A Journey into Reciprocal Space : A crystallographer’s perspective

274 Pages·2021·42.274 MB·English

by  Emeritus Professor Anthony Michael Glazer

A Journey into Reciprocal Space
(Second Edition)
A crystallographer’s perspective

A Journey into Reciprocal Space
(Second Edition)
A crystallographer’s perspective
Anthony Michael Glazer
Physics Department, Oxford University, Oxford, UK
and
Jesus College, Oxford, UK
IOP Publishing, Bristol, UK
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ISBN 978-0-7503-3875-2(ebook)
ISBN 978-0-7503-3873-8(print)
ISBN 978-0-7503-3876-9(myPrint)
ISBN 978-0-7503-3874-5(mobi)
DOI 10.1088/978-0-7503-3875-2
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Contents
Preface ix
Acknowledgments x
Author biography xi
1 Direct space 1-1
1.1 What are crystals? 1-1
1.2 Miller indices 1-2
1.3 Symmetry operations and elements 1-4
1.4 Point-group symmetry 1-5
1.4.1 Symmetry operations of the first kind 1-5
1.4.2 Symmetry operations of the second kind 1-8
1.4.3 Point groups 1-12
1.5 Translational symmetry 1-14
1.5.1 Lattices 1-15
1.5.2 Unit cells 1-17
1.5.3 Planes and directions 1-19
1.5.4 Crystal systems 1-20
1.5.5 Crystallographic restriction theorem 1-21
1.5.6 Bravais lattices 1-22
1.6 Crystal structures 1-26
1.6.1 Convolution 1-27
1.6.2 Convolution applied to crystals 1-27
1.6.3 Examples of simple crystal structures 1-28
1.7 Space groups 1-34
1.7.1 Symmorphic space groups 1-35
1.7.2 Non-symmorphic space groups 1-38
1.7.3 Types 1-41
1.7.4 Subgroups and supergroups 1-43
References 1-44
2 The reciprocal lattice 2-1
2.1 A brief history 2-1
2.2 Definition of the reciprocal lattice 2-2
2.3 Construction 2-4
AJourneyintoReciprocalSpace(SecondEdition)
2.4 Geometrical calculations 2-8
2.4.1 Metric tensors 2-8
2.4.2 Interplanar distances 2-8
2.4.3 Example calculations 2-10
References 2-11
3 Diffraction 3-1
3.1 Introduction 3-1
3.2 Laue equations 3-1
3.3 Bragg’s Law 3-3
3.4 The Ewald sphere 3-7
3.5 Lost in reciprocal space? 3-7
3.5.1 Stationary crystal 3-8
3.5.2 Oscillating and rotating crystal 3-10
3.5.3 Polycrystalline powder 3-12
3.5.4 Laue diffraction 3-14
3.5.5 Energy-dispersive diffraction 3-17
3.5.6 Confusion over sub and super? 3-19
3.6 Imaging 3-21
3.6.1 Fourier transformation 3-24
3.6.2 A simple view of crystal diffraction 3-26
3.6.3 Lattice diffraction 3-30
3.7 Form factors 3-32
3.7.1 X-rays 3-32
3.7.2 Neutrons 3-35
3.7.3 Electrons 3-36
3.8 Structure factors 3-38
3.9 Thermal scattering 3-41
3.10 Intensities of reflections 3-45
3.11 Laue classes 3-50
3.12 Anomalous dispersion 3-51
3.13 Solution of crystal structures 3-54
3.13.1 The phase problem 3-54
3.13.2 Fourier synthesis 3-56
3.13.3 The Patterson method 3-59
3.13.4 Charge flipping 3-61
3.13.5 Rietveld refinement 3-62
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AJourneyintoReciprocalSpace(SecondEdition)
3.13.6 Total scattering analysis 3-64
3.13.7 What’s new? 3-67
3.14 Aperiodic crystals 3-69
3.15 Disordered and partially-ordered crystals 3-74
References 3-78
4 Dynamical diffraction 4-1
4.1 Multiple scattering 4-1
4.2 Renninger effect 4-3
4.3 Darwin’s dynamical theory 4-4
4.4 Bloch’s theorem 4-7
4.5 Two-beam approximation in electron diffraction 4-10
4.6 Pendellösung or thickness fringes 4-15
References 4-17
5 Waves in a periodic medium 5-1
5.1 Waves in space 5-1
5.2 Periodic boundary conditions 5-2
5.3 Brillouin zones 5-4
5.4 Wigner–Seitz cell 5-5
5.5 Higher-order Brillouin zones 5-10
5.6 Density of states 5-12
References 5-15
6 Thermal and electronic properties 6-1
6.1 Heat capacity of solids 6-1
6.1.1 Einstein model 6-2
6.1.2 Debye model 6-3
6.2 Vibrations of atoms 6-7
6.2.1 One-dimensional monatomic chain 6-8
6.2.2 One-dimensional diatomic chain 6-12
6.2.3 Lattice dynamics 6-19
6.3 Heat conduction 6-22
6.3.1 Normal processes 6-22
6.3.2 Umklapp processes 6-23
6.4 Measurement of phonon dispersion 6-24
6.4.1 Absorption spectroscopy 6-25
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AJourneyintoReciprocalSpace(SecondEdition)
6.4.2 Inelastic scattering of light 6-26
6.4.3 Inelastic scattering of neutrons 6-29
6.5 Free electrons in a metal 6-30
6.6 Tight-binding and nearly-free electrons 6-32
6.6.1 Tight-binding 6-33
6.6.2 Nearly-free electron 6-34
6.7 Metal or insulator? 6-37
6.7.1 Sodium 6-39
6.7.2 Calcium 6-39
6.7.3 Diamond, silicon, germanium 6-40
References 6-40
7 Distortion modes 7-1
7.1 Introduction 7-1
7.2 Atomic displacements 7-2
7.3 Octahedral tilting 7-5
7.4 Group representations 7-7
7.5 Distortion modes 7-10
References 7-17
Appendices
Appendix A A-1
Appendix B B-1
Appendix C C-1
Appendix D D-1
Appendix E E-1
Appendix F F-1
Appendix G G-1
Appendix H H-1
Index: With reference to Section Headings I-1
viii
Preface
Theconceptofreciprocalspaceisover100yearsoldandhasbeenofparticularuse
for crystallographers to understand the patterns of spots seen on a detector when
x-raysarediffractedbycrystals.However,ithasamuchmoregeneraluse,especially
inthephysicsofthesolidstate.Inordertounderstandwhatitis,howtoconstructit,
andhowtomakeuseofit,itisfirstnecessarytostartwiththeso-calledrealordirect
spaceandthenshowhowreciprocalspaceisrelatedtoit.Directspacedescribesthe
objects we see around us, especially regarding crystals, their physical shapes and
symmetries, and the arrangements of atoms within: the so-called crystal structure.
Reciprocal space, on the other hand, deals with the crystals as seen through their
diffraction images. Indeed, crystallographers are accustomed to working backward
from the diffraction images to the crystal structures, which we call crystal-structure
solution. In solid-state physics, one usually works the other way, starting with
reciprocal space to explain various solid-state properties, such as thermal and
electrical phenomena.
In this book, I begin with the crystallographer’s point of view of direct and
reciprocal space and then develop this in a form suitable for solid-state physics
applications. Note that, while for the crystallographer, reciprocal space is a handy
means of dealing with diffraction, for the solid-state physicist, it is thought of as a
way to describe the formation and motion of waves, in which case the physicist
thinks of reciprocal space in terms of momentum or wave-vector k-space. This is
because, for periodic structures, a characteristic of normal crystals, elementary
quantumexcitations,e.g.,phononsandelectrons,canbedescribedbothasparticles
and waves. The treatment given here will be, by necessity, brief, but I would hope
that this will suffice to lead the reader to build upon the concepts described. I have
tried to write this book in a suitable form for both undergraduate and graduate
students of what today we call ‘condensed matter physics’.
ix