Foundations of Experimental Physics [1 ed.] 0367819961, 9780367819965

Table of contents :
Cover
Half Title
Title
Copyright
Preface
About the Authors
Contents
Preface
About the Authors
1. Introduction
2. Improving Signal-to-Noise Ratio
2.1 Introduction
2.2 Noise in Electronic Devices
2.3 Hardware Devices for Noise Reduction
2.4 Software Methods to Improve Signal-to-Noise Ratio
2.5 Summary
3. Vacuum Science and Technology
3.1 Introduction
3.2 Significance of Vacuum in Experimental Physics
3.3 Basic Laws and Terms in Vacuum Physics
3.4 Vacuum Pumps
3.5 Measurement of Vacuum: Vacuum Gauges
3.6 Vacuum Materials and Accessories
3.7 Ultra-High Vacuum (UHV) System
3.8 Summary
4. Photons and Electrons: Sources, Monochromators and Detectors
4.1 Introduction
4.2 Photon Sources
4.3 Monochromators or Wavelength Selectors
4.4 Spectral Line Shape
4.5 Intensity of Spectral Lines
4.6 Optical Fibres
4.7 Photon Detectors
4.8 Electrons as an Incident Beam for Characterisation
4.9 Electron Sources or Electron Gun
4.10 Motion of Electrons in the Electrostatic Field
4.11 Electron Energy Analysers
4.12 Electron Detectors
4.13 Summary
5. Nuclear Accelerators and Detectors
5.1 Introduction
5.2 Cockroft-Walton Accelerator/Generator
5.3 Van de Graaf Generator
5.4 Sloan and Lawrence Type Linear Accelerator
5.5 Cyclotron
5.6 Synchrotron
5.7 Particle Detectors
5.8 Neutron Detection
5.9 Summary
6. Diffraction Methods
6.1 Introduction
6.2 X-Ray Diffraction
6.3 X-Ray Diffractometers
6.4 Electron Diffraction
6.5 Neutron Diffraction
6.6 Summary
7. Microscopy
7.1 Introduction
7.2 Optical Microscopes
7.3 Field Microscopes
7.4 Electron Microscopes
7.5 Scanning Probe Microscopes
7.6 Summary
8. Electron and Optical Spectroscopy
8.1 Introduction
8.2 Electron Spectroscopy Techniques for Solids
8.3 Optical Spectroscopy Techniques
8.4 Summary
9. Magnetic Spectroscopy, Nuclear Spectroscopy and Magnetisation Measurements
9.1 Introduction
9.2 Nuclear Magnetic Resonance Spectroscopy
9.3 Electron Spin Resonance (ESR)
9.4 Mössbauer Spectroscopy Technique
9.5 Vibrating Sample Magnetometer Technique
9.6 Superconducting Quantum Interference Device (SQUID)
9.7 Rutherford Backscattering Spectrometry
9.8 Gamma Ray Spectroscopy
9.9 Summary
10. Cryogenic Temperature Methods
10.1 Quest for Low Temperature
10.2 Thermoelectric Cooling
10.3 Cooling by Latent Heat of Evaporation
10.4 Liquid He and Superfluidity
10.5 Dilution Refrigerator
10.6 Pomeranchuk Cooling
10.7 Adiabatic Demagnetisation
10.8 Laser Cooling
10.9 Low Temperature Measurements
10.10 Summary
11. Error Analysis and Statistical Methods
11.1 Introduction
11.2 Quantitative Estimation of the Errors
11.3 Statistical Handling of Data
11.4 Distribution of Data and Its Properties
11.5 Principle of Maximum Likelihood
11.6 Fitting of Data
11.7 Covariance
11.8 Coefficient of Linear Correlation
11.9 χ2 Test for the Distribution
11.10 Summary
Further Reading
Index

Citation preview

Foundations of

Experimental Physics

Foundations of

Experimental Physics

Shailaja Mahamuni Adjunct Professor, S. P. Pune University, Pune, India Deepti Sidhaye Assistant Professor, Department of Physics S. P. Pune University, Pune, India Sulabha Kulkarni INSA Senior Scientist, Centre for Materials for Electronics Technology (CMET). Pune, India

First edition published 2021 by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 © 2021 Shailaja Mahamuni, Deepti Sidhaye, Sulabha Kulkarni and Capital Publishing Company CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other PHDQVQRZNQRZQRUKHUHDIWHULQYHQWHGLQFOXGLQJSKRWRFRS\LQJPLFUR¿OPLQJDQG recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www. copyright.comor contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered WUDGHPDUNV DQG DUH XVHG RQO\ IRU LGHQWL¿FDWLRQ DQG H[SODQDWLRQ ZLWKRXW LQWHQW WR infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this book has been requested Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan or Bhutan) ISBN: 978-0-367-81996-5 (hbk) ISBN: 978-1-003-09774-7 (ebk) Typeset in Times New Roman by Innovative Processors, New Delhi

Preface

Experiments are the core essence of Science as they serve the foundation for FUHDWLRQRIVFLHQWL¿FNQRZOHGJH2EVHUYDWLRQVUHFRUGHGOHGWRIRUPXODWLRQRI laws and sometimes the theories were put forward which needed experimental proofs. Thus, validation of the laws, over time, comes through experiments. In a nutshell, experiments play a pivotal role in the progress of science and therefore learning about experimental methods is of paramount importance for science education as well as for pursuit of science research. While the knowledge about experimental methods is indispensable for researchers involved in experimental work, those interested in theoretical aspects of science also need to know what can be measured and with what accuracy and limitation. This necessity of the knowledge of experimental methods, for both students and researchers alike, formulates the prime motivation for our book. We are aware that a lot of literature exists on developments of new techniques, experiments and instruments. Even some specialized journals and books are published to emphasize the experimental techniques in science. The uniqueness of our book lies in the fact that it caters to certain unaddressed needs. This book has been an outcome of informal discussions with several undergraduate and postgraduate teachers from many universities as well as interactions with the students. Many of them have emphasized on the requirement of a single book, which would cover various methodologies of solid-state experimental techniques on geometrical and electronic structure, microscopy in different environmental or ambient conditions. The discussions also highlighted the need for a book helping the readers in understanding the sources of errors, how much improvement can be expected and how to handle the data after making experiments. Keeping all this in mind, the present book thus gives details required to be understood by the experimental methods of solid state regarding fundamental topics such as improving signal, photon and electron sources, detectors, electron energy analysers, statistical methods to handle the data, basics of low temperature physics and vacuum science. The book covers diffraction methods including photon, electron and neutron sources for bulk solids as ZHOODVWKLQ¿OPVDQGQDQRPDWHULDOV7KHERRNDOVRLQWURGXFHVWKHWHFKQLTXHV of electron spectroscopy like Photoelectron Spectroscopy (XPS, UPS), Auger Electron Spectroscopy (AES), Nuclear Magnetic Resonance (NMR), Electron Spin Resonance (ESR) and Mössbauer spectroscopy. Along with these,

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Foundations of Experimental Physics

various optical spectroscopies like UV-Vis-NIR, Fourier Transform InfraRed (FTIR) Spectroscopy, Raman Spectroscopy are described in the book. Details of various optical, electron (SEM, FESEM,TEM) and probe microscopy techniques (STM, AFM, SNOM) and more are also a part of the book. It is believed that this century will be of interdisciplinary sciences. This book will therefore not only be useful to students of Physics but also to the students of Chemistry and to some extent to Biology students. The book is meant for masters as well as Ph.D. students of universities. It would also be useful to the undergraduate students as a reference book for their experimental projects. :HKRSHWKDWWKLVERRNZLOOEHQH¿WDODUJHQXPEHURIVWXGHQWVE\QRWRQO\ generating but also sustaining their interest in science. Shailaja Mahamuni Deepti Sidhaye Sulabha Kulkarni

About the Authors

Dr. Shailaja Mahamuni presently Adjunct Professor and formerly Professor at the Department of Physics, S. P. Pune University. She has taught Experimental Physics to post graduate students and her area of research interests include physics of nanophase materials and optical behaviour of quantum dots and she has published over 100 research papers. Dr. Deepti Sidhaye is currently working in the department of Physics, S. P. Pune University, Pune, India. Her professional work comprises of Science Education, Research and Outreach. A merit holder throughout her academic career and recipient of many awards, she completed Ph.D. from National Chemical Laboratory, India. Apart from her well-cited research work regarding nanomaterials, she is working on developing innovative methods of teaching Science. She is presently Joint Secretary, Indian Physics Association (Pune Chapter). Her Science Education related contributions include organization of activities like Summer school in Physics for undergraduates, delivering invited talks and writing science articles in newspapers, magazines. She has travelled to European countries for collaborative work and for academic events like Lindau Nobel Laureates meeting in Germany and Physics workshops in ICTP, Italy. She pursues Science Popularization by combining her keen interests in Science and Arts through creative writing and performances in different languages. Dr. Sulabha Kulkarni is presently an INSA, senior scientist at the Centre for Materials for Electronics Technology (CMET) Pune, India. Dr. Kulkarni has a long research and teaching career of over 45 years. Before joining CMET, Pune, she has worked at IISER, Pune, Banasthali University Rajasthan (as Pro-Vice Chancellor) and University of Pune. She has over 300 peer-reviewed research journal publications and has supervised 40 Ph.D. students. She has also authored six books. Dr. Kulkarni has widely travelled in Germany, France, U.K., Italy, Japan, Korea, China, Taiwan, Singapore and many other countries either for collaborations or conferences. Dr. Sulabha Kulkarni is a Fellow of Indian National Science Academy (Delhi), Indian National Academy of 6FLHQFH $OODKDEDG  ,QGLDQ$FDGHP\ RI 6FLHQFH %HQJDOXUX $VLD 3DFL¿F Materials Academy of Materials (APMAM) and Maharashtra Academy of Sciences.

Contents

Preface About the Authors

v

vii

1. Introduction

1

2. Improving Signal-to-Noise Ratio 2.1 Introduction 2.2 Noise in Electronic Devices 2.3 Hardware Devices for Noise Reduction 2.4 Software Methods to Improve Signal-to-Noise Ratio 2.5 Summary 3. Vacuum Science and Technology 3.1 Introduction   6LJQL¿FDQFHRI9DFXXPLQ([SHULPHQWDO3K\VLFV 3.3 Basic Laws and Terms in Vacuum Physics 3.4 Vacuum Pumps 3.5 Measurement of Vacuum: Vacuum Gauges 3.6 Vacuum Materials and Accessories 3.7 Ultra-High Vacuum (UHV) System 3.8 Summary 4. Photons and Electrons: Sources, Monochromators

and Detectors 4.1 Introduction 4.2 Photon Sources 4.3 Monochromators or Wavelength Selectors 4.4 Spectral Line Shape 4.5 Intensity of Spectral Lines 4.6 Optical Fibres 4.7 Photon Detectors 4.8 Electrons as an Incident Beam for Characterisation 4.9 Electron Sources or Electron Gun 4.10 Motion of Electrons in the Electrostatic Field 4.11 Electron Energy Analysers

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10

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Foundations of Experimental Physics

4.12 Electron Detectors 4.13 Summary

110

112

5. Nuclear Accelerators and Detectors 5.1 Introduction 5.2 Cockroft-Walton Accelerator/Generator 5.3 Van de Graaf Generator 5.4 Sloan and Lawrence Type Linear Accelerator 5.5 Cyclotron 5.6 Synchrotron 5.7 Particle Detectors 5.8 Neutron Detection 5.9 Summary

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6. Diffraction Methods 6.1 Introduction 6.2 X-Ray Diffraction 6.3 X-Ray Diffractometers 6.4 Electron Diffraction 6.5 Neutron Diffraction 6.6 Summary

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7. Microscopy 7.1 Introduction 7.2 Optical Microscopes 7.3 Field Microscopes 7.4 Electron Microscopes 7.5 Scanning Probe Microscopes 7.6 Summary

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212

8. Electron and Optical Spectroscopy 8.1 Introduction 8.2 Electron Spectroscopy Techniques for Solids 8.3 Optical Spectroscopy Techniques 8.4 Summary

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9. Magnetic Spectroscopy, Nuclear Spectroscopy

and Magnetisation Measurements 9.1 Introduction 9.2 Nuclear Magnetic Resonance Spectroscopy 9.3 Electron Spin Resonance (ESR) 9.4 Mössbauer Spectroscopy Technique 9.5 Vibrating Sample Magnetometer Technique 9.6 Superconducting Quantum Interference Device (SQUID) 9.7 Rutherford Backscattering Spectrometry

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Contents xi

9.8 Gamma Ray Spectroscopy 9.9 Summary

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282

10. Cryogenic Temperature Methods 10.1 Quest for Low Temperature 10.2 Thermoelectric Cooling 10.3 Cooling by Latent Heat of Evaporation   /LTXLG+HDQG6XSHUÀXLGLW\ 10.5 Dilution Refrigerator 10.6 Pomeranchuk Cooling 10.7 Adiabatic Demagnetisation 10.8 Laser Cooling 10.9 Low Temperature Measurements 10.10 Summary

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11. Error Analysis and Statistical Methods 11.1 Introduction 11.2 Quantitative Estimation of the Errors 11.3 Statistical Handling of Data 11.4 Distribution of Data and Its Properties 11.5 Principle of Maximum Likelihood 11.6 Fitting of Data 11.7 Covariance   &RHI¿FLHQWRI/LQHDU&RUUHODWLRQ 11.9 F2 Test for the Distribution 11.10 Summary

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 351

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Further Reading

355

Index

357

1

Introduction

Observations are the heart of every branch of science. The theories and laws are developed to explain such observations. Sometimes theories are proposed which need to be supported or proved by the experiments. Therefore, experiments play a central role in science. The advancement of science needs sophisticated experimental techniques, i.e. equipment, probes and detectors. The ability and reliability of experimental techniques are decided by the advancements in technology, which in turn depend upon the current knowledge of science. In general, science and technology grow hand-in-hand. Technology is also driven by the demands of science. To do good science, one is required to understand basic principles of experimental techniques, the advantages and limitations thereof. This book as the title suggests is about the experiments in solid-state physics, which is a vast branch of physics. It is not possible therefore to cover each and everything but intention is to introduce techniques essential to understand basic solid-state material properties. It should be remembered that solid-state physics is essentially about material properties, their internal structure, electrical, electronic, magnetic, optical, mechanical and other properties. It tries to explain the properties of matter by laying out the essential foundation of physics. This book spreads over 11 chapters. How the solid material properties predicted by theories can be experimentally observed is emphatically discussed. The experimental techniques of solid-state matter use different probes like electrons, photons or ions to reveal their properties. Figure 1.1 shows schematically a generalised scheme of an experiment. Essentially, the probes DUHVRPHSDUWLFOHVRU¿HOGVZKLFKLQWHUDFWZLWKPDWWHUJHWPRGL¿HGDQGJLYH RXWRUHYHQDEVRUEHG YDULRXVRXWSXWSDUWLFOHVRUSURGXFH¿HOGVZKLFKDUH then detected. The output is just in the form of scattered radiation. But, the LQWHUDFWLRQRISUREHZLWKPDWWHUDQGKRZLWJHWVPRGL¿HGLVWKHPRVWLPSRUWDQW as this output is essentially used as data to understand the properties of solid PDWWHU 7KH QDWXUH RI SUREH LQWHUDFWLRQV ZLWK PDWWHU DQG ¿QDOO\ GHWHFWLRQ decide a particular technique. Today, there is huge variety of techniques depending upon the permutations and combinations of all three. Additionally, the experimental conditions of matter, viz. temperature, vacuum etc. also need to be considered. These conditions determine the properties of matter and are discussed. Often the probe, material and detector

2

Foundations of Experimental Physics

Fig. 1.1: Schematics of what is essential in an experiment.

also need to be arranged in some special fashion to be more effective to collect the data. Probe, material under investigation and detector may need to be housed in a chamber of glass, steel or other solid. This forms a complete experimental set up. Before collecting the data, in order to come to the knowledge of properties of matter, it is important to plan the experiment. Given a choice one should also decide the required precision, and possible sources of errors. Not all the sources of errors can be eliminated but some can be avoided! One can draw proper conclusions based on certain statistical methods and are often used by the physicists. Most of the data is accompanied by the noise produced from various sources. Notably, careful statistical analysis of noise led to the discovery of new elementary particle! Moreover, the noise thermometry is also used to estimate low temperature of the matter. We therefore start with ‘improving signal-to-noise ratio’ in Chapter 2. In the modern experiments, physical quantities are converted in to the electrical signal. One and all HOHFWULFDO JDGJHWV DGG QRLVH WR WKH VLJQDO DQG WKH VRXUFHV RI QRLVH DUH ¿UVW discussed. The basic principles of measurements of signal-to-noise ratio and improving signal to noise ratio using hardware and software techniques are further discussed in this chapter. Noise measurements or estimating signal­ to-noise ratio thus gains primary importance. One can see in this chapter, use RI YDULRXV VLPSOH WHFKQLTXHV OLNH GLIIHUHQFH DPSOL¿HU WR ORFNLQDPSOL¿HU depending upon the sophistication level of an experiment. Some software methods also need to be accompanied by hardware used for noise reduction

Introduction 3

and is discussed in the chapter. In brief, most commonly used hardware and software devices in the noise reduction are elaborated in the chapter. A number of experiments in modern physics require vacuum. This is true, when electron, ion or atom beam is used as a probe or detected beam consists of either of these particles. Further to it, sometimes the instruments producing above particles as probes and detectors may require sub-atmospheric environment. Even for analysing the surfaces of solids or investigating the interactions of gases with clean solid surfaces, the solids may be required to be in vacuum. Chapter 3 gives details of vacuum science, pumps and gauges. The chapter begins by laying down the foundation of vacuum physics. This includes the discussion on necessity of vacuum in experimental techniques, listing historical developments pertaining to vacuum technology and FODVVL¿FDWLRQEDVHGRQGLIIHUHQWUDQJHVRIYDFXXP7KHQH[WSDUWRIWKHFKDSWHU is about how to create vacuum and measure it. Different vacuum pumps and vacuum gauges required to create and measure the vacuum respectively are discussed. Besides the basic principle of quadrupole mass spectrometer, which is useful in measuring the ultra-high vacuum and also useful in leak detection is discussed. The vacuum accessories and their materials are also discussed. The chapter ends with the discussion of a typical high vacuum set up and an ultra-high vacuum set up. In Chapter 4, the collectively referred to as probes or sources of photons and electrons along with their monochromators (an instrument capable of making or separating the probes as per their wavelength or energy) and detectors are discussed. Underlying principles behind different photon and electron sources and detectors are explained. The photon sources are most widely used in various techniques because of their traditional use as black body radiators. Just heat an object above some critical temperature and it will start ‘glowing’ or radiating light. For example, heat a tungsten wire, depending XSRQLWVWHPSHUDWXUHLWZLOOEHFRPH¿UVWUHGGLVKWKHQUHGDQGWKHQZKLWHOLJKW (photons with different wavelengths) would come out. However, this simple blackbody radiation as a photon source has many drawbacks. Developments in understanding and instrumentation have evolved many photon sources like discharge lamps, light emitting diodes, lasers, X-rays and synchrotron sources. Some sources like lasers may produce monochromatic light but often in many cases it becomes essential to make them monochromatic using prism or grating. The photons in a very wide range of wavelengths from far infra-red to X-rays can be detected using various instruments called ‘detectors’. Detectors detect the photons and produce electrical signal, which can be analysed and recorded as data. The detectors such as photodiodes, photomultipliers and charge coupled devices are discussed. This is followed E\VRXUFHVRIHOHFWURQVRUµHOHFWURQJXQV¶(OHFWURQVFDQEHHDVLO\GHÀHFWHG using a magnet. But, magnets are bulky and if electromagnets are used, they also need power. Therefore, electrons are energetically sorted out by means RIHOHFWURQHQHUJ\DQDO\VHUVOLNHUHWDUGLQJ¿HOGDQDO\VHUF\OLQGULFDOPLUURU

4

Foundations of Experimental Physics

DQDO\VHU KHPLVSKHULFDO DQDO\VHU DQG VHFWRU DQDO\VHU XVLQJ HOHFWULF ¿HOG Electrons can be detected by Faraday cup, channeltrons and channelpates, which are described here. 6FLHQWL¿F GHYHORSPHQWV DW WKH HQG RI WKH QLQHWHHQWK FHQWXU\ DQG WKH onset of twentieth century saw a rise in the understanding of atomic structure. Discovery of radioactivity and famous alpha particle experiment by Ernest Rutherford (1871-1937), who was a leading scientist of his time, triggered the curiosity of understanding the structure of an atom. Initial estimate of Ernest Rutherford to smash the atoms by energetic particle like proton was few MeV. This was a big task around 1920. But, theoretical calculations by Gurney and Gamow in 1928 predicted that atom like lithium should be smashed at about 500 keV, a much lower energy than predicted by Rutherford. However, experimentally achieving even 400-500 keV energy was a big task at that time. Particularly in Europe and America, this challenge of achieving high energy charged particles was undertaken by many scientists. Cockcraft and Walton succeeded soon in building the 500 kV generator and subsequently splitting lithium atoms. This marks the beginning of ‘Accelerators’, the machines capable of increasing the energy of charged particles from few keV to GeV. Initially, the accelerators were built by nuclear physicists. Idea was to thereby understand the structure of atoms and nuclei. But, soon solid-state physicists MRLQHG WKH UDFH DV WKH\ VDZ FRQVLGHUDEOH EHQH¿WV RI XQGHUVWDQGLQJ VROLGV using accelerators, like synchrotrons. Therefore, this book includes Chapter 5 on accelerators. It gives in the beginning a small introduction and then gives basic principles of Cokcraft-Waltot, Van de Graaf and linear accelerators. The scientists went on building bigger and bigger accelerators of higher and higher energy, occupying large space. The concept in accelerator design changed with the availability of high-power radio frequency generators and new compact designs of accelerators like cyclotron, microtron etc. became more successful. These too are discussed in this chapter. However, discussion of yet more advanced Large Hadron Collider (LHC), which is indeed the largest energy accelerator to date falls beyond the scope of this book. Growth RI QXFOHDU DQG SDUWLFOH SK\VLFV ÀRXULVKHG ZLWK WKH GHYHORSPHQW RI YDULRXV particle detectors. In 1927 ‘Wilson cloud chamber’, the simplest detector was designed by Wilson who received the Nobel Prize for the same discovery. Wilson Cloud chamber was further improved by Blacket. A further progress is made by bubble chamber, hydrogen bubble chamber and multi-wire proportional chamber. Each detector has its own importance and has played a crucial role in the nuclear physics. Basic principles of particle detectors are also briefed in Chapter 5, along with nuclear accelerators. Chapter 6 is about X-ray, electron and neutron diffraction methods HVVHQWLDO LQ DQDO\VLQJ WKH VWUXFWXUH RI EXON WKLQ ¿OPV DQG QDQRPDWHULDOV The origin of diffraction experiments stems from an early experiment by Max von Laue with his colleagues in 1912. Interestingly, their immediate experiment in 1912 was successful for which Laue received the Nobel Prize

Introduction 5

in 1914. The idea was that if X-rays were of short wavelength comparable to the distances between atoms in a solid (estimated to be few angstroms) then they should diffract like light gets diffracted by slits, well known light experiments. Indeed, with few trials, Laue and his colleagues obtained clear signature of diffraction. Inspired by this discovery, Braggs (father and son), both physicists, gave a simple expression which is known as ‘Bragg’s Law’. It forms the basic principle now in analysing the lattice constants of materials. It effectively meant that electrons and neutrons (discovered by James Chadwick in 1932) also should produce diffraction patterns at appropriate energies. These interesting and logical developments were accompanied by progress in the precision instruments and electronics. The chapter on diffraction is aimed DW PDNLQJ WKH UHDGHUV KDYH D ÀDYRXU RI WKHVH VFLHQWL¿F PHWKRGV 7KXV LW outlines the diffraction methodology and various X-ray, electron and neutron diffractometers. The advantages and limitations are discussed as appropriate. It is said that ‘seeing is believing’. Chapter 7 is about various microscopes. Microscopic techniques are not only important in physics for exploring morphology of solids, two-dimensional layers, but are also, indispensable in chemistry, biology and other sciences. Scattering of radiation is a basis of microscopy and gives information regarding the crystal structure, morphology and grain size. A large number of optical and electron microscopes are currently available. The optical microscopy is the oldest in this group. For a long time, it was believed that resolution of the optical microscope is restricted E\WKHZDYHOHQJWKRIOLJKWURXJKO\ȝP$EEH¶VFRQGLWLRQ (OHFWURQDQG ion microscopes as well as probe microscopes were developed, which allow even atomic resolution by changing the energy of the particles (and hence the de Broglie wavelength). Abbe’s diffraction limit was circumvented even by optical microscopes and optical nanoscope was built recently. The discovery attracted the Nobel Prize in 2014. The excitement of nanoscope is also shared in the present chapter. Realisation that solids are composed of atoms (or molecules in organic and bio materials) ignited the need to observe (or image) them using a microscope! The optical lenses and even instruments based on WKHPZHUHQRWVXI¿FLHQWWRPDJQLI\WKHPLFURVFRSLFGHWDLOV,Q(UQVW 5XVND DQG 0D[ .QROO GHYHORSHG ¿UVW HOHFWURQ PLFURVFRSH 0LFURVFRSHV GHSHQGLQJRQWKHLUSURELQJWRRORUUDGLDWLRQDUHLGHQWL¿HGDVRSWLFDOHOHFWURQ ¿HOG DQG SUREH PLFURVFRSHV 'HYHORSPHQWV LQ HDFK PLFURVFRS\ WHFKQLTXH due to the realisation of technological advancements in instrumentation and newly made discoveries led to the extent that it is possible with some of these microscopes such as Field Ion Microscope, Atom Probe Microscope, Scanning Tunnelling Microscope, Atomic Force Microscope and Scanning Near Field Optical Microscope to observe the images of small particles, atoms or obtain atomic resolution. The possibility of moving precisely the samples or probes with as high precision as sub-nanometer is the key to this success. But this is not the end of the story! The modern microscopes also include possibility of exciting the atoms with X-rays and produce spectra

6

Foundations of Experimental Physics

due to transitions of electrons in the excited atoms. This reveals the identity of atoms. The technique is known as Energy Dispersive Analysis of X-rays and is an integral part of most of the modern electron microscopes. Electron microscopes can also produce electron diffraction patterns. Interestingly, RSWLFDOPLFURVFRSHVDOVRFDQJLYHµ¿QJHUSULQWLQJ¶RUFKHPLFDOLGHQWLW\ZLWK ÀXRUHVFHQFH DQG 5DPDQ VSHFWURVFRSHV 2WKHU PRGHUQ RSWLFDO PLFURVFRSHV are confocal microscope and nanoscope or Stimulated Emission Depletion PLFURVFRSHZKLFKDUHQRWOHVVH[FLWLQJ%LRORJLVWV¿QGWKHPH[WUHPHO\XVHIXO and use them routinely now. The chapter on microscopy covers starting with WKHSULQFLSOHRIRSWLFDOOHQVUHVROXWLRQPDJQL¿FDWLRQPRVWRIWKHLPSRUWDQW microscopes principles and working. Chapter 8 is organised into two parts. Electron spectrometers came into use in late 1970s after the development of X-ray Photoelectron Spectrometer by Kai Siegbahn. He showed that using such an electron spectrometer it is SRVVLEOHWRQRWRQO\PDNHWKHFKHPLFDOLGHQWL¿FDWLRQEXWLGHQWLI\HOHFWURQLF structure of molecules, charge transfers and various excitations in solids. The instrumentation of the spectrometer is usually rather complex due to need of vacuum requirement of X-ray source, sample and detector, which is discussed in the chapter. As the intense, white X-rays (large band of wavelengths) became available at synchrotrons, many new experiments to VWXG\GLIIHUHQWDVSHFWVRIVROLGVWKLQ¿OPVZHUHLQLWLDWHG8VLQJ89VRXUFH instead of X-rays enables to reveal valence band structure of solids. In fact, observation of band structure, direct-indirect band gaps, density of states was possible due to photoelectron spectrometer. Another interesting technique known as Auger Electron Spectroscopy also is described in this chapter. It LV EDVHG RQ D SKHQRPHQRQ NQRZQ DV µ$XJHU (IIHFW¶ ¿UVW REVHUYHG E\ /L] Meitner in 1922 and independently by P. Auger in 1923, much earlier than X-ray Photoelectron Spectrometer. However, it came into practice almost at the same time as X-ray Photoelectron Spectrometer due to late development of vacuum techniques as well as electronics needed. It should be remembered here that X-ray (or UV) Photoelectron or Auger spectroscopy are surface techniques. Why is it so will be found in Chapter 8. The basic principles and schematic diagrams of the layouts of the spectrometers are also described. We discuss rather conventional techniques like UV-Vis-NIR, luminescence, FTIR and then Raman spectroscopy. These are able to reveal band structure, defects in solids etc. The simple UV-Vis-NIR spectrometer even throws light on some complex phenomena like plasmons, collective excitations of electrons in nanomaterials. Indeed, just by locating the spectral positions it is possible to know whether solids are spherical nanoparticles or are nanorods or have more FRPSOH[JHRPHWU\899LV1,5VSHFWURPHWHULVDOVRXVHGLQ¿QGLQJRXWEDQG JDSVLQVHPLFRQGXFWRUV7KHOXPLQHVFHQFHRUÀXRUHVFHQFHVSHFWURPHWHULVDQ important spectrometer of modern science as many research efforts are made in developing light emitting devices. The quenching of light emission enables to understand defects in solids. Fourier Transform Infra-Red and Raman

Introduction 7

spectrometer are in a way complementary to each other due to their selection UXOHV ZKLFK VKRZ ¿QJHUSULQWV RI FHUWDLQ PROHFXOHV 5DPDQ VSHFWURPHWHU is further used in the analysis of advanced materials like diamonds, carbon nanotubes, graphene and so on. In continuation with Chapter 8, Chapter 9 is also on spectroscopy. But, instead of the properties of materials based on the interactions with the outer electrons of an atom, it is about the interactions with the nucleus or investigations of magnetic properties. In this chapter, nuclear magnetic resonance (NMR), electron spin resonance spectroscopy (ESR), Mossbauer spectroscopy, Rutherford back scattering spectroscopy and gamma-ray spectroscopy are covered. The two non-spectroscopy techniques, viz. Vibrating Sample Magnetometry (VSM) and Superconducting Quantum Interference Device (SQUID) which are widely used in studying the magnetic properties of materials are also included here. Magnetic interaction of electrons and nuclei are useful to explore the structures of molecules and solids. Magnetic resonance spectroscopy is largely used to determine the structure and bonding in molecules. The nuclear magnetic resonance (NMR) was developed extremely fast as routine analysis technique due to wide applications as witnessed by award of six Nobel Prizes to the topic from 1943 WKURXJKLQSK\VLFVFKHPLVWU\DQGPHGLFLQH105LQPROHFXOHVZDV¿UVW observed and described by Rabi in 1938, for which he was awarded Nobel Prize in 1946. Applicability of NMR to liquids and solids is demonstrated by Bloch and Purcell, for which they received Nobel Prize in 1952. First, commercial NMR unit appeared in the market in 1952 itself (Varian). Electron Spin Resonance (ESR) technique is useful to detect free radicals and assess paramagnetic samples. The chapter contains information about a variety of facets of these techniques such as the principle, instrumentation, spectral analysis, applications etc. The techniques have a huge number of applications. Note that the well-known medical technique of magnetic resonance imaging (MRI) is an extension of NMR. After discussing these techniques, other important techniques of Mössbauer spectroscopy, Rutherford back scattering spectroscopy and gamma-ray spectroscopy are described. Each one of these KDV LWV RZQ VLJQL¿FDQFH Mössbauer spectroscopy is sometimes referred to as nuclear gamma resonance spectroscopy. It has applications in diverse ¿HOGVVXFKDVELRORJ\PHGLFLQHPHWDOOXUJ\PLQHUDORJ\HWF5XWKHUIRUGEDFN scattering is observed due to collisions between monoenergetic ions with the target atoms. The chapter also includes description of Vibrating Sample Magnetometer (VSM) technique and Superconducting Quantum Interference Device (SQUID) technique used for magnetisation measurements. As the name suggests, in Vibrating Sample Magnetometer, a sample is vibrated physically sinusoidally and magnetic momenta are measured as a function RI PDJQHWLF ¿HOG DQG WHPSHUDWXUH 'XH WR YLEUDWLRQV WKH PDJQHWLF ¿HOG RI the sample changes and according to Faraday’s law, concurrently generates HOHFWULF ¿HOG LQ WKH FRLO 648,' LV D PRUH VHQVLWLYH PDJQHWRPHWHU WKDQ

8

Foundations of Experimental Physics

VSM. ItFDQPHDVXUHVPDOOHUPDJQHWLF¿HOGVZLWKPRUHSUHFLVLRQ,WUHTXLUHV -RVHSKVRQMXQFWLRQVLQDVXSHUFRQGXFWLQJULQJ1XFOHDUUDGLDWLRQVSHFL¿FDOO\ gamma rays are primarily used in medicine (radiation therapy), astrophysics JDPPDUD\VSHFWURVFRS\ JHRSK\VLFVWRPDSURFNVDQG¿QGK\GURFDUERQV and mineral deposits in soil), besides in material science. Gamma ray spectroscopy is also outlined in chapter 9. 2EVHUYDWLRQV RI VXSHUFRQGXFWLYLW\ VXSHUÀXLGLW\ DQG VXSHUVROLG SKDVH VROLG ZKLFK ÀRZV OLNH ÀXLG ZLWKRXW YLVFRVLW\  DW ORZ WHPSHUDWXUH DUH milestones in condensed matter physics. Studies on condensed matter phase are extended at low temperature to explore phase transitions, magnetic properties, and superconducting behaviour. Low temperature methods are also required in the spectroscopic methods to attain higher resolution than at room temperature. The answers to questions such as, ‘How to achieve low temperature by the latent heat of evaporation, thermoelectric effect, dilution cooling, and demagnetisation?’ and ‘How to measure the low temperature?’ etc. are addressed in Chapter 10. The chapter starts with the interesting developments, viz. experiments of Kammerlingh Onnes, who was studying low temperature gases with a motto “knowledge through measurements”. He managed to liquefy helium gas in 1908, which enabled him to observe VXSHUFRQGXFWLYLW\LQZKLFKHOHFWURQVÀRZZLWKRXWUHVLVWDQFHLQPHUFXU\DW .LQWKH\HDU2QQHVZRQWKH1REHO3UL]HLQ6XSHUÀLXLGLW\ LQZKLFKDWRPVÀRZZLWKRXWUHVLVWDQFHZDVGHPRQVWUDWHGE\WKHWZRSDSHUV SXEOLVKHG E\ D $OOHQ DQG 0LVHQHU DQG E  .DSLW]D LQ  6XSHUÀXLGLW\ can be understood on the basis of Bose-Einstein condensation, the process in which the Bosons condense in the single quantum state. Low temperature is often essential to observe many quantum mechanical phenomena. This chapter not only describes how to achieve low temperature but also gives a discussion on low temperature measurements. Finally, the error analysis in any experiment like understanding the origin of noise and its minimisation (discussed in Chapter 2) is very important and forms the basis of Chapter 11. Similar to noise, errors cannot be completely eliminated from any experiment. Even if all others taken care off, the error DULVLQJGXHWRSUHFLVLRQOLPLWRIDQLQVWUXPHQWRU¿QDOO\GXHWRµXQFHUWDLQW\ principle’ cannot be eliminated. The uncertainty observed in the measurements LVUHTXLUHGWREHVSHFL¿HGH[SOLFLWO\LQVFLHQWL¿FUHSRUWV7KHDQVZHUWRWKH questions like: How to quantify error in the experimental data? How error propagates while evaluating the parameters from the data? How to present the GDWD"+RZWRFDUU\RXW¿WWLQJRIWKHGDWD"+RZWRMXVWLI\WKH¿WWLQJRIWKHGDWD" etc. and statistical methods of data handling are to be found in Chapter 11. To summarise, this book discusses various experimental techniques required for the understanding of solid matter, viz. structure, morphology, electronic structure as well as data analysis and methods to improve the data. ,WDOVRJLYHVVXI¿FLHQWGHWDLOVDERXWPDWHULDOVQHHGHGIRUWKHH[SHULPHQWVDQG techniques to create special environment for such an analysis.

2 Improving Signal-to-Noise Ratio

2.1 Introduction Noise, simply put, is any unwanted signal that interferes with the detection RIWKHGHVLUHGVLJQDO)RUH[DPSOHZKLOHOLVWHQLQJWRDVSHFL¿FVRXQGRWKHU EDFNJURXQGVRXQGVVXFKDVSHRSOHWDONLQJWUDI¿FHQYLURQPHQWDOVRXQGVHWF  act as distractors and therefore can be considered as acoustic noise. Someone walking in front of you while you are watching TV can be considered as YLVXDOQRLVH7KHFUDFNOLQJVRXQGWKDW\RXPD\KHDUZKLOHOLVWHQLQJWRUDGLR is caused by system noise. In this chapter, we will focus on the system noise, particularly that in an electrical system. ,QWRGD\¶VH[SHULPHQWVDSK\VLFDOTXDQWLW\LVFRQYHUWHGLQWRDQHOHFWULFDO VLJQDODQGLVTXDQWLWDWLYHO\PHDVXUHGLQWHUPVRIHLWKHUWKHHOHFWULFDOFXUUHQW RUWKHYROWDJH7KHVHPHDVXUHPHQWVDUHQRWIUHHRIQRLVHLQGXFHGHUURUV7KH PHDVXUHGYDOXHRIDTXDQWLW\s can be represented as  s = x + δx  where xLVWKHDFWXDOYDOXHRIWKHTXDQWLW\DQGį[ is the measurement error. :KHQ HYHU\ PHDVXUHPHQW LV FRQVLVWHQWO\ VPDOOHU RU ODUJHU WKDQ WKH DFWXDO YDOXH LW LV FRQVLGHUHG DV D µV\VWHPDWLF HUURU¶ RU DQ µRIIVHW¶ 1RLVHLQGXFHG HUURUVKRZHYHUWHQGWREHUDQGRPRUXQSUHGLFWDEOH)RUUHOLDEOHDQGDFFXUDWH HVWLPDWLRQ RI WKH DFWXDO YDOXH LW LV QHFHVVDU\ WR PLQLPLVH WKH LQÀXHQFH RI UDQGRPQRLVHLQGXFHGHUURUVLQWKHPHDVXUHGYDOXH$VLPSOHDQGWKHPRVW FRPPRQZD\WRUHGXFHWKHFRQWULEXWLRQRIUDQGRPQRLVHLVWRDYHUDJHPXOWLSOH measurements of the same quantity. The underlying assumption is that the DFWXDOYDOXHRIWKHTXDQWLW\UHPDLQVFRQVWDQWDFURVVPHDVXUHPHQWVZKLOHWKH UDQGRP QRLVH LV DYHUDJHG RXW 7KXV WKH HPSLULFDO HVWLPDWLRQ RI WKH DFWXDO YDOXHREWDLQHGIURPN measurements is 1 N  ∑ xi  N i=1 7KH VWDQGDUG GHYLDWLRQ LQ WKH PHDVXUHG YDOXH FDQ EH FRQVLGHUHG DV D measure of noise amplitude. S = μx =

δS = σ x =

1 N

N

∑ (xi − μ x ) i=1

2





10

Foundations of Experimental Physics

7KHVLJQDOWRQRLVHDPSOLWXGHUDWLRFDQWKHQEHGH¿QHGDVIROORZV S μx =   δS σ x ,QFUHDVLQJWKHYDOXHRINQXPEHURIPHDVXUHPHQWV ZRXOGLQFUHDVHWKH VLJQDOWRQRLVH UDWLR S/N  DV UDQGRP YDOXH RI QRLVH ZLOO EH QXOOL¿HG )RU WKHUHOLDELOLW\RIVLJQDOGHWHFWLRQWKHVLJQDOWRQRLVHUDWLRLVDPRUHUHOHYDQW SDUDPHWHUWKDQWKHVLJQDOOHYHO)RULQVWDQFHWDONLQJRQDEXV\URDGDOWKRXJK ORXGHUPD\EHOHVVGHWHFWDEOHWKDQZKLVSHULQJLQDQLQWHQVLYHFDUHXQLWLQD KRVSLWDO7KXV IRU HYHU\ H[SHULPHQW RQH KDV WR VHW D OLPLW DQG DQVZHU WKH TXHVWLRQµ+RZODUJHWKHVLJQDOWRQRLVHUDWLRVKRXOGEHWRFDOOLWDVLJQL¿FDQWO\ detectable signal?’ $ERYH FULWHULRQ RI VLJQDO DQG QRLVH OHYHO HVWLPDWLRQ LV YHU\ VLPSOLVWLF EDVHGRQVHYHUDODVVXPSWLRQV)RUH[DPSOHWKHQRLVHLQDQRQOLQHDUV\VWHP PD\EHPXOWLSOLFDWLYHLQVWHDGRIDGGLWLYHDVH[SUHVVHGLQHTXDWLRQ 7KH QRLVHFDQDOVREHFRUUHODWHGWRWKHVLJQDO,QVXFKFDVHVDYHUDJLQJPXOWLSOH PHDVXUHPHQWVPD\QRWDOZD\VLPSURYHWKHVLJQDOWRQRLVHUDWLR0RUHRYHULW may not be always possible to measure the physical quantity multiple times. For example, in a communication system, the signal is transmitted only once, thus JLYLQJWKHUHFHLYHURQO\RQHRSSRUWXQLW\WRGHWHFWWKHVLJQDOWKDWLVFRUUXSWHG by the transmission channel noise. In such situations, a priori knowledge of channel and noise characteristics, such as spectrum or bandwidth, can be used WRLPSURYHWKHVLJQDOWRQRLVHUDWLR ,Q WKH PRVW VLWXDWLRQV LPSURYLQJ WKH VLJQDOWRQRLVH UDWLR EHFRPHV LPSHUDWLYHDVLWOLPLWVWKHSHUIRUPDQFHRIWKHPHDVXULQJV\VWHP7KHNH\WR reduce noise is to understand the sources of noise and treat for each source separately. Variety of software and hardware solutions can be employed for WKLV SXUSRVH ,Q WKLV FKDSWHU ZH ZLOO WU\ WR XQGHUVWDQG YDULRXV WHFKQLTXHV to tackle a weak signal. Basic principle of each technique is elaborated, and XVDJH JXLGHOLQHV DUH JLYHQ ,Q SUDFWLFH RQH QHHGV WR FULWLFDOO\ HYDOXDWH LWV DGYDQWDJHVDQGOLPLWDWLRQVLQRUGHUWRDSSO\RQHRUPRUHRIWKHVHWHFKQLTXHV +HUH VLJQDO LV GH¿QHG DV D SDUW WKDW LV WR EH FRQVLGHUHG ZKLOH QRLVH LV XQZDQWHGVLJQDOZKLFKLQHYLWDEO\DSSHDUVDQGFDQQRWEHDYRLGHG It may be interesting to note that the measurement of noise can be used to estimate temperature of the system in cryogenic regime. r=

2.2 Noise in Electronic Devices $OOPRGHUQPHDVXUHPHQWV\VWHPVKDYHHOHFWURQLFFRPSRQHQWV$OOHOHFWURQLF components produce noise, as its origin is thermodynamical and quantum PHFKDQLFDO ,Q PRVW RI WKH H[SHULPHQWV DEVROXWH YDOXH RI QRLVH UHPDLQV constant. The accuracy and precision of the measurements become problematic when signal intensity is about the same as that of the noise. Therefore, better nomenclature would be signal (S WRQRLVHN UDWLRLIS/N becomes less than

,PSURYLQJ6LJQDOWR1RLVH5DWLR 11

 VLJQDO JHWV HQLJPDWLF ,I RQH UHSHDWV WKH H[SHULPHQW VHYHUDO WLPHV WKH VLJQDOWRQRLVHUDWLRLV S average signal amplitude x = =  N average noise amplitude σ



where x  LV DYHUDJH RI VLJQDO DPSOLWXGH DQG VWDQGDUG GHYLDWLRQ 1 σ= ∑ (xi − x)2 . N i $QRWKHUZD\WRGH¿QHWKHVLJQDOWRQRLVHUDWLRLVWRFRQVLGHUTXRWLHQWRI signal power to noise power and it is indicated in decibel. V2 S signal power = = 10 log10 s2  Vn N noise power



where VsLVPDJQLWXGHRIVLJQDOYROWDJHDQGVnLVPDJQLWXGHRIQRLVHYROWDJH &RPSDULVRQ RI VLJQDOV KDYLQJ GLIIHUHQW PDJQLWXGH RI YROWDJH LV WKHQ feasible. Signal is usually associated with the noise, which has a different origin. If the noise is uncorrelated, resultant noise can be obtained by simply adding WKHVTXDUHVRIUHVSHFWLYHFRPSRQHQWVV1, V2, V3«VD\ $GGLQJWRWDOYROWDJHV WRJHWKHUJLYHV 2 Vtotal = V12 + V22 + "

Therefore, Vtotal = V12 + V22 + " 



2QWKHRWKHUKDQGWZRFRUUHODWHGQRLVHYROWDJHVDUHDGGHGDVIROORZV Vtotal = V12 + V22 + 2γV1V2 ZLWKFRUUHODWLRQFRHI¿FLHQWJ



The real problem is to understand the correlation, which is not a simple task. Noise originating from the same source could be correlated, whereas if origin is from different sources, they may not be correlated. DC signal or low frequency signal has to compete with substantial low IUHTXHQF\QRLVH,QSULQFLSOHQRLVHFDQEHFODVVL¿HGDV WKHUPDORU-RKQVRQ QRLVH VKRWQRLVH ÀLFNHUQRLVHDQG HQYLURQPHQWDOQRLVH+RZHYHU LQSUDFWLFHRQHFDQQRWFOHDUO\GLVWLQJXLVKQRLVHLQWRYDULRXVFDWHJRULHV

2.2.1 Johnson Noise 7KHUPDO RU -RKQVRQ QRLVH DSSHDUV HYHQ LQ DEVHQFH RI FXUUHQW (OHFWURQLF FRPSRQHQWV DUH LQ FRQWDFW ZLWK WKH HQYLURQPHQW DW ¿QLWH WHPSHUDWXUH 7KH SUREDELOLW\RI¿QGLQJWKHQXPEHURIHOHFWURQVLQDJLYHQVSDFHYDULHVUDQGRPO\ VLPLODUWRWKH%URZQLDQPRWLRQRIWKHPROHFXOHVLQDSHUIHFWJDV 6XFKD UDQGRPPRWLRQRIHOHFWURQVJLYHVULVHWRWKHWKHUPDOQRLVH,Q-RKQVRQ IRXQGWKDWQRQSHULRGLFSRWHQWLDOH[LVWVLQHYHU\FRQGXFWRUDQGPDJQLWXGHRI

12

Foundations of Experimental Physics

the potential is temperature dependent. Nyquist subsequently described the noise mathematically. Thermal noise as well as shot noise is called as white QRLVHDVLWGRHVQRWGHSHQGRQWKHDEVROXWHYDOXHRIWKHIUHTXHQF\ Equipartition theorem of statistical mechanics applies to the electrons. $V\VWHPKDVDQDYHUDJHHQHUJ\ kBT (kB is a Boltzmann constant and T is 2 WHPSHUDWXUHLQ.HOYLQ IRUHDFKTXDGUDWLFWHUPDSSHDULQJLQLWV+DPLOWRQLDQ (DFKGHJUHHRIIUHHGRPKDVÀXFWXDWLRQVZLWKDQDYHUDJHHQHUJ\ kBT . For 2 instance, resistors and diodes produce noise due to the random thermal ÀXFWXDWLRQV DQG UDQGRP WUDQVPLVVLRQ HYHQWV UHVSHFWLYHO\ ,Q SDUWLFXODU HOHFWURQVFDQWUDQVPLWRUUHÀHFWUDQGRPO\ ,QFDVHRIHOHFWURPDJQHWLFZDYHHDFKPRGHFDQEHWUHDWHGDVRQHGHJUHH of freedom per electromagnetic mode. 1 1 ε0 E 2 + B 2 = kBT  2 2 μ0



ZKHUHİ0DQGȝ0DUHSHUPLWWLYLW\DQGSHUPHDELOLW\RIIUHHVSDFHDQGE and B are PDJQLWXGHVRIHOHFWULF¿HOGDQGPDJQHWLF¿HOGYHFWRUVRIWKHHOHFWURPDJQHWLF ZDYH 5RRWPHDQVTXDUHHPIQRLVH HOHFWURPDJQHWLFPRGHGHQVLW\LQFHUWDLQ EDQGZLGWK 2 emf noise en = 4kBTR Δ bandwidth 



where RLVDYDOXHRIUHVLVWDQFH

7KHUPDO SRZHU LQ D FHUWDLQ IUHTXHQF\ EDQGZLGWK ǻf is P ( f )Δf = kBT Δf . Consider a resistance R HOHFWURQV MLWWHU DURXQG WKH UHVLVWDQFH VLPLODU WR WKH %URZQLDQ PRWLRQ RI PROHFXOHV GXH WR DYHUDJH WKHUPDO HQHUJ\ 7KH UDQGRP PRWLRQ RI HOHFWURQV JHQHUDWHV QRLVH HYHQ WKRXJK QHW FXUUHQW DQG KHQFH YROWDJH GURS  DW DQ\ LQVWDQW LV ]HUR DFURVV WKH UHVLVWDQFH 1RQ]HUR YDOXHRIYDULDQFHOHDGVWR-RKQVRQQRLVH δV = V 2 .

2

2QDQDYHUDJHWKHQRLVHSRZHULQWKHUHVLVWDQFHLV P = V , which is also R DYHUDJHWKHUPDOHQHUJ\SHUHOHFWURQGXULQJPHDVXUHPHQWWLPHt. V 2 kB T = R time +RZHYHU IUHTXHQF\ UDQJH ǻf GXULQJ ZKLFK VLJQDO LV PHDVXUHG P=

1 = 2Δf . time δ 2V = 2kBTRΔf

,PSURYLQJ6LJQDOWR1RLVH5DWLR 13

0RUHDFFXUDWHFDOFXODWLRQVOHDGWRWKHPDJQLWXGHRIQRLVHDVIROORZV δ 2V = 4kBTRΔf 



$PSOLWXGH RI -RKQVRQ QRLVH YROWDJH DW DQ\ SDUWLFXODU PRPHQW LV XQSUHGLFWDEOH EXW YROWDJH PDJQLWXGH REH\V WKH *DXVVLDQ EHKDYLRXU 7R PLQLPLVH WKHUPDO QRLVH VHQVLWLYH GHWHFWRUV DUH RIWHQ FRROHG WR ORZ WHPSHUDWXUH$QRWKHURSWLRQLVWRQDUURZGRZQWKHIUHTXHQF\EDQGZLGWKRI WKHGHWHFWRU+RZHYHUWKHWLPHWDNHQIRUUHFRUGLQJWKHPHDVXUHPHQWVPD\ prolong in this case, and may not be always acceptable.

2.2.2 Shot Noise 6KRW QRLVH LV GXH WR WKH VWDWLVWLFDO ÀXFWXDWLRQV &RQVLGHU DQ H[SHULPHQW RI PHDVXULQJȕSDUWLFOHVIURPDUDGLRDFWLYHVRXUFH1XPEHURIȕSDUWLFOHVUHDFKLQJ WKHGHWHFWRUGXULQJVVD\LVQRWFRQVWDQW7KHÀXFWXDWLRQLQVWDWLVWLFVLVD shot noise. Shot noise is also encountered when electrons or other charged SDUWLFOHVFURVVDMXQFWLRQDQGLWHVVHQWLDOO\DULVHVGXHWRWKHSDUWLFXODWHQDWXUH RI WKH HOHFWURQV ,Q PRVW FDVHV WKH MXQFWLRQ LVpn MXQFWLRQ /HW XV DVVXPH WKDWFXUUHQWWKURXJKWKHUHVLVWRUFRPSULVHVRIVWHDGLO\PRYLQJLQGHSHQGHQW XQFRUUHODWHGHOHFWURQV(OHFWURQVFDQEHUDQGRPO\UHÀHFWHGRUWUDQVPLWWHGDQG the randomness in its motion generates the noise with magnitude, in2 in2 = 2q I EDQGZLGWK



where q is the charge, in is the instantaneous current, and I is the current ÀRZLQJWKURXJKpnMXQFWLRQ &RQVLGHUDQRWKHUH[DPSOHRIODVHUOLJKW/DVLQJDFWLRQLVUDQGRP,IWKH ODVHUVSRWLVEULJKWHQRXJKVOLJKWÀXFWXDWLRQVLQQXPEHURISKRWRQVHPLWWHG by the laser per unit time per unit solid angle remain more or less constant. 7KHUDQGRPYDULDWLRQLVZHDNHIIHFWLQWKLVFDVH2QWKHRWKHUKDQGLIWKHODVHU LVKDYLQJDZHDNLQWHQVLW\VPDOOÀXFWXDWLRQVLQSKRWRQHPLVVLRQUDWHZRXOG PDWWHU 6XEVHTXHQWO\ WKH ODVHU LQWHQVLW\ ZRXOG UHYHDO FKDQJHV 6LPLODUO\ IRUODUJHUFXUUHQWVVPDOOÀXFWXDWLRQVLQWKHQXPEHURIHOHFWURQVSDVVLQJSHU VHFRQGSHUXQLWWLPHPD\QRWH[KLELWDQ\HIIHFW+RZHYHULIRQHLVGHDOLQJ ZLWKVPDOOFXUUHQWDQGIRUYHU\VPDOOGXUDWLRQRIWLPHVXEWOHFKDQJHVLQWKH HOHFWURQÀRZUDWHGXHWRUDQGRPUHÀHFWLRQVZRXOGPDNHDVXEVWDQWLDOSK\VLFDO GLIIHUHQFH7KLVUDQGRPYDULDWLRQLVWHUPHGDVWKHVKRWQRLVH ,W FDQ EH SDUWLDOO\ VXSSUHVVHG LI WUDQVPLVVLRQ HYHQWV DUH FRUUHODWHG DQG QRW FRPSOHWHO\ UDQGRP ,W LV GXH WR WKH FKDUJH WUDQVIHU VD\ IURP SVLGH WR QVLGHRIWKHMXQFWLRQ7KHPDJQLWXGHRIWKHVKRWQRLVHLVPXFKVPDOOHUWKDQ the thermal noise. Shot noise also depends on the frequency bandwidth for the measurements. 6KRWQRLVHLVGLVWLQFWIURPYROWDJHDQGFXUUHQWÀXFWXDWLRQVLWRFFXUVHYHQ ZLWKRXWDQ\DSSOLHG'&YROWDJHRUFXUUHQWÀRZLQJ6LQFHWKHVKRWQRLVHLV WKHÀXFWXDWLRQLQWKHÀRZRIXQFRUUHODWHGHOHFWURQVLWLVJLYHQE\D3RLVVRQ distribution.

14

Foundations of Experimental Physics

7KHDYHUDJHYDOXHRIFXUUHQWQXPEHURI1HOHFWURQVZLWKFKDUJHeÀRZLQJLQ eN time T LVJLYHQE\ I = T 2 I noise = 2e I (Df )



6KRW QRLVH VKRXOG EH GLVWLQJXLVKHG IURP WKH -RKQVRQ QRLVH ZKLFK DSSHDUV XQGHU WKH HTXLOLEULXP FRQGLWLRQV HYHQ LQ WKH DEVHQFH RI FXUUHQW 7KHVKRWQRLVHUHIHUVWRWKHDYHUDJHÀXFWXDWLRQVLQWKHFXUUHQWDQGLQFUHDVHV with the magnitude of the current. It can only be minimised by narrowing the frequency bandwidth. This is an important noise contribution not only in the fundamental physics, but also in electronics and telecommunication.

2.2.3 Flicker Noise Flicker noise or 1/f noise is not due to the fundamental properties of the matter. Its origin is not well understood. It is also called as contact noise in context of the detectors and excess noise when referred to the resistors. Flicker noise is DIUHTXHQF\GHSHQGHQWQRLVHDQGFDQEHJLYHQDVIROORZV KI 2   f where K depends on the resistor material and geometry. I is current through the circuit and f is the frequency. )OLFNHU QRLVH LV PDLQO\ REVHUYHG IRU ORZ IUHTXHQF\ VLJQDOV IURP '& OHYHO WKDW LV  IUHTXHQF\ WR DERXW  +]  $YRLGLQJ ORZ IUHTXHQFLHV IRU PHDVXUHPHQWVLVWKHEHVWUHPHG\)OLFNHUQRLVHLQVHQVLWLYHDPSOL¿HUV\VWHPV LVFDOOHGDVGULIW$VGH¿QHGDERYHLWGHSHQGVRQWKHIUHTXHQF\DQGKHQFHLV called as 1/f noise. Vav =

2.2.4 Environmental Noise 7UDQVIHURIHQHUJ\IURPVXUURXQGLQJWRWKHPHDVXUHPHQWV\VWHPLVGH¿QHGDV WKHHQYLURQPHQWDOQRLVH+RZHYHURQHPD\FRPHDFURVVQRWRQO\WHUUHVWULDO but also extraterrestrial noise. Each instrument acts like an antenna, and stray SLFNXSDFWVDVDQLQWHUIHULQJQRLVH,WW\SLFDOO\RFFXUVDWVSHFL¿FIUHTXHQF\RU narrow frequency bandwidth depending on the source. 0RVW FRPPRQ VRXUFHV RI DQ HQYLURQPHQWDO QRLVH DUH WKH HOHFWULF DQG PDJQHWLF¿HOGVSURGXFHGE\WUDQVPLVVLRQOLQHVDQGLWVKDUPRQLFVWHOHYLVLRQ VWDWLRQV PRWRUV DUFEDVHG GHYLFHV FKDQJHV LQ LRQRVSKHUH DQG OLJKWHQLQJ 2WKHU VRXUFHV OLNH UHÀHFWHG UDGLDQW HQHUJ\ PHFKDQLFDO YLEUDWLRQ DQG WKH electrical interaction between different instruments also contribute to the HQYLURQPHQWDO QRLVH 5HGXFWLRQ RI HQYLURQPHQWDO QRLVH UHTXLUHV SURSHU VKLHOGLQJ RI WKH FLUFXLWV DQG ZLUHV *URXQGLQJ RI WKH LQVWUXPHQWV LV DOVR LPSRUWDQWWRUHGXFHWKHHQYLURQPHQWDOQRLVH)RUWUDQVPLVVLRQRIWKHVLJQDO XVHRIIUHTXHQF\GLIIHUHQWIURPHQYLURQPHQWDOQRLVHLQFDVHRIWUDQVPLVVLRQ OLQH+] LVQHFHVVDU\

,PSURYLQJ6LJQDOWR1RLVH5DWLR 15

6RIWZDUH DV ZHOO DV KDUGZDUH GHYLFHV DUH UHTXLUHG WR GLVUHJDUG QRLVH +DUGZDUH GHYLFHV DUH HPEHGGHG LQ WKH LQVWUXPHQW XVHG IRU PHDVXUHPHQWV DQGUHMHFWQRLVHDWWKHLQLWLDOOHYHODQGKHQFHSUHYHQWLWIURPEHLQJDPSOL¿HG at later stages. :KLWHQRLVHUHGXFWLRQFDQEHDFKLHYHGE\FDUHIXOO\PDWFKLQJLPSHGDQFH and reducing temperature and bandwidth. If experimentally feasible, reduce FDSDFLWLYHLQWHUIHUHQFHDQGUHGXFHWKHUHVLVWDQFHLQWKHFLUFXLW,QFDVHRIYHU\ low source impedance, one can use impedance matching transformer that LPSURYHV WKH VLJQDO ,PSHGDQFH PDWFKLQJ LV DOVR KHOSIXO IRU WKH DPSOL¿HU which works at a higher input resistance and hence its noise factor reduces. 3DUWLFXODUO\DWKLJKHUIUHTXHQFLHVLQRUGHUWRDYRLGWKHUHÀHFWLRQLPSHGDQFH PDWFKLQJLVHVVHQWLDO%DQGZLGWKFDQDOVREHUHGXFHG$OWHUQDWLYHO\RQHFDQ DOVRDYHUDJHWKHVLJQDOPDQ\WLPHV Shot noise can be reduced by minimising the current through the circuit. :KHQLQGLYLGXDOHOHFWURQVRUSKRWRQVDUHPHDVXUHGPXOWLFKDQQHOVFDOLQJRU JDWHGLQWHJUDWLRQLVDGYDQWDJHRXV$PSOL¿HULQWURGXFHVWKHÀLFNHURUf noise, which can be reduced by changing the frequency at which the measurements are being carried out.

2.3 Hardware Devices for Noise Reduction ,PSURYLQJ WKH VLJQDOWRQRLVH UDWLR LV FUXFLDO ZKHQ KLJK VHQVLWLYLW\ DQG DFFXUDF\ LV UHTXLUHG +DUGZDUH QRLVH UHGXFWLRQ SURFHVVHV LQYROYH DGGLWLRQ RI VKLHOGV ¿OWHUV V\QFKURQRXV GHWHFWRUV SULRU WR WKH DPSOL¿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spurious signal, the main signal remains unaffected. Such a shielded cable has WREHFRXSOHGWR%D\RQHW1HLO&RQFHOPDQ%1& FRQQHFWRU+LJKO\VHQVLWLYH GHYLFHVDUHSURWHFWHGE\PHWDOOLFZLUHKRXVLQJNQRZQDV)DUDGD\FDJH3URSHU VKLHOGLQJRIWHQUHTXLUHVVHYHUDOWULDOH[SHULPHQWV ,PSHGDQFH PDWFKLQJ )LJ   LV UHTXLUHG WR DYRLG VLJQDO UHÀHFWLRQ LQ RUGHUWRSUHVHUYHWKHVLJQDOLQWHJULW\+LJKHUWKURXJKSXWRIWKHGDWDOHDGVWR WKHUHGXFWLRQLQQRLVHLQFRUSRUDWLRQLQWKHVSHFWUD$WHFKQLTXHRILPSHGDQFH matching is adopted to transfer maximum signal and minimise the losses so DVWRPDLQWDLQVLJQDOOHYHODWKLJKYDOXH7KHHOHFWULFFLUFXLWVDQGGHYLFHVDUH used to establish the condition in which the impedance of a load is equal to the internal impedance of the source. This condition of impedance matching leads to transfer of the maximum power from the source to the load. The

16

Foundations of Experimental Physics

maximum power transfer theorem of electric network theory states that at DQ\ JLYHQ IUHTXHQF\ WKH PD[LPXP SRZHU LV WUDQVIHUUHG IURP WKH VRXUFH WR WKHORDGZKHQWKHORDGLPSHGDQFHLVHTXDOWRWKHFRQMXJDWHRIWKHJHQHUDWRU LPSHGDQFH:KHQWKHVHFRQGLWLRQVDUHVDWLV¿HGWKHSRZHULVGHOLYHUHGZLWK HI¿FLHQF\WKDWLVDVPXFKSRZHULVGLVVLSDWHGLQWKHLQWHUQDOLPSHGDQFH RIWKHJHQHUDWRUDVLVGHOLYHUHGWRWKHORDG,QJHQHUDOWKHORDGLPSHGDQFHZLOO QRWEHWKHSURSHUYDOXHIRUPD[LPXPSRZHUWUDQVIHU$QHWZRUNFRPSRVHGRI inductors and capacitors may be inserted between the load and the generator WKDW LV FRQMXJDWH RI WKH JHQHUDWRU LPSHGDQFH VHH )LJ   ,PSHGDQFH PDWFKLQJ PD[LPLVHV WKH SRZHU WUDQVIHU DQG PLQLPLVHV WKH UHÀHFWLRQ IURP the load. If a source has complex impedance ZSi* and load impedance is Z/R, then * maximum power transfer takes place when ZSi = Z Lo . 0LQLPXPUHÀHFWLRQLVREWDLQHGZKHQZSi Z/R.

Fig. 2.1: Typical impedance matching circuit.

$QRWKHUFRPPRQZD\WRUHMHFWQRLVHLVWRXVHRIGLIIHUHQFHDPSOL¿HUV ,QWKHLQLWLDOVWDJHVRIWKHFLUFXLWYROWDJHIURPWUDQVGXFHUFDQEHIHGWRWKH GLIIHUHQFH DPSOL¿HU 7KH GLIIHUHQFH DPSOL¿HU DPSOL¿HV GLIIHUHQFH EHWZHHQ WKHWZRLQSXWVWKHUHE\UHMHFWLQJQRLVHJHQHUDWHG (OHFWURQLF¿OWHULQJXVLQJSDVVLYHFRPSRQHQWV)LJ LVDOVRFRPPRQO\ HPSOR\HGWRUHMHFWQRLVH7KHUPDOQRLVHDQGVKRWQRLVHSUHVHQWLQ'&VLJQDO FDQ EH HIIHFWLYHO\ UHPRYHG E\ XVLQJ ORZ SDVV ¿OWHUV 2Q WKH FRQWUDU\ ORZ IUHTXHQF\ ÀLFNHU QRLVH FDQ EH UHMHFWHG E\ WKH KLJK SDVV ¿OWHU:KLWH QRLVH FDQEHUHGXFHGE\FKRRVLQJDQDUURZIUHTXHQF\EDQGZLGWK(QYLURQPHQWDO QRLVHFDQEHUHGXFHGE\FKRRVLQJDSURSHUIUHTXHQF\%DQGSDVV¿OWHUFDQEH GHVLJQHGXVLQJWKHRSHUDWLRQDODPSOL¿HUDVZHOO +LJKSDVV¿OWHUDOORZVVLJQDOWRWUDQVPLWRQO\LQKLJKIUHTXHQF\UHJLPH /RZSDVV¿OWHUOHWVWKHVLJQDOSDVVRQO\DWORZHUIUHTXHQFLHV $FWLYH RU EDQG SDVV ¿OWHUV DOORZ WR SDVV VLJQDO RQO\ LQ SDUWLFXODU IUHTXHQF\EDQG. 7KH ¿OWHUV FRXOG EH VLPSOH /&5 ,QGXFWDQFH / &DSDFLWDQFH & DQG 5HVLVWDQFH 5  FLUFXLWV RU PD\ FRPSULVH RI RSHUDWLRQDO DPSOL¿HUV 7KH

,PSURYLQJ6LJQDOWR1RLVH5DWLR 17

output is recorded across a capacitor or a resistor. One can choose a signal in a particular frequency regime by appropriate selection of inductance, FDSDFLWDQFHDQGUHVLVWDQFHYDOXHV7KHHQYLURQPHQWDODQGÀLFNHUQRLVHFDQEH HDVLO\UHMHFWHGE\FKRRVLQJDIUHTXHQF\RIRXWSXWVLJQDODZD\IURPHOHFWULF SRZHUOLQHIUHTXHQF\RU+] DQGE\FKRRVLQJDVLJQDODWKLJKHURXWSXW frequency. 7KHEDVLFSULQFLSOHRISDVVLYH¿OWHUVFRPSULVLQJRIDGGLWLRQDOFDSDFLWRU CLQWKHUHVLVWLYHFLUFXLW LVVLPSO\EDVHGRQSKDVHFKDQJHVWKDWRFFXULQWKH YROWDJH DFURVV WKH FDSDFLWRU ZKLFK ODJV EHKLQG WKDW RI FXUUHQW DQG KHQFH YROWDJHDFURVVWKHUHVLVWRU ,QFDSDFLWLYHFLUFXLWV2KPVODZWDNHVWKHIRUPVp IpXC (with Vp as DSHDNYROWDJHDQGIpDVDSHDNFXUUHQW ZKHUHWKHFDSDFLWLYHUHDFWDQFHDW 1 DQJXODUIUHTXHQF\ȦLV X C = , which is frequency dependent. ωC $W ORZ IUHTXHQF\ RU ]HUR IUHTXHQF\ XC EHFRPHV YHU\ ODUJH DQG incorporation of capacitor acts as open circuit condition. On the other hand, at KLJKIUHTXHQF\UHDFWDQFHEHFRPHVYHU\VPDOODQGDOPRVWDFWVWRVKRUWFLUFXLW condition. Parallel analogy for inductor can be found in standard textbooks. The reactance offered by the inductor LLVDOVRIUHTXHQF\GHSHQGHQWDQGLVJLYHQ by X/ ȦL. 'LIIHUHQWFRQ¿JXUDWLRQVIRUIUHTXHQF\UHJLRQRILQWHUHVWDUHGHSLFWHGLQ Fig. 2.2.

 'LIIHUHQFH$PSOL¿HU In quite a few experiments, it is required to measure a signal with respect to a UHIHUHQFHVLJQDORUWRRYHUFRPHWKHGLIIHUHQFHEHWZHHQJURXQGRIWKHYROWDJH VRXUFHDQGWKHDPSOL¿HU,QVXFKFDVHVWKHQRLVHJHQHUDWHGLQWKHWUDQVGXFHU FLUFXLW FDQ EH PLQLPLVHG E\ XVLQJ WKH GLIIHUHQFH DPSOL¿HU SULRU WR WKH ¿UVW DPSOL¿FDWLRQVWDJH6LJQDODQGUHIHUHQFHDUHFRQQHFWHGWRWKHQRQLQYHUWLQJ DQGLQYHUWLQJWHUPLQDORI2S$PS)LJ 7KHGLIIHUHQFHEHWZHHQWKHWZR VLJQDOVLVDPSOL¿HG1RLVHLVWKXVUHMHFWHGGHSHQGLQJRQWKHFRPPRQPRGH UHMHFWLRQUDWLR&055  7KHGLIIHUHQFHEHWZHHQWZRLQSXWVLJQDOVLVDPSOL¿HGDQ\VLJQDOFRPPRQ WRERWKLQSXWVZRXOGEHUHMHFWHG1RWHWKDWERWKWKHLQSXWUHVLVWRUVDUHLGHQWLFDO DQGIHHGEDFNDVZHOODVUHVLVWRUEHWZHHQQRQLQYHUWLQJWHUPLQDODQGFRPPRQ DUHDOVRLGHQWLFDO$SSOLFDWLRQRI2KP¶VODZJLYHV I1 =

V1 − v− R1

I2 =

v− − V0 R2

'XHWRWKHKLJKLPSHGDQFHRIWKHRSHUDWLRQDODPSOL¿HUI1 I2

18

Foundations of Experimental Physics

Fig. 2.2:9DULRXVFRQ¿JXUDWLRQRIWKHHOHFWURQLF¿OWHUVXVHGIRUUHGXFLQJWKHQRLVH D ORZSDVV¿OWHUE KLJKSDVV¿OWHUDQGF EDQGSDVV¿OWHU

Fig. 2.3:7\SLFDOFLUFXLWXVHGLQWKH¿UVWVWDJHDPSOL¿HUWRUHMHFWQRLVH

Therefore, 6ROYLQJIRUY_, 

V1 − v− v− − V0 = R1 R2 Y_(R1 + R2  V1R1 + V0R1

Therefore,

v− =

V1 R2 + V0 R1 (R1 + R2 )

9ROWDJHGLYLGHUFRQVLGHUDWLRQ\LHOGV

,PSURYLQJ6LJQDOWR1RLVH5DWLR 19

v+ = V2

R2  (R1 + R2 )



7KHRSHUDWLRQDODPSOL¿HUZLWKQHJDWLYHIHHGEDFNORRSOHDGVWRY+ Y_ v− =

Therefore,

V1 R2 + V0 R1 R2 = v+ = V2  (R1 + R2 ) (R1 + R2 )



R2 (V2 − V1 ) = V2 − V1 , if R1 = R2 R1 2QHRIWKHLQSXWVVKRXOGEHUHIHUHQFHYROWDJHDQGDQRWKHUYROWDJHVKRXOG be the one for which measurements are undertaken. Difference between the WZRVLJQDOVLVDPSOL¿HG$Q\RWKHUYROWDJHDWWKHLQSXWWHUPLQDOVZRXOGQRW EHDPSOL¿HG 7KHFRPPRQPRGHUHMHFWLRQUDWLRLQWHUPVRIJDLQALVGH¿QHGDVIROORZV This implies,

V0 =

CMRR =

Adiffernece  Acommon



7KHKLJKHUFRPPRQPRGHUHMHFWLRQUDWLRH[FOXGHVFRPPRQPRGHQRLVH ZKLFKLVLQSKDVHDWLQYHUWLQJDQGQRQLQYHUWLQJWHUPLQDOV

2.3.2 Integrator Integration can reduce white noise but not 1/f noise or pink noise. The coherent RU QRQUDQGRP VLJQDO DGGV GLUHFWO\ ZLWK UHVSHFW WR WKH LQWHJUDWLRQ WLPH ZKHUHDVUDQGRPQRLVHDGGVDVWKHVTXDUHURRWRILQWHJUDWLRQWLPH+HQFH61 S S tSi LPSURYHVDVVTXDUHURRWRILQWHJUDWLRQWLPH$VDUHVXOW f = = t i Ni Nf t Ni 7\SLFDOFLUFXLWWKDWFDQEHXVHGDVLQWHJUDWRULVJLYHQLQ)LJ

Fig. 2.4:,QWHJUDWRUFLUFXLWWKDWFDQEHXVHGSULRUWRDPSOL¿HU

2XWSXW YROWDJH LV SURSRUWLRQDO WR LQWHJUDO RI WKH LQSXW YROWDJH RYHU WKH WLPHVSHFL¿HGE\µ5&¶WLPHFRQVWDQW7KHPHDQYDOXHRIQRLVHEHLQJ]HURLW ZRXOGEHUHMHFWHG:RUNLQJRIWKHFLUFXLWLVJLYHQLQVKRUWDVIROORZV /HW ii be the input current while if EH WKH FXUUHQW ÀRZLQJ WKURXJK WKH capacitor.

20

Foundations of Experimental Physics

When the capacitor C begins to charge, the current through the capacitor would be, if = − C +HQFH

dV0 V and ii = i dt R

dV0 Vi = dt R Vi Therefore, dt dV0 = − RC Integrating one obtains, −C

V0 = −

1 Vi dt  RC ∫



,QWHJUDWLRQLVFDUULHGRXWE\¿UVWRSHQLQJWKHVZLWFKDQGWKHQFORVLQJWKH switch.

2.3.3 Modulation When the signal from transducer is in the form of DC signal or if it is in low IUHTXHQF\UHJLPHTXLWHRIWHQLWLVSHUWXUEHGE\WKHÀLFNHUf QRLVHRUGULIW ,QVXFKDFDVHLWLVDGYDQWDJHRXVWRPRGXODWHWKHVLJQDOWRWKHKLJKIUHTXHQF\ UHJLPH7KHPRGXODWHGVLJQDOLVDOORZHGWRSDVVWKURXJKKLJKSDVV¿OWHUWR UHMHFWfQRLVH7KHQH[WVWHSLVWRGHPRGXODWHDQGUHFRYHUWKHVLJQDO  /RFNLQ$PSOL¿HU )LOWHUV DUH XVHIXO IRU UHMHFWLQJ QRLVH LQ D SDUWLFXODU IUHTXHQF\ UHJLPH 7KH ORFNLQDPSOL¿HULVEDVHGRQWKHSKDVHVHQVLWLYHGHWHFWLRQ36' ZKLFKFDQ EHXVHGWRLPSURYHWKHVLJQDOWRQRLVHUDWLRFRQVLGHUDEO\HYHQLIWKHVLJQDOLV buried completely in the noise. Thereby, it has many applications, particularly ZKHQVLJQDOLVYHU\ZHDN3KDVHVHQVLWLYHGHWHFWLRQLVHVVHQWLDOO\GHPRGXODWLRQ RIWKHVLJQDOKDYLQJH[DFWO\VDPHIUHTXHQF\DQGSKDVHRIWKHUHIHUHQFHZDYH $GYDQWDJHLVWKDWWKH36'UHMHFWVVLJQDODWDOORWKHUIUHTXHQFLHVDQGZKLFKDUH not in phase with the reference signal. $ ORFNLQDPSOL¿HU multiplies the input signal by the reference signal HLWKHU SURYLGHG IURP WKH LQWHUQDO oscillator RU DQ H[WHUQDO VRXUFH  DQG LQWHJUDWHVLWRYHUDVSHFL¿HGWLPHXVXDOO\RIWKHRUGHURIPLOOLVHFRQGVWRD few seconds. The resulting signal is DC signal, where the contribution from any signal that is not at the same frequency as the reference signal is attenuated essentially to zero. Similarly, RXWRISKDVHFRPSRQHQW of the signal that has the same frequency as the reference signal (because sine functions are orthogonal WRWKHFRVLQHIXQFWLRQVRIWKHVDPHIUHTXHQF\ LVQXOOL¿HGWKDWLVZK\DORFN LQDPSOL¿HULVDSKDVHVHQVLWLYHGHWHFWRU

,PSURYLQJ6LJQDOWR1RLVH5DWLR 21

)LUVW OHW XV GLVFXVV WKH PDWKHPDWLFDO IRXQGDWLRQ RI DQ LGHDO ORFNLQ DPSOL¿HU,QSXWVLJQDOLV$&/HWXVGHQRWHWKHLQSXWVLJQDOE\ I (t ) = Acos (ωt + φ) + n; ω = 2πf LVDQJXODUIUHTXHQF\



3KDVHLVJLYHQE\ȦWI and n represents noise at frequency f. Noise at DQ\RWKHUIUHTXHQF\FDQEHUHMHFWHGE\XVLQJEDQGSDVVHOHFWURQLF¿OWHU1RWH that signal detection means the measurement of its amplitude and phase. /HWXVDVVXPHWKDWWKHUHIHUHQFHVLJQDODWIUHTXHQF\f is multiplied with WKHLQSXWVLJQDO)LJ 5HIHUHQFHVLJQDOLVJLYHQE\ r (t ) = 2 cos (ωt ) and r '(t ) = 2sin (ωt ) 



Thereby, the output signal of the multiplier is O(t ) = I (t ) × r (t ) and O '(t ) = I (t ) × r '(t ) 



8VLQJWKHWULJRQRPHWULFLGHQWLWLHV 2 cos (x )cos (y ) = cos (x + y ) + cos (x − y ) 2sin (x)cos (y ) = sin (x + y ) + sin (x − y ) O(t ) = I (t ) × r (t ) = A cos (φ) + Acos (2ωt + φ) + 2n cos (ωt ) 



O '(t ) = I (t ) × r '(t ) = Asin (φ) + Asin (2ωt + φ) + 2n sin (ωt ) 



and

where A cos (I DQGA sin (I DUHWKHIUHTXHQF\LQGHSHQGHQWFRPSRQHQWV 2QHFDQXVHORZEDQGSDVV¿OWHUVRWKDWIUHTXHQF\GHSHQGHQWFRPSRQHQWV DUHUHMHFWHGDQGIUHTXHQF\LQGHSHQGHQWFRPSRQHQWVVXUYLYH /RFNLQDPSOL¿HU UHFRYHUV X = Acos (φ) and Y = A sin (φ) by knowing I(t IUHTXHQF\f and Ic(t DPSOLWXGHDQGSKDVHFDQEHGHWHUPLQHG ⎛Y⎞ A = ( X 2 + Y 2 )1 2 φ = arctan ⎜ ⎟  ⎝ X ⎠



$OWHUQDWLYHO\RQHFDQUHZULWH A exp (iφ) = X + i Y .

The multiplication and subtraction are carried out electronically.

9HU\ FRPPRQO\ ORFNLQDPSOL¿HU LV XVHG LQ HOHFWURQ LQGXFHG $XJHU electron spectroscopy as well as in low intensity light detection. Consider the example of light detection. For the sample with a high optical density, the WUDQVPLWWHGLQWHQVLW\LVYDQLVKLQJO\ORZZKHUHLWLVRILPPHQVHXVH /HWXVHODERUDWHWKHFRQFHSWRIXVLQJORFNLQDPSOL¿HUIRUOLJKWGHWHFWLRQ &RQVLGHU DQ H[SHULPHQWDO VHWXS ZKLFK LV XVHG IRU OLJKW GHWHFWLRQ KDYLQJ photodiode as the light sensor. In order to measure the weak signal, one LQFRUSRUDWHVFKRSSHUH[WHUQDOIUHTXHQF\JHQHUDWRU DQGWKHORFNLQDPSOL¿HU

22

Foundations of Experimental Physics

Fig. 2.5:%ORFNGLDJUDPLOOXVWUDWLQJZRUNLQJRIORFNLQDPSOL¿HU

Chopper allows light to pass for stipulated duration and blocks the light for remaining duration as illustrated in Fig. 2.6.

Fig. 2.6: Typical optical chopper used for modulating the light source.

Signal is modulated using optical chopper comprising of wheel with dark and transparent regions. It periodically interrupts the light. For example, FRQVLGHUDFKRSSHUUXQQLQJDW+])LJ 6LJQDODPSOL¿HULVIROORZHG E\DVZLWFKRSHUDWHGE\WKHZDYHGHULYHGIURPWKHFKRSSHU$VZLWFKLVFORVHG DQG FRQQHFWV WKH RXWSXW RI WKH DPSOL¿HU WR D ORZ SDVV 5& ¿OWHU ZKHQ WKH VLJQDOOHYHOIURPFKRSSHULVKLJKDQGYLFHYHUVD:KHQWKHFKRSSHUEORFNV the signal, the switch is opened or disconnects the circuit. In this case, the ORZSDVV5&¿OWHUDOORZVVLJQDOWRSDVVWKURXJKZKHQSKDVHRIWKHZDYHIRUP FRQWUROOLQJWKHVZLWFKLVVDPHDVWKDWRILQSXW$&VLJQDO:KHQWKHVZLWFK LVFORVHGWKHQRLVHSDVVHVWKURXJK5&¿OWHUZKLFKVPRRWKHQVWKHVDPHWR DYHUDJHRXWWKHPHDQYDOXHWREH]HUR,QHVVHQFHWKHGHYLFHLVDEDQGSDVV ¿OWHU1RWDEO\WKHHIIHFWLYHEDQGZLGWKDQGKHQFHWKHQRLVHUHMHFWLRQFDSDFLW\

,PSURYLQJ6LJQDOWR1RLVH5DWLR 23

LVJRYHUQHGE\WKH5&WLPHFRQVWDQWIRUWKHWXQHGDPSOL¿HUWKHTXDOLW\IDFWRU Q FHQWUDOIUHTXHQF\RI¿OWHUEDQGZLGWKQDUURZHULVWKHEDQGZLGWKJUHDWHU LVQRLVHUHMHFWLRQ7\SLFDOO\IRU5&WLPHFRQVWDQWT VEDQGZLGWKLV +] 7KHFHQWUDOIUHTXHQF\LVORFNHGE\WKHFKRSSHU¶VIUHTXHQF\,QRUGHUWR UHFRYHUFRPSOHWHVLJQDOWZRUHFWL¿HUVDUHXVHGLQSDUDOOHODQGDUHRSHUDWHGLQ SRVLWLYHH[FXUVLRQDQGQHJDWLYHH[FXUVLRQRIWKHZDYH7ZRUHFWL¿HUVWRWDNH DFFRXQWRISRVLWLYHDVZHOODVQHJDWLYHH[FXUVLRQDUHQRWVKRZQLQWKH¿JXUH 7KHVLJQDOFDQQHYHUGULIWRXWVLGHWKHEDQGSDVVRIWKH¿OWHU1RLVHLVHTXDOO\ GLVWULEXWHGDERXWWKHPHDQYDOXHWKDWRQHREWDLQVE\GHPRGXODWLRQ,WGRHVQRW JLYHULVHWRD'&VLJQDO,QFUHDVLQJ5&WLPHFRQVWDQWUHGXFHVLWVPDJQLWXGH ,QVWHDGRIDVLQJOHVZLWFKRQHFDQLQFRUSRUDWHWZRVZLWFKHVRQHFRQQHFWVWKH 5&FLUFXLWLQSRVLWLYHKDOIRIWKHZDYHZKLOHRWKHUDOORZVLWWRSDVVWKURXJK GXULQJQHJDWLYHH[FXUVLRQ

Fig. 2.7:6FKHPDWLFRYHUYLHZRIZRUNLQJRIORFNLQDPSOL¿HU

$GYDQWDJHRIXVLQJORFNLQDPSOL¿HURUµSKDVHVHQVLWLYHGHWHFWRU¶LQVWHDG RIUHFWL¿HUIROORZHGE\WKH¿OWHU LVLQWKLVFDVHWKHVLJQDOFDQEHVKLIWHGWR WKHKLJKHUIUHTXHQF\E\PRGXODWLRQ$VDUHVXOWORZIUHTXHQF\µf noise’ is HIIHFWLYHO\UHMHFWHG%DQGZLGWKEHLQJH[WUHPHO\QDUURZEHWWHUQRLVHUHMHFWLRQ LVDWWDLQHG,IRQHXVHVMXVWD¿OWHULWPD\GULIWRQWKHIUHTXHQF\VFDOHOHDGLQJ WRODUJHFKDQJHLQWKHRXWSXW0RUHRYHUWKHUHFWLI\LQJFLUFXLWDOVRUHVSRQGV WRQRLVHDQGLWUHFWL¿HVWKHQRLVHDORQJZLWKWKHVLJQDO2XWSXWZRXOGEHDQ undesirable combination of signal and noise!

2.3.5 Gated Integrator 3KDVHVHQVLWLYHGHWHFWRUVDUHVXSHULRULQZRUNLQJIRUWKHVWHDG\DQGVORZO\ YDU\LQJVLJQDOV)RUVZLIWO\YDU\LQJIXQFWLRQVV\QFKURQRXVRUJDWHGLQWHJUDWRU LVUHTXLUHG7KHJDWHGLQWHJUDWRULQWHJUDWHVDQDQDO\WLFDOVLJQDOLQDQDORJIRUP  RYHUD¿[HGWLPHZLQGRZ)LJ ,ILQSXWLVSXOVHGWKHLQWHJUDWRUZLQGRZ LVV\QFKURQLVHGZLWKWKHDQDO\WLFDOWULJJHU7KHPHWKRGLPSURYHVVLJQDOWR QRLVHUDWLRE\LQWHJUDWLQJRQO\ZKHQWKHVLJQDOLVSUHVHQW)LJ 7KHUHJLRQ LQZKLFKQRLVHLVSUHVHQWJDWHGLQWHJUDWRUGRHVQRWZRUNDQGKHQFHUHMHFWV QRLVHHI¿FLHQWO\%R[FDURUJDWHGLQWHJUDWRUZRUNVLQHLWKHUWKHVLQJOHSRLQW mode or the scan mode. In the single point mode, certain part of the transient

24

Foundations of Experimental Physics

UHSHWLWLYH VLJQDO LV VDPSOHG DQG VWRUHG 6DPSOLQJV DUH DYHUDJHG RXW ,Q WKH VFDQQHG PRGH VDPSOLQJ SRLQW LV PRYHG WR FRYHU WKH HQWLUH VLJQDO 5HWULHYLQJ WKH VLJQDO FROOHFWHG SRLQW E\ SRLQW ZRXOG \LHOG WKH LPSURYHG VLJQDOWRQRLVHUDWLR %R[FDUDYHUDJLQJUHIHUVWRDYHUDJLQJRIRXWSXWRIWKHJDWHGLQWHJUDWRU

Fig. 2.8:7\SLFDOFLUFXLWRIWKHJDWHGLQWHJUDWRUWRLPSURYHVLJQDOWRQRLVH ratio of analog signal.

5&ORZSDVV¿OWHULVJDWHGE\VZLWFKZKLFKFORVHVV\QFKURQRXVO\ZLWK WKHLQSXWVLJQDODQGDYHUDJHVWKHVLJQDOGXULQJWKHJDWHGWLPH

Fig. 2.9: 2SHUDWLRQRIER[FDUDYHUDJHUWKHDYHUDJLQJWDNHVSODFHRQO\ZKHQ

WKHJDWHLVRSHQHG*DWHZLGWKDVZHOODVWULJJHUGHOD\FDQEHDGMXVWHGDVSHU

the requirements.

2.4 Software Methods to Improve Signal-to-Noise Ratio Software methods or digital data processors are simple personal computers or special digital processors to perform numerical calculations on the data. 6RIWZDUHPHWKRGVDUHPRUHÀH[LEOHWKDQKDUGZDUHPHWKRGV)XUWKHUFKDQJH

,PSURYLQJ6LJQDOWR1RLVH5DWLR 25

of computer algorithms is much easier and less tedious than altering hardware PHWKRGVWRLPSURYHWKHVLJQDOWRQRLVHUDWLR,IWKHGDWDLVLQDQDORJXHIRUP WKHQLWQHHGVWREHGLJLWLVHGXVLQJDQDORJXHWRGLJLWDO$'& FRQYHUWHU$VD UHVXOWVLJQDOLVUHSUHVHQWHGE\VHTXHQFHRIQXPEHUVUDWKHUWKDQYROWDJHRU FXUUHQWHWF'LJLWDO¿OWHULQJFDQSHUIRUPDOOWKH¿OWHULQJIXQFWLRQVDVORZSDVV ¿OWHUKLJKSDVV¿OWHUDQGEDQGSDVV¿OWHU 2QHRIWKHH[DPSOHVLVJLYHQLQ)LJ

Fig. 2.10:8VHRIDGLJLWDO¿OWHUIRUVLJQDOSURFHVVLQJ

'LJLWDO¿OWHUVDUHHVVHQWLDOO\SURJUDPVVWRUHGLQWKHFRPSXWHU7KHVH¿OWHUV DUHYHU\VWDEOHDVDIXQFWLRQRIWLPHDQGWHPSHUDWXUHGRQRWGULIW &RPSDUHG WRDQDORJXH¿OWHUVWKHGLJLWDO¿OWHUVDUHDFFXUDWHYHUVDWLOHDQGFRPSDFW 'LJLWDO ¿OWHULQJ LQFOXGHV HQVHPEOH DYHUDJLQJ GDWD VPRRWKHQLQJ DQG )RXULHUWUDQVIRUP6PRRWKHQLQJRIWKHGDWDFDQLPSURYHWKHVLJQDOWRQRLVH UDWLR6LPLODUWRWKHER[FDUDYHUDJLQJFHUWDLQQXPEHURIFRQVHFXWLYHSRLQWV VD\¿YHRUVHYHQRUQLQHDUHDYHUDJHGDWDWLPHDQGZLOOEHQRWHGE\DVLQJOH point. The line connecting these points is rather smooth and has less noise.

2.4.1 Ensemble Averaging &RPSOHWHGDWDLVUHFRUGHGPXOWLSOHWLPHVDQGLWVDYHUDJHLVWDNHQWRUHSUHVHQW WKHPHDVXUHPHQWV)LJXUH ,IVDPHVHWRIPHDVXUHPHQWVLVUHSHDWHGn times then, ⎛S⎞ ⎛S⎞ ⎜⎝ ⎟⎠ = n ⎜⎝ ⎟⎠  N i N f



(QVHPEOHDYHUDJLQJLVWKHEHVWWHFKQLTXHLIWKHVLJQDOLVFKDQJLQJUDSLGO\ )RU VORZO\ YDU\LQJ VLJQDO FRPSOHPHQWDU\ WHFKQLTXH RI ER[FDU DYHUDJLQJ can be employed.

2.4.2 Box-Car Averager %R[FDU DYHUDJHU RU JDWHG GHWHFWRU DOORZV UHFRYHU\ RI SXOVHG QRLV\ VLJQDO often encountered in biological measurements and pulsed laser experiments, ZKLFK LV LQ GLJLWDO IRUP ,W DYHUDJHV RU LQWHJUDWHV GDWD GXULQJ SDUWLFXODU GXUDWLRQ RI WLPH FDOOHG DV JDWLQJ WLPH RU JDWLQJ GXUDWLRQ )LJ  $V LW does not accept signal in remaining off time, thereby, noise in that duration is

26

Foundations of Experimental Physics

Fig. 2.11:7\SLFDOHQVHPEOHDYHUDJHGVSHFWUD

HIIHFWLYHO\UHMHFWHG7KHJDWLQJWLPHRUJDWLQJGXUDWLRQFDQEHSUHGHWHUPLQHG E\ WULJJHULQJ VLJQDO 'LVWDQFH EHWZHHQWZR ZLQGRZV RII WLPH  LV DOVR SUH GHWHUPLQHG 0RUHRYHU WKH GDWD FROOHFWHG GXULQJ VQDSVKRW WLPH LV DYHUDJHG RXW7KHUHE\QRLVHSUHVHQWGXULQJWKHVKRUWJDWLQJWLPHLVDOVRDYHUDJHGRXW ,IǻT is gating time and TLVSHULRGWLPHODSVHEHWZHHQWZRZLQGRZV WKH LPSURYHPHQWLQWKHVLJQDOWRQRLVHUDWLRLVJLYHQE\ Signal T   = Noise ΔT /HWXVDVVXPHWKDWVLJQDOUHFRUGHGEHV. It comprises Vsignal and Vnoise/HWXVWDNHVsignalDVPHDQYDOXHRIWKHVLJQDO and VnoiseDVVWDQGDUGGHYLDWLRQRIQRLVHVLJQDO6LQFHQRLVHGXHWRGLIIHUHQW sources is uncorrelated, one can take 2 2 2 Vnoise = Vnoise,1 + Vnoise,2 +" 12

Therefore,

Vnoise

⎤ 1 ⎡ 2 = ⎢ ∑ Vnoise,i ⎥ N ⎣ i ⎦

,PSURYLQJ6LJQDOWR1RLVH5DWLR 27

0RUHRYHUDYHUDJHYDOXHRIVLJQDOLVVsignal =

⎤

1 ⎡

⎢ ∑ Vsignal ⎥  N ⎣ i ⎦



12

Therefore,

⎤ ⎤ 1 ⎡

1 ⎡ 2 V  N ⎢ ∑ Vsignal ⎥ + N ⎢ ∑ Vnoise,i ⎥ ⎣ i ⎦

⎣ i ⎦

= V av.signal +

Vsignal Vnoise

=

1

12

2 ⎡Vav.noise ⎤ ⎣ ⎦ N

Vsignal 12 1 ⎡ 2 ⎤ V av.noise ⎦ N ⎣

∝ N 



Fig. 2.12:,OOXVWUDWLRQRIER[FDUDYHUDJLQJIRUGLJLWDOVLJQDO$YHUDJLQJLV FDUULHGRXWLQDVPDOOLQWHUYDOFDOOHGDVER[

7KHER[FDUGHWHFWRUVFDQEHGHVLJQHGWRSURFHVVWKHVLJQDOVIURPFKRSSHG OLJKWEHDPGHWHFWHGE\SKRWRPXOWLSOLHURUSKRWRYROWDLFGHWHFWRUV7KHYDU\LQJ UHVSRQVHWLPHVRIDOWRJHWKHUGLIIHUHQWGHWHFWRUVLVDFFRPPRGDWHGDVµRQ¶WLPH DVZHOODVµRII¶WLPHFDQEHYDULHG&RQVHTXHQWO\LWLVH[WUHPHO\ÀH[LEOHHYHQ FRPSDUHGWRWKHORFNLQDPSOL¿HU

2.4.3 Fourier Transforms 8VHRI)RXULHUWUDQVIRUPVWRLPSURYHWKHVLJQDOWRQRLVHUDWLRLVYHU\FRPPRQ LQ VSHFWURVFRS\ :H ZLOO ¿UVW UHYLHZ WKH FRQFHSW RI )RXULHU WUDQVIRUP E\ VROYLQJIHZPDWKHPDWLFDOH[DPSOHV Example:)RXULHUWUDQVIRUPRIDQRQSHULRGLFSXOVHRUER[FDUIXQFWLRQ Π a (x ) = 0

if , − ∞ < x < − (a 2 )

=1

if , − (a 2 ) < x < (a 2 )

=0

if , (a 2 ) < x < ∞

28

Foundations of Experimental Physics

)RXULHUWUDQVIRUP +∞

Φ ( p) =

∫

Π a ( x) e2πipx dx

−∞ +a 2

=

∫ e

2πipx

dx

−a 2

=

1 ⎡eπipa − e−πipa ⎤ ⎦ 2πip ⎣

⎡sin πpa ⎤ =a⎢ ⎥ ⎣ πpa ⎦

= a sin c (πpa)

)XQFWLRQVȆa(x DQGĭp ZKLFKDUH)RXULHUWUDQVIRUPVRIHDFKRWKHUDUH depicted in Fig. 2.13.

Fig. 2.13:D $IXQFWLRQUHSUHVHQWLQJDSXOVHDQGE LWV)RXULHUWUDQVIRUP

Fourier analysis is a mathematical technique to add two or more sine and/ RUFRVLQHRVFLOODWLRQVWRSURGXFHWKHUHVXOWDQW25GHFRQYROXWHWKHUHVXOWDQW LQWRGLIIHUHQWVLQHDQGFRVLQHZDYHV $IXQFWLRQE(t SHULRGLFLQt with a period T, where 1/T f can be written as Fourier series expansion, E (t ) =

n=∞

∑

Cn e i 2πnft

n= −∞

with Cn =

1 T

to + T

∫

E (t ) e−i 2 π nft dt

t0

Cn are complex and t0 are arbitrary. 25 IRU D QRQSHULRGLF IXQFWLRQ E(t  DQG G(f  DUH FDOOHG DV )RXULHU pairs if

,PSURYLQJ6LJQDOWR1RLVH5DWLR 29

E (t ) = ∫ G ( f ) ei 2 πnf df G ( f ) = ∫ E (t ) e−i 2πnt dt /HW XV DSSO\ RXU PDWKHPDWLFDO XQGHUVWDQGLQJ RI )RXULHU WUDQVIRUP WR UHFRYHU WKH ZHDN VLJQDO IURP D QRLV\ EDFNJURXQG 1RWH WKDW WLPH GRPDLQ contains the same information as that of frequency domain. Consider the data LQWLPHGRPDLQDQGIUHTXHQF\GRPDLQDVIROORZV )LJXUHD VKRZVIUHTXHQFLHVRIWZRPRQRFKURPDWLFVRXUFHVf1 and f2. The radiant power P(f LVDOVRSORWWHGDVDIXQFWLRQRIIUHTXHQF\LQHDFK case. ,QWKHVHFRQGSDUW)LJE WKHLQVWDQWDQHRXVSRZHUDVDIXQFWLRQ of time is plotted from a source radiating two frequencies f1 and f2. The plot H[KLELWVWKHSHULRGLFLW\RUEHDWDVWKHWZRZDYHVJRLQDQGRXWRISKDVH7KH difference in frequency is approximately 10%.

Fig. 2.14: ([DPSOHRIDVSHFWUXPLQWLPHGRPDLQDQGLWV)RXULHUWUDQVIRUPD IRU VLQJOHIUHTXHQF\DQGE IRUWZRIUHTXHQFLHV

7KHWLPHGRPDLQVLJQDOLVGHULYHGIURPWKHIUHTXHQF\GRPDLQVLJQDOE\ WKHHTXDWLRQ P(t ) = k cos (2πf1t ) + k cos (2πf 2 t ) 



where k is a constant and t is the time. 7KHLQWHUFRQYHUVLRQRIWLPHDQGIUHTXHQF\LVFRPSOH[ZKHQPRUHWKDQ two frequencies are concerned. ,QWKH)RXULHUWUDQVIRUPVSHFWURVFRS\WLPHGRPDLQVSHFWUDDUHFRQYHUWHG LQWR ZDYHOHQJWK GRPDLQ XVLQJ WKH 0LFKHOVRQ LQWHUIHURPHWHU 7KH RSWLFDO frequency is high (of the order of 1012 to 1015+] 1RWUDQVGXFHULVDYDLODEOH WKDWPHDVXUHVWLPHYDULDWLRQRIWKHSRZHULQWKLVIUHTXHQF\UHJLPH$YDLODEOH

30

Foundations of Experimental Physics

GHWHFWRUVZRXOGUHFRUGDYHUDJHSRZHU7RREWDLQWLPHGRPDLQVLJQDOVWKHUHIRUH UHTXLUHV D PHWKRG RI FRQYHUWLQJ RU PRGXODWLQJ D KLJKIUHTXHQF\ VLJQDO WR one of measurable frequency without distorting the time relationship carried E\ WKH VLJQDO 'HSHQGLQJ RQ WKH ZDYHOHQJWK UHJLPH GLIIHUHQW PRGXODWLRQ techniques are required. +HUH ZH RXWOLQH PRGXODWLQJ RSWLFDO UDGLDWLRQ E\ WKH 0LFKHOVRQ LQWHUIHURPHWHU $ JUDSK FDOOHG DV FRVLQH FXUYH JLYLQJ LQWHQVLW\ IURP WKH 0LFKHOVRQLQWHUIHURPHWHUDVDIXQFWLRQRIGLVWDQFHWUDYHUVHGE\WKHPRYDEOH PLUURULVSORWWHG)UHTXHQF\RIWKHFRVLQHFXUYHLVVXEVWDQWLDOO\VPDOOHUWKDQ the optical frequency that can be used to change the domain from frequency to time. $VZHNQRZWKH0LFKHOVRQLQWHUIHURPHWHUVSOLWVEHDPLQWHQVLW\LQWZR KDOYHVRIQHDUO\HTXDOSRZHU6SHFWURPHWHUUHFRPELQHVWKHPLQVXFKDZD\ WKDWLQWHQVLW\YDULDWLRQRIWKHFRPELQHGEHDPFDQEHPHDVXUHGDVDIXQFWLRQ of difference in lengths of the paths of two beams. )LJXUHGHSLFWVWKHVFKHPDWLFOD\RXWRIWKH0LFKHOVRQLQWHUIHURPHWHU %HDPVSOLWWHUWUDQVPLWVKDOIRIWKHUDGLDWLRQDQGUHÀHFWVWKHRWKHUKDOI7KH UHVXOWLQJWZLQEHDPVDUHUHÀHFWHGIURPWKH¿[HGPLUURUDQGPRYDEOHPLUURU The beams again meet at the beam splitter and half of each beam is GLUHFWHG WRZDUGV WKH VDPSOH DQG GHWHFWRU +RUL]RQWDO PRWLRQ RI WKH PLUURU FDXVHV SRZHU RI WKH UDGLDWLRQ WKDW UHDFKHV WKH GHWHFWRU WR ÀXFWXDWH LQ D predictable manner. If the two mirrors are equidistant from the splitter, two beams recombine SUHFLVHO\LQSKDVHFDXVLQJFRQVWUXFWLYHLQWHUIHUHQFH0RWLRQRIWKHPLUURULQ HLWKHUGLUHFWLRQE\ȜFKDQJHVWKHSDWKOHQJWKRIWKHEHDPE\ȜOHDGLQJWR WKHGHVWUXFWLYHLQWHUIHUHQFH

Fig. 2.15:6FKHPDWLFGLDJUDPRID0LFKHOVRQLQWHUIHURPHWHUXVHGIRUUHFRUGLQJ Fourier transform.

,PSURYLQJ6LJQDOWR1RLVH5DWLR 31

7KH GLIIHUHQFH LQ SDWK OHQJWKV LQ WZR EHDPV LV FDOOHG UHWDUGDWLRQ į$ plot of output power P(t  DV D IXQFWLRQ RI į KDV FRVLQH IRUP DQG FDOOHG DV interferogram. The radiation striking the detector after passing through a 0LFKHOVRQLQWHUIHURPHWHUZLOOJHQHUDOO\EHPXFKORZLQIUHTXHQF\DFRXVWLF UDQJH  2QHF\FOHRIVLJQDORFFXUVZKHQPLUURUPRYHVE\GLVWDQFHȜ,IDPLUURU LVPRYLQJZLWKDFRQVWDQWYHORFLW\YmirrorDQGLIWLPHUHTXLUHGDPLUURUWRPRYH WKHGLVWDQFHȜLVJLYHQE\IJWKHQ vmirror τ = the frequency f =

λ  2

1 v mirror  = τ λ 2

 

7KHZDYHQXPEHU v of the radiation is f = 2vmirror ν . The relation between the optical frequency of radiation and the frequency c of the interferogram is obtained by λ = ν v   c Interferogram frequency is directly proportional to the optical frequency. 7KHFRVLQHZDYHRIWKHLQWHUIHURJUDPFDQEHGHVFULEHGE\ f = 2vmirror

1 P( ν) cos 2πf   2 $PSOLWXGHRISRZHURILQWHUIHURJUDPLQWLPHGRPDLQLVSURSRUWLRQDOWR the radiant power of the beam in frequency domain. One has to take into account the fact that beam splitter will not split WKHLQWHQVLW\H[DFWO\LQKDOIDQGWKDWWKHGHWHFWRUUHVSRQVHDQGWKHDPSOL¿HU EHKDYLRXUDUHIUHTXHQF\GHSHQGHQW Variable B( ν) takes care of these aspects and depends on P( ν)  P(δ ) =

P(δ ) = B( ν) cos 2πft P(δ ) = B( ν) cos 4πvm νt 7KHPLUURUYHORFLW\LQWHUPVRIUHWDUGDWLRQLV vmirror =

δ 2t

+HQFH P(δ ) = B( ν) cos 2πδν 



,Q WKH VHFRQG ¿JXUH WKH LQWHUIHURJUDP FDQ EH GHVFULEHG E\ WZR WHUPV ZLWKGLIIHUHQWIUHTXHQFLHV)LJ  P(δ ) = B1 ( ν) cos 2πδν1 + B2 ( ν) cos 2πδν2

32

Foundations of Experimental Physics

Fig. 2.16: Comparison of different interferograms. The right hand side spectra are JLYHQLQWHUPVRIZDYHQXPEHUDQGQRWIUHTXHQF\

For a continuum source, the interferogram can be represented as sum of LQ¿QLWHQXPEHURIFRVLQHWHUPV +∞

P (δ ) =

∫

B( ν) cos 2πδνd ν

−∞

The Fourier transform of this integral is +∞

B( ν) =

∫

P (δ ) cos 2πδνd δ 



−∞

$ FRPSOHWH )RXULHU WUDQVIRUP UHTXLUHV ERWK UHDO DQG LPDJLQDU\ components. Optical Fourier transform spectroscopy consists of recording Pį  DV D IXQFWLRQ RI į DQG WKHQ PDWKHPDWLFDOO\ WUDQVIHUV WKH VDPH LQ IUHTXHQF\ VSHFWUXP)LJ  ,QSUDFWLFHLQ¿QLWHYDOXHRIįFDQQRWEHVFDQQHG7KHRXWSXWLVVDPSOHG SHULRGLFDOO\DQGRQO\D¿QLWHVL]HRIVDPSOLQJFDQEHGLJLWLVHG7KHFRQVWUDLQW leads to limited resolution and restricting the frequency range.

2.4.4 Resolution The resolution of Fourier transform spectrometer can be described as Δ( ν) = ν1 − ν2

,PSURYLQJ6LJQDOWR1RLVH5DWLR 33

5HIHUWRWKHLQWHUIHURJUDPRIWZRIUHTXHQFLHVLQ)LJE RQHKDVWR record the interferogram from maximum “A” of retardation and maximum of “B” of retardation. The maximum B occurs only when δ ν2 is greater than δ ν1  δν2 − δν1 = 1 Δ( ν) = ν2 − ν1 =

1  δ



Intensity as a function of įLVPHDVXUHG7KHUHVROXWLRQLQZDYHQXPEHUV ZLOO LPSURYH LQ SURSRUWLRQ WR WKH UHFLSURFDO RI WKH GLVWDQFH WKDW WKH PLUURU WUDYHOV 8VXDOO\WZRVSHFWUDDUHUHFRUGHGRQHZLWKWKHVDPSOHSODFHGLQWKHSDWK RIWKHOLJKWDQGDQRWKHURQHZLWKRXWWKHVDPSOH7KHWZRVSHFWUDDUHGLYLGHG SRLQWE\SRLQWDQGWKHUHVXOWLV)RXULHUWUDQVIRUPHGWRREWDLQWKHIUHTXHQF\ GHSHQGHQWWUDQVPLVVLRQRIWKHVDPSOH7KHUHÀHFWLYLW\RIWKHVDPSOHFDQEH measured similarly. 5HVROXWLRQLVLQYHUVHO\SURSRUWLRQDOWRWKHSDWKGLIIHUHQFHEHWZHHQWZR RSWLFDOEHDPV Δ( ν) = ν2 − ν1 =

1 δ

7KH SRVLWLRQ RI WKH PRYLQJ PLUURU LV PHDVXUHG SUHFLVHO\ E\ FRXQWLQJ WKHLQWHUIHUHQFHIULQJHVE\+H1HODVHU6LQFHPLUURUSRVLWLRQLVNQRZQYHU\ accurately, the frequency can be measured accurately. 7KH FKDUDFWHULVWLFV RI )7,5 DUH KLJK WKURXJKSXW RI WKH VSHFWURPHWHU OHDGLQJWRWKHKLJKUDGLDQWSRZHUWRWKHGHWHFWRUDQGWKHUHE\LPSURYLQJWKH VLJQDOWRQRLVHUDWLR$WWKHVDPHWLPHWKHUHVROXWLRQLVKLJKDVVOLWVDUHQRW LQYROYHGDQGWKHUHVROYLQJSRZHUUHPDLQVFRQVWDQWWKURXJKRXWWKHVSHFWUXP $VGDWDDFTXLVLWLRQWLPHRIWKHVSHFWURPHWHULVOHVVWKHDYHUDJLQJGDWDQXPEHU RIWLPHVLVSRVVLEOHWRREWDLQVDPHVLJQDOWRQRLVHUDWLR

2.4.5 Advantages of Fourier Transform Spectrometry )RXULHU WUDQVIRUP LQVWUXPHQWV KDYH IHZ RSWLFDO FRPSRQHQWV DQG QR VOLWV WR attenuate the signal, thereby the throughput is high. This is known as high WKURXJKSXWRU-DTXLQRWDGYDQWDJH ([WUHPHO\KLJKUHVROYLQJSRZHUDQGZDYHOHQJWKUHSURGXFLELOLW\'O/O for the closest lines is about 6 ppm. $OO HOHPHQWV RI WKH VRXUFH UHDFK WKH GHWHFWRU VLPXOWDQHRXVO\ (QWLUH spectrum can be recorded in one second or so. Notable decrease in the time required to acquire data allows one to record the multiple scans and take HQVHPEOHDYHUDJHWRLPSURYHVLJQDOWRQRLVHUDWLR7KLVLVFDOOHGDVPXOWLSOH[ RU)HOOJHWWDGYDQWDJH.

34

Foundations of Experimental Physics

Notably, for the most optical instruments, the decrease in the resolution LVDFFRPSDQLHGE\DQLQFUHDVHLQWKHVOLWZLGWKWKDWGHFUHDVHVWKHVLJQDOWR noise ratio. ,W LV UDWKHU XQFRPPRQ WR XVH )RXULHU WUDQVIRUP WHFKQLTXHV LQ 899,6 region. In UV and VIS region, the noise is not the detector noise, but it is VKRWQRLVHDQGÀLFNHUQRLVHIURPWKHVRXUFH)RUÀLFNHUDQGVKRWQRLVHWKH magnitude of noise increases as the radiant power of the source increases. $YHUDJLQJLQFUHDVHVWKHVLJQDOWRQRLVHUDWLRIRUVWURQJSHDNVDQGGHJUDGHV LW IRU ZHDNHU SHDNV ,I WKH VRXUFH RI QRLVH LV ÀLFNHU QRLVH HQFRXQWHUHG LQ VRXUFHVWKHQRLVHLVQRWUHGXFHGE\DYHUDJLQJ)RUÀLFNHUQRLVHDULVLQJIURP the background radiation of the sources, S/N ratio degrades for all the peaks.

2.5 Summary 1RLVH LV LQHYLWDEOH LQ WKH H[SHULPHQWDO PHDVXUHPHQWV GXH WR LWV TXDQWXP mechanical origin. One requires data with reasonably good signal to noise ratio (S/NWREHDERXWLVWUHDWHGDVDQH[FHOOHQWYDOXH 9DULRXVW\SHVRIQRLVHFDQ be generated while making the measurements with the electronic gadgets such DV-RKQVRQRU7KHUPDOQRLVH6KRWQRLVH)OLFNHUQRLVHDQGHQYLURQPHQWDO noise. Thermal noise at temperature TLQDEVROXWH DQGIUHTXHQF\EDQGǻf is

dV = (4k BTR Df ) ZKLOHWKHVKRWQRLVHLV in2 = 2q I Df and the Flicker noise is Van = (KI 2 / f ) indicating that only Flicker noise is frequency GHSHQGHQW,PSURYLQJS/N is crucial, especially if the signal is poor. 6LPSOH KDUGZDUH GHYLFHV FDQ EH DWWDFKHG SULRU WR SUHDPSOLI\LQJ WKH VLJQDO OLNH HOHFWURQLF ¿OWHUV GLIIHUHQWLDO DPSOL¿HU LQWHJUDWRU HWF 7KH EDVLF SULQFLSOHV RI RSHUDWLRQ DUH RXWOLQHG LQ WKH SUHVHQW FKDSWHU 0RGXODWRU GHPRGXODWRUFDQDOVREHHPSOR\HGWRLPSURYHWKHVLJQDOWRQRLVHUDWLRWRJHW rid of the Flicker or 1/fQRLVH)XUWKHULPSURYHPHQWLVIHDVLEOHZLWKWKHDLGRI /RFNLQDPSOL¿HUZKLFKLVDSKDVHVHQVLWLYHGHWHFWRUZKLFKGHPRGXODWHVWKH VLJQDOKDYLQJH[DFWO\WKHVDPHIUHTXHQF\DQGSKDVHRIWKHUHIHUHQFHVLJQDO DQG KHQFH FDQ UHMHFW QRLVH DW DOO RWKHU IUHTXHQFLHV YHU\ HI¿FLHQWO\ *DWHG integrator is a complementary technique which can be used for the periodic VLJQDORUVZLIWO\YDU\LQJVLJQDOLQFRQWUDVWWRWKH/RFNLQDPSOL¿HUZKLFKLV XVHIXOIRUVWHDG\DQGVORZO\YDU\LQJVLJQDO 6RIWZDUH WHFKQLTXHV WR LPSURYH VLJQDO WR QRLVH UDWLR LQFOXGH HQVHPEOH DYHUDJLQJ IRU UDSLGO\ YDU\LQJ VLJQDO ER[FDU DYHUDJLQJ IRU VORZO\ YDU\LQJ VLJQDODQGXVHRI)RXULHUWUDQVIRUP)RXULHUWUDQVIRUPFDQEHXVHGIRULQIUD UHG VSHFWURVFRS\ 5DPDQ VSHFWURVFRS\ DQG QXFOHDU PDJQHWLF UHVRQDQFH spectroscopy. Use of Fourier transform yields higher throughput of the VSHFWURPHWHU WKDW OHDGV WR LPSURYHG S/N ratio. Concurrently, the resolution LVKLJKDVOLPLWLQJUHVROXWLRQE\¿QLWHZLGWKRIVOLWVLVQRWDQREVWDFOH7LPH UHTXLUHG IRU UHFRUGLQJ WKH GDWD EHLQJ YHU\ VPDOO RQH FDQ UHSHDW WKH GDWD QXPEHURIWLPHVDQGFDUU\RXWHQVHPEOHDYHUDJLQJRIWKHVLJQDOWRLPSURYH S/N ratio.

3

Vacuum Science and Technology

3.1 Introduction The word ‘vacuum’ originates from the Latin word ‘vacuus’ which means ‘vacant’ or ‘empty’. Simply put, it is a general term for a space with less particle density and subsequently lower pressure than that of atmosphere. Perfect vacuum or absolute vacuum will denote total emptiness. However, in reality, vacuum is a partially empty space. Degree of vacuum is inversely proportional to the density of particles. Higher the degree of vacuum, lesser the number of particles. In our daily life, creation of partial vacuum enables us to carry out various processes. First example that comes to mind is that of ‘vacuum cleaners’ which suck up dust by lowering the pressure above the object that is being cleaned. Other examples include very crucial processes like respiration to the FRPSDUDWLYHO\ OHVV VLJQL¿FDQW RQHV VXFK DV VXFNLQJ OLTXLG WKURXJK D VWUDZ 9DFXXP LV DQ LQWHJUDO SDUW RI GHYLFHV VXFK DV WKHUPRV ÀDVNV ,Q VFLHQWL¿F research, vacuum chambers are required to carry out certain experiments such as those utilizing beam of electrons as a probe. Modern electron accelerators, colliders, photoelectron spectrometers and electron microscopes operate in ultra-high vacuum conditions. For generating vacuum, a space is made partially empty by removing some of the gas molecules occupying it. Instruments used for creation of vacuum are commonly known as vacuum pumps while those used for measuring vacuum are called vacuum gauges. These form the main part of ‘vacuum technology’. Foundation for vacuum technology is ‘vacuum science’ and it is known to comprise of phenomena and laws governing generation and sustenance of vacuum. In order to understand terms pertaining to vacuum science, consider a vessel containing particles in a constant random motion. These particles collide with each other and with the walls of the container and exert a force on its surface. Number and intensity of particle collisions constitute ‘force’ and subsequently force per unit area is called ‘pressure’. Atmosphere, i.e. the blanket of gases surrounding the earth exerts a pressure. At sea level, magnitude of this pressure is taken as one atmosphere and is equivalent to 760 mm of mercury column height. Therefore, keeping this in mind, removal of air from the chamber to create pressure less than atmospheric pressure will be termed as creating vacuum. Pressure corresponding to vacuum will always be less than 1 atmosphere.

36

Foundations of Experimental Physics

History of the concept of vacuum goes back to the time of Greek philosophers. However, vacuum related experiments were performed much later namely in 17th century. While Galileo had initiated this work, his DVVRFLDWH 7RUULFHOOL SURGXFHG YDFXXP H[SHULPHQWDOO\ LQ  +H ¿OOHG D glass tube closed at one end with mercury and submerged its open end in a pool of mercury. He also showed that irrespective of dimensions, shape or degree of tilt of the tube, the mercury column was always 760 mm above WKHOHYHOLQWKHSRRO7KLVZDVWKH¿UVWTXDQWLWDWLYHHVWLPDWLRQRISUHVVXUHRI DWPRVSKHULFDLU7RUULFHOOLDOVRLQYHQWHGEDURPHWHU±WKHVFLHQWL¿FLQVWUXPHQW widely used for measuring atmospheric pressure and the unit ‘Torr’ is coined DIWHUKLVQDPH$PDMRUFRQWULEXWLRQWRWKH¿HOGRIYDFXXPSK\VLFVZDVWKHQ made by the scientist Blaise Pascal; the SI unit of pressure (Pa) is named after KLP 1H[W PLOHVWRQHV LQ WKH ¿HOG RI YDFXXP SK\VLFV ZHUH WKH LQYHQWLRQ RI manometer and construction of vacuum pumps. Otto von Guericke is credited for the modifying water pumps to air pumps. He is also remembered for his demonstration wherein he joined two copper bowls (Magdeburg hemispheres) together to form a hollow sphere, which, after removal of air through them, could not be pulled by horses until many horses were made to pull the same.

 6LJQL¿FDQFHRI9DFXXPLQ([SHULPHQWDO Physics In experimental physics, one needs to generate vacuum or reduce the pressure for various purposes. Some of them are listed below:   ,QWKLQ¿OPGHSRVLWLRQJDVHVIURPWKHFRQWDLQHUVKRXOGEHUHPRYHGE\ evacuation in order to avoid reactions such as oxidation. 2. In surface sensitive techniques such as photoelectron spectroscopy, vacuum is required to avoid interference of impurities in determination of structure and composition.   7RDYRLGR[LGDWLRQRI¿ODPHQWVLQODPSV 4. To decrease the thermal or electrical energy transfer.   7RUHPRYHDGVRUEHGJDVHVRUYRODWLOHÀXLGIURPWKHEXON 6. To increase the mean free path (distance between colliding particles) in accelerators, vacuum coating units etc. 7. To create clean surfaces for studies useful in semiconductor devices, integrated circuits, catalysis, corrosion etc. 8. To store hygroscopic and readily oxidisable materials. In general, many of the research branches require basic knowledge of vacuum technology. One needs to understand the advantages and limitations of each method to choose a proper technique. For example, instruments like microscopes cannot tolerate any vibrations and prefer silent pumps over PHFKDQLFDOSXPSV6LPLODUO\RQHDYRLGVSXPSVZKLFKXVHRUJDQLFÀXLGVLQ situations where ultra-clean vacuum is needed to avoid back-streaming and

Vacuum Science and Technology 37

contamination of the system. In this chapter, we will see how techniques based on various physical phenomena can be used to create and monitor vacuum. We will begin with the description of basic concepts in vacuum science.

3.3 Basic Laws and Terms in Vacuum Physics 8QGHUVWDQGLQJ RI NLQHWLF WKHRU\ RI JDVHV LV FUXFLDO IRU WKH ¿HOG RI YDFXXP science and technology. Let us look at some fundamental quantities entailed in it.  ‡ 3UHVVXUH is expressed as force exerted on a surface per unit area. Its SI unit is Pascal or N/m2, which is equal to 10 dyne/cm2 in cgs unit. Apart from these, pressure can be expressed in terms of different units in different contexts, e.g. the unit ‘bar’ is used in meteorology to report atmospheric pressures while ‘Torr (or mm Hg)’, a more convenient unit for low pressures, is used in high-vacuum physics and engineering. The conversion factors between various units of pressure are tabulated as follows: 7DEOH Conversion factors of pressure units

1 Pa

Pascal (Pa) or N/m2

Torr (Torr)

Atmosphere (atm)

Pound per square inch (psi)

Bar

1

7.56 × 10í

9.869 × 10í

1.45 × 10í

10í

1 Torr

1.33 × 102

1

1.316 × 10í

1.934 × 10í

1.33 × 10–3

1 atm

1.01 × 10

760

1

14.696

1.013

1 psi

6.894 × 10

51.715

1

6.894 × 10–2

1 bar

1 × 105

7.56 × 102

14.504

1

5 3

1 Pa = 1

N m

2

6.804 × 10

í

0.98692

= 9.8692 ×10−6 atm = 10−5 bar = 7.501 ×10−3 Torr

(3.1)

 ‡ 9ROXPHVolume of a container is its capacity and its SI unit is litre.  ‡ 7HPSHUDWXUH It is the degree of hotness or coldness of a substance but more importantly, it is the average kinetic energy of the particles contained in it. Its SI unit is Kelvin.  ‡ *UDP0ROHFXOHRU0ROH Quantity of gas (or any substance) having a mass equal to its molecular weight in to grams. A mole of a gas contains gas atoms equal to Avogadro’s number which is equal to 6.023 × 1023.  ‡ Gram molecular volume is volume occupied by a gram molecule of gas. %R\OH¶VODZgives the relationship between pressure P and volume V for D¿[HGDPRXQWRIJDVDWDFRQVWDQWWHPSHUDWXUH PV = C where C is constant, P is pressure and V is volume.

(3.2)

38

Foundations of Experimental Physics

 ‡ &KDUOHV¶V /DZ gives the relationship between volume (V) and temperature (T) for a given mass of gas at a constant pressure. V = constant T

(3.3)

 ‡ ,GHDO*DV/DZ PV = nRT

(3.4)

where n is number of moles (amount of substance) and R is gas constant. Pressure can also be written as (3.5) p = nkBT where kB is Boltzmann constant  ‡ 7KHUPDOFRQGXFWLYLW\ is given by K=

1 nmvλCv 3

(3.6)

where ȞLVDYHUDJHYHORFLW\ȜLVPHDQIUHHSDWKn is number of particles per unit volume and Cv is thermal conductivity  ‡ 'DOWRQ¶VODZRISDUWLDOSUHVVXUH Total pressure P exerted by a mixture of gases is equal to sum of their partial pressures (P1, P2, P3 etc.) where, partial pressure is the pressure exerted by one particular gas in the gas mixture.  ‡ $YRJDGUR¶VODZ Equal volume of all ideal gases measured at the same temperature and pressure contain 6.023 × 1023 molecules where 6.023 × 1023 is denoted as Avogadro’s number (NA).  ‡ /RVFKPLGWQXPEHUIt is the number of molecules per cubic centimeter of gas at 760 Torr and at 0°C. Its value is 2.637 × 1019 and can be obtained from Avogadro’s number.  ‡ 0HDQ IUHH SDWK Molecules in a gas undergo collisions with other molecules. The average distance travelled by a molecule between two such consecutive collisions is called mean free path and is given by the following mathematical expression. O=

1

1

2 πnδ 2

(3.7)



 ZKHUH į LV PROHFXODU GLDPHWHU DQG n is number of molecules per unit volume.  ‡ *DV)ORZDQG3XPSLQJ)ORZRIJDVFDQEHTXDQWL¿HGE\FRQVLGHULQJ the conductance determined by number of gas molecules passing through a cross section of a pipe. Let QEHWKHTXDQWLW\RIJDVÀRZLQJWKURXJKDSLSH Q= p

dV Torr l s-1 dt

(3.8)

Vacuum Science and Technology 39

To quantify, one uses Reynold number and Knudsen number.  ‡ 5H\QROGQXPEHU5HIt is a dimensionless number denoted by the ratio of inertial and viscous force. ρvD (3.9) η   ZKHUHȡLVGHQVLW\v is velocity of molecules, D is diameter of pipe and ȘLVYLVFRVLW\RIJDV   7KHYDOXHRI5H\QROGQXPEHUGHWHUPLQHVWKHW\SHRIJDVÀRZHJDYHU\ high (greater than 4000) value of Reynolds number is associated with WXUEXOHQWÀRZ  ‡ .QXGVHQQXPEHUKQ It is a dimensionless number denoted by the ratio of mean free path of molecules of a gas to the characteristic dimension of DYHVVHOWKURXJKZKLFKLWLVÀRZLQJ Re =

λ (3.10) D   ZKHUHȜLVPHDQIUHHSDWKDQGD is characteristic dimension, e.g. if the gas was enclosed in a cubic box, D will be length of the box.   .QXGVHQQXPEHULVXVHGIRUGHWHUPLQLQJZKHWKHUDÀXLGLVLQDFRQWLQXXP state. Kn < 0.1 is a continuum state. Kn > 1 characterises the molecular state.  ‡ 3XPSLQJVSHHGS It is the volume (V) or amount of a gas removed by the vacuum pump in time t. Kn =

V t Effective speed depends on conductance and is given by

S=

(3.11)

1 1 1

= + Se S C

(3.12)

where C is conductance and S is actual speed of the pump.  ‡ 7KURXJKSXWPumping speed (S) refers only to the volume of gas being pumped in a certain time, but throughput (Q) involves the pressure term as well. The formula for throughput is Q=

PV = PS t

(3.13)

Throughput can also be called ‘Gas load’. It can be used to determine the QHWPDVVRIJDVÀRZLQJDWSDUWLFXODUVHFWLRQRIDV\VWHP Pressure in the chamber is a dynamic balance between gas removal and gas injected in the system. In fact, Gas load is a result of several processes such as outgassing due to gases that desorb from the surfaces under vacuum

40

Foundations of Experimental Physics

in unit time, gases escaping through leakages, back streaming of gases due WR WKHUPDO ÀXFWXDWLRQV DQG LRQ LPSDFW HWF 6R KLJK YDFXXP LV HVWDEOLVKHG E\YDULRXVPROHFXODUÀRZSURFHVVHVZKLFKFRXQWHUHDFKRWKHU6FKHPDWLFRI gas load is shown in Fig. 3.1. Let us understand this quantitatively. For that purpose, let us make the following denotations: Gas compensated by outgassing from the wall: Qo, Permeation of gases: Qp, Leakage from microscopic cracks and pores: QL, Back streaming: QB, Effective speed: Se, Volume: V, Pressure: P Then, net change in the amount of gas is V

dp = − Se p + Q0 + QP + QL + QB dt

(3.14)

)LJ Schematic depicting gas load.

 ‡ 9DFXXP6\VWHPMajor components necessary to create vacuum are a chamber, piping, and one or more pump (or pumps) while the minor FRPSRQHQWVFRPSULVHRIYDOYHVÀDQJHVVHDOLQJVWUDSHWF7KHJDXJHV are used to measure the vacuum.  ‡ 3DUDPHWHUV RQ ZKLFK DFKLHYHPHQW RI YDFXXP GHSHQGV Ultimate vacuum achievable in a system depends on the pumping speed, gas desorption rate, gas penetration rate, gas leakage through the walls of the system and further through the mechanical joints, the inner surface ¿QLVKLQJ RI WKH V\VWHP PDWHULDOV XVHG WR SUHSDUH WKH YDFXXP V\VWHP and the procedure required to be followed to build the vacuum system.  ‡ 5DQJHRIYDFXXPRange of vacuum required differs depending on the parameters used. Following table shows salient ranges of vacuum.

Vacuum Science and Technology  7DEOH Range of vacuum Vacuum Low Medium

Expressed in torr 750 to 25 25 to 7.5×10-4

High

7.5×10-4 to 7.5×10-7

Very High

7.5×10-7 to 7.5×10-9

Ultra High

7.5×10-9 to 7.5×10-13

It may be noted that, generation of high vacuum requires appropriate design of system, vacuum pumps, and measuring gauges and also using appropriate materials for different components.

 9DFXXP3XPSV A vacuum pump is an equipment that removes molecules from a container RUDFRQ¿QHGVSDFHLQRUGHUWRFUHDWHYDFXXPLQLW9DFXXPSXPSVRSHUDWH using different mechanisms as follows (i) Molecules from the concerned container are displaced or transferred by repetitive mechanical action. These are then compressed for creating pressure gradient leading to expulsion of the molecules through the outlet via synchronized action of valves. (ii) A high­ VSHHGÀXLGMHWRUPRYLQJVROLGVXUIDFHJXLGHVWKHPRWLRQRIJDVPROHFXOHV to the outlet of the container. (iii) Binding the molecules chemically or via processes like adsorption, condensation so that they are entrapped. Hence, WKH YDFXXP SXPSV FDQ EH EURDGO\ FODVVL¿HG RQ WKH EDVLV RI WKHLU ZRUNLQJ mechanism as gas transfer type and gas binding or entrapment pumps. Following bar diagram summarizes the capability of different pumps. Type and number of vacuum pumps to be used is dictated by the nature of the applications and the degree of vacuum required for them. Applications such as vacuum molding, freeze drying and semiconductor manufacturing need increasing degrees of vacuum. Ultimate vacuum will depend on the design and construction of a vacuum system and gas load. Some of the most commonly used vacuum pumps are described in this section.

 5RWDU\9DQH3XPS This type of pump creates vacuum by mechanical displacement of air and belongs to positive displacement subtype of gas transfer vacuum pumps. These pumps have coordinated action of valves and moving parts. Initially, a volume of gas/molecules from the container to be evacuated is drawn into one side of the pump and is sealed off from the vessel. It is then compressed mechanically so that its pressure rises above atmosphere. Once this happens, the gas gets ejected out using a pressure-relief valve. Ratio of pressures corresponding to discharge and intake is called the compression ratio. In short, a small volume of molecules is transferred from the container to the atmosphere. This process



Foundations of Experimental Physics

continues cyclically. Every cycle results in further lowering of the pressure in the container. Schematic of a rotary pump is illustrated in Fig. 3.2. It shows an off-centrically moving disk-type rotor inside a steel stator. Blades called YDQHVDUH¿WWHGRQWKHURWRUGLDPHWULFDOO\*DVPROHFXOHVIURPWKHFRQWDLQHUWR be evacuated enter through the inlet. They are then swept by the rotary vanes and compressed creating pressure to open the outlet valve. The gas molecules are then driven out through the outlet.

)LJ Schematic of a rotary pump.

The pump is immersed in oil for lubrication, sealing and cooling. Gas swept away to the atmosphere passes through oil and hence oil vapour may come out from exhaust end of the pump. Gas ballast valve provided with an oil sealed rotary pump is a one-way valve which allows air to be admitted in the pump when needed to reduce condensation of water associated with humid gas. Gas ballast valve closes immediately after compressed pressure becomes higher than one atmosphere. One may also use ‘double stage or two stage’ rotary pump. In double stage rotary pump, two rotary pumps are attached in series. An output or exhaust HQGRIWKH¿UVWSXPSLVFRQQHFWHGWRWKHLQSXWRIWKHVHFRQGSXPS5RWDU\ pumps are sub-divided according to the pumping speed. The pressure range of operation for these pumps is atmospheric pressure up to 10–2 Torr (single stage) and up to 10–4 Torr (two stage). Gas ballasting is required to take care of the contaminants along with air. For example, sometimes air containing water vapour is pumped. Due to compression, this water vapour may condense into liquid form. Usually, a controlled amount of air from outside the pump is allowed into the vacuum system using a gas ballast.This air prevents the condensation of air.

Vacuum Science and Technology 43

The main advantage offered by rotary vane pumps is that its operation starts from atmospheric pressure. These pumps are simple, inexpensive and can be used as a roughing pump to back other pumps like diffusion or turbo molecular pump. In terms of disadvantages, the lowest achievable vacuum is limited to 10-4 Torr by rotary vane pump. Oil seal used between stator and rotor further reduces ultimate achievable vacuum due to back streaming. These vacuum pumps cause noise due to the vibrations. The oil used can enter the vacuum chamber being pumped.

 7XUER0ROHFXODU3XPS Turbo Molecular Pump is a gas transfer pump. In this pump, the gas molecules are taken away from the container into the atmosphere guided by a fast-moving solid surface namely, rapidly spinning multiply blended structure resembling a turbine.The velocity and hence kinetic energy of these blades is passed on to the gas molecules colliding with them. This momentum transfer drives the gas molecules through the outlet. Unlike rotary pumps, here, gas transfer takes places without compression so, the volume of the gas remains the same. So, as shown in Fig. 3.3, typically, a turbomolecular pump consists of a stator and a rotor. The rotor has turbine type blades that rotate between corresponding blades of the stator. In other words, alternate axial rotating discs are cut with slots that rotate in stationary slotted plates at high speed. The average velocity of gas molecules can be determined on the basis of Kinetic theory of gases. Pumping effect is the function of blade velocity and hence the impact process between the molecules and the rotor blades. Pumping velocity and the molecular mass determine the compression ratio.

)LJ Schematic of a Turbomolecular pump.

44

Foundations of Experimental Physics

Turbo molecular pumps are easy to operate and assure clean, oil free vacuum. When relatively large quantity of gas is to be pumped out, these pumps are preferred due to their high pumping speed. These pumps are advantageous as they have a long life and low running cost. Modern ultra-high vacuum systems are pumped by these pumps. Unlike Rotary vane pumps, Turbomolecular pumps do not operate at outlet pressures of one atmosphere instead they require a pressure of a few thousand times below atmosphere. Due to this, they are backed by a secondary pump producing vacuum of the order of 10-3 Torr. In these pumps, rotor moves at very high speed, thereby inducing vibrations. Back streaming of hydrogen gas can be caused due to its small mass and hence minimal momentum transfer. The turbomolecular pumps have high initial cost due to the mechanical precision required in fabrication of rotor/stator system.

 'LIIXVLRQ3XPS Diffusion pumps are another important kind of vacuum pumps. Here, gas molecules are driven from the inlet towards the outlet using streams of hot oil vapour. The basic pumping action is initiated by the diffusion of gases in the vapour jet. Fast jets of boiling oil draw gas molecules from the container with them. So, evacuation is attained by momentum transfer from streaming oil molecules to diffusing molecules. Interaction between gas and oil vapour is based on diffusion. Thus, diffusion is responsible for dragging the gas molecules towards backing region. Pump operates in the pressure range of 10–3 Torr to 10–10 Torr. Vacuum of the order of 10–10 Torr is achieved only with the aid of liquid nitrogen trap and baking the system for long duration. In this pump, special high molecular weight (400 to 500 amu) and low vapour pressure oil is heated. It is passed through nozzle forming high speed jets which impart large momentum to gas molecules. Oil vapour exerts a pressure of 1-2 mbar. Gas molecules are forced in downward direction and pumped by a roughing pump (rotary pump). Oil is cooled by water cooled pump body and returns to oil reservoir for cyclic heating. Schematic of Diffusion pump is seen in Fig. 3.4. Instead of oil, mercury can also be used. But mercury vapours are hazardous and form alloy with metal containers. Thereby, glass pumps are to be used which are fragile. Various traps are connected between the diffusion pump and the vacuum system so as to avoid entering of oil vapours in the system. Diffusion pumps are able to produce much higher pumping speed at low pressure as compared to a mechanical pump under similar conditions such as weight and size. These pumps have simple construction hence the maintenance cost is low. Also, the operation is silent. Main disadvantage of this type of pump is the necessity of a roughing pump like rotary pump. Back streaming of gases is possible in this type of pumps. Oil vapour can enter the vacuum chamber. Oil used is usually expensive. Continuous water cooling of the pump body is necessary.

Vacuum Science and Technology 45

)LJ Schematic of diffusion pump.

 *HWWHU3XPS An important mechanism of attaining vacuum is by removing gas molecules using ‘Gettering’ or chemisorption. Transition metals such as titanium are used for gettering. It forms stable compounds with molecules of gases like hydrogen, nitrogen, carbon monoxide, carbon dioxide etc. The pressure range associated with this type of pump is around 10-4 Torr to 10-10 Torr. One important type of getter pumps is sublimation pump. Its schematic is shown in Fig. 3.5. This category of pumps involves evaporation of a getter PDWHULDO DQG LWV VXEVHTXHQW ¿OP IRUPDWLRQ RQ WKH LQQHU ZDOOV RI WKH SXPS Generally, titanium is the preferred material for a sublimation pump. A wire made of titanium based alloy is heated by passing current through it. Here, a power supply is used to pass current through a titanium in order to heat it in vacuum system to temperatures greater than 1000°C.This leads to sublimation RIWLWDQLXPDQGIRUPDWLRQRID¿OPRQWKHZDOO:KLOHWKH¿OPJHWVGHSRVLWHG on the wall, the gases get adsorbed or react to form a product that condenses on the walls. The pumping speed of the pump is directly proportional to the VWLFNLQJ FRHI¿FLHQW RI WKH JDVHV 7KH VWLFNLQJ FRHI¿FLHQW GLIIHUV IURP JDV to gas. It is around 0.4 for carbon monoxide but it is only around 0.001 for hydrogen gas. Simple operation and high pumping speed are advantages of this type of SXPS,WDOVRKDVVRPHGLVDGYDQWDJHVVXFKDVEUHDNDJHRI¿ODPHQWVDQGLV mainly used as an auxiliary pump that assists other pumps.

46

Foundations of Experimental Physics

)LJ Schematic of titanium sublimation pump.

 6RUSWLRQ3XPS As indicated by the term ‘sorption’, physical adsorption is the basis of this vacuum pump. The gas molecules are adsorbed on the surface of porous molecular sieves. Since these molecular sieves are maintained at the 77 K i.e. the temperature of liquid nitrogen, the adsorbed gas molecules are effectively trapped in the condensate state. As shown in Fig. 3.6, a typical sorption pump consists of stainless-steel cylinder containing zeolites or molecular sieves. In order to maintain the molecular sieves at low temperature, this cylinder is immersed in a liquid nitrogen container. The container is usually made of polymers. The gas molecules enter through the intake port. The zeolite molecular sieves are porous solids typically composed of alkali metal aluminosilicates. As evident from the name, the pore size is of the order of molecular dimensions and therefore the gas molecules can get trapped in the pores.  ,RQ3XPS The name ‘ion pump’ of this vacuum pump stems from the fact that it works on the principle of ionization. In an ion pump, gas molecules are ionized by DSSO\LQJ HOHFWULF DQG PDJQHWLF ¿HOGV LQ PXWXDOO\ SHUSHQGLFXODU GLUHFWLRQV Residual gases in the pump form a discharge, which sputters Ti based cathode DQG IRUPV IUHVK ¿OP RQ WKH VXUIDFH RI WKH SXPS ZDOO ,RQL]HG PROHFXOHV SK\VLVRUE RU FKHPLVRUE RQ IUHVKO\ IRUPHG 7L ¿OP DQG KHQFH DUH SXPSHG Electrical as well as chemical cleanup is involved. Stray electrons in a vacuum system can be accelerated by simultaneous application of electric and

Vacuum Science and Technology 47

)LJSchematic of a sorption pump. (Ref: S.K. Kulkarni, Nanotechnology: Principles and Practices, 3rd Ed., Springer)

PDJQHWLF¿HOG$SSUR[LPDWHO\N9YROWDJHDQG.*DXVVPDJQHWLF¿HOG of a permanent magnet is employed in mutually perpendicular direction. Force on electron moving with velocity vLQPDJQHWLF¿HOGB is dv = ev × B dt Electron moves in a spiral path F=m

(3.15)

mv 2 = evB r

(3.16)

having radius mv (3.17) eB So, as seen from the equations, electrons travel spirally towards the anode and on the way, ionize the gas molecules. Positively charged ions are accelerated towards cathode; where they sputter electrode material which is usually titanium. Titanium is highly reactive and is a getter material. It can react with gas molecules and condense on anode or even trap some gas molecules on anode. In cathode, noble gas ions like Ar are embedded. But in subsequent ion bombardment, Ar ions are released. This creates instability in pressure or pressure bursts occur. Typically, this can happen in ion pumps ZLWKGLRGHFRQ¿JXUDWLRQ7RUHGXFHWKHVHHIIHFWVVORWWHGFDWKRGHVDUHPDGH or triode ion pumps are designed. Ion pumps with diode as well as triode geometry are depicted in Fig. 3.7. Pumping speed is maximum at about 10–6 to 10-7 Torr and range of operation is 10–4 Torr to 10–10 Torr. One important advantage of ion pump is that it is a clean pump as no ÀXLGVDUHXVHG6LPLODUO\VLQFHWKHUHDUHQRPHFKDQLFDOPRYLQJSDUWVLWLV r=

48

Foundations of Experimental Physics

)LJSchematics of a diode and triode ion pumps. (Ref: S.K. Kulkarni, Nanotechnology: Principles and Practices, 3rd Ed., Springer)

devoid of vibrations, noise and back streaming. This pump itself can be used as a vacuum gauge as ion current is a measure of pressure of gas. Also, this pump does not require a continuous backing pump but a roughing pump is necessary to get initial vacuum of the order of 10-3 Torr. The pump also has some disadvantages. Starting a sputter ion pump at higher pressure leads to KHDY\FXUUHQWWKURXJKWKHSXPSZKLFKUHGXFHVLWVOLIH0DJQHWLF¿HOGDQG high voltage are required for the pump. The vacuum pump is heavy and has limited life.

 0HDVXUHPHQWRI9DFXXP9DFXXP*DXJHV Instruments used for carrying out quantitative measurements of vacuum are called vacuum gauges. As the entire vacuum range is really wide - almost 15 orders of magnitude, one vacuum gauge cannot cover the entire range of vacuum. Hence, a series of vacuum gauges, each having a characteristic measuring range can be used to cover the entire range. Along with the desired vacuum range, the operating condition is another factor that determines the selection of vacuum gauge.

Vacuum Science and Technology 49

7KHYDFXXPJDXJHVFDQEHFODVVL¿HGRQWKHEDVLVRIPHDVXULQJWHFKQLTXH involved. At lower vacuum, corresponding higher pressures can be measured directly using gauges based on direct force measurements and hence called ‘direct gauges’. However, higher vacuum or corresponding lower pressure needs to be measured in terms of some physical quantities which vary in proportion to the number density of molecules. So, the measurements could be done in terms of thermal conductivity, ionization, viscosity etc. and is followed by calibration. Gauges employing such methodology can be called ‘indirect gauges’. U-tube manometer and McLeod gauge are examples of direct gauges. ‘Pirani Gauge’ is an example of an indirect gauge as it makes use of thermal conductivity or transport. Similarly, an indirect gauge called ‘hot cathode ionization gauge’ utilizes the property of ionization. More than one kind of vacuum gauge can be used if a large range of vacuum needs to be covered especially in the cases where more than one vacuum pumps are used. For direct vacuum gauges, the measurement is independent of composition of the gases. Even if there is a mixture of gases, total pressure of such a mixture is measured. In case of indirect gauges, however, identity of the gases is important since properties such as ionization differ from gas to gas. Partial pressures corresponding to residual gas mixtures are tackled by residual gas analyzers that work on the basis of mass spectroscopy and monitor vacuum. The chosen vacuum gauge needs to be mounted at a suitable location preferably in close proximity of the system where the vacuum needs to be measured. Some of the commonly used direct and indirect vacuum gauges are described below.

3.5.1 U-tube Manometer U-tube manometer is one of the simplest possible vacuum gauges. Fig. 3.8 shows the schematic of a manometer. It is a direct gauge consisting of transparent U-shaped tubing usually made up of glass. The glass tubing is SDUWLDOO\¿OOHGZLWKDOLTXLGKDYLQJORZYDSRXUSUHVVXUHDQGNQRZQGHQVLW\ȡ Mostly, mercury is employed for this purpose. Pressure exerted on this liquid is utilized for the measurement of vacuum. One end of the U-tube is open to the atmosphere while the other end is connected to vacuum system. Before connecting it to the system under evacuation, there is no difference in height of both the columns. On connecting it, a pressure difference causes the liquid level in the arm connected to the vacuum system to go up causing height difference ‘h’ in the meniscus levels in the two columns. Therefore, this height difference is in proportion with the pressure difference. The relation between the two is expressed through the following equation. p = hȡg

(3.18)

where p is pressure in the vacuum system, h LVKHLJKWRIWKHOLTXLGFROXPQȡ is density of liquid and g is acceleration due to gravity.

50

Foundations of Experimental Physics

)LJSchematic of a manometer.

A sealed version of manometer is also available, where one end can be VHDOHGRIIDIWHUHYDFXDWLQJWKHJODVVWXEHDQGOLTXLGLV¿OOHGWRDGHVLJQDWHG level to eliminate the effects due to changes in atmospheric pressure. Vacuum of the order 10–2 Torr can be measured using a U-tube manometer. The accuracy in the measurement of the order of 5×10–4 Torr can be achieved in SUHVVXUH PHDVXUHPHQW 5DQJH FDQ EH H[WHQGHG XVLQJ RWKHU PRGL¿FDWLRQ RI manometer; namely, inclined manometer or differential manometer. Accuracy FDQ EH LPSURYHG E\ PDJQL¿FDWLRQ$ VLPSOH FRQYH[ OHQV RQ WKH VFDOH FDQ serve the purpose. Simple construction is the prime advantage of manometer as one can read the difference in height directly. U-tube manometer has some disadvantages as well. Liquid manometers use glass tubes and hence are cumbersome and fragile. The problems like refraction of light in glass, dissolution of gases in liquid, sticking of liquid to glass wall, expansion of liquid due to temperature can result into errors. Long height of glass tube is another limiting factor.

3.5.2 McLeod Gauge It is also known as a compression gauge as it works on the principle of compression of gas following Boyle’s law. As seen earlier, according to Boyle’s law, the product of initial pressure and initial volume is equal to the SURGXFWRI¿QDOSUHVVXUHDQG¿QDOYROXPH Pi × Vi = Pf ×Vf

(3.19)

Vacuum Science and Technology 

Consider an isolated gas with volume Vi and pressure Pi. When this volume is compressed to volume Vf, then this decrease in the volume will lead to a concomitant increase in the pressure Pf . On the basis of Boyle’s law, the ratio of initial and compressed value will yield the magnitude of pressure. Figure 3.9 shows the schematic of McLeod gauge. The system consists of interconnected glass bulb and two identical capillary tubes and a glass bulb ¿OOHGZLWKPHUFXU\WKDWDFWVDVDPHUFXU\UHVHUYRLU,QLWLDOO\JDVDQGEXOEDUH at pressure P and volume of the gas is equal to the volume of the bulb which is V. Reservoir of mercury is lifted up so that the mercury reaches top of capillary. Gas in the capillary of diameter d and bulb is compressed. If A is area and h is the height then, V = Ah

(3.20)

where A is the cross-sectional area of the capillary and h is the difference in height of the mercury column. Therefore, Pi Vi = Pi (Ah)I

(3.21)

Pressure can be written in terms of height since 1 Torr of pressure is 1 mm of height of mercury column, hence, Pi Vi = Ah2

(3.22)

P ∝ h2

(3.23)

Consequently,

)LJSchematic of a McLeod gauge. (Ref: S.K. Kulkarni, Nanotechnology: Principles and Practices, 3rd Ed., Springer)



Foundations of Experimental Physics

Thus, pressure in vacuum system can be measured or calibrated in terms of height of mercury column. Using this gauge, vacuum in the range atmosphere to 10–4 Torr can be measured. Absolute measurement of pressure as well as calibration of other gauges is possible with the aid of McLeod gauge. It is an inexpensive and portable gauge but needs to be operated skillfully. Due to usage of glass, this type of gauge is breakable, clumsy to use. Another disadvantage is the use of mercury leads to toxic vapours. The height measurement needs to be done accurately.

 7KHUPRFRXSOH*DXJH Thermocouple gauge is a thermal conductivity based gauge. If electric current LV SDVVHG WKURXJK D ¿ODPHQW ZLUH  LW JHWV KHDWHG ,I LW FRPHV LQ FRQWDFW with gas molecules from the sample, some heat will be taken away by these molecules on the basis of thermal conductivity. The density of molecules will be proportional to the pressure. Hence, the amount of heat lost will be a function of the pressure. If a large number of molecules are present in an area, the pressure generated will be high, density will also be high which means the KHDWORVWZLOOEHKLJKDQGVRWKHZLUHZLOOEHFRROHGIRUDJLYHQFXUUHQWÀRZ through it. Figure 3.10 shows the schematic of a thermocouple gauge.

)LJ Schematic of a thermocouple gauge.

As is evident from the name, in a thermocouple gauge, a thermocouple is used to record the temperature of a wire exposed to vacuum. A thermocouple consists of two dissimilar metal wires fused together to form two junctions out of which one is at a reference temperature and the other junction is at the temperature to be measured. Thermocouple works on the basis of Seebeck effect according to which a temperature difference between the two junctions results in a voltage. In a thermocouple gauge, temperature recorded by the thermocouple will be a function of the pressure and hence vacuum will be

Vacuum Science and Technology 53

measured in terms of heat conducted. Using this gauge, vacuum in the range atmosphere to 10–4 Torr can be measured. This is a robust indirect gauge. Unlike, U-tube manometer and McLeod gauge, toxic materials like mercury are not used in it but sensitivity of a thermocouple can be a limiting factor.

3.5.4 Pirani Gauge Pirani gauge is also an indirect gauge in which measurement is done on the basis of thermal conductivity but in this gauge, temperature is not measured by a thermocouple. Instead, it is measured in terms of resistance recorded by :KHDWVWRQHEULGJHE\LQFRUSRUDWLQJD¿ODPHQWLQWRLW Wheatstone bridge is the most important feature in the construction of a Pirani gauge. Figure 3.11 shows the schematic of Pirani gauge. Two arms of the Wheatstone’s bridge comprise of standard resistance. Third arm of the bridge has a reference wire. A wire with similar resistance is located in the fourth arm. Bulb of the fourth arm is exposed to vacuum. Typically, a SODWLQXPZLUHLVXVHGDVD¿ODPHQW

)LJSchematic of a Pirani gauge.

Working of a Pirani gauge is as follows. Both the wires are heated by passing a constant current. The standard resistors are adjusted to zero current through the central ammeter while the sensing wire is evacuated with 10–4 Torr. At atmospheric pressure, resistance bridge is in balanced state. Now, as the system starts getting evacuated, number of gas molecules starts decreasing. 7KLVGHFUHDVHVQXPEHURIFROOLVLRQVGXHWRZKLFKWKHKHDWHG¿ODPHQWVWDUWV losing its heat. This, in turn, leads to change in resistance which disturbs the balance of Wheatstone bridge. Current recorded by the ammeter in the central arm can be calibrated as a function of pressure.

54

Foundations of Experimental Physics

One of the primary advantages of a Pirani Gauge is the rugged nature of these gauges. But this type of gauge has a limitation that it can only be used to record vacuum better than 10-3 Torr.

3.5.5 Cold Cathode Ionization Gauge The phenomenon of ionization is exploited to create indirect type of vacuum gauges namely ionization gauges. In this kind of gauges, gas density and pressure are measured indirectly by measuring electrical current which culminates from the collection of the positive ions created inside the gauge. Depending on the mode of operation, there are two kinds of ionization gauges namely cold cathode ionization gauges and hot cathode ionization gauges. These are typically used for the vacuum range of 10–2 to 10–10 Torr. Penning gauge is the most widely used cold cathode gauge. Like an ion pump, principle of Penning gauge is also based on the ‘Penning discharge’. Here, anode maintained at a very high voltage (~2 kV) is placed symmetrically between two cathode plates. An external magnet is used to generate a strong PDJQHWLF ¿HOG 7KLV PDJQHWLF ¿HOG FXUEV WKH WHQGHQF\ RI HOHFWURQV WR JHW accelerated towards the anode. Instead they move spirally. So, the applied HOHFWULFDQGPDJQHWLF¿HOGVOHDGWRLRQL]DWLRQRIJDVDQGJHQHUDWLRQRISODVPD The number of gas molecules will dictate the amount of gas being ionized. The amount of gas ionized will constitute the ion current. The number density of gas molecules will also determine the pressure generated. In this manner, the ion current will be a measure of the pressure generated. Figure 3.12 shows the schematic of a Penning gauge. $EVHQFH RI KRW ¿ODPHQW DQG VLPSOH FRQVWUXFWLRQ DUH WKH SULPDU\ advantages of Penning gauge. It is robust in nature and insensitive to exposure to air. Penning gauge also has some limitations pertaining to discharge such as QRQOLQHDUUHODWLRQEHWZHHQWKHGLVFKDUJHFXUUHQWDQGSUHVVXUH,WLVGLI¿FXOWWR initiate discharge at low pressure.

)LJSchematic of Penning gauge. (Ref: S.K. Kulkarni, Nanotechnology: Principles and Practices, 3rd Ed., Springer)

Vacuum Science and Technology 55

3.5.6 Hot Cathode Ionization Gauges ,QDKRWFDWKRGHJDXJHHOHFWURQVHPLWWHGWKHUPLRQLFDOO\E\D¿ODPHQWLRQL]H the gas molecules generating positive ions and hence plasma current. The rate of ionization is a function of number of the gas molecules. Since these molecules constitute the pressure, plasma current is proportional to the pressure. Therefore, hot cathode ionization gauges measure pressure in terms of the plasma current. Design of hot cathode ion gauge (Fig. 3.13) was improved to create Bayard-Alpert (B.A.) Gauge (Fig. 3.14) in order to address lower range of pressure or higher range of vacuum.

)LJ Schematic of an ionization gauge. (Ref: S.K. Kulkarni, Nanotechnology: Principles and Practices, 3rd Ed., Springer)

,QFDVHRIWULRGHKRWFDWKRGHLRQJDXJHWKHWULRGHFRQVLVWVRID¿ODPHQW VXUURXQGHGE\JULG(OHFWURQVWKHUPLRQLFDOO\HPLWWHGIURPDKRW¿ODPHQWDUH attracted towards the surrounding grid maintained at potential of the order of 200 V. Barring some electrons that impinge on the grid, the other electrons pass through it and travel towards the cylindrical ion collector surrounding the grid. In the process, they collide with residual gases and ionize them. The positively charged ions and the electrons are collected by the ion collector and the grid respectively. The ion current is proportional to the number of gas atoms and hence the pressure. Usually emission current is in the range 1–10 mA leading to ion current of the order of 10–8 – 10–7 A at pressure 10–6 Torr. Frequent collisions result into loss of energy and hinder the ionization process. Production of soft X-rays can also be another hinderance that can affect the calibration and hence the measurement.

56

Foundations of Experimental Physics

)LJ Schematic of a Bayerd-Alpert gauge. (Ref: S.K. Kulkarni, Nanotechnology: Principles and Practices, 3rd Ed., Springer)

In case of Bayerd-Alpert (B.A.) gauge, the basic principle remains the VDPH EXW WKH FRQVWUXFWLRQ RI WKH GHYLFH LV GLIIHUHQW +HUH WKH ¿ODPHQW LV outside the grid. The collector is positioned at the center of the grid and is in the form of a thin wire instead of the large area collector thereby reducing the production of soft X-rays that affect the measurement. B.A. gauge can measure vacuum of the order of 10-10 Torr which is better than that measured by a triode ionization gauge. Unlike cold cathode LRQL]DWLRQJDXJHYHU\KLJKYROWDJHRUPDJQHWLF¿HOGLVQRWUHTXLUHGIRUWKH hot cathode ionization gauges. But there can be limitations pertaining to the ¿ODPHQWDVLWKDVOLPLWHGOLIHDQGQHHGVWREHGHJDVVHG

 4XDGUXSROH0DVV6SHFWURPHWHU406 Sometimes gauges are required to measure partial pressures exerted by individual constituent gases in a mixture. Quadrupole mass spectrometer (QMS) is usually employed for this purpose. It can measure pressures up to 10–14 Torr. It can also be used for leak detection in UHV systems. In a QMS, gas molecules are ionized through impact of electrons emitted IURPDKHDWHGHOHPHQW7KHVHLRQVDUHWKHQ¿OWHUHGRXWRQWKHEDVLVRIPDVVWR charge ratio by the ‘quadrupole’. The word ‘quadrupole’ refers to the ‘four’ SDUDOOHO URGV SUHVHQW LQ WKH PDVV ¿OWHU RI 4XDGUXSROH 0DVV 6SHFWURPHWHU These can be seen in the schematic of QMS in Fig. 3.15. These conducting cylindrical rods are arranged in a square array. They are electrically connected

Vacuum Science and Technology 57

)LJSchematic of quadrupole mass spectrometer.

in such a way that opposite pairs will be at the same potential. The potential is a combination of dc component and an alternating component. If the dc component is denoted by A and the alternating component is denoted as (B FRVȦt), then, one pair of the rods will be at potential (A + (B FRVȦt)) while the other pair will be at the potential –(A + (B FRVȦt)). Let us see how QMS will work as a mass analyzer. AC and DC voltages given to the rods are increased with ratio remaining constant. At a particular combination of voltages, paths taken by the ions will depend on their being lighter or heavier which essentially is a result of their mass to charge ratio. So, at any given moment, depending on the values of dc and alternating voltages, ions with only particular value of m pass through the quadrupole e ¿OWHUDQGUHDFKWKHGHWHFWRU7KHUHVWRIWKHLRQVGHYLDWHIURPWKHLUSDWKVWULNH the rods and neutralize molecules. Simply put, a set of ions originating from D FRQVWLWXHQW JDV DUH µ¿OWHUHG¶ RQ WKH EDVLV RI PDVV WR FKDUJH UDWLR DQG DUH then detected.The detector output will be a mass spectrum comprising of ion current values as a function of mass number. Magnitude of ion currents will be in proportion to the quantities of gases and hence it can then be utilized for partial pressure measurement.

3.6 Vacuum Materials and Accessories 3.6.1 Vacuum Materials Different components of a vacuum system such as valves, pipes etc. play a crucial role in achieving the vacuum required for any application. In order that these components function properly, it is necessary to build them with suitable materials.

58

Foundations of Experimental Physics

As the main aim of any vacuum system is creating and maintaining vacuum, the materials used need to be non-porous in nature to minimize outgassing. They also need to be mechanically strong, should be able to withstand conditions such as extreme working temperatures and corrosion. 9DSRXU SUHVVXUH RI D PDWHULDO LV DOVR VLJQL¿FDQW ,W KDV WR EH LQYHUVHO\ proportional to the degree of vacuum required. Since vacuum components have a variety of shapes, it is desirable to have materials offering ease of fabrication. In accordance with these requirements, austenitic stainless steel is the most popular material for vacuum systems due to the plethora of desirable properties it offers. It is mechanically strong, can be easily welded, is corrosion resistant. It can be used for building walls of the vacuum chambers as well as other devices in the vacuum system. Aluminum, copper, nickel, titanium, gold, tungsten, molybdenum are some of the other metals used for vacuum systems. Alloys of aluminum are preferred due to low outgassing. Electric leads, electronic tube parts can be made of copper. Oxygen free highly conducting (OFHC) copper and gold are used for sealing joints. Chambers made of glass has been used in vacuum systems since a long time due to desirable characteristics of glass such as chemical stability, low outgassing UDWHVKLJKHOHFWULFDOUHVLVWLYLW\ORZFRHI¿FLHQWRIWKHUPDOH[SDQVLRQ$SDUW from glass, some other insulating materials are also used in vacuum systems e.g. insulating material used in the electrical feed throughs is alumina. Plastics have high outgassing rate so they are usually avoided in vacuum systems but WKHUHDUHH[FHSWLRQVVXFKDVSRO\HWK\OHQH7HÀRQHWF

3.6.2 Vacuum Accessories 6RPHRIWKHSULPHDFFHVVRULHVXVHGLQYDFXXPV\VWHPDUHÀDQJHVJDVNHWV 2ULQJV EHOORZV DQG YDOYHV (DFK RQH RI WKHP SOD\V D VSHFL¿F UROH LQ WKH vacuum system. A vacuum system contains several parts that need to be coupled or joined. Flanges are crucial for demountable joints. They are mechanical devices used for connecting vacuum chamber, vacuum pump, tubing to each other. Gaskets especially O-rings are utilized as vacuum seals. 7KH\DUHSODFHGEHWZHHQWKHÀDQJHV)ODQJHVDUHWLJKWHQHGWROHDGWRYDFXXP WLJKW VHDOV %HOORZV DUH ÀH[LEOH FRQQHFWLQJ HOHPHQWV DQG FDQ EH PDGH RI many diaphragms fastened together. They are convoluted and can be opened and closed. They can be used to separate vacuum chamber from mechanical SDUWVRUDOVRDVDIHHGWKURXJK9DOYHVDUHXVHGWRFRQWUROWKHÀRZRIJDV Each of these vacuum accessories can be of different kinds and a suitable kind is chosen depending upon the application and the degree of vacuum required for it. Some types of these accessories are as follows:  ‡ 'LIIHUHQWNLQGVRIÀDQJHVDUHGHVLJQDWHGE\VWDQGDUGRUJDQL]DWLRQVDQG manufacturers. Some typical kinds would be Quick Flange (QF), Klein )ODQJH.) &RQÀDW&)NQLIHHGJH )ODQJH&DPSRUEROW,62ÀDQJH HWF2XWRIWKHVH&RQÀDWÀDQJHVVHDOHGXVLQJFRSSHUJDVNHWVDUHSUHIHUUHG

Vacuum Science and Technology 59

for ultra high vacuum systems. Connectors such as tees, crosses etc. can EHXVHGWRFRQQHFWÀDQJHVRIGLIIHUHQWVL]HV  ‡ )ODQJHVPD\FRQWDLQJURRYHVWRPRXQW2ULQJJDVNHWV'HSHQGLQJXSRQ the range of vacuum, suitable O-ring gaskets are chosen. O-ring gaskets used in high vacuum system can be made up of elastomers such as neoprene but in case of ultra high vacuum system, gaskets made up of soft metals are used. Flanges with large sizes are sealed with indium or gold O-rings. High vacuum chambers are usually sealed with neoprene or viton O-rings.  ‡ 7RMRLQWKHFRPSRQHQWVDQGLQPRWLRQIHHGWKURXJKVEHOORZVPDGHRI stainless steel are used. In small feed throughs, convoluted bellows made up of thin metal sheet are used. Edge welded bellows can be used for different purposes such as an expansion joint, vibration isolator etc.  ‡ 'LIIHUHQW W\SHV RI YDOYHV DUH DYDLODEOH 7KHUH DUH QHHGOH YDOYHV JDWH YDOYHV ¿QH OHDN YDOYHV JDV DGPLVVLRQ YDOYHV HWF 7KHLU QDPHV DUH suggestive of their function. For example, gas admission valves are needed for letting gas inside the vacuum system.

 8OWUD+LJK9DFXXP8+9 6\VWHP 8OWUD KLJK YDFXXP LV QHFHVVDU\ IRU ¿HOGV VXFK DV SDUWLFOH SK\VLFV semiconductor technology etc. Two types of UHV systems are most commonly used. In case the measurements in UHV need to be carried out under vibration free conditions, it is required to use the vacuum pumps without any moving part. In this case, low vacuum can be achieved by cryosorption pump, while high vacuum can be attained by the sputter ion pump along with the titanium sublimation pump (TSP). On the contrary, if gas load in the vacuum system is large, one needs to choose the pumps which actually remove the molecules from UHV system and throw it out. In such cases, turbomolecular pump, diffusion pump backed E\URWDU\SXPSLVDSUHIHUUHGFRQ¿JXUDWLRQ7RDFKLHYHXOWLPDWHYDFXXPRQH can additionally use sputter ion pump and TSP. UHV systems are usually built with stainless steel. Also, bakeable components are preferred in case of UHV systems to be used for experiments requiring clean solid surfaces. Gaskets used in ultra high vacuum conditions would be made up of soft metals like OFHC copper.

3.8 Summary This chapter is devoted to an important aspect of experimental physics namely vacuum science and technology. Initially, basic concepts in vacuum science are described. Then, we move on to two important aspects of vacuum technology – the attainment and measurement of vacuum. Vacuum is attainment achieved using vacuum pumps. Some important vacuum pumps such as rotary vane

60

Foundations of Experimental Physics

pump, turbomolecular pump, diffusion pump, getter pump, sorption pump and sputter ion pump have been discussed in this chapter. The discussion reveals that Rotary pump and Diffusion pumps are used as roughing pump and high vacuum pumps respectively and when ultrahigh vacuum is required, pumps like Turbo molecular pump, Ion Pumps are used. The next part of the chapter is devoted to the tools ‘instrumental’ for measurement of vacuum namely vacuum gauges. Brief accounts of vacuum gauges such as U-tube manometer, McLeod Gauge, Thermocouple Gauge, Ionization Gauges and Quadrupole mass spectrometer have been given. Last part of the chapter describes vacuum materials, accessories and ultra high vacuum system.

4

Photons and Electrons: Sources,

Monochromators and Detectors

4.1 Introduction Earlier, our perspective of the world was mostly dominated by the information obtained by using visible radiation as our eyes work as detectors in this spectral regime. The scope widened with the advent of instrumentation to emit and detect: (i) photons corresponding to wide region of the electromagnetic spectrum apart from the visible radiation, (ii) electrons and (iii) ions. Due to interactions encompassing a wider range of energies, additional information about the sample (which was earlier inaccessible) became vivid. It is known that every submicroscopic effect is associated with a certain range of energies. Apart from the energy of interaction, the nature of the interaction such as whether it is elastic or inelastic also plays an important role in determining what information is obtained. Interaction can also be seen from the fact that a charged particle traversing through a matter can lose its energy in different manner such as excitation, ionisation of atoms along its path, Bremsstrahlung, Cerenkov radiation, transition radiation etc. The circumstances and magnitude of energy determine the prevalence of either of them, e.g. interaction of high HQHUJ\HOHFWURQEHDPZLWKWKHHOHFWULF¿HOGRIQXFOHLUHVXOWVLQ%UHPVVWUDKOXQJ i.e. deceleration of electrons results into an emission of photons. Change of circumstances will result in other phenomena such as excitation and ionisation. For quantitative estimation, the order of magnitude of energies corresponding to representative submicroscopic effects is tabulated in Table 4.1. Table 4.1: Representative energies for submicroscopic effects (order of magnitude) Rotational energies of molecules Vibrational energies of molecules or phonons Energy between outer electron shells Binding energy of valence electrons Binding energy of core electrons Energy to ionise atom or molecule

10í eV 0.1 eV 1 eV 1- 10 eV 10-10,000 eV 1-100 eV

While using photons as probe, one needs to note that the sources and detectors useful in different spectral regions of the electromagnetic spectrum

62

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are not identical. Thus, not only information gained from various spectral regions is different, but also, instrumentation required tends to differ. Both photons and electrons have their set of advantages as well DV GLVDGYDQWDJHV 6SHFL¿F DGYDQWDJH RI XVLQJ SKRWRQV IRU D YDULHW\ RI spectroscopic and microscopic techniques is that, mostly, one can carry out the experiment in air. Situation is not the same for the use of electron beam to understand the matter. In these experiments, which require high vacuum conditions, the use of different components may be restricted. Moreover, the instrumentation is order of magnitude expensive. Secondly, the objects far away from the spectrometer (as in the case of astrophysics) can also be explored using photons. However, the cross-section of interaction of matter with photons is weaker compared to that of electrons. As a result, in solid state, generally, bulk related details can be probed by a beam of photons, while electrons yield information regarding top few layers. Electron gun useful at low energy (say 100 eV) is construction wise same as that of the high energy (>80 keV) electron gun used in transmission electron microscopy. The comparison is mere representative. Other details will be evident as we discuss the nuances pertaining to each of the probes in the following sections.

4.2 Photon Sources Photons can be emitted as a result of natural as well as man-made processes. 6XQOLJKW LV WKH PRVW IDPLOLDU DQG VLJQL¿FDQW H[DPSOH RI SKRWRQV REWDLQHG QDWXUDOO\ ,Q WKH VFLHQWL¿F H[SHULPHQWV D EHDP RI SKRWRQV LV DUWL¿FLDOO\ generated as per the requirement. Table 4.2 gives the comparison of brightness of various sources of photons. One can clearly see that some of the man-made photon sources are brighter than the Sun. In fact, a photon source would be chosen on the basis of requirement of intensity. In this section, we will introduce few commonly XVHGSKRWRQVRXUFHVLQVFLHQWL¿FUHVHDUFKODERUDWRULHV Table 4.2: Comparison of brightness of various older photon sources with the modern photon sources Brightness photons/s/mm2/ mrad2/0.1% BW

Source

Greater than 1024

The synchrotron

Greater than 10

Undulator radiation

22

10

20

1010

The Sun

10

X-ray tube

10

Tungsten wire lamp (60 W)

10

Candle

9 6

Photons and Electrons: Sources, Monochromators and Detectors 63

4.2.1 Black Body Radiation The term ‘black body’ radiation was coined by the German physicist Gustav Kirchhoff in 1860s while proposing Kirchhoff’s law of spectral analysis. The name originates from the fact that if a body absorbs all the visible light falling on it, it will appear black. A black body is supposed to be an ideal absorber as well as emitter of incident radiation. Black body radiation is the oldest manmade light source and is used in spectroscopy even to date. A tungsten wire lamp (at about 2700 K) can be used as black body radiation. Planck’s radiation law describing the spectral energy distribution paved way for the birth of quantum mechanics. Planck’s Black body radiation law, UHODWLQJGHQVLW\RISKRWRQVȡv RIIUHTXHQF\ȞZLWKWHPSHUDWXUHT is given by, ρ (ν) d ν =

8πhν3 c

3

dν exp (hν k BT ) − 1

(4.1)

where h is Planck’s constant, c is speed of light in vacuum, kB is Boltzmann constant and T is temperature in Kelvin. Figure 4.1 shows spectral distribution of a black body radiation maintained at different temperatures. The distribution KDVPD[LPXPDWDVSHFL¿FZDYHOHQJWKGHSHQGLQJRQWKHWHPSHUDWXUHRIWKH black body. With increasing temperature, it is seen to shift towards higher energy or shorter wavelength. Also, area under the curve increases with increasing temperature. Keeping this in mind, a completely enclosed cavity maintained at a uniform temperature could be an ideal black body source. Practically, it will have a hole or an aperture and the value of emissivity will deviate slightly from unity. The most common source for visible and near infra-red light is tungsten ¿ODPHQWODPSKHDWHGXSWR.7KHHQHUJ\GLVWULEXWLRQRIWKHVRXUFHLV DSSUR[LPDWHO\OLNHDEODFNERG\ȝP ([WHQGLQJWKHOLIHRIWXQJVWHQ

Fig. 4.1: Spectral distribution of black body radiation shifts towards shorter wavelength or higher frequency at higher temperature.

64

Foundations of Experimental Physics

¿ODPHQW ODPS DQG DYRLGLQJ HYDSRUDWLRQ RI WXQJVWHQ ¿ODPHQW LV DFKLHYHG WKURXJKDGGLWLRQRIEURPLQHLQWKHODPS%URPLQHJDVUHDFWVZLWKWKH¿ODPHQW Condensation of compound is not possible due to the high temperature of WKH JODVV HQYHORSH :KHQ FRPSRXQG PROHFXOHV VWULNH WKH ¿ODPHQW LW JHWV GLVVRFLDWHGOHDYLQJEHKLQGWXQJVWHQDWRPRQWKH¿ODPHQWZKLOHKDORJHQJDV is released back in the bulb to have another halogen cycle. Glowbar or silicon carbide rod (SiC) is heated by passing the current through it to reach the temperature in the range 1000-1600 K to give the IR radiation in 4-10 ȝP wavelength. Resistively heated, closely wound coils of nichrome and kanthanal wire (which are commonly used in home appliances and industrial heaters), can also be used as infra-red sources. Emissivity is poor than glowbars though. Nernst glower is a rare earth oxide (ZrO2, Y2O, Er2O ¿ODPHQWDERXW cm in length, which is heated by passing current through it to emit the radiation LQWKHUDQJHWRȝP,QRUGHUWREHFRPHFRQGXFWLQJLWUHTXLUHVSUHKHDWLQJ GXH WR D ODUJH QHJDWLYH WHPSHUDWXUH FRHI¿FLHQW RI UHVLVWDQFH 6LJQL¿FDQW LQWHQVLW\EHORZȝPLVDQDGYDQWDJHRI1HUQVWJORZHURYHUJORZEDU

4.2.2 Microwave Sources 0LFURZDYH VRXUFHV RI ; EDQG DERXW  *+]  DQG 4 EDQG  *+]  DUH most commonly used in the electron spin resonance spectroscopy (ESR). Magnetrons can generate high power and high frequency signal however, it suffers from instability in the frequency. As a result, magnetrons are not used in ESR. Klystrons (vacuum tubes) are used as microwave source in ESR spectroscopy. Klystrons can generate microwaves in a wide range of frequencies from radio frequency to high microwave region. Klystrons convert energy of the accelerated electrons to electromagnetic radiation. A klystron consists of an HOHFWURQ JXQ D FDYLW\ ZLWK JULGV DW SRVLWLYH ELDV DQG D UHSHOOHU RU UHÀHFWRU plate maintained at a negative potential than the cathode. Velocity of electrons gets modulated by oscillations in cavity. Bunching of electrons due to SRVLWLYHO\ELDVHGJULGVDQGUHÀHFWRURUUHSHOOHULVUHVSRQVLEOHIRUJHQHUDWLRQRI WKHHOHFWURPDJQHWLF¿HOG.O\VWURQWXEHVZKLFKFDQJHQHUDWHPRQRFKURPDWLF PLFURZDYHUDGLDWLRQLQ;EDQGZLWKIUHTXHQF\*+]DUHFRPPRQO\XVHG in ESR. Higher frequency bands can also be generated with the aid of travelling wave tubes. 4.2.3 Discharge Lamps Discharge lamps are the most common photon sources. In this family of photon sources, photons are generated by voltage induced electric discharge, SURGXFHG LQ JDV ¿OOHG WUDQVSDUHQW HQFORVXUHV 'HSHQGLQJ RQ WKH ¿OOHG JDV light of different frequencies can be generated. For example, light generated in mercury lamp is in visible range and hence appears white. Hydrogen and deuterium generate ultra-violet light. Xe gas lamp acts as a good source in

Photons and Electrons: Sources, Monochromators and Detectors 65

visible and near infra-red regime. In a discharge lamp, radiation originates from electrons returning to the ground state after having been excited into a higher orbital. Recombination at the ground state occurs with certain life time DQGZLWKZHOOGH¿QHGSUREDELOLW\RIHPLWWLQJSKRWRQV One can use high pressure as well as low pressure gas in the lamps. High pressure discharge lamps are useful when higher radiant power is desirable. 7KHOLJKWLVJHQHUDWHGE\JDVGLVFKDUJHLQWKHDUFWXEHE\FXUUHQWÀRZLQJLQ two electrodes enclosed in high-quality quartz. The tungsten pin electrodes DUHVHDOHGLQWKHGLVFKDUJHWXEHDORQJZLWK¿OOLQJPDWHULDOVVXFKDV+JRU1D vapour (Fig. 4.2). Initially, a high voltage pulse is required to ignite the lamp. Warm up time of few minutes is required for vapourisation of Hg or Na and generation of pressure of 1-10 bars. The electrical conductivity is produced in the discharge. Power conversion in high pressure lamps is higher than low pressure lamps. Gas discharge produces isolated spectral lines corresponding to atomic transitions. These lines can be directly used. Or, one may coat the discharge lamp by phosphor from inside to convert light into desired spectral regime. Notably, the discharge lamps also cover wide range of photon energy.

Fig. 4.2: Illustration of the operation of mercury high pressure DERXWRIDWPRVSKHULFSUHVVXUH GLVFKDUJHODPS

High-pressure discharge lamps are the expensive sources of photons compared to the low-pressure discharge lamps. The high-pressure discharge lamps need to cool down completely prior to re-starting them. Longer warm up times and re-starting times make it unsuitable for certain applications. Discharge is sustained as cold discharge or hot discharge at low pressure. Other DGYDQWDJHVRIORZSUHVVXUHGLVFKDUJHODPSV)LJ DUHKLJKHUFRQYHUVLRQ HI¿FLHQF\RIDERXWFRPSDUHGWRLQFDQGHVFHQWODPSVHI¿FLHQF\a  longer life, and low heat production (or wastage of energy). Frequent (on and off) switching of discharge lamps limits life. If mercury lamps are broken accidently, hazardous mercury vapours could be of great concern. Cold as well as hot cathode discharge lamps are being used as the laboratory sources.

66

Foundations of Experimental Physics

Fig. 4.3: Schematics of low-pressure sodium discharge lamp. Na vapour LV¿OOHGDWORZSUHVVXUHDERXWRIDWPRVSKHULFSUHVVXUH +RWFDWKRGH emits electrons, which, in turn ionise Na vapour. The visible photons are HPLWWHGZKLOH1DDWRPUHWXUQVWRWKHJURXQGVWDWH7KLQLQGLXPR[LGH¿OP

FRDWHGRQZDOOUHÀHFWVLQIUDUHGUDGLDWLRQDQGWUDQVPLWVYLVLEOHUDGLDWLRQ

A continuum source of ultraviolet light is deuterium or hydrogen discharge lamp, operated at low pressure. Low pressure plasma is sustained by application of high voltage across cathode and anode. In modern lamps, DUFLVIRUPHGDWORZYROWDJHEHWZHHQR[LGHFRDWHGDQGKHDWHG¿ODPHQWDQGD PHWDOHOHFWURGH7KHKHDWHG¿ODPHQWSURYLGHVHOHFWURQVDQGSRWHQWLDORIWKH order of 40 V is applied between two electrodes. The continuous output from 160 to 400 nm is feasible. The reaction of deuterium is, D2 + Ee o D*2 o Dc + Ds + hv; where, D*2 represents the excited molecule, D’ and D’’ are singly ionised and doubly ionised atoms. Gas discharge also emits a continuous spectrum because electrons are accelerated and deaccelerated by scattering processes in the gas phase. Higher temperature can be reached by the gas discharge lamps. For instance, mercury discharge lamps can have temperatures as high as 6000 K. Notably, the smaller VL]HODPSVDUHDOVRDYDLODEOH0RUHRYHUHI¿FLHQF\RIOLJKWSRZHURXWSXWWR LQSXW HOHFWULFDO HQHUJ\ LV DERXW  FRPSDUHG WR WKDW RI WXQJVWHQ ¿ODPHQW ODPSVZKLFKLVDERXW Very often, UV radiation produced by the transition from excited He or Ne gas while returning to the ground state is used in spectroscopy. Ionisation source having a cold cathode capillary discharge is used to produce ultraviolet light, He-I and He-II for the ultra-violet photoelectron spectroscopy. He-I line LVWKHSKRWRQEHDPHPLWWHGIURPWKHQHXWUDODWRPVDWQPZKLOH+H,,LV DSKRWRQEHDPHPLWWHGIURPWKHVLQJO\LRQLVHGDWRPVDWQP

4.2.4 Light Emitting Diodes Light emitting diode (LED) is a special type of diode in which injection of carriers across a forward-biased junction causes emission of incoherent light. Or, it is a device which converts electrical energy to light energy (injection electroluminescence). It is used as a light source especially for short distances. The prime advantage of LED over an incandescent lamp is that its operating temperature is much lower than an incandescent lamp. Besides this, other

Photons and Electrons: Sources, Monochromators and Detectors 67

advantages are long life, robustness and low cost. Figure 4.4 depicts a schematic diagram of typical LED. A transparent window on top of LED emits the generated light. On receiving energy greater than the band gap, the electrons can undergo transition from valence to conduction band leaving behind holes. On applying forward bias, barrier height reduces. The carriers are concentrated at the junction boundary. The carriers in every side combine releasing energy in the form of photons. The intensity of emitted light is directly proportional to the number of carriers injected, which in turn will depend on the applied bias.

Fig. 4.4: Schematic diagram forward biased p-n junction LED. Electron-hole recombination in the depletion region effectively generate photons.

Sandwich formation of the type p-AlGaAs/p-GaAs/n-AlGaAs, KHWHURVWUXFWXUH KHOSV LQ FRQ¿QLQJ WKH FKDUJH FDUULHUV LQ *D$V DFWLYH OD\HU Probability of recombination is then enhanced. Along with radiative, transition, non-radiative transitions also occur generating heat. Furthermore, light travels LQ DOO GLUHFWLRQV RI WKH MXQFWLRQ 7KH OLJKW FDQ EH JDWKHUHG HI¿FLHQWO\ E\ IRUPLQJ OLJKW UHÀHFWLQJ OD\HUV DERYH DQG EHQHDWK WKH OLJKW HPLWWLQJ OD\HUV 7KHSULPHFKDOOHQJHIDFHGLQDWWDLQLQJKLJKHUHI¿FLHQF\RI/('LVWRDYRLG re-absorption of photons so that photons can escape out from the device. Typically, LEDs are composed of direct band gap materials such as gallium phosphide (GaP having Eg H9 JDOOLXPQLWULGH*D1KDYLQJEg H9 LQGLXPDQWLPRQLGH,Q6EKDYLQJEg = 0.18 eV), gallium arsenide phosphide (GaAsP having Eg about 1.9 eV), gallium arsenide (GaAs having Eg H9 HWF2XWRIWKHVH*D3DQG*D$V3HPLWYLVLEOHUHGJUHHQ\HOORZ  light while GaAs emits IR light. In 2014, Nobel Prize in physics was awarded WR$NDVDNL$PDQRDQG1DNDPXUDZKRPDGHWKH¿UVWEOXH/('VLQWKHHDUO\ V7KLVZDVDVLJQL¿FDQWVWHSLQIDEULFDWLQJZKLWH/('OLJKWZKLFKLV important for applications in computer and smart phone screens, requiring light with low electricity consumption. The latest blue light emitting diodes are made of indium gallium nitride (InGaN). Band gap engineered epitaxial layers are grown on semiconductor wafers to form hetero-structures that can withstand high level of carrier injection in LEDs. Indirect band gap materials like silicon and germanium are known for non-radiative transitions leading to generation of heat, hence they are not preferred for LEDs.

68

Foundations of Experimental Physics

4.2.5 Lasers 7RZQHV FRQFHLYHG WKH EDVLF LGHD RI 0$6(5 LQ  DQG DOVR FRQVWUXFWHG WKH¿UVWGHYLFHIRUFRKHUHQWDPSOL¿FDWLRQQDPHO\0LFURZDYH$PSOL¿FDWLRQ by Stimulated Emission of Radiation (MASER) later with his co-workers. Based on the theory of stimulated emission, proposed by Einstein, Maiman built a similar device but with electromagnetic radiation of higher frequency range. LASER turned out to be one of the most important discoveries due to the multitude of applications it offers. To date, Laser physics is evolved as a separate branch. An extremely narrow spectral width is one of the important reasons for the preference. The light emission could be spontaneous or VWLPXODWHG)LJ 7KHVSRQWDQHRXVHPLVVLRQFDQEHZHOOXQGHUVWRRGRQ the basis of quantum electrodynamics. Quantitative treatment is beyond the VFRSHRIWKHSUHVHQWERRN,QVLPSOHZRUGVWKHÀXFWXDWLRQVLQWKHDPELHQW HOHFWURPDJQHWLF ¿HOG DUH UHVSRQVLEOH IRU WKH VSRQWDQHRXV HPLVVLRQ 'XH WR WKH ]HURSRLQW HQHUJ\ RI WKH HOHFWURPDJQHWLF ¿HOG ÀXFWXDWLRQV RFFXU HYHQ ZLWKRXW DSSOLFDWLRQ RI WKH ¿HOG 7KHVH ÀXFWXDWLRQV LQGXFH WKH VSRQWDQHRXV HPLVVLRQIURPWKHH[FLWHGVWDWHV7KHEDVLFFRQGLWLRQWKDWQHHGVWREHVDWLV¿HG for stimulated emission is ‘population inversion’, and is discussed in the following section.

Fig. 4.5: Illustration of optical absorption, spontaneous emission and stimulated emission.

Lasers are popular line sources. Earlier, use of lasers was limited due to availability of only few wavelengths. Now, dye lasers can give various bands of spectral regions from UV through visible to infra-red. To date, lasers are world’s most sophisticated light sources for spectroscopy pertaining to coherent radiation. It may be worthwhile to note the conditions for coherence which are: two sources of radiation must have identical frequencies (and hence wavelength). The phase difference between two beams must remain constant with time. Lasers are high intensity, narrow spectral bandwidth (0.01 nm or OHVV  SRODULVHG DQG FRKHUHQW OLJKW VRXUFHV 3KDVH RI WKH ZDYH¿HOG IURQW LV maintained for large distance. The underlying basic principle is stimulated HPLVVLRQ RI UDGLDWLRQ /LJKW $PSOL¿FDWLRQ E\ 6WLPXODWHG (PLVVLRQ RI Radiation (LASER) is triggered by incident photons. The number of photons

Photons and Electrons: Sources, Monochromators and Detectors 69

produced by stimulated emission exceeds the number lost by absorption. This in turn needs a larger number of particles in the excited state than in the ground state. Emitted photon travels in exactly the same direction and is precisely in phase with the photon that caused the emission. Pumping, meaning reversal of population of active species is attained by excitation of electrons by means of electrical discharge, passage of electric current, or exposure to an intense radiant source. Due to pumping, the higher energy levels are populated in active species. For lasing action to occur, the excited states need to have longer life time (1 ms or longer). Note that the lifetime of the vibrational excited state is 10±to 10–14V$VGHSLFWHGLQ)LJOHWE0 and E1 be the ground and the ¿UVWH[FLWHGVWDWHZLWKFDUULHUSRSXODWLRQN0 and N1 respectively. If N0 > N1, radiation is absorbed and spontaneous radiation dominates. On the contrary, if N1 > N0, most atoms occupy energy level E1 that results into weak absorption. Stimulated emission predominates in this case and the light LVDPSOL¿HG Laser functions as an oscillator (resonator). Radiation produced by the lasing action passes back and forth through the medium (Fig. 4.6). Additional SKRWRQV DUH JHQHUDWHG WKURXJK HDFK SDVVDJH$Q HQRUPRXV DPSOL¿FDWLRQ LV thus feasible. Repeated passage also produces highly parallel beam. Non­ parallel rays escape from the sides of the medium. One of the mirrors is SDUWLDOO\UHÀHFWLQJWRDOORZSDUWRIWKHUDGLDWLRQWRHVFDSHIURPWKHFDYLW\

Fig. 4.6: Schematic diagram to explain working of a LASER.

The essential components of the laser are: active medium, excitation source or pump and UHÀHFWLQJPLUURUVRUUHVRQDWRU Population of the level is described by the temperature of the system via Boltzmann statistics N = N0 exp (–E/kBT) (4.2) where N is number of particles occupying energy level E at temperature T, N0 is the occupancy of the ground level and kB is the Boltzmann constant. In order to understand the lasing concepts, let us recall our understanding of the emission process.

70

Foundations of Experimental Physics

6SRQWDQHRXV HPLVVLRQ LV GH¿QHG DV UDGLDWLYH WUDQVLWLRQ ZLWK SKRWRQ emission having energy, hv = E2 – E0; from state E2 to the ground state E0, in order to attain the thermal equilibrium. The thermal or Boltzmann population is given by; ⎛ E − E0 ⎞ N 2 = N 0 exp ⎜ − 2  ⎟ ⎝ kT ⎠ The mean life time of the atom in the excited state is about 10–8V$W¿QLWH temperature, matter radiates thermal or black body radiation. The probability of transition from an upper state to the lower state is called as Einstein $FRHI¿FLHQWGHQRWHGE\A20 and is a measure of spontaneous depopulation of energy state 2. The stimulated emission can be understood as follows. If photons of energy hv20 are incident on the system, it pushes system in the state E0 from E2 or leads to the transition of the system from E2 to the lower energy level E0. Einstein further proved that probability of transition from the excited to the ground state and from the ground to the excited state is exactly same. It is called as Einstein-BFRHI¿FLHQW&RHI¿FLHQWRIVSRQWDQHRXVHPLVVLRQLV UHODWHGWRWKHFRHI¿FLHQWRIVWLPXODWHGHPLVVLRQ Let the spectral energy density of the electromagnetic radiation be U(v). Let energy E2 be greater than E1 (E2 > E1). Let the occupancy of energy level E1 be N1 atoms; and that of energy level E2 be N2 atoms. The transition rate depends on U(v) and on the electric dipole moment matrix element, which is given by μ fi ≡

∫ ψ f (er )ψ i d τ *

(4.4)

where \fLVWKH¿QDOZDYHIXQFWLRQ\i is the initial wave function, and (er ) is the electric dipole moment operator. Now, the total probability of absorption is given by R1→ 2 = B12 ρ( ν) The total probability of emission is R2→1 = A21 + B21 ρ (ν)

 (4.6)

Emitted photon energy hȞ E2 – E1, which gives rise to, N1 = exp (E2 − E1 kT ) = exp (hν kT ) due to the Boltzmann factor. N2 Under equilibrium, the total absorption rate for the system must be equal to the total emission rate. Therefore, N1Rĺ = N2Rĺ

(4.7)

which leads to, N1B12ȡȞ  N2A21 + N2B21ȡȞ 



As a result, (N1B12 – N2 B21 ȡȞ  N2A21

(4.9)

Photons and Electrons: Sources, Monochromators and Detectors 71

Or, in other words, ρ (ν) =

N 2 A21 (N1 B12 − N 2 B21 ) ρ (ν) =

Substituting

A21 / B21 ⎛ N1 B12 ⎞ ⎜⎝ N B ⎟⎠ − 1 2 21

(4.10)

(4.11)

N1 = exp (E2 − E1 kT ) = exp (hν kT ) gives N2 A21 / B21

ρ (ν) =

⎛ B12 (hν kT ) ⎞ (4.12) − 1⎟ ⎜⎝ B e ⎠ 21 7KHHTXDWLRQRIWKHVSHFWUDOHQHUJ\GHQVLW\RIUDGLDWLRQRIIUHTXHQF\ȞLQ thermal equilibrium at temperature T must be consistent with Planck’s law for black body radiation: ρ (ν) =

8πhν3

)URPHTXDWLRQV DQG 

c

3

1  exp (hν kT ) − 1



A 8πhν3 B12 = 1 and 21 = B21 B21 c3

7KLVLVDUHODWLRQEHWZHHQ(LQVWHLQ¶VFRHI¿FLHQWV 1RWHWKDWWKHFRHI¿FLHQWVRIVWLPXODWHGHPLVVLRQDQGVWLPXODWHGDEVRUSWLRQ are equal. A And 21 ∝ ν3 implies the larger is the energy difference between the B21 energy levels, spontaneous emission is more likely compared to the stimulated emission. A21 ν kT Rearrangement of terms leads to = e(h ) − 1 . B21ρ (ν) This equation implies, under thermal equilibrium, and at temperature leading to hȞ!!kT, the spontaneous emission is far more probable. On the other hand, the stimulated emission becomes possible only if, hȞ§ kT. In other words, lasing action can be understood as follows: N A + N 2 B21ρ (ν) ⎡ A21 ⎤ N 2 emission rate = 2 21 = ⎢1 + ⎥ absorption rate N1 B12ρ (ν) B221ρ (ν)⎦ N1 ⎣ If hȞkT or E2 – E1kT then N emission rate = 2  absorption rate N1

(4.14)



72

Foundations of Experimental Physics

Under thermal equilibrium, one expects N2  N1. However, under the non-equilibrium condition, if by some means, population inversion is attained, (N1N2), then emission would exceed absorption. E − E1 This means that the applied radiation of frequency ν = 2 will be h DPSOL¿HGLQLQWHQVLW\GXHWRWKHLQWHUDFWLRQSURFHVV Number of photons emitted would exceed incoming number of photons. In order to continue such a situation, one must maintain the population inversion of the states. Optical pumping must continue. In a laser, inducing radiation is coherent as atoms radiate in phase because their charge oscillations are in phase with the incident radiation. Note that the Laser is a coherent parallel EHDPIRFXVHGE\UHÀHFWLRQEHWZHHQWKHHQGPLUURUVRIWKHUHVRQDQWFHOO7KH emitted photons are in phase and move coherently. The resulting intensity is the square of the constructively combined amplitudes, and therefore, high. The states between which the transitions occur, are an upper metastable state with UHODWLYHO\ORQJOLIHWLPHDQGWKHJURXQGVWDWH7KHJURXQGVWDWHKDVLQ¿QLWHO\ ORQJOLIHWLPH,QDFFRUGDQFHZLWKWKHXQFHUWDLQW\UHODWLRQǻEîǻt = ƫ long OLIHWLPHRIXSSHUVWDWHPHDQVǻtLVODUJHDQGWKHUHIRUHǻE will be narrow. The monochromatic radiation would be emitted. Laser is unidirectional. As indicated earlier, the photons not traveling axially, escape through the sides of the cavity. In practice, lasing action does not take place in two level system, as population inversion is not feasible. The Ruby laser is known to mankind as WKHYHU\¿UVWODVHUZKLFKLVWKUHHOHYHOV\VWHP,WPD\EHQRWHGWKDWODVLQJLV possible in three level system, but requires more energy. Four level laser can work on the modest energy. The most common four level laser is Nd:YAG laser, which operates at low threshold compared to three level system. It is easy to achieve population inversion even by pumping a relatively small number of ions into the upper level. Solid state lasers are versatile and typically characterised by large range of average peak energies, varying pulse width, varying pulse repetition rate and wavelength. Size and shape of active material can be easily chosen to achieve required performance. Moreover, different active materials have varying wavelength, gain, and energy storage. The output energy can be LQFUHDVHGE\DGGLQJDPSOL¿HUV,QDVROLGVWDWHODVHUDFWLYHFHQWUHVDUH¿[HG dopants in the dielectric crystal or glassy material. Usually crystal is shaped in WKHIRUPRIDURGDQGSXPSLQJLVGRQHE\WKHÀDVKODPSRUGLRGHODVHU0RVW of the incident energy gets converted into heat and hence the lasing system needs to be water cooled. 7KH DFWLYH UHJLRQ RI WKH UHVRQDQW FDYLW\ LV ¿QLWH 7KH VWDQGLQJ ZDYH SDWWHUQ IRUPHG E\ ZDYHV FRQ¿QHG LQ D FDYLW\ JHQHUDWHV PRGHV /DVHU oscillates in number of transverse and longitudinal modes. In a longitudinal c mode, frequency separates; Δν ~ , where d is a length of the cavity and c 2d is velocity of light.

Photons and Electrons: Sources, Monochromators and Detectors 73

Let us see typical examples of solid state Lasers. The Ruby laser (Cr­ doped Al2O KDVDQHPLVVLRQZDYHOHQJWKRIQP$O2O is a wide band JDSPDWHULDO7KHWUDQVLWLRQPHWDOLRQ&ULVGRSHGXSWR LQLWWRIRUP ruby. Transition metal ion generates metastable levels within the forbidden gap which is responsible for lasing action (Fig. 4.7 (a)). E0 is a ground state having occupancy N0. Under equilibrium conditions, N2N1 N0, N1 and N2 being occupancy of excited states E1 and E2 respectively. Optical pumping with 400 nm, leads to the transition from E0 to E2. Life time of E2LVQV$V a result, electrons from state E2 rapidly undergo the transition to Ec which is a PHWDVWDEOHVWDWHZLWKOLIHWLPHRIDERXWPVDWURRPWHPSHUDWXUHFDXVLQJWKH SRSXODWLRQLQYHUVLRQ6WLPXODWHGHPLVVLRQDWQPLVGXHWRWKHWUDQVLWLRQ from the metastable levels to E0. Four level laser as Nd:YAG is more popular as pumping is easier in the V\VWHP7KHVH ODVHUV DUH SXPSHG E\ ÀDVK ODPS RU GLRGH ODVHU ,Q 1G“’@DQG>±’’@ Value from standard tables is 0.68269. In other words, the area between [X – ıX + ı@LV On the parallel grounds, the probability P within pıRIX is P(within pı  =

1 2π

+ p

∫

e −u

2

/2

du

(11.66)

−p

For Gaussian, FWHM = 2σ 2 ln 2 .

Fig. 11.10: Illustrative diagram showing probability of a measurement which is proportional to the area under the curve for sector X – pıDQGX + pıUHVSHFWLYHO\

Error Analysis and Statistical Methods 341

11.5 Principle of Maximum Likelihood Improbable Data can be Rejected Combination of two or more independent measurements of same physical quantity poses a question to select the most probable value. For instance, consider the data giving measurement of melting temperature of certain metal E\ WZR GLIIHUHQW WKHUPRFRXSOHV :KLFK RQH LV D EHWWHU HVWLPDWH" +RZ WR GHFLGHLW" One considers what is called as ‘weighed average’ facilitating the importance to be given to precise measurements. Let the two different sets of measurements be x = xa ± σ a x = xb ± σb If the difference xa − xb LVJUHDWHUWKDQERWKıaDQGıb, one can conclude that the measurements are inconsistent. It is then required to perform the measurements again. Assume discrepancy xa − xb is not large. One cannot x + xb simply take the best estimate xbest (of true value X) as a . 2 Precise measurement should receive higher weightage. The probability of obtaining the value xa is PX ( xa ) ∝

1 −( xa − X )2 / 2σ2a e σa

(11.67)

1 −( xb − X )2 / 2σb2 e σb

(11.68)

and that of xb getting observed is PX ( xb ) ∝

The probability of obtaining both the values PX (xa) in one set and PX (xb) in another set is PX ( xa , xb ) = PX ( xa ) PX ( xb ) ∝

2 1 e− χ / 2 σ a σb

2

where

⎛x − X⎞ ⎛x − X⎞ +⎜ b χ =⎜ a ⎟ ⎝ σb ⎟⎠ ⎝ σa ⎠ 2

(11.69) 2

(11.70)

The best estimate of (unknown true value) X is the value for which xa and xb are most likely. 7KH PD[LPXP YDOXH RI WKH SUREDELOLW\ UHTXLUHV Ȥ2 to be minimum. Therefore, differentiate the expression for probability with respect to X and equate it to 0.

342

Foundations of Experimental Physics

2

xa − X σ 2a

xa

Therefore,

σ 2a

−

X σ 2a

+2 +

xb − X

xb σb2

σb2 −

X σb2

=0

(11.71)

=0

(11.72)

⎛ 1 1 ⎞

= X + (11.73)

⎜ ⎟ σ 2a σb2 ⎝ σ a2 σb2 ⎠ The solution of this equation is xe, the value close to the actual value of variable x. xa

And hence,

+

xb

⎡ x x ⎤ ⎡ 1

1 ⎤

xe = ⎢ a2 + b2 ⎥ ⎢ 2 + 2 ⎥ ⎣ σ a σ b ⎦ ⎣ σ a σ b ⎦

(11.74)

1 1 /HWXVGH¿QHZHLJKW wa = 2 and wb = 2

σb σa The weighed average xe =

wa xa + wb xb wa + wb

σ a = σb , xbest =

If

(11.75)

(11.76)

xa + xb 2

(11.77)

N

In general, xe =

∑ wi xi

i =1 N

∑ wi

, where wi =

1 σi2

and error σ x = (∑ ωi )−1 2 (11.78)

i =1

Following example illustrates the use of weighed average. Example: In 1984, Wohl et al. reported the following precise values of the PDVVRIFKDUJHGʌPHVRQV:KLFKYDOXHZRXOG\RXFKRRVHDVFRUUHFWYDOXH IRUIXUWKHUWUHDWPHQW" 0DVVRIʌPHVRQV NH9ʌPHVRQVRUSLRQVDUHVXEDWRPLFVKRUWOLYHG particles from hadron family): S. no.

Value

Standard error

1

139569

8

2

139571

10

3

139568.6

2.0

4

139566.7

2.4

5

139565.8

1.8

6

139567.5

0.9

Error Analysis and Statistical Methods 343 N

xbest =

∑ wi xi

i =1 N

∑ wi

, where wi =

1 σi2

.

i = 1

Express each value of mass as m = (139560 + x “ǻx keV. Weigh each value inversely as the square of its standard error: x

ǻx

w ǻx)2

wx

9

8

2

18

11

10

1

11

8.6

2.0

25

215

6.7

2.4

17

114

5.8

1.8

31

180

7.5

0.9

123

923

N

xe =

∑ wi xi

i =1 N

∑ wi

⎛ 1461⎞ =⎜ = 7.3 ⎝ 199 ⎟⎠

and

the

best

estimate

for

error

is

i = 1

σ x = (∑ ωi )−1 2

. Calculate the standard error; weight of 199 corresponds to error of 0.7. Thus, m = 139567.3 ± 0.7 keV.

11.6 Fitting of Data Quite often, trend of the change in the physical quantity is an important issue WKDQDFWXDOYDOXHVRIWKHGDWDSRLQWV&RQVLGHU¿WWLQJRIWKHGDWDLQDVWUDLJKW line. For a freely falling body, 

Ȟ Ȟ0 + gt

(11.79)

Or, consider an ideal gas equation; PV = nRT, where pressure and temperature vary linearly. The equation that in general represents a linear relation is y = mx + c

(11.80)

The pair of points (x, y) should be close to the line or should be equidistant from the line.

344

Foundations of Experimental Physics

In general, if y and xDUHOLQHDUO\UHODWHGRQHKDVWR¿QGRXWWKHOLQHWKDW EHVW¿WVWKHGDWD

11.6.1 Linear Regression or Least Square Fit to the Line Let us assume that the measurement of each yi is governed by the Gaussian or QRUPDOGLVWULEXWLRQZLWKWKHVDPHZLGWKSDUDPHWHUıy for all the measurements. (true value of yi) = mxi + c ………if m and c are known The measurement of yi is governed by the normal distribution centered on WUXHYDOXHZLWKWKHZLGWKıy. Assume that the uncertainty in x is very small and hence one can neglect the same.

Fig. 11.11: Fitting of a data in a linear curve for variable x without any uncertainty and variable y normally distributed about Y and standard deviation ıy.

The probability of obtaining the observed value yi P( yi ) ∝

1 −( yi − c − mxi )2 / 2σ2y e σy

(11.81)

Probability depends on values of m and c. The probability of obtaining the complete set of measurements y1, y2, ...., yn is the product of individual probabilities. Pm, c ( y1 , y2 , ..., yn ) = Pm, c ( y1 ) × Pm, c ( y2 ) × " × Pm, c ( yn ) ∝

(11.82)

1 −χ2 / 2 e , Nσ y

2 where the exponent is given by χ =

n

( yi − c − mxi ) 2

i =1

σ 2y

∑

(11.83)

Error Analysis and Statistical Methods 345

The values of m and c can be estimated for which the probability P is PD[LPXPRUIRUZKLFKȤ2 is minimum. ∂χ 2 ⎛ −2 ⎞ n = ⎜ ⎟ ∑ ( yi − c − mxi ) = 0 ∂m ⎝ σ 2y ⎠ i = 1

(11.84)

∂χ 2 ⎛ −2 ⎞ n = ⎜ ⎟ ∑ xi ( yi − c − mxi ) = 0 ∂c ⎝ σ 2y ⎠ i = 1

(11.85)

These can be written as simultaneous equations in m and c as cN + m∑ xi = ∑ yi

(11.86a)

c ∑ xi + m∑ xi2 = ∑ xi yi

(11.86b)

⎡N ∑ xi ⎥⎤ ⎡m⎤ = ⎢⎡∑ yi ⎥⎤ ⎢ ⎢ ⎥ ⎢⎣ ∑ xi ∑ xi2 ⎥⎦ ⎣c ⎦ ⎢⎣ ∑ xi yi ⎥⎦ Therefore, least square constants m and c are c= m=

(∑ xi2 )(∑ yi ) − (∑ xi )(∑ xi yi ) Δ

N (∑ xi yi ) − (∑ xi )(∑ yi ) Δ

⎛ ⎞ Δ = N ⎜ ∑ xi2 ⎟ − (∑ xi )2 ⎝ i ⎠

(11.87)

(11.88) (11.89)

(11.90)

Uncertainty in the measurement of yi is σ 2y =

1 N

∑ ( yi − c − mxi )2

(11.91)

11.6.2 Least Square Fit to the Other Curves Variable y can be expressed as polynomial of a second variable x: y = a + bx + cx2

(11.92)

We have series of measurements (xi, yi) with all yi equally uncertain, all xi are exact. Assume that yi are governed by normal distribution; a, b, c are unknown and need to be determined. The probability of obtaining the observed value yi is P( yi ) ∝

1 −( yi − a − bxi − cxi2 )2 / 2σ2y e σy

(11.93)

346

Foundations of Experimental Physics

χ2 =

n

( yi − a − bxi − cxi2 ) 2

i =1

σ 2y

∑

(11.94)

a, b, c are those values for which PLVWKHODUJHVWȤ2 is minimum. Differentiate Ȥ2 with respect to a, b, c and setting the derivatives equal to zero, the equations can be solved by iterations. In principle, one should show two curves: one actual data points and model curve as smooth curve. Second plot should show residuals. 3RRU¿WVFDQRFFXULQLWHUDWLRQVEHFDXVH (1) initial guess is not good, (2) the program has exceeded the built in limit on the number of iterations. Consider an exponential function y = aebx To linearise take a logarithm, ln y = ln a + bx

(11.95) (11.96)

If we have series of measurements (xi, yi), then for each yi we can calculate zi = ln yi

(11.97)

The pairs (xi, zi VKRXOGOLHRQWKHOLQH7KHOLQHFDQEHEHVW¿WWHGE\WKH method of least squares to give best estimates for the constants ln a and b. (YHQ LI ıy the uncertainty in yi is the same for all measurements, the σy dz σy = , and hence uncertainty in zi = ln yi is not same. In fact, σ z = dy y HYHQLIıyLVVDPHIRUDOOPHDVXUHPHQWVız is larger if y is smaller. In this case, one has to use a weighted least square method.

11.6.3 Data to Fit the Straight Line Consider two physical variables x and y connected by a linear relation y = mx + c (11.98) The measurement of each yi is governed by the Gaussian distribution with WKHVDPHZLGWKSDUDPHWHUıi for all measurements. In case, m and c are known, true value of yi can be computed as (11.99) (true value of yi) = mxi + c The measurement of yi is governed by the Gaussian distribution with ZLGWKıy. The probability of obtaining the observed value yi is 1 − ( yi − c − mxi )2 / 2σ2y Pm, c ( yi ) ∝ e (11.100) σy where the subscripts m and c indicate that this probability depends on values of m and c.

Error Analysis and Statistical Methods 347

The probability of obtaining the complete set of measurements y1, y2, ...., yN is the product: Pm, c ( y1 , y2 , ..., y N ) = Pm, c ( y1 ) ... Pm, c ( y N ) ∝ where the exponent is given by χ = 2

N

∑

1 σ Ny

2

e− χ

/2

(11.101)

2

(yi − c − mxi )

(11.102) σ 2y Use the principle of maximum likelihood to determine the uncertainty in VWDQGDUGGHYLDWLRQıy. i =1

The value of ıy, for which the probability P may be maximum when ⎤ ⎡N ∂p N + 3) = 0 = σ −( ⎢ ∑ ( yi − c − mxi ) 2 − Nσ 2 ⎥ exp(− χ 2 / 2) y ∂c ⎥⎦ ⎣⎢i = 1

(11.103)

1 N (11.104) ∑ ( yi − c − mxi )2 N i =1 Some measurements are discrete. The distribution function is essentially a list of probabilities P(xi) associated with each of the possible values of xi. The Poisson distribution is a discrete distribution and arises in counting experiments. ,IWKHDYHUDJHQXPEHURIHYHQWVLVȝWKHQWKHSUREDELOLW\RIREVHUYLQJx events is 2 And therefore, (σ y ) =

μ x −μ e (11.105) x! ,IȝWKH3RLVVRQGLVWULEXWLRQORRNVOLNHGHFD\LQJH[SRQHQWLDO,Iȝ! WKH3RLVVRQGLVWULEXWLRQLVSHDNHGDQGORRNVOLNHWKHQRUPDOGLVWULEXWLRQı2 is the sample variance. A plot of normal distribution is also called as Gaussian distribution or a bell-shaped curve. Pp ( x, μ) =

11.7 Covariance Covariance is a measure of how changes in one variable are associated with the changes in another variable. Let us consider the measurement of two quantities x and y in order to determine the function z(x, y). Uncertainty in z is δz =

∂z ∂z δx + δy ∂x ∂y

348

Foundations of Experimental Physics

If errors in x and y are independent and random, 2

⎛ ∂z ⎞ ⎛ ∂z ⎞ δz = ⎜ δx⎟ + ⎜ δy⎟ ⎝ ∂x ⎠ ⎝ ∂y ⎠

2

If the measurements of x and y are governed by independent normal GLVWULEXWLRQVZLWKVWDQGDUGGHYLDWLRQVıxDQGıy, then the values of z(x, y) are normally distributed with standard deviation: 2

⎛ ∂z ⎞ ⎛ ∂z ⎞ σz = ⎜ σx ⎟ + ⎜ σ y ⎟ ⎝ ∂x ⎠ ⎝ ∂y ⎠ 2 The standard deviation of N measurements σ x =

ıx the width of Gaussian function; 1 σ x 2π

e−( xi − X )

2

2

1 N ∑ (xi − X )2 , where N i =1

/ 2σ2x

7R ¿QG z(x, y), measure N pairs of data (x1, y1)....(xn, yn). Compute the mean value x DQGWKHVWDQGDUGGHYLDWLRQıx: zi = z ( xi , yi ) ∂z ∂z (xi − x ) + ( yi − y) (11.106) ∂x ∂y with assumption that the standard deviation is small. Partial derivatives are taken at x = x and y = y . With this approximation the mean becomes ≈ z( x , y ) +

z= =

1 N

∑ zi

N i =1 ⎤ 1 N ⎡ ∂z ∂z z ( x , y ) + (xi − x ) + ( yi − y)⎥ ∑ ⎢ N i = 1⎣ ∂x ∂y ⎦

(11.107)

≈ z( x , y ) N

N

i =1

i =1

∑ (xi − x ) = 0 = ∑ ( yi − y)

as

(11.108)

Consequently, the task is to only calculate the function, z (x , y ). The standard deviation of N values z1, z2, ...., zn:

σ 2q =

1 N

∑ (zi − z )2 =

⎤ 1 N ⎡ ∂z ∂z ∑ (xi − x ) + ∂y ( yi − y)⎥ N i = 1 ⎢⎣ ∂x ⎦

2

(11.109)

Error Analysis and Statistical Methods 349

Expanding one yields, 2

2

⎛ ∂z ⎞ 1 ⎛ ∂z ⎞ 1 σ 2q = ⎜ ⎟ (xi − x )2 + ⎜ ⎟ ∑ ∑ ( yi − y)2 ⎝ ∂x ⎠ N ⎝ ∂y ⎠ N +2

∂z ∂z 1 N ∑ (xi − y)( yi − y) ∂x ∂y N i = 1

/HWXVGH¿QHFRYDULDQFH σ xy = One can then write, σ 2z

1 N ∑ (xi − x )(yi − y ) N i =1

(11.110) (11.111)

2

2

⎛ ∂z ⎞ ∂z ∂z ⎛ ∂z ⎞ = ⎜ ⎟ σ 2x + ⎜ ⎟ σ 2y + 2 σ xy ⎝ ∂x ⎠ ∂x ∂y ⎝ ∂y ⎠

(11.112)

If the measurements of x and y are independent, then ıxy approaches 0; whatever is the value of yi, the difference xi − x could be negative or positive. After many measurements, it balances. If the measurements of x and y are not independent, then the covariance ıxy need not be zero. In fact, when the covariance is not zero, the errors in x and y are correlated. 7KHFRYDULDQFHVDWLV¿HVWKH6FKZDU]LQHTXDOLW\ σ xy ≤ σ x σ y

(11.113)

Let us prove the Schwarz inequality. Let us take some arbitrary value of t. For the arbitrary value of tZHFDQGH¿QHA(t) as 1 2 [(xi − x ) + t ( yi − y)] ∑ N 1 = ∑ ⎡⎣( xi − x ) 2 + t 2 ( yi − y) 2 + 2t ( xi − x )( yi − y ) ⎤⎦ N

A(t ) =

Therefore, A(t ) = σ 2x + t 2 σ 2y + 2tσ xy

(11.114) (11.115)

Also note that, A(t • /HWXV¿QGPLQLPXPYDOXHRIA(t) which will also be greater than zero. 7R¿QGWKHPLQLPXPYDOXH at and

∂A = 0 = 2σ xy + 2tσ 2y ∂t

(11.116)

2 Amin, σ xy = − tσ y

Amin • For minimum value of A,



350

Foundations of Experimental Physics

A(t ) ≥ 0 σ 2x − 2t 2 σ 2y + t 2 σ 2y ≥ 0 σ 2x − t 2 σ 2y ≥ 0 t≤

σx σy

(11.118)

Substituting this value in the expression for A(t), we get, A(t ) = σ 2x + t 2 σ 2y + 2tσ xy = σ 2x +

σ 2x σ 2y

= 2σ 2x + 2 However,

σ 2y + 2

σx σ xy σy

σx σ xy σy

A(t) t 0 σx σ xy ≥ 0 σy

which leads to

2σ 2x + 2

as a result,

σ x σ y ≥ σ xy

(11.119) (11.120) (11.121) (11.122)

 &RHI¿FLHQWRI/LQHDU&RUUHODWLRQ Covariance for linear curve estimates how well the data points (x1, y1)… (xn, yn) are linearly related. The extent to which the set of points (x1, y1)…(xn, yn) support a linear UHODWLRQVKLSLVPHDVXUHGE\WKHOLQHDUFRUUHODWLRQFRHI¿FLHQW r= Therefore, r=

σ xy σxσ y

∑ (xi − x )( yi − y ) 12 ⎡ ∑ (x − x )2 ∑ ( y − y )2 ⎤ i i ⎣ ⎦

(11.123)

(11.124)

Prove that |r| d 1. Let us assume y = mx + c, which leads to yi = mxi + c. Therefore,

y = mx + c

(11.125)

which implies,

( yi − y) = m( xi − x )

(11.126)

Error Analysis and Statistical Methods 351

Therefore, r =

m∑ (xi − x ) 2

2 ⎤1 2

⎡ ∑ (xi − x ) 2 m 2 ∑ (xi − x ) ⎣ ⎦

=

m = ±1 m

(11.127)

11.9 F2 Test for the Distribution If measurements are repeated several times, it would be expected that it will follow one or more statistics. Discrete variables may obey binomial statistics, multinomial statistics or Poisson statistics. If the variable is continuous, one has to use either Gaussian function, or Lorentzian function or in some cases, exponential decay 1 e−t τ . One can predict regarding the probability τ distribution. However, one needs to check whether predicted distribution is correct or not. The F2GLVWULEXWLRQLVXVHGWRWHVWWKHJRRGQHVVRI¿WRIWKHFDOFXODWHGGDWD with the experimental data. For continuous variable, Gaussian function is f ( x) =

1 σ x 2π

e − ( xi − X )

2

/ 2 σ 2x

(11.128)

and Lorentzian function is γN

(11.129) 2 γ 4π 2 ( ν − νo )2 + N 4 While exponential decay that is typically seen in the radioactive nuclei I ( ν) =

f ( x) = e − x a

(11.130)

For discrete variable, one observes Binomial distribution, which is given by probability of v successes in n trials: p( ν) =

n! p ν q1 − ν ν!(n − ν)!

(11.131)

where q = 1  p. Poisson distribution is given by p( ν) =

μ ν −μ e ν!

(11.132)

ȝLVWKHPHDQDQGvFRXQWVLQGH¿QLWHLQWHUYDO How to decide whether actual experimental results are governed by the expected limiting distribution? For instance, assume that we have measured certain quantity several times. Determine the average as the best estimate for X.

352

Foundations of Experimental Physics

x=

1 N

∑ xi

'HWHUPLQHVWDQGDUGGHYLDWLRQı σ=

1 N

∑ ( xi − x )2

Choose the bins. Typically, bins can be divided as ... x − 2σ, x − σ, x, x + σ .... Count the number of measurements O that fall p in bin p. Count the number of measurements that fall in bin p; say Ep assuming the Gaussian or normal distribution. ap + 1

Ep = N

∫

dx e − x

2

2σ 2

(11.133)

ap

For a Gaussian distribution as shown in Fig. 11.12, the values of probabilities P1, P2, P3, and P4 are 16%, 34%, 34%, and 16% respectively.

Fig. 11.12: Fraction of probability in sectors of the Gaussian curve.

Probability that one measurement falls in the interval, a”x ”b is given by area under the curve of Gaussian distribution. How well does the observed number Op compare with the expected number Ep",IWKHK\SRWKHVLVWKDWWKHPHDVXUHPHQWVDUHQRUPDOO\GLVWULEXWHG is correct, then Op – Ep would be small. How large Op – Ep would be acceptable to us" Op should have an average value EpDQGZRXOGEHH[SHFWHGWRÀXFWXDWH

around Ep with standard deviation /HWXVGH¿QH

2 χ =

Ep .

n

(O p − E p ) 2

p =1

Ep

∑

(11.134)

2 2 If χ = 0 , then agreement is perfect. If χ ≤ n , the observed and expected 2 distribution agrees. In case, χ ≥ n total disagreement is the case, where n is number of bins.

Error Analysis and Statistical Methods 353

11.10 Summary

Standard deviation σ =

∑ (xi − μ)2 i

N

(measured value of x) = xe ± δx δx Fractional uncertainty = x e Propagation of uncertainties:

For sums and differences δz ≈ δx + δy + " + δu XSSHUERXQGRQįz) δz = (δx)2 + (δy)2 + ...... (independent random errors) Product and quotient

δz δx δy ≈ + + ..... XSSHUERXQGRQįz) z x y 2

2

⎛ δy ⎞ δz ⎛ δx ⎞ = ⎜ ⎟ + ⎜ ⎟ + ... (independent and random errors) ⎝ x⎠ ⎝ y⎠ z If z = bx, where b is known exactly, δz = b δx . dz If z is a function of one variable, z(x), then δz = δx . dx δz δx = n If z is a power, z = xn, then . z x If z is any function of several variables, x, y, 2

2

⎛ ∂z ⎞ ⎛ ∂z ⎞ Then, δz = ⎜ δx⎟ + ⎜ δy⎟ + " (for independent random errors) ⎝ ∂x ⎠ ⎝ ∂y ⎠ For any limiting distribution, f(x) for measurement of a continuous variable x: f(x)dx = probability that any one measurement will give an answer between x and x + dx. b

∫a

f ( x) dx = probability that any one measurement will give an answer

between x =a and x = b. The normal distribution is f ( x) = PZLWKLQı  

1 σ 2π

e −( x − X )

2

/ 2σ2

Weighed averages:

If x1, x2, ... are measurements of the same quantity x, with known uncertainties,

ı1ı2, ... then the best estimate for x and error are

354

Foundations of Experimental Physics

∑ wi xi i

xe =

∑ wi i

and

σ x, e = (∑ ωi )−1 2 /HDVWVTXDUH¿WWRDVWUDLJKWOLQHy = mx + c.

Least square constants m and c are

c=

(∑ xi2 )(∑ yi ) − (∑ xi )(∑ xi yi ) Δ

, m =

N (∑ xi yi ) − (∑ xi )(∑ yi ) Δ

⎛ ⎞ Δ = N ⎜ ∑ xi2 ⎟ − (∑ xi )2 ⎝ i ⎠ The covariance and correlation: 'H¿QHFRYDULDQFH σ xy =

1 N ∑ (xi − x )( yi − y) N i =1

The extent to which the set of points (x1, y1)... (xn, yn) support a linear UHODWLRQVKLSLVPHDVXUHGE\WKHOLQHDUFRUUHODWLRQFRHI¿FLHQW r=

σ xy σxσ y

Therefore, r =

∑ (xi − x )( yi − y) 12 ⎡ ∑ (xi − x )2 ∑ ( yi − y)2 ⎤ ⎣ ⎦

7KHUHVXOWVRIDQ\UHSHDWHGPHDVXUHPHQWVFDQEHJURXSHGLQELQVk = 1, 2, …, n. Let Op GHQRWH QXPEHU RI PHDVXUHPHQWV REVHUYHG LQ ELQ k. While Ep denotes the number expected in bin p, based on some assumed distribution. 'H¿QHFKLVTXDUHDV χ2 =

n

(O p − E p ) 2

p =1

Ep

∑

The reduced χ 2 = χ 2 / d , dLVQXPEHURIGHJUHHVRIIUHHGRP 2 2 If reduced χ = χ / d  1 DJUHHPHQWEHWZHHQOk and Ek is bad. 2 2 If χ = χ / d  1 WKHDJUHHPHQWLVVDWLVIDFWRU\

Further Reading

1. Analysis and Presentation of Experimental Results, C. Christodoulides and G. Christodoulides Springer, Basel (2017). 2. An Introduction to Error Analysis, J.R. Taylor, Oxford University Press, University Science Books, Mill Valley, CA (1982). 3. Characterization of Materials, J.B. Watchman and Z.H. Kalman, Butterworth-Heinemann, Oxford (1993). 4. Experimental Physics, R.A. Dunlap, Oxford University Press, Oxford (1988). 5. Fundamentals of Molecular Spectroscopy, C.N. Banwell and E.M. McCash, McGraw-Hill International Limited (1996), 4th Edition. 6. Fundamentals of Vacuum Technology, W. Umrath, Leybold Vacuum (2007). 7. Instrumental Methods of Analysis, H.H. Willard, L.L. Merritt, J.A. Dean and F.A. Settle, CBS Publishers and Distributers, Delhi (1986). 8. Instrumental Analysis, D.A. Skoog, F.J. Holler and S.R. Crouch, Cengage Learning, Delhi (2007), 11th Indian reprint, (2012). 9. Laser Cooling, D. Wineland and W. Itano, Phys.Today (1987), 34. 10. Matter and Methods at Low Temperature, F. Pobell, Springer Verlag (2007), 3rd Edition. 11. Modern Spectroscopy, J.M. Hollas, John Wiley and Sons Limited (2004), 4th Edition. 12. Modern Vacuum Physics, A. Chambers, Chapman and Hall/CRC (2005), 1st Edition. 13. Nanotechnology: Principles and Practices, S.K. Kulkarni, Springer, Capital Publishing Company, New Delhi (2015), 3rd Edition. 14. Nanotribology and Nanomechanics, B. Bhushan, Springer Verlag (2011). 15. Photoelectron and Auger Spectroscopy, T.A. Carlson, Plenum Press, New York (1975). 16. Physical Principle of Electron Microscopy, F. Edger, Springer Science (2005). 17. Physics of Semiconductor Devices, S.M. Sze and K.K. Ng, John Wiley and Sons (2007). 18. Practical Physics, G.L. Squires, Cambridge University Press, Cambridge (1985), 3rd Edition.

356

Foundations of Experimental Physics

19. Principles of Lock in Detection and The State of Art, Zurich Instruments (2016). 20. Radiation Detection and Measurements, G.F. Knoll, John Wiley and Sons (2010), 4th Edition. 21. Solid State Electronic Devices, B.G. Streetman and S.K. Banerjee, Phi Learning Pvt. Ltd. (2009). 22. Solid State Physics, N.W. Aschroft and N. Mermin, Cengage Learning, Delhi (1976). 23. Solid Surfaces, Interfaces, and Thin Films, H. Luth, Springer (2010), 5th Edition. 24. Superconducting Quantum Interference Device, Instruments and Applications, R.L. Fagaly, Rev. Sci. Instrum. 77, 101101 (2011). 25. Techniques for Nuclear and Particle Physics Experiments, W.R. Leo, Springer Verlag (1994), 2nd Edition.

Index

Abbe’s Criterion 179, 182, 185, 211 Accelerator 4, 113, 114, 115, 119, 120, 123, 126, 141, 176 Accuracy 10, 15, 50, 141, 312, 315, 318, 320, 322 Accurate Results 322 Active Medium 69 Acoustic Gas Thermometry 313 Adiabatic Demagnetisation 287, 302­ 308 Airy Disk 182, 183 Angle Resolved Ultraviolet Photoelectron Spectroscopy (ARUPS) 223 Anharmonic Oscillator 238 Anode 47, 54 Anti-Stokes Lines 241, 242 Aperture 63, 88, 104, 108, 111, 182, 183, 186, 194, 196, 200, 202, 211 Atom Probe Microscope 5, 180, 193, 195 Atomic Force Microscope 5, 180, 187, 204, 207, 208 Atomic Scattering Factor 148, 149, 150, 174 Auger Peaks 218, 220, 226 Auger Process 76, 198, 226 Auger Spectroscopy 6, 168, 213 Average Value 14, 27, 131, 352 Avogadro’s Law 38 %DQGSDVV¿OWHU 104, 106 Bandwidth 10, 12, 13, 14, 15, 16, 22, 23, 68, 78, 83, 230 Bayard-Alpert (B.A.) Gauge 55 BCS Theory 274 Bin Size 331

Binomial Distribution 129, 335, 351 Black Body Radiation 63, 70, 71, 112, 312 Boltzmann Constant 12, 38, 63, 69, 89, 172, 199, 293, 314 Boltzmann Factor 70, 87, 242, 304 Box-car Averaging 24, 25, 27, 34 Boyle’s Law 37, 50, 51 Bragg-Brentano Diffractometer 155, 156 Bragg’s Law 5, 96, 146, 147 Bremsstrahlung 61, 75, 97, 127, 128, 140 Bright Field Image 202 Broadening 84, 86, 91, 156, 157, 253 Calorimetry 140 Cantilever 207-209 Cathodoluminescence 190, 197, 234 Cerenkov Detectors 139, 142 Cerenkov Radiation 61, 127, 139 Channel Electron Multiplier 111 Channeltron 110, 111, 112 Characteristic Times 130 Characteristic X-rays 75, 198, 200 Charge Coupled Device (CCD) 91-95, 111, 112, 182, 230 Charle’s Law 38 Chi square test 160, 354 Chopper 21, 22, 23 Clausius-Clapyron equation 297, 301 Cockroft-Walton Accelerator 114-117 Cockcroft-Walton Multiplier 114-115 Cockcroft-Walton Tension Multiplier 115-116 Cockcroft -Walton’s First Accelerator 116-117 &RHI¿HFLHQWRI/LQHDU&RUUHODWLRQ 350-351

358

Foundations of Experimental Physics

Cold Cathode Discharge Lamp 112 Cold Cathode Ionization Gauge 54, 56 Collision Broadening 85 Common Mode Rejection Ratio 17, 19 Composition Determination 218 Compton Effect 281 Compton Scattering 75, 128 138, 140, 146, 281 Compound Microscope 180, 182-184 Concentric Hemispherical Analyser 109-110, 168 Confocal Microscope 180, 186-187, 189 Contact Mode 209 Cooling Power 295, 296, 297, 299, 300, 306, 320 Cosine Function 20 Covariance 347, 349, 350, 354 Cryogenic Temperature 285-320 Cyclotron 103, 113, 114, 124-125, 126, 141 Cylindrical Mirror Analyser 108-109, 227 Dark Field Image 202, 203 Debye-Scherrer 152-154, 155, 174 Density of States 207, 223, 232, 305 Detectors 14, 15, 23, 30, 61, 112, 128, 130, 135, 136, 152, 153, 213, 230, 278 'LIIHUHQFH$PSOL¿HU Diffraction Grating 144, 230, 236 Diffraction Methods 143-177 Diffractometer 149-166 Debye-Scherrer X-ray Diffractometer 152-155 Diffusion Pump 44, 45, 59, 60 Digital Filter 25 Dilution Refrigerator 286, 292, 294­ 301, 302, 306, 320 Diode Laser 72, 73, 74 Discharge Lamps 64-66, 112, 224 Dispersing Elements 81-84 Distribution 63, 106, 128-129 Doppler Broadening 86 (I¿FLHQF\ (LQVWHLQ¶V&RHI¿FLHQWV

Elastic Scattering 145-146 Electrical Resistance Devices 316-318 Electroluminescence 66, 185, 233-234 Electromagnetic Calorimeter 140 Electromagnetic Radiation 78, 112 Electron Diffraction 96, 97, 105, 166­ 172 Electron Detectors 110, 200, 201 Electron Energy Analyser 104-110, 112 Electron Energy Loss Spectroscopy 97, 204 Electron Gun 62, 64, 95, 97-99 Electron Microscope 35, 97, 104, 113, 180, 192, 195-204, 212 Electron Spin 64, 247, 248, 259, 260, 262 Electron Spin Resonance 258-265, 283 Electronic Filter 16, 18, 21, 34 (OHFWURVWDWLF'HÀHFWLRQ$QDO\VHU Energy Level Diagram 185, 188, 189, 214, 220, 225, 264, 265, 269 Ensemble Averaging 25, 34 Environmental Noise 14, 15, 16, 34 Ewald Construction 148, 151, 152, 167 Extended X-ray Absorption Fine Structure 204 Fabry Perot Etalons 80 Faraday Cup 110-111 Fermi Level 74, 75, 205, 206, 207, 215, 218, 227, 244 Field Emission 98, 99 Field Emission Microscope 190, 191, 192 Field Emission Scanning Electron Microscope 200-202 Field Ion Microscope 180, 192-193 Filters 15, 16, 17, 20, 25, 153, 236 Fitting of Data 343-347 Flicker Noise 14, 16, 17, 34 Fluorescence 184, 185, 188, 231, 236, 237, 265-266, 267-268 Fluorescence Microscope 184-186, 187, 188 Flux Quantisation 275, 276 Fourier Transform 25, 27-32, 33-34, 159, 168, 240, 245, 249, 252

Index 359 Fourier Transform Infra Red

Spectroscopy (FTIR) 33, 94, 239,

240, 245

Fowler-Nordheim Equation 191

Gamma Radiation 265, 267, 270, 282

Gamma Rays 266, 267, 268, 269, 270,

281, 282

Gamma Ray Detectors 268

Gamma Ray Spectroscopy 247, 267,

281-282

Gas Flow 38, 39

Gas Thermometry 312, 313

Gated Integrator 23-24, 34

Gaussian Distribution 129, 335, 336,

337, 347, 352

Gaussian Function 86, 335, 348, 351

Geiger Muller Counter 130, 131, 133,

134, 135, 153, 336

Getter Pump 45, 60

Glow Bar 64

GM Counter 135, 136

Golay Cell 94

Grating Monochromator 81-84

Grazing Angle X-ray Diffraction

(GIXRD) 161-166

Grazing Incidence 150, 161, 162, 164,

170, 172

Harmonic Oscillator 85, 237, 238

Helium Gas Vapour Pressure

Thermometry 314-315

High Energy Neutron 138, 140, 141

High Pressure Discharge Lamps 65

Holographic Filter 80

Hot Cathode Ionization Gauge 49, 54,

55, 56

Ideal Gas Law 38

Impedance Matching 15-16

Integrated Sphere 230, 231

Integrator 19-20, 23, 24, 34

Intensity of Diffraction 148-149

Interference Filter 79-80

Ion Pump 46-48, 54, 59, 60

Ionisation 61, 66

Ionisation Chamber 131

Ionisation Counter 132-134, 135

Ionisation Gauge 54, 55, 56

IR Detectors 94

Johnson Noise 11-13, 14

Josephson Junction 272, 276, 277

Klystron 64, 261

Knudsen Number 39

Koopman’s Frozen Orbitals 217

Lambert-Beer Law 228

Laser 68-75

Laser Cooling 286, 287, 308-310

Lasing Action 13, 69, 71, 72, 73

Latent Heat of Evaporation 285, 288­ 292, 297, 302

Laue Diffractometer 149-152

Laws of Thermodynamics 288-290

Least Square Fit 344, 345, 354

LEED 105, 166, 167-170

Lennard Jones Potential 290

Light Emitting Diode (LED) 66-67,

89, 233

Linear Fitting 344

Linear Proton Accelerator 123-124

Linear Regression 344-345

Liquefaction of Gases 290-292

/RFNLQDPSOL¿HU 106

Locschmidt Number 38

Lorentzian Distribution 85, 351

Lorentzian Function 85, 351

Low Energy Electron Diffraction 105,

166, 167

Low Energy Neutron 138, 140

Low Pass Filter 16, 18, 24, 25

Low Pressure Discharge Lamps 65

Low Temperature 95, 285-287

Luminescence 66, 137, 138, 188, 197,

231-237, 319

Luminescence Spectrometer 235-237

Magnetic Cooling 303, 308

Magnetic Materials 175, 242, 277

Magnetic Moment 172, 175, 259

Magnetic Spectroscopy 247-283

Magnetic Susceptibilty 314, 319, 320

Magnetization Measurements 247, 270,

283

360

Foundations of Experimental Physics

Magnetron 64, 260, 304 0DJQL¿FDWLRQ McLeod Gauge 49, 50-52, 53, 60 Mean Free Path 36, 38, 39, 134, 223, 326 Meissner Effect 273, 274 Michelson Interferometer 29, 30, 31, 239, 245 Microscopy 62, 179-212 Microwaves 64, 260, 261 Modulation 20, 23, 30, 105, 106, 209, 210 Molecular Flow 40 Molecular Vibrations 61, 237, 245 Monochromator 61, 78-84, 112, 229, 230, 235, 236-237 Mössbauer Spectroscopy 247, 265-270, 283 Multiplet Splitting 218, 219-220 Multi Wire Proportional Counter 139­ 140 Nanoscope 180, 187-188 Nd : YAG Laser 72, 73 Near Edge X-ray Absorption Fine Structure 204 Nernst Glower 64, 239 Neutron Detection 140-141 Neutron Diffraction 145, 172-176 1HXWURQ5HÀHFWLYLW\ Noise 9, 10, 34 Noise Thermometry 314 Normal Distribution 327, 335, 336, 344, 345, 347, 348, 352, 353 Nuclear Accelerators 113-142 Nuclear Adiabatic Cooling 286, 320 Nuclear Magnetic Cooling 308 Nuclear Magnetic Resonance 34, 247, 248, 249, 258, 283 Nucleus-Neutron Interaction 173 Nuclear Spectroscopy 248-258 Nuclear Spin 249-250, 252, 255, 258, 262, 263, 264, 292, 294, 296, 308 Numerical Aperture 88, 182, 183 Nyquist Noise 12, 312 O-ring 58, 59 Offset 9

Optical Fibre 87-88, 211, 318, 319 Optical Filter 79-80 Pair Production 128, 138, 140, 281 Particle Detectors 127-140 Pelletron 120-121 Peltier Effect 287 Peltier Element 287 Penning Gauge 54 Periodic Lattice 174 Phonons 76, 88, 241, 268, 274, 293, 296, 303, 304, 316, 317 Phosphorescence 184-185, 231, 245 Photodetector 89, 91, 186, 199, 208, 234, 235 Photodiode 21, 89-90, 94, 112, 186, 230, 236 Photoelectric Effect 90, 128, 138, 140, 215, 225, 327 Photoluminescence Spectroscopy 213, 232 Photomultiplier Tube 90-91, 111, 136, 137, 186, 230 Photon 62 Photon Detectors 88-89, 90, 94 Photon Source 62-78, 112, 222, 223 Pirani Gauge 49, 53-54, 322, 323 Pixel 92-93 Planck’s Constant 63, 84, 144, 206, 260 Planck’s Radiation Law 63 Plasmon Loss 218, 221 Platinum Resistance Thermometer 317 Poisson Distribution 13, 129, 336, 347, 351 Pomeranchuk Cooling 301-302, 320 Population Inversion 68, 72, 73, 74, 75 Position Sensitive Detectors 195 Powder Diffraction 157, 158, 160, 161, 177 Precise Results 323 Precision 113, 276, 321, 322 Pressure 37 Principle of Maximum Likelihood 341­ 343, 347 Prism Monochromator 81

Index Probability 65, 67, 70, 97, 99 Probe Microscope 180, 187, 188, 195 Propagation of Errors 324-329 Propogation of Uncertainty 72, 84, 85, 128, 292, 316, 321, 326, 336 Proportional Counter 130, 131, 132, 134-135, 139-140, 269 Pump 42-43 Pumping 38, 44, 69 Pumping Speed 39, 40, 42, 44, 45, 47 Q Band Microwaves 64 Quadrupole Mass Spectrometer 56-57, 60 Radiation 30, 31, 34, 61, 63, 64, 65 Radiation Detectors 128, 141, 269 Radioactivity 113, 117, 130, 142, 233 Radio Frequency Source 64, 121, 122, 248, 253, 254 Raman Depolarization Ratio 241 Raman Effect 189, 242, 244 Raman Microscopy 189, 245 Raman Spectroscopy 34, 112, 189, 241-245 Random Error 321, 323, 330, 331, 335 Rayleigh’s Criterion 181 Rayleigh Scattering 237, 242 RBS 247, 277, 278, 279, 280, 283 5HÀHFWHG+LJK(QHUJ\(OHFWURQ Diffraction (RHEED) 167, 170­ 172 Resolution 32-33 Resonant Cavity 72, 74, 261, 262 Resonant Frequency 123, 208, 209, 252, 254, 255 Resonance Spectroscopy 247-248, 249 Resonator 69, 73 Retarding Field Analyser 105-106, 108, 112, 168, 227 Reynold Number 39 Rietveld Method 157-161 Rotary Vane Pump 41-43, 44 Rotons 293 Ruby Laser 72, 73 Satellites 220

361

Scanning Electron Microscope 180, 196, 199-200 Scanning Near Field Optical Microscope 188, 210-211 Scanning Probe Microscope 195, 204­ 211, 212 Scanning Tunnelling Spectroscopy 207 Scintillation 116, 127, 134, 268 Scintillation Detector 134, 136-138, 153 Seebeck Effect 52, 95, 315 Seeman-Bohlin Geometry of XRD 156-157 Secondary Electrons 91, 95, 96, 105, 106, 111, 198, 201, 202 Semiconductor Detectors 138, 139 Semiconductor Devices 36, 88, 91, 318-319 Semiconductor Laser 74 Sensitive Photodetectors 186 Sensitivity 90, 91, 129-130 SERS 243, 244 Shot Noise 13-14, 15, 16, 34, 89, 105, 108, 314 Seemann-Bohlin 156 Signal 9, 10, 11 Signal-to-noise Ratio 9-11, 15, 20, 23, 24, 25, 26, 27, 33, 34, 236, 240, 285 Simple Microscope 181-182 Sloan and Lawrence Type Linear Accelerator 114, 121, 122, 123 Solid State Laser 72, 73, 74, 112 Small Angle Neutron Scattering 175­ 176 Small Angle X-ray Scattering (SAXS) 161, 164-166 Sonoluminescence 234-235 Sorption Pump 46, 47 Spectral Distortion 237 Spectral Lineshape 265 Spectroscopy 63, 64, 78, 91 Spin 172, 247 Spin Orbit Splitting 219 Spontaneous Emission 68, 70, 71, 188, 189 SQUID 7, 247, 270, 272-277, 283, 314, 319

362

Foundations of Experimental Physics

Tandem Van De Graaf Accelerator 119-120, 141 Tapping Mode 209 Thermal Detector 94 Thermal Neutron 138, 141, 173 Thermal Noise 11, 12, 13, 16, 34, 89, 285, 320 Thermionic Emission 75, 98, 99 Thermistor 95, 318 Thermocouple 52, 53, 95, 315, 316, 341 Thermocouple Gauge 52-53 Thermoelectric Cooling 285, 286, 287­ 288 Thermoelectric Detectors 95 Thermoelectric Devices 315-316 Thermoluminescence 233 Thermometer 312, 313, 314, 315, 317, 318, 319, 321 Ultrahigh Vacuum System 60, 192, 217, 222 UV-vis-NIR Spectroscopy 6, 213, 228-231 Uncertainty 8, 72, 84, 85, 128, 316, 321, 324, 326, 328, 329, 336, 337 Uncertainty Principle 8, 292 Undulator 126 U-tube Manometer 49-50, 53, 60

Vacuum Materials 57-59, 60 Vacuum Pump 35, 36, 39, 41-57 Valence Band 74, 218, 219, 222, 223, 232, 233 Van de Graaf Generator 17-121 Van der Waal Forces 207, 210, 289, 290, 292, 295, 312 Vapour Pressure Thermometer 314, 315 Variance 128, 129, 132, 347 Vibrating Sample Magnetometer 7, 247, 270-271, 283 Wavelength Selector 78-84 Weighed Average 342 Wiggler 126 X Band microwaves 64 X-ray Diffraction 144, 145-149, 150, 152, 156, 157-158, 161, 167, 168, 174 X-ray Diffraction Patterns, analysis of 157-161 X-ray Photoelectron Spectroscopy 214, 215-222 ;UD\5HÀHFWLYLW\;55  X-ray Tube 75, 76, 97, 113, 153, 155, 164 Yield of Auger Electron 226

Vacuum 35-36 Vacuum Accessories 58-59 Vacuum Gauge 35, 48-57, 60

Zeeman Effect 260, 303