for example, in the study of lightsensitive proteins), chemical mixing, or applied electric fields.

4. RADIATION DAMAGE LIMITS RESOLUTION It has been clear since the early 20th century that radiation damage places a fundamental limit on resolution in practically all imaging and diffraction methods in biology (Owen, Rudino-Pinera, & Garmen, 2006) (neutron diffraction excepted). As we have seen, single-particle (SP) cryo-EM deals with this issue by merging many real-space images of similar molecules, each of which receives less than the very low critical “damage dose,” which is too small to allow a useful image to be formed from one molecule alone. This dose is a function of resolution, in which fine detail is destroyed first. Detailed measurements and theory for the resolution dependence of dose are given elsewhere (Howells et al., 2009). By comparison with similar XFEL SP data-merging algorithms, the cryo-EM real-space images present no

X-Ray Lasers in Biology: Structure and Dynamics

117

phase problem, and do not possess the additional Friedel symmetry which is present in diffraction patterns in the absence of multiple scattering. Imaging thus solves the phase problem. In 1970, shortly after Crewe’s paper, a second paper appeared (Breedlove & Trammell, 1970), showing that single-atom imaging of molecules should never be possible using any form of scattered radiation (except perhaps neutrons and helium atoms, for which bright sources did not exist), because the radiation dose needed to do so would destroy the molecule. This follows from the ratio of cross sections for useful image-forming elastic scattering, to damaging inelastic scattering, over a range of beam energies and types of radiation. A fuller analysis (Henderson, 1995) used this ratio multiplied by the average amount of energy deposited in the sample by inelastic scattering to compare damage and resolution for electrons and X-ray diffraction (XRD). This average deposited energy is about 20 eV for TEM and is approximately equal to the full X-ray beam energy for XRD, where photoelectrons are created by an inelastic event in which a photon is annihilated. Since both the scattering cross section ratio and the amount of energy deposited favor electrons, this paper concludes that electron microscopy provides more information per unit damage than XRD. The Breedlove and Trammel paper also contains the sentence “…this does not prevent X-ray molecular microscopy if the observations are made sufficiently rapidly…within 1013 seconds.” This estimate of 100 fs for damage-free imaging has turned out to be remarkably prescient—recent XFEL crystallography using 50-fs pulses has shown 0.2 nm resolution scattering at huge doses from crystals (where the images are periodically averaged) and 0.59 nm hard X-ray scattering (limited by an aperture) from individual virus particles at the LCLS, summed over many shots (Munke et al., 2016), and about 0.2 nm resolution scattering from single shots, on one dehydrated tobacco-mosaic virus in vacuum, supported on graphene. This idea that one could “outrun” radiation damage, by collecting the useful elastic scattering for image formation before significantly damaging inelastic scattering had occurred, was then further explored by Solem (1986), and in detail, in response to the promise of the XFEL with its high-intensity femtosecond pulses, in molecular dynamics simulations by Neutze, Wouts, van der Spoel, Weckert, and Hajdu (2000). (Recall that X-rays are scattered by the atomic electron cloud alone, rather than the nuclei, whose positions are tracked in molecular dynamics simulations. Electron beams are scattered by both electrons and nuclei.) Since damage processes in crystallography occur on timescales as long as a second (longer than synchrotron exposure times), the idea of outrunning damage was not

118

John C.H. Spence

entirely new and had been appreciated in the synchrotron community, but the conceptual breakthrough here was to realize that with if laser amplification allowed, an almost unlimited number of X-ray photons to be packed into an arbitrarily brief pulse, one could break the nexus between resolution, radiation damage, and sample size (Howells et al., 2009), and so achieve damage-free atomic resolution from arbitrarily small samples, such as a single virus, if a beam could be focused down to those dimensions. And one could use samples in their native, room-temperature environment, avoiding the need to freeze samples to reduce damage. The first experimental evidence for this “diffract-then-destroy” mechanism came at lower resolution using the V-UV laser Flash at DESY in 2006 (Chapman et al., 2006), suggesting the possibility of high-resolution damage-free movies (Spence, 2008). High resolution (0.8 nm) results from protein nanocrystals using a 1.8 kV XFEL beam were first published in 2011 (Chapman et al., 2011), together with the first single-particle XFEL results (Seibert et al., 2011). Following initial elastic scattering, for samples larger than the inelastic mean free path of ejected photoelectrons, the photoelectrons thermalize, taking the sample temperature to perhaps 500,000 K and vaporization. For samples smaller than this size, the photoelectrons escape, leaving a charged sample which undergoes a coulomb explosion. Fig. 7 shows the fading of high-angle scattering (corresponding to the finest detail in the sample) with increasing XFEL pulse duration at 1.8 kV,

Fig. 7 Bragg peak intensity in merged SFX from Photosystem I (Chapman et al., 2011) as a function of inverse resolution (in inverse nm) for several different X-ray pulse durations, normalized to the result at 50 fs (Barty et al., 2012). Fine detail is destroyed first, and the effective pulse duration is set by the time taken to attenuate the high-order Bragg peaks.

X-Ray Lasers in Biology: Structure and Dynamics

119

for Bragg diffraction from Photosystem I protein (Barty et al., 2012). For the longest pulses, late-arriving X-rays are diffracting from already damaged crystal. As a rule of thumb, 1 μm3 of protein crystal (without heavy atoms) produces about 106 scattered hard X-rays at the tolerable “safe dose” damage limit of 30 MGy, but much higher doses are tolerable if the pulse is shorter than about 20 fs. The incident pulse may contain about 1011 hard X-ray photons, over 98% (at 12 kV) of which pass through a protein crystal without interaction. Of the remaining 2%, 84% are annihilated in the production of photoelectrons, 8% are scattered by the Compton process, and 8% are Bragg scattered. For a 40-fs, 2-keV pulse with irradiance 1017 W/cm2, 10% of the carbon atoms in a protein crystal absorb a photon. A cascade of photoelectrons and Auger electrons releases this energy, followed by a cascade of low-energy electrons caused by secondary impact or field ionizations taking place on a 10–100-fs timescale. Coulomb repulsion of the ions and increase in electron temperature then causes displacement of both atoms and ions during the pulse. This heating lead to vaporization of the sample if the secondary electrons cannot escape, as the temperature can rise to over 500,000 K. Simulations by both molecular dynamics (Hau-Riege, 2012) and hydrodynamic codes (Caleman et al., 2011) predict that 0.5 nm motions of the ions can occur in less than 100 fs, and that pulses as short as 10 fs may be required to achieve atomic resolution with one particle per shot, a more demanding requirement than that for SFX, which benefits from periodic averaging and the “Bragg boost”—the squaring effect due to coherent amplification mentioned previously. It is found that doses of up to a thousand times greater than the Garman–Henderson “safe dose” can be used (Owen et al., 2006) for similar resolution, if sufficiently brief XFEL pulses are used (Redecke et al., 2013). More specifically, if the “safe dose” is about 30 MGy for cooled samples at synchrotrons (or 0.2 MGy at RT), then it is estimated to be about 700 MGy for an XFEL using 70 fs pulses (see Chapman, Caleman, & Timneanu, 2014 for a full discussion). Recently, site-specific damage effects have been imaged in density maps around Fe metal clusters in ferredoxin using XFEL data (Nass et al., 2015), and compared with synchrotron results. A submicron beam focus was used at maximum XFEL intensity, with beam energy above the iron K edge. This work and supporting simulations (Hau-Riege & Bennion, 2015) suggest that pulse durations of 20 fs or less may be needed to minimize some types of sitespecific damage when using the smallest beam focus for highest intensity in single-particle (SP) imaging, particularly if heavy atoms, which produce a strong local shower of photoelectrons, are present.

120

John C.H. Spence

Spot-fading studies (Fig. 7) show how the disappearance of the outer Bragg reflections “gates” the time resolution of the process— the effective pulse duration which matters is the time taken for these spots to fade, destroying translational symmetry before the pulse ends, not the duration of the pulse (Barty et al., 2012). For single particles, the onset of damage is more difficult to determine from the continuous distribution of scattering in the patterns, and will need to be studied by modeling of known structures, once reliable high-resolution data are obtained from monodispersed particles. At present, the resolution of 3D reconstructions from SP data (about 10 nm) is not sufficient to see these damage-limiting effects on resolution.

5. SERIAL CRYSTALLOGRAPHY AT XFELs FOR STRUCTURAL BIOLOGY Why use an XFEL rather than a synchrotron for protein crystallography? By comparison with the highly stable modern synchrotrons, the SASEmode XFEL would appear to be highly unsuitable for crystallography, with its 15% shot-to-shot intensity variation (from amplified noise), 0.1% bandwidth, noisy time structure (which is different for each shot), and a focused intensity which immediately drills holes in steel, so that each shot must come from a new sample. In summary, therefore, every shot at a XFEL must be considered a separate “experiment,” producing data which must be carefully scaled and analyzed and perhaps somehow “deconvoluted” against different instrumental effects before merging. However, the use of the diffractionbefore-destruction mode and the new algorithms which have been developed to deal effects such as beam intensity fluctuations, variations in crystal size during serial crystallography, and errors in crystal orientation determination, and XFEL crystallography have been found to offer the following advantages: (i) The reduction in radiation damage observed when using the shortest pulses allows crystallography at room temperature without the need for cooling, and in a controlled chemical environment, from the smallest (e.g., submicron) crystals, from which useful data cannot readily be obtained at synchrotrons. This opens the way to the study of dynamics at room temperature. The important point is that, when studied at room temperature, the correct thermal energy, representative of physiological conditions, is available to drive the chemical reactions as they are observed, in the correct thermal bath, which may not be the case for the quenched samples used in trapping experiments.

X-Ray Lasers in Biology: Structure and Dynamics

121

(ii) Showers of microcrystals are frequently observed during crystal growth trials, yet it may take months or years to find the conditions required to grow crystals large enough for conventional crystallography. Time-consuming screening trials can be avoided by direct injection of these microcrystals in a liquid jet or similar sample delivery device. Since good diffraction patterns have been obtained from nanocrystals just a few dozen molecules on a side, research into the identification of “invisible” protein nanocrystals too small to be detected by optical microscopy continues, using methods such as SONICC (Haupert & Simpson, 2011) and Brownian motion tracking using the NanoSight instrument, from Malvern Instruments Ltd. Methods of growing the required nanocrystals are under continuous development—these include growth in LCP (Liu et al., 2013) and growth in living cells, either extracted from the cells or with the cells themselves (containing nanocrystals of Bacillus thuringiensis) injected into the XFEL beam (Gallat et al., 2014). In cases such as the small-molecule amyloid crystals important for Alzheimer’s disease, the buildup of strain in the crystals limits the crystal size (Sawaya et al., 2016). (iii) The improved time resolution possible using an XFEL. (iv) Noncyclic reactions can be studied (since each sample is destroyed), rather than requiring cyclic low-dose stroboscopic conditions on the same sample region. (v) When using submicron crystals, the optical absorption length for pump lasers is comparable with the crystal dimensions, allowing saturated pumping. (vi) For diffraction studies of nanocrystals reacting with a substrate, discussed in more detail below, diffusive mixing is possible, since the diffusion time of the substrate into the crystals is short (Schmidt, 2013; Wang, Weierstall, Pollack, & Spence, 2014). (vii) In several cases, resolution appears to be better at XFELs than synchrotrons for similar protein crystals; however, detailed tests of these claims with full control of crystal quality, dose, temperature factors, and beam diameter remain to be done and will be difficult. (viii) Inner-shell X-ray absorption (Mitzner et al., 2013) and emission (Kern et al., 2014; Kroll et al., 2016) spectra may be collected in synchrony with snapshot X-ray scattering from nanocrystals, allowing the chemical state of heavy atoms to be tracked in time through a chemical reaction, in correlation with density maps, using pumpprobe or mixing experiments.

122

John C.H. Spence

Soon after the initial SFX results were obtained at LCLS in late 2009 (limited in resolution by the soft X-ray beam energy), high-resolution results were obtained in February 2011 (Boutet et al., 2012). Since then, more than 100 structures determined using an XFEL have been deposited in the protein data bank (PDB). Recent SFX examples include the GPCR angiotensin receptor (important for drugs which control hypertension) ˚ resolution (Zhang et al., 2015) and rhodopsin bound to arrestin at 2.9 A (important for the first event in human vision and other light-sensitive proteins) (Kang et al., 2015), among many others. Serial crystallography itself has also been developed at synchrotrons, where the source brightness and detector speed are sufficient to freeze any crystal rotation during an exposure if the sample is delivered in a viscous medium, such as LCP, in which it can be grown (Nogly et al., 2015). An important recent advance has been the realization that, in crystals for which disorder consists solely of rigid-body displacements of proteins from the ideal lattice, the Debye theory (Debye, 1913) of scattering from crystals with thermal motion predicts that the strong diffuse scattering seen between Bragg reflections in these crystals is just the single-particle diffraction pattern from one unit cell, loosely described as the molecular transform (except around the origin of reciprocal space). Since this anisotropic scattering extends well beyond the Bragg reflections (and is not subject to thermal damping), this effect has now been used to extend the resolution of density maps of Photosystem II from 0.45 to 0.35 nm (Ayyer et al., 2016). Because it provides “oversampled” data (intensity running between the Bragg spots), this continuous scattering can also assist in solving the phase problem and opens up the possibility of solving imperfect crystals. Ideally, for this approach, one wants something like a liquid crystal, which possesses orientational order but not translational symmetry.

6. MOLECULAR MACHINES AND SINGLE-PARTICLE IMAGING In this section we consider the unique insights which the XFEL’s time and spatial resolution, and other characteristics, provide for our understanding of the molecular machines of life, and briefly summarize progress in the relevant single-particle initiative at LCLS. Single-particle methods (one particle per shot) are reviewed in Ekeberg et al. (2015), Bostedt et al. (2016), and Liu and Spence (2016).

X-Ray Lasers in Biology: Structure and Dynamics

123

Studies of the folded state of proteins in crystals have revolutionized structural biology. However, increasingly it has been realized that under physiological conditions, proteins sample a large ensemble of conformations around this average structure due to the availability of thermal energy, and that this dynamic behavior, consisting of near-equilibrium fluctuations, is crucial to their function. At higher temperatures, many proteins therefore switch rapidly between substates, but are inactive at the temperatures around 100 K where most crystal structures are now determined. The conformational states sampled by proteins are difficult to access by crystallography, which averages over the entire ensemble of states present in the crystals. Additional controlling conditions include solvent chemistry (e.g., pH), pressure, local electric fields, and ligand binding. Protein function is then the result of a complex interplay between thermal motions and their chemical and physical environment (Henzler-Wildman & Kern, 2007). The modern description of protein function is therefore based on a multidimensional energy landscape (Wales, 2003) that defines the relative probabilities of the conformational states (the thermodynamics), the energy barriers between them (kinetics), and the work cycle. This landscape concept was taken from the original Eyring transition-state theory in chemical dynamics. Historically, studies on myoglobin have led the way (see Fenimore, Frauenfelder, McMahon, & Young, 2004 and references therein). Molecular processes depend on alterations in rates and populations in an ensemble, such as the enzymes facilitating reactions, or the changes in intracellular ion concentrations which trigger neurological processes. Very large rate increases can be achieved by very small changes in free energy (a few kT), so that breaking a few hydrogen bonds or van der Waals contacts in a protein containing thousands of such interactions can turn on a signaling cascade or catalyze a chemical reaction. Intrinsic protein dynamics occur only in this free energy range of several kT. As suggested elsewhere (Henzler-Wildman & Kern, 2007), the conformational substates sampled by a protein, and the pathways between them, are not random, but rather a result of the evolutionary selection of states that are needed for protein function. Signal transduction, enzyme catalysis, and protein–ligand interactions occur as a result of the binding of specific ligands to complementary preexisting states of a protein and the consequent shifts in the equilibria. In other words, the energy landscape is an essential, intrinsic property of a protein, encoded in its fold, and is central to its function: the ligand does not induce the formation of a new structure but, instead, selects from preexisting structures, according to one recent school of thought.

124

John C.H. Spence

Several classes of protein dynamics may be distinguished, according to driving force, reversibility, speed, cyclic nature, and thermodynamics. Certain molecular motors convert chemical energy, provided by ATP hydrolysis (12 kcal/mol, or 20 kT at 300 K, or 0.52 eV/molecule), into mechanical motion. The molecular machines of life are otherwise mostly driven by thermal fluctuations (together with input of chemical energy), operating on timescales longer than microseconds. They might be thought of as molecular structures which focus Brownian motion. In equilibrium, buffeted by surrounding water molecules, these machines (such as the ribosome and kinesin) may be said to be idling. (An example is Kinesin, which in equilibrium is equally likely to move to left or right on tubules, but which moves only in one direction when provided with chemical energy.) A third class of systems is light-sensitive proteins, responding to large photon energies. Note that while time-sequence information is not needed to map out the energy landscape, it is needed to determine the way the path adopted by a particular driven system is traversed. Relative energies for molecular machines operating with energies around kT can be obtained from the ratio n1/n2 of occurrence of two particular conformations in equilibrium, since this ratio is given by a Boltzmann factor. In this way, if we assume that we need to observe only n1 ¼ 1 example of the most extreme highest-energy conformation quenched from an ensemble, we may find that energy difference for any given total number N ¼ n1 + n2 of reconstructed density maps from the ensemble (Dashti et al., 2014). A recent example of the determination of an energy landscape, for the ribosome, using cryo-EM data, can be found in Dashti et al. (2014). Thus the much larger amount of data (larger N) obtainable in XFEL SP experiments (especially when using the new high repetition rate machines) will give access, to those much rarer, larger-energy and larger conformational changes, not seen in cryo-EM imaging, which may be important to physiological function and rate limiting (and which go beyond the harmonic approximation commonly made in molecular dynamics simulations or seen from crystallography). For future data sets obtainable from the European XFEL over a few days, these energy differences may extend up to about 18 kT, greater than the energy available from ATP per molecule. These very large conformational changes would then be visible at a moderate resolution of, perhaps, 1 nm. In this way one may go beyond the small conformational changes imposed by the study of proteins which can be crystallized, which can only provide a periodic average over all conformers in the crystal. Pump-probe SP studies would also have the important advantage of providing time-sequence information.

X-Ray Lasers in Biology: Structure and Dynamics

125

Since most biochemical reactions occur on a microsecond or longer timescale, the value of XFEL imaging on the femtosecond timescale (other than for reduction of damage) in biology has been questioned. But in fact, as has been pointed out for quantum-chemical reactions, all timescales, from the excited-state lifetimes of the initial electronic excitation onward, are relevant (Moffat, 2014). Enzymes, for example, rely on fluctuations that are much faster than the enzymatic constants. The binding of charged ligands can be electrostatically steered over very small diffusional distances, and therefore over very short times. Other reactions depend upon the formation of the correct cluster of ionizing groups with extremely short lifetimes. The crucial initial stages of light-driven processes such as human vision and photosynthesis clearly play out in the femtosecond regime. It is useful also to distinguish sub-ns processes on the atomic scale (involving electronic excitations) as chemical reaction dynamics, to be distinguished from the kinetics described by rate equations and final states of thermodynamic equilibrium. All these insights deepen our understanding of biochemistry, and improved time and spatial resolution can also provide more accurate refinement of atomic potentials used in molecular dynamics simulations. In summary, some of the Grand Challenge problems in structural biology that might be addressed using XFELs include the following: (i) Image water molecules at protein surfaces in the hydration shell, with picosecond time resolution. These surface waters may play a critical role in molecular recognition. (ii) Refine atomic potentials used in simulations against high time and spatial resolution data. (iii) Trace photochemical reactions back to their origins in electronic excitation. (iv) Image rare, large, conformational changes (perhaps at modest resolution) which are accessible because of the very large SP data sets which will become available with new XFELs. (v) Image, at atomic resolution, the water-splitting event in photosynthesis with high time resolution. (vi) Image, at high spatial resolution and microsecond time resolution, the mechanism of enzyme catalysis. New mixing jet designs offer an excellent prospect of achieving this goal. The Single-Particle Initiative (SPI) program at LCLS sets aside dedicated noncompetitive beamtimes over a multiyear period to systematically trace, identify, and rectify the resolution-limiting factors in SP diffraction at LCLS. Steady progress has been made in reducing beamline background scattering

126

John C.H. Spence

and improving detector performance, to the extent that the hard X-ray SP ˚ in diffraction patterns facility (CXI) has demonstrated scattering at 3.4 A from single virions. For soft X-ray data, where the scattering is stronger, data collected under different conditions at the LAMP chamber (at the AMO experimental station at LCLS) and merged and phased for three-dimensional reconstruction showed about 10 nm resolution images of a virus capsid. With continued progress it is reasonable to expect that 1 nm resolution or better will be achieved before long, with much larger amounts of SP data available soon from the new DESY facility.

7. TIME-RESOLVED SERIAL CRYSTALLOGRAPHY, OPTICAL PUMP-PROBE METHODS, AND PHOTOSYNTHESIS Some of the most important successes of time-resolved high-resolution imaging using an XFEL have resulted from the method of optically pumped, time-resolved serial femtosecond diffraction (TR-SFX). In these experiments (see Fig. 3A) on light-sensitive proteins, micron-sized protein crystals are excited in a liquid jet upstream of the X-ray pulse, where their snapshot is recorded. The time delay between excitation and X-ray interaction (which corresponds to one frame of a movie) may be determined either (most accurately) by timing electronics (with pump illumination spatially extended along the flow) or by the flow time in the liquid stream (less accurate, but allowing longer delays). Thousands of snapshots are required (with the crystals in random orientations) for each delay (movie frame) to build up a three-dimensional diffraction data set. The first TR-SFX results were obtained by Aquila et al. (2012), for Photosystem I—ferredoxin. More recent examples, incorporating many advances in instrumentation and the 1000-fold improvement over synchrotrons in time resolution when using an XFEL, can be found in Barends et al. (2015) (for myoglobin), Kupitz et al. (2014) (for Photosystem II), Tenboer et al. (2014), and Pande et al. (2016) (for photoactive yellow protein), and Young et al. (2016) and Suga et al. (2017) (for Photosystem II) (see Kern, Yachandra, & Yano, 2015 for a review). Steady improvements in SFX data analysis algorithms (discussed below) beyond simple Monte Carlo averaging can now resolve the small changes of a few percent in structure factor magnitudes due to optical illumination of a micron-sized protein crystal, despite the scaling problems due to the continuous variation in crystal size and orientation, while working with partial reflections. It should be noted that the simplified description of a pump-probe “molecular movie” glosses over multiple issues. For example, in a protein

X-Ray Lasers in Biology: Structure and Dynamics

127

crystal excited by a femtosecond optical pulse with two reaction paths around a cycle, molecules in different unit cells have certain probabilities of either not being excited at all, or initiating a reaction on either path, and further branching is possible. Each path, described by chemical rate equations, will produce different intermediate species with different rate constants. Measurement of lattice constants, temperature factors, and overall resolution provide assurance that the crystal remains intact during the cycle, and that the outer envelope and contact points between molecules in different unit cells are little affected. (Destruction of the crystal by the photoelectron cascade comes later.) The observables, from a stream of nanocrystals, are the Bragg beams, which, after phasing, provide a periodic spatial average of charge density from the average of all illuminated crystals at one time point (the pump-probe delay). The method therefore requires accurate knowledge of the unilluminated (dark) ground-state structure from prior crystallography at the highest resolution, so that methods such as singular value decomposition (or modeling using molecular dynamics) can be used to separate the time-resolved charge densities along each path. From this, the amounts of intermediate species which come and go during the reaction cycle can be extracted, based on the rate equations describing the reaction kinetics. Fig. 8 shows the work of Pande et al. (2016), who achieved 200 fs time resolution over a 3 ps range in their remarkable TR-SFX study of photoactive yellow protein, sufficient to provide several frames of a 0.15-nm resolution movie of the trans/cis isomerization reaction which results from photon

Fig. 8 Trans to cis isomerization in PYP. Weighted DED maps in red (3σ) and blue (3σ). Reference structure in yellow; structures before transition in excited state in pink; after transition in ground state in green. Important negative feature denoted α, positive feature β. Pronounced changes at arrows. (A) Representative time delay before transition. Dashed line: direction of C2]C3 double bond, feature β1. Dotted lines: hydrogen bonds of the ring hydroxyl to Glu46 and Tyr42. (B) Chromophore configuration from 100 to 400 fs pump-probe delay. (C) Chromophore configuration at 799 fs after transition. (D) Chromophore configuration at longer times from 800 to 1200 fs. (E) 3 ps chromophore configuration; dashed line: direction of β (Pande et al., 2016).

128

John C.H. Spence

absorption in this light-sensitive protein. The mechanism is the same as that which occurs in the first event in human vision, when photons strike rhodopsin at the retina, involving a conical intersection (a degeneracy in nuclear coordinates for excited and ground states; Schoenlein, Peteanu, Mathies, & Shank, 1991). This was done using the GDVN liquid injection system in Fig. 4. Laser illumination, simulating the effect of sunlight on a plant or organism, causes a small change in structure factors, which can be phased by the molecular replacement method, to produce a difference density map between the bright and dark states. This project followed earlier work on the same system over a longer time interval using the same method at lower time resolution (Tenboer et al., 2014). It is clear that much more accurate results could be obtained if the Laue method, previously adopted for this work, could be used (e.g., Schotte et al., 2003). Here, a wide energy spread in the beam is used to provide a “thicker” Ewald sphere which spans the full angular profile of the Bragg peak, allowing each snapshot to record full reflections (for a single projection) at each time point and eliminating the need to scale Bragg peaks between different sized crystals of different partiality. The required large energy spread dE/E is, however, not normally possible using the highly monochromatic X-ray laser (for the LCLS, dE/E  0.001). Moffat (2014) finds that to provide angleintegrated intensities from a crystal with mosaic disorder dϕ ¼ 102 and Bragg angle β, one required dE/E > ϕ cot ϕ. For a high-angle reflection with β ¼ 0.3 this requires dE ¼ 260 eV at E ¼ 8 kV, or less for more perfect crystals, and more for low-angle reflections. The suggestion has been made that the submicron-sized crystals sometimes used for SFX are more perfect, since their size is smaller than one mosaic block of the traditional model. However, this model may not apply to many proteins, whose defect structure is not well known (Snell, Bellamy, & Borgstahl, 2003). The use of a “chirped” beam (which changes energy during the pulse) and the use of “two-color” methods have also been proposed. Here, the XFEL generates pairs of pulses with a tunable femtosecond-scale delay, at slightly different wavelengths. For an analysis of errors in SFX using two colors, see Li, Schmidt, and Spence (2015). A promising approach is the use of genetic engineering to create lightsensitive protein domains within a system of interest, known as optogenetics. If nanocrystals can be grown, this would provide a general method of studying protein dynamics (Moffat, 2014); however, the structural change induced by the light must be small enough to be accommodated by the crystal. New approaches to XFEL time-resolved diffraction are reviewed in Spence (2014), including the use of attosecond pulses of duration Δt. Here,

X-Ray Lasers in Biology: Structure and Dynamics

129

the unavoidable broadening of the energy spread ΔE (eV) ¼ 4.14/Δt (fs) in a band-limited beam could provide just the conditions needed for Laue diffraction, since 14 as pulses would provide 3% bandwidth at 10 kV. In addition, the temporal coherence allows Bragg beams from different reflections, excited at different wavelengths but diffracted into the same direction to interfere briefly (for the duration of the beating period), contributing to solution of the phase problem by providing three-phase invariants (Spence, 2014). An important recent development has been the achievement of pump-probe time-resolved SFX using the lipid cubic phase (LCP) injectors, which require a far smaller amount of protein (Nango et al., 2016; Nogly et al., 2016).

8. TIME-RESOLVED SFX FOR SLOWER PROCESSES: MIXING JETS AND OTHER EXCITATIONS A considerable literature exists on solution scattering experiments at synchrotrons which provide snapshot diffraction from a mixture of solutions during a chemical reaction (see Van Slyke, Wang, Singh, Pollack, & Zipfel, 2014, for example, for work on the binding kinetics of RNA). The reaction may be triggered in some way, or result from mixing, which normally needs to be sped up using turbulence, since this mixing, prior to chemical reaction of the species, determines the time resolution of the method. In biology, because the entropy term in the free energy is large, slow entropic processes are important, and slower diffusive processes may also be rate limiting, so that reactions can occur on the microsecond, millisecond, or longer timescales. These reactions can be triggered by the photoelectrons generated by the X-ray beam itself, as in the well-studied case of horseradish peroxidase (Berglund et al., 2002). The high brightness of modern synchrotrons and fast detector speeds have therefore recently enabled serial crystallography methods to provide “molecular movies” of enzyme mechanisms, triggered by the beam, in which the radiation dose is kept well below the Garman–Henderson “safe dose,” and resolution loss during the reaction is minimal (Horrell et al., 2016). Using an XFEL in the serial crystallography mode, it becomes possible to use micron-sized crystals, so that rapid diffusive mixing into the crystals becomes possible with such small crystals, turbulent mixing is not needed, and the crystals provide atomic resolution data. (The diffusion time for glucose into a 1-μm crystal of lysozyme, for example, is about 20 ms.) Furthermore, radiation damage can be almost eliminated, thus disentangling the effects of damage from the chemical reaction. Most importantly, the chemical reaction can then be imaged by snapshot XRD at room temperature, under

130

John C.H. Spence

physiological conditions, where the correct thermal energy is available to take part (with other driving forces) in driving these reactions. A description of the first double-focusing GDVN “mixing jet” for XFEL sample delivery is given in Wang et al. (2014)—a more recent design can be found in Calvey, Katz, Schaffer, and Pollack (2016) and Wang et al. (2014), and these have now been tested successfully at LCLS. Fig. 9 shows the results of such a time-resolved mixing experiment at an XFEL (Kupitz et al., 2016). Here, the reaction between the enzyme betalactamase and a small drug molecule CEF (boxed in the figure) has run to completion, using solution of the drug molecule and enzyme microcrystals which were mixed before delivery to the GDVN jet. The density map shows the drug bound into the enzyme ring at two locations. A four-frame movie of the drug molecule during binding has also been obtained and submitted for publication by this group. A second example can be found in Stagno et al. (2017) for the adenine riboswitch RNA aptamer, where a 10-s delay after mixing captures the structure of an intermediate phase. We can foresee now a much wider range of methods being used to trigger reactions for imaging dynamics at XFELs in the near future. These might include terahertz pumping (of the hydration shell around proteins, which couples via a dipole interaction), temperature jump, and temperature equilibrium measurements, and, particularly, caged molecule release experiments (Schlichting, 2000), including pH changes driven by optical pumping of proton-release cages (e.g., Lommel et al., 2013) and other photolabile compounds.

9. FAST SOLUTION SCATTERING AND ANGULAR CORRELATION FUNCTION METHODS Does the XFEL offer advantages for the small- (and wide-) angle X-ray scattering methods, SAXS and WAXS? (We might refer to XFEL WAXS as “fast solution scattering” or FSS. Other names include fluctuation X-ray scattering or correlated fluctuation scattering). Certainly the welldocumented radiation damage effects in these techniques might be minimized, and superior time resolution obtained. As a result, we have seen remarkable studies of both the phase transitions in water at low temperature (Nilsson, Schreck, Perakis, & Pettersson, 2016) and of photosensitive protein molecules studied by time-resolved pump-probe XFEL solution scattering (Arnlund et al., 2014). In that important study of the Blastochloris viridis reaction center, 500 fs time resolution and about 0.4 nm spatial resolution were obtained in the difference maps between the optically pumped

X-Ray Lasers in Biology: Structure and Dynamics

131

Fig. 9 Electron density in the catalytic cleft of BlaC. (a) Refined model of the entire tetramer (σ ¼ 1.1) in the asymmetric unit after mixing. The electron density (2Fo  Fc) is shown in blue in the binding pockets. Subunits A and C contain phosphate; subunits B and D have a bound ceftriaxone, with D being slightly more strong. (b) Enlarged section of the Apo (red electron density) subunit D binding pocket showing a phosphate ED. (c) Enlarged section of the mixed (blue electron density) subunit D binding pocket showing ceftriaxone ED. b and c show slightly different views of the same subunit binding pocket; however, there are minimal changes to the ligand binding sphere. From Kupitz, C., Olmos, J. L., Holl, M., Tremblay, L., Pande, K., Pandey, S., . . . Schmidt, M. (2016). Structural enzymology using X-ray free electron lasers. Structural Dynamics, 4(4), 044003. http://dx.doi.org/10.1063/1.4972069.

and dark states, allowing a molecular movie to be obtained following photon excitation. (Prior crystallography provided an accurate dark state structure, allowing extensive modeling by molecular dynamics.) The time-dependent diffraction provided details of the “quake” mechanism responsible for dissipating energy, which prevents unfolding of the protein following absorption of the large photon energy (2.5 eV ≫ kT). Time constants were

132

John C.H. Spence

obtained for both the initial quake motion (7 ps) at lower scattering angles and the later high-q heating process (14 ps). The epicenter of the “quake” was seen to occur at the chlorophyll cofactors. In a similar way, Levantino et al. (2015) have published TR-FSS studies of myoglobin, using LCLS data to observe light-induced structural rearrangement of a photo-excited chromophore, and resulting “quake” motions. They see damped oscillations with a 3.6-ps time period. The formation of a photoexcited gold trimer complex has been studied by fast solution scattering (FSS) to reveal details of the chemical covalent bond formation with 500 fs time resolution (Kim et al. (2015)). The 3D structure of reaction intermediates were also determined with sub-Angstrom resolution. It has been pointed out that solution scattering from molecules frozen in time or space should be anisotropic, containing speckles (additional to the effect of coherent interparticle scattering), unlike synchrotron WAX data which are isotropic, because the molecules rotate during the exposure. This is easily understood for the case of just a few particles per shot, if the spatial coherence of the beam spans only the particle size, so that the intensity of scattering from each randomly orientated particle must be added at the detector. Furthermore, a method exists for extracting the electron density map (image) of one particle, using this anisotropic scattering with many identical, randomly oriented particles per shot in solution (Kam, 1977). Clearly, such two-dimensional FSS patterns contain more information than one-dimensional WAXS patterns, facilitating inversion to three-dimensional images. The FSS patterns are nevertheless deficient in the information needed for three-dimensional reconstruction (Elser, 2011). An excellent tutorial review of this Kam theory and its history can be found in Kirian (2012). The concept can be understood in the simple case of 2D identical objects lying flat on a plane normal to the beam, which differ only by random rotations about the beam direction. Then the 2D angular correlation function (ACF) for each particle will be independent of its orientation, allowing them to be added together. (The ACF is the autocorrelation function of the diffracted intensity taken around each resolution ring in the diffraction pattern.) With many particles per shot, it can be shown that this anisotropic ACF formed from diffraction patterns with many particles per shot consists of the one-particle ACF added to a conventional WAX background (Kirian, 2012), which can be subtracted because it is isotropic. In principle, the resulting ACF can then be inverted to become a real-space image by phasing and Fourier transforming the data twice: once to convert the ACF to the diffracted intensity, and a second phasing and transform to give the real-space image (Saldin et al., 2011).

X-Ray Lasers in Biology: Structure and Dynamics

133

This anisotropy in FSS has been observed in X-ray scattering from colloidal glasses (Wochner et al., 2009) and from randomly oriented gold nanorods lying flat on a membrane (Saldin et al., 2011). These data were inverted using the Kam theory to provide an experimental image of a typical nanorod. For proteins in solution, the anisotropy in XFEL FSS data (with a recording time much shorter than the rotational diffusion time of the molecules) is usually swamped by other experimental artifacts causing anisotropy (e.g., scattering from the host liquid boundaries and stray X-ray sources in the beamline). Success has, however, been achieved using two-dimensional lithographed structures (Pedrini et al., 2013) at low resolution, and from data in the PDB for a ligand-gated ion channel (pLGIC) using an important new development of the Kam approach (Donatelli, Zwart, & Sethian, 2015) which provides inversion to an image with a single phasing step. A significant theoretical finding is that the results of this method, as originally suggested by Kam, are independent of the number of particles per shot (Kirian, Schmidt, Wang, Doak, & Spence, 2011); however, experimental resolution (in the absence of modeling) appears to be better using the single-particle method (with one particle per shot) inverted by the EMC approach. It is difficult to improve on the SP mode, with a direct hit and the beam diameter matched to the particle diameter; however, experimental impact parameters are rarely zero, and hit rates are low (e.g., 1% or less), whereas FSS (many particles per shot) has 100% hit rate. Thus the optimum number of particles per shot (and analysis method) remains to be determined for real experimental conditions, including background scattering and variations in impact parameter. The FSS method should be particularly powerful for known structures when detecting differences between ground and excited-state structures in solution scattering, where many sources of error are eliminated in these difference measurements (Pande, Schwander, Schmidt, & Saldin, 2014). Striking experimental FSS results have been obtained from polymer dumbbells in solution at LCLS using the Kam angular correlation method to reconstruct an image of one dumbbell (Starodub et al., 2012). Here, the difference in sample density from the host solution is small, as for protein. This paper, together with Kirian (2012), provides an excellent introduction to this fascinating approach to single-particle imaging.

10. DATA ANALYSIS 10.1 Serial Crystallography The new features of SFX diffraction patterns have required new algorithms for data analysis, while the high spatial coherence of the beam has provided

134

John C.H. Spence

new opportunities for solving the phase problem. In general there are four cases of XRD in the transmission geometry, differing in boundary conditions and coherence conditions. With a beamwidth smaller than the crystal, one may consider (i) large perfect crystals (such as semiconductor silicon), (ii) large mosaic crystals, or (iii) the case of a spatially coherent beam focused at the diffraction limit onto a crystal over an area smaller than one mosaic block. Finally, (iv) there is the case of a nanocrystal immersed in a wide coherent beam. This last case is the situation for most serial crystallography at XFELS, where, as shown in Fig. 10, for the smallest crystals, one finds (N  2) interference fringes, for a crystal containing N planes along direction g, running between Bragg reflections in direction g. This is akin to the (N  2) subsidiary maxima seen between the principle maxima in the optical transmission diffraction pattern from a grating of N slits. These fringes, running in several directions, therefore give the size of the crystal and may be used to solve the phase problem (Spence et al., 2011). The third case of a diffraction-limited coherent beam of nanometer dimensions will soon arise experimentally at diffraction-limited synchrotrons and XFELS, and raises

Fig. 10 XFEL snapshot at 2 kV from Photosystem I submicron crystal, showing diffraction from liquid jet (vertical thick streak through origin) and 17 subsidiary interference fringes between origin and first-order Bragg reflection, indicating that the crystal consists of 19 crystal planes between facets normal to this direction. From Chapman et al. (2011).

X-Ray Lasers in Biology: Structure and Dynamics

135

many interesting issues in diffraction physics which have been analyzed for the analogous situation in the fully coherent scanning transmission electron microscope (STEM) (see Spence, 2013). If the beam divergence angle is larger than the Bragg angle, these coherent diffraction orders overlap at the detector, producing interference fringes which depend on the absolute position of the beam with respect to the crystal lattice, and may be analyzed according to the theory of ptychography for periodic samples (Spence, Zatsepin, & Li, 2014). The effects of such a beam striking the side of a nanocrystal, and of the use of these patterns for solving the phase problem, are also discussed in Spence et al. (2014). Since crystals (destroyed by each shot) cannot be rocked through the Bragg condition to provide the angular integration needed for a full estimate of a structure factor (so that goniometers are rarely used), new algorithms which address the scaling issues created by beam intensity fluctuations, variations in crystal size, and the precise determination of crystal orientation from diffraction pattern intensities and geometry had to be developed. Expressions for XFEL diffraction by protein nanocrystals were first derived from first principles by Kirian et al. (2010). Since then, software has been developed and made available by several groups, notably White (“CrystFEL,” White et al., 2016) and Sauter (“cppxfel,” Ginn et al., 2015; Ginn, Evans, Sauter, & Stuart, 2016). New features of the patterns include the interference fringes between Bragg spots, which correspond to the “shape transform,” or Fourier transform of the external shape of the nanocrystal. This function, laid down around every Bragg peak, has an angular width in reciprocal space of approximately λ/D for a crystal of width D, and makes one contribution to mosaicity for larger crystals, which may be considered as a set of slightly misaligned blocks (see Snell et al., 2003 for a review of mosaicity in protein crystals). A scattering vector must first be assigned to every Bragg spot, and this then provides the rotation matrix which must be determined for each shot, between crystal and laboratory frame. Subsequent indexing has mostly been achieved using standard crystallography software (e.g., MOSFLM, see Winn et al., 2011), allowing the data from many nanocrystals to be merged into a three-dimensional diffraction volume; however, newer algorithms developed specially for SFX data can now index patterns using fewer spots (about five) and do not require large unit cells (Li, Zatsepin, Li, Liu, & Spence, 2017). Indexing ambiguities can be resolved using the expectation maximization and compression (EMC) method (Brehm & Diederichs, 2014; Liu & Spence, 2014). A simplified version of this algorithm is implemented in CrystFEL. Initially,

136

John C.H. Spence

full reflections were obtained using a Monte Carlo approach, which relies on recording and merging randomly oriented crystals whose orientations span and adequately sample the rocking curve for every Bragg reflection. The resulting error in structure factor measurement (e.g., Rsplit) then falls off inversely as the square root of the number of patterns, as errors are added in quadrature due to variations in crystal size, orientation, and a combination of impact parameter and shot-to-shot variations in beam intensity, as shown in Fig. 11. This proportionality constant has fallen dramatically over the past 5 years, as algorithms have improved and sources of error (especially associated with detector metrology, crystal size scaling, and beam energy bandwidth) have been estimated or reduced. Nevertheless, this Poisson scaling does mean that 100 times more data are needed to add one significant figure. The method, which avoids the use of a goniometer at preset measured orientations, amounts to “shooting first, and asking questions later,” as Rossmann has commented. Much research has focused on the very difficult measurement of partiality or “postrefinement” (Bolotovsky, Steller, & Rossmann, 1998). The fraction of a full reflection which is intercepted 0.2 Rsplit y = 18.5/sqrt(N)

Rsplit

0.15

0.1

0.05

0

0

50

100

150

200

Number of patterns (× 1000)

Fig. 11 The experimental reduction in scattering factor error measurement (Rsplit) with increasing number N of diffraction patterns follows a Poisson error law Rsplit ¼ k/sqrt(N). For this SFX analysis of Photosystem II complex (PDB 3WU2), k ¼ 18.5. Progress in SFX algorithm development, partial reflection analysis, and scaling is measured by the reduction in k in recent years.

X-Ray Lasers in Biology: Structure and Dynamics

137

by the Ewald sphere and the precise deviation of each reflection from the exact Bragg condition defines partiality, as described in White et al. (2016), Uervirojnangkoorn et al. (2015), Kabsch (2014), and Sauter (2015). This is complicated by the fact that, for a given range of energies, the “thickened” Ewald sphere spans a wider range at high angle than low. A significant advance has been the method of Ginn et al. (2015), which has provided 0.175-nm resolution structures from a few thousand protein nanocrystals. A histogram showing the number of reflections predicted as a function of X-ray wavelength is used to refine the orientation matrix until a sharp peak is found in the histogram, which gives the beam energy spread. Partiality is based on a model angular profile for the Bragg peak and spot locations are refined. This field of algorithm development for SFX data, including iterative refinement of experimental parameters (including, particularly the wavelength distribution in each X-ray pulse, variations in crystal size and diffraction conditions, and modeling of the Bragg profile), remains an active area of research which is producing large payoffs by reducing the amount of beamtime and protein needed to obtain an accurate structure. Additional experimental parameters might include variations in sampleto-detector distance, background due to stray X-ray scattering, readout noise, and pixel saturation. A comparison of XFEL and synchrotron data from similar lysozyme crystals has been given by Boutet et al. (2012). The phasing of SFX data has been achieved mainly by the molecular replacement method (Rupp, 2010), which uses a protein with similar sequence and fold in the PDB as a model structure. Recently, success has also been obtained in several cases with the SAD method used for native sulfur phasing of XFEL data, demonstrating the increasing accuracy of SFX data analysis (see Batyuk et al., 2016) for sulfur phasing of a GPCR membrane protein, and references therein). New de novo approaches for experimental phase measurement include measurement of the intensity dependence of scattering factors from heavy atoms (including sulfur), which are predicted to saturate. By sorting the data according to pulse intensity, an analysis similar to SAD or isomorphous replacement may then be applied (Son, Chapman, & Santra, 2011). It has also been shown that the interference fringes between Bragg reflections in the smallest crystals shown in Fig. 10 (due to the “shape transform”) provide the “oversampling” needed to solve the phase problem (Spence et al., 2011). For an experimental demonstration of this approach, and additional references, see Kirian et al. (2015). For membrane proteins, in which more than 50% of the volume of the unit cell may contain solvent, one already has the oversampling conditions needed

138

John C.H. Spence

for low-resolution phasing (He, Fang, Miller, Phillips, & Su, 2016; Millane & Lo, 2013) which in principle provide a powerful approach to phase measurement, avoiding the possibility of model bias in methods such as molecular replacement, which are based on modeling from known structures in the protein database. For a diffraction-limited coherent X-ray beam of nanometer dimensions, the situation is analogous to that in the fully-coherent STEM (Spence, 2013). If the beam divergence angle is larger than the Bragg angle, these coherent diffraction orders overlap at the detector, producing interference fringes which depend on the absolute position of the beam with respect to the crystal lattice, and may be analyzed according to the theory of ptychography for hard X-rays (Spence et al., 2014). In the normal diffract-and-destroy mode of serial crystallography, it is not clear that ptychography, which requires many diffraction patterns from the same sample, is possible.

10.2 Single Particles For SP (single-particle) data analysis, with one particle, such as a virus, per shot, the methods of coherent diffractive imaging (CXDI) have been adapted for XFEL data, including the hybrid input–output (HIO) algorithm and its variants (see Marchesini, 2007; Spence, 2017 for reviews, and Millane & Lo, 2013 for a review of related iterative phasing methods in crystallography and the important constraint ratio concept). Unlike the CXDI problem, the orientational relationship between successive diffraction patterns must first be determined, using randomly oriented particles of unknown structure (and requiring a certain minimum number of detected photons), whose accuracy may limit resolution, prior to solution of the phase problem. Approaches to these problems include manifold embedding (Yoon et al., 2011) and the remarkable EMC algorithm (Loh & Elser, 2009; Sigworth, 1998) widely used in cryo-EM, which has been applied to SP XFEL data (see Ekeberg et al., 2015 and references therein). As a model sample, in order to get an estimate of scattered intensity per shot, we first consider a sphere of uniform electron density. The far-field coherent diffraction pattern of a sphere of radius R, embedded in a medium, is given by (Starodub et al., 2008)   sin ðRqÞ  Rq cos ðRqÞ 2 I ðqÞ ¼ I0 re2 ΔΩjΔρj2 4π q3

X-Ray Lasers in Biology: Structure and Dynamics

139

where I0 is the incident fluence, Δρ is the difference of the complex electron density of the sphere to the medium, and q ¼ (4π/λ) sin (θ) is the photon momentum transfer for a scattering angle (pixel location) 2θ. The term in brackets is the 3D Fourier transform of a uniform sphere. The difference electron density can be written in terms of complex refractive index n  2 2π 2 2 re jΔρj ¼ 2 jΔnj2 ; λ 2    2  2 jΔnj2 ¼ nport  nwater  ¼ δport  δwater + βport  βwater where δ and β are the optical constants and the subscripts refer to the protein and the water medium. Far from absorption edges Δρ is independent of wavelength. The intensity pattern I(q) consists of circular rings (similar to Airy rings) with minima spaced in q approximately by π/R. The maximum intensity of the rings is given by Imax ðqÞ ¼ I0 re2 ΔΩjΔρj2 16π 2 R2 =q4 : This q4 dependence leads to the familiar problem in coherent diffractive imaging of having to simultaneously record strong intensity at low angles and much weaker intensities at higher angles, with the range of intensities often exceeding the dynamic range of the detector. The fastest variation of diffracted intensity across the detector will be due to interference between X-rays scattered by points in the object located furthest from each other: the sphere diameter, in other words. Measuring this maximum fringe frequency requires at least two detector pixels per period, or a so-called Shannon sampling of Δq ¼ π=ð2RÞ. In this case the pixel solid angle is ΔΩ ¼ ðλ=2π Þ2 Δq2 so that the previous equation becomes Imax ðqÞ ¼ I0 re2 jΔρj2 π 2 λ2 q4 which is independent of sphere radius. It may seem that for merging of thousands of diffraction patterns from similar randomly oriented single particles (such as a virus), the same methods as used in the cryo-EM community could be used. Here, noisy low-dose projection images of many copies of a particle, lying in many random orientations, are recorded within the field of view of each image, and must be merged to produce a three-dimensional image (Spence, 2013). However,

140

John C.H. Spence

the XFEL diffraction patterns also require solution of the phase problem, and, unlike real-space cryo-EM images, there is no requirement for correction of electron lens aberrations, while an enantiomorphous ambiguity arises from the Friedel symmetry of low-resolution patterns, not present for realspace images. In addition, diffraction patterns have an origin, unlike images, and the background due to ice in cryo-EM images must be treated differently from the background in a XRD pattern due to diffraction from a water jacket surrounding the particle. Building on previous work on iterative phasing of continuous diffraction patterns, two main approaches have been developed for the reconstruction of a three-dimensional image (density map) from many randomly oriented snapshot single-particle XRD patterns, and for dealing with the associated problems of particle inhomogeneity. We will give here only a brief outline of the general principles of these methods focusing on key issues. The manifold embedding approach (Yoon et al., 2011) is illustrated in Fig. 12, simplified for the case of a three-pixel (x,y,z) detector and single-axis rotation of a particle, in order to illustrate the principle of the method. With this simplification, a snapshot diffraction pattern can be represented as a three-dimensional vector, with each component representing the scattered intensity value at a pixel. These vectors (the diffraction snapshots) arrive in a random sequence, but rotation of a particle traces out a loop

A

B F F

Fig. 12 Simplified manifold embedding approach for a sample which can rotate only about one axis, and a three-pixel detector. One vector in this three-dimensional space represents a diffraction pattern, each axis is a pixel, and each coordinate value is an intensity for that pixel. Rotation of the molecule causes the vector to trace out a loop as the particle returns to its original orientation, while neighboring points on the loop represent similar diffraction patterns with small vectors χ (the least-squares difference, Euclidean metric) between their ends. Patterns recorded from molecules in random orientations can then be sequenced for a movie by identifying the loop path.

X-Ray Lasers in Biology: Structure and Dynamics

141

(a one-dimensional manifold) in this three-dimensional space of intensities. Determining this manifold allows one to assign an orientation to each snapshot. In general the detector has N pixels and particle rotation about three axes generates a 3D manifold in the N-dimensional Hilbert space of pixel intensities. The manifold is seen to be parameterized by a three-dimensional latent space defined by the three Euler angles defining the particle orientation. Many practical difficulties arise, including the transformation from angular increment to coordinate change in N dimensions, and the effects of noise and conformational changes. In the simplest case, a second conformation would define a second distinct loop; however, the effects of noise thicken the manifolds so that they may overlap. The key issue of distinguishing changes in particle orientation from conformational changes (essential in order to make a 3D “molecular movie”) is resolved using the fact that the operations associated with conformational change commute, while those associated with the rotation group do not. Conformational changes alter the internal structure of a particle, unlike rotations. An important feature of this approach is that all the data are used for all the analyses, rather than selecting subclasses (e.g., of orientation or conformation) for successive analysis. However, even in the absence of noise, a minimum number of scattered counts are needed to identify a particular orientation, proportional to the number of distinct orientations sought. The computational demands of this approach are considerable and set the limit on the size of the largest molecule, which can be analyzed. A second approach has been based on the principle of expectation maximization (Ho et al., 2016; Loh & Elser, 2009). The method is most simply explained in two dimensions (2D), for the case of a set of noisy 2D pictures I(k) of the same nonsymmetric object, which are known to lie in any one of four orientations i ¼ 1,4 differing by a 90 degree rotation about their normal. Here, k is the image index and extends over the N
N pixels of the pictures. A model is first assumed, which may consist of random values, and is generated in each of the four orientations i (expansion). Assuming Poisson noise, the probability Pi(k) is calculated that an experimental image I(k) came from each model in orientation i. To avoid the occurrence of extremely small numbers in the first iteration, these probabilities are normalized to unity. The process is repeated (maximization) for each image, giving a set of coefficients Pi(k). Four new models are then formed from the weighted sum M(i) ¼ ΣkPi(k)I(k). Since the four initial orientation-generating operations applied to the model are known, it is

142

John C.H. Spence

then possible to return the four new models to the same orientation, average them, and use the average as a new estimate of the model. Iterations then continue from the first step. An experimental demonstration of the method using low-resolution 2D X-ray shadow images has been demonstrated using as few as 2.5 photons per image (Ayyer, Philipp, Tate, Elser, & Gruner, 2014). The method has some similarity to cryo-EM methods based on crosscorrelation between experimental patterns to find similar orientations, but has the advantage that cross-correlations are computed between experimental patterns and a model, rather than between every possible experimental pair, so that the computation time is linear, rather than quadratic, in the number of patterns. Like the GTM method, this approach uses the entire body of data in each update of the model parameters; however, the GTM method does not constrain the data to fit a three-dimensional model density. At least a thousand counts per image are needed to assign patterns to a particular orientation class. In both these methods, solution of the noncrystallographic phase problem (reviewed in Spence, 2017) may be integrated with the problem of orientation determination. Particle inhomogeneity (which increases with particle size) is the most important problem for single-particle XFEL imaging, and may be solved in principle by the ability of the above methods to distinguish conformations, if sufficient high-quality data are available. A method for obtaining a three-dimensional reconstruction from a single shot is described in Schmidt et al. (2008), using multiple incident beams split apart by a beamsplitter and arriving at the sample from orthogonal directions. Several authors have pointed out that the curvature of the Ewald sphere provides limited three-dimensional information from a single shot. Bergh, Huldt, Tıˆmneanu, Maia, and Hajdu (2008) describe other possibilities for extracting three-dimensional information from a single shot, such as Laue diffraction using harmonics, coherent convergent beam diffraction, and multiple-pinhole Fourier transform holography. Fig. 13 shows the diffraction pattern obtained from a single Chlorella virus (one particle per shot). The rings extend clearly to a resolution of 12 nm. The 3D reconstruction of a mimi virus from similar patterns is described in Ekeberg et al. (2015). A database for SFX and single-particle data has been established at http:// cxidb.org/index.html, where published data can be found and used to evaluate new algorithms. This site also makes available the HAWK program for EMC analysis of XFEL SP data.

X-Ray Lasers in Biology: Structure and Dynamics

143

Fig. 13 XFEL diffraction pattern from a single PBCV icosohedral virus (Chlorella) of 190 nm diameter. Single LCLS shot using 1.8 kV X-rays showing 12 nm resolution. pnCCD detector. From Liu, H., & Spence, J. C. H. (2016). XFEL data analysis for structural biology. Quantitative Biology, 4(3), 159–176. With permission of Springer.

11. SUMMARY This is an exciting time for projects aimed at imaging protein dynamics using XFELs. From the above we can see at least three “Grand Challenge” projects which could be identified for structural biology: (i) imaging the water-splitting event in photosynthesis; (ii) imaging single-pass enzyme dynamics to expose the mechanisms involved, as a basis for structural enzymatics (Johnson, Yukl, Klema, Klinman, & Wilmot, 2013); and (iii) imaging large, rare, high-energy conformational changes by single-particle methods by making use of the large amount of data which will become available. A wide range of methods are now being developed to trigger reactions, from THz radiation to caged-release compounds, to optogenetics and spectroscopy, while new modes of imaging continue to be developed, from solution scattering, simultaneous snapshot diffraction, and correlated fluctuation analysis, to mixing jet experiments based on diffusive mixing. All these

144

John C.H. Spence

benefit from the unique capability of the EXFEL to allow atomic- resolution time-resolved imaging under physiological condition with minimal radiation damage. As should be clear from this review, this new field of science is at that very exciting early stage, where many new ideas and techniques flourish, and the mature methodologies of the future have yet to be established. As Humphrey Davey commented in 1806 “Nothing promotes the advancement of Science so much as the invention of a new instrument.”

ACKNOWLEDGMENTS This work was supported by NSF STC award 1231306 and NSF ABI award 1565180. Thanks to N. Zatsepin and A. Shevchuck for their helpful comments.

REFERENCES Aquila, A., Hunter, M. S., Doak, R. B., Kirian, R. A., Fromme, P., White, T. A., … Chapman, H. N. (2012). Time-resolved protein nanocrystallography using an X-ray free-electron laser. Optics Express, 20(3), 2706–2716. http://dx.doi.org/10.1364/ Oe.20.002706. Arnlund, D., Johansson, L. C., Wickstrand, C., Barty, A., Williams, G. J., Malmerberg, E., … Neutze, R. (2014). Visualizing a protein quake with time-resolved X-ray scattering at a free-electron laser. Nature Methods, 11(9), 923–926. http://dx.doi.org/10.1038/ Nmeth.3067. Awel, S., Kirian, R. A., Eckerskorn, N., Wiedorn, M., Horke, D. A., Rode, A. V., … Chapman, H. N. (2016). Visualizing aerosol-particle injection for diffractive-imaging experiments. Optics Express, 24(6), 6507–6521. http://dx.doi.org/10.1364/ Oe.24.006507. Ayyer, K., Philipp, H. T., Tate, M. W., Elser, V., & Gruner, S. M. (2014). Real-Space x-ray tomographic reconstruction of randomly oriented objects with sparse data frames. Optics Express, 22(3), 2403–2413. http://dx.doi.org/10.1364/OE.22.002403. Ayyer, K., Yefanov, O. M., Oberthur, D., Roy-Chowdhury, S., Galli, L., Mariani, V., … Chapman, H. N. (2016). Macromolecular diffractive imaging using imperfect crystals. Nature, 530(7589), 202–206. http://dx.doi.org/10.1038/nature16949. Barends, T. R. M., Foucar, L., Ardevol, A., Nass, K., Aquila, A., Botha, S., … Schlichting, I. (2015). Direct observation of ultrafast collective motions in CO myoglobin upon ligand dissociation. Science, 350(6259), 445–450. http://dx.doi.org/10.1126/science.aac5492. Barty, A., Caleman, C., Aquila, A., Timneanu, N., Lomb, L., White, T. A., … Chapman, H. N. (2012). Self-terminating diffraction gates femtosecond X-ray nanocrystallography measurements. Nature Photonics, 6(1), 35–40. http://dx.doi.org/ 10.1038/Nphoton.2011.297. Batyuk, A., Galli, L., Ishchenko, A., Han, G. W., Gati, C., Popov, P. A., … Cherezov, V. (2016). Native phasing of x-ray free-electron laser data for a G protein–coupled receptor. Science Advances, 2(9), e1600292. Bergh, M., Huldt, G., Tıˆmneanu, N., Maia, F. R. N. C., & Hajdu, J. (2008). Feasibility of imaging living cells at subnanometer resolutions by ultrafast X-ray diffraction. Quarterly Reviews of Biophysics, 41(3–4), 181–204. http://dx.doi.org/10.1017/ S003358350800471X. Berglund, G. I., Carlsson, G. H., Smith, A. T., Szoke, H., Henriksen, A., & Hajdu, J. (2002). The catalytic pathway of horseradish peroxidase at high resolution. Nature, 417(6887), 463–468. http://dx.doi.org/10.1038/417463a.

X-Ray Lasers in Biology: Structure and Dynamics

145

Bolotovsky, R., Steller, I., & Rossmann, M. G. (1998). The use of partial reflections for scaling and averaging X-ray area detector data. Journal of Applied Crystallography, 31, 708–717. http://dx.doi.org/10.1107/S0021889898004361. Bonifacio, R., Pellegrini, C., & Narducci, L. M. (1984). Collective instabilities and high-gain regime in a free electron laser. Optics Communications, 50(6), 373–378. http://dx.doi.org/ 10.1016/0030-4018(84)90105-6. Bostedt, C., Boutet, S., Fritz, D. M., Huang, Z. R., Lee, H. J., Lemke, H. T., … Williams, G. J. (2016). Linac coherent light source: The first five years. Reviews of Modern Physics, 88(1), 015007. http://dx.doi.org/10.1103/RevModPhys.88.015007. Boutet, S., Lomb, L., Williams, G. J., Barends, T. R. M., Aquila, A., Doak, R. B., … Schlichting, I. (2012). High-resolution protein structure determination by serial femtosecond crystallography. Science, 337(6092), 362–364. http://dx.doi.org/10.1126/ science.1217737. Breedlove, J. R., & Trammell, G. T. (1970). Molecular microscopy—Fundamental limitations. Science, 170(3964), 1310–1313. http://dx.doi.org/10.1126/science.170. 3964.1310. Brehm, W., & Diederichs, K. (2014). Breaking the indexing ambiguity in serial crystallography. Acta Crystallographica Section D: Biological Crystallography, 70, 101–109. http://dx. doi.org/10.1107/S1399004713025431. Caleman, C., Bergh, M., Scott, H. A., Spence, J. C. H., Chapman, H. N., & Timneanu, N. (2011). Simulations of radiation damage in biomolecular nanocrystals induced by femtosecond X-ray pulses. Journal of Modern Optics, 58(16), 1486–1497. http://dx.doi.org/ 10.1080/09500340.2011.597519. Calvey, G. D., Katz, A. M., Schaffer, C. B., & Pollack, L. (2016). Mixing injector enables time-resolved crystallography with high hit rate at X-ray free electron lasers. Structural Dynamics, 3(5), 054301. http://dx.doi.org/10.1063/1.4961971. Chapman, H. N., Barty, A., Bogan, M. J., Boutet, S., Frank, M., Hau-Riege, S. P., … Hajdu, J. (2006). Femtosecond diffractive imaging with a soft-X-ray free-electron laser. Nature Physics, 2(12), 839–843. http://dx.doi.org/10.1038/nphys461. Chapman, H. N., Caleman, C., & Timneanu, N. (2014). Diffraction before destruction. Philosophical Transactions of the Royal Society, B: Biological Sciences, 369(1647), 20130313. http://dx.doi.org/10.1098/rstb.2013.0313. Chapman, H. N., Fromme, P., Barty, A., White, T. A., Kirian, R. A., Aquila, A., … Spence, J. C. H. (2011). Femtosecond X-ray protein nanocrystallography. Nature, 470(7332), 73–77. http://dx.doi.org/10.1038/nature09750. Crewe, A. V., Wall, J., & Langmore, J. (1970). Visibility of single atoms. Science, 168(3937), 1338–1340. http://dx.doi.org/10.1126/science.168.3937.1338. Dashti, A., Schwander, P., Langlois, R., Fung, R., Li, W., Hosseinizadeh, A., … Ourmazd, A. (2014). Trajectories of the ribosome as a Brownian nanomachine. Proceedings of the National Academy of Sciences of the United States of America, 111(49), 17492–17497. http://dx.doi.org/10.1073/pnas.1419276111. Debye, P. (1913). Interference of x rays and heat movement. Annalen Der Physik, 43(1), 49–95. Derosier, D. J., & Klug, A. (1968). Reconstruction of 3 dimensional structures from electron micrographs. Nature, 217(5124), 130–134. http://dx.doi.org/10.1038/217130a0. Donatelli, J. J., Zwart, P. H., & Sethian, J. A. (2015). Iterative phasing for fluctuation X-ray scattering. Proceedings of the National Academy of Sciences of the United States of America, 112(33), 10286–10291. http://dx.doi.org/10.1073/pnas.1513738112. Eckerskorn, N., Li, L., Kirian, R. A., Kupper, J., DePonte, D. P., Krolikowski, W., … Rode, A. V. (2013). Hollow Bessel-like beam as an optical guide for a stream of microscopic particles. Optics Express, 21(25), 30492–30499. http://dx.doi.org/10.1364/ Oe.21.030492.

146

John C.H. Spence

Ekeberg, T., Svenda, M., Abergel, C., Maia, F. R. N. C., Seltzer, V., Claverie, J. M., … Hajdu, J. (2015). Three-dimensional reconstruction of the giant mimivirus particle with an X-ray free-electron laser. Physical Review Letters, 114(9), 098102. http://dx.doi.org/ 10.1103/PhysRevLett.114.098102. Elser, V. (2011). Strategies for processing diffraction data from randomly oriented particles. Ultramicroscopy, 111(7), 788–792. http://dx.doi.org/10.1016/j.ultramic.2010.10.014. Fenimore, P. W., Frauenfelder, H., McMahon, B. H., & Young, R. D. (2004). Bulk-solvent and hydration-shell fluctuations, similar to alpha- and beta-fluctuations in glasses, control protein motions and functions. Proceedings of the National Academy of Sciences of the United States of America, 101(40), 14408–14413. http://dx.doi.org/10.1073/pnas.0405573101. Gallat, F. X., Matsugaki, N., Coussens, N. P., Yagi, K. J., Boudes, M., Higashi, T., … Chavas, L. M. G. (2014). In vivo crystallography at X-ray free-electron lasers: The next generation of structural biology? Philosophical Transactions of the Royal Society, B: Biological Sciences, 369(1647), 20130497. http://dx.doi.org/10.1098/rstb.2013.0497. Ginn, H. M., et al. (2015). A revised partiality model and post-refinement algorithm for XFEL data. Acta Crystallographica Section D: Biological Crystallography, 71, 1400. Ginn, H. M., Evans, G., Sauter, N., & Stuart, D. (2016). On the release of cppxfel for processing X-ray free-electron laser images. Journal of Applied Crystallography, 49, 1. Glaeser, R. M. (2007). Electron crystallography of biological macromolecules. Oxford; New York: Oxford University Press. Graves, W. S., K€artner, F. X., Moncton, D. E., & Piot, P. (2012). Intense superradiant X rays from a compact source using a nanocathode array and emittance exchange. Physical Review Letters, 108(26), 263904. Hajdu, J., Acharya, K. R., Stuart, D. I., Barford, D., & Johnson, L. N. (1988). Catalysis in enzyme crystals. Trends in Biochemical Sciences, 13(3), 104–109. http://dx.doi.org/ 10.1016/0968-0004(88)90051-5. Hantke, M. F., Hasse, D., Maia, F. R. N. C., Ekeberg, T., John, K., Svenda, M., … Andersson, I. (2014). High-throughput imaging of heterogeneous cell organelles with an X-ray laser. Nature Photonics, 8(12), 943–949. http://dx.doi.org/10.1038/ Nphoton.2014.270. Haupert, L. M., & Simpson, G. J. (2011). Screening of protein crystallization trials by second order nonlinear optical imaging of chiral crystals (SONICC). Methods, 55(4), 379–386. http://dx.doi.org/10.1016/j.ymeth.2011.11.003. Hau-Riege, S. P. (2012). Photoelectron dynamics in X-ray free-electron-laser diffractive imaging of biological samples. Physical Review Letters, 108(23), 238101. http://dx.doi. org/10.1103/PhysRevLett.108.238101. Hau-Riege, S. P., & Bennion, B. J. (2015). Reproducible radiation-damage processes in proteins irradiated by intense x-ray pulses. Physical Review E, 91(2). 022705. http://dx.doi. org/10.1103/PhysRevE.91.022705. He, H., Fang, H., Miller, M. D., Phillips, G. N., Jr., & Su, W. P. (2016). Improving the efficiency of molecular replacement by utilizing a new iterative transform phasing algorithm. Acta Crystallographica. Section A: Foundations and Advances, 72(5), 23–31. Henderson, R. (1995). The potential and limitations of neutrons, electrons and X-rays for atomic resolution microscopy of unstained biological molecules. Quarterly Reviews of Biophysics, 28(2), 171–193. http://dx.doi.org/10.1017/S003358350000305X. Henderson, R., & Unwin, P. N. T. (1975). 3-dimensional model of purple membrane obtained by electron-microscopy. Nature, 257(5521), 28–32. http://dx.doi.org/10.1038/257028a0. Henzler-Wildman, K., & Kern, D. (2007). Dynamic personalities of proteins. Nature, 450(7172), 964–972. http://dx.doi.org/10.1038/nature06522. Ho, P. J., Knight, C., Tegze, M., Faigel, G., Bostedt, C., & Young, L. (2016). Atomistic three-dimensional coherent x-ray imaging of nonbiological systems. Physical Review A, 94(6), 063823.

X-Ray Lasers in Biology: Structure and Dynamics

147

Horrell, S., Antonyuk, S. V., Eady, R. R., Hasnain, S. S., Hough, M. A., & Strange, R. W. (2016). Serial crystallography captures enzyme catalysis in copper nitrite reductase at atomic resolution from one crystal. IUCrJ, 3, 271–281. http://dx.doi.org/10.1107/ S205225251600823x. Howells, M. R., Beetz, T., Chapman, H. N., Cui, C., Holton, J. M., Jacobsen, C. J., … Starodub, D. (2009). An assessment of the resolution limitation due to radiation-damage in X-ray diffraction microscopy. Journal of Electron Spectroscopy and Related Phenomena, 170(1–3), 4–12. http://dx.doi.org/10.1016/j.elspec.2008.10.008. Johnson, B. J., Yukl, E. T., Klema, V. J., Klinman, J. P., & Wilmot, C. M. (2013). Structural snapshots from the oxidative half-reaction of a copper amine oxidase: Implications for O2 activation. Journal of Biological Chemistry, 288(39), 28409–28417. http://dx.doi. org/10.1074/jbc.M113.501791. Kabsch, W. (2014). Processing of X-ray snapshots from crystals in random orientations. Acta Crystallographica. Section D: Biological Crystallography, 70, 2204–2216. http://dx.doi.org/ 10.1107/S1399004714013534. Kam, Z. (1977). Determination of macromolecular structure in solution by spatial correlation of scattering fluctuations. Macromolecules, 10(5), 927–934. http://dx.doi.org/10.1021/ ma60059a009. Kang, Y. Y., Zhou, X. E., Gao, X., He, Y. Z., Liu, W., Ishchenko, A., … Xu, H. E. (2015). Crystal structure of rhodopsin bound to arrestin by femtosecond X-ray laser. Nature, 523(7562), 561–567. http://dx.doi.org/10.1038/nature14656. Kern, J., Tran, R., Alonso-Mori, R., Koroidov, S., Echols, N., Hattne, J., … Yachandra, V. K. (2014). Taking snapshots of photosynthetic water oxidation using femtosecond X-ray diffraction and spectroscopy. Nature Communications, 5, 4371. http://dx. doi.org/10.1038/ncomms5371. Kern, J., Yachandra, V. K., & Yano, J. (2015). Metalloprotein structures at ambient conditions and in real-time: Biological crystallography and spectroscopy using X-ray free electron lasers. Current Opinion in Structural Biology, 34, 87–98. Kim, H. K., Kim, J. G., Nozawa, S., Sato, T, Oang, K. Y., Kim, T. W., … Adachi, S. (2015). Direct observation of bond formation in solution with femtosecond X-ray scattering. Nature, 518, 385. Kirian, R. A. (2012). Structure determination through correlated fluctuations in x-ray scattering. Journal of Physics B: Atomic Molecular and Optical Physics, 45(22), 223001. http://dx.doi.org/10.1088/0953-4075/45/22/223001. Kirian, R. A., Bean, R. J., Beyerlein, K. R., Barthelmess, M., Yoon, C. H., Wang, F. L., … Chapman, H. N. (2015). Direct phasing of finite crystals illuminated with a freeelectron laser. Physical Review X, 5(1), 011015. http://dx.doi.org/10.1103/ PhysRevX.5.011015. Kirian, R. A., Schmidt, K. E., Wang, X. Y., Doak, R. B., & Spence, J. C. H. (2011). Signal, noise, and resolution in correlated fluctuations from snapshot small-angle x-ray scattering. Physical Review E, 84(1), 011921. http://dx.doi.org/10.1103/PhysRevE.84.011921. Kirian, R. A., Wang, X. Y., Weierstall, U., Schmidt, K. E., Spence, J. C. H., Hunter, M., … Holton, J. (2010). Femtosecond protein nanocrystallography—Data analysis methods. Optics Express, 18(6), 5713–5723. http://dx.doi.org/10.1364/Oe.18.005713. Kondratenko, A. M., & Saldin, E. L. (1980). Generation of coherent radiation by a relativistic electron beam in an ondulator. Particle Accelerators, 10(3–4), 207–216. Kroll, T., et al. (2016). X-ray absorption spectroscopy using a self-seeded soft X-ray free-electron laser. Optics Express, 24, 22469–22480. Kupitz, C., Basu, S., Grotjohann, I., Fromme, R., Zatsepin, N. A., Rendek, K. N., … Fromme, P. (2014). Serial time-resolved crystallography of photosystem II using a femtosecond X-ray laser. Nature, 513(7517), 261–265. http://dx.doi.org/10.1038/ nature13453.

148

John C.H. Spence

Kupitz, C., Olmos, J. L., Holl, M., Tremblay, L., Pande, K., Pandey, S., … Schmidt, M. (2016). Structural enzymology using X-ray free electron lasers. Structural Dynamics, 4(4), 044003. http://dx.doi.org/10.1063/1.4972069. Landau, E. M., & Rosenbusch, J. P. (1996). Lipidic cubic phases: A novel concept for the crystallization of membrane proteins. Proceedings of the National Academy of Sciences of the United States of America, 93(25), 14532–14535. Levantino, M., Schiro, G., Lemke, H. T., Cottone, G., Glownia, J. M., Zhu, D. L., … Cammarata, M. (2015). Ultrafast myoglobin structural dynamics observed with an X-ray free-electron laser. Nature Communications, 6, 6772. http://dx.doi.org/10.1038/ ncomms7772. Lewandowski, J. R., Halse, M. E., Blackledge, M., & Emsley, L. (2015). Direct observation of hierarchical protein dynamics. Science, 348(6234), 578–581. http://dx.doi.org/ 10.1126/science.aaa6111. Li, C. F., Schmidt, K., & Spence, J. C. (2015). Data collection strategies for time-resolved X-ray free-electron laser diffraction, and 2-color methods. Structural Dynamics, 2(4), 041714. http://dx.doi.org/10.1063/1.4922433. Li, C., Zatsepin, N., Li., X., Liu, H., & Spence, J. C. H. (2017). SPIND. A new algorithm for automated indexing of sparse diffraction patterns. Structural Dynamics, Submitted. Liu, H. G., & Spence, J. C. H. (2014). The indexing ambiguity in serial femtosecond crystallography (SFX) resolved using an expectation maximization algorithm. IUCrJ, 1, 393–401. http://dx.doi.org/10.1107/S2052252514020314. Liu, H., & Spence, J. C. H. (2016). XFEL data analysis for structural biology. Quantitative Biology, 4(3), 159–176. http://dx.doi.org/10.1007/s40484-016-0076-z. Liu, W., Wacker, D., Gati, C., Han, G. W., James, D., Wang, D. J., … Cherezov, V. (2013). Serial femtosecond crystallography of G protein-coupled receptors. Science, 342(6165), 1521–1524. http://dx.doi.org/10.1126/science.1244142. Loh, N. T. D., & Elser, V. (2009). Reconstruction algorithm for single-particle diffraction imaging experiments. Physical Review E, 80(2), 026705. http://dx.doi.org/10.1103/ PhysRevE.80.026705. Lommel, K., Schafer, G., Grenader, K., Ruland, C., Terfort, A., Mantele, W., & Wille, G. (2013). Caged CO2 for the direct observation of CO2-consuming reactions. Chembiochem, 14(3), 372–380. http://dx.doi.org/10.1002/cbic.201200659. Lyubimov, A. Y., Murray, T. D., Koehl, A., Araci, I. E., Uervirojnangkoorn, M., Zeldin, O. B., … Berger, J. M. (2015). Capture and X-ray diffraction studies of protein microcrystals in a microfluidic trap array. Acta Crystallographica. Section D: Biological Crystallography, 71, 928–940. http://dx.doi.org/10.1107/S1399004715002308. Madey, J. M. J. (1971). Stimulated emission of bremsstrahlung in a periodic magnetic field. Journal of Applied Physics, 42(5), 1906–1913. http://dx.doi.org/10.1063/1.1660466. Marchesini, S. (2007). Invited article: A unified evaluation of iterative projection algorithms for phase retrieval (vol 78, art no 011301, 2007). Review of Scientific Instruments, 78(4), 049901. http://dx.doi.org/10.1063/1.2736942. Millane, R. P., & Lo, V. L. (2013). Iterative projection algorithms in protein crystallography. I. Theory. Acta Crystallographica Section A, 69, 517–527. http://dx.doi.org/10.1107/ S0108767313015249. Mitzner, R., Rehanek, J., Kern, J., Gul, S., Hattne, J., Taguchi, T., … Yano, J. (2013). Ledge X-ray absorption spectroscopy of dilute systems relevant to metalloproteins using an X-ray free-electron laser. Journal of Physical Chemistry Letters, 4(21), 3641–3647. http:// dx.doi.org/10.1021/jz401837f. Moffat, K. (2014). Time-resolved crystallography and protein design: Signalling photoreceptors and optogenetics. Philosophical Transactions of the Royal Society, B: Biological Sciences, 369(1647), 20130568. http://dx.doi.org/10.1098/rstb.2013.0568. Motz, H. (1951). Applications of the radiation from fast electron beams. Journal of Applied Physics, 22(5), 527–535. http://dx.doi.org/10.1063/1.1700002.

X-Ray Lasers in Biology: Structure and Dynamics

149

Muller, E. W. (1951). Das Feldionenmikroskop. Zeitschrift Fur Physik, 131(1), 136–142. http://dx.doi.org/10.1007/Bf01329651. Munke, A., Andreasson, J., Aquila, A., Awel, S., Ayyer, K., Barty, A., … Zook, J. (2016). Coherent diffraction of single Rice Dwarf virus particles using hard X-rays at the Linac Coherent Light Source. Scientific Data, 3, 160064. http://dx.doi.org/10.1038/sdata.2016.64. Nango, E., Royant, A., Kubo, M., Nakane, T., Wickstrand, C., Kimura, T., … Arima, T. (2016). A three-dimensional movie of structural changes in bacteriorhodopsin. Science, 354(6319), 1552–1557. Nass, K., Foucar, L., Barends, T. R. M., Hartmann, E., Botha, S., Shoeman, R. L., … Schlichting, I. (2015). Indications of radiation damage in ferredoxin microcrystals using high-intensity X-FEL beams. Journal of Synchrotron Radiation, 22, 225–238. http://dx.doi. org/10.1107/S1600577515002349. Nelson, G., Kirian, R. A., Weierstall, U., Zatsepin, N. A., Farago´, T., Baumbach, T., … Heymann, M. (2016). Three-dimensional-printed gas dynamic virtual nozzles for x-ray laser sample delivery. Optics Express, 24(11), 11515–11530. http://dx.doi.org/ 10.1364/OE.24.011515. Neutze, R., Wouts, R., van der Spoel, D., Weckert, E., & Hajdu, J. (2000). Potential for biomolecular imaging with femtosecond X-ray pulses. Nature, 406(6797), 752–757. http://dx.doi.org/10.1038/35021099. Nilsson, A., Schreck, S., Perakis, F., & Pettersson, L. G. M. (2016). Probing water with X-ray lasers. Advances in Physics: X, 1(2), 226–245. http://dx.doi.org/10.1080/23746149.2016.1165630. Nogly, P., James, D., Wang, D. J., White, T. A., Zatsepin, N., Shilova, A., … Weierstall, U. (2015). Lipidic cubic phase serial millisecond crystallography using synchrotron radiation. IUCrJ, 2, 168–176. http://dx.doi.org/10.1107/S2052252514026487. Nogly, P., Panneels, V., Nelson, G., Gati, C., Kimura, T., Milne, C., … Standfuss, J. (2016). Lipidic cubic phase injector is a viable crystal delivery system for time-resolved serial crystallography. Nature Communications, 7, 12314. http://dx.doi.org/10.1038/ncomms12. Oberthuer, D., Knosˇka, J., Wiedorn, M. O., Beyerlein, K. R., Bushnell, D. A., Kovaleva, E. G., … Mariani, V. (2017). Double-flow focused liquid injector for efficient serial femtosecond crystallography. Scientific Reports, 7, 1. Oghbaey, S., et al. (2016). Fixed target combined with spectroscopy mapping: Approaching 100% hit rate for serial crystallography. Acta Crystallographica Section D, Biological Crystallography, 72, 944. Owen, R., Rudino-Pinera, E., & Garmen, E. F. (2006). Experimental determination of the radiation dose limit for cryocooled protein crystals. Proceedings of the National Academy of Sciences of the United States of America, 103, 4912. Pande, K., Hutchison, C. D. M., Groenhof, G., Aquila, A., Robinson, J. S., Tenboer, J., … Schmidt, M. (2016). Femtosecond structural dynamics drives the trans/cis isomerization in photoactive yellow protein. Science, 352(6286), 725–729. http://dx.doi.org/10.1126/ science.aad5081. Pande, K., Schwander, P., Schmidt, M., & Saldin, D. K. (2014). Deducing fast electron density changes in randomly orientated uncrystallized biomolecules in a pump–probe experiment. Philosophical Transactions of the Royal Society, B: Biological Sciences, 369(1647), 20130332. Pauling, L. (1960). The nature of the chemical bond and the structure of molecules and crystals: An introduction to modern structural chemistry (3rd ed.). . Ithaca, NY: Cornell University Press. Pedrini, B., Menzel, A., Guizar-Sicairos, M., Guzenko, V. A., Gorelick, S., David, C., … Abela, R. (2013). Two-dimensional structure from random multiparticle X-ray scattering images using cross-correlations. Nature Communications, 4, 1647. http://dx.doi.org/ 10.1038/ncomms2622. Pellegrini, C. (2012). The history of X-ray free-electron lasers. The European Physical Journal H, 37(5), 659–708. http://dx.doi.org/10.1140/epjh/e2012-20064-5. Redecke, L., Nass, K., DePonte, D. P., White, T. A., Rehders, D., Barty, A., … Chapman, H. N. (2013). Natively inhibited Trypanosoma brucei cathepsin

150

John C.H. Spence

B structure determined by using an X-ray laser. Science, 339(6116), 227–230. http://dx. doi.org/10.1126/science.1229663. Ribic, P. R., & Margaritondo, G. (2012). Status and prospects of x-ray free-electron lasers (X-FELs): A simple presentation. Journal of Physics D: Applied Physics, 45(21), 213001. http://dx.doi.org/10.1088/0022-3727/45/21/213001. Roedig, P., Duman, R., Sanchez-Weatherby, J., Vartiainen, I., Burkhardt, A., Warmer, M., … Meents, A. (2016). Room-temperature macromolecular crystallography using a micro-patterned silicon chip with minimal background scattering. Journal of Applied Crystallography, 49, 968–975. http://dx.doi.org/10.1107/S1600576716006348. Roessler, C. G., Agarwal, R., Allaire, M., Alonso-Mori, R., Andi, B., Bachega, J. F., … Zouni, A. (2016). Acoustic injectors for drop-on-demand serial femtosecond crystallography. Structure, 24(4), 631–640. http://dx.doi.org/10.1016/j.str.2016.02.007. Rupp, B. (2010). Biomolecular crystallography: principles, practice, and application to structural biology. New York: Garland Science. Saldin, D. K., Poon, H. C., Bogan, M. J., Marchesini, S., Shapiro, D. A., Kirian, R. A., … Spence, J. C. H. (2011). New light on disordered ensembles: Ab initio structure determination of one particle from scattering fluctuations of many copies. Physical Review Letters, 106(11), 115501. http://dx.doi.org/10.1103/PhysRevLett.106.115501. Sauter, N. K. (2015). XFEL diffraction: Developing processing methods to optimize data quality. Journal of Synchrotron Radiation, 22, 239–248. http://dx.doi.org/10.1107/ S1600577514028203. Sawaya, M. R., Rodriguez, J., Cascio, D., Collazo, M. J., Shi, D., Reyes, F. E., … Eisenberg, D. S. (2016). Ab initio structure determination from prion nanocrystals at atomic resolution by MicroED. Proceedings of the National Academy of Sciences of the United States of America, 113(40), 11232–11236. http://dx.doi.org/10.1073/pnas.1606287113. Schlichting, I. (2000). Crystallographic structure determination of unstable species. Accounts of Chemical Research, 33(8), 532–538. http://dx.doi.org/10.1021/ar9900459. Schlichting, I. (2015). Serial femtosecond crystallography: The first five years. IUCrJ, 2, 246–255. http://dx.doi.org/10.1107/S205225251402702x. Schlichting, I., Rapp, G., John, J., Wittinghofer, A., Pai, E. F., & Goody, R. S. (1989). Biochemical and crystallographic characterization of a complex of c-Ha-ras p21 and caged GTP with flash photolysis. Proceedings of the National Academy of Sciences of the United States of America, 86(20), 7687–7690. Schmidt, M. (2013). Mix and inject: Reaction initiation by diffusion for time-resolved macromolecular crystallography. Advances in Condensed Matter Physics, 2013, 10. http://dx. doi.org/10.1155/2013/167276. Schmidt, K. E., Spence, J. C. H., Weierstall, U., Kirian, R., Wang, X., Starodub, D., … Doak, R. B. (2008). Tomographic femtosecond X-ray diffractive imaging. Physical Review Letters, 101(11), 115507. Schoenlein, R. W., Peteanu, L. A., Mathies, R. A., & Shank, C. V. (1991). The 1st step in vision—Femtosecond isomerization of rhodopsin. Science, 254(5030), 412–415. http:// dx.doi.org/10.1126/science.1925597. Schotte, F., Lim, M. H., Jackson, T. A., Smirnov, A. V., Soman, J., Olson, J. S., … Anfinrud, P. A. (2003). Watching a protein as it functions with 150-ps time-resolved X-ray crystallography. Science, 300(5627), 1944–1947. http://dx.doi.org/10.1126/science.1078797. Seddon, E., Clarke, J., Dunning, D., Masciovecchio, C., Milne, C., Parmigiani, F., … Wurth, W., (2017) Short wavelength free electron laser sources and science: A review. Reports on Progress in Physics, (In press). Seibert, M. M., Ekeberg, T., Maia, F. R. N. C., Svenda, M., Andreasson, J., Jonsson, O., … Hajdu, J. (2011). Single mimivirus particles intercepted and imaged with an X-ray laser. Nature, 470(7332), 78–81. http://dx.doi.org/10.1038/nature09748. Shapiro, D. A., Chapman, H. N., DePonte, D., Doak, R. B., Fromme, P., Hembree, G., … Weierstall, U. (2008). Powder diffraction from a continuous microjet of

X-Ray Lasers in Biology: Structure and Dynamics

151

submicrometer protein crystals. Journal of Synchrotron Radiation, 15, 593–599. http://dx. doi.org/10.1107/S0909049508024151. Sierra, R. G., Gati, C., Laksmono, H., Dao, E. H., Gul, S., Fuller, F., … DeMirci, H. (2016). Concentric-flow electrokinetic injector enables serial crystallography of ribosome and photosystem II. Nature Methods, 13(1), 59–62. http://dx.doi.org/10.1038/nmeth.3667. Sigworth, F. J. (1998). Maximum-likelihood approach to the partilce-alignment problem. Biophysical Journal, 74, A226. Snell, E. H., Bellamy, H. D., & Borgstahl, G. E. O. (2003). Macromolecular crystal quality. Macromolecular Crystallography, Part C, 368, 268–288. http://dx.doi.org/10.1016/S00766879(03)68015-8. Solem, J. C. (1986). Imaging biological specimens with high-intensity soft X-rays. Journal of the Optical Society of America B: Optical Physics, 3(11), 1551–1565. Son, S. K., Chapman, H. N., & Santra, R. (2011). Multiwavelength anomalous diffraction at high X-ray intensity. Physical Review Letters, 107(21), 218102. http://dx.doi.org/ 10.1103/PhysRevLett.107.218102. Spence, J. (2008). X-ray imaging—Ultrafast diffract-and-destroy movies. Nature Photonics, 2(7), 390–391. http://dx.doi.org/10.1038/nphoton.2008.115. Spence, J. C. H. (2013). High-resolution electron microscopy (4th ed.). Oxford: Oxford University Press. Spence, J. C. H. (2014). Approaches to time-resolved diffraction using an XFEL. Faraday Discussions, 171, 429–434. Spence, J. C. H. (2017). Outrunning damage: Electrons vs X-rays - timescales and mechanisms. Structural Dynamics. In press. Spence, J. C. H., & Chapman, H. N. (2014). The birth of a new field. Philosophical Transactions of the Royal Society, B: Biological Sciences, 369(1647), 1. http://dx.doi.org/ 10.1098/rstb.2013.0309. Spence, J. C. H., & Doak, R. B. (2004). Single molecule diffraction. Physical Review Letters, 92(19), 198102. http://dx.doi.org/10.1103/PhysRevLett.92.198102. Spence, J. C. H., Huang, Y. M., & Sankey, O. (1993). Lattice trapping and surface reconstruction for silicon cleavage on (111). Ab-initio quantum molecular-dynamics calculations. Acta Metallurgica et Materialia, 41(10), 2815–2824. http://dx.doi.org/ 10.1016/0956-7151(93)90096-B. Spence, J. C. H., Kirian, R. A., Wang, X. Y., Weierstall, U., Schmidt, K. E., White, T., … Holton, J. (2011). Phasing of coherent femtosecond X-ray diffraction from size-varying nanocrystals. Optics Express, 19(4), 2866–2873. http://dx.doi.org/10.1364/Oe.19.002866. Spence, J. C. H., Weierstall, U., & Chapman, H. N. (2012). X-ray lasers for structural and dynamic biology. Reports on Progress in Physics, 75(10). 102601. http://dx.doi.org/ 10.1088/0034-4885/75/10/102601. Spence, J. C. H., Zatsepin, N. A., & Li, C. (2014). Coherent convergent-beam time-resolved X-ray diffraction. Philosophical Transactions of the Royal Society, B: Biological Sciences, 369(1647), 20130325. Stagno, J. R., et al. (2017). Structures of riboswitch RNA reaction states by mix-and-inject XFEL serial crystallography. Nature, 541, 242. Stan, C. A., Milathianaki, D., Laksmono, H., Sierra, R. G., McQueen, T. A., Messerschmidt, M., … Boutet, S. (2016). Liquid explosions induced by X-ray laser pulses. Nature Physics, 12(10), 966–971. http://dx.doi.org/10.1038/nphys3. Starodub, D., Aquila, A., Bajt, S., Barthelmess, M., Barty, A., Bostedt, C., … Bogan, M. J. (2012). Single-particle structure determination by correlations of snapshot X-ray diffraction patterns. Nature Communications, 3, 1276. http://dx.doi.org/10.1038/ncomms2288. Starodub, D., Rez, P., Hembree, G., Howells, M., Shapiro, D., Chapman, H. N., … Spence, J. C. H. (2008). Dose, exposure time and resolution in serial X-ray crystallography. Journal of Synchrotron Radiation, 15(1), 62–73. http://dx.doi.org/10.1107/ S0909049507048893.

152

John C.H. Spence

Suga, M., Akita, F., Sugahara, M., Kubo, M., Nakajima, Y., Nakane, T., … Nakano, T. (2017). Light-induced structural changes and the site of O= O bond formation in PSII caught by XFEL. Nature, 543(7643), 131–135. Tenboer, J., Basu, S., Zatsepin, N., Pande, K., Milathianaki, D., Frank, M., … Schmidt, M. (2014). Time-resolved serial crystallography captures high-resolution intermediates of photoactive yellow protein. Science, 346(6214), 1242–1246. http://dx.doi.org/ 10.1126/science.1259357. Uervirojnangkoorn, M., Zeldin, O. B., Lyubimov, A. Y., Hattne, J., Brewster, A. S., Sauter, N. K., … Weis, W. I. (2015). Enabling X-ray free electron laser crystallography for challenging biological systems from a limited number of crystals. eLife 4, e05421. http://dx.doi.org/10.7554/eLife.05421. Van Slyke, A. L., Wang, J., Singh, A., Pollack, L., & Zipfel, W. R. (2014). Fast binding kinetics of RNA aptamers measured using a novel microfluidic mixer. Biophysical Journal, 106(2), 796a. Wales, D. J. (2003). Energy landscapes. Cambridge, UK; New York: Cambridge University Press. Wang, D. J., Weierstall, U., Pollack, L., & Spence, J. (2014). Double-focusing mixing jet for XFEL study of chemical kinetics. Journal of Synchrotron Radiation, 21, 1364–1366. http:// dx.doi.org/10.1107/S160057751401858x. Weierstall, U. (2014). Liquid sample delivery techniques for serial femtosecond crystallography. Philosophical Transactions of the Royal Society, B: Biological Sciences, 369(1647). 20130337. Weierstall, U., Doak, R. B., Spence, J. C. H., Starodub, D., Shapiro, D., Kennedy, P., … Chapman, H. N. (2008). Droplet streams for serial crystallography of proteins. Experiments in Fluids, 44(5), 675–689. http://dx.doi.org/10.1007/s00348-007-0426-8. Weierstall, U., James, D., Wang, C., White, T. A., Wang, D. J., Liu, W., … Cherezov, V. (2014). Lipidic cubic phase injector facilitates membrane protein serial femtosecond crystallography. Nature Communications, 5, 3309. http://dx.doi.org/10.1038/ncomms4309. Weierstall, U., Spence, J. C. H., & Doak, R. B. (2012). Injector for scattering measurements on fully solvated biospecies. Review of Scientific Instruments, 83(3). 035108. http://dx.doi. org/10.1063/1.3693040. White, T. A., Mariani, V., Brehm, W., Yefanov, O., Barty, A., Beyerlein, K. R., … Chapman, H. N. (2016). Recent developments in CrystFEL. Journal of Applied Crystallography, 49, 680–689. http://dx.doi.org/10.1107/S1600576716004751. Winn, M. D., Ballard, C. C., Cowtan, K. D., Dodson, E. J., Emsley, P., Evans, P. R., … Wilson, K. S. (2011). Overview of the CCP4 suite and current developments. Acta Crystallographica. Section D: Biological Crystallography, 67, 235–242. http://dx.doi.org/ 10.1107/S0907444910045749. Wochner, P., Gutt, C., Autenrieth, T., Demmer, T., Bugaev, V., Ortiz, A. D., … Dosch, H. (2009). X-ray cross correlation analysis uncovers hidden local symmetries in disordered matter. Proceedings of the National Academy of Sciences of the United States of America, 106(28), 11511–11514. http://dx.doi.org/10.1073/pnas.0905337106. Yoon, C. H., Schwander, P., Abergel, C., Andersson, I., Andreasson, J., Aquila, A., … Ourmazd, A. (2011). Unsupervised classification of single-particle X-ray diffraction snapshots by spectral clustering. Optics Express, 19(17), 16542–16549. http://dx.doi. org/10.1364/Oe.19.016542. Young, I. D., Ibrahim, M., Chatterjee, R., Sauter, N. K., Kern, N., Yachandra, V. K., & Yano, J. (2016). Structure of photosystem II and substrate binding at room temperature. Nature, 540, 453–457. Zhang, H. T., Unal, H., Gati, C., Han, G. W., Liu, W., Zatsepin, N. A., … Cherezov, V. (2015). Structure of the angiotensin receptor revealed by serial femtosecond crystallography. Cell, 161(4), 833–844. http://dx.doi.org/10.1016/j.cell.2015.04.011. Zuo, J. M., Kim, M., O’Keeffe, M., & Spence, J. C. H. (1999). Direct observation of d-orbital holes and Cu-Cu bonding in Cu2O. Nature, 401(6748), 49–52. http://dx. doi.org/10.1038/43403.

INDEX Note: Page numbers followed by “f ” indicate figures, and “t” indicate tables.

A Aharonov–Bohm effect, application of, 72–73 Anamorphotic plate, 68–70, 69f Anisotropic scattering, 122, 132 Astigmatism of objective lens, 39 testing for, 48

B Bragg boost, 105–106, 119 Bright-field imaging, 13 in TEM, 63, 74 zone plates for resolution and contrast improvement in, 23

C Carbon films, through-focus series circular hole in, 53–55, 54f with specimen-space cooling, 53–55, 55f of thicker carbon film, 55–56, 56f Cartesian 2D transform, 96 Coherent diffractive imaging (CXDI), 138 Compact XFEL (CXFEL), 109 Continuous phase transition disc object with, 85–86 weak-phase disc object with, 86 Coulomb repulsion, 119 Cryoelectron microscopy (cryo-EM), 104–105, 138, 142 Cryomicroscopy, survey methods for, 63

D Dark-field images, 13 zone plates for resolution and contrast improvement in, 23 Destructive readout method, 111–112 Diffraction-limited coherent X-ray beam, of nanometer dimensions, 138 Diffraction theory, 75 Diffract-then-destroy mechanism, 117–118

Disc objects object function, 78–79 phase profile form of, 78f straight-edged plates, on cylindrical axis with half-plane phase-changing (Hilbert) plate, 90–92 with opaque (Foucault) plate, 89–90 and Zernike plates with continuous phase transition, 85–86 loss function for, 83 step phase change, 83–85 Double-focusing mixing jet, for XFEL, 129–130

E EELS. See Energy loss spectroscopy (EELS) Einzel lens, 71–72, 71f Electromagnetic objective, specimen stage for, 36f Electron beams, 63–67, 117–118 energy for modern XFELs, 109 and resolution limit, single-field condenser objective, 26–32 properties of, 5–6 Electron microscope development of, 3–6 narrow aperture in, 23 resolving power of, 39–40, 48–49, 58 single-field condenser objectives in, 24 Embedding approach, 140–141, 140f Energy loss spectroscopy (EELS), 63–67 Enzyme catalysis, 105 Ewald sphere, 128, 135–137, 142

F Fast solution scattering (FSS), 112–115, 112f, 130–132 anisotropy in, 133 First-order Bragg-diffracted beams, 115 153

154 Fluorescent screens properties of, 18–19 resolution on, 37 Foucault plates, 68, 70 disc object with, 89–90 plate function, postspecimen plates theory, 80–81 Fourier series, 75 Foucault/Hilbert plates, 81 2D transform, 75–76 in cylindrical coordinates, 95 Free-electron laser, 110–111 invention of, 107 FSS. See Fast solution scattering (FSS)

G Gas-dynamic virtual nozzle (GDVN), 111–115, 113f Gaussian image plane dark-field images, 13 resolution limit for two points radiating incoherently in, 8 GDVN. See Gas-dynamic virtual nozzle (GDVN) GTM method, 142

H Half-plane phase-changing (Hilbert) plate, disc object with, 90–92 Hankel transform, 75 illumination function, 77 object function, disc object, 78–79 spherical object, 79 2D transform, in cylindrical coordinates, 95–96 Helical phase object, object function, 79 High-angle scattering, fading of, 118–119, 118f Hilbert plates, 68, 71 disc object with, 90–92 magnetic equivalent of, 73–74 plate function, postspecimen plates theory, 80–81 Hole-free phase plate, 68–72

Index

I Illumination function, postspecimen plates theory, 76–77 Interference fringe system, 12, 17–18

J Japanese SACLA XFEL, 110–111

K Kam theory, 132–133 Knife edges device, 62

L Lattice planes of gold single crystal, 5f tremolite crystals, 52f Laue method, 105–106, 128 LCLS. See Linac coherent light source (LCLS) LCP jet. See Lipid cubic phase (LCP) jet Light microscope history of, 3–5 resolution limit of, 2–3, 7–9 resolving power of, 5–6 Light microscopical theory, 2–3 Linac coherent light source (LCLS), 107–109 at SLAC, 110–111, 111f SPI program, 125–126 Linear lattices, in partially coherent illumination due to spherical and chromatic aberrations, resolution limit of, 12–23 single-field condenser, 33–34 Lipid cubic phase (LCP) jet, 112–115, 114f Listeria monocytogenes, 49, 50f

M Magnetic imaging systems, 21 Molecular machines, XFEL, 122–126

N NMR method, 104–105 Nonperiodic specimen structures resolution limit of transmission electron microscope in, 7–8

155

Index

resolution of, 3–5 resolving power for, 6 sharpness of, 13

O Object function, postspecimen plates theory, 77–79 disc object, 78–79 helical phase object, 79 spherical object, 79 Opaque (Foucault) plate, disc object with, 89–90 Optical pump-probe methods, 126–129 Optimum electron objective, 24 Optogenetics, 128 Orbital angular momentum (OAM), 63–67, 65f generating technique for, 67

P Phase-contrast transfer function (PCTF), 74 Phase objects, 62, 75–76 Phase plates, 62 for transmission electron microscopy postspecimen plates, 68–81 straight-edged plates, on cylindrical axis, 89–92 Zernike plates, 81–89 Photoactive yellow protein (PYP), trans/cis isomerization in, 127–128, 127f Photographic emulsions for linear lattices, 21–22 properties of, 18–19 Photographic exposure image movement, transmission electron microscope electron image, due to electric and magnetic fields, 37–38 and heating of specimen, 35–36 Photosynthesis, XFEL, 126–129 Plate function, postspecimen plates theory Foucault/Hilbert plate, 80–81 intercepting and phase changing, 80 Zernike plate geometries, 80–81 Platinum–iridium evaporated film evaporated coating of, 55–56, 57f micrographs of, 4f Prespecimen phase plates geometry of, 68–70

phase shift controlling methods, 70–74 principle of, 62 theory and practice of, 63 theory for, 74–81 illumination function, 76–77 object function, 77–79 plate function, 80–81 Protein dynamics classes of, 124 intrinsic, 123 Protein function, 123 Pump-probe method, 105–106 Pure phase objects, 75–76

Q Quantum mechanical theory, 110–111

R Radiation-chemical process, 40–41, 44

S SASE. See Self-amplified spontaneous emission (SASE) Seeding method, 107–109 Self-amplified spontaneous emission (SASE), 107–109 Serial crystallography, 111–112, 122 data analysis of, 133–138 Serial femtosecond X-ray diffraction (SFX), 112–115, 112f, 122 database for, 142 data, phasing of, 137–138 Single-field condenser objective axial field strength distribution, measurement comparison, 29f beam paths for imaging specimen area and diffraction pattern, 25f cross section of, 26f linear lattices, in partially coherent illumination, 33–34 parameters of, 28, 30t pole-piece system of, 27f measurements of, 31–32, 32t principle, ray paths, and construction, 23–26 properties of, 27–28 selected area aperture in electron microscope with, 28f

156 Single-field condenser objective (Continued ) two incoherently radiating points electron beam energy and resolution limit for, 26–32 permissible disturbances, nature and magnitude of, 32–33 Single-particle imaging, XFEL, 122–126 Single-Particle Initiative (SPI) program, 125–126 Single-particles (SP) data analysis, 138–142 diffraction, 112–115, 112f Small-angle X-ray scattering methods (SAXS), advantages for, 130–132 Specimen-space cooling method, 44–49, 53 Spherical objects object function, 79 phase profile form of, 78f and Zernike plates, weak-phase spherical object, 86–89 SPI program. See Single-Particle Initiative (SPI) program Spiral zone plate, 63–67, 66f Star Wars program, 107–109 Straight-edged plates, on cylindrical axis, 89 disc object, with opaque (Foucault) plate, 89–90

T Thin-film techniques, 71 Through-focus series, carbon films circular holes imaged with specimenspace cooling, 53–55, 54–55f of thicker carbon film, 55–56, 56f Time-consuming screening trials, XFEL, 121 Time-resolved serial femtosecond diffraction (TR-SFX), 126–129 mixing jets and excitations, 129–130 Topatron mass spectrometer, 42–44 ion currents measurements, 43f Transmission electron microscopy (TEM), resolution limit of bright-field imaging, 63, 74 zone plates for improving resolution and contrast in, 23 changes in specimen contamination of specimen, 40

Index

specimen cooling, 49–52 specimen, electron beam, and residual gases interaction, 39–44 specimen-space cooling, residual gas during electron irradiation, 44–49 dark-field images, zone plates for improving resolution and contrast in, 23 diffraction and spherical aberration combination, 7–9 estimation of, 2–5 lack of sharpness, in image lenses instability, 38–39 lenses noncircularity, 39 linear lattices, in partially coherent illumination, 12–23 permissible disturbances, nature and magnitude of, 10–11 phase plates for postspecimen plates, 68–81 straight-edged plates, on cylindrical axis, 89–92 Zernike plates, 81–89 photographic exposure image, movement of electron image, due to electric and magnetic fields, 37–38 and heating of specimen, 35–36 self-structure of supporting films, 53–58 single-field condenser objective linear lattices, in partially coherent illumination, 33–34 principle, ray paths, and construction, 23–26 two incoherently radiating points, 26–33 sources of errors, 5–6 Tremolite crystals behavior of, 49, 52f contamination of, 49, 51f TR-SFX. See Time-resolved serial femtosecond diffraction (TR-SFX) Tulip plate, 70, 70f 2D angular correlation function (ACF), 132

157

Index

2D transform, in cylindrical coordinates conversion from Cartesian form, 96–97 Fourier series, 95 Hankel transform, 95–96 inverse transform, 98

V Vortex beams, 63–67, 74

W Weak-phase disc object, with continuous phase transition, 86 Weak-phase spherical object with ramped Zernike plate, 87–89 with stepped plate, 86–87 Wide-angle X-ray scattering methods (WAXS), advantages for, 130–132

X XFEL. See X-ray free-electron laser (XFEL) X-ray beams bandwidth of, 110 of nanometer dimensions, diffractionlimited coherent, 138 time-average power of, 110 X-ray diffraction (XRD), 117 X-ray free-electron laser (XFEL), 103–104, 107–111 Bragg boost, 105–106, 119 crystallography, advantages, 120–122 data analysis serial crystallography, 133–138 single particles, 138–142 diffraction pattern, PBCV icosohedral virus, 143f expectation maximization principle, 141–142 fast solution scattering and angular correlation function methods, 130–133

history of, 107–109 molecular machines and single-particle imaging, 122–126 to molecular movies, 105–106 optical pump-probe methods, 126–129 photosynthesis, 126–129 Photosystem I submicron crystal, 134f radiation damage limits resolution, 116–120 structural biology applications to, 106 data acquisition modes and sample delivery for, 111–116 Grand Challenge problems, 125 serial crystallography at, 120–122 time-consuming screening trials, 121 time-dependent diffraction, 130–132 time-resolved diffraction, 128–129 time-resolved mixing experiment, 130 time-resolved serial femtosecond diffraction, 126–129 time-resolved SFX, 129–130 X-ray scattering, 115–116 XRD. See X-ray diffraction (XRD)

Z Zach phase plate, 68–70, 69f Zernike phase plates, 62–63, 67–68, 71 geometries, 80–81 images with disc object and, 83–86 rotationally symmetric object and, 82–83 size parameter, 81–82 spherical object and, 86–89 linear transition, 81 Spirochete bacteria comparison, 64f step transition, 80–81