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Inelastic Scattering Inelastic Scattering Revision 1.0 5 Mar. 2020 Inelastic Scattering Brent Fultz Barbara and Stanley Rawn, Jr., Professor of Materials Science and Applied Physics with Tim Kelley, Mike McKerns, Jiao Lin, JaeDong Lee, Hillary Smith, Olivier Delaire Dept. Applied Physics and Materials Science, AP hˆ MS California Institute of Technology h j j i Pasadena, California 91125 USA © 2020 by B. Fultz Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all information and its use. The author and publisher have attempted to find the copyright holders of all images reproduced in this book, and apologize to copyright holders if permission was not obtained. If copyrighted materials were not acknowledged properly, please write to us so we can do so in future reprints. Except as permitted under U.S. Copyright Law, no part of this book may be reproduced by any means, elec- tronic, magnetic, chemical, mechanical, or other methods not yet invented, without written permission from the publisher. The use of general descriptive names, registered names, trademarks, etc., in this publication does not imply that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting and production by the author. Cover design by the author. This book is dedicated to the new user – may you benefit from our mistakes. VI Preface The bright future has arrived for inelastic scattering studies of materials, molecules, and condensed matter. The international science community and national sci- ence agencies have invested heavily in new instruments and new sources of neutrons and x-rays that are overcoming many of the historical limitations of inelastic scattering research. The newest generation of transmission electron microscopes offers highly monochromatic electrons that enable measurements of thermal excitations that would otherwise be buried beneath the elastic line. A new family of inelastic laser scattering experiments are now possible, such as those that couple vibrations of the cavity itself to photons in from laser. Higher intensities and instrument sensitivities are major and obvious im- provements. Experimental inelastic scattering has been particularly constrained by low countrates, forcing experimental compromises in energy resolution and intensity. The ARCS inelastic neutron spectrometer with its location at the high-power target station of the Spallation Neutron Source and with its high detection efficiency, for example, provides unprecedented experimental pro- ductivity, overcoming many of the restrictions caused by the low countrates that have accompanied inelastic neutron scattering experiments to date. As features in the data from inelastic scattering experiments become more visible, and averaging processes are less necessary, more information can be extracted from the experimental measurements. The goal of this book is to describe the underlying scattering physics and dynamic processes in materi- als, and show opportunities to elevate the level of science done with inelastic spectrometers. A large body of specialized knowledge is required to design modern experiments for inelastic instruments such as time-of-flight chopper spectrometers. This body of knowledge will only grow as new capabilities become available. Unfortunately, the underlying concepts are scattered over many disciplines. For example, the books on the theory of thermal neutron scattering by S. W. Lovesey and G. L. Squires are superb. Similarly, excellent solid-state physics texts by J. M. Ziman, C. Kittel, U. Rossler,¨ and N. W. Ashcroft and N. D. Mermin are available for understanding the principles of excitations in condensed matter. Today the concepts from neutron scattering and con- densed matter physics are not connected well by existing written texts. It is a challenge to organize a coherent presentation of this wide body of knowledge, connected by the needs of experimental inelastic scattering. This document did not begin as a textbook. The original draft was a manual VII VIII Preface of specifications for the ARCS data analysis software. Defining specifications is a major step in planning a software project and setting its scope. Writing a manual of specifications forces a high degree of detailed planning of classes and modules. As we struggled with these details on software structure, it became obvious that they should parallel as closely as possible the science and practice of experimental inelastic neutron scattering research. Besides the challenge of organizing the higher-level intellectual concepts, practical problems with notation became apparent almost immediately. (Should the scattering vector be Q~, κ~ or ∆~k? What about its sign?) The present book is intended for a spectrum of readers spanning from grad- uate students beginning their doctoral research in inelastic neutron scattering, researchers who need to learn how to use chopper spectrometers and their data analysis, and ourselves, the authors, who need a reference manual. Our heartfelt concern, however, is for the graduate student who enters the field of inelastic neutron scattering with no experience with instruments and only a sketchy understanding of the scientific principles. This text was designed to help the reader understand the principles and methods of inelastic scattering, and do so efficiently. Explanations are presented at the minimum level of detail required to un- derstand physical concepts. There is some sacrifice of the care in development found in The Theory of Neutron Scattering from Condensed Matter by S. W. Lovesey and Introduction to the Theory of Inelastic Neutron Scattering by G. L. Squires, but our goal was to provide explanations that are “best buys,” producing the most physical insight for the amount of intellectual effort required to understand them. Another goal was to present the field of inelastic scattering as a codified intellectual discipline, showing inter-relationships between different topics. To do so, the notation from other books has been altered in places, for example ~ ~ Q~, defined as ki kf was selected for the scattering vector so that κ could be consistent with its− usage in The Theory of Lattice Dynamics in the Harmonic Ap- proximation by A. A. Maradudin, et al. (It is an unfortunate, but well-established convention that the scattering vector in the diffraction literature is of opposite ~ ~ ~ sign, ∆k kf ki.) Graduate≡ − students learning the concepts presented in this book are assumed to have some understanding of scattering experiments – a good understanding of x-ray diffraction would be suitable preparation. The student should have some competence with the manipulation of Patterson functions in Fourier space to the level developed, for example, in Transmission Electron Microscopy and Diffractometry of Materials by B. Fultz and J. W. Howe, and should have some understanding of solid-state physics at the level of Principles of the Theory of Solids by J. M. Ziman or Solid-State Physics by H. Ibach and H. Luth.¨ Brent Fultz Pasadena April, 2020 Contents 1 Scattering 1 1.1 A Sampler of Scattering Mechanisms . 2 1.2 Coherence and Incoherence . 5 1.3 Momentum Transfer and Phase Shifts . 14 1.4 Born Approximation . 18 1.5 Correlation Function for Elastic Scattering – The Patterson Func- tion . 23 2 Inelastic Scattering 43 2.1 Correlation Function for Inelastic Scattering – The Van Hove Function . 44 2.2 Autocorrelation Functions . 54 2.3 Essence of Coherent Inelastic Neutron Scattering . 60 3 Inelastic Electron Scattering and Spectroscopy 69 3.1 Inelastic Electron Scattering . 69 3.2 Electron Energy-Loss Spectrometry (EELS) . 70 3.3 Plasmon Excitations . 79 3.4 Core Excitations . 83 3.5 Energy-Filtered TEM Imaging (EFTEM) . 99 3.6 Energy Dispersive X-Ray Spectrometry (EDS) . 104 3.7 Quantitative EDS . 115 Further Reading . 122 Problems . 123 4 General Formulation of Thermal Neutron Scattering 129 4.1 Quantum Behavior . 137 4.2 Practical Expressions for Phonon Scattering . 143 4.3 Magnetic Scattering . 147 Further Reading . 154 Problems . 155 IX X CONTENTS 5 Dynamics of Materials and Condensed Matter 157 5.1 Phonon Thermodynamics . 157 5.2 Heat Capacity . 163 5.3 Lattice Dynamics . 170 5.4 Computer Simulations of Lattice Dynamics . 174 5.5 Group Theory and Lattice Dynamics . 177 5.6 Spin Dynamics in Solids . 188 5.7 Simulations of Spin Dynamics . 197 5.8 Interactions between Thermal Excitations of Electrons and Phonons205 Further Reading . 210 6 Chopper Spectrometers 213 6.1 Concept of a Chopper Spectrometer . 213 6.2 Neutron Sources . 216 6.3 Neutron Guides . 219 6.4 Fermi Choppers . 227 6.5 Detectors . 229 6.6 Energy Resolution . 232 6.7 Q Resolution . 234 6.8 Optimization for ∆Q=Q in Elastic Scattering . 236 6.9 Optimization of ∆Q=Q for Inelastic Scattering . 238 6.10 Background . 243 6.11 Sample Design . 245 6.12 Sample Design: Worked Example of LiFePO4 . 250 6.13 Sample Design: Sachets for Powders . 251 Further Reading . 252 7 Essential Data Processing 257 7.1 Steps to Transforming Data into a Function of Energy and Mo- mentum . 260 7.2 Transformations and Information . 267 7.3 Absorption . 273 7.4 Neutron Weighting . 275 7.5 Calculation of Multiphonon Scattering . 275 7.6 Corrections for Simultaneous Multiple Scattering and Multi- phonon Scattering . 284 Further Reading . 291 8 Computational Scattering Science 293 8.1 Software for Inelastic Scattering . 293 8.2 Simulation of Neutron Scattering Experiments with MCViNE . 296 8.3
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