Mirror Symmetry: The Mother of all Crystal Symmetries (Springer Series in Solid-State Sciences, 200) [1st ed. 2024] 9819983606, 9789819983605

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Springer Series in Solid-State Sciences 200

M. A. Wahab

Mirror Symmetry The Mother of all Crystal Symmetries

Springer Series in Solid-State Sciences Volume 200

Series Editors Manuel Cardona, Abt. Experimentelle Physik, MPI für Festkörperforschung, Stuttgart, Germany Peter Fulde, MPI für Physik komplexer Systeme, Dresden, Sachsen, Germany Thierry Giamarchi, Department of Quantum Matter Physics, University of Geneva, Geneva, Switzerland Armen Gulian, Institute for Quantum Studies, Chapman University, Ashton, MD, USA Bernhard Keimer, Max Planck Institute for Solid State Research, Stuttgart, Germany Yoshio Kuramoto, Sendai, Miyagi, Japan Roberto Merlin, Department of Physics, University of Michigan, Ann Arbor, MI, USA Hans-Joachim Queisser, MPI für Festkörperforschung, Stuttgart, Germany Sven Rogge, Physics, UNSW, Sydney, NSW, Australia Horst Störmer, New York, NY, USA Klaus von Klitzing, Max Planck Institute for Solid State Research, Stuttgart, Germany

The Springer Series in Solid-State Sciences features fundamental scientific books prepared by leading and up-and-coming researchers in the field. They strive to communicate, in a systematic and comprehensive way, the basic principles as well as new developments in theoretical and experimental solid-state science. We welcome submissions for monographs or edited volumes from scholars across this broad domain. Topics of current interest include, but are not limited to: • • • • • • • • •

Semiconductors and superconductors Quantum phenomena Spin physics Topological insulators Multiferroics Nano-optics and nanophotonics Correlated electron systems and strongly correlated materials Vibrational and electronic properties of solids Spectroscopy and magnetic resonance

M. A. Wahab

Mirror Symmetry The Mother of all Crystal Symmetries

M. A. Wahab Department of Physics Jamia Millia Islamia New Delhi, Delhi, India

ISSN 0171-1873 ISSN 2197-4179 (electronic) Springer Series in Solid-State Sciences ISBN 978-981-99-8360-5 ISBN 978-981-99-8361-2 (eBook) https://doi.org/10.1007/978-981-99-8361-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

This book deals with the concepts that are extremely important to understand the crystallographic problems: many aspects that are directly related to mirror symmetry and many others that are based on “mirror combination scheme.” Investigations reveal that mirror symmetry is the only fundamental symmetry in crystalline solids (this is similar to the term fundamental units in physics), discovered for the first time in crystallographic history. All other symmetries, such as rotations, inversions, translational periodicity and their compatible combinations that the crystals exhibit, can be easily derived from suitable combinations of mirrors. Hence, these are termed “derived symmetries” (similar to the term “derived units” in physics). It is true that this book is the result of motivation/inspiration from some fundamental concepts of mirrors, intersection of two plane mirrors at different angles and the following two fundamental sources that are truly the ideal representations of mirror combination scheme: (i) the principle of ray optics capable of producing perfectly periodic and infinite images of an object placed between the two parallel mirrors and (ii) the inherently primitive Wigner–Seitz and Brillouin–Zone cells containing the only atom at their centres, surrounded by many pairs of parallel planes (acting as mirrors) capable of producing the perfectly periodic direct and reciprocal lattices in 1D, 2D and 3D, respectively, without involving translational symmetry of any kind. In addition to the lattice generations, the mirror combination scheme is helpful in understanding many other crystallographic concepts which could not be understood otherwise, so far. Mirror combination scheme is inherently centrosymmetric (because of the presence of multiple pair of mirrors with a lattice point at the centre) and hence spherically symmetric in nature. Accordingly, the Wigner–Seitz unit cells, Brillouin–Zones and diffraction patterns (obtained from crystals) provide spherically symmetric results, which allow the rotation and inversion symmetries and forbid translational symmetries of any kind (either microscopic or macroscopic) in crystalline solids. Historically, the topic is as old as the subject of crystallography itself, as each and every crystal shape is inherently related to its symmetries. Further, the mirror being the only fundamental symmetry in crystalline solids, the topic is part and parcel of all the crystallography courses. Therefore, because of its fundamental nature and the v

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importance of the topic, it was decided to get this fundamental work published in the form of a book so as to reach the same to a larger audience. Also, the basic aim of this publication is to make everyone aware of the simplicity of crystallography as a subject, which at present is not so, to many students, teachers and even sometimes to experts. As a result, the crystallography is becoming a rare subject in the university curricula. On the contrary, due to its inherent interdisciplinary character, I personally believe that the crystallography as a subject should have been one of the most popular and favourite subjects in the universities all over the world. Therefore, it is for the crystallographers (particularly those at the decision-making positions have greater responsibilities) to look into the pertinent question of how and why the crystallography as a subject is getting difficult for all and hence becoming unpopular in the university system, day by day. In the book, I have tried my best to keep it simple. The title of the book itself suggests that the mirror as a fundamental symmetry and the mirror combination scheme as a concept are going to play many important roles in understanding different crystallographic problems. Therefore, from the application point of view, this text will have its own importance. It gives us a first-hand information about the terms fundamental and derived symmetries in crystallography. It helps us to correctly assign the point group symmetry to a crystal lattice of a given crystal system, and it helps us to remove the discrepancy existing in the number of space lattices and the corresponding Laue symmetries and many others, persisted for a long time. Besides resolving many crystallographic discrepancies and ambiguities, the most important feature of the mirror combination scheme is that it helps us to understand the crystal diffractions in terms of zone boundaries of a Brillouin zone and the relationship between the zone boundaries and the equivalent sets of Braggs’ planes. This work will certainly enhance the understanding of students/researchers and other readers who are working on various diffraction processes, and they will certainly feel the simplicity in determining the crystal structures to a great extent. The subject matters of the book have been divided into nine chapters, where each chapter is comprehensive and complete in itself. Presentations of mathematical treatment and the text have been kept easy. Also, they have been supported by a large number of illustrations (some of them appear for the first time in crystallographic history) to make the subject understandable to even undergraduate students. I sincerely hope that this book will be equally beneficial to students, teachers as well as to all other interested readers connected to crystallography and the subjects of related disciplines. Chapter 1 deals with the concepts related to plane mirrors. This includes the process of image formation through reflection, a concept quite wisely used in the derivation of Braggs’ equation, the change of handedness of the objects by plane mirrors, image formation from two inclined plane mirrors and two intersecting plane mirrors. Chapter 2 deals with the newly developed concept of mirror as the only “fundamental symmetry” and all others as “derived symmetries” in crystals. Keeping this concept in view, the geometrical model exhibiting the rotational, inversion and other symmetries is presented. To substantiate the results obtained from the geometrical

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model, the matrix model of intersecting mirrors by using orthogonal axes is also presented. Chapter 3 deals with the role of mirror combination scheme in understanding the concept of Wigner–Seitz unit cells (alternative to direct unit cells) in 1D, 2D and 3D lattices in direct space. Further, the derivation of correct number of symmorphic space groups with the help of mirror combination scheme has been discussed. The subject matters related to reciprocal space have been discussed in Chap. 4. Different ways of constructing reciprocal lattice, the concept of Ewald sphere and the importance of the limiting sphere have been discussed. Constructions of Brillouin zones in 1D, 2D and 3D, determination of higher order Brillouin zones and the relationships between diffraction geometry, Braggs’ planes, zone boundaries and centro-symmetry are also discussed. Motivated from the Braggs’ equation because of the inverse relationship between Braggs’ angle and interplanar spacing, Chap. 5 deals with the calculation of interplanar spacing and Braggs’ angle for cubic crystal system and some other elemental crystals to verify the results obtained from diffraction data of the corresponding crystals in reciprocal space. In Chap. 6, results obtained from some cubic crystals of elements and MX systems have been analysed in the light of mirror combination scheme. Chapter 7 deals with the aspects which provide us some logical explanations that do not favour the existence of any form of translational symmetry (either microscopic or macroscopic) in crystals. Chapter 8 deals with the resolutions of some ambiguities, discrepancies and confusions that existed in different forms in crystallography, by using the mirror combination scheme during last few years by us. The last chapter (Chap. 9) deals with the topics related to fundamental crystallography. Although proper care has been taken during the preparation of the manuscript, still some errors are expected to creep in. Any form of omissions, errors and suggestions brought to the knowledge of the author will be appreciated and thankfully acknowledged. I sincerely admit and acknowledge that I learnt the art of systematic presentation of the research work or other related scientific matters from my research supervisor Prof. G. C. Trigunayat (late), Department of Physics, University of Delhi, India. I also sincerely acknowledge the work of my two sons Mr. Khurram Mujtaba Wahab and Mr. Shad Mustafa Wahab for making the required diagrams and formatting the manuscript. My special thanks to other members of my family for their continuous support and encouragement during the preparation of the manuscript. I am indeed grateful to all the authors and publishers of books and journals mentioned in the reference/bibliography for freely consulting them and even borrowing some ideas during preparation of this manuscript. I am also grateful to Springer Nature for the timely publication of this book. New Delhi, India

M. A. Wahab

Contents

1 Fundamental Concepts Related to a Plane Mirror . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Plane Mirror and Ray Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Laws of Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Types of Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Image Formation by a Plane Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Effect of Mirror Rotation on Reflected Ray . . . . . . . . . . . . . . . . . . . . 1.7 Mirror Reflection and Handedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Handedness and Enantiomorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Image Formation by Two Inclined Plane Mirrors . . . . . . . . . . . . . . . 1.10 Number of Images Formed by Two Inclined Plane Mirrors . . . . . . 1.11 Image Formation by Two Intersecting Plane Mirrors Under Limiting Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Horizontal, Vertical and Dihedral Mirror Planes . . . . . . . . . . . . . . . . 1.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 3 4 6 7 10 11 14

2 Mirror: The Only Fundamental Symmetry in Crystals . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Analysis of Intersecting Mirrors Using Orthogonal Axes . . . . . . . . 2.3 Derivation of Rotational Symmetries from Mirror Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Crystallographic and Non-crystallographic Rotational Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Derivation of Inversion Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Generation of 1-D, 2-D and 3-D Point Groups . . . . . . . . . . . . . . . . . 2.7 Point Group Minimum (PGM) and Minimum Symmetry Form (MSF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Role of Mirror and Rotation in Crystals of Different Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Symmetry Verification of 2-D Lattices . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22

16 16 17 19

23 28 29 30 33 35 37

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2.10 Symmetry Verification of Low Symmetry 3-D Lattices . . . . . . . . . . 2.11 Allocation of Correct Point Groups to Crystal Lattices . . . . . . . . . . 2.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 43 45 46 46 50

3 Mirror Combination Scheme in Direct Lattice . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Procedure to Construct a Wigner–Seitz Unit Cell . . . . . . . . . . . . . . . 3.3 Combination of Two Parallel Mirrors and 1-D Lattice . . . . . . . . . . . 3.4 Mirror Combination Scheme in W–S of 1-D Lattice . . . . . . . . . . . . 3.5 Mirror Combination Scheme in W–S of 2-D Lattices . . . . . . . . . . . 3.6 Constructions of HCP, RCP Lattices and Their W–S Unit Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Mirror Combination Scheme in W–S of 3-D Lattices . . . . . . . . . . . 3.8 Derivation of Correct Symmorphic Space Groups . . . . . . . . . . . . . . 3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 55 56 57 57 63 65 67 68

4 Mirror Combination Scheme in Reciprocal Lattice . . . . . . . . . . . . . . . . 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 The Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Construction of Reciprocal Lattice (First Approach) . . . . . . . . . . . . 72 4.4 Construction of Reciprocal Lattice (Second Approach) . . . . . . . . . 78 4.5 Construction of Reciprocal Lattice According to Ewald . . . . . . . . . 79 4.6 Interpretation of Braggs’ Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.7 Diffraction Geometry and Ewald Sphere . . . . . . . . . . . . . . . . . . . . . . 85 4.8 Vector Form of Braggs’ Equation from Ewald Construction . . . . . 87 4.9 The Ewald Sphere and the Limiting Sphere . . . . . . . . . . . . . . . . . . . 89 4.10 Construction of Brillouin Zones in 1-D, 2-D and 3-D Lattices . . . . 92 4.11 Determination of Higher Order Brillouin Zones in Cubic Crystal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.12 Results from Comparison of Unit Cell Data of Cubic Crystal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 Importance of d-Spacing in Diffraction of Crystals . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Braggs’ Equation and Interplanar d-Spacing . . . . . . . . . . . . . . . . . . . 5.3 Equivalence of Braggs’ Equation and Laue Equations . . . . . . . . . . 5.4 Derivation of d-Spacing Formulae of Different Crystal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Using Cartesian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Using General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 105 108 110 110 112

Contents

Calculation of d-Spacing and Braggs’ Angle in Cubic Crystal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Calculation of d-Spacing in Primitive Lattices of Other Crystal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Diffraction Results of Cubic Crystals of Some Elements and MX-System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Diffraction Geometry, Braggs’ Planes and Zone Boundaries . . . . . 5.9 Diffraction Patterns, Brillouin Zones and Centro-symmetry . . . . . . 5.10 Diffraction Patterns, Brillouin Zones and Translational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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113 121 126 133 135 136 139 142

6 Study of Diffraction Results of Some Cubic Crystals . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A Brief Survey of Diffraction Conditions and Related Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Interpretation of Cubic XRD Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Indexing Procedure of Cubic XRD Powder Patterns . . . . . . . . . . . . 6.5 Analytical Study of Diffraction Results of Some Elemental Cubic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Analytical Study of Diffraction Results of Some MX-Cubic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143

7 Possibility of Translational Symmetry (if Any) in Crystals . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 A Brief Historical Background of Crystallographic Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Translational Symmetries/Translational Periodicity in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Earlier Proposed Systematic Absences in Crystals . . . . . . . . . . . . . . 7.5 Role of Lattice Centering (if Any) in Systematic Absences . . . . . . 7.6 Role of Screw Axes (if Any) in Systematic Absences . . . . . . . . . . . 7.7 Role of Glide Planes (if Any) in Systematic Absences . . . . . . . . . . 7.8 Reality of Enantiomorphous and Some Other Pairs in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Important Observations Against Translational Symmetries in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Some Obvious Reasons for Reduction in the Number of Space Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Some Obvious Inconsistencies/Contradictions About Screw Axes/Glide Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169

143 146 151 153 160 167 168

169 171 172 173 176 178 180 182 183 184 186 188

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8 Resolution of Existing Discrepancies, Ambiguities and Confusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Necessity of HCP and RCP Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Discrepancy in the Representation of RCP and CCP . . . . . . . . . . . . 8.4 Discrepancy in the Number of Space Lattices 14 or 16 . . . . . . . . . . 8.5 Ambiguity in 16 Space Lattices and 11 Laue Groups . . . . . . . . . . . 8.6 Discrepancies in the Allocation of Point Groups to 2-D and 3-D Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Resolution of Symmorphic Space Groups . . . . . . . . . . . . . . . . . . . . . 8.8 Confusion Over Centro-symmetric Nature of Diffraction Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Confusion Over Translational Symmetries in Crystals . . . . . . . . . . 8.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Fundamental Crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Crystals and Different Polyhedral Shapes . . . . . . . . . . . . . . . . . . . . . 9.3 Crystal Polyhedra and the Concept of Lattice . . . . . . . . . . . . . . . . . . 9.4 Lattice, Basis and the Crystal Structures . . . . . . . . . . . . . . . . . . . . . . 9.5 Crystal Dimension and Related Symmetries . . . . . . . . . . . . . . . . . . . 9.6 Miller Indices to Represent Points, Directions and Planes in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Representation of Directions and Planes in Cubic Crystal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Crystal Parameters in 2-D and 3-D Systems . . . . . . . . . . . . . . . . . . . 9.9 Crystal Systems and Axial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 189 190 191 192 193 194 195 195 196 197 198 199 199 199 200 204 208 211 217 219 220 221 224

About the Author

M. A. Wahab is Former Professor and Head of the Department of Physics, Jamia Millia Islamia, New Delhi, India, where he joined as Lecturer in 1985. He completed his Ph.D. (Physics) from the University of Delhi, India, and M.Sc. (Physics) from Aligarh Muslim University, India. Earlier, he served as Lecturer at the P. G. Department of Physics and Electronics, University of Jammu, Jammu and Kashmir, India, from 1981. During these years, he taught electrodynamics, statistical mechanics, theory of relativity, advanced solid-state physics, crystallography, physics of materials, growth and imperfections of materials and general solid-state physics. Author of five books, Solid State Physics, Essentials of Crystallography, and Numerical Problems in Solid State Physics, Numerical Problems in Crystallography and Symmetry Representations of Molecular Vibrations (the last two books from Springer Nature), Prof. Wahab has also contributed more than 100 research papers in several journals of repute and supervised 15 Ph.D. theses during his career at Jamia Millia Islamia. He has published the discovery of hexagonal close packing (HCP) and rhombohedral close packing (RCP) as the two new space lattices, along with his son (Mr. Khurram Mujtaba Wahab), as their first joint paper after his retirement.

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Chapter 1

Fundamental Concepts Related to a Plane Mirror

1.1 Introduction Had the fundamental nature of mirror been discovered earlier, many of its different concepts could have been used for better understanding of crystallographic problems. However, since it has already been discovered now and some crystallographic confusions/ambiguities have already been explained/resolved successfully using the concept ‘mirror combination scheme’, it is obligatory to review the fundamental concepts related to mirror in somewhat details not only just to refresh our knowledge but they also turn out to be quite useful for understanding the crystallography related problems in different ways and will be clear in other chapters. Keeping this in view, we have decided to present a brief review of the concepts related to mirror in a 2-D system in this chapter. This review is expected not only to provide us the foundation for rest of the topics to be discussed in other chapters but will also help us to understand the intricate issues in crystallography which could not have been possible otherwise.

1.2 Plane Mirror and Ray Optics A plane mirror is in general considered to be highly smooth and polished surface which reflects almost the entire light falling on it. This is made by polishing or silvering one side of a glass plate as shown in Fig. 1.1 [1]. When a ray of light is allowed to strike a reflecting surface such as a plane mirror at a certain angle with the normal, it is called the ‘incident ray’ and the angle which the incident ray makes with the normal at the point of incidence is called the ‘angle of incidence’. It is denoted by the letter ‘i’. Similarly, the ray of light obtained after reflection from the plane surface, in the same medium in which the incident ray is travelling, it is called the ‘reflected ray’ and the angle which the reflected ray makes

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. A. Wahab, Mirror Symmetry, Springer Series in Solid-State Sciences 200, https://doi.org/10.1007/978-981-99-8361-2_1

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1 Fundamental Concepts Related to a Plane Mirror

Fig. 1.1 Representation of a plane mirror

with the normal at the point of incidence is called the ‘angle of reflection’. It is denoted by the letter ‘r’. The perpendicular drawn at the point of incidence to the surface of the mirror is called the ‘normal’. The process of reflection through a plane mirror is illustrated in Fig. 1.2. Certain characteristics of the image formed by a plane mirror are: 1. Size of the image is same as the size of the object. 2. Image is laterally inverted (i.e. image appears to be inverted only on the vertical axis). 3. Image formed by a plane mirror is upright. 4. Image formed by a plane mirror is virtual. 5. Image formed by a plane mirror is as far behind the mirror as the object in front of the mirror.

Fig. 1.2 Reflection at a plane surface

1.4 Types of Reflection

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Fig. 1.3 A normal incidence and corresponding normal reflection

1.3 Laws of Reflection There are two laws of reflection. They are: 1. The incident ray, the reflected ray and the normal at the point of incidence, lie in the same plane as shown in Fig. 1.2. 2. The angle of incidence ‘i’ is equal to the angle of reflection ‘r’, i.e. ∠i = ∠r. The simplest example of the second law of reflection is if the angle of incidence is 0°, which is the case of a normal incidence. In this case, the angle of reflection is also 0°, this is the corresponding normal reflection. We see this situation, when we look straight into a mirror. The normal incidence and normal reflection are shown in Fig. 1.3.

1.4 Types of Reflection The Laws of Reflection are true for any given surface. However, this does not mean that all the parallel rays that approach a surface, then all the reflected rays should also be parallel. In fact, the direction of the reflected rays will depend on the texture of the reflecting surface (i.e. whether the texture of the given surface is smooth or rough). Based on this, the reflections can be put into two different categories. They are called: (i) Regular or specular reflection, and (ii) Irregular or diffused reflection. Regular or Specular Reflection When parallel rays of light are allowed to incident on a flat surface which is smooth and even, they are reflected as parallel rays in some other direction following the laws of reflection, this type of reflection is called the regular or specular reflection as shown in Fig. 1.4a. The regular or specular reflection occurs only when the light rays are reflected from a smooth and shiny surface. On such a surface, the direction of the normal to the surface remains the same at every point on it. Therefore, the parallel

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1 Fundamental Concepts Related to a Plane Mirror

Smooth Surface

Rough Surface

(a)

(b)

Fig. 1.4 a Specular reflection, and b diffused reflection

incident rays comes back as parallel reflected rays in the same plane. For example, the image formed while looking at a mirror is due to a regular or specular reflection. Reflections taking place from the objects like looking glass, still water, transparent oil, highly polished metal surfaces and crystal planes are some examples of this category. Regular or specular reflection is useful in image formation of a given object. Irregular or Diffused Reflection On the other hand, when parallel rays of light are allowed to incident on a rough surface containing bumps and curves, they get reflected in various possible directions and therefore an irregular or a diffused reflection will occur as shown in Fig. 1.4b. In such cases, the reflected rays are not parallel because each point on such a surface will have its normal in different direction. This means the angle of incidence is different at each point. Then according to the second law of reflection, each angle of reflection is also different. Reflections taking place from ground, walls, trees, suspended particles in air, polycrystalline samples are some examples of irregular or diffused reflections. It helps in the general illumination of places and helps us to see things around us.

1.5 Image Formation by a Plane Mirror Let us consider an object ‘O’, which is placed in front of a plane mirror MM1 . A ray of light is supposed to start from the point ‘O’ perpendicularly is reflected back along the same path as shown in Fig. 1.5. Another ray which moves along OB is reflected along BC by obeying the laws of reflection such that BN acts as normal at the point B. When we produce OA and CB backward such that they meet at a point I, then ‘I’ becomes the image of the object ‘O’. Therefore, we have to prove that the magnitude OA = AI. From the laws of reflection, we know that the angle of incidence is equal to the angle of reflection, i.e. ∠i = ∠r . Also, from the Fig. 1.5, we have

1.5 Image Formation by a Plane Mirror

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Fig. 1.5 Image formation by a plane mirror

∠i = ∠1

(i)

It is because they are pair of alternate angles. Similarly, the pair of corresponding angles is given by ∠r = ∠2

(ii)

Keeping in view the laws of reflection and comparing the angles in (i) and (ii), we have ∠1 = ∠2 Now, considering the following two s BAI and BAO as shown in figure, we have ∠1 = ∠2 Similarly, the angles, ∠3 = ∠4 It is because each angle is 90°. Further, the side BA is common to both. Therefore, the two triangles are equal, i.e. BAI = BAO. Thus, in particular it can be easily said that OA = AI.

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1 Fundamental Concepts Related to a Plane Mirror

1.6 Effect of Mirror Rotation on Reflected Ray In order to check the effect of rotation of a mirror (through a certain angle) on the reflected ray while keeping the position of the incident ray unchanged, let us consider a ray of light AB, which is made to incident on a plane mirror MM, in a position such that BN is the normal and BC is the reflected ray, where the angle ∠ABN = ∠CBN = ∠i, and ∠ABC = 2∠i

(i)

Because the angle of incidence is equal to the angle of reflection, i.e. ∠i = ∠r . Now, let the given mirror MM is rotated through an angle ‘θ’ about the point B while keeping the position of incident ray unchanged, such that M1 M1 is the new position of the mirror and BN1 is the new position of the normal. As the position of the incident ray remains the same, the new angle of the incidence becomes ∠ABN1 whose magnitude is (i + θ ). Further, let BD is the new reflected ray, such that ∠DBN1 is the new angle of reflection (i + θ ). From Fig. 1.6, we have ∠ABD = ∠ABN1 + ∠DBN1 = ∠(i + θ ) + ∠(i + θ ) = 2∠i + 2∠θ Subtracting Eqs. (i) from (ii), we obtain ∠CBD = ∠ABD − ∠ABC = 2∠i + 2∠θ − 2∠i = 2∠θ

Fig. 1.6 Effect of mirror rotation on reflected ray

(ii)

1.7 Mirror Reflection and Handedness

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Thus, for a given incident ray, if the plane mirror is rotated through a certain angle (say θ), then the reflected ray gets rotated through twice that angle (i.e., 2θ). This is an important concept of mirror optics and similarly very useful in crystallography.

1.7 Mirror Reflection and Handedness From the preceding section, we find that a reflection operation in a plane mirror reverses the direction of the axis perpendicular to it, but leaves the axes parallel to it unchanged. As a result, the mirror reflection changes the handedness of an object (symmetrical or asymmetrical) along the axis perpendicular to the mirror (whereas the translation and rotation operations do not affect the handedness). This suggests that there is a direct relationship between the axis inversion and handedness of the image. This becomes obvious when a right handed axial system is placed in front of a vertical (y–z) plane mirror such that the mirror reverses the direction of the x-axis and makes the axial system of the image, a left handed. This is the image of the right handed system in front of the mirror with the change of handedness, as shown in Fig. 1.7. This illustrates the change of handedness taking place in a single axis inversion in a plane mirror [2]. The handedness of a coordinate system can also be represented by simple mathematical treatment in terms of the cross-product of its Cartesian unit vectors. For a right-handed coordinate system, we can write xˆ × yˆ = zˆ and for a left handed system Fig. 1.7 Change of a right handed system into a left handed system

(1.1)

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1 Fundamental Concepts Related to a Plane Mirror

xˆ × yˆ = −ˆz

(1.2)

Now suppose that the original coordinate system (associated with the object) is a right handed system, represented by Eq. 1 and illustrated in Fig. 1.7. Inverting only the x-axis, this will correspond to the coordinate transformation: x → −x  , y → y  , z → z  . In Cartesian unit vectors form, this can be written as −xˆ  × yˆ  = zˆ  or xˆ  × yˆ  = −ˆz 

(1.3)

which represents a left handed system. Therefore, in general, the handedness of an image is related to the number of axes inverted by a mirror. We can also use another mathematically important treatment from crystallographic point of view. That is the matrix form (for details, refer next chapter). The above mirror operation can be equivalently represented as: ⎛

⎞ −1 0 0 ⎝ 0 1 0 ⎠ = m[100] 0 01

(1.4)

This simply shows that the mirror plane is perpendicular to x-axis, which changes the handedness of the object (Fig. 1.7). When the two axes (suppose x and y) of the axial system are inverted, this corresponds to the coordinate transformation: x → −x  , y → −y  , z → z  . In Cartesian unit vectors form, this can be written as −xˆ  × − yˆ  = zˆ  or xˆ  × yˆ  = zˆ 

(1.5)

which is a right handed system. Implying that the combined operation keeps the handedness of the original system preserved. In the matrix form, the above operations is equivalent to a twofold rotation with respect to z-axis and can be represented as: ⎛

⎞ −1 0 0 ⎝ 0 −1 0 ⎠ = 2[001] 0 0 1

(1.6)

Finally, inverting all the three axes, the corresponding coordinate transformation will be:

1.7 Mirror Reflection and Handedness

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x → −x  , y → −y  , z → −z  . In Cartesian unit vectors form, this can be written as −xˆ  × − yˆ  = −ˆz  or xˆ  × yˆ  = −ˆz 

(1.7)

which is again a left handed system. In matrix form, the above combined operation is equivalent to an inversion, located at the origin of the coordinate system and can be represented as: ⎛

⎞ −1 0 0 ⎝ 0 −1 0 ⎠ = i 0 0 −1

(1.8)

On the basis of the above discussed simple exercises, one can conclude that: an odd number of axes inversion, change (or invert) the handedness and an even number of axes inversion, retain (or preserve) the handedness. The simple odd number of axes inversion rule appear to have no effect on any symmetrical object like an atom or a crystal made of identical atoms or from a number of different symmetrical atoms, because the diffraction patterns of all crystals are found to exhibit center of symmetry. However, in molecular crystals, it will depend on the symmetry or asymmetry of the individual molecule. A right handed system attached to a crystal unit cell is shown in Fig. 1.8.

Fig. 1.8 A right handed system attached to a unit cell

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1 Fundamental Concepts Related to a Plane Mirror

1.8 Handedness and Enantiomorphy A mirror/reflection operation has the property to change the character (or handedness) of the motif, it will change a right handed object into a left handed and vice-versa. It happens due to lateral inversion. This is a phenomenon due to which the image of an object turns through an angle of 180° w.r.t the vertical axis, such that the image of a right handed object appears left handed behind the mirror, and vice-versa. For example, the change of handedness (i.e. a right handed and left handed forms of the letter P, and the words such as ATOM and PHYSICS) are illustrated in Fig. 1.9. Mirror coincident figures of this type are known as enantiomorphous pairs of objects and the phenomenon is known as enantiomorphism. Mirror operation has the following other characteristic features: (a) The mirror operation defines bilateral symmetry about its own plane (called mirror plane or reflection plane). (b) For every point at a distance r along the normal to the mirror plane, there exists an equivalent point at a distance –r. (c) For a point (x, y, z), reflection across a mirror plane σxz takes the point into (x, –y, z), similarly others. (d) A mirror plane has only one operation associated with it, since σ2 = E. In the matrix form, the mirror plane σxz is given by ⎛

σx z

⎞ 1 0 0 = ⎝ 0 −1 0 ⎠ 0 0 1

and the product σ. σ = σ2 is given by

Fig. 1.9 Reflection of an object by a plane mirror exhibiting enantiomorphism

(1.9)

1.9 Image Formation by Two Inclined Plane Mirrors

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⎛

⎞⎛ ⎞ ⎛ ⎞ 1 0 0 1 0 0 100 σ 2 = ⎝ 0 −1 0 ⎠⎝ 0 −1 0 ⎠ = ⎝ 0 1 0 ⎠ = E 0 0 1 0 0 1 001

(1.10)

1.9 Image Formation by Two Inclined Plane Mirrors Above, we studied some important aspects of a mirror such as the image formation, change of handedness of the object, etc. by a single plane mirror. However, if we place another plane mirror inclined to the first mirror at a certain angle, then more than one image of the object will be formed by the combination, depending on the angle between them. Because of multiple reflections, sometimes it becomes difficult to keep the track of what orientation each image will have. These multiple reflections actually lead to the formation of multiple images. For simplicity, let us consider below some simpler cases. In order to make a comparative study of the kinds of images formed by the two mirrors, two extreme types of objects are considered in the present study, one a symmetric object (such as a spherical ball) and the other an asymmetric object (such as the capital letter R). They are respectively placed between the two mirrors as shown in different cases by varying the angle between them. Case I: Angle between the two mirrors is 120° Let us consider two mirrors at an angle of 120° to each other as shown by solid lines in Fig. 1.10. A small spherical ball is placed in between the two mirrors. The images, I1 and I2 are formed exactly on the opposite side of each mirror, at the same distance from the respective mirror, just as we would expect due to reflection from a single mirror. The two mirrors, yield an image each. We also observe that the two images, I1 and I2 lie at equal distance from a dotted line (not a mirror line) behind the two mirrors, they prima facie appear to be the image of each other across the dotted line in Fig. 1.10a. A similar experiment can be performed by using an asymmetric alphabet, for example say a capital letter R. Let us suppose that the original capital letter R is a right handed object, then we observe the two images in the form of left handed R behind the two mirrors (Fig. 1.10). As they both are left handed (with similar handedness), they cannot be the image of each other across the dotted line, is clearly evident. Looking at the two considered cases when the angle between the mirrors is 120°, we find a very important difference between the two results. In the first case, where we considered a spherical ball as an object, which is spherically symmetric like an atom in a crystal, the two formed images are found to be indistinguishable with the object in terms of handedness. On the other hand, the letter R is an asymmetric object, and its two images are clearly distinguishable from the object in terms of handedness. The two different observations provide us an important information from

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1 Fundamental Concepts Related to a Plane Mirror

Fig. 1.10 Image of an object when two mirrors are at an angle of 120°

crystallographic point of view. As we know that the atoms (constituting a crystal) are spherically symmetric in nature, causing no handedness issue and hence the crystal irrespective of which system it belongs, will always remain centro-symmetric as exhibited by its diffraction pattern. However, in molecular crystals, the situation may be slightly different, its final symmetry will depend on the symmetry or asymmetry of its unit cell. On the other hand, for an asymmetric case (at present the letter R), the handedness of the two images are clearly opposite to that of the object. Case II: Angle between the two mirrors is 90° When the angle between the two mirrors is decreased to 90°, number of images are increased. One image of the symmetric ball will be formed from a single reflection from each mirror. This way, we obtain the two images I1 and I2 . These two reflections will then create the third image. In other words, the image I3 is obtained as a composite reflection from both the images I1 and I2 as shown in Fig. 1.11a. Therefore, in this case in all, three indistinguishable images are formed. However, when the object is asymmetric such as the letter R, two different types of images are formed, two of the directly formed images are found to be left handed and the third one (in the form of composite image) is found to be the right handed, clearly distinguishable from each other in terms of handedness (Fig. 1.11b). Looking again at the two considered cases when the angle between the two mirrors is 90°, we find another very important result. That is, when the object has a spherical shape like a ball or an atom, the result is same as in the first case discussed above. However, when the object is asymmetric such as the letter R, two different types of images are formed, where two of the three images are found to be left handed and the third one is a right handed, clearly distinguishable from each other in terms of handedness (Fig. 1.11b). This observation provides us another important information

1.9 Image Formation by Two Inclined Plane Mirrors

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Fig. 1.11 Image of an object when two mirrors are at an angle of 90°

from the point of view that the nature of images of an asymmetric object depends on the angle between the two intersecting mirrors. Case III: Angle between the two mirrors is 60° In this case, the two mirrors are placed such that the angle between them is 60°. The initial two images I1 and I2 of the spherically symmetric object (the ball) are obtained by the direct reflections from the object and hence obtained from a single reflection. Further, the image I3 is obtained from a double reflection and can be seen to be formed due to reflection of I2 through an extension of the top (horizontal) mirror. Similarly, the image I4 is obtained from the second reflection of I1 through the angled (slanting) mirror. Finally, the image I5 is obtained as a composite (third) reflection from both the images I3 and I4 . The whole process is illustrated in Fig. 1.12a.

Fig. 1.12 Image of an object when two mirrors are at an angle of 60°

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However, when the object is asymmetric such as the letter R, two different types of images are formed. The first two of the directly formed images (from first reflection) are found to be left handed as above in Case II. The next two images (from second reflection) are found to be the right handed, and the fifth and final one in the form of a composite image (from third reflection) is found to be the left handed, clearly distinguishable from each other in terms of handedness (Fig. 1.12b). Accordingly, three left handed images and two right handed images are obtained. Looking again at the two considered cases when the angle between the two mirrors is 60°, we find a similar result as discussed in Case II. When the object has a spherical shape like a ball or an atom, the result is same as in the two cases discussed above. However, when the object is asymmetric such as the letter R, two different types of images are formed, three of the five images (first, third and fifth) are left handed and the two images (second and fourth) are right handed, clearly distinguishable from each other in terms of handedness (Fig. 1.12b). This observation provides us two important information: (i) The number of images of both symmetric and asymmetric objects, depend on the angle between the two intersecting mirrors, while (ii) The nature of images of an asymmetric object will depend on whether the number of reflection is odd or even, i.e., the odd reflections will produce a left handed image and the even reflections will produce a right handed image, respectively. In other words, the odd reflections will change the handedness of the object and the even reflection will retain (or preserve) the handedness of the object. An alternative method of the process explained above for the angle 60° between the two inclined mirrors, is based on a concept, called the circle concept. In this case, all the images that are formed will lie on the circumference of a circle. Here, the center of the circle is the same as the point of intersection of the two inclined mirrors, and the radius of the circle is equal to the distance of object from the center, as illustrated in Fig. 1.13. However, the number of left handed and the right handed images for asymmetric objects will remain the same as discussed above.

1.10 Number of Images Formed by Two Inclined Plane Mirrors We discussed above the formation of different number of images when two plane mirrors are inclined to each other with an object in between them. The number of images formed by the two inclined mirrors depends on the angle (say θ) and also the position of the object between them. One can easily determine the number of images if the angle and the actual position of the object between the two mirrors are known, by using the following formula:

1.10 Number of Images Formed by Two Inclined Plane Mirrors

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Fig. 1.13 Circle concept of image formation

◦

◦

= an odd number ; then the number of images formed is n = 360 , when 1. If 360 θ θ the object is placed asymmetrically between the two mirrors. ◦ ◦ 2. If 360 = an odd number ; then the number of images formed is n = 360 − 1, θ θ when the object is placed symmetrically between the two mirrors. ◦ ◦ 3. If 360 = an even number ; then the number of images formed is n = 360 − 1, θ θ no matter how the object is placed between the two mirrors. ◦ 4. If 360 = an integer ; then the number of images formed = nearest even integer. θ Numeral one is subtracted in 2 and 3 because of the loss of one image due to overlapping of the images. Let us consider some specific examples: (a) When θ = 90° The number images formed are: n=

360◦ 360◦ −1= −1=4−1=3 90◦ θ

(b) When θ = 72° The number images formed are: n=

360◦ 360◦ −1= −1=5−1=4 θ 72◦

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1.11 Image Formation by Two Intersecting Plane Mirrors Under Limiting Case Let us consider the limiting case of intersection of two plane mirrors. In this case, the angle between the two intersecting mirrors is, θ = 0. This means that the two plane mirrors are parallel to each other and the number of images obtained from such a mirror combination is theoretically infinity. The process of formation of a few images is shown in Fig. 3.2. Also, the important aspects from crystallographic points of view are discussed in Chap. 3. In mathematical form, one can show that when θ = 0°, the number images formed are: n=

360◦ 360◦ −1= ◦ −1=∞ θ 0

However, there may be a question that the infinite images are not actually seen when the two plane mirrors are parallel and facing each other? The simple answer of this, is the following: 1. After every successive reflection, some amount of light energy is absorbed. Thus luminosity of the image goes on decreasing, till they are no longer visible. 2. As the distance of the images goes on increasing from the eye, it is unable to resolve far off images.

1.12 Horizontal, Vertical and Dihedral Mirror Planes Mirror planes can acquire three different orientations in crystals and molecular systems. They are horizontal, vertical and dihedral, they are symbolically represented as σh , σv and σd , respectively. They can be defined as: (a) A horizontal mirror plane σh is defined as perpendicular to the principal axis of rotation. (b) Both σv and σd , mirror planes are defined such that they contain the principal axis of rotation, while the principal axis is perpendicular to σh mirror plane. (c) When both σv and σd mirror planes happen to be in the same system, they are differentiated by defining σv mirror plane to contain the greater number of atoms or to contain a principal axis of a reference Cartesian coordinate system (x or y axis). (d) Any σd mirror planes typically will contain bond angle bisectors. (e) Taking the example of a square planar MX4 molecule, the five mirror plane symmetries are shown in the form of three classes as σh , 2σv and 2σd in Fig. 1.14.

1.13 Summary

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Fig. 1.14 Examples of horizontal, vertical and dihedral mirror planes in a molecular system

1.13 Summary 1. A plane mirror is in general considered to be highly smooth and polished surface which reflects almost the entire light falling on it. This is made by polishing or silvering one side of a glass plate as shown in Fig. 1.1. 2. Certain characteristics of the image formed by a plane mirror are: (a) Size of the image is same as the size of the object. (b) Image is laterally inverted (i.e. image appears to be inverted only on the vertical axis). (c) Image formed by a plane mirror is upright. (d) Image formed by a plane mirror is virtual. (e) Image formed is as far behind the mirror as the object in front of the mirror. 3. There are two laws of reflection. They are: (a) The incident ray, the reflected ray and the normal at the point of incidence, lie in the same plane as shown in Fig. 1.2.

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1 Fundamental Concepts Related to a Plane Mirror

(b) The angle of incidence ‘i’ is equal to the angle of reflection ‘r’, i.e. ∠i = ∠r. 4. Reflections can be put into two broad categories based on the texture of the reflecting surface, i.e. whether the surface is smooth or rough: (i) the regular or specular reflection, and (ii) the irregular or diffused reflection. 5. Keeping the incident ray fixed, if the plane mirror is rotated through a certain angle (say θ), then the reflected ray gets rotated through twice the incident angle (i.e., 2θ). 6. A reflection operation in a plane mirror reverses the direction of the axis perpendicular to it, but leaves the axes parallel to it unchanged. As a result, the mirror reflection changes the handedness of an object (symmetrical or asymmetrical) along the axis perpendicular to the mirror. 7. There exists a direct relationship between axis inversions and handedness of the image. An odd number of axes inversion, changes (or inverts) the handedness of the object and an even number of axes inversion, retains (or preserves) the handedness of the object. 8. When two plane mirrors are inclined to each other with an object in between them, many images are formed. The number of images formed by the two inclined mirrors will depend on the angle (say θ) and the position of the object between them. 9. If the angle and the actual position of the object between the two mirrors are known, one can easily determine the number of images that are formed by using the following formula: ◦

◦

= an odd number ; then the number of images formed is n = 360 , (a) If 360 θ θ when the object is placed asymmetrically between the two mirrors. ◦ (b) If 360 = an odd number ; then the number of images formed is n = θ 360◦ − 1, when the object is placed symmetrically between the two mirrors. θ 360◦ (c) If θ = an even number ; then the number of images formed is n = 360◦ − 1, no matter how the object is placed between the two mirrors. θ ◦ (d) If 360 = an integer ; then the number of images formed = nearest even θ integer. (e) If the angle between the two mirrors, θ = 0°, the number images formed are theoretically infinity. 10. Mirror planes can acquire three different orientations in crystals and molecular systems. They are horizontal, vertical and dihedral, they are symbolically represented as σh , σv and σd , respectively (Fig. 1.4).

References

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References 1. Morgan, K.: Reflection and Planar Mirrors—m260. Project Physnet—Michigan State University, USA (2000) 2. DeWeerd, A.J., Eric Hill, S.: Reflections on Handedness, The Physics Teacher, vol. 42. University of Redlands, Redlands, CA (2004)

Chapter 2

Mirror: The Only Fundamental Symmetry in Crystals

2.1 Introduction In the present day crystallography, the translational symmetry is considered to be the most fundamental symmetry and all other symmetries such as reflection, rotation, inversion and their compatible combinations are supposed to comply with the translation for their validity. However, a recent study provides us enough evidence to suggest that mirror symmetry is the only fundamental symmetry in crystals, and all other symmetries such as rotation, inversion, rotoreflection, rotoinversion and their compatible combinations can be easily derived from the same, while translational periodicity is the natural outcome of all symmetry combinations. In fact, the latest work is the result of motivation from the following two fundamental sources, providing the natural representation of mirror combination scheme: (i) the principle of ray optics capable of producing infinite images of an object placed between the two parallel mirrors as in one dimensional periodic lattice, and (ii) the inherently primitive Wigner–Seitz and Brillouin-Zone cells containing the only atom at their center and surrounded by many pairs of parallel planes acting as mirrors, and capable of producing the perfectly periodic direct/reciprocal lattices, respectively without involving any translational symmetry (microscopic or macroscopic). Accordingly, we need to start this chapter with the development of mathematical formulation of intersecting mirrors in 3-D using orthogonal axes, on the basis of which we can easily understand the concept of fundamental and derived symmetries for the first time in the crystallographic history and then use the same to derive various symmetry elements or symmetry operations possible (including 32 point group symmetries) in crystalline solids. This will, in turn, prove the fundamental nature of the ‘mirror symmetry’.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. A. Wahab, Mirror Symmetry, Springer Series in Solid-State Sciences 200, https://doi.org/10.1007/978-981-99-8361-2_2

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2 Mirror: The Only Fundamental Symmetry in Crystals

2.2 Analysis of Intersecting Mirrors Using Orthogonal Axes In a recent study on symmetry by Wahab [1], has shown enough evidence to suggest that mirror symmetry is the only fundamental symmetry in crystals. In order to understand the same, let us first consider the case of two vertical plane mirrors intersecting along the z-axis at an angle, θ < 45°. Let the first mirror (mx ) is placed parallel to (x–z) plane which changes the axis y into − y (Fig. 2.1). The corresponding matrix is given by ⎛

⎞ 1 0 0 (m x ) = ⎝ 0 −1 0 ⎠ 0 0 1

(2.1)

The second mirror (mxy ) is at an angle, θ < 45° w.r.t the mirror (mx ) as shown in Fig. 2.1. This mirror (mxy ) throws the axis x to 2θ degree away, so that x changes into x, . Simple resolution of x, into parallel and perpendicular components will provide us x, = x cos2θ + y sin2θ

Fig. 2.1 Two intersecting mirrors on vertical axis

(2.2)

2.3 Derivation of Rotational Symmetries from Mirror Combinations

23

The mirror (mxy ) also throws the axis y to 2(90 − θ) degree away (Fig. 2.1), so that y changes into y, . Simple resolution of y, into parallel and perpendicular components will provide us y, = − y cos2θ + x sin2θ = x sin2θ − y cos2θ

(2.3)

The two mirrors mx and mxy make no change in the z-axis, i.e. z, = z

(2.4)

Combining Eqs. 2.2, 2.3 and 2.4, the corresponding matrix (mxy ) is obtained as ⎛ ⎞ cos2θ sin2θ 0 ( ) m x y = ⎝ sin2θ −cos2θ 0 ⎠ 0 0 1

(2.5)

The result obtained in Eq. 2.5 gets modified due to the presence of the mirror (mx ), therefore taking the product of the two matrices, we obtain the final result as ⎛

cos 2θ (mxy ) × (mx ) = ⎝ sin 2θ 0 ⎛ cos 2θ = ⎝ sin 2θ 0

⎞ ⎛ ⎞ sin 2θ 0 1 0 0 − cos 2θ 0 ⎠ × ⎝ 0 −1 0 ⎠ 0 1 0 0 1 ⎞ − sin 2θ 0 cos 2θ 0 ⎠ = Cn 0 1

(2.6)

The matrix obtained in Eq. 2.6 represents the general form of rotational symmetry when two mirrors intersect at an angle θ. Different angles at the intersection of two mirrors will provide different rotational symmetries (as discussed in chapter one), where the axis of rotational symmetry coincides with the axis of intersection of two mirrors. Substituting the value of intersection angle, the folds of rotational symmetry can be easily determined.

2.3 Derivation of Rotational Symmetries from Mirror Combinations We know that in crystals only limited numbers of rotational symmetries are possible. This is defined according to the relation θ=

2π n

24

2 Mirror: The Only Fundamental Symmetry in Crystals

where n = 1, 2, 3, 4 and 6. We also know that two mirrors intersecting at an angle θ, together produce a rotation of 2θ. Accordingly, if the angle at the intersection of two mirrors is known the fold of rotational symmetry can also be determined directly using the relation n=

2π π 180◦ = = 2θ θ θ

(2.7)

Equations 2.6 and 2.7 will provide the same result. Let us assume that the value of θ is such that n is a whole number. This limit suggests that the maximum value of θ is 180° and the minimum 1°. They will correspond to 1- and 180-fold rotations, respectively. The possible crystallographic symmetries can be derived when the mirrors are placed at different intersecting angles. (i) The angle at the Intersection is 180° This implies that the two mirrors are parallel and together act as a single mirror. Substituting the value in Eq. 2.6, the matrix becomes ⎛

⎞ cos 2θ − sin 2θ 0 C(θ = 180◦ ) = ⎝ sin 2θ cos 2θ 0 ⎠ 0 0 1 ⎛ ⎞ ⎛ ⎞ ◦ cos 360 − sin 360◦ 0 100 = ⎝ sin 360◦ cos 360◦ 0 ⎠ = ⎝ 0 1 0 ⎠ = E 0 0 1 001

(2.8)

The identity matrix represents onefold (Fig. 2.2) rotation. (ii) The angle at the Intersection is 90° This implies that the two mirrors are perpendicular to each other and intersect at 90°. Substituting the value in Eq. 2.6, the matrix becomes ⎛

⎞ cos 2θ − sin 2θ 0 C(θ = 90◦ ) = ⎝ sin 2θ cos 2θ 0 ⎠ 0 0 1 ⎛ ⎞ ⎛ ⎞ ◦ cos 180 − sin 180◦ 0 −1 0 0 = ⎝ sin 180◦ cos 180◦ 0 ⎠ = ⎝ 0 −1 0 ⎠ = C2 0 0 1 0 0 1

(2.9)

The last matrix in Eq. 2.9 represents the twofold (Fig. 2.3) rotation w.r.t the principal axis.

2.3 Derivation of Rotational Symmetries from Mirror Combinations Fig. 2.2 One vertical mirror

Fig. 2.3 Two intersecting mirrors at 90°

25

26

2 Mirror: The Only Fundamental Symmetry in Crystals

(iii) The angle at the Intersection is 60° This implies that the mirrors are placed such that they intersect at an angle of 60° with each other. Substituting the value in Eq. 2.6, the matrix becomes ⎛

⎞ cos 2θ − sin 2θ 0 C(θ = 60◦ ) = ⎝ sin 2θ cos 2θ 0 ⎠ 0 0 1 √ ⎛ ⎞ ⎛ ◦ cos 120 − sin 120◦ 0 −1/2 − 3/2 √ = ⎝ sin 120◦ cos 120◦ 0 ⎠ = ⎝ 3/2 −1/2 0 0 0 0 1

⎞ 0 0 ⎠ = C3 1

(2.10)

The last matrix in Eq. 2.10 represents the threefold (Fig. 2.4) rotation w.r.t the principal axis. (iv) The angle at the Intersection is 45° This implies that the mirrors are placed such that they intersect at an angle of 45° with each other. Substituting the value in Eq. 2.6, the matrix becomes ⎛

⎞ cos 2θ − sin 2θ 0 C(θ = 45◦ ) = ⎝ sin 2θ cos 2θ 0 ⎠ 0 0 1 Fig. 2.4 Three intersecting mirrors at 60°

2.3 Derivation of Rotational Symmetries from Mirror Combinations

27

⎛

⎞ ⎛ ⎞ cos 90◦ − sin 90◦ 0 0 −1 0 = ⎝ sin 90◦ cos 90◦ 0 ⎠ = ⎝ 1 0 0 ⎠ = C4 0 0 1 0 0 1

(2.11)

The last matrix in Eq. 2.11 represents the fourfold (Fig. 2.5) rotation w.r.t the principal axis. (v) The angle at the Intersection is 30° This implies that the mirrors are placed such that they intersect at an angle of 30° with each other. Substituting the value in Eq. 2.6, the matrix becomes ⎛

⎞ cos 2θ − sin 2θ 0 C(θ = 30◦ ) = ⎝ sin 2θ cos 2θ 0 ⎠ 0 0 1 Fig. 2.5 Four intersecting mirrors at 45°

28

2 Mirror: The Only Fundamental Symmetry in Crystals

Fig. 2.6 Six intersecting mirrors at 30°

√ ⎞ ⎛ 3/2 cos 60◦ − sin 60◦ 0 1/2 − √ = ⎝ sin 60◦ cos 60◦ 0 ⎠ = ⎝ 3/2 1/2 0 0 1 0 0 ⎛

⎞ 0 0 ⎠ = C6 1

(2.12)

The last matrix in Eq. 2.12 represents the sixfold (Fig. 2.6) rotation w.r.t the principal axis.

2.4 Crystallographic and Non-crystallographic Rotational Symmetries As pointed out above, if n is a whole number then the value of θ will lie in the range 180◦ ≤ θ ≤ 1◦

2.5 Derivation of Inversion Center

29

Table 2.1 Angle at the intersection and corresponding rotational symmetry S. N

Angle at the intersection of two mirrors, θ (in degree)

The fold of rotation, n =

1

180

1

2

90

2

3

60

3

4

45

4

5

36

5

6

30

6

7

20

9

8

18

10

9

15

12

10

12

15

11

10

18

12

9

20

13

6

30

14

5

36

15

4

45

16

3

60

17

2

90

18

1

180

2π 2θ

=

π θ

=

180◦ θ

Substituting different possible values of θ in Eq. 2.7, the values of n can be easily obtained. They are provided in Table 2.1. Under the above condition, Table 2.1 suggests that in total there are 18 rotational symmetries possible including 5 crystallographic (1, 2, 3, 4 and 6 possible in crystals) and 13 others possible in quasi-crystals (called non-crystallographic). However, inclusion of fractional values of θ will give rise to infinite values of n.

2.5 Derivation of Inversion Center Center of inversion is the symmetry element possible only in 3-D systems, such as crystals. This symmetry element can be visualized by placing a horizontal mirror w.r.t the vertical mirrors used to create n-fold rotations. By this process, a holohedral symmetry element (such as 2/m, 3/m, etc.) containing inversion center corresponding to each rotation axis can be obtained. From this, other subgroup symmetry elements can be easily obtained. For example, when a twofold rotation (which itself is obtained from two interacting mirrors at 90°) is combined with a perpendicular mirror, an

30

2 Mirror: The Only Fundamental Symmetry in Crystals

Fig. 2.7 Creation of center of inversion

inversion center is produced (Fig. 2.7). Here, 2˜ (called 2-tilde) represents a twofold rotation combined with a perpendicular mirror reflection, in a rotoreflection system. In matrix notation, this combined operation can be written as ⎛

⎞⎛ ⎞ ⎛ ⎞ −1 0 0 10 0 −1 0 0 2[001] × m[001] = ⎝ 0 −1 0 ⎠⎝ 0 1 0 ⎠ = ⎝ 0 −1 0 ⎠ = 1 0 0 1 0 0 −1 0 0 −1 Similarly, other inversion centers, such as 3/m, 4/m, etc. can be obtained in other cases too. Further, other remaining symmetries such as rotoreflection, rotoinversion, and hence all the point groups can also be obtained by suitable combinations of mirrors.

2.6 Generation of 1-D, 2-D and 3-D Point Groups After deriving all the rotational symmetry elements possible in crystalline solid from different mirror combinations, it is not very difficult to derive all point groups (also called crystal classes) that are possible in 1-, 2- and 3-dimensions: Let us first, check the point groups that are associated with the figures in between 2.2, 2.3, 2.4, 2.5 and 2.6, they provide us the following point groups: 1 and m These two point groups belong to 1-D crystal system, as mentioned in the International Table [2]. Now, for a 2-D crystal system, the ten point groups are: m and 1; mm2 (or 2mm) and 2; 3m and 3; 4mm and 4; 6mm and 6

2.6 Generation of 1-D, 2-D and 3-D Point Groups

31

These ten point groups belong to 2-D (provided in Table 2.2) as well as 3-D crystal systems. In a 2-dimensional form, they are represented as illustrated in Fig. 2.8. Now, let us derive the remaining 22 point groups, exclusively belong to threedimensional crystal systems. For the purpose, first of all consider Fig. 2.7 which provides us the point groups 2/m and 1, respectively. Adding these two, we have a total of 12 point groups in all. Further, let us derive others: 1. By placing a horizontal mirror in the form of a plane in each figure from 2.3, 2.4, 2.5 and 2.6 as shown in Fig. 2.9a–d, we can obtain the following point groups: 2/m (already counted above), 222, mmm; 3/m = 6, 32; 4/m, 422, 4/mmm; 6/m, 622, 6/mmm. Adding them to the previous one, the total number of point groups, so far becomes 22. 2. Now performing similar operations like Fig. 2.7 for other rotoreflection axes to obtain the following point groups: 3, 4, 6 shown in Fig. 2.9e–g. This makes the total number of point groups to be equal to 24 (6 is already counted above). 3. The point groups’ 3m, 42m and 6m2 can be obtained from the combination of mirrors with suitable improper axes as shown in Fig. 2.9h–j. Including them, the total point groups become 27. 4. The remaining five point groups, viz. 23, m3, 432, 43m and m3m for cubic crystal system can be obtained by placing a set of 4 mirrors along each x, y and z axis Table 2.2 Classification of 2-D point groups as per mirror combination scheme Rotation ≡ Intersecting mirrors

Mirrors intersecting at different angles

1

m

2

mm2

3

3m

4

4mm

6

6mm

Fig. 2.8 Ten crystallographic point groups in 2-D

32

2 Mirror: The Only Fundamental Symmetry in Crystals

(a)

(b)

(e)

(h)

(c)

(f)

(i)

(d)

(g)

(j)

(k)

Fig. 2.9 Illustrations for 32 point group symmetries

such that their intersections coincide with the three fourfold axes as shown in Fig. 2.9k.

2.7 Point Group Minimum (PGM) and Minimum Symmetry Form (MSF)

33

2.7 Point Group Minimum (PGM) and Minimum Symmetry Form (MSF) From the fundamental crystallography, we know that there are respectively 2, 10 and 32 point groups in 1-D, 2-D and 3-D crystal systems. Also, the classification of crystal systems is based on the geometry and the symmetry of the unit cells, where the first crystal system is always the least symmetric and the last crystal system is the most symmetric, respectively. Further, the first crystal system has only one lattice (such as a point lattice in 1-D, an oblique lattice in 2-D and a triclinic lattice in 3-D), it is primitive and least symmetric in each (1-D, 2-D and 3-D) crystal system. On the other hand, other crystal systems can have one or more non-primitive lattices. The symmetries in the eight crystal systems in 3-D are arranged in an increasing order (Table 2.9) and in general will lie in the range [3], Essential symmetry < intermediate symmetry < holosymmetry where, the essential symmetry actually represents the ‘minimum symmetry form’ from each symmetry category (e.g., rotations, mirrors and inversions) in the given crystal system. Now, let us consider the 1-D, 2-D and 3-D cases separately. Case I: 1-D Crystal System We know that mirror being the most fundamental, it is the only symmetry element/ operation possible in a 1-D crystal system, it is also clear from Table 2.4. Case II: 2-D Crystal System In a 2-D crystal system, the symmetry operations of a given unit cell are either proper rotations (mono axial) or mirrors, where different orders of rotations and mirrors are: (a) Proper rotations: onefold, twofold, threefold, fourfold, sixfold. (b) Mirror symmetries: m, mm2, 3m, 4mm, 6mm. The above symmetry information suggest that the minimum symmetry form (MSF) under each category of symmetry operations in 2-D is: MSF of proper rotation symmetry is 1, and MSF of mirror symmetry is m Further, the concept of minimum symmetry form (MSF) as stated above is found to be equivalent to another similar term, called the ‘point group minima’ (PGM) defined when 10 point groups are classified in an increasing order of symmetry in its own category (i.e. either proper rotation or mirror reflection). Also, according to group theory, a member of the PGM is simply the smallest subgroup of its corresponding super groups. This can be clearly understood by a careful observation of the distribution of 10 point groups under two different symmetry categories as provided in Table 2.2. They are also shown in Fig. 2.10 in the form of subgroup and super group hierarchy in two different crystallographic notations.

34

2 Mirror: The Only Fundamental Symmetry in Crystals

Fig. 2.10 Subgroup and super group hierarchy of ten 2-D point groups

Case III: 3-D System Similarly, in a 3-D crystal system, the non-translational symmetry operations (i.e. the 32 point groups) belong to proper rotations, mirrors and inversions, where different orders of rotations, mirrors and inversions are: (a) Proper rotations: onefold, twofold, threefold, fourfold, sixfold (b) Mirror symmetries: m, mm2, 3m, 4mm, 6mm, etc. (c) Inversion symmetries: 1, 2/m, 3/m, 4/m, 6/m, etc. The above symmetry information suggest that the minimum symmetry form (MSF) under each category of symmetry operations in crystalline solids is: MSF of proper rotational symmetry is 1 MSF of mirror symmetry is m, and MSF of inversion symmetry is 1. Therefore, applying similar argument to a 3-D system which we applied to 2-D system, the 32 point groups can be classified in terms of proper rotation, rotoreflection and rotoinversion, respectively. This can be clearly understood by a careful observation of their distribution under three different symmetry categories [3] as provided in Table 2.3, where the point groups marked with asterisks are not counted toward the total. From the above discussion, we observe that the three members 1, m and 1 which in any case represent the minimum symmetry form (MSF) in each symmetry category that is used in representing crystal systems. Further, we observe that 3-D lattices belong to any of the following three categories of point groups: least symmetric, intermediate symmetric or high (or holo) symmetric [3]. Accordingly, the least symmetric 2-D (oblique) lattice will contain 1 and m, two compatible MSF as its members (and not 1 and 2 as prevalent in literature; proof follows), similarly the least symmetric 3-D (triclinic) lattice will contain 1, m and 1, three compatible MSF as its members (and not only 1 and 1 as prevalent in literature; proof follows), respectively. Like

2.8 Role of Mirror and Rotation in Crystals of Different Dimensions

35

Table 2.3 Classification of 3-D point groups as per mirror combination scheme Rotation ≡ intersecting mirrors

Mirrors intersecting at different angles

A mirror perpendicular to the intersecting mirrors

1

m

2

mm2

1 ( )* 1

3

(m)*

3

4

4

6

6

222

(m)*

4 m 6 m 2 m,

32(2)

3m

3m

422

4mm, 42m

622

6mm, 62m

4 mmm 6 mmm

23(3)

(m)*

m3

432

43m

m3m

* Not

mmm

counted towards the total

Table 2.4 Symmetry system present in different crystal dimensions

Crystal dimension

Symmetry forms

One (line)

Mirror

–

–

Two (plane)

Mirror

Rotation

–

Three (space)

Mirror

Rotation

Inversion

2-D, in 3-D also the subgroup and super group hierarchy of thirty two point groups is shown in Fig. 2.11.

2.8 Role of Mirror and Rotation in Crystals of Different Dimensions According to the latest study on symmetry conducted by Wahab [1], has shown that mirror symmetry is the only fundamental symmetry in crystals and other symmetries, such as proper rotation, inversion, rotoreflection, rotoinversion and their compatible combinations can be easily derived from the same, while translational periodicity is the natural outcome of all symmetry combinations. The latest work is the result of motivation from the following two fundamental sources based on mirror combination scheme: (i) the principle of ray optics capable of producing infinite images of an object placed between the two parallel mirrors [4] as in a one dimensional lattice, and (ii) the inherent primitive Wigner–Seitz and Brillouin-zone cells containing the only atom at the center, surrounded by many pairs of parallel planes acting as mirrors [5–8] are capable of producing the perfectly periodic direct and reciprocal lattices,

36

2 Mirror: The Only Fundamental Symmetry in Crystals

Fig. 2.11 Subgroup and super group hierarchy of thirty two 3-D point groups

respectively without involving any translational symmetry (microscopic or macroscopic). However, the fundamental role of mirror symmetry in crystalline solid can also be visualized from the fact that mirror is the only symmetry which is found to exist in crystals of all (three) dimensions as has been presented in Table 2.4. Based on simple geometrical representation, we can visualize three different situations of a mirror under three different crystal dimensions as summarised in Table 2.5. We observe not only a change in the shape of the mirror in different crystal dimensions but also its role appears to be different! In a similar manner, we can visualize two different situations of rotation axis. This is a line in 3-D and becomes a point in 2-D, while it loses its existence in 1-D, Table 2.5 Varying shapes and roles of a mirror

Crystal dimension

Shape of mirror

Act as

Three (space)

Plane

Normal mirror

Two (plane)

Line

Rotation (axis)

One (line)

Point

Inversion

2.9 Symmetry Verification of 2-D Lattices

37

Table 2.6 Varying shapes and roles of rotation axis Crystal dimension

Shape of rotation axis

Act as

Three (space)

Line

Normal rotation (axis)

Two (plane)

Point

Inversion

One (line)

Does not exist

–

as provided in Table 2.6. Like the case of mirror, we observe not only a change in the shape of the rotation axis in different crystal dimensions but also its role appears to be different! With these information in hand, along with the information related to mirror combination scheme, we can easily understand and analyse the symmetry behaviour of a given unit cell, the assignment of proper point groups to a lattice, and the distribution of overall point groups in different crystal systems, particularly the low symmetry systems of 2-D and 3-D lattices in the following sections.

2.9 Symmetry Verification of 2-D Lattices As far as the least symmetric unit cells, the oblique lattice of 2-D and triclinic lattice of 3-D are concerned, it is sufficient to consider the point group symmetries they get as their share according to MSF criterion, i.e., two compatible MSF members (1 and m) and three compatible MSF members (1, m and 1), respectively. However, for further satisfaction of the reader, let us use some geometric models to understand the same, little elaborately. In this respect, from the available information about three different categories of symmetries analyzed in Tables 2.4, 2.5 and 2.6, we come to know that a mirror plane in 3-D, becomes a mirror line in 2-D and hence acts as a rotation axis along the edges and diagonals of the unit cells in the given plane. Let us apply these results and draw images across: (i) edges, (ii) diagonals and the middle lines, (iii) bisectors of edges and diagonals of each unit cell of 2-D crystal lattice and check the shape and symmetry of the corresponding unit cells. Case I: Images across edges First of all, let us consider the relatively complicated case of an oblique lattice. An oblique unit cell with an angle of 60° is considered. Inscribe a small circle at its center and mark it as primary unit cell for identification, obtain four images across the four edges of the unit cell as shown in Fig. 2.12. First image is obtained from the left mirror line (left edge of the primary unit cell), assign the number 1 to it. Similar process is carried out from other mirror lines (from other edges of the primary unit cell) and the corresponding images are obtained in each case, one by one in a clockwise manner, they are numbered as 2, 3 and 4, respectively. This completes the process of imaging

38

2 Mirror: The Only Fundamental Symmetry in Crystals

Fig. 2.12 Illustration of 2-D lattice images by considering the unit cell edge as mirror line

Fig. 2.13 Illustration of 2-D lattice images by considering the unit cell edge as mirror line

the oblique lattice. The images obtained from the four different edges as exhibited in Fig. 2.12 are combined together and shown in Fig. 2.13b. Further, for a better clarity of the situation, it was decided to consider the oblique lattice with some other angles just to see the effect of angle (if any) on the resulting unit cell shape. For the purpose, the oblique lattice with two different angles such as 50° and 70° were selected, so that the angle 60° lies in between them. Images obtained corresponding to each of these three angles as above are separately combined and shown in Fig. 2.13 in decreasing value of angle. First image in three considered cases is formed at double the assigned angle to the primary oblique lattice, i.e. at 140°, 120° and 100° as shown in Fig. 2.13a–c. The illustrations for three different situations provided in Fig. 2.13 show that the resulting shapes of the oblique lattice are asymmetric in nature. However, Fig. 2.13b (corresponding to the oblique unit with angle 60°) appears to be more symmetric among the three cases but still it does not exhibit anything to show the presence of any twofold symmetry in it. Now following the same procedure described above for oblique lattice, the images for other three lattices (i.e. rectangular, square and hexagonal) are obtained as shown in Fig. 2.14. Looking at the resulting illustrations, we observe that these three lattices exhibit complete symmetry, fulfilling the requirement of their respective rotation axes (and mirror lines). However, the correct point groups to the rectangular and centered rectangular is 2 and mm2 and not m and 2mm as assigned to them earlier.

2.9 Symmetry Verification of 2-D Lattices

39

Fig. 2.14 Illustrations obtained from rectangular, square and hexagonal lattices

The above exercise clearly suggests the asymmetric nature of the oblique unit cell drawings and hence the twofold rotation has been wrongly assigned to it. It is actually not compatible to such a system. According to mirror combination scheme, this is possible only when two mirror lines intersect at 90° (hence, this operation is correctly written as mm2), as in rectangular and other higher symmetry systems. Hence, associating the point group ‘2’ to an oblique lattice as per the earlier assignment is not correct. Accordingly, the twofold rotational symmetry needs to be replaced by ‘m’ in oblique crystal lattice. Similarly, the point groups 2 and mm2 should be assigned to both rectangular and centered rectangular lattices in place of ‘m’ and 2mm which are found to fulfil the symmetry requirements. Further, when two mirrors intersect at 45° and 30°, they will give rise to fourfold and sixfold rotations, as exhibited by square lattice and hexagonal lattice, respectively [1]. There lies no problems with these lattices. Case II: Images across diagonals and middle lines In this case, consider an oblique unit cell with any arbitrary angle (in the present case 65°) and draw diagonals (Fig. 2.15a) to check for the presence of a twofold by taking the image of its one half, treating the diagonals as mirror lines (or equivalently assuming the rotation of the same half through 180° by treating a diagonal as the rotation axis lying in the same plane). The image obtained by carrying out the said operation on one diagonal shown in Fig. 2.15b does not support the existence of twofold as it does not replicate (overlap) the other half (however, the line divides the figure into two geometrically equal parts). Carrying out a similar exercise on the rectangular unit cell (Fig. 2.15c), we observe a similar result through diagonal mirror lines (Fig. 2.15d). However, the mirror lines passing through the middle of the edges do produce twofold rotation shown in Fig. 2.15c (as they replicate one half to other half on mirror/rotation). This is the reason, two perpendicular mirror lines produce mm2 point group symmetry. Similar diagonals (or middle lines) are drawn in square and hexagonal 2-D lattices to check for the respective rotational symmetries as illustrated in Fig. 2.16. In them, each half perfectly replicates (overlaps) the other half.

40

2 Mirror: The Only Fundamental Symmetry in Crystals

Fig. 2.15 a An oblique unit cell with diagonals, b Image after one rotation Fig. 2.16 Showing diagonals and middle lines in square and hexagonal unit cells

Case III: Images across the bisector of edge and diagonal In this case, let us first fix the origin of the unit cell and draw perpendicular bisectors to two edges and one diagonal of each 2-D lattice as shown in Fig. 2.17. These perpendicular bisectors act as mirrors and through reflections complete the unit cell. This indicates that such mirror lines are able to replicate unit cell without any problem in all 2-D lattices. In fact, this case is equivalent to a Wigner–Seitz cell of each lattice. Thus analyzing the above exercises, it is easy to conclude that none of the three ways discussed above show the existence of twofold rotation in oblique lattice and hence the point group 2 assigned to it must be replaced by ‘m’. Nature of the resulting images of the unit cells after mirror/rotation operations are summarized in Table 2.7, where the word symmetric represents both mirror and rotation operations in first two cases and only mirror in the last. At the end of this discussion, it is necessary to point out that while assigning the twofold symmetry to an oblique lattice (such as a parallelogram), there is a possibility of committing a mistake (and most likely, a similar situation might have led to the existing discrepancy in the oblique lattice), that inadvertently we make it a 3-D system by considering an imaginary vertical axis during its rotation as illustrated in Fig. 2.18, which is not justified. Actually for a 2-D system, the rotation axis has to be in the plane of the parallelogram itself, this is the reason that a mirror line and

Fig. 2.17 Perpendicular bisectors in 2-D lattices

2.10 Symmetry Verification of Low Symmetry 3-D Lattices

41

Table 2.7 Resulting image of the unit cell after mirror/rotation operation in 2-D Unit cell →

Resulting image of the unit cell after mirror/rotation operation in 2-D

Image across↓

Oblique

Rectangular

Square

Hexagonal

Edge (mirror/ rotation)

Asymmetric to mirror/rotation

Symmetric

Symmetric

Symmetric

Diagonal/middle (mirror/rotation)

Asymmetric to mirror/rotation

Asymmetric to Symmetric diagonals symmetric to middle lines

Symmetric

Bisectors (only mirrors)

Symmetric

Symmetric

Symmetric

Symmetric

Fig. 2.18 Two successive rotations of 180° bring the parallelogram to self-coincidence in 3-D

Table 2.8 Classification of crystal system and associated point groups Crystal system

Point groups according to Previous assignment

Present assignment

Oblique

1, 2

1, m

Rectangular

m, 2mm

2, mm2

Centered rectangular

m, 2mm

2, mm2

Square

4, 4mm

4, 4mm

Hexagonal

3, 3m, 6, 6mm

3, 3m, 6, 6mm

a rotation axis act in an identical manner in a 2-D system as clear from Table 2.5. Previous assignment of point groups and the present results have been summarised in Table 2.8, where the last column shows the required changes in the form of present findings, based on mirror combination scheme and MSF criterion.

2.10 Symmetry Verification of Low Symmetry 3-D Lattices In principle, we can apply the same procedure to low symmetry 3-D lattices (particularly to triclinic lattice) as discussed above for 2-D cases, and obtain the required images through six faces of the primary unit cell. However, an alternative and relatively simpler procedure can be adopted to get the correct result when lattice conditions are somewhat relaxed. In such cases, it is sufficient to put three extended mirror

42

2 Mirror: The Only Fundamental Symmetry in Crystals

Fig. 2.19 Illustrations for a mm2 and b 2/m symmetries

planes on three (of the six) faces of the primary unit cell slightly beyond the three crystallographic axes (as will be shown in latter figures) and look for the twofold symmetry along the axis of their intersections. However, it is necessary to keep in mind the criterion based on the mirror combination scheme, according to which two mirrors intersecting at 90° will produce the symmetry mm2 with a twofold rotational symmetry along the axis of their intersection (as shown in Fig. 2.19a) and three mutually perpendicular mirrors will produce the symmetry 2/m (as shown in Fig. 2.19b), respectively. Hence, while analysing the figures resulting from low symmetry 3-D lattices in terms of their geometry and symmetry we must follow the two mentioned criteria so that we can obtain the correct result. Let us study the triclinic, monoclinic and orthorhombic lattices separately one by one. Triclinic Lattice Since the angular relationship in a triclinic lattice is α /= β /= γ /= 90◦ , it is the least symmetric among all the space lattices and known as a general lattice of 3-D system. This provides us three pairs of different oblique planes similar to that encountered in 2-D oblique lattice. Therefore, in principle same procedure can be applied to triclinic lattice. Accordingly, a similar asymmetric resulting figure could be obtained and a similar end result is expected. Therefore, it will consist of only three lowest order symmetries, i.e. 1, m and 1, the members of the minimum symmetric form (MSF) of rotation, mirror and inversion, to a 3-D system. Monoclinic Lattice The process of symmetry verification in monoclinic lattice is comparatively easier because of some relaxed lattice conditions. The presence of two right angles (α = γ = 90◦ /= β) in its unit cell indicates the existence of two (one pair) oblique faces and two pairs of rectangular faces. This allows us to put three extended mirror planes,

2.11 Allocation of Correct Point Groups to Crystal Lattices

43

Fig. 2.20 Illustration obtained for monoclinic system

one on the lower face and one each on two side faces of the primary unit cell and look for the twofold proper rotational symmetry along three crystallographic axes, which are passing through three different intersections of mirrors. Looking at the illustration shown in Fig. 2.20, we observe that the monoclinic lattice does exhibit two twofold proper rotational axes, one each along the indicated direction. This gives us two sets of (mm2) symmetry systems but not any single (2/m) symmetry system (because the third plane is perpendicular to none of the two perpendicular axes, which is clear from the angles at the top), however somehow wrongly assigned point group is prevalent in literature. Orthorhombic Lattice The process of symmetry verification in orthorhombic lattice is further simplified because of the presence of three right angles (α = β = γ = 90◦ ) in its unit cell. This allows us to put an extended mirror plane on each of the three faces of the primary unit cell and look for the possible twofold symmetry along the axis of their intersections. Looking at the illustration shown in Fig. 2.21, we clearly observe three mutually perpendicular twofold axes along the indicated directions. This gives us a system where three mutually perpendicular mirrors produce ‘2/m’ along all the three crystallographic directions. Therefore, in addition to 2/m there are other two wellknown symmetries such as 222 and mmm (which is correctly written as m2 m2 m2 ). These three point groups are the genuine members of orthorhombic system. Therefore, the point group 2mm must be replaced by 2/m in orthorhombic crystal system.

2.11 Allocation of Correct Point Groups to Crystal Lattices As per earlier classifications, thirty two point groups were classified into 14 space lattices, and in turn the 14 space lattices into seven crystal systems. However, according to a recent study of geometry and symmetry of the structures generated from the close packing of identical atoms, Wahab and Wahab [9] discovered HCP

44

2 Mirror: The Only Fundamental Symmetry in Crystals

Fig. 2.21 Illustration obtained for orthorhombic system

(hexagonal close packing) and RCP (rhombohedral close packing) as the two new and independent space lattices. The construction of their primitive unit cells, W-S, B-Z, etc. are briefly discussed in Chap. 3. In order to accommodate the two newly discovered lattices, the eighth crystal system came into existence. Based on the geometry and symmetry, the HCP and the trigonal lattices were found to share the same set of point groups (seven in all) and hence are placed in the same crystal system while the RCP is found to share only five of them and remained as an independent lattice in the form of rhombohedral crystal system. Here, it is to be pointed out that according to Philips [10] the point groups 6 and 62m earlier belonged to the trigonal crystal system but later on for some unknown reasons shifted to hexagonal system. However, from symmetry point of view they are found to be genuinely belong to the trigonal system (although they are subgroup of hexagonal system) and hence we have brought them back to their right place. Keeping in view the proper geometry and symmetry of the lattice and taking into account the mirror combination scheme, the 32 point groups were classified into 16 space lattices (called Bravais and Wahab lattices) and these 16 space lattices were then classified into eight crystal systems, based on symmetry. Further, taking into account the corrections in the point groups of triclinic, monoclinic and orthorhombic crystal systems, the final classification in corrected form with proper distribution of point groups is provided in Table 2.9.

2.12 Summary

45

Table 2.9 Distribution of point groups according to mirror combination scheme S. No

No. of mirrors

Arrangement of mirrors

Crystal system

Distribution of point groups

1

One (simple)

Any ways

Triclinic

m, 1, 1

2

Two, (m, m)v

Intersect at 90°

Monoclinic

( ) 2, mm2, 1

3

Three

One horizontal mirror added to the above pair

Orthorhombic

222,

4

Three, (m, m, m)v

Intersect at 60°

RCP

3, 3, 32, 3m, 3m

5

Three, (m, m, m)v

Intersect at 60° One horizontal mirror added to the above trio

Trigonal/HCP

3, 3, 32, 3m, 3m, 6, 62m

Two sets of mutually ⊥ two mirrors One horizontal mirror added to the above sets

Tetragonal

4, 422, 4mm, 4 4, m4 , 42m , mmm

Four 6

(m, m)v mh

(m, m, m)v mh

Four, (m, m)v (m, m)v (m, m)v (m, m)v mh

7

Six (m,m, m)v (m, m,m)v Seven (m, m, m)v (m,m, m)v mh

8

*v

Twelve (m,m,m,m)v (m,m,m,m)v (m,m,m,m)v

2 m , mmm

Two sets of Hexagonal mutually ⊥ three mirrors One horizontal mirror added to the above trios

6, 622, 6mm, 6 6 m , mmm

Three sets of mutually ⊥ four mirrors on each axis

23, 432, 43m, m3, m3m

Cubic

( ) = vertical, h = horizontal, 1 not counted toward total

2.12 Summary 1. The analysis of intersecting mirrors at an angle θ using orthogonal axes, provides us the matrix representing the general form of rotational symmetry in 3-D crystal system, is given by the relation ⎛

⎞ cos 2θ − sin 2θ 0 Cn (θ ) = ⎝ sin 2θ cos 2θ 0 ⎠ 0 0 1 2. If the angle at the intersection of two mirrors is known, the fold of rotational symmetry can also be determined directly using the relation

46

2 Mirror: The Only Fundamental Symmetry in Crystals

n=

π 180◦ 2π = = 2θ θ θ

3. If n is a whole number in above equation, then the value of θ will lie in the range 180◦ ≤ θ ≤ 1◦

4. 5.

6.

7.

8.

9. 10.

This will provide us the whole range of rotational symmetries including the crystallographic and non-crystallographic (including quasi-crystalline). Centres of inversion (including the holohedries) can be created by placing a horizontal mirror w.r.t the vertical mirrors used to create n-fold rotations. With the help of mirror combination scheme, the derivation of 2, 10 and 32 crystal classes (point groups) for 1-D, 2-D and 3-D, respectively is extremely easy. Mirror combination scheme provides us the concept such as minimum symmetry form (MSF) or point groups’ minima (PGM) representing the smallest sub group element of different symmetry maxima. Presence of mirror symmetry in all three crystal dimensions shows its fundamental nature. Its role in all three different crystal dimensions appears to be different! Mirror combination scheme in different crystal dimensions has become the source of origin of the concept of minimum symmetry form (MSF) or point group minima (PGM), which brought about the inevitable changes in the allocation of point groups in some low symmetry crystal systems (such as oblique and rectangular in 2-D, and triclinic, monoclinic and orthorhombic in 3-D). Mirror combination scheme is extremely useful in classification of crystal systems with correct point group allocation, in 1-D, 2-D and 3-D crystal systems. Need of the HCP (hexagonal close packing) and RCP (rhombohedral close packing) space lattices were overdue in crystallography. With their inclusion as independent lattices, the closed packed elements and compounds have got their due (proper and independent) representations in crystal system table.

Appendix 1 See Tables 2.10 and 2.11.

Appendix 2 See Tables 2.12 and 2.13.

Appendix 2

47

Table 2.10 Generating elements and their matrices (orthogonal axes) ⎞ ⎞ ⎛ ⎛ −1 0 0 100 ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ Identity1 = ⎜ Inversion1 = ⎜ ⎝ 0 −1 0 ⎠ ⎝0 1 0⎠ 0 0 −1 001 ⎛ ⎞ ⎞ ⎛ −1 0 0 10 0 ⎜ ⎟ ⎜ ⎟ ⎟ ⎟ 2[001] = ⎜ 2[001] = m[001] = ⎜ ⎝ 0 −1 0 ⎠ ⎝0 1 0 ⎠ 0 0 1 0 0 −1 ⎞ ⎞ ⎛ ⎛ −1 0 0 1 0 0 ⎟ ⎟ ⎜ ⎜ ∗ ⎟ ⎟ 2∗ [010] = ⎜ 2 [010] = m[010] = ⎜ ⎝ 0 1 0 ⎠ ⎝ 0 −1 0 ⎠ ⎛ ⎜ 3[001] = ⎜ ⎝

0 0 −1

− 21 −

√

√

3 2

3 2 1 −2

0

0 ⎞

⎛

0 0 1 ⎞

0

√

⎞

3 1 0 ⎟ ⎟ ⎜ 2√ 2 3 1 ⎟ ⎟ ⎜ 3[001] = , 0⎠ ⎝− 2 2 0 ⎠ 0 0 −1 1

0

0 −1 0 ⎟ ⎜ ⎟ 4[001] = ⎜ ⎝1 0 0⎠ 0 0 1 ⎞ ⎛ √ 1 − 23 0 2 ⎟ ⎜√ 3 1 6[001] = ⎜ 0⎟ ⎠ ⎝ 2 2 0

⎛

1

⎛

⎞

001 ⎜ ⎟ ⎟ 3∗ [111] = ⎜ ⎝1 0 0⎠ 010 ⎞

⎛ 0 1 ⎜ 4[001] = ⎜ −1 0 ⎝ 0 0 ⎛ −1 ⎜ √2 ⎜ 6[001] = ⎝ − 23 0

0

⎟ 0 ⎟ ⎠ −1 √

3 2 − 21

0

⎞

⎟ 0 ⎟ ⎠ 0 −1

48

2 Mirror: The Only Fundamental Symmetry in Crystals

Table 2.11 Generating elements and their matrices (crystallographic axes) ⎞ ⎞ ⎛ ⎛ −1 0 0 100 ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ Identity 1 = ⎜ Inversion 1 = ⎜ ⎝ 0 −1 0 ⎠ ⎝0 1 0⎠ 0 0 −1 001 ⎛ ⎞ ⎞ ⎛ −1 0 0 10 0 ⎜ ⎟ ⎜ ⎟ ⎟ ⎟ 2[001] = ⎜ 2[001] = m[001] = ⎜ ⎝ 0 −1 0 ⎠ ⎝0 1 0 ⎠ 0 0 1 0 0 −1 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 001 0 −1 0 0 1 0 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ∗ [111] = ⎜ ⎟ ⎟, 3[001] = ⎜ −1 1 0 ⎟ 3[001] = ⎜ 3 1 −1 0 ⎠ ⎠ ⎝1 0 0⎠ ⎝ ⎝ 0 0 −1

0 0 1 ⎛

0 −1 0

⎞

010 ⎛

⎞ 0 1 0

⎜ ⎟ ⎟ 4[001] = ⎜ ⎝1 0 0⎠

⎜ ⎟ ⎟ 4[001] = ⎜ ⎝ −1 0 0 ⎠ 0 0 −1

0 0 1 ⎞ ⎛ −1 0 0 ⎟ ⎜ ⎟ H → 2∗ [010] = ⎜ ⎝ −1 1 0 ⎠ 0 0 −1 ⎛ ⎞ 1 −1 0 ⎜ ⎟ ⎟ 6[001] = ⎜ ⎝1 0 0⎠

⎞ 1 0 0 ⎟ ⎜ ∗ ⎟ H → 2 [010] = m[010] = ⎜ ⎝ 1 −1 0 ⎠ 0 0 1 ⎛ ⎞ −1 1 0 ⎜ ⎟ ⎟ 6[001] = ⎜ ⎝ −1 0 0 ⎠ 0 0 −1

0 0 1 *Supplementary generating elements

Table 2.12 Crystallographic direction and symmetry elements

Direction

Symmetry elements

[000]

1

[100]

2, 2(H), 2(≡ m), 4, 4

[010]

2, 2(H), 2, 4, 4

[001]

2, 2, 3, 3, 4, 4, 6, 6

2, 2

2, 2, 3, 3

2, 2

⎛

Appendix 2

49

Table 2.13 Symmetry elements, directions and matrices Symmetry element

Directions

1

[000]

2

[100], [010], [001]

H[100], H[010]

[110], [101], [011]

[110], [110], [011]

H[210], H[120]

3

[011], [111]

Matrices ⎞ ⎛ 100 ⎟ ⎜ ⎜0 1 0⎟ ⎠ ⎝ 001 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 −1 0 0 −1 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 −1 0 ⎟, ⎜ 0 1 0 ⎟, ⎜ 0 −1 0 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 −1 0 0 −1 0 0 1 ⎞ ⎛ ⎞ ⎛ 1 −1 0 −1 0 0 ⎟ ⎜ ⎟ ⎜ ⎜ 0 −1 0 ⎟, ⎜ −1 1 0 ⎟ ⎠ ⎝ ⎠ ⎝ 0 0 −1 0 0 −1 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 01 0 0 0 1 −1 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 1 0 0 ⎟, ⎜ 0 −1 0 ⎟, ⎜ 0 0 1 ⎟ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 0 0 −1

1 0 0 0 10 ⎞ ⎛ ⎞ ⎛ ⎞ 0 −1 0 0 0 −1 −1 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −1 0 0 ⎟, ⎜ 0 −1 0 ⎟, ⎜ 0 0 −1 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 −1 −1 0 0 0 −1 0 ⎛ ⎞ ⎛ ⎞ 1 1 0 −1 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ 1 −1 0 ⎟, ⎜ 0 1 0 ⎟ ⎝ ⎠ ⎝ ⎠ 0 0 −1 0 0 −1 ⎞ ⎞ ⎛ ⎛ 001 0 −1 0 ⎟ ⎟ ⎜ ⎜ ⎜ 1 −1 0 ⎟, ⎜ 1 0 0 ⎟ ⎠ ⎠ ⎝ ⎝ ⎛

010 ⎞ ⎛ ⎞ ⎛ ⎞ −1 0 0 −1 0 0 1 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ 0 1⎟ ⎠, ⎝ 0 0 −1 ⎠, ⎝ 0 0 −1 ⎠ 0 0 1 0 0 −1 0 0 ⎞ ⎛ ⎞ 10 010 ⎟ ⎜ ⎟ ⎜0 0 1⎟ , 0 0⎟ ⎠ ⎝ ⎠

0 0 1 [111], [111], [111]

32

[011], [111]

[111], [111], [111]

⎛ 0 ⎜ ⎜ 0 ⎝ −1 ⎛ −1 ⎜ ⎜ −1 ⎝ 0 ⎛ 0 ⎜ ⎜ −1 ⎝ 0

01

100 ⎞ ⎛ ⎞ ⎛ ⎞ 0 −1 0 0 1 0 0 −1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 0 ⎟ ⎠, ⎝ −1 0 0 ⎠, ⎝ 1 0 0 ⎠ 1 0 0 −1 0 0 −1 0 (continued)

50

2 Mirror: The Only Fundamental Symmetry in Crystals

Table 2.13 (continued) Symmetry element

Directions

3–5 (S6 )

[111], [111], [111]

4

[100], [010], [001]

43

[100], [010], [001]

6

[001]

65

[001]

6–5

[001]

Matrices ⎛ ⎞ 1 −1 0 ⎜ ⎟ ⎜1 0 0 ⎟ ⎝ ⎠ 0 0 −1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 0 0 −1 0 −1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 1 ⎟, ⎜ 0 1 0 ⎟, ⎜ 1 0 0 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛

0 −1 0

1 0 ⎜ ⎜0 0 ⎝ 0 −1 ⎛ 1 −1 ⎜ ⎜1 0 ⎝ 0 0 ⎛ 0 1 ⎜ ⎜ −1 1 ⎝ 0 0 ⎛ 0 −1 ⎜ ⎜ 1 −1 ⎝ 0 0

10 0 ⎞ ⎛

0 0 1 ⎞ ⎛

⎞ 0 0 −1 0 −1 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1⎟ ⎠, ⎝ 0 1 0 ⎠, ⎝ 1 0 0 ⎠ 0 10 0 0 0 1 ⎞ 0 ⎟ 0⎟ ⎠ 1 ⎞ 0 ⎟ 0⎟ ⎠ 1 ⎞ 0 ⎟ 0 ⎟ ⎠ −1 0

* 1.

Matrix of an inverse symmetry element can be obtained by simply changing the sign of the digits of the starting matrix 2. 42 ≡ 2, 62 ≡ 3, 63 ≡ 2, and 64 ≡ 32 3

5

3. 3 ≡ S65 , 3 ≡ S63 = 1(i ), 3 ≡ S6 2

4

4. 6 ≡ S35 , 6 ≡ S34 = 3(C3 ) = 3 ≡ S62 ( ) 3 4 2 5 5. 6 ≡ S33 = m h (σh ); 6 ≡ S32 = 32 C32 = 3 ≡ S64 ; 6 ≡ S3

References 1. Wahab, M.A.: The mirror: mother of all symmetries in crystals. Adv. Sci. Technol. Med. 12 (2020). 2. Henry, N.F.M., Lonsdale, K. (ed.): Symmetry Groups, vol. I, International Tables for X-ray Crystallography. The Kynoch Press, Birmingham (1965) 3. Wahab, M.A.: Essentials of Crystallography, 2nd edn. Narosa Publishing House, New Delhi (2014) 4. Morgan, K.: Reflection and Planar Mirrors – m260. Project Physnet- Michigan State University, USA (2000) 5. Turrel, G.: Infrared and Raman Spectra of Crystals. Academic Press, USA (1972) 6. Koster, G.F.: Space Groups and Their Representations. Academic Press, USA (1957)

References

51

7. Leslie, A.: Crystals, Symmetry and Space Groups. MRC Laboratory of Molecular Biology (LMB) Crystallography Course, United Kingdom (2013) 8. Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Harcourt Asia PTE Ltd., Singapore (2001) 9. Wahab, M.A., Wahab, K.M.: Resolution of ambiguities and the discovery of two new space lattices. ISST J. Appl. Phys. 6(1), 1 (2015) 10. Philips, F.C.: An Introduction to Crystallography, ELBS and Longman Group Limited, 4th edn. The University Press, Glasgow (1971)

Chapter 3

Mirror Combination Scheme in Direct Lattice

3.1 Introduction In the first two chapters, we studied some important conceptual aspects of plane mirrors and the formation of different types and numbers of images with same or different handedness. Also, the mathematical formulation of intersecting mirrors was used to generate rotational and inversion symmetries. Further, 2 (1-D), 10 (2-D) and 32 (3-D) point groups have been determined using mirror combination scheme to justify the fundamental nature of mirror symmetry. Also, making use of a newly developed concept of minimum symmetry form (MSF) or point groups minima (PGM), it has been possible to make some inevitable changes in the wrongly allotted point groups in low symmetry crystal systems, such as oblique and rectangular in 2-D and triclinic, monoclinic and orthorhombic in 3-D. In this chapter, we are going to discuss the aspects that originate from two fundamental sources which are the true representatives of mirror combination scheme. The first source is related to the fundamentals of optics, according to which the combination of two parallel mirrors with an object atom between them, can form its infinite images as in a 1-D crystal exhibiting perfect translational periodicity (without the help of any translational symmetry). On the other hand, the second source is related to crystals of all three dimensions, where one or many pairs of parallel lines/planes acting as mirrors with the object atom at the center, present in every Wigner–Seitz cell (an alternative cell, equivalent in length/area/volume of the corresponding conventional unit cell of the direct space lattices), which can respectively produce the desired 1-D, 2-D and 3-D crystal lattices in direct space (without the help of translational symmetry) which exhibit perfect translational periodicity, while keeping intact the symmetries of the corresponding conventional lattices. Thus, we come to know that the generation of infinite periodic crystal lattice with the help of mirror combination scheme in all three dimensions definitely contradicts any role of translational symmetry, while the same is considered to be responsible for the generation of crystals of different dimensions in the present day crystallography. Further, a brief

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. A. Wahab, Mirror Symmetry, Springer Series in Solid-State Sciences 200, https://doi.org/10.1007/978-981-99-8361-2_3

53

54

3 Mirror Combination Scheme in Direct Lattice

description of the construction of HCP and RCP unit cells is provided to introduce them to maintain the continuity of discussion of possible lattices and crystal systems in 3-D. The consideration of HCP and RCP certainly helps us to derive the correct symmorphic space groups of different crystal dimensions based on the mirror combination scheme and by using the correct crystallographic data obtained after applying the necessary corrections in the point groups of different lattices in 2-D and 3-D crystal systems in the last chapter. This exercise will indirectly verify the authenticity of inclusion of the two new (HCP and RCP) space lattices and the point group corrections carried out in the last chapter.

3.2 Procedure to Construct a Wigner–Seitz Unit Cell A Wigner–Seitz unit cell is an alternative way of selecting a primitive unit cell of equal length, equal area or equal volume in a given 1-D, 2-D or 3-D crystal system. Thus in general, the Wigner–Seitz cell is defined as primitive cell such that for a fixed lattice point, the W–S cell is the length, area or volume surrounding it, that is closest to that point. Length, area and volume of the respective Wigner–Seitz unit cells (Fig. 3.1) of general 1-D, 2-D and 3-D lattices are: Length of a 1-D unit cell = a

Ar ea o f a 2 − D unit cell = a⭢ × b⭢ = ab sinγ

3.3 Combination of Two Parallel Mirrors and 1-D Lattice

55

Fig. 3.1 Illustrations of length in 1-D, area in 2-D and volume in 3-D general lattices

V olume of a 3 − D unit cell = a⭢ · b⭢ × c⭢ ( )1/2 = abc 1 − cos 2 α − cos 2 β − cos 2 γ + 2 cos α cos β cos γ A Wigner–Seitz unit cell is constructed around a lattice point according to the following procedure: 1. Select a reference point in a given lattice and draw lines to connect this point with all other neighbouring lattice points. 2. At mid points of each line segment between the reference point and the neighbouring lattice points, draw a vertical line (in 1-D and 2-D) and a plane (in 3-D) as perpendicular bisector. 3. Smallest length (in 1-D), area (in 2-D) or volume (in 3-D) enclosed around the reference lattice point gives the required Wigner–Seitz unit cells (mentioned above). 4. The characteristic feature of a Wigner–Seitz unit cell is that it retains the symmetry of the original lattice.

3.3 Combination of Two Parallel Mirrors and 1-D Lattice Visualization of infinite images formed by a pair of parallel mirrors hanged on the opposite walls of a barber’s shop is a common phenomenon. This is possible due to simple principle of ray optics [1]. The fundamental principle of image formation of an object, such as an atom placed between two parallel mirrors is shown in Fig. 3.2. Using the above principle of parallel mirrors with the object such as an atom in between them, infinite images of this in the form of a one-dimensional chain of atoms (as a 1-D lattice) can be obtained. This suggests that actual translation of

56

3 Mirror Combination Scheme in Direct Lattice

Fig. 3.2 Principle of image formation of an object by two parallel mirrors

the object point is neither required nor physically (or practically) possible to get the (infinite) array, actually it is the combined effect of the parallel mirrors and the object point between them to generate an infinite periodic array without the involvement of any form of translation or the so called ‘translational symmetry’. This implies that the lattice array is not the result of any physical translation of the object point infinite times in space (as in general associated with the translation symmetry) but it is the lattice array (lattice and the associated basis) which exhibits the translational periodicity due to the combined effect of two parallel mirrors and the object between them.

3.4 Mirror Combination Scheme in W–S of 1-D Lattice Construction of a W–S cell in a 1-D lattice is the easiest one. Simply select a reference point in the given lattice, connect this point with two other neighbouring lattice points. Draw a vertical line at mid points of each line segment as a perpendicular bisector. The region between the two vertical lines is the required W–S cell. This has the same length as the length between two points in the original lattice as shown in Fig. 3.3. The characteristic feature of the two parallel lines (acting as mirror lines) with a lattice point in between them, and that the said arrangement can produce infinite images as encountered above in 1-D lattice without the involvement of any kind of translation. It is interesting to note that the W–S cell retains the symmetry of the original lattice. Considering an inverse process, with the help of two parallel mirror lines and an object (atom) between them, the formation of 1-D crystal lattice can be obtained in an extended form exhibiting perfect translational periodicity (Fig. 3.3b).

3.6 Constructions of HCP, RCP Lattices and Their W–S Unit Cells

57

Fig. 3.3 a A direct unit cell, and b a Wigner–Seitz unit cell in 1-D

3.5 Mirror Combination Scheme in W–S of 2-D Lattices Wigner–Seitz cells of 2-D lattices (obtained by using the procedure mentioned above) are shown in Fig. 3.4 [2]. Conversely, using the basic principle of formation of 1-D infinite lattice with the help of two parallel mirrors and an object between them, the formation of 2-D crystal lattices can be understood in their extended forms. Here, each Wigner–Seitz cell possesses several pairs of parallel lines (called ‘mirror lines’), each pair produces infinite 1-D images. Combining together the different pairs of parallel mirror lines of the given Wigner–Seitz cell will produce the desired infinite 2-D lattice exhibiting perfect translational periodicity. Further, each W–S cell retains the symmetry of the corresponding original cell (lattice). Here, it is to be pointed out that in Fig. 3.4c, there seemed to appear the glide lines as mentioned in the literature, as long as the lattice remained in the conventional form. However, as soon as the Wigner–Seitz cell is constructed, the impact of glide line disappears.

3.6 Constructions of HCP, RCP Lattices and Their W–S Unit Cells It is well known from the history of crystallography that Bravais worked out a mathematical theory of crystal symmetry based on the concept of the crystal lattice and proposed 14 space lattices in 1848, then classified them into 7 crystal systems [3]. However, due to persistent ambiguity in the Bravais representation of trigonal, rhombohedral and hexagonal structures, Wahab and Wahab [4] in their studies on close packing of identical atoms (spheres) discovered that the hexagonal close packing (HCP) and rhombohedral close packing (RCP) actually represent independent space lattices and they called them ‘Wahab lattices’ by constructing their primitive unit

58

3 Mirror Combination Scheme in Direct Lattice

Fig. 3.4 Wigner–Seitz unit cells of 2-D lattices

cells, Wigner–Seitz unit cells, Brillouin zones, supported by mathematical calculations. In fact, these two lattices were very much needed to explain many ambiguous crystallographic anomalies. For example, the existing ambiguity in the number of symmorphic space groups: 14 space lattices give only 61 symmorphic space groups [5, 6] and the remaining 12 symmorphic space groups are obtained from these two to make the total 73, it is the same number as has been proposed according to arithmetic groups. Further, based on the geometry and symmetry of the unit cells, Wahab and Wahab [4] classified the 16 space lattices into 8 crystal systems. Indeed, the new

3.6 Constructions of HCP, RCP Lattices and Their W–S Unit Cells

59

format of 8 crystal systems, 16 space lattices and 32 point groups is more appealing and provides us greater symmetry system in crystallography. Below, we provide only a brief description of the construction of HCP and RCP unit cells (for details, refer [4]). We start with a close packed layer of identical spheres of radius R (in the present case R = 50 mm) of A—type as shown in Fig. 3.5a. An identical layer is placed over the reference layer such that the upper spheres just touch the tips of the lower spheres (so that a = b = c, a special case) to get a simple hexagonal structure as shown in Fig. 3.5b (in general a = b /= c is a standard consideration for SH structures). However, if the identical layer is displaced with respect to the reference layer such that the upper spheres just fit into B voids, the resulting structure is a tilted hexagonal close packed (HCP) structure as shown in Fig. 3.5c (alternatively when the upper spheres just fit into C voids, Fig. 3.5d is obtained). The lower and upper spheres are joined properly to complete the unit cell of simple hexagonal (SH) and the hexagonal close packed (HCP) unit cells in two possible orientations. In a similar manner, if the spheres in third and fourth layers are respectively placed on C and A voids without changing the direction of packing, we get rhombohedral and cubic close packed (RCP/CCP) unit cells as shown in Fig. 3.6a (where CCP unit is simply the second order rhombohedron and hence has no independent identity). For a comparison, the formation of RCP unit in a FCC unit cell is shown in Fig. 3.6b.

Fig. 3.5 Construction of a a single close packed layer, b SH unit, c HCP unit AB (cyclic), d HCP unit AC (anti-cyclic)

60

3 Mirror Combination Scheme in Direct Lattice

Fig. 3.6 Formation of a RCP (blue lines) and CCP (red lines), b RCP in FCC

After giving a brief description of unit cell construction, we also provide the results obtained from calculations and the overall results in brief in the form of points. Results obtained from various geometrical constructions and the crystallographic calculations have been summarised in the Tables 3.1 and 3.2. 1. A simple hexagonal lattice (SH) is well known and defined as a = b = c, α = β = 90° and γ = 120°, while a hexagonal close packed (HCP) lattice is defined for the first time with a = b = c = 2R and α = 60◦ , β = γ = 120◦ . Table 3.1 Comparison of unit cell shape and axes for SH, HCP and FCC lattices Lattice type

Direct lattice Primitive lattice

SH (special) SH a = b = c = 2R a = b = c = 2R α = β = 90°, γ = 120°

Reciprocal lattice Wigner–Seitz unit cell

Reciprocal lattice

Brillouin zone

SH

SH a* = b* /= c* Axes (x and y) rotate thorough 30°

SH

Rhombohedron

Truncated octahedron

HCP Rhombic RCP a = b = c = 2R a = b = c = 2R dodecahedron α = 60°, α = β = γ = 60° β = γ = 120° FCC a=b=c α=β=γ= 90°

Rhombic RCP Dodecahedron a, = b, = c, = 2R α = β = γ = 60°

√

a* =b* =c* = √3

( 2π ) a

2

α* = 109.47°, β* = γ* = 70.53° Rhombohedron √

a* =b* =c* = √3 2

( 2π )

α* = β* = γ* = 109.47°

a,

Truncated octahedron

3.6 Constructions of HCP, RCP Lattices and Their W–S Unit Cells

61

Table 3.2 Comparison of different volumes for SH, HCP and FCC unit cells Lattice type

SH HCP FCC/CCP

Direct lattice Conventional lattice √ 12 3R3 √ 12 2R3 √ 16 2R3

Reciprocal lattice Primitive lattice √ 4 3R3 √ 4 2R3 √ 4 2R3

Wigner–Seitz unit cell* √ 4 3R3 √ 4 2R3 √ 4 2R3

Reciprocal lattice ( 2π )3 1 √ 4 3

1 √ 4 2 1 √

4 2

R

( 2π )3 R

( 2π )3 R

Brillouin zone* ( 2π )3 1 √ 4 3

1 √ 4 2 1 √

4 2

R

( 2π )3 R

( 2π )3 R

√ √ 3 2 l l ∗ For rhombic dodecahedron, = √ ; for truncated octahedron, = √ R R 2 3 (where l is the side of the polyhedron)

2. One SH unit is made up of three parallelepiped units, each with a = b = c = 2R, α = β = 90° and γ = 120°, while one HCP unit is made up of three rhombohedral close packed (RCP) units, each with a = b = c = 2R, α = β = γ = 60°. That is, the primitive unit of SH remains SH, while the primitive unit of HCP is RCP. In both cases, three primitive units are completely contained within their respective parent unit cells. 3. The axis of the simple hexagonal unit is the same as the principal axis, i.e. [along] [001] direction, while the axis of the cyclic AB unit of HCP is tilted along 113 direction. The angle of tilt with respect the principal axis is given by (√ ) 2 cos √ = 35.26◦ 3 -1

4. Both, the constructions as well as calculations of Wigner–Seitz units and the Brillouin zones corresponding to two unit cells (SH and HCP) suggest that the resulting shape for SH remains simple hexagonal, while for HCP they are rhombic dodecahedron and truncated octahedron, respectively (Table 3.1). 5. The primitive unit, the Wigner–Seitz unit and the Brillouin zone of HCP are found not only identical to their respective counterpart of FCC in shape but also equal in volume during comparison of HCP and FCC units. The same was supported by unit cell calculations. Table 3.2 provides a comparison of volumes for SH, HCP and FCC unit cells. 6. Three layer atomic packing of FCC, HCP and RCP exhibit the same coordination number 12, but their orientations are found to be different. (i) In FCC, the close packed planes are arranged in a body diagonal manner along the [111] direction (Fig. 3.6b), while in CCP (known to be the second order rhombohedron) and RCP, they are arranged along the principal axis (Fig. 3.6a).

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3 Mirror Combination Scheme in Direct Lattice

Fig. 3.7 Coordination in a FCC, b HCP and c RCP

(ii) In Wigner–Seitz unit, the twelve identical bonds of FCC follow 4:4:4 configurations involving three planes of atoms, while in HCP and RCP, they follow 3:6:3 configurations involving three planes of atoms (Fig. 3.7) but with different orientations. In HCP, the close packed planes arrange in AB/A… sequence while in RCP, they follow ABC/A… sequence. (iii) In the Brillouin zone of HCP, four hexagonal axes pass through the hexagonal faces of the truncated octahedron, while in the Brillouin zone of FCC, the three cubic axes pass through the square faces of the truncated octahedron (Fig. 3.8). Equivalence of the Wigner–Seitz/Brillouin zone volume of HCP and FCC is also supported by simple crystallographic calculations. These results are not only very interesting but also very useful and have been observed for the first time in crystallographic history. They are understandable because of the fact that FCC, HCP and RCP are close packed, have equal packing efficiency (74%) and have same coordination numbers (12) in a system of identical atoms (mono atomic system), however as shown above, the orientations of atoms in them are different (Fig. 3.7).

Fig. 3.8 B–Z of a HCP and b FCC

3.7 Mirror Combination Scheme in W–S of 3-D Lattices

63

7. A trigonal unit is inherently associated with SH throughout (or in general other non-close packed structures in direct lattice) while a rhombohedral unit is inherently associated with HCP as shown above (or in general other close packed structures such as FCC in direct lattice), respectively. 8. The study of symmetry related to SH and HCP suggests that the characteristic symmetry of simple hexagonal structure is a sixfold pure rotation only, that of trigonal and HCP must be a 3- or 6-fold rotation, and that of a rhombohedral structure must be a 3-fold rotation, while a threefold is common to all. Further, a trigonal structure can be only non-close packed while HCP and RCP must be close packed only. 9. The symmetry analysis of SH, Trigonal, HCP and RCP units suggests that the number of symmetry elements (point groups) belonging to trigonal and SH together is the same as that of HCP and RCP combined, i.e. Symmetry elements of (Trigon + SH) = Symmetry elements of (HCP + RCP) = 7 + 5 = 12 10. It is interesting that Trigonal and HCP exhibit identical symmetry elements (point groups) in terms of both numbers and nature, while SH and RCP exhibit identical number but of different nature. The above results related to geometry and symmetry of SH, Trigonal, HCP and RCP unit cells clearly remove all existing ambiguities and confusions related to their representations. Now, they can be represented quite distinctly and clearly in terms of their geometry and symmetry.

3.7 Mirror Combination Scheme in W–S of 3-D Lattices In Sects. 3.4 and 3.5, we discussed the constructions of Wigner–Seitz cells of 1-D and 2-D lattices in detail. In this section, let us discuss the same for 3-D space (or Bravais and Wahab) lattices, this terminology was used first by Wahab and Wahab [4]. We can easily construct the W–S cells of all sixteen lattices by following the set procedure enumerated above. However, in Figs. 3.9 and 3.10 respectively, we only show the constructions of W–S cells of three well known cubic lattices (on which our emphasis has been throughout in this book) and the two newly found HCP and RCP lattices (as they are not available in literature). Here, the W–S cell of simple cubic lattice exhibits 6 equivalent {100} square faces (or three equivalent pairs of mirror planes) in the form of a cube as shown inside Fig. 3.9a. Similarly, the W–S cell of bcc lattice exhibits 14 faces in all, out of which 8 faces represent equivalent {111} hexagonal faces (four equivalent pairs of mirror planes) and 6 faces represent equivalent {200} square faces (or three equivalent pairs of mirror planes), respectively. The shape of the W–S of bcc lattice is truncated octahedron shown inside the cube in Fig. 3.9b. Finally, the W–S cell of fcc lattice

64

3 Mirror Combination Scheme in Direct Lattice

Fig. 3.9 Wigner–Seitz unit cells of sc, bcc and fcc

Fig. 3.10 Wigner–Seitz unit cells of a HCP and b RCP

exhibits 12 equivalent {110} rhombus faces (or six pairs of mirror planes). The shape of the W–S of fcc lattice is rhombic dodecahedron shown inside the bigger cube in Fig. 3.9c. Further, each W–S cell retains the symmetry of the corresponding original cell (lattice). To our surprise, the W–S of HCP and RCP unit cells are found to be similar in shape to that of the W–S of FCC (Fig. 3.9), except a slight difference in their orientations, it is due to the difference in the orientations of the first nearest neighbour coordinating atoms in three different close packed structures as shown in Fig. 3.7. We know that in a conventional lattice, the primitive cell has only one atom, while a non-primitive cell has more than one atom per unit cell, respectively. On the other hand, a Wigner–Seitz cell is a specific type of primitive cell, it contains the only atom at its center enclosed by many pairs of parallel planes (acting as mirrors) as shown

3.8 Derivation of Correct Symmorphic Space Groups

65

in Fig. 3.4 for 2-D and Figs. 3.8, 3.9 and 3.10 for 3-D lattices, respectively. The Wigner–Seitz unit cell has a very special feature in it, in real space, it acts simply as a mathematical construct but in momentum space (or k-space) it defines a Brillouin zone. From fundamental crystallography, we know that FCC, HCP and RCP all belong to close packed crystal system but happen to represent different crystal structures. In mono atomic system, they have the same coordination number (12), their Wigner– Seitz cells bound to have the same shape (however, the corresponding conventional unit cells have different shapes) but oriented differently due to different ways of packing of atoms in them (i.e. 4:4:4 coordination in FCC and 3:6:3 both in HCP and RCP, respectively. However, HCP and RCP also differ between them in their orientations due to different arrangements of close packed layers in their structures, such as AB/A… and ABC/A… as shown in Fig. 3.7 and hence they represent different crystal structures. This provides us three different orientations of close packing of identical atoms which have only one W–S/B–Z shape. This actually provides the answer of 14 geometrical shapes (as proposed by Bravais simply on mathematical basis) and 16 space lattices demonstrated by Wahab and Wahab [4] on the basis of the derivation of lattice parameters, constructing their primitive cells, Wigner–Seitz cells and Brillouin zones, and well supported by mathematical calculations for the first time in the history of crystallography (this is of course cannot be explained simply on the basis of conventional lattices). It appears that this aspect has been missed by all earlier crystallographers including Bravais himself. The beauty of the W–S/B–Z representation of periodically arranged atoms in crystalline material is that irrespective of the primitive or non-primitive conventional unit cell, the corresponding W–S/B–Z unit cell is always primitive, where the only atom situated at the center is surrounded by different pairs of equivalent parallel mirrors (in a centro-symmetric manner) and the periodic arrangement of atoms obtained from them, they define the direct and reciprocal space lattices with perfect translational periodicity. The centrosymmetric nature of diffraction pattern and the Brillouin zone with a defined origin (000) allows the symmetries such as mirror, rorational, rotoreflection, rotoinversion and translational periodicity but forbids any form of translational symmetries, whether microscopic or macroscopic [7].

3.8 Derivation of Correct Symmorphic Space Groups Based on arithmetic groups, the proposed symmorphic space groups in 1-D, 2-D and 3-D crystal systems are 2, 13 and 73, respectively. However, the discrepancy about the number of symmorphic space groups in literature for 3-D is quite common. Majority of the standard text books (e.g. [5, 6], and many others) mention only 61. However, some of the texts also mention 65 and 73, but their explanations for additional numbers are extremely vague, hypothetical and qualitative in nature. Because of the persisting ambiguities and confusions related to symmorphic space groups, in the following we discuss this longstanding issue and look for its resolution.

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3 Mirror Combination Scheme in Direct Lattice

As we know through literature that the proposed 73 symmorphic space groups or the arithmetic groups is based on the crystal systems and the associated point groups with them. However, due to the inevitable changes in the assignment of point groups to some 2-D (such as oblique and rectangular) and 3-D (e.g. triclinic, monoclinic, orthorhombic, trigonal and hexagonal) crystal systems as discussed in the last chapter, and the discovery of two independent and new space lattices (the HCP and RCP) by Wahab and Wahab [4], we expect a minor change in the total number of arithmetic groups/symmorphic space groups. Let us check the same through calculation, using the basic formula given by, Symmorphic space group in a crystal system = (No. of lattices × No. of point groups) in that crystal system Therefore, in order to determine the symmorphic space groups, it is necessary to know the number of lattices in a crystal system and the number of point groups associated with each lattice of that crystal system. Taking into account only the 14 space lattices and the point groups associated to seven crystal systems (according to earlier classification), only 61 symmorphic space groups could be obtained as mentioned by Ashcroft and Mermin, and Martin T. Dove, etc. However, taking into account the two new space lattices, HCP and RCP in 3-D and the calculations carried out based on the corrected crystallographic data (provided in the Tables 2.8 and 2.9 with no change in 1-D) related to 1-D, 2-D and 3-D are summarized in Table 3.3. The HCP contains 7 point groups and RCP 5, and together they contribute 12, the number that is required in 3-D to make the total symmorphic space groups to be 61 + 12 = 73. This, in turn, justifies the consideration of HCP and RCP as the two new and independent lattices. The difference of one (i. e. 73 from arithmetic groups and 74 from present calculation) that occurs, may be due to the inevitable reshuffle of point groups in different crystal systems as discussed in the last chapter. The crystal system, the point groups and the number of symmorphic space groups for 3-D are provided in Table 3.4, where the consideration of two symmorphic space groups P(1), B(1) has the same basis as that of 10 symmorphic space groups, 5 each belonging to trigonal and HCP, and common with rhombohedral system. Table 3.3 Crystallographic Data to obtain Symmorphic Space Groups Dimension

Lattice types

Point groups (no change)

Symmorphic space groups

Previous

Revised

Previous

Revised

One

1

1

2

2

2

Two

5

5

10

13

12

Three

14

16

32

61

(73) 74

3.9 Summary

67

Table 3.4 Three dimensional symmorphic space groups Crystal system Lattice Point groups type

Symmorphic space groups

Triclinic

Pm, P1, P1

Monoclinic

Orthorhombic

P

m, 1, 1 ( ) P 2, mm2, 1 B or C P C, A or B I F

2 m

( ) P2, Pmm2,P 1 ( ) B2, Bmm2,B 1 P2 m , P222, Pmmm C2 m , C222, Cmmm I2 m , I222, Immm F2 m , F222, Fmmm

222 mmm

Rhombohedral P 3, 3, 32, 3m, 3m (CP) (RCP)

R3, R3, R32, R3m, R3m

Trigonal hexagonal (CP)

P HCP

3, 3, 32, 3m, 3m, 6, 62m P3, P3, P32, P3m, P3m, P6, P62m

Tetragonal

P I

4, 4,

CP3, CP3, CP32, CP3m, CP3m, CP6, CP62m 4 m,

422, 4mm, 42m,

Simple hexagonal

P

Cubic

P I F

*

6mm,

I4,

4 mmm

6, m6 , 622,

P4 P4 m , P422, P4mm, P42m, mmm I4 P4, I4 m , I422, I4mm, I42m, mmm

P4, P4,

P6,

P6 m,

P622, P6mm,

P6 mmm

6 mmm

23, m3, 432,

P23, Pm3, P432, P43m, Pm3m

43m, m3m

I23, Im3, I432, I43m, Im3m

( ) 1 not counted towards the total

F23, Fm3, F432, F43m, Fm3m

3.9 Summary 1. Based on the principle of mirror combination scheme, using two parallel mirrors with an object atom in between them, infinite images of the same can be obtained in the form of a one-dimensional crystal lattice. Infinite translations of the object/ unit cell are neither required nor possible (practical) in physical sense. 2. Combinations of parallel mirrors in 1-D, 2-D and 3-D as found in Wigner– Seitz unit cells, produce corresponding direct space lattices, all exhibiting perfect translational periodicity. 3. Formation of 2-D and 3-D Wigner–Seitz unit cells can be understood in terms of mirror combination scheme in direct space lattices (real crystals). In turn, the formation of infinite images by 2-D and 3-D Wigner–Seitz unit cells exhibiting perfect translational periodicity, clearly negates the requirement (or existence) of translational symmetry in crystals. 4. Each W–S cell retains the symmetry of the corresponding direct lattice in all crystal dimensions.

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3 Mirror Combination Scheme in Direct Lattice

5. A comparison of the primitive unit, Wigner–Seitz unit and the Brillouin zone of HCP and FCC is found to show identical shape and identical volume. The same result is obtained from unit cell calculations. 6. Three layer atomic packing of FCC, HCP and RCP exhibit the same coordination number 12, but their orientations are found to be different. 7. In Wigner–Seitz unit, the twelve identical bonds of FCC follow 4:4:4 configurations involving three planes of atoms, while in HCP and RCP, they follow 3:6:3 configurations involving three planes of atoms (Fig. 3.7) but with different orientations. In HCP, the close packed planes arrange in AB/A… sequence while in RCP, they follow ABC/A… sequence. 8. Inclusion of the HCP (hexagonal close packing) and RCP (rhombohedral close packing) space lattices in 3-D crystal system helps us correctly determine the symmorphic space groups as predicted by the arithmetic group. 9. Mirror combination scheme is extremely useful in deriving the correct symmorphic space groups, both in 2-D and 3-D crystal systems.

References 1. Morgan, K.: Reflection and Planar Mirrors—m260. Project Physnet- Michigan State University, USA (2000) 2. Wahab, M.A.: Essentials of Crystallography, 2nd ed. Narosa Publishing House, New Delhi (2014) 3. Schuh, C.P.: Mineralogy and Crystallography: On the History of These Sciences From Beginnings Through 1919, Tucson, Arizona (2007) 4. Wahab, M. A., Wahab, K. M.: Resolution of Ambiguities and the Discovery of Two New Space Lattices. ISST. J.Appl.Phys. 6(1) 1 (2015) 5. Ashcroft, N. W., Mermin, N. D.: Solid State Physics, Harcourt Asia PTE Ltd., Singapore (2001) 6. Dove T. M.: Structure and Dynamics (An Atomic View of Materials). Oxford Master Series in Condensed Matter Physics, Oxford University Press, Oxford (2003) 7. Leslie, A.: Crystals, Symmetry and Space Groups, MRC Laboratory of Molecular Biology (LMB) Crystallography Course, United Kingdom (2013)

Chapter 4

Mirror Combination Scheme in Reciprocal Lattice

4.1 Introduction In the last chapter, we studied about the construction of Wigner–Seitz cells in 1-D, 2-D and 3-D lattices and the related aspects. The Wigner–Seitz cells are found to possess certain sets of equivalent parallel mirrors (and hence work in accordance with mirror combination scheme) which are capable of producing infinitely extended 1-D, 2-D and 3-D lattices, where each of them exhibits perfect translational periodicity. Significantly, a Wigner–Seitz cell retains the symmetry of its corresponding conventional form of linear, planar and space lattice. A direct lattice and its corresponding reciprocal lattice are known to be inversely related to each other. Accordingly, both Wigner–Seitz cells and Brillouin zones are inversely related to each other. In fact, a Brillouin zone is simply the Wigner–Seitz cell of the given reciprocal lattice. Therefore, both direct and reciprocal lattices exhibit similar symmetries and hence can be represented by the same coordinate axes. Further, like the Wigner–Seitz cells, the Brillouin zones are primitive in nature with the only atom at their center and hence they are inherently centro-symmetric. Similarly, like the Wigner–Seitz cells, the Brillouin zones are based on mirror combination scheme which can produce the corresponding reciprocal lattice exhibiting perfect translational periodicity in all three crystal dimensions. It is important to note that all forms of diffraction process are taking place in reciprocal space and the resulting diffraction pattern of a given crystal structure corresponds to different sets of Braggs’ planes; they actually represent different zone boundaries of different Brillouin zones. Therefore, considering the diffraction results of different cubic crystals in general, the first diffraction peak observed in a powder pattern (or the set of nearest reflections in the form of spots in other single crystal diffraction patterns) corresponds to the first Brillouin zone of a simple cubic or bcc unit cell. This is obtained from the widest set of parallel planes of the given crystal structure, because they are identified to have the least reciprocal lattice vector and hence diffract at the lowest Braggs’ angle, first. On the other hand, the

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. A. Wahab, Mirror Symmetry, Springer Series in Solid-State Sciences 200, https://doi.org/10.1007/978-981-99-8361-2_4

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70

4 Mirror Combination Scheme in Reciprocal Lattice

first two diffraction peaks are required to complete the first Brillouin zones for fcc and DC, each. Higher order diffraction peaks, similarly correspond to the different sets of parallel planes of progressively increasing size of reciprocal lattice vectors (or decreasing interplanar spacing in direct lattice) to produce other higher order Brillouin zones for a given crystal structure.

4.2 The Reciprocal Lattice From the study of fundamental crystallography, we are familiar with the structures of different crystal systems in ordinary or coordinate space (also called the direct or real space). These provide us the positions of atoms in the crystal lattice (also called the direct lattice). However, to determine the direct space lattice structure, the most popular experimental technique that we use is some forms of diffraction (which include X-rays, electrons, and neutrons). The particles in the beams that are used in these techniques have dual properties and behave as matter waves through the de Broglie relationship, λ = h/p, where λ is the wavelength, h the Planck’s constant (= 6.63 × 10−34 J·s), and p the momentum of the particle, respectively. Further, the propagation of a wave is described by the advancement of its wave front, where the wave vector k⭢in is perpendicular to the plane wave front, and the relationship between the vectors p⭢ and k⭢ is given by p⭢ = [h/(2π )]k⭢ where h/(2π) = è. ⭢ one obtains Further, from the following two relationships, λ = h/ p⭢ and p⭢ = è k, ⭢ = 2π/λ |k| Here, it is to be noted that the wave vector k⭢in has a unit of inverse length. Therefore, it is convenient to define a reciprocal space lattice in the momentum space that is related to the real space. However while dealing with a crystal, we know that every crystal has two lattices associated with it: the crystal lattice, and the reciprocal lattice. Thereby, we mean that a given crystal lattice is a reciprocal lattice, either of itself (such as the case of a simple cubic lattice) or of some other appropriate lattice. Therefore, the symmetry of a real space lattice and the symmetry of its reciprocal space lattice is related to each other. Table 4.1 provides the direct and the corresponding reciprocal lattices of 3-D crystal systems [1, 2]. On the other hand, the reciprocal lattice associated with a crystal is also known as a lattice in the Fourier space. Further, the diffraction pattern of a crystal is in fact a representation (or a map) of the reciprocal lattice of the crystal, and not a direct representation (or a map) of the crystal lattice and each point in the reciprocal lattice represents a set of equidistant parallel planes in the direct lattice. The simple cubic

4.2 The Reciprocal Lattice Table 4.1 Direct and corresponding reciprocal lattice in different crystal systems

71

Crystal system

Direct lattice

Reciprocal lattice

Triclinic

P

P

Monoclinic

P, C

P, C

Orthorhombic

P, A or B or C, I, F P, A or B or C, F, I

Rhombohedral (RCP) R

I

Trigonal/HCP

P/I

P/HCP

Tetragonal

P, I

P, I

Hexagonal

P

P

Cubic

P, I, F

P, F, I

lattice is a good example of self-dual, because the shape and symmetry of the cube remain the same in both real space and in reciprocal space. The fundamental concepts of reciprocal lattice can be understood broadly in two different ways [3-5]: 1. First approach is based on the definition of reciprocal lattice (unit cell) vectors ⭢ c⭢, using simple a⭢ ∗ , b⭢∗ , c⭢∗ in terms of the direct lattice (unit cell) vectors a⭢ , b, vector algebra and simple geometry. With the help of this, several other useful relations can be derived. For example, refer next section. 2. The second approach is based on the notion that crystals are made of a large number of different families (or sets) of equidistant parallel planes, where each set of planes can be represented by a common normal drawn from a point (taken as the origin) as a one dimensional line, and setting its length to be equal to 2π times the reciprocal of interplanar spacing dhkl (the spacing between the two consecutive planes) as shown in Fig. 4.1. This is specified as a reciprocal lattice vector and the first reciprocal lattice point R1 is marked at the end of the first vector, resulting reciprocal lattice points along a* axis is shown in Fig. 4.1b. In a similar manner, other reciprocal lattice vectors for the remaining sets of planes can be obtained and their terminal points can be used to define the pattern of points called the reciprocal lattice points. 3. After generating the reciprocal lattice net, the Brilluoin zone is simply defined as the Wigner–Seitz unit cell of the reciprocal lattice: B-Z = R-L + W-S It is important to note that FCC, HCP and RCP in a monoatomic system represent close packed structures in direct space with 74% packing efficiency. However, Table 4.1 suggests that in reciprocal space, they all represent the body centered lattice (as expected) with 68% packing efficiency; this belongs to a non-close packed system.

72

4 Mirror Combination Scheme in Reciprocal Lattice

Fig. 4.1 a A common normal to a set of equidistant parallel planes, b reciprocal lattice row

4.3 Construction of Reciprocal Lattice (First Approach) The dimension wise relationships between the direct lattice and reciprocal lattice are: Case I: 1-D Lattice The relationship between the reciprocal lattice translation vector ‘⭢ a ∗ ’ for a given 1-D a ’ is given by direct lattice (primitive) with the translation vector ‘⭢ a⭢ ∗ · a⭢ = a⭢ · a⭢ ∗ = 2π or ⭢a∗ =

2π a⭢

This indicates that the reciprocal lattice translation has the same direction as that of the direct lattice but its unit is inverse, such as (length−1 ). Reciprocal character of these two lattices are shown in Fig. 4.2. Case II: 2-D Lattice (a) Plane Rectangular Lattice For a given plane rectangular lattice with a /= b and γ = 90◦ , the following relationships between the reciprocal lattice translation vectors a⭢ ∗ , b⭢∗ and the direct lattice (primitive) translation vectors a⭢ , b⭢ exist:

4.3 Construction of Reciprocal Lattice (First Approach)

73

Fig. 4.2 Direct and reciprocal lattice vectors in 1-D exhibiting reciprocal character

a⭢ ∗ · a⭢ = a⭢ · a⭢ ∗ = 2π b⭢∗ · b⭢ = b⭢ · b⭢∗ = 2π a⭢ ∗ · b⭢ = b⭢∗ · a⭢ = 0 so that ⭢a∗ =

2π a⭢

2π ∗ and b⭢ = b⭢ These relationships suggest that ⭢ or a⭢ ∗ is || to a⭢ a⭢ ∗ is ⊥ to b, and b⭢∗ is ⊥ to a⭢ , or b⭢∗ is || to b⭢ They are shown in Fig. 4.3. (b) Oblique Lattice For a general plane lattice with a /= b and γ = γ ◦ , the relationships between the direct and reciprocal lattices are not straightforward as for the orthogonal axes. Here, the angle between a* and b* will be 180° − γ as shown in Fig. 4.4. Here, their magnitudes are: γ = 90 − γ ∗ + 90 − γ ∗ + γ ∗ = 180 − γ ∗ Therefore, γ ∗ = 180 − γ

74

4 Mirror Combination Scheme in Reciprocal Lattice

Fig. 4.3 Direct and reciprocal lattice vectors in 2-D (rectangular lattice) exhibiting reciprocal character Fig. 4.4 Direct and reciprocal lattice vectors in 2-D (oblique lattice)

Further, d⭢10 = a⭢ sin (180◦ − γ ) = a⭢ sin γ , and a⭢ ∗ =

2π 2π 2π = = ◦ ⭢ a ⭢ sin (180 − γ ) a ⭢ sin γ d10

Similarly, d⭢01 = b⭢ sin (180◦ − γ ) = b⭢ sin γ , and 2π 2π 2π = = b⭢∗ = ◦ ⭢ ⭢ ⭢ b sin(180 − γ ) b sin γ d01 Figure 4.4 shows the relations for a⭢ − b⭢ plane of an oblique unit cell, where both the vectors d⭢10 and d⭢01 , respectively meet the lines (10) and (01) at 90°, because the angle γ between the directions a⭢ and b⭢ is not 90°. Actually, the lattice parameters a⭢ and d⭢10 are not equal in magnitude or direction, but are related by the sin of the angle ∗ and a⭢ ∗ between the two axes. This means that the reciprocal lattice parameters d⭢10 will involve the sin of the inter-axial angle. In a similar manner, the reciprocal lattice ∗ and b⭢∗ will also involve the sin of the inter-axial angle, as considered parameters d⭢01 above.

4.3 Construction of Reciprocal Lattice (First Approach)

75

Fig. 4.5 Direct lattice (on left), reciprocal lattice (on right) of five 2-D lattices

So far, we have discussed above two isolated cases of 2-D lattices. However, in a similar manner reciprocal lattices for all other direct lattices can be obtained. Direct and reciprocal of all five 2-D lattices are shown in Fig. 4.5 [6, 7]. Case III: 3-D Lattice The extension from two dimensions to three dimensions is simple and straightforward. For a 3-D space lattice, the relationships between the direct and reciprocal lattices in general are obtained as follows: Let us consider a unit cell of general lattice (Fig. 4.6) whose b-c plane has an area, ⭢ c|sin α, A = b⭢ × c⭢ = |b||⭢ then, its volume is V = Area of the b-c plane × height = A × d100 Similarly, V = Area of the a-c plane × height = B × d010 = Area of the a-b plane × height = C × d001 Therefore, based on the above for a given space lattice, the following relationships between the reciprocal lattice translation vectors a⭢ ∗ , b⭢∗ , c⭢∗ and the direct lattice ⭢ c⭢ exist: (primitive) translation vectors a⭢ , b,

76

4 Mirror Combination Scheme in Reciprocal Lattice

Fig. 4.6 A general 3-D lattice

Fig. 4.7 Relationships between reciprocal unit cells of a larger and b smaller real cells

4.3 Construction of Reciprocal Lattice (First Approach)

77

Table 4.2 Relationship between direct and reciprocal lattice parameters and inter-axial angles Triclinic a∗ =

K bc sin α ; V

b∗ =

K ca sin β ; V

c∗ =

K ab sin γ V

( )1/2 V = abc 1 − cos 2 α − cos 2 β − cos 2 γ + 2 cos α cos β cos γ = 2abc{sin s · sin(s − α) · sin(s − β) · sin(s − γ )}1/2 ; V ∗ =

1 V

2s = α + β + γ , K = 2π cosα ∗

=

cos β cos γ −cos α ; sin β sin γ

cos β ∗ =

cos γ ∗ = 1st setting a ∗ =

K a sin γ

cos γ cos α−cos β ; sin γ sin α

cos α cos β−cos γ sin α sin β

Monoclinic K ∗ b sin γ ; c =

; b∗ =

K c

; α ∗ = β ∗ = 90◦ ;

γ ∗ = 180◦ − γ 2nd setting

a∗

K a sin β ; β ∗ = 180◦

b∗ =

=

a∗ =

; b∗ =

K a

K b

;

c∗ =

K c sin β ;

α ∗ = γ ∗ = 90◦ ;

−β K b

Orthorhombic ; c∗ = Kc ; α ∗ = β ∗ = γ ∗ = 90◦ Tetragonal

a ∗ = b∗ =

K a

; c∗ =

K c

; α ∗ = β ∗ = γ ∗ = 90◦

Cubic ; α ∗ = β ∗ = γ ∗ = 90◦

a ∗ = b∗ = c∗ =

K a

a∗

c∗

Hexagonal =

b∗

=

2K √ ; a 3

=

K c

; α ∗ = β ∗ = 90◦ ; γ ∗ = 60◦

Rhombohedral a ∗ = b∗ = c∗ =

K ·a 2 sin α , V

( )1/2 wher e V = a 3 1 − 3cos 2 α + 2cos 3 α

cos α ∗ = cos β ∗ = cos γ ∗ =

a⭢ ∗ =

cos 2 α−cos α sin 2 α

cos α = − (1+cos α)

2π b⭢ × c⭢ 2π 2π A = = 2π = a⭢ d100 V a⭢ · b⭢ × c⭢

Similarly, c⭢ × a⭢ 2π 2π 2π B = 2π = = b⭢∗ = d010 V a⭢ · b⭢ × c⭢ b⭢ c⭢∗ =

a⭢ × b⭢ 2π 2π 2π C = = 2π = c⭢ d001 V a⭢ · b⭢ × c⭢

An interesting example exhibiting the relationships between the reciprocal unit cells of a larger and a smaller real cell, respectively are shown in Fig. 4.7 [8].

78

4 Mirror Combination Scheme in Reciprocal Lattice

The exact relationships between the direct and reciprocal lattice parameters and inter-axial angles of primitive unit cells of various crystal systems are provided in Table 4.2.

4.4 Construction of Reciprocal Lattice (Second Approach) Since, for a given direct lattice (irrespective of its dimension) there is a corresponding reciprocal lattice, therefore to obtain the reciprocal lattice for a given direct lattice, we adopt the following procedure (refer Fig. 4.1): 1. From a common origin, draw a normal to each crystal plane (in 3-D) or line (in 2-D) or a point in 1-D. 2. Set the length of each normal equal to 2π times the reciprocal of the interplanar spacing, dhkl (separation between two consecutive planes in 3-D, lines in 2-D and points in 1-D). 3. Mark a point at the end of each normal, and call it the reciprocal lattice point. 4. A collection of such points obtained in this way for all possible sets of parallel planes, is known as reciprocal lattice. Using a graphical construction, the relation between a given 2-D direct lattice and its corresponding reciprocal lattice net is obtained as shown in Fig. 4.8, where only a ∗ are shown by dotted lines. The direct lattice points few reciprocal lattice vectors, d⭢hkl are represented by circles while the reciprocal lattice points are shown by dots. Here, this is to be noted that the coordinates of reciprocal lattice points are usually denoted by the symbol hkl or hk0 (without brackets), they represent (hkl) plane or (hk0) line of the direct 3-D or 2-D lattice, respectively. In fact, a reciprocal lattice Fig. 4.8 Graphical representation of reciprocal lattice for monoclinic system

4.5 Construction of Reciprocal Lattice According to Ewald

79

Table 4.3 Modifications required for non-primitive unit cells Direct lattice parameter

Direct lattice volume

Lattice type P and R

a*, b*, c*

V*

a, b, c

V

A

a*, 2b*, 2c*

4V*

B

2a*, b*, 2 c*

4V*

C

2a*, 2b*, c*

4V*

F

2a*, 2b*, 2c*

8V*

I

2a*, 2b*, 2c*

8V*

Reciprocal lattice parameters

Reciprocal lattice volume

preserves all the characteristic features of the given planes in 3-D (or lines in 2-D) of the lattice they represent. Therefore, it can be stated that: 1. The direction of the reciprocal lattice point from the origin preserves the orientation of the (parallel) planes (or lines or points). 2. The distance of the reciprocal lattice point from the origin preserves the interplanar spacing (or separation between two consecutive parallel planes) of the set of planes in 3-D (or lines in 2-D or points in 1-D), it represents in the direct lattice. If the direct lattice of the conventional unit cell is non-primitive, then the above procedure can be applied to obtain the reciprocal lattice after making necessary modifications as provided in the Table 4.3.

4.5 Construction of Reciprocal Lattice According to Ewald From the structural aspect of crystallography, construction of reciprocal lattice is the most useful way to describe the diffraction phenomenon. The concept of reciprocal lattice was first introduced by Ewald [9], quickly it became an important tool in illustrating and understanding the diffraction geometry and the related relevant mathematical relationships. Later on, he also introduced the concept of Ewald sphere (also known as sphere of reflection) for a better visual representation of the diffraction phenomenon. The procedure suggested by him makes use of the reciprocal of the interplanar spacing, dhkl to fabricate a geometrical construction (as discussed above for a 2-D system) which then serves as a very effective way of understanding the diffraction results. Similarly, in the process of setting up an arrangement of X-ray source, specimen and detector, it is useful to predict the motions that will have to be applied to see the particular diffraction effects. On the other hand, while using the Braggs’ equation for finding the position required for diffraction, one must determine all possible values of interplanar spacing dhkl . For this purpose, we will make use of the values of dhkl obtained for simple cubic crystal in the next chapter and provided in Table 5.2, where each dhkl represents a

80

4 Mirror Combination Scheme in Reciprocal Lattice

Fig. 4.9 d111 represents a vector with magnitude d meeting the plane at right angle

vector whose magnitude is equal to the separation between two consecutive parallel planes, d, in angstroms and oriented in a direction from the origin of the unit cell such that meeting the nearest plane at right angle as shown in Fig. 4.9 for a (111) plane. Here, it is to be noted that the orientation problem is related to the fact that the diffracting Braggs’ planes are inherently three dimensional while restricting our analysis to a 2-D plane within the lattice, does not actually reflect the 3-D effect. However, to simplify the problem let us consider the case of a simple cubic unit cell and represent the two dimensional plane as a vector d⭢111 (by removing a dimension from this) and define the same as the perpendicular distance from the origin of the unit cell to the first plane in the (111) family as illustrated in Fig. 4.9. While this step removes a dimensional element, but the choice of common origin makes it evident that the sheaf of d⭢hkl vectors representing different set of lattice planes in the conventional unit cell become extremely dense near the origin as illustrated in Fig. 4.10 (only for one quadrant), which may not ultimately be very useful. Therefore, in order to remove such an accumulation of vectors near the origin and getting a better result, Ewald proposed that instead of plotting the d⭢hkl vectors as such, the reciprocal of these vectors should be plotted from the same origin. The reciprocal vector is defined as: 1 ∗ ≡ d⭢hkl d⭢hkl

(4.1)

Based on the Ewald’s proposal, the inverse of each dhkl value (taken from Table 5.2) has been calculated as provided in Table 4.4 along with other data. ∗ instead Figure 4.10 can now be reconstructed by plotting the reciprocal vectors d⭢hkl ⭢ of the dhkl as shown in Fig. 4.11 (with a magnification of 66.6%). The units are in reciprocal angstroms and the space is therefore a reciprocal space.

4.5 Construction of Reciprocal Lattice According to Ewald

81

Fig. 4.10 Graphical representation of some d⭢hkl in a simple cubic crystal

Table 4.4 The d-spacing in direct and reciprocal space in simple cubic crystal

( ) N = h2 + k2 + l2

Plane (hkl)

d-spacing in simple cubic ( −1 ) ∗ dhkl (Å) Å dhkl

1

(100)

6.000

0.166

2

(110)

4.243

0.236

3

(111)

3.464

0.289

4

(200)

3.000

0.333

5

(210)

2.683

0.373

6

(211)

2.449

0.408

8

(220)

2.121

0.471

9

(221)

2.000

0.500

10

(310)

1.897

0.527

11

(311)

1.809

0.553

12

(222)

1.732

0.577

It is interesting to note that the points in this case repeat at perfectly periodic intervals defining a space lattice, called the reciprocal lattice. The repeating translations are called reciprocal lattice vectors a⭢ ∗ , b⭢∗ and c⭢∗ . The inter-axial reciprocal angles are α*, β* and γ* where the reciprocal of an angle is defined as its complement, or 180° minus the real-space angle. For orthogonal systems (such as orthorhombic, tetragonal and cubic) the angular relations are quite simple. However, for non-orthogonal systems (such as triclinic, monoclinic and hexagonal), the angular relations are more complex. They are provided in Table 4.2.

82

4 Mirror Combination Scheme in Reciprocal Lattice

Fig. 4.11 Graphical representation of a⭢ ∗ , b⭢∗ for cubic crystal system

The reciprocal lattice makes the visualization of Braggs’ planes very easy. To establish the index of any point in the reciprocal lattice, simply count the number of repeat units in the a⭢ ∗ , b⭢∗ and c⭢∗ directions. Figure 4.11 shows only the hk0 plane of the reciprocal lattice of cubic crystal system, but the lattice is fully three dimensional. When connected, the innermost points in the lattice will define a threedimensional shape that is directly related to the shape of the real-space unit cell. Thus the symmetry of the real space lattice propagates into the reciprocal lattice. Any vector in the lattice represents a set of Braggs’ planes, mathematically; it can be resolved into its components as: ∗ = h a⭢ ∗ + k b⭢∗ + l c⭢∗ d⭢hkl

where a⭢ ∗ , b⭢∗ and c⭢∗ are defined in Table 4.2.

(4.2)

4.6 Interpretation of Braggs’ Equation

83

4.6 Interpretation of Braggs’ Equation In order to understand the concept of Ewald sphere (or sphere of reflection), let us first understand the concept of reciprocal lattice vector with the help of some simple representations (such as graphical representation and interpretation) of Braggs’ equation. For this purpose, it is conventional to incorporate the order of diffraction, integer ‘n’ appearing in the Braggs’ equation (2d sin θ = nλ) with the d-spacing and angle θ . After incorporating n with d-spacing and the angle θ , the general form of the Bragg’s equation becomes (for details see Chap. 5), 2dhkl sin θhkl = λ

(4.3)

This indicates that the order of diffraction is defined by the specific set of {hkl} planes (often called the Braggs’ planes) or the diffraction caused at a particular angle θhkl (often called the Braggs’ angle) by the specific set of {hkl} planes. Equation 4.3 can further be written as sin θhkl =

1/dhkl λ = 2/λ 2dhkl

(4.4)

Now let us try to represent the quantities in Eq. 4.4 as different components of a triangle inside a circle whose diameter is 2/λ (or radius 1/λ) by making use of the knowledge of simple geometry, according to which a triangle inscribed inside a circle is a right-angled triangle when the diameter of the circle is taken as the hypotenuse of the triangle. The perpendicular component is 1/dhkl and the opposite angle is θhkl as shown in Fig. 4.12a. This is simply a graphical representation of the Braggs’ law. However, to understand the physical meaning of the geometrical representation, let us assume that the horizontal diameter AO is the direction of incident X-ray

Fig. 4.12 Geometrical representation of Braggs’ equation

84

4 Mirror Combination Scheme in Reciprocal Lattice

beam. Since the line AP makes an angle θhkl with respect to the incident beam, it has the slope of a crystal plane (hkl) (as shown at the center of the circle in Fig. 4.12b). Further, since OP is normal to the crystal plane as also to AP and 1 ∗ = dhkl = σhkl = G, called the reciprocal lattice vector. Also, has the length dhkl ∠OC P = 2∠O A P = 2θhkl so that CP is the direction of diffracted beam from the set of planes {hkl}. Thus, Fig. 4.12b is a graphical interpretation of Braggs’ law in terms of reciprocal lattice vector σhkl . The whole process can be stated briefly in the following steps: 1. Imagine the crystal with a specific set of planes {hkl} to be at the center C of the circle (sphere in three dimensions) of radius 1/λ. 2. Point O where direct X-ray beam leaves the circle (after passing through the crystal) is the origin (000) of the reciprocal lattice net. 3. Whenever a reciprocal lattice vector intersects the circle (sphere in three dimensions), the Braggs’condition (Eq. 4.3) is satisfied, then a diffracted beam passes through the point of intersection and hence the X-ray diffraction becomes possible. 4. The locus of a point where the diffracted beam and reciprocal lattice point intersect the circle (sphere in three dimensions) of radius 1/λ is called the Ewald sphere or the sphere of reflection. The angle between the incident X-ray beam with the specific set of crystal planes {hkl} is the Braggs’ angle θhkl , and the angle between the incident X-ray beam with the diffracted X-ray beam is 2θhkl ; this is the reason that 2θ is used as a measurement convention in the X-ray diffraction data. Therefore, the relationship between the angle of incidence θ and the experimentally measured angle of diffraction 2θ, i.e. the relationship θ/2θ is usually maintained in many spectrometers, as shown schematically in Fig. 4.13. Fig. 4.13 Relationship between the angle of incidence θ and the diffraction angle 2θ

4.7 Diffraction Geometry and Ewald Sphere

85

4.7 Diffraction Geometry and Ewald Sphere Now following the above-mentioned steps, let us construct a sphere of radius 1/λ (in actual sense only an imaginary sphere) with the crystal at its center as shown in Fig. 4.14. As pointed out above, the point O (000), where direct X-ray beam leaves the circle/sphere (after passing through the crystal) represents the origin of the reciprocal lattice net. The reciprocal lattice associated with the crystal’s lattice is viewed as tangent to the sphere at this point. The Ewald sphere contains all that is needed to visualize the diffraction geometrically. A rotation or oscillation of the crystal (and its associated real-space lattice) will also rotate or oscillate the reciprocal lattice (because the reciprocal lattice is defined in terms of the real-space lattice). The arrangement at the specific time when a point from the reciprocal lattice net, say the reflections, such as 010 and 020 brought into contact with the sphere by rotating the reciprocal lattice net are shown in Fig. 4.15b and c, respectively. Combining this with the diffraction geometry shown in Fig. 4.12b, we can write, CO =

d∗ 1 and O P = 020 λ 2

Fig. 4.14 The Ewald sphere with the crystal at its center and associated reciprocal lattice net

86

4 Mirror Combination Scheme in Reciprocal Lattice

Fig. 4.15 Rotation of reciprocal lattice net to intersect reflections with the Ewald sphere

Hence, sin θ020 =

d ∗ /2 OP = 020 1/λ CO

or λ=

2 sin θ020 ∗ d020

Making use of the definition of the reciprocal vector, dhkl ≡

1 ∗ dhkl

we can write: λ = 2d020 sin θ020

4.8 Vector Form of Braggs’ Equation from Ewald Construction

87

Similarly, one can show λ = 2d010 sin θ010 This is the Braggs’ equation corresponding to the d-spacing, d020 (d010 ) and Braggs’ angle, θ020 (θ010 ). Since each point in the reciprocal lattice net represents a d*-value, when any one intersect (such as the reflection 010 or 020 as shown above) the sphere during rotation/oscillation, the diffraction condition is met and that particular diffraction is said to occur. It is therefore, a crystal is made to rotate or oscillate sometimes to facilitate more and more points in the reciprocal lattice net to touch (intersect) the sphere and give rise to the diffraction effect. The construction of Ewald sphere is very useful in explaining the diffraction in a manner that avoids any complicated calculations. It allows us to visualize the diffraction effect using a pictorial and mental model, and permits to use a simple analysis of otherwise complex relationships among the crystallographic axes and planes.

4.8 Vector Form of Braggs’ Equation from Ewald Construction From above discussion, it is clear that the relationship between the space lattice (in real space) and the corresponding reciprocal lattice (in reciprocal space) is completely symmetrical in the sense that the set of planes in one system can be represented as perpendicular to the rows of points in another system. However, to make the quantities in reciprocal space numerically as well as dimensionally identical to the quantities in real space, a factor of 2π is introduced (which a crystallographer prefers to omit). As a result, for example the reciprocal lattice vector will now be equal to 2π times the reciprocal of the spacing of (hkl) planes of the crystal lattice. Therefore, multiplying all the vectors appearing in Fig. 4.12b by a factor of 2π, the resulting Ewald construction will appear to be as shown in Fig. 4.16. Fig. 4.16 Ewald construction in vector form

88

4 Mirror Combination Scheme in Reciprocal Lattice

Here, two possibilities may arise: 1. If the circle (sphere in three dimensions) does not pass through any lattice point, it will indicate that the particular wavelength in question will not be diffracted by the crystal in that orientation and hence a reorientation of the crystal is needed. Further, if the magnitude of the vector |OA| 2a. On the other hand, it may be seen from the Fig. 4.12b that the longer the vector OC, the shorter the wavelength, greater is the possibility of circle’s intersecting a reciprocal lattice point and hence the occurrence of diffraction is more probable. 2. Further from Fig. 4.12, if the circle passes through a reciprocal lattice point P, then OP is the reciprocal lattice vector normal to the set of lattice planes, e.g. AP. Hence, OP = |σ | = |G| = 1/d = d ∗ . Let k = OC and k , = CP be the incident and the reflected wave vector, respectively, then the disposition of vectors require that k, = k + G

(4.5)

Equation 4.5 shows that: (i) the scattering changes only the direction of k, and hence (ii) the scattered wave differs from the incident wave by a reciprocal lattice vector G (Fig. 4.16). Since for an elastic scattering, there will be no change in the magnitude of the vector, i.e. | | 2π |k| = |k , | = λ

(4.6)

Now, squaring Eq. 4.5, we obtain k ,2 = (k + G)2 = k 2 + G 2 + 2k · G Making use of Eq. 4.6, we have G 2 + 2k · G = 0

(4.7)

This is the Bragg’s equation in vector form. Representation of some low index planes is shown on Ewald sphere in vector form in Fig. 4.17.

4.9 The Ewald Sphere and the Limiting Sphere

89

Fig. 4.17 Representation of hkl-planes on Ewald sphere in vector form

4.9 The Ewald Sphere and the Limiting Sphere In the last two sections, we studied about the construction of Ewald sphere (also known as sphere of reflection), where a reflection is obtained from the crystal when a reciprocal lattice vector intersects the surface of the Ewald sphere. In a similar manner, other reflections are obtained if other reciprocal lattice vectors are made to intersect the surface of the Ewald sphere, either by oscillation or rotation of the reciprocal lattice net (this is equivalent to the oscillation or rotation of a crystal at the center of the Ewald sphere) w.r.t the origin of the reciprocal lattice where the outgoing X-ray beam intersects the Ewald sphere. This way we can obtain only a part of all possible reflections. However, maximum possible reflections can be mapped, if the crystal is made to rotate in various possible ways to bring all the families of planes to reflection (i.e. to make all possible reciprocal lattice points to intersect the Ewald sphere). In other words, if the Ewald sphere itself is made to rotate in all three dimensions about (000), the origin of the reciprocal lattice so that the Ewald sphere grows into a larger sphere (of radius 2/λ) which contains all the reflections that are possible to be collected using the same wavelength of the X-rays as shown in Fig. 4.18a. One such rotation about the vertical axis passing through the origin is shown in Fig. 4.18b. The construction of such a sphere whose origin is the same as the origin of the reciprocal lattice point and a radius of 2/λ, is known as the ‘Limiting sphere’. It defines the complete data set, and any reciprocal lattice point outside this sphere cannot be observed unless the wavelength of the X-rays is changed. Therefore using Ewald’s sphere, we can readily notice that the wavelength defines the maximum resolution: ∗ dmax =

1 2 = dmin λ

∗ The value of dmax = 2/λ is imposed by Eq. 4.3, since the scattering angle cannot exceed 2θ = 180° (or θ = 90°).

90

4 Mirror Combination Scheme in Reciprocal Lattice

Fig. 4.18 The Ewald sphere and the Limiting sphere

A rough estimate [8] provides the number of lattice points that will hold by the limiting sphere is equal to the volume of the limiting sphere divided by the volume V* of one reciprocal cell (BZ). That is, N=

4πr 3 3V ∗

(4.8)

Since, radius of the limiting sphere is 2/λ, so the number of possible reflections is equal to N=

32π 33.5 33.5V = ∗ 3 = 3V ∗ λ3 V λ λ3

(4.9)

Equation 4.9 shows that the number of available reflections depends only on V and λ, i.e. the unit-cell volume and the wavelength of the X-rays. Therefore, it is important to note that: 1. The shorter the wavelength of the X-ray used, the larger the Ewald sphere and the more reflections may be seen (in theory). 2. Larger the direct unit cells, mean smaller the reciprocal unit cells, which again populate the limiting sphere more densely and hence increase the number of measurable reflections. However, the concept of mirror combination scheme and the fundamental principle on which the relationship of direct and reciprocal lattice based is that the set of planes in one system can be represented as perpendicular to the rows of points in another system. Implications of these should be taken into account in the calculation.

4.9 The Ewald Sphere and the Limiting Sphere

91

Further, according to Leslie [10], we know that the diffraction pattern cannot possess any translational symmetry, because it has a defined origin. Taking this important conclusion into consideration and comparing the same with the situation observed in limiting sphere [which too has a defined origin (000) as the origin of the reciprocal lattice and containing all possible reflections on the surface of the sphere of fixed distance 2/λ], we can similarly conclude that no translational symmetry of any kind (microscopic or macroscopic) is possible in crystals. A simple comparison can also be drawn between the limiting sphere discussed above and the spherical projection of a crystal structure by projecting various crystal planes on the surface of a unit sphere around the crystal (is at the center), where the projection of a plane is realized when the plane normal vectors intersect the unit sphere. This leads to an unambiguous 3-D projection of the crystal structure. Figure 4.19 shows the spherical projection for a simple cubic crystal structure, where only the low index planes such as {100}, {110} and {111} in partial form of projections are considered.

Fig. 4.19 The spherical projection of low index planes of a simple cubic crystal structure

92

4 Mirror Combination Scheme in Reciprocal Lattice

4.10 Construction of Brillouin Zones in 1-D, 2-D and 3-D Lattices A Brilloiun zone is defined as the region in k-space/reciprocal space in 1-D, 2-D or 3-D that the electrons/phonons occupy to move freely without being diffracted. Because of its connection with the k-space, the first Brillouin zone is equivalent to a Wigner–Seitz unit cell in reciprocal space, i.e. B-Z = W-S cell of R-L The first B-Z and all other (higher) zones are symmetrical about k = 0. The construction of the first B-Z for any given (direct) lattice can be made according to the following procedure. 1. Obtain the reciprocal lattice net of the given (direct) lattice as per the procedure given in Sects. 4.3/4.4. 2. Construct a W-S unit cell in the resulting reciprocal lattice according to the procedure given in Sect. 3.4. This will represent the first B-Z corresponding to the given reciprocal lattice. Since the exact nature of the region in the form of line, plane or space within the first B-Z (according to its unit cell dimension) is governed by the diffraction condition, therefore let us start by using the Braggs’ diffraction condition to understand the construction of B-Z in 1-D, 2-D and 3-D lattices, one by one. (a) One-Dimensional Lattice We know that the general (or vector) form of Braggs’ diffraction condition is given by Eq. 4.7. However for a 1-D lattice, the components of the reciprocal lattice ⭢ and the wave vector k⭢ are given by vector G ⭢ = G

(

) 2π ˆ ˆ x in x and k⭢ = ik a

Substituting these values in Eq. 4.7, we obtain [(

) ]2 ( ) 2π ˆ 2π ˆ ˆ in x + 2ik x · in x = 0 a a 4π 4π 2 n x kx = 0 or 2 n 2x + a a π or k x = − n x a

(4.10)

wher e n x = ±1, ±2, . . . will provide us various 1-D Brillouin zones as shown in Fig. 4.20a. A simpler form of B-Z of a linear lattice is shown in Fig. 4.20b.

4.10 Construction of Brillouin Zones in 1-D, 2-D and 3-D Lattices

93

Fig. 4.20 One-dimensional Brillouin zones

(b) Two-Dimensional (Square) Lattice For simplicity of the problem, let us take the case of a square lattice with a = b and γ = 90◦ . However for a 2-D lattice, the corresponding components of the ⭢ and the wave vector k⭢ are given by reciprocal lattice vector G ( ) ( ) ˆ x + jn ˆ x + jk ˆ y and k⭢ = ik ˆ y ⭢ = 2π in G a Substituting these values in Eq. 4.7, we obtain [

) 2π (ˆ ˆ y in x + jn a or

]2

) [ 2π ( )] ( ˆ ˆ ˆ ˆ in x + jn y = 0 + 2 ik x + jk y · a

) ) 4π ( 4π 2 ( 2 2 n =0 n + k + n k + n x x y y x y a a2

ˆ etc. = 0. where iˆ · iˆ = jˆ · jˆ, etc. = 1 and iˆ · jˆ = jˆ · k, Further simplifying the above equation, we obtain n x kx + n y k y = −

) π( 2 n + n 2y a x

(4.11)

where n x and n y are integers for diffraction by the vertical columns and horizontal rows of atoms. For the first zone, one integer (say n x ) is ±1 and other integer (say n y ) is zero. Therefore, the zone boundaries for the first zone are: For n x = ±1, n y = 0 ±k x = −

π π gi ving k x = ± a a

Similarly, for n x = 0, n y = ±1 ±k y = −

π π gi ving k y = ± a a

94

4 Mirror Combination Scheme in Reciprocal Lattice

Fig. 4.21 Two-dimensional Brillouin zones

This is the required first B-Z in 2-D as illustrated in Fig. 4.21. In the region k < ± πa of the first zone, electrons/phonons move freely without being diffracted. However, at k = ± πa , they are prevented from moving in the x or ( ) y direction due to diffraction. As the value of k exceeds πa , the number of possible directions of motion decreases gradually, until when π k= = a sin45◦

√ 2π a

This implies that the electrons/phonons are diffracted even when they move diagonally (at 45° with k x and k y axes) inside the square. It is here the first zone ends and the second zone begins. For the second zone, both the integers (n x and n y ) in Eq. 4.11 are equal to ±1, i.e. ±k x ± k y =

2π a

4.10 Construction of Brillouin Zones in 1-D, 2-D and 3-D Lattices

95

This gives us four equations representing zone boundaries: n x = +1, n y = +1 gi ving k x + k y =

2π (line F E) a

n x = −1, n y = +1 gi ving − k x + k y =

2π (line F G) a

n x = −1, n y = −1 giving − k x − k y =

2π (line G H ) a

n x = +1, n y = −1 gi ving k x − k y =

2π (line E H ) a

The second B-Z contains electrons/phonons with k values from π/a (that do not fit into the first zone) to 2π/a, or between the square ABCD and EFGH, are also shown in Fig. 4.21. The above constructions up to third zones for a square lattice is shown in a simpler form in Fig. 4.22a, where it can be verified that all Brillouin zones have the same area by translating the parts (or fragments) of higher zones into the first zone through suitable reciprocal lattice vectors of integral multiple, separately shown in Fig. 4.22b– d for first three Brillouin zones. Representation of various zones into a single zone like this is known as reduced zone scheme [3]. Here, it is to be pointed out that different B-Z may have different number of Braggs’ planes (or zone boundaries or fragments) as shown clearly in Fig. 4.22c (II B-Z) and d (III B-Z), they are 4 and 8, respectively. Still higher zones may give rise to higher number of fragments. Same principle is followed in 3-D. (c) Three-Dimensional (Simple Cubic) Lattice To keep the problem simple, let us take the case of a simple cubic lattice with a = b = c and α = β = γ = 90◦ . However for a 3-D lattice, the components of the ⭢ and the wave vector k⭢ are given by reciprocal lattice vector G

Fig. 4.22 First three Brillouin zones of a square lattice shown together and separate

96

4 Mirror Combination Scheme in Reciprocal Lattice

( ) ( ) ˆ x + jn ˆ x + jk ˆ y + kn ˆ z and k⭢ = ik ˆ y + kk ˆ z ⭢ = 2π in G a Substituting these values in Eq. 4.7, we obtain [

)] )]2 ) [ 2π ( ( 2π (ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ in x + jn y + kn z = 0 in x + jn y + kn z + 2 ik x + jk y + kk z · a a or

) ) 4π ( 4π 2 ( 2 n x kx + n y k y + n z kz = 0 n x + n 2y + n 2z + 2 a a

where iˆ · iˆ = jˆ · jˆ = kˆ · kˆ = 1 and iˆ · jˆ = jˆ · kˆ = kˆ · iˆ = 0. Further simplifying the above equation, we obtain n x kx + n y k y + n z kz = −

) π( 2 n x + n 2y + n 2z a

(4.12)

This equation could have been obtained by simply generalizing the Eq. 4.11 obtained for 2-D. However, from Eq. 4.12, it follows that the first Brillouin zone for a simple cubic lattice is also simple cubic, where the only atom of the cube lies at its center and whose walls (zone boundaries) intersect the kx , ky and kz axes at the ± π/a as shown in Fig. 4.23a. All six {100} faces of the cube represent equivalent Braggs’ planes. The second Brillouin zone of the simple cubic lattice is obtained by adding a pyramid (like a triangle in 2-D) to each face of the first zone. The resulting diagram is shown in Fig. 4.23b.

Fig. 4.23 a First B-Z, and b first and second B-Z of simple cubic crystal

4.11 Determination of Higher Order Brillouin Zones in Cubic Crystal System

97

4.11 Determination of Higher Order Brillouin Zones in Cubic Crystal System Knowing the set of Braggs’ planes associated with the first Brillouin zone, other set of Braggs’ planes associated with second, third, etc. Brillouin zones can be determined manually by applying simple ‘addition rule’, a process used in crystallography as follows: Case I: Simple Cubic Crystal To start with, we have six {100} equivalent Braggs’ planes of first B-Z in simple cubic crystal. They are: (100), (010), (001), (100), (010) and (001) evident from Fig. 4.23a. Now, applying the addition rule properly, the zone boundaries (Braggs’ planes) of next higher order can be obtained. 100 + 010 = 110 100 + 001 = 101 010 + 001 = 011 100 + 010 = 110 100 + 001 = 101 010 + 001 = 011

100 + 010 = 110 100 + 001 = 101 010 + 001 = 011 100 + 010 = 110 100 + 001 = 101 010 + 001 = 011

These are twelve (six pairs of equivalent) second order planes obtained as second zone boundaries; where each plane settles on each of the twelve edges of the first zone cube such that parts of the second zone boundary take the pyramidal shape as shown in Fig. 4.23b. In a similar manner, the Braggs’ planes involved with the third Brillouin zone can be determined by considering {100} and {110} sets of planes and applying the addition rule properly. 100 + 011 = 111 100 + 011 = 111 100 + 011 = 111 100 + 011 = 111 100 + 011 = 111 010 + 101 = 111 100 + 011 = 111 100 + 011 = 111 There are eight members of {111} family as third order Braggs’ planes obtained as the third zone boundaries; where each of which settles on each of the eight corners of the first B-Z (not shown). In a similar manner, the set of Braggs’ planes can be

98

4 Mirror Combination Scheme in Reciprocal Lattice

Fig. 4.24 a, c First B-Z, and b, d First and Second B-Z of bcc, and fcc, respectiely

obtained for higher order zones. Same procedure can be adopted to obtain the sets of Braggs’ planes needed to complete various higher order zones in other cubic crystal systems. Case II: Body Centered Cubic Crystal In this case, to start with we have twelve {110} equivalent Braggs’ planes of first B-Z. They are: (110), (011), (101), (110), (110), (011), (011), (101), (101), (110), (011) and (101) as shown in Fig. 4.24a [11]. Now, applying the addition rule properly, the zone boundaries (Braggs’ planes) of next higher order zone can be obtained. 110 + 110 = 200 110 + 110 = 020 011 + 011 = 002 110 + 110 = 200 110 + 110 = 020 011 + 011 = 002 These are six (three pairs of equivalent) second order planes; where each plane settles on the edge of a pair of first zone boundaries such that each zone face has a share of half the plane. This way all the twelve {110} planes of the first zone are covered as shown in Fig. 4.24b. Similarly, with the help of six {200}: (200), (020), (002), (200), (020) and (002) planes and twelve {110}: (110), (011), (101), (110),

4.11 Determination of Higher Order Brillouin Zones in Cubic Crystal System

99

(110), (011), (011), (101), (101), (110), (011) and (101) planes, the following 24 next order planes can be obtained. 002 + 110 = 112 200 + 011 = 211 020 + 101 = 121 002 + 110 = 112 002 + 110 = 112 200 + 011 = 211

200 + 011 = 211 020 = 101 = 121 020 + 101 = 121 002 + 110 = 112 200 + 011 = 211 020 + 101 = 121

002 + 110 = 112 200 + 011 = 211 020 + 101 = 121 002 + 110 = 112 002 + 110 = 112 200 + 011 = 211

200 + 011 = 211 020 + 101 = 121 020 + 101 = 121 002 + 110 = 112 200 + 011 = 211 020 + 101 = 121

These are 24 (twelve pairs of equivalent) third order planes settles appropriately on the Fig. 4.24b to get the resulting third B-Z (not shown). In a similar manner, the process can be continued to obtain the set of Braggs’ planes involved with other higher zones. Case III: Face Centered Cubic Crystal In this case, to start with we have eight {111} equivalent planes and six {200} equivalent planes, all related to first B-Z. They are: (111), (111), (111), (111), (111), (111), (111) and (111); and (200), (020), (002), (200), (020) and (002), respectively as shown in Fig. 4.24c. Now, applying the addition rule properly, the planes of next order can be obtained. With the help of {111} family members, we obtain the following zone boundaries (Braggs’ planes): 111 + 111 = 220 111 + 111 = 202 111 + 111 = 022 111 + 111 = 220 111 + 111 = 202 111 + 111 = 022

111 + 111 = 220 111 + 111 = 202 111 + 111 = 022 111 + 111 = 220 111 + 111 = 202 111 + 111 = 022

In a similar manner, with the help of {200} family members, we obtain the following zone boundaries (Braggs’ planes): 200 + 020 = 220 200 + 002 = 202 020 + 002 = 022 200 + 020 = 220 200 + 002 = 202 020 + 002 = 022

200 + 020 = 220 200 + 002 = 202 020 + 002 = 022 200 + 020 = 220 200 + 002 = 202 020 + 002 = 022

It is interesting to note that both {111} and {200} yield the same result. These are twelve (six pairs of equivalent) second order Braggs’ planes, they appropriately

100

4 Mirror Combination Scheme in Reciprocal Lattice

settle on the first zone faces to take the resulting shape as shown in Fig. 4.24d. In a similar manner, the set of Braggs’ planes for higher order zones can be obtained (not shown).

4.12 Results from Comparison of Unit Cell Data of Cubic Crystal System Table 4.1 provides us the relationships between direct and its corresponding reciprocal lattices for different crystal systems. Taking this information into account and the studies carried out in the last chapter and this, we observe that the unit cells shown in Fig. 4.25 represent the first B-Z cells of the three well known cubic lattices. Based on the studies made in these two chapters, a simple comparison of the data related to conventional unit cells, Wigner–Seitz cells and the Brillouin zones provide us the following important information: 1. In the conventional form, all the three cubic lattices, the unit cells of sc, bcc and fcc are represented by a cube with six faces each. 2. The resulting shapes of W-S cells of sc, bcc and fcc, are simple cubic, cubooctahedron and rhombic dodecahedron, respectively. They contain 6, 14 and 12 faces, respectively. 3. The first B-Z cell of simple cubic lattice is also a simple cubic with 6 (three pairs of) equivalent {100} planes; they all represent the first order Braggs’ planes. 4. The first B-Z cell of bcc lattice is a rhombic dodecahedron with 12 (six pairs of) equivalent {110} planes; they all represent the first order Braggs’ planes. 5. The first B-Z of fcc lattice is a cubo-octahedron with 8 (four pairs of) equivalent {111} planes (they represent the first order Braggs’ planes) and 6 (three pairs of) equivalent {200} planes (represent the second order Braggs’ planes), respectively. All the 14 Braggs’ planes in this case are the members of the first Brillouin zone of fcc. 6. Similarly, the construction of other higher Brillouin zones can be explained. However, this is to be kept in mind that the shape of the first Brillouin zone and the number of zone boundaries in it, is different for different crystal lattice and hence the resulting Brillouin zone may contain one set of Braggs’ planes or more.

Fig. 4.25 Brillouin zones of sc, bcc and fcc

4.13 Summary

101

Table 4.5 Data related to conventional/W-S/B-Z unit cells of cubic crystal system Lattice type

Shape of the unit cell Crystal lattice (CL)

Wigner–Seitz (W-S)

Brillouin zone (B-Z)

Planes involved with three cubic cases CL W-S B-Z

Simple cubic

Simple cube

Simple cube

Simple cube

06

06

06

Body centered cubic

Simple cube

Cubo-octahedron

Rhombic dodecahedron

06

14

12

Face centered cubic

Simple cube

Rhombic dodecahedron

Cubo-octahedron

06

12

14

7. We note that the data related to the shape of conventional/W-S/B-Z unit cells and the number of Braggs’ planes involved with them is completely different from the data of the direct space lattices of bcc and fcc. These are provided in Table 4.5. 8. These are significant information and clearly suggest that the diffraction (of crystals) recognises only the W-S/B-Z unit cells and not the conventional direct lattice unit cells, particularly for structural work. Important Conclusions: The first Brillouin zone is sufficient to determine the crystal structure of any crystal system of a given material, because of the following reasons: 1. Only the first B-Z is found to be sufficient to study the physical and other structure related properties of crystals, and 2. All B-Z’s have same length, area or volume in 1-D, 2-D or 3-D crystal systems, respectively and have the same properties.

4.13 Summary 1. The concept of reciprocal lattice can be understood in two different ways: First, write the reciprocal lattice (unit cell) vectors a⭢ ∗ , b⭢∗ , c⭢∗ in terms of the direct ⭢ c⭢, using simple vector algebra and simple geomlattice (unit cell) vectors a⭢ , b, etry. The second is based on the notion that in crystals, the families of parallel planes can be represented by a common normal (from the same origin) as one dimensional lines and setting their lengths to be equal to 2π times the reciprocal of their interplanar spacing dhkl , one can obtain the reciprocal lattice points. 2. The concept of reciprocal lattice was first introduced by Ewald [9], quickly it became an important tool in illustrating and understanding the diffraction geometry and the related relevant mathematical relationships. Later on, he also introduced the concept of Ewald sphere (also known as sphere of reflection) for a better visual representation of the diffraction phenomenon.

102

4 Mirror Combination Scheme in Reciprocal Lattice

3. The concept of Ewald sphere (or sphere of reflection) can be understood with the help of graphical representation and graphical interpretation of Braggs’ law in terms of reciprocal lattice vector. A reflection is obtained from the crystal only when the reciprocal lattice vector intersects the surface of the Ewald sphere. 4. The construction of Ewald sphere is very useful in explaining the diffraction in a manner that avoids any complicated calculations. It allows us to visualize the diffraction effect simply by using a pictorial and mental model, and without the use of any complex relationships among the crystallographic axes and planes. 5. The maximum possible reflections can be mapped if the crystal is made to rotate in all possible ways to bring the maximum families of planes to reflection (i.e. to make all possible reciprocal lattice points to intersect the Ewald sphere). In other words, if the Ewald sphere itself is made to rotate in all three dimensions about (000), the origin of the reciprocal lattice so that the Ewald sphere grows into a larger sphere of fixed radius 2/λ (known as limiting sphere), which contains all the reflections that are possible to be collected using the same wavelength of the X-rays. 6. Limiting sphere defines the complete data set, and any reciprocal lattice point outside this sphere cannot be observed unless the wavelength of the X-rays is changed. 7. According to Leslie [10], we know that the diffraction pattern cannot possess any translational symmetry, because it has a defined origin. Taking this important conclusion into consideration and comparing the same with the situation observed in limiting sphere which too has a defined origin (000) as the origin of the reciprocal lattice and containing all possible reflections on the surface of the sphere of fixed radius 2/λ, one can similarly conclude that no translational symmetry of any kind (microscopic or macroscopic) is possible in crystals. 8. Another simple comparison can be made between the limiting sphere and the spherical projection of a crystal structure by projecting various crystal planes on the surface of a unit sphere around the crystal (at the center), where the projection of a plane is realized when the plane normal vectors intersect the unit sphere. This leads to an unambiguous 3-D projection of the crystal structure. 9. Like Wigner–Seitz unit cells, the Brillouin zones are also based on mirror combination scheme and similarly produce the reciprocal lattice exhibiting perfect translational periodicity. The Brilluoin zone is simply defined as the Wigner–Seitz unit cell of a reciprocal lattice: B-Z = R-L + W-S 10. It is important to note that all kinds of diffraction work are in reciprocal space and the diffraction patterns providing different sets of Braggs’ planes actually represent the zone boundaries of different Brillouin zones. 11. The exhibition of center of symmetry in diffraction experiments by all crystals (irrespective of centro-symmetric or not) is only due to the spherically symmetric distribution of Braggs’ planes in different Brillouin zones (containing an atom at its center as defined origin) in reciprocal space (Figs. 4.21,

References

103

4.22, 4.23, 4.24 and 4.25). So far, this phenomenon was known to be due to Friedel’s law. This historic result definitely supports the idea of space lattice to be represented by W-S/B-Z as far as the diffraction of crystals is concerned. 12. The spherically symmetric nature of diffraction patterns in reciprocal space allows the mirror, rotational and inversion symmetries while forbids the presence of translational symmetries of any kind (microscopic or macroscopic) in crystals. This is an extremely important result and must be taken into account during crystal structure work. 13. Knowing the set of Braggs’ planes associated with the first Brillouin zone, other set of Braggs’ planes associated with second, third, etc. Brillouin zones can be determined manually by applying simple ‘addition rule’, a process used in crystallography. 14. A simple comparison of the data related to conventional unit cells, Wigner–Seitz cells and the Brillouin zones clearly suggest that the diffractions of crystals recognise only the W-S/B-Z unit cells and not the conventional direct lattice unit cells as far as the crystal structure determination is concerned.

References 1. Radaelli, P.G.: Symmetry in Crystallography, Understanding the International Tables, IUCr. Oxford Science Publication, Oxford University Press, USA (2011) 2. Wahab, M.A.: The Mirror: Mother of all Symmetries in Crystals, Adv. Sci. Eng. Med. 12289– 313 (2020) 3. Wahab, M.A.: Solid State Physics: Structure and Properties of Materials, 3rd edn. Narosa Publishing House, New Delhi (2015) 4. Wahab, M.A.: Essentials of Crystallography, 2nd ed. Narosa Publishing House, New Delhi (2014) 5. Verma, A.R., Srivastava, O.N.: Crystallography for Solid State Physics, Wiley Eastern Limited (1987) 6. Connolly, J. R.: Introduction to X-ray Powder Diffraction (2012) 7. Hammond, C.: The basics of Crystallography and Diffraction, IUCr. 2nd ed. Oxford University Press, USA (2001) 8. Rhodes, G.: Crystallography Made Crystal Clear, 3rd ed., Elsevier, Academic Press, Amsterdam (2006) 9. Ewald, P.P.: The “reciprocal lattice” in structure theory. Z Kristallogr 56, 129–156 (1921) 10. Leslie, A.: (Andrew Leslie), Crystals, Symmetry and Space Groups, MRC Laboratory of Molecular Biology (LMB) Crystallography Course, United Kingdom (2013) 11. Ashcroft,N. W., Mermin,N. D.: Solid State Physics, Harcourt Asia PTE Ltd., Singapore (2001)

Chapter 5

Importance of d-Spacing in Diffraction of Crystals

5.1 Introduction Braggs’ equation suggests that the interplanar d-spacing is an important and integral part of a diffraction process. Therefore, keeping in mind the fact that B–Z is the W–S of reciprocal lattice, the interplanar spacing calculations for low index planes for an arbitrary value of cell dimension in three cubic crystal lattices; sc, bcc and fcc and similarly in primitive cases of some elemental and MX-crystals have been made to check the nature of occurrence of Braggs’ planes in them. The results obtained from these calculations are found not only to be in consonance with the experimental data available for crystals of metallic elements and MX-systems but they also appear to take care of the ‘systematic absences’ arising due to the so called ‘lattice centring’, screw axes and glide planes, without actually taking them into consideration. This allows us to conclude that the set of parallel planes (of atoms) of crystals with the least interplanar spacing diffract first (by using the beam of X-rays, electrons or neutrons) due to its shortest reciprocal lattice vector and then continue in a systematic and spherically symmetric manner with increasing order of interplanar spacing in reciprocal space. This newly found criterion is extremely important for the work of crystal structure determination using any diffraction technique.

5.2 Braggs’ Equation and Interplanar d-Spacing Braggs’ equation has been derived on the basis of following important considerations: (i) That a crystal is made of a number of different sets of equidistant parallel planes (of atoms), where the d-spacing of each set of parallel planes is different, and one such set of (hkl) planes is shown in Fig. 5.1.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. A. Wahab, Mirror Symmetry, Springer Series in Solid-State Sciences 200, https://doi.org/10.1007/978-981-99-8361-2_5

105

106

5 Importance of d-Spacing in Diffraction of Crystals

Fig. 5.1 Diffraction of X-rays by equidistant parallel crystal planes

(ii) Following the law of mirror reflection that the angle of incidence is equal to the angle of reflection, the incident waves are reflected specularly from the parallel planes of atoms in crystals, where each crystal plane reflects an appreciable fraction of the radiation, as if the same is taking place from a normal silvered mirror. (iii) That the path difference covered by the rays between any two consecutive parallel planes of atoms is equal to an integral multiple of the wavelength used, and they diffract in phase. Based on the above considerations, Braggs’ formulated a simple equation by using a 2-D crystal model of identical and parallel planes of atoms shown in Fig. 5.1, it is given by 2dhkl sin θ = nλ

(5.1)

where dhkl is the interplanar spacing, i.e. the distance between any two consecutive parallel (hkl) planes in a given crystal, λ is the wavelength of light (or radiation used), θ is the Braggs’ angle and n is the order of diffraction. Order of the Braggs’ equation depends on two variables. 1. The interplanar d-spacing between different sets of planes, such as d1 , d2 , d3 , etc. 2. The Braggs’ angle made by different sets of planes with the incident or the diffracted beam, such as θ1 , θ2 , θ3 , etc. where d1 > d2 > d3 , … and θ1 < θ2 < θ3 , … in a given crystal. Experimental results related to a crystal diffraction suggest that these two parameters vary together in reciprocal manner, that is, when d-spacing is large, the Braggs’ angle is small, and vice versa. However, for labelling the reflections, we need to use the Miller indices of planes. For example, a beam corresponding to a value of n > 1 could be identified by a statement such as ‘the nth-order reflections from the (hkl) planes’. The higher order ‘n’ is conventionally associated with d, then rewriting the

5.2 Braggs’ Equation and Interplanar d-Spacing

107

Braggs’ equation in this form, we have: (

dhkl 2 n

) sin θ = λ

(5.2)

This makes the nth order diffraction of (hkl) planes of spacing—‘dhkl ’ look like the first-order diffraction planes of spacing—(dhkl /n). Diffraction planes of this reduced spacing will have the Miller indices (nh nk nl). The modified form of Eq. 5.2 can be written as 2dnhnknl sin θ = λ

(5.3)

( ) . where, dnhnknl = dhkl n Further, for successive peaks, there is an increase in the angular orientation of the planes. Keeping this in view, there is a need to associate ‘n’ with θ also, Eq. 5.3 can be rewritten as 2dhkl sin(nθ ) = λ

(5.4)

Now, absorbing ‘n’ with θ, Eq. 5.4 can be written in terms of Miller indices, hkl as 2dhkl sin θhkl = λ

(5.5)

where dhkl and θhkl uniquely define (or represent) the separation between two consecutive crystal planes belonging to a particular set of {hkl} planes, and the corresponding Braggs’ angle, respectively for a given diffraction peak. The formula given by Eq. 5.5 is the representation of Braggs’ equation in proper order, which is compatible with the diffraction process. Because sin θ cannot be > 1 (as the Braggs’ angle θ is always ≤ 90°), the wavelength of the X-ray limits the number of diffraction peaks that can appear. Further, from the point of view of configuration of atoms in a layer, the above considered equidistant parallel planes of atoms (under the first consideration) may be: (i) Similar (or identical), containing identical atoms (as in elemental crystals). (ii) Different, containing different types of atoms (as in MX or any other polyatomic crystals). These aspects will be discussed further in next chapter. Now, let us write Eq. 5.5 in the form: sin θhkl =

λ 2dhkl

(5.6)

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5 Importance of d-Spacing in Diffraction of Crystals

Constructive interference

Destructive interference

Fig. 5.2 Constructive and destructive interference

From Eq. 5.6, it is clear that the Braggs’ angle and the interplanar spacing have inverse relationships. This is of great utility in reciprocal space where the crystal diffraction happens to take place. Each peak position in a powder pattern is found to correspond to a set of equidistant parallel atomic planes with certain (hkl) values for a fixed d-value and fixed angle as predicted by the Braggs’ law. However, it is to be noted that the Braggs’ equation provides only a necessary condition and not the sufficient condition for diffraction by real crystals. The Braggs’ reflections are found to occur only when: 1. In the simplest form of Braggs’ equation 2d sin θ = nλ, the wavelength of the incident beam follows the condition nλ ≤ 2d, because |sin θ | ≤ 1. However, in the (hkl) form using Eq. 5.5, it is given by. λ ≤ 2dhkl , because |sin θhkl | ≤ 1. This is the reason that visible light cannot be used for diffraction of a crystal. Further, no diffraction will occur if this condition is not satisfied. 2. The conditions for constructive interference must be fulfilled. The constructive interference enhances the resulting intensity of the diffracted beam. On the other hand, the destructive interference decreases the resulting intensity of the diffracted beam. The extreme situations of the two cases are shown in Fig. 5.2.

5.3 Equivalence of Braggs’ Equation and Laue Equations Although Braggs’ equation is the result of the fundamental periodic arrangement of crystal planes of atoms, but in true sense it does not refer to the actual arrangement of atoms associated with them. On the other hand, the Laue equations have been derived keeping in view the simple static atomic model of a crystal and appear to be more general because three Laue equations take care of three rows of atoms along three crystallographic axes, as far as the diffraction is concerned. However, despite the difference in the mode of derivations of Braggs’ and Laue equations, the Braggs’ equation can be shown to be equivalent and also a consequence of the more general Laue equations. Laue equations are represented by a set of three equations, where each equation provides the diffraction effect corresponding to a row of atoms along one of the three

5.3 Equivalence of Braggs’ Equation and Laue Equations

109

crystallographic axes, but all three equations need to be satisfied simultaneously for a diffraction of the crystal to occur. Three Laue equations are given by a(cos α − cos α0 ) = eλ b(cos β − cos β0 ) = f λ

(5.7)

c(cos γ − cos γ0 ) = gλ where a, b, c are the axial lengths along a-row, b-row and c-row of atoms, respectively. Similarly, e, f, g are integers, one each for a row, α0 , β0 , γ0 are the angles made by the incident plane wave front and α, β, γ are the angles made by the diffracted plane wave front from each row of atoms and λ is the wavelength of the X-ray used. Now, in order to prove that the Braggs’ equation is equivalent to Laue equations, let us consider a simple system of arrangement of atoms, such as a simple cubic crystal system with a = b = c and the angle between the incident and diffracted beam to be equal to 2θ. Then squaring and adding the three Laue equations, we obtain [ a 2 cos2 α + cos2 β + cos2 γ + cos2 α0 + cos2 β0 + cos2 γ0 ( ) − 2(cos α cos α0 + cos β cos β0 + cos γ cos γ0 ) = λ2 e2 + f 2 + g 2

(5.8)

Further, from solid geometry we know that the law of direction cosines states that cos2 α + cos2 β + cos2 γ = cos2 α0 + cos2 β0 + cos2 γ0 = 1 and cos α cos α0 + cos β cos β0 + cos γ cos γ0 = cos 2θ Making use of these results, Eq. 5.8 reduces to ) λ2 ( 2 e + f 2 + g2 2 a ) λ2 ( 2 2 or sin θ = 2 e + f 2 + g 2 4a

2(1 − cos 2θ ) =

(5.9)

Similarly, squaring the Braggs’ equation and rearranging various terms, we have sin2 θ =

) n 2 λ2 ( 2 h + k2 + l2 4a 2

where, for a simple cubic crystal system d=(

a h2 + k2 + l 2

)1/2

(5.10)

110

5 Importance of d-Spacing in Diffraction of Crystals

Now, comparing Eqs. 5.9 and 5.10, we find that e = nh, f = nk and g = nl This means that a diffracted beam defined by the integers e, f, g in Laue’s treatment may be interpreted as the nth order diffraction from a set of (hkl) planes in Braggs’ treatment. The order of diffraction n is simply equal to the largest common factor of the numbers e, f, g. This proves the equivalence of Braggs’ and Laue equations and also suggests that the Braggs’ equation is a consequence of the more general Laue equations.

5.4 Derivation of d-Spacing Formulae of Different Crystal Systems Two methods are in use to derive the formula of interplanar d-spacing between two consecutive parallel planes in a crystal of a given crystal system. They are briefly described below.

5.4.1 Using Cartesian Geometry In this case, we shall limit our discussion to the unit cells which are expressed in terms of the orthogonal coordinate axes, in which simple Cartesian geometry is applicable. Thus let us consider three mutually perpendicular axes Ox, Oy and Oz, and assume that an (hkl) plane (which is parallel to a plane passing through the origin) makes intercepts a/h, b/k and c/l on the three axes at A, B and C, respectively as shown in Fig. 5.3. Further, let OP (= dhkl , the interplanar spacing) be normal to the plane drawn from the origin and respectively makes angles α, β and γ with the three orthogonal axes. Therefore, we can write: OA = a/h, OB = b/k, OC = c/l and OP = dhkl . From triangle OPA etc., we have cos α =

dhkl OP dhkl OP dhkl OP = , cos β = = and cos γ = = OA a/h OB b/k OC c/l

Now, making use of the direction cosine, which states that cos2 α + cos2 β + cos2 γ = 1 and substituting the values of cos α, cos β and cos γ in Eq. 5.11, we obtain 2 dhkl d2 d2 + hkl 2 + hkl 2 = 1 2 (a/h) (b/k) (c/l)

(5.11)

5.4 Derivation of d-Spacing Formulae of Different Crystal Systems

111

Fig. 5.3 (hkl) plane intercepting x, y, z axes at A, B and C, respectively

or [ 2 dhkl

] h2 l2 k2 =1 + + b2 c2 a2

So that [ dhkl =

l2 h2 k2 + 2+ 2 2 b c a

]−1/2 (5.12)

This is a general formula and applicable to the ‘primitive lattices’ of orthorhombic, tetragonal and cubic crystal systems, where for orthorhombic crystal system the axial relationship is given by a /= b /= c. Further, for: Tetragonal crystal system: a = b /= c, the above Eq. 5.12, reduces to [

dhkl

l2 h2 + k2 + = a2 c2

]−1/2

Cubic crystal system: a = b = c, the above Eq. 5.12, reduces to )−1/2 ( dhkl = a h 2 + k 2 + l 2 or

112

5 Importance of d-Spacing in Diffraction of Crystals

dhkl = (

a h2

+

k2

+ l2

)1/2

5.4.2 Using General Method Simple Cartesian geometry is used to determine the interplanar spacing in the unit cells involving orthogonal coordinate system. However, for the unit cells involving oblique system, simple Cartesian geometry is not very helpful. Therefore, let us use the reciprocal lattice concept to get the general expression for interplanar spacing in complex cases. Accordingly, let us write 1 = σhkl · σhkl d2hkl ) ( ) ( = ha∗ + kb∗ + lc∗ · ha∗ + kb∗ + lc∗ = h2 a∗ · a∗ + hka∗ · b∗ + hla∗ · c∗ + khb∗ · a∗ + k2 b∗ · b∗ + klb∗ · c∗ + lhc∗ · a∗ + lkc∗ · b∗ + l2 c∗ · c∗

(5.13)

Making use of the identities a*·a* = a∗2 , etc., a*·b* = b*·a* = a* b* cosγ*, etc. and collecting the like terms, Eq. 5.13 becomes 1 d2hkl

= h2 a∗2 + k2 b∗2 + l2 c∗2 + 2hka∗ b∗ cos γ ∗ + 2klb∗ c∗ cos α ∗ + 2lhc ∗ a∗ cos β ∗

( Substituting the values of a* = become 1 2 dhkl

=

b×c V

(5.14)

) , b*, c*, etc. and simplifying Eq. 5.14 will

1 [ 2 2 2 2 h b c sin α + k 2 c2 a 2 sin2 β + l 2 a 2 b2 sin2 γ V2

+ 2hkc2 ab(cos α cos β − cos γ ) + 2lhb2 ac(cos γ cos α − cos β)] + 2kla 2 bc(cos β cos γ − cos α)

(5.15)

( ) where V 2 = a 2 b2 c2 1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ This is a general expression, valid for triclinic crystal system. The equations for other crystal systems can be obtained by substituting the respective lattice parameters (axes and angles). They are provided in Table 5.1.

5.5 Calculation of d-Spacing and Braggs’ Angle in Cubic Crystal System

113

Table 5.1 Interplanar d-spacing in various crystal systems Crystal system Cubic Tetragonal

dhkl )−1/2 ( a h2 + k2 + l 2 ] [ 2 2 2 −1/2 h +k + cl 2 a2

Orthorhombic

[

Hexagonal

[

Rhombohedral Monoclinic

Triclinic

h2 a2

+

k2 b2

+

l2 c2

]−1/2

] 2 −1/2 4/3(h 2 +hk+k 2 ) + cl 2 a2 ( )1/2 a 1−3cos 2 α+2cos 3 α

[(h 2 +k 2 +l 2 )sin 2 α+2(hk+kl+lh)(cos 2 α−cos α )]1/2 [ 2 2 ]−1/2 h a2

+ l 2 − (2hlcosβ) ac c

sin 2 β

⎡ | | | ⎢ | h/a ⎢h| ⎢ | k/b ⎢a| ⎣ | | | l/c

+

k2 b2 | |

| | | 1 h/a | | k| |+ | cos γ k/b | b| | | | | | | cos β l/c | | | 1 cos γ | | | cos γ 1 | | | | cos β cos α

cos γ cos β || 1

cos α

cos α

1

| |

| | | 1 | | l| | cos α |+ c || cos γ | | | | 1 | | cos β |−1/2 | cos β || | cos α || | | 1 |

cos β ||

|⎤−1/2 |

cos γ h/a ||⎥ 1

|⎥

k/b ||⎥ ⎥

cos α l/c

|⎦ | |

5.5 Calculation of d-Spacing and Braggs’ Angle in Cubic Crystal System The reciprocal relationship between the interplanar d-spacing and the Braggs’ angle shown by Bragg’s formula in Eq. 5.6 is clearly evident. Further, the intimate connection among the diffraction patterns, the equivalent sets of Braggs’ planes and the zone boundaries of a Brillouin zone (the W–S cell of the reciprocal lattice) discussed in the last chapter provides us a great opportunity to understand them better in 3 dimensions. Keeping this in view, it was decided to calculate the interplanar d-spacing and the Braggs’ angle in the well-studied and most symmetric three cubic crystal systems:) ( sc, bcc and fcc for some low index crystal planes ranging from N = h 2 + k 2 + l 2 = 1 to 12 excluding the non-possible values such as 7, 15, 23, etc. [where 1 ≡ (100), 2 ≡ (110), 3 ≡ (111), 4 ≡ (200), 5 ≡ (210), 6 ≡ (211), 8 ≡ (220), 9 ≡ (221) or (300), 10 ≡ (310), 11 ≡ (311) and 12 ≡ (222)] using the well-known formula [1], dhkl = (

a h2

+

k2

+ l2

)1/2

(5.16)

Since, Eq. 5.16 is strictly valid for simple cubic (primitive) lattice, therefore while using this equation for bcc and fcc, we need to modify the same by taking into account the positions of the planes that originate due to centering (additional

114

5 Importance of d-Spacing in Diffraction of Crystals

atoms). The presence of additional planes in bcc and fcc (as compared to sc) can be easily verified for some low index planes, such as (100), (110), (111), (200), (220) and (222) from the unit cells of three cubic crystal systems shown in Fig. 5.4 [1, 2]. For simplicity, the side of the cube in all three cases has been taken as, a = 6 Å. The corresponding Braggs’ angle is given by θhkl = sin

−1

(

λ 2dhkl

)

where λ = 1.54 Å for CuKα radiation.

Fig. 5.4 Representation of some low index planes in cubic crystal systems

(5.17)

5.5 Calculation of d-Spacing and Braggs’ Angle in Cubic Crystal System

115

Case I: Simple Cubic System Equation 5.16 is applicable to simple cubic crystal system as such, therefore substituting the above permissible values of hkl one by one, we can obtain 6 d100 = ( )1/2 = 6 2 2 1 + 0 + 02 Similarly, substituting the other values of h, k and l in Eq. 5.16, we obtain 6 6 6 6 d110 = √ = 4.243, d111 = √ = 3.464, d200 = = 3, d210 = √ = 2.683, 2 2 3 5 6 6 6 d211 = √ = 2.449, d220 = √ = 2.121, d221 = d300 = √ = 2.000, 6 9 2 2 6 6 6 d310 = √ = 1.897, d311 = √ = 1.809, d222 = √ = 1.732 10 11 2 3 Since no additional plane appears in between any of the three low index planes in the unit ( cell of simple ) cubic structure (Fig. 5.4), therefore the calculated values of N = h 2 + k 2 + l 2 is continuous except the values 7, 15, 23, etc. which are not possible due to structural conditions. Now making use of Eq. 5.17, the corresponding Braggs’ angles for different possible values of N can be determined as:

θ111 θ210 θ220 θ310 θ222

(

) 1.54 = 7.373◦ , 2×6 ( ) 1.54 = 26.396◦ , = sin−1 3.464 ( ) 1.54 = 35.029◦ , = sin−1 2.683 ( ) 1.54 = 46.558◦ , = sin−1 2.121 ( ) 1.54 = 54.273◦ , = sin−1 1.897 ( ) 1.54 = 62.766◦ = sin−1 1.732

θ100 = sin−1

θ200 θ211 θ300 θ311

(

) 1.54 = 21.282◦ , 4.243 ( ) 1.54 = 30.886◦ , = sin−1 3 ( ) 1.54 = 38.964◦ , = sin−1 2.449 ( ) 1.54 = 50.354◦ , = sin−1 2.000 ( ) 1.54 = 58.353◦ , = sin−1 1.809

θ110 = sin−1

However, to observe more reflections of higher order (or to accommodate the set of Braggs’ planes of smaller d-values), the X-rays of larger wavelengths could be used. In contrary, to observe more reflections of lower order (or to accommodate the set of Braggs’ planes of higher d-values), the X-rays of smaller wavelengths could be used.

116

5 Importance of d-Spacing in Diffraction of Crystals

Case II: Body Centered Cubic System In Fig. 5.4, additional planes are observed between (100), (111) and some other higher order planes (not shown here) in the body centered cubic system as compared to simple cubic system. Therefore, Eq. 5.16 needs to be modified accordingly wherever needed. Thus, after taking this fact into account, substitute the proper hkl values to obtain the required d-values. Hence, d100 =

1 a 6 (d100 )o f sc = . ( ) =3 2 2 2 2 1 + 0 + 02 1/2

Similarly, substituting the proper values of h, k and l and taking into account the presence of additional planes due to centring in Eq. 5.16, we obtain 6 6 6 6 d110 = √ = 4.243, d111 = (d111 )o f sc = √ = 1.732, d200 = = 3, 2 2 2 3 2 6 6 6 d210 = (d210 ) o f sc = √ = 1.342, d211 = √ = 2.449, 2 6 2 5 6 6 3 d220 = √ = 2.121, d221 = d300 = (d221 ) o f sc = √ = 1.000, 2 2 2 9 6 6 6 d310 = √ = 1.897, d311 = (d311 ) o f sc = √ = 0.905, 2 10 2 11 6 d222 = √ = 1.732 2 3 Because of the presence of additional planes between (100), (111) and some other higher order planes (Fig. 5.4) in bcc, the calculated values of some dhkl are found to match with their corresponding higher order planes. For example, we observe that the values of d-spacing d100 = d200 , d111 = d222 , etc., because similar situations are found to occur in many other higher order cases. This implies that the lower order planes [such as (100) and (111)] become the members of their corresponding higher order family of planes (200) and (222), etc., respectively. They are shown bold in the Table 5.2, while for such other higher order planes, spaces are left blank. Further, making use of Eq. 5.17, the corresponding Braggs’ angles for different possible values of N can be determined as:

5.5 Calculation of d-Spacing and Braggs’ Angle in Cubic Crystal System

117

Table 5.2 The d-spacing in three cubic crystal systems ( ) d-spacing Plane (hkl) N = h2 + k 2 + l 2 sc

bcc

fcc

1

(100)

6.000

3.000

3.000

2

(110)

4.243

4.243

2.121

3

(111)

3.464

1.732

3.464

4

(200)

3.000

3.000

3.000

5

(210)

2.683

–

–

6

(211)

2.449

2.449

–

8

(220)

2.121

2.121

2.121

9

(221)

2.000

–

–

10

(310)

1.897

1.897

–

11

(311)

1.809

–

1.809

12

(222)

1.732

1.732

1.732

θ100 θ111 θ210 θ220 θ310 θ222

( ) ) 1.54 1.54 ◦ −1 = 21.282◦ , = 30.886 , θ110 = sin = sin 4.243 3 ( ( ) ) 1.54 −1 ◦ −1 1.54 = 62.766 , θ200 = sin = 30.886◦ , = sin 1.732 3 ( ( ) ) 1.54 1.54 −1 −1 = N P, θ211 = sin = 38.964◦ , = sin 1.342 2.449 ( ( ) ) 1.54 1.54 = 46.558◦ , θ300 = sin−1 = N P, = sin−1 2.121 1.000 ( ( ) ) 1.54 1.54 = 54.273◦ , θ311 = sin−1 = N P, = sin−1 1.897 0.905 ( ) 1.54 = 62.766◦ = sin −1 1.732 −1

(

where NP represents the diffractions that are not possible. Again, because of the presence of additional planes between (100), (111) and some other higher order planes in bcc, the calculated values of some θhkl are found to match with their corresponding higher order angles. For example, we observe that the angles, θ100 = θ200 , θ111 = θ222 , etc., because similar situations are found to occur in many other higher order cases. This is a similar result as obtained above for the family of planes. Further, we also observe that for angles, such as θ210 , θ300 and θ311 , the diffractions are not possible because (λ ≤ 2dhkl ) condition is not fulfilled at these angles. Case III: Face Centered Cubic System Similar to bcc, additional planes are also observed in between (100), (110) and some other higher order planes in the face centered cubic system (Fig. 5.4) as compared

118

5 Importance of d-Spacing in Diffraction of Crystals

to simple cubic system. Therefore, only after taking this fact into account, substitute the proper hkl values in Eq. 5.16 to obtain the required d-values. Hence, 1 a 6 (d100 ) o f sc = . ( ) =3 2 2 12 + 02 + 02 1/2 6 6 = (d110 ) o f sc = √ = 2.121, 2 2 2 6 6 = √ = 3.464, d200 = = 3, 2 3 6 6 = (d210 )o f sc = √ = 1.342, 2 2 5 6 6 6 = (d211 ) o f sc = √ = 1.225, d220 = √ = 2.121, 2 2 6 2 2 6 3 = d300 = (d221 ) o f sc = √ = 1.000, 2 9 6 6 = (d310 )o f sc = √ = 0.949, 2 2 10 6 6 = √ = 1.809, d222 = √ = 1.732 11 2 3

d100 = d110 d111 d210 d211 d221 d310 d311

Like bcc, because of the presence of additional planes in between (100), (110) and some other higher order planes (Fig. 5.4) in fcc, the calculated values of some dhkl are found to match with their corresponding higher order planes. For example, we observe that the values of d-spacing, d100 = d200 , d110 = d220 , etc., because similar situations are found to occur in many other higher order cases. This implies that the lower order planes become the members of their corresponding higher order family of planes. They are shown bold in the Table 5.2, while for such other higher order planes, spaces are left blank. Now making use of Eq. 5.17, the corresponding Braggs’ angles for different possible values of N can be determined as:

5.5 Calculation of d-Spacing and Braggs’ Angle in Cubic Crystal System

( ) ) 1.54 1.54 = 30.886◦ , θ110 = sin−1 = 46.558◦ , 3 2.121 ( ( ) ) 1.54 1.54 = 30.886◦ , = 26.396◦ , θ200 = sin−1 = sin−1 3 3.464 ( ( ) ) 1.54 1.54 = N P, θ211 = sin−1 = N P, = sin−1 1.342 1.225 ( ( ) ) 1.54 1.54 = 46.558◦ , θ300 = sin−1 = N P, = sin−1 2.121 1.000 ( ( ) ) 1.54 1.54 = N P, θ311 = sin−1 = 58.353◦ , = sin−1 0.949 1.809 ( ) 1.54 = 62.766◦ = sin−1 1.732

θ100 = sin−1 θ111 θ210 θ220 θ310 θ222

119

(

where NP represents the diffractions that are not possible. Again, because of the presence of additional planes between (100), (110) and some other higher order planes in fcc, the calculated values of some θhkl are found to match with their corresponding higher order angles. For example, we observe that the angles, θ100 = θ200 , θ110 = θ220 , etc., because similar situations are found to occur in many other higher order cases. This gives us similar results as obtained above for the family of planes. Further, we also observe that for angles, such as θ210 , θ211 , θ300 and θ310 , the diffractions are not possible because (λ ≤ 2dhkl ) condition is not fulfilled at these angles. The d-spacing data obtained from above calculations for three cubic crystal systems are summarized in Table 5.2. They are found to be decreasing with increasing value of N in all three cubic cases. From Table 5.2, the following important observations are made: 1. The fundamental interplanar equation for cubic system given by Eq. 5.16 (which usually represents all three forms of cubic systems as mentioned in different text books) is strictly valid only to primitive or simple cubic crystal system and the equation to be used for bcc or fcc must be modified according to the presence of new planes due to additional centered atoms (as compared to simple cubic), as has been done in the above calculations according to the illustrations of some low index planes in cubic crystal system provided in Fig. 5.4. 2. The values of d-spacing d100 of bcc and both d100 and d110 of fcc (including some higher order values, both in bcc and fcc) are equal to their next higher order spacing and automatically merge with them. For example, in bcc, the (100) plane loses its independent identity as it becomes a member of the (200) family. Similar situations arise with some other higher order planes of bcc and fcc. 3. The d-spacings’d100 , d110 and d111 in sc, bcc and fcc, respectively are found to be the widest in their own category. 4. In a given crystal system, different families of equidistant parallel planes have different d-spacing. For example, an exhibition of this feature is illustrated in a 2-D system in Fig. 5.5. It shows that higher the Miller indices, the smaller the

120

5 Importance of d-Spacing in Diffraction of Crystals

Fig. 5.5 Families of equidistant parallel planes in a 2-D system

d-spacing, and the lower the atomic density. Same principle is applied in a 3-D system. 5. In a simple cubic case, all the d-spacings’ are in decreasing order, they perfectly match with the order of planes observed from powder diffraction patterns [3–5] of simple cubic element (Po) polonium, MX-compound (CsCl) cesium chloride and other simple cubic crystals. 6. Similar to sc, in bcc and fcc, the values of d-spacings’ perfectly match with the planes observed from the powder diffraction patterns of their respective elements or compounds. They prima facie appear to take care of the conditions for respective systematic absences and others (such as screw axes and glide planes) arising due to the lattice centring as provided in Tables 7.3–7.5! These aspects are taken up in Chap. 7 [6]. 7. Unlike the present notion regarding systematic absences in bcc and fcc (and in general all other centered lattices), the data from Table 5.2 suggest that no reflections are actually absent (or no planes are missing) but they appear to be so, because of identical d-spacing the lower order planes merge with their next higher order parallel planes and together they give rise to the diffraction peaks. The calculated values of d-spacing from Table 5.2 suggest that the planes {100}, {110} and {111} have widest separations (the planes of widest spacing are also known to be the planes of closest packing), respectively in sc, bcc and fcc in direct space. This implies that in reciprocal space, these planes will correspond to the least separations ∗ ≡ 1/d⭢hkl ) in their own category and will (because of shortest lattice vector, as d⭢hkl give rise to the first diffraction peaks at their lowest diffraction angles according to

5.6 Calculation of d-Spacing in Primitive Lattices of Other Crystal Systems

121

Table 5.3 The Braggs’ angle in three cubic crystal systems ( ) Plane (hkl) Angle, θhkl N = h2 + k 2 + l 2 sc

bcc

fcc

1

(100)

7.373°

30.886°

30.886°

2

(110)

21.282°

21.282°

46.558°

3

(111)

26.396°

62.766°

26.396°

4

(200)

30.886°

30.886°

30.886°

5

(210)

35.029°

NP

NP

6

(211)

38.964°

38.964°

NP

8

(220)

46.558°

46.558°

46.558°

9

(221) or (300)

50.354°

NP

NP

10

(310)

54.273°

54.273°

NP

11

(311)

58.353°

NP

58.353°

12

(222)

62.766°

62.766°

62.766°

Braggs’ Eq. 5.6. This, further implies that the subsequently decreasing d-spacings will provide the diffraction peaks of higher orders at subsequently increasing Braggs’ angles in different cubic systems. This is how the diffraction process will continue in cubic and all crystalline materials of other crystal systems. In a similar manner, the θhkl data obtained from above calculations for three cubic crystal systems are summarized in Table 5.3. They are found to be increasing with increasing value of N in all three cubic cases.

5.6 Calculation of d-Spacing in Primitive Lattices of Other Crystal Systems Below we provide the calculations of d-spacing corresponding to some low index (hkl) planes for primitive cases of crystals belonging to elements of crystal systems other than cubic and triclinic, using the standard formulae and actual structural (axial and angular) data to have an idea in advance about the nature of occurrence of the combination of d-spacing and Braggs’ planes in Tables 5.4, 5.5, 5.6, 5.7, 5.8. For similar reasons, we also provide the calculated values of d-spacing corresponding to some low index (hkl) planes of some well-known crystals belonging to an MX cubic crystal systems, using the standard formulae and actual structural (axial and angular) data in Tables 5.9, 5.10. These calculations will in turn, substantiate the results obtained from the three cubic crystal systems that we took up above for d-spacing and angle calculations.

122

5 Importance of d-Spacing in Diffraction of Crystals

Table 5.4 d-Spacing in Se monoclinic crystal

S. No.

N

Plane (hkl)

( ) d-spacing [dhkl Å ]

1

1

(100)

12.83

2

1

(001)

09.32

3

1

(010)

08.07

4

2

(101)

07.34

5

2

(110)

06.83

6

4

(200)

06.42

7

2

(011)

06.09

8

3

(111)

05.43

9

4

(002)

04.66

10

4

(020)

04.04

11

8

(202)

03.67

12

8

(220)

03.42

13

8

(022)

03.05

14

12

(222)

02.77

Case I: Crystals of Some Elements 1. Monoclinic Crystal System Making use of the formula given below, let us make the calculation of d-spacing: [ dhkl =

h2 a2

+

l2 c2

−

(2hl cos β) ac

sin2 β

k2 + 2 b

]−1/2

Given: Monoclinic crystal of Selenium (Se) with lattice parameters: a = 12.85 Å, b = 8.07 Å, c = 9.31 Å; β = 93◦ , 8, = 93.13◦ ; SG = P21 /a 2. Orthorhombic Crystal System Making use of the formula given below, let us make the calculation of d-spacing: [

dhkl

h2 k2 l2 = 2 + 2+ 2 a b c

]−1/2

Given: Orthorhombic crystal of Neptunium (Np) with lattice parameters: a = 4.72 Å, b = 4.88 Å, c = 6.66 Å; SG = Pmcn

5.6 Calculation of d-Spacing in Primitive Lattices of Other Crystal Systems Table 5.5 d-Spacing in Np orthorhombic crystal

123

S. No.

N

Plane (hkl)

( ) d-spacing [dhkl Å ]

1

1

(001)

06.66

2

1

(010)

04.88

3

1

(100)

04.72

4

2

(011)

03.94

5

2

(101)

03.85

6

2

(110)

03.39

7

4

(002)

03.33

8

3

(111)

03.02

9

5

(012)

02.75

10

5

(102)

02.72

11

4

(020)

02.44

12

6

(112)

02.38

13

4

(200)

02.36

14

5

(021)

02.29

15

5

(201)

02.22

16

5

(120)

02.17

17

5

(210)

02.12

18

6

(121)

02.06

19

6

(211)

02.02

3. Tetragonal Crystal System Making use of the formula given below, let us make the calculation of d-spacing: Table 5.6 d-Spacing in Cl tetragonal crystal

S. No.

N

Plane (hkl)

d-spacing ( ) [dhkl Å ]

1

1

(100), (010)

08.56

2

1

(001)

06.12

3

2

(110)

06.05

4

2

(101), (011)

04.98

5

3

(111)

04.30

6

4

(200), (020)

04.28

7

5

(210), (120)

03.83

8

5

(201), (021)

03.51

9

6

(211), (121)

03.25

10

6

(112)

02.73

124

5 Importance of d-Spacing in Diffraction of Crystals

Table 5.7 d-Spacing in Te simple hexagonal crystal

S. No.

N

Plane (hkl)

( ) d-spacing [dhkl Å ]

1

1

(001)

05.92

2

1

(100), (010)

03.85

3

2

(101), (011)

03.23

4

4

(002)

02.96

5

5

(102), (012)

02.35

6

2

(110)

02.23

7

3

(111)

02.08

8

4

(200), (020)

01.93

9

5

(201), (021)

01.83

10

6

(112)

01.78

11

5

(210), (120)

01.46

12

6

(211)

01.41

[ dhkl =

l2 h2 + k2 + 2 2 a c

]−1/2

Given: Tetragonal crystal of Chlorine (Cl) with lattice parameters: a = b = 8.56 Å, c = 6.12 Å; SG = P4/ncm 4. Hexagonal Crystal System Making use of the formula given below, let us make the calculation of d-spacing: [

dhkl

4/3(h 2 + hk + k 2 ) l2 = + a2 c2

]−1/2

Given: Simple Hexagonal crystal of Tellurium (Te) with lattice parameters: a = b = 4.45 Å, c = 5.92 Å; SG = P31 21 5. Rhombohedral (RCP) Crystal System Making use of the formula given below, let us make the calculation of d-spacing: dhkl

)1/2 ( a 1 − 3 cos2 α + 2 cos3 α = [( ) ( )]1/2 h 2 + k 2 + l 2 sin2 α + 2(hk + kl + lh) cos2 α − cos α

Given: Rhombohedral crystal of Arsenic (As) with lattice parameters: a = b = c = 4.13 Å; α = 54◦ , 10, = 54.17◦ ; SG = R3m

5.6 Calculation of d-Spacing in Primitive Lattices of Other Crystal Systems Table 5.8 d-Spacing in As rhombohedral crystal

125

S. No.

N

Plane (hkl)

( ) d-spacing [dhkl Å ]

1

3

(111)

03.43

2

1

(100), (010), (001)

03.11

3

2

(110), (101), (011)

02.75

4

6

(211), (121), (112)

02.02

5

9

(221), (212), (122)

01.74

6

12

(222)

01.71

7

5

(210), (201), (021)

01.65

8

4

(200), (020), (002)

01.55

9

8

(220), (202), (022)

01.37

From the results of above calculations, we can make another important and general conclusion that irrespective of a single crystal diffraction or a powder diffraction, the number of diffraction peaks (N) decreases with the increase in symmetry of the crystal system, i.e., Triclinic < Monoclinic < Orthorhombic < Tetragonal < Cubic. Case II: Some MX-Crystals (i) CsCl: Simple Cubic Crystal System Given: Simple cubic crystal of Cesium Chloride (CsCl) with lattice parameters: Table 5.9 d-Spacing in CsCl crystal

S. No.

N

Plane (hkl)

( ) d-spacing [dhkl Å ]

1

1

(100)

4.123

2

2

(110)

2.915

3

3

(111)

2.380

4

4

(200)

2.062

5

5

(210)

1.844

6

6

(211)

1.683

7

8

(220)

1.458

8

9

(221)

1.374

9

10

(310)

1.304

10

11

(311)

1.243

11

12

(222)

1.190

12

13

(320)

1.144

13

14

(321)

1.102

14

16

(400)

1.031

126

5 Importance of d-Spacing in Diffraction of Crystals

Table 5.10 d-Spacing in NaCl crystal

S. No.

N

Plane (hkl)

( ) d-spacing [dhkl Å ]

1

1

(100)

4.123

2

2

(110)

2.915

3

3

(111)

2.380

4

4

(200)

2.062

5

5

(210)

1.844

6

6

(211)

1.683

7

8

(220)

1.458

8

9

(221)

1.374

9

10

(310)

1.304

10

11

(311)

1.243

11

12

(222)

1.190

12

13

(320)

1.144

13

14

(321)

1.102

14

16

(400)

1.031

a = 4.123 Å , dhkl = (

a h2

+

k2

+ l2

)1/2

(ii) NaCl: Face Centered Cubic Crystal System Given: Face Centered Cubic Crystal of Sodium Chloride (NaCl) with lattice parameters: a = 5.638 Å We know that CsCl and NaCl are members of MX-category and respectively belong to simple cubic and face centered cubic categories. Tables 5.9, 5.10 provide the sequence of (hkl) planes that are found to be compatible to their respective monoatomic counterparts as in Table 5.2.

5.7 Diffraction Results of Cubic Crystals of Some Elements and MX-System We know that cubic crystal system belongs to the most symmetric category because of its simplest structural conditions (i.e., a = b = c; α = β = γ = 90◦ ). As a result, the crystal structure determination of this system is relatively easier as compared to other crystal systems. In this section, we are going to present the diffraction results

5.7 Diffraction Results of Cubic Crystals of Some Elements and MX-System

127

of some real cases of relevant monoatomic elements belonging to sc, bcc, fcc and some MX-systems (such as CsCl, NaCl, etc.) in the form of worked out examples. Example 1: Calculated and observed data of d-spacings and intensities for αPolonium are provided in Table 5.11 for different possible planes. Obtain the average nearest value of its axial parameter, a. Solution: Given: α-Polonium is the only known simple cubic crystal in the periodic Table, calculated and observed data of d-spacing and intensities [7], value of its axial parameter, a = ? Making use of the formula, √ ) ( a = d N, wher eN = h2 + k2 + l2 . We can easily obtain the value of ‘a’ as provided in the Table 5.11. The average calculated value of axial length of polonium cubic unit cell, a = 3.345 Å. Example 2: Molybdenum is a cubic crystal and gives a diffraction pattern that yields the following seven largest d-spacings: 2.225, 1.574, 1.285, 1.113, 0.995, 0.907 and 0.841 Å. The wavelength of the X-ray used is λ = 1.54 Å. Index the diffraction data and determine the side of the unit cell. Also, determine the value of θ100 and show that the corresponding (100) plane does not exist. Solution: Given: Molybdenum is a cubic crystal, seven largest d-spacings are: 2.225, 1.574, 1.285, 1.113, 0.995, 0.907 and 0.841 Å, λ = 1.54 Å, a = ?, θ100 = ? Obtain various Braggs’ angles using the equation Table 5.11 Calculated and observed d-spacing and intensities for α-Polonium ( 2 2 2) 1/d 2 Intensity Axial length ‘a’ h +k +l (hkl) (Å) Calculated

Observed

Calculated

Observed

Cal

Obs

1 (100)

0.0894

0.0896

100

100

3.345

3.341

2 (110)

0.1787

0.1795

91

80

3.345

3.337

3 (111)

0.2681

0.2692

38

40

3.345

3.338

4 (200)

0.3575

0.3595

21

25

3.345

3.336

5 (210)

0.4469

0.4486

62

55

3.345

3.338

6 (211)

0.5362

0.5371

50

45

3.345

3.342

8 (220)

0.7150

0.7169

18

10

3.345

3.341

9 (221)

0.8045

0.8065

42

30

3.345

3.341

10(310)

0.8937

0.8978

32

25

3.345

3.337

11(311)

0.9831

0.9866

32

15

3.345

3.339

128

5 Importance of d-Spacing in Diffraction of Crystals

θhkl = sin−1

d (Å)

θ◦

sin θ

sin2 θ

2.225

20.258

0.346

0.1199

1.574

29.304

0.489

0.2396

1.285

36.842

0.599

0.3595

1.113

43.799

0.692

0.4790

0.995

50.706

0.774

0.907

57.984

0.841

66.323

(

λ 2dhkl

)

a= √ d N

I(f)

2

(110)

3.146

100.0

4

(200)

3.148

16.0

6

(211)

3.147

31.0

8

(220)

3.148

9.0

0.5989

10

(310)

3.146

14.0

0.848

0.7189

12

(222)

3.142

3.0

0.916

0.8387

14

(321)

3.147

24.0

0.0599

N=

sin2 θ c· f

(hkl)

Common factor (c.f)

Thus the average value of side of the unit cell, a = 3.146 Å. In order to find the value of θ100 , we make use of the Braggs’ equation, λ = 2d100 sinθ100 , or sinθ100 = λ/(2d100 ) = 1.54/(2 × 3.146) = 0.245 Therefore, θ100 = 14.182◦ This reflection is not found as the first order reflection in the data table. However, it has been found as the second order reflection from (100) planes, which is supposed to be equivalent to the first order reflection from the (200) planes. Let us find out whether this reflection is observed or not. For the purpose, we obtain, sinθ200 = 2λ/(2d200 ) = 1.54/3.146 = 0.4895. Therefore, (θ100 )2nd order ~ (θ200 )1st order = 29.31°. This implies that the first order reflection from the (200) planes is observed, which can be verified from the Table at 29.304°. This shows that Molybdenum crystal belongs to a body centred cubic. Example 3: Diffraction data of a cubic crystal of an element show peaks at 2θ angles 20.20°, 28.72°, 35.36°, 41.07°, 46.19°, 50.90°, 55.28° and 59.42°. If the wavelength

5.7 Diffraction Results of Cubic Crystals of Some Elements and MX-System

129

of X-ray is 0.7107 Å using Mo target, determine crystal structure, axial length and identify the element. Solution: Given: Crystal is cubic, 2θ angles: 20.20°, 28.72°, 35.36°, 41.07°, 46.19°, 50.90°, 55.28° and 59.42°, λ = 0.7107 Å (Mo target), crystal structure = ?, a = ?, element = ? From the given angles, calculate the d-spacing by using the formula, d = λ/(2 sin θ), we have 2θ

θ

sin θ

sin2 θ

d = λ/(2 sin θ ) (Å)

20.20

10.10

0.1770

0.0313

2.0076

28.72

14.36

0.2480

0.0615

1.4328

35.36

17.68

0.3037

0.0922

1.1700

41.07

20.53

0.3507

0.1229

1.0132

46.19

23.09

0.3922

0.1538

0.9060

50.90

25.45

0.4297

0.1846

0.8269

55.28

27.64

0.4639

0.2152

0.76660

59.42

29.71

0.4956

0.2456

0.7170

From first table, make use of the values of θ and sin2 θ, obtain the common factor (c.f) and add into other column of the next table. Line

θ (°)

sin2 θ

1

10.10

0.0308

2

14.36

0.0615

3

17.68

0.0922

4

20.53

0.1229

8

5

23.09

0.1538

10

6

25.45

0.1846

12

7

27.64

0.2152

14

8

29.71

0.2456

16

Common factor (c.f)

N= 2 4

0.0154

6

sin2 θ c. f

√

N

√ 2 √ 4 √ 6 √ 8 √ 10 √ 12 √ 14 √ 16

√ a=d N 2.840 2.866 2.866 2.866 2.865 2.864 2.868 2.868

The values of N for allowed reflections give us that the unknown cubic structure is the body centered cubic (bcc). Finally, the average value of ‘a’ from the Table is found to be 2.86 Å. The above data indicates that the element is iron. Clearly, it has bcc structure. Example 4: The powder diffraction data of the element Ni show its 2θ peaks at angles 43.67°, 50.84°, 74.76°, 90.74°, 96.08°, 118.24°, 138.66° and 147.26°, respectively, assumed to have a cubic structure. If the wavelength of X-ray used is 1.54 Å, determine its crystal structure and axial length.

130

5 Importance of d-Spacing in Diffraction of Crystals

Solution: Given: Structure is cubic, 2θ angles: 43.67°, 50.84°, 74.76°, 90.74°, 96.08°, 118.24°, 138.66° and 147.26°, λ = 1.54 Å, crystal structure ?, a = ? As above, let us prepare the data sheet in tabular form and calculate the d-spacing by using the formula, d = λ/(2 sin θ), we have 2θ

θ

sin θ

sin2 θ

d = λ/(2 sin θ) (Å)

43.67

21.84

0.3720

0.1384

2.069

50.84

25.42

0.4293

0.1842

1.794

74.76

37.38

0.6071

0.3686

1.268

90.74

45.37

0.7117

0.5065

1.082

96.08

48.04

0.7436

0.5530

1.036

118.24

59.12

0.8582

0.7366

0.897

138.66

69.33

0.9356

0.8754

0.823

147.26

73.63

0.9595

0.9206

0.803

From first table, make use of the values of θ and sin2 θ, obtain common factor (c.f) and add into other column of the next table. Line

θ (°)

sin2 θ

1

21.84

0.1384

2

25.42

0.1842

3

37.38

0.3686

8

4

45.37

0.5065

11

5

48.04

0.5530

12

6

59.12

0.7366

16

7

69.33

0.8754

19

8

73.63

0.9206

20

Common factor (c.f)

N= 3

0.0461

4

sin2 θ c. f

√

N

√ 3 √ 4 √ 8 √ 11 √ 12 √ 16 √ 19 √ 20

√ a=d N 3.584 3.588 3.586 3.588 3.588 3.588 3.587 3.591

The values of N for allowed reflections give us that the unknown cubic structure is face centered cubic (fcc). Finally, the average value of ‘a’ from the Table is found to be 3.588 Å. The above data indicates that the crystal structure is fcc and the element is Nickel. Example 5: Aluminium powder gives a diffraction pattern that yields the following eight largest d-spacings: 2.338, 2.024, 1.431, 1.221, 1.169, 1.0124, 0.9289 and 0.9055 Å. Aluminium has a cubic close packed structure and the wavelength of the X-ray used is λ = 1.5405 Å. Index the diffraction data. Solution: Given: Eight largest d-spacings: 2.338, 2.024, 1.431, 1.221, 1.169, 1.0124, 0.9289 and 0.9055 Å, λ = 1.5405 Å, ‘a’ = ? To obtain θhkl corresponding to each d-spacing, let us use Eq. 5.17

5.7 Diffraction Results of Cubic Crystals of Some Elements and MX-System

θhkl = sin−1

(

λ 2dhkl

131

)

Further, aluminium is known to have fcc lattice and the only reflections observed are( those with )all even or all odd indices. The only values of sin2 θ in sin2 θ = A h 2 + k 2 + l 2 allowed are: 3A, 4A, 8A, 11A, 12A, 16A, 19A and 20A for the first eight reflections. d/Å

θ

sin θ

sin2 θ

2.338

19.208

0.329

0.1082

2.024

22.395

0.381

1.431

32.547

0.538

1.221

39.124

0.631

0.3981

1.169

41.223

0.659

0.4343

a= √ d N

3

111

4.049

0.1451

4

200

4.048

0.2894

8

220

4.047

11

311

4.049

12

222

1.0124

49.552

0.761

4.049

0.5791

16

400

4.049

0.9289

55.996

0.9055

58.321

0.829

0.6872

19

331

4.049

0.851

0.7242

20

420

4.049

0.03606

N=

sin2 θ c. f

hkl

Common factor (c.f)

Thus the average value of lattic parameter, a = 4.049 Å = 4.05 Å. Example 6: Diamond powder diffraction pattern yields the following five dspacings: 2.060, 1.261, 1.0754, 0.8916, and 0.8186 Å and corresponding 2θ angles at: 43.915°, 75.302°, 91.415°, 119.521° and 140.587°, respectively. The wavelength of the X-ray used is λ = 1.5405 Å. Index the diffraction data and determine the lattice parameter, ‘a’. Solution: Given: Five largest d-spacings are: 2.060, 1.261, 1.0754, 0.8916 and 0.8186 Å, 2θ angles are: 43.915°, 75.302°, 91.415°, 119. 521° and 140.587°, ‘a’ =? d/Å

θ

sin θ

sin2 θ

Common factor (c.f)

N=

sin2 θ c. f

hkl

a= √ d N

2.060

21.957

0.3739

0.1398

3

111

3.568

1.261

37.651

0.6108

0.3731

8

220

3.567

1.0754

45.747

0.7163

0.5130

11

311

3.567

0.8916

59.761

0.8639

0.7464

0.8186

70.294

0.9414

0.8863

0.0466

16

400

3.566

19

331

3.568

Thus the average value of lattic parameter, a = 3.567 Å.

132

5 Importance of d-Spacing in Diffraction of Crystals

Example 7: Cesium chloride powder gives a diffraction pattern that yields the following ten largest d-spacings: 4.13, 2.915, 2.380, 2.062, 1.844, 1.683, 1.458, 1.374, 1.304 and 1.243 Å and the corresponding 2θ angles (in degrees) at 21.5, 30.7, 37.8, 43.9, 49.4, 54.5, 63.8, 68.2, 72.4 and 76.5, respectively. CsCl has a simple cubic structure. The wavelength of the X-ray used is λ = 1.5405 Å. Determine the value of lattice parameter ‘a’. Solution: Given: Ten largest d-spacings are: 4.123, 2.915, 2.380, 2.062, 1.844, 1.683, 1.458, 1.374, 1.304 and 1.243 Å, corresponding 2θ angles (in degrees) are: 21.5, 30.7, 37.8, 43.9, 49.4, 54.5, 63.8, 68.2, 72.4 and 76.5, λ = 1.5405 Å, value of lattice parameter ‘a’ = ? In order to determine the value of lattice parameter ‘a’, let us calculate other required data as provided in the data Table. Data of CsCl hkl

√ a=d N

1

100

4.123

0.069

2

110

4.122

0.105

3

111

4.122

4

200

4.124

0.175

5

210

4.123

0.209

6

211

4.122

0.528

0.279

8

220

4.124

34.10

0.561

0.315

9

300 or 221

4.122

1.304

36.20

0.591

0.349

10

310

4.124

1.243

38.25

0.620

0.384

11

311

4.123

d/Å

θ

sin θ

sin2 θ

4.123

10.75

0.186

0.035

2.915

15.25

0.263

2.380

18.90

0.324

2.062

21.95

0.374

0.139

1.844

24.70

0.418

1.683

27.25

0.458

1.458

31.90

1.374

Common factor (c.f)

0.035

N=

sin2 θ c. f

Thus the average value of lattic parameter, a = 4.123 Å. Example 8: Sodium chloride powder gives a diffraction pattern that yields the following nine largest d-spacings: 3.25, 2.82, 1.99, 1.70, 1.63, 1.41, 1.29, 1.26, 1.15 Å and the corresponding 2θ angles (in degrees) at 27.42, 31.70, 45.54, 53.55, 56.40, 65.70, 76.08, 84.11 and 89.94, respectively. Sodium chloride has face centered cubic structure, wavelength of the X-ray used is λ = 1.5405 Å. Determine the value of lattice parameter ‘a’. Solution: Given: Nine largest d-spacings are: 3.25, 2.82, 1.99, 1.70, 1.63, 1.41, 1.29, 1.26 and 1.15 Å, corresponding 2θ angles (in degrees) are: 27.42, 31.70, 45.54, 53.55, 56.40, 65.70, 76.08, 84.11 and 89.94, λ = 1.5405 Å, value of lattice parameter ‘a’ =?

5.8 Diffraction Geometry, Braggs’ Planes and Zone Boundaries

133

In order to determine the value of lattice parameter ‘a’, let us calculate other required data as provided in the data Table. Data of NaCl hkl

√ a=d N

3

111

5.629

4

200

5.640

8

220

5.628

11

311

5.638

12

222

5.646

0.2942

16

400

5.640

0.6162

0.3797

19

331

5.623

0.6698

0.4487

20

420

5.634

0.7067

0.4990

24

422

5.634

d/Å

θ

sin θ

sin2 θ

3.25

13.71

0.2370

0.0562

2.82

15.85

0.2731

0.0746

1.99

22.77

0.3870

0.1498

1.70

26.98

0.4536

0.2058

1.63

28.20

0.4725

0.2233

1.41

32.85

0.5424

1.29

38.04

1.26

42.06

1.15

44.97

Common factor (c.f)

0.018724

N=

sin2 θ c. f

Thus the average value of lattic parameter, a = 5.635 Å.

5.8 Diffraction Geometry, Braggs’ Planes and Zone Boundaries It is interesting to note that all forms of diffraction (including X-rays, electrons and neutrons) provide their results in reciprocal space after satisfying Braggs’ diffraction condition and that the Brillouin zone is simply the Wigner–Seitz unit cell of the reciprocal lattice. Since the diffraction geometry and the Bragg planes or the Brillouin zones and their zone boundaries are obtained from diffraction results, as they all originate from reciprocal lattice and therefore they are expected to have close relationships. This can be understood easily with the help of Braggs’ equation taken in vector form as given in Eq. 4.7 (Chap. 4). To construct a Braggs’ plane, it is convenient to replace the wave vector k⭢ by −k⭢ in the equation so that both k⭢ and g⭢ begin from the same origin, 000, of the reciprocal lattice. Hence, the modified equation can be written in the form [5, 8–10]: g⭢ g⭢ g⭢ k⭢ · = · 2 2 2

(5.18)

This equation is valid for all kinds of waves including that of electron waves travelling in crystals. According to this, when the electrons with wave vector k⭢ reaching a Brillouin zone boundary, they will suffer strong Braggs’ diffraction (reflection) by the crystal planes.

134

5 Importance of d-Spacing in Diffraction of Crystals

Now, we construct a plane normal to g⭢ at its midpoint g2⭢ , then it will mean that any vector k⭢ drawn from the origin, 000, to any position on this plane will satisfy the Braggs’ diffraction condition as shown in Fig. 5.6. Such a plane is termed as a Braggs’ plane. A two-dimensional view of this concept is illustrated in the first Brillouin zone obtained from a square lattice as shown in Fig. 5.7, where the four zone boundaries of the square represent the four equivalent Braggs’ planes, i.e. {10} ≡ (10), (01), (10) and (01) in 2-D. In Sect. 4.10, we already encountered a similar situation in 2-D and 3-D. A simple view of this concept is illustrated in the first Brillouin zone obtained from a simple cubic lattice as shown in Fig. 4.20a, where the six zone boundaries of the cube represent the six equivalent Braggs’ planes, i.e. {100} ≡ (100), (010), (001), (100), (010) and (001) in 3-D. Here, it is to be noted that the center of the zone does not exclusively belong to the first zone only, as it has to be a part of all other zones for higher order diffractions (where the number of zones will depend on the nature of Miller indices of equivalent Braggs’ planes) distributed in spherically symmetric manner. Fig. 5.6 Relationship between Braggs’ equation and diffraction plane

5.9 Diffraction Patterns, Brillouin Zones and Centro-symmetry

135

Fig. 5.7 Braggs planes in a square lattice

5.9 Diffraction Patterns, Brillouin Zones and Centro-symmetry The intimate connection between Braggs’ planes (diffraction geometry) and zone boundaries and other basic principles discussed above for 2-D are also applicable to 3-D. This strongly suggests that the diffraction of X-rays, electrons and neutrons recognizes only the W–S/B–Z cells and not the conventional unit cells of the space lattice. Accordingly, the diffraction process first of all selects an atom from the closest packed plane of the given crystal structure to be at the center (acting as a defined origin) in accordance with the diffraction geometry satisfying the Braggs’ equation and then: 1. The diffraction process selects the first nearest neighbour set of closest packed planes (of atoms) of smallest reciprocal lattice vector to construct the zone boundaries of the first Brillouin zone around the central atom (taken as the origin) on satisfying the Braggs’ condition as illustrated in Figs. 4.21. This is the most important criterion for the first order diffraction to take place. 2. Following the same criterion, the diffraction process selects the next nearest neighbour set of planes (of atoms) with increasing reciprocal lattice vectors (or in other words, the next less close packed planes with less widely separated in direct space) one by one, each time and completes the construction of higher zones (as per the Braggs’ condition), in spherically symmetric manner (Figs. 4.21–4.24). The creation of successive Brillouin zones around the central atom by the sets of planes (of atoms) of increasing interplanar spacing takes place in a similar manner in 3-D as illustrated in Fig. 5.8 for a square lattice in 2-D.

136

5 Importance of d-Spacing in Diffraction of Crystals

Fig. 5.8 Exhibition of center of symmetry by a diffraction pattern

3. The exhibition of center of symmetry in diffraction patterns of all crystals (whether centro-symmetric or not) is only due to the spherically symmetric distribution of Braggs’ planes in the Brillouin zone (containing an atom at its center as defined origin) in reciprocal space (Figs. 4.21–4.24). So far, this phenomenon was known to be due to Friedel’s law [11,12]. This historic result definitely supports the idea of space lattice to be represented by W–S/B-Z as far as the diffraction of crystals is concerned.

5.10 Diffraction Patterns, Brillouin Zones and Translational Symmetry The above discussion about diffraction patterns and Brillouin zones suggests two extremely important historic results:

5.10 Diffraction Patterns, Brillouin Zones and Translational Symmetry

137

(i) The distribution of Braggs’ planes in different Brillouin zones (as exhibited by diffraction patterns) is found to be spherically symmetric around an atom at the center acting as defined origin. The spherically symmetric nature of diffraction patterns in reciprocal space allows the mirror, rotational and inversion symmetries while forbids the presence of translational symmetry in crystals [13]. This is an extremely important result and must be taken into account during crystal structure work. (ii) Spherically symmetric distribution of Braggs’ planes (indexed one quadrant only) for some lower order Brillouin zones obtained from 2-D square lattice is shown in Fig. 5.8. Similar distributions can be shown in case of other 2-D and 3-D lattices. (iii) Different Brillouin zones (obtained from the given diffraction pattern) may exhibit same/different number of Braggs’ planes, called multiplicity [3]. The multiplicity (or the number of crystallographic equivalent planes) exhibited in low index hkl values by three cubic crystal systems along with DC are provided in Table 5.12. Table 5.12 suggests that the maximum multiplicity exhibited by a cubic crystal is 48 which corresponds to the Braggs’ planes {hkl} ≡ {321} with N = 14, there may be some other possible sequences with the condition (h > k > l). In this case, there are eight possible combinations of positive and negative hkl indices and six possible permutations among them. Complete list of the 48 planes in the form of hkl indices are given below: hkl

hkl

hkl

hkl

h kl

hkl

hkl

hkl

klh

klh

klh

klh

k lh

kl h

klh

klh

lhk

lhk

lhk

lhk

l hk

lh k

lhk

lhk

khl

khl

khl

khl

k hl

kh l

khl

khl

hlk

hlk

hlk

hlk

h lk

hl k

hlk

hlk

lkh

lkh

lkh

lkh

l kh

lk h

lkh

lkh

In a similar manner, the multiplicity list for other hkl indices can be obtained by using the following simple rules: 1. If the two indices are equal, say for the planes of hhl type (such as 112, or 223) then hkl and khl becomes identical. Therefore, the value of M will reduce to 24. 2. If an index is zero, say for the planes of hk0 type (such as 120, or 230) then the positive and negative values of the third index are identical. Therefore, the value of M will reduce to 24. 3. If the first two indices are equal and the third index is zero such as hh0 type (i.e. 110, 220, etc.), then the value of M will reduce to 12. 4. If the three indices are equal, say for the planes of hhh type (such as 111, or 222) then the value of M reduces to 8. 5. If two indices are zero, say for the planes of h00 type (such as 100, or 200) then the value of M reduces to 6, the minimum value.

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5 Importance of d-Spacing in Diffraction of Crystals

Table 5.12 Multiplicity exhibited in low index hkl by cubic crystals ) ( 2 ) Σ( 2 {hkl} planes Multiplicity Diffracting {hkl} planes h + k2 + l2 h + k2 + l2 1

{100}

2

{110}

3

{111}

4

{200}

5

{210}

6

{211}

7

…

8

{220}

9

{300}

( ( ( ( ( (

{310}

11

{311}

12

{222}

13

{320}

14

{321}

15

…

16

{400}

17

{410}

18

{411}

19

{331}

20

{420}

+ 02

+ 02

12 + 12 + 02 12

+ 12

+ 12

22 + 02 + 02 22

+ 12

+ 02

22 + 12 + 12

) ) ) ) ) )

… ( 2 ) 2 + 22 + 02 ( 2 ) 3 + 02 + 02 ( 2 ) 2 + 22 + 12 ( 2 ) 3 + 12 + 02 ( 2 ) 3 + 12 + 12 ( 2 ) 2 + 22 + 22 ( 2 ) 3 + 22 + 02 ( 2 ) 3 + 22 + 12

{221} 10

12

… ) ( 2 4 + 02 + 02 ( 2 ) 4 + 12 + 02 ) ( 2 4 + 12 + 12 ( 2 ) 3 + 32 + 12 ) ( 2 4 + 22 + 02

sc

bcc

fcc

dc

06

100

–

–

–

12

110

110

–

–

08

111

–

111

111

06

200

200

200

–

24

210

–

–

–

24

211

211

–

–

…

…

…

…

12

220

220

220

220

06

300

–

–

–

24

221

–

–

–

24

310

310

–

–

24

311

–

311

311

08

222

222

222

-

24

320

–

–

–

48

321

321

–

–

…

…

…

…

06

400

400

400

400

24

410

–

–

–

24

411

411

–

–

24

331

–

331

331

24

420

420

420

–

Using the above mentioned simple rules, one can prepare the multiplicity list for various hkl values in different crystal systems. They are provided in Table 5.13.

Table 5.13 Multiplicity table for different crystal systems Crystal system

hkl

hhl

hh0

0kk

hhh

hk0

h0l

0kl

h00

0k0

00l

Cubic

48

24

12

12

08

24

24

24

06

06

06

Tetragonal

16

08

04

08

08

08

08

08

04

04

02

Hexagonal

24

12

06

12

12

12

12

12

06

06

02

Orthorhombic

08

08

08

08

08

04

04

04

02

02

02

Monoclinic

04

04

04

04

04

04

02

04

02

02

02

Triclinic

02

02

02

02

02

02

02

02

02

02

02

5.11 Summary

139

Fig. 5.9 Hexahedron

Family of Planes in Cubic Crystals In orthogonal crystal systems, the choice of labelling the three axes, i.e. x-, y- and z-axes is entirely arbitrary. However, based on symmetry in cubic crystals the (100) plane is physically equivalent to the mathematically distinct (010) and (001) planes. This leads to the grouping of various numbers of planes into different sets, or families. (110)(011)(101) . . . . . . (111)(1 1 1) . . . . . . (210)(120)(201)(102)(021)(012) . . . . . . (321)(123)(231)(132)(213)(312) . . . . . . (100)(010)(001)(00)(00)(00) . . . . . .

3 × 2 × 2 = 12 2×2×2=8 6 × 2 × 2 = 24 6 × 2 × 2 × 2 = 48 3×2=6

The multiplicity 48 represents the most symmetric form in cubic crystal system. Hexahedron form (Fig. 5.9) exhibits this symmetry. The sets of planes are denoted by {hkl}.

5.11 Summary 1. Braggs’ equation has been derived on the basis that a crystal is made of a number of different sets of equidistant parallel planes (of atoms), where the d-spacing of each set of parallel planes is different, and one such set of planes is shown in Fig. 5.1. The general form of the equation is: 2d sin θ = nλ. 2. Order of the Braggs’ equation depends on two variables:

140

5 Importance of d-Spacing in Diffraction of Crystals

(i) In crystals, the d-spacing between different sets of planes are different, such as d1 , d2 , d3 , etc., where d1 > d2 > d3 , … in a crystal. (ii) The Braggs’ angle made by different sets of planes with the incident or the diffracted beam, such as θ1 , θ2 , θ3 , etc., where θ1 < θ2 < θ3 , … in a crystal. 3. Absorbing the term ‘n’ representing the order of Braggs’ equation with d and θ, the modified form of Braggs’ equation in terms of hkl indices can be written as: 2dhkl sin θhkl = λ

4.

5.

6.

7.

where dhkl and θhkl uniquely define a crystal plane, and similarly the corresponding Braggs’ angle, respectively for a given diffraction peak. This formula is the representation of Braggs’ equation in proper order which is compatible with the diffraction results. From the point of view of configuration of atoms in a layer plane, the equidistant parallel planes of atoms (under consideration) may be: (i) Similar or identical, only when they contain identical atoms (as in elemental crystals), and (ii) different, when containing different types of atoms (as in MX or any other polyatomic crystals). The Braggs’ equation (Eq. 5.6), makes it clear that the Braggs’ angle and the dspacing have reciprocal relationships. This is of great utility in reciprocal space where the crystal diffraction happens to take place. Each peak position in a powder pattern is found to correspond a set of equidistant parallel atomic planes with certain (hkl) values with a fixed d-value and fixed angle as predicted by the Braggs’ law. However, it is to be noted that the Braggs’ equation provides a necessary and not sufficient condition for diffraction by real crystals. The Braggs’ reflections are found to occur only when: nλ ≤ 2d, because |sin θ| ≤ 1. Laue equations are represented by a set of three equations, where each equation provides the diffraction effect corresponding to a row of atoms along one of the three crystallographic axes, but all three equations need to be satisfied simultaneously for a diffraction of crystal to occur. Three Laue equations are given by a(cos α − cos α0 ) = eλ b(cos β − cos β0 ) = f λ c(cos γ − cos γ0 ) = gλ where a, b, c are the axial lengths along a-row, b-row and c-row of atoms, respectively. Similarly, e, f, g are integers, one each for a row, α0 , β0 , γ0 are the angles made by the incident plane wave front and α, β, γ are the angles made by the diffracted plane wave front from each row of atoms and λ is the wavelength of the X-ray used.

5.11 Summary

141

8. For a simple cubic system, the Braggs’ equation and the Laue equations provide us the relationships e = nh, f = nk and g = nl

9.

10.

11.

12.

13.

This means that a diffracted beam defined in the Laue’s treatment by the integer e, f, g may be interpreted as the nth order diffraction from a set of (hkl) planes in the Braggs’ treatment. The order of diffraction ‘n’ is simply equal to the largest common factor of the numbers e, f, g. This proves the equivalence of Braggs’ and Laue equations and also suggests that the Braggs’ equation is a consequence of the more general Laue equations. There are two different modes of d-spacing calculations using: (i) Cartesian geometry: valid for orthogonal crystal systems, and (ii) General method: valid for all crystal systems. The results obtained from the calculations of d-spacings’/angles made for sc, bcc and fcc, suggest that they perfectly match with the experimentally observed values obtained from the powder (and others) diffraction patterns of the respective elements or compounds. They also prima facie appear to take care of the conditions for respective systematic absences and others (such as screw axes and glide planes) arising due to the lattice centring as provided in Tables 7.3–7. 5! The exhibition of center of symmetry in diffraction experiments by all crystals (irrespective of centro-symmetric or not) is only due to the spherically symmetric distribution of Braggs’ planes in the Brillouin zone (containing an atom at its center as defined origin) in reciprocal space (Figs. 4.21–4.24). So far, this phenomenon was known to be due to Friedel’s law. This historic result definitely supports the idea of space lattice to be represented by W-S/B-Z as far as the diffraction of crystals is concerned. The spherically symmetric nature of diffraction patterns in reciprocal space allows the mirror, rotational and inversion symmetries while forbids the presence of translational symmetries of any kind (microscopic or macroscopic) in crystals. This is an extremely important result and must be taken into account during crystal structure work. Different Brillouin zones containing different zone boundaries (obtained from the given diffraction pattern) may exhibit same or different number of Braggs’ planes (termed as multiplicity). For example, the maximum multiplicity exhibited by a cubic crystal corresponding to the set of Braggs’ planes {123} with N = 14 is 48.

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5 Importance of d-Spacing in Diffraction of Crystals

References 1. Wahab, M.A.: Solid State Physics: Structure and Properties of Materials, 3rd edn. Narosa Publishing House, New Delhi (2015) 2. Wahab, M.A.: The Mirror: Mother of all Symmetries in Crystals. Advance Science, Engineering and Medicine 12, 289–313 (2020) 3. Hebbar, K.R.: Basics of X-ray Diffraction and its Applications. I. K. International Publishing House Pvt. Ltd., New Delhi, India (2011) 4. Dent Glasser, L.S.: Crystallography and its Applications. Van Nostrand Reinhold Company Limited, New York (1977) 5. Radaelli, P.G.: Symmetry in Crystallography, Understanding the International Tables. IUCr. Oxford Science Publication, Oxford University Press, USA (2011) 6. Clegg, W.: X Ray Crystallography, Oxford University Primers, 2nd edn. Oxford University Press, Oxford (2015) 7. Maxwell, C.R.: Physical Properties and Crystal Structure of Polonium. Iowa State University, Capstones (1946) 8. Koster, G.F.: Space Groups and Their Representations, Academic Press, USA, 1957 9. Cullity, B.D., Stock, S.R.: Elements of X-Ray Diffraction, Prentice Hall, USA (2001) 10. Chatterjee, S. K.: Crystallography and the World of Symmetry, Springer-Verlag Berlin Heidelberg, Germany (2008) 11. Wahab, M.A.: Essentials of Crystallography, 2nd edn. Narosa Publishing House, New Delhi (2014) 12. Fornasini, P.: Basic Crystallography, Slides for the 3rd African School and Workshop on X-rays in Materials. University of Trento, Dakar (2012) 13. Leslie, A.: Crystals Symmetry and Space Groups. MRC Laboratory of Molecular Biology (LMB) Crystallography Course, Cambridge (2013)

Chapter 6

Study of Diffraction Results of Some Cubic Crystals

6.1 Introduction In earlier chapters, we studied about the mirror as the only fundamental symmetry in crystalline solid because all other symmetries can be easily derived from suitable combinations of mirrors. We also studied the geometry of crystal lattices in the primitive forms of Wigner–Seitz cells and Brillouin zones, respectively in direct and reciprocal spaces (they perfectly represent the mirror combination scheme), importance of the intimate relationships that exists between the reciprocal of d-spacing and the Braggs’ angle and its use in understanding the diffraction results. Keeping in view the studies made so far, we plan to make an analytical study of different aspects of diffraction results for the most symmetric cubic crystal system (i.e. sc, bcc and fcc) along with the results of the element DC and some MX-crystals such as CsCl, NaCl, etc., obtained from different diffraction techniques (particularly the powder technique) to understand them better, in this chapter. The conclusive results obtained from the same will be used to understand, analyse and to resolve the existing ambiguities in conventional lattices, particularly those containing lattice centering, screw axes and glide planes (these aspects as such will be taken up in the next chapter).

6.2 A Brief Survey of Diffraction Conditions and Related Aspects Diffraction (of X-rays, electrons, neutrons, etc.) requires two important conditions to be met: (a) Coherent waves (with wavelength λ), and (b) Crystalline array of atoms (or molecules) with interatomic spacing of the order of ~ λ.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. A. Wahab, Mirror Symmetry, Springer Series in Solid-State Sciences 200, https://doi.org/10.1007/978-981-99-8361-2_6

143

144

6 Study of Diffraction Results of Some Cubic Crystals

Here, we provide some important aspects of the survey related to Braggs’ law, interference taking place in the scattered waves (and some other related aspects) and diffraction results in brief. Diffraction of Waves of Different Kinds 1. To satisfy the Braggs’ law for constructive interference, the path difference between any two consecutive waves among the waves scattered by a set of lattice planes must be equal to an integral multiple of the wavelength used. 2. Different kinds of waves could be: (a) electromagnetic waves (light waves, Xrays…), (b) matter waves (electrons, neutrons…) or (c) mechanical waves (sound waves, waves on water surface…). 3. Diffraction is treated as a special case of constructive and destructive interference. In other words, diffraction is also treated as a coherent reinforced scattering (or reinforced scattering of coherent waves). 4. Constructive interference is the result of synchronized diffracted waves that are added together to increase its intensity. On the other hand, the destructive interference is the result of diffracted waves which cancel out each other to decrease its intensity. The processes of constructive and destructive interference are illustrated in Fig. 6.1. When the two waves are perfectly in phase, we have δ = nλ, n = 0, 1, 2, . . . When the two waves are perfectly out of phase, we have Fig. 6.1 Constructive and destructive interference of waves

6.2 A Brief Survey of Diffraction Conditions and Related Aspects

145

δ = nλ, n = 1/2, 3/2, . . . where δ is the phase difference. 5. In diffraction results, the so called systematic absences are an extreme case of destructive interference between the X-ray beams diffracted by individual plane of atoms. In principle, they can arise only when one set of atoms (in one plane) diffracts X-rays that are exactly out-of-phase with those diffracted by a second set of atoms of the same type or different type (in the next plane). Two conditions must be met simultaneously for systematic absences. Alternatively, the two conditions are that, equal number of diffracted beams must be: (i) out-of-phase by λ/2 or π, and (ii) of the same amplitude (determined by form factors, f). In cases, where the destructive interference is not found to be complete and some reduced intensities are observed, then it is believed that one or both of these conditions is not met properly. Diffraction by Crystalline Array 1. The Braggs’ law predicts the angles (or the corresponding d-spacings) at which the peaks will be diffracted, but not the intensities. 2. In a diffraction pattern, the Peak position provides the information about crystal structure or d-spacing in a crystalline phase. A number of other information may also be obtained from peak shape and symmetry. 3. The peak intensity in the diffraction pattern provides the information about the scattering power of the d-spacing; this is in turn related to the arrangement of constituent atoms in the structure and abundance of phases in a mixture. 4. Diffraction intensities are influenced by the atomic number (Z) of the atoms in the structure, by the shape and size of the specimen, and by other factors related to the machine. 5. The peak intensities are used to determine the positions of atoms located in the unit cell. Some Other Important Aspects 1. Because each mineral material is different from one another, either in terms of its chemistry or the geometric pattern of its atomic arrangement in space, therefore each PXRD pattern is a fingerprint. 2. Intuitively, we might expect that a plane of atoms with larger ‘Z’ value will give more intense reflections, because the heavier the atom, the better at scattering of X-rays. 3. The physical basis for the diffraction of neutron beam or electron beam is the same as that for the diffraction of X-rays, the only difference is in the mechanism of scattering.

146

6 Study of Diffraction Results of Some Cubic Crystals

4. Structures with lighter elements (having smaller ‘Z’ values) can be studied using neutron diffraction. Since the neutrons are scattered by the nucleus of an atom, therefore their scattering varies less from element to element. On the other hand, the X-rays are scattered by electron cloud, and therefore the lighter elements barely re-emit them. 5. Electrons are charged particles and interact strongly with all atoms, so an electron beam of low energy, such as a few eV would be completely absorbed by the specimen. Therefore, to make the electron beam to penetrate into the specimen, it is necessary that a beam of very high energy (~50 keV to 1 meV) is used. Similarly, because of the less penetrating power of an electron beam, a thin (~100–1000 nm) specimen is preferred. 6. X-ray scattering from a single molecule is very weak and extremely difficult to detect above the noise level. On the other hand, in a crystal in which, a huge number of atoms or molecules are arranged in an identical orientation, the scattered waves can add up and raise the signal to a measurable level. Therefore, a crystal acts as an amplifier.

6.3 Interpretation of Cubic XRD Data We know that in a cubic crystal system, there can be only four possibilities including diamond lattice, i.e., sc, bcc, fcc and DC, their unit cells are shown in Fig. 6.2. Customarily, one uses a simple method to identify whether the given (unknown) cubic structure belongs to one of these, when the X-ray diffraction data/patterns are available. The simple cubic structure does not really pose any major problem because of its primitive nature. However, even for others, we can easily identify the set of principal diffracting {hkl} planes and the corresponding 2θ values related to four cubic cases, from the obtained diffraction data (particularly the powder diffraction data) as they are vastly studied crystal materials. Further, for cubic structures the Braggs’ law gives us sin2 θ =

) λ2 ( 2 h + k 2 + l2 2 4a

Fig. 6.2 Four well known cubic crystal structures

(6.1)

6.3 Interpretation of Cubic XRD Data

147

where the axial length ‘a’ and the wavelength of the X-ray beam, λ are constants for a given crystal and target material, respectively. λ2 On the other hand, 4a 2 is a constant for a given diffraction pattern. This gives us, ) ( sin2 θ ∝ h2 + k2 + l2 = N

(6.2)

Since, the Miller indices) h, k and l are always integers; it is possible to find out ( the values of h2 + k2 + l2 in the increasing order for different cubic crystals by taking suitable integer values of h, k and l into considerations. Then the ratio of initial two sin2 θ values in each case will give us a number related to that particular cubic structure, i.e. sin 2 θ1 h21 + k21 + l21 = sin 2 θ2 h22 + k22 + l22

(6.3)

where θ1 and θ2 are the two diffracting angles associated with the principal diffracting planes {h1 k1 l1 } and {h2 k2 l2 }, respectively in a particular crystal system. Case I: From Table 5.2, we observe that the first two principal diffracting planes for sc structure are (100) and (110). Substituting the indices of these planes in Eq. 6.3, we obtain sin 2 θ1 12 + 02 + 02 = = 0.5 sin 2 θ2 12 +12 + 02 Thus, if the crystal structure of the unknown cubic metal is sc, the ratio of sin2 θ values corresponding to the initial two diffracting principal planes will be 0.5. However, for an inverse ratio, the numerical value is 2. Case II: Similarly, from Table 5.2 we observe that the first two principal diffracting planes for bcc structure are (110) and (200). Substituting the indices of these planes in Eq. 6.3, we obtain sin 2 θ1 12 + 12 + 02 = = 0.5 sin 2 θ2 22 + 02 + 02 Thus, if the crystal structure of the unknown cubic metal is bcc, the ratio of sin2 θ values corresponding to the initial two diffracting principal planes will be 0.5 (for an inverse ratio, the numerical value is 2). Here, we observe that the ratio of sin2 θ values for sc and bcc are the same, which may become a cause of confusion sometimes, we will discuss this aspect in the next section. Case III: Further, from Table 5.2 we observe that the first two principal diffracting planes for fcc structure are (111) and (200). Substituting the indices of these planes in Eq. 6.3, we obtain

148

6 Study of Diffraction Results of Some Cubic Crystals

sin 2 θ1 12 + 12 + 12 = = 0.75 sin 2 θ2 22 + 02 + 02 This indicates that if the crystal structure of the unknown cubic metal is fcc, the ratio of sin2 θ values corresponding to the initial two diffracting principal planes will be 0.75 (for an inverse ratio, the numerical value is 1.33). Case IV: Finally, from other available sources we observe that the first two principal diffracting planes for a DC structure are (111) and (220). Substituting the indices of these planes in Eq. 6.3, we obtain sin 2 θ1 12 +12 +12 = = 0.375 sin 2 θ2 22 +22 +02 This indicates that if the crystal structure of the unknown cubic metal is DC, the ratio of sin2 θ values corresponding to the initial two diffracting principal planes will be 0.375 (for an inverse ratio, the numerical value is 2.66, double the value of fcc). A general observation of diffraction data suggests that the first XRD peak for a given crystal lattice (also valid for cubic crystal system) is due to diffraction from the planes having lowest Miller indices. It is because of the fact that the corresponding lattice planes are found to represent the closest packed planes with greater atomic density (as also widely spaced set of planes) in the given crystal system. The concept of large d-spacing is well supported by Braggs’ equation. The atomic densities in three low index planes of three cubic crystal structures are shown in Fig. 6.3. According to the atomic density criterion, the set of equivalent planes for the first XRD peak in three cubic cases are: (i) {100} planes in sc, where h2 + k2 + l2 = 1. (ii) {110} planes in bcc, where h2 + k2 + l2 = 2. (iii) {111} planes in fcc, where h2 + k2 + l2 = 3. ) ( Further from Eq. 6.2, we have N = h 2 + k 2 + l 2 where N is the sum of the squares of the Miller indices of the given plane. The permitted value of N for sc, bcc and fcc have been obtained from the calculation and the values are provided in Table 5.2. In a similar manner, one can find for DC. These values can be represented in simple form in a line diagram as shown in Fig. 6.4. They actually represent the peak positions in their respective powder diffraction patterns; below the line diagram the angles are mentioned. From the figure, we observe that in simple cubic and bcc structures, the peaks are approximately equally spaced. In an fcc structure, the peaks appear alternatively in pairs and a single. Finally, in the diamond cubic structure, the peaks are alternatively more widely and less widely spaced.

6.3 Interpretation of Cubic XRD Data

149

Fig. 6.3 Atomic density in three low index planes of three cubic crystal structures

An important conclusion that can be drawn from the above discussion and the study carried out in the preceding chapter that irrespective of a single crystal diffraction or a powder diffraction, the number of diffraction peaks (N) in the XRD spectrum of different cubic crystal system decreases with the increase in number of atoms in the unit cell, progressively from sc to DC. In other words, this can be written in terms of a decrease in the number of diffraction peaks in cubic crystal systems in the order, N = S = Distance on the film or chart

Fig. 6.4 Comparison of peak positions (and values for N) of sc, bcc, fcc and DC

150

6 Study of Diffraction Results of Some Cubic Crystals

Diamond cubic < face centered cubic < body centered cubic < simple cubic. Further, from the above structural discussion, we observe an important similarity in the formation of bcc and DC structures. While a bcc unit cell is understood to

Fig. 6.5 Formation of a bcc from sc and b DC from fcc unit cells, with a diatomic basis

6.4 Indexing Procedure of Cubic XRD Powder Patterns

151

have been formed by two interpenetrating simple cubic unit cells, a DC unit cell can be understood to have been formed by two interpenetrating face centered cubic unit cells. Alternatively, a bcc unit cell can be understood to be a simple cubic with a diatomic basis; similarly, a DC unit cell can be understood to be an fcc unit with a diatomic basis as shown in Fig. 6.5.

6.4 Indexing Procedure of Cubic XRD Powder Patterns The structure of a given crystal can be determined by studying its diffraction pattern obtained by exposing the same to a beam of X-rays, electrons or neutrons. The diffracted beam can take only certain specific directions, along which the Braggs’ conditions are satisfied. Therefore, by measuring the directions of the diffracted beams and their intensities, one can determine the structure of the given crystal. However, in the present section, our primary concern will be to determine the size and shape of the unit cell from XRD peak positions/reflections and keeping the intensity of the peaks for the moment as secondary. Therefore, when we intend to index a diffraction pattern, we need to assign the correct Miller indices to each peak or reflection, because that decides the set of Braggs’ planes involved with each peak (or a set of equivalent reflections) and hence the shape of the resulting Brillouin zone, knowing which the crystal structure can be easily determined. Hence, an XRD pattern is said to be properly indexed when all the peaks in the diffraction pattern are labelled correctly and no any expected peak is supposed to be missing for the particular structure. For an example, a properly indexed diffraction powder pattern is shown in Fig. 6.6. Here, all the peaks are correctly indexed and accounted for, one only needs to assign the correct cubic lattice to it and calculate the lattice parameters. In order to index a given PXRD pattern of a cubic crystal, we follow the steps given below: 1. 2. 3. 4. 5.

Identify the peaks. Note down the diffraction angle, 2θ and then determine sin2 θ. Calculate the ratio sin2 θ/sin2 θmin and multiply by the appropriate integers. Select the result from step (3) that yields h2 + k2 + l2 as an integer. Compare the results, with the sequences of h2 + k2 + l2 values to identify the cubic lattice. 6. Calculate lattice parameters. It is to be noted that the step 4 yields a list of integers that represent various h2 + k2 + l2 values. Different cubic crystal systems (vide sc, bcc, fcc and DC) exhibit different order of increasing h2 + k2 + l2 integer values due to different kinds of centerings in them. The sequence of different values of integers for different cubic crystals is given below. On comparison of the experimentally obtained sequence with them, one can identify the correct cubic lattice.

152

6 Study of Diffraction Results of Some Cubic Crystals

Fig. 6.6 An example of properly indexed powder pattern

SC: BCC: FCC: DC:

1 : 2 : 3 : 4 : 5 : 6 : 8 : 9 : 10 : 11 : 12 : 13 : 14 2 : 4 : 6 : 8 : 10 : 12 : 14 : 16 : 18 : 20 3 : 4 : 8 : 11 : 12 : 16 : 19 : 20 : 24 3 : 8 : 11 : 16 : 19 : 24

Based on the above sequences of h2 + k2 + l2 values, one can easily identify the structure of the diffraction pattern shown in Fig. 6.6, its structure is diamond cubic. Now, comparing the sequences of h2 + k2 + l2 values of sc and bcc, we observe a similarity between the two. When dividing the sequence of bcc by 2, we obtain a new sequence of the type: 1 : 2 : 3 : 4 : 5 : 6 : 7 : 8 : 9 : 10 : 11 : 12 : 13 : 14 This looks similar to the sequence of sc except the presence of the digit 7 in it. If the diffraction pattern happens to contain only initial six peaks, and the ratio of sin2 θ are taken as 1, 2, 3, 4, 5, and 6 for these peaks (reflections), then the space lattice may be identified as a simple cubic or a body centered cubic; it is not possible to distinguish between the two, unambiguously. In other words, the confusion is inevitable. In order to avoid the confusion, some authors have rightly pointed out that there must be at least seven diffraction peaks to ascertain whether the given structure belongs to sc or bcc [1]. It is because the digit 7 can appear only in bcc (in the form of its double, i.e. 14) and not in sc, whereas there is a gap in between the digits 6 and 8 in sc due to absence of 7. However, having said all this, it is also a fact to keep in mind that the first sc plane with indices (100) in Miller index form,

6.5 Analytical Study of Diffraction Results of Some Elemental Cubic Crystals

153

Table 6.1 Extinction rules for cubic crystals Crystals

Allowed reflections

SC

For all values of (h2 + k2 + l2 )

BCC

For even values of (h + k + l)

FCC

When h, k and l are all odd or all even

DC

When h, k and l are all odd or all even, (h + k + l) should be divisible by four

is represented by the digit 1, while the first bcc plane with indices (110) needs to be represented by the digit 2 and not by 1 as pointed out above in Sect. 6.3. Therefore, the sequence of bcc structure must always start from the digit 2 and not 1, and hence keeping this fact into consideration, in principle, there should not be any confusion between the two crystal systems. Based on the experimental observations of diffraction patterns of elements and some other MX cubic crystals, some empirical rules have been formulated (commonly known as extinction rules) for cubic crystal structures. They are provided in Table 6.1.

6.5 Analytical Study of Diffraction Results of Some Elemental Cubic Crystals Keeping in view our main concern of determining the size and shape of the unit cell from XRD peak positions/reflections, we are going to present in this section, the analysis of diffraction results (particularly the powder diffraction, also supported by others wherever necessary) of some important crystals of those cubic systems for which the ambiguities arise due to centering in space lattices (such as bcc, fcc and DC) by taking into account the following fundamental aspects as the qualitative criterions: 1. The fundamental principle developed in Sect. 4.5 that the plane corresponding to least d-spacing vector in reciprocal space (or densest low index planes in direct space, see Sect. 6.3) will diffract first and then others will follow in an increasing order of d-spacing. 2. In a crystal, the set of planes which are parallel, equidistant and identical in composition, diffract (constructively) in phase and give rise a peak of high intensity. 3. In a crystal, the set of planes which are parallel and equidistant but not identical in composition, diffract (destructively) out of phase and give rise a peak of low intensity. Let us consider the case of different cubic crystal lattices separately, one by one.

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6 Study of Diffraction Results of Some Cubic Crystals

Case I: Simple Cubic Crystal As far as the simple cubic crystal structure is concerned, the sequence of diffracting planes obtained in Table 5.2 is found to match very well with the respective observed patterns in metal Po (polonium) and CsCl crystal under MX category (see also Table 6.3). From above discussion, we observe that in a primitive or simple cubic crystal, all integral values of the indices h, k, and l are possible, except 7, 15, 23, etc. (see 5.2) also ( 2 Table ) because no such integral values can satisfy the quadratic form of h + k2 + l2 sequence. Therefore, we can infer that: 1. It is easy to identify a primitive( or simple cubic ) crystal lattice by a simple inspection of the digits present in the h2 + k2 + l2 sequence. However, a proper care is needed while assigning the Miller indices to each reflection to avoid any mix-up with bcc. 2. The cubic unit cell dimension ‘a’ can be obtained from the first peak position of a powder pattern (or the first set of indexed reflections from single crystal patterns). However, in general, an average value obtained from all the peaks is preferred. 3. Since the experimental error is relatively constant, a powder peak/reflections (from others) with largest Braggs’ angle may be chosen to minimize the error in the lattice parameter calculation, or alternatively perform the least squares on all powder peaks or single crystal reflections. 4. In a simple cubic crystal, all integral values of the indices h, k, and l (except 7, 15, 23, etc.) are allowed peaks/reflections. They exhibit a number of different family of planes as provided in Table 6.2. For example, three simple cubic unit cells shown in Fig. 6.7, exhibit 100, 200, 300 families of (hkl) planes. In these illustrations, we observe that only the extreme planes contain atoms while the intermediate planes (in second and third) lying within the unit cells have no atoms. This suggests that when we speak of a 200 family of planes or a 300 family of planes, etc., the 100 planes become an integral part of 200, 300, etc. families of (hkl) planes and contribute to the allowed reflections for them. Since, only the 100 planes actually contribute to the diffracted intensity in such cases (these planes come under criterion 3, which states that equidistant and parallel but not identical planes), the overall intensity is expected to go down progressively (comparatively fast) with increasing order of the Braggs plane/Braggs’ angle. The resulting effect is, I100 > I200 > I300 , . . . or Iθ (100) > Iθ (200) > Iθ (300) , . . . Similar conditions will apply to other families of planes of sc and primitive unit cells of other crystal systems. 5. The intensity pattern observed in α-Polonium (Tables 5.11 and 6.3), the only element in the periodic table to exhibit simple cubic structure, shows a systematic decrease with the decrease of planar atomic density.

6.5 Analytical Study of Diffraction Results of Some Elemental Cubic Crystals

155

Fig. 6.7 (100) and imaginary (200), (300) planes in simple cubic unit cells

Case II: Body Centered Cubic Crystal As far as the bcc structure is concerned, the sequence of diffracting planes obtained in Table 5.2 matches very well with the respective patterns observed in metal (viz. molybdenum, iron, etc.) and CsBr MX crystals. The observed pattern of (hkl) planes (in Table 5.2) for the body-centered cubic is slightly different from that of the simple cubic case. One fundamental difference between the two cubic cases is related to the series of (n00) reflections. As observed above, in a simple cubic case all permissible values of n (as per Braggs’ condition) contribute to diffraction peaks, while in bcc it is not so. To understand this issue clearly, let us first check the reasons cited in literature and then try to analyse the case, respectively in view of the Fig. 5.4d for 200 plane and Fig. 5.4f for 222 plane, Table 6.2 Family of planes in cubic crystal system S. No.

General form (hkl)

Miller indices of family of planes

( 2 ) h + k2 + l2 value

1

h00

100, 200, 300, 400, 500, 600, …

1, 4, 9, 16, 25, 36, …

2

hk0

110, 220, 330, 440, …

2, 8, 18, 32, …

3

hkl

111, 222, 333, 444, …

3, 12, 27, 48, …

4

2hk0

210, 420, 630, …

5, 20, 45, …

5

2hkl

211, 422, 633, …

6, 24, 54, …

6

3hk0

310, 620, …

10, 40, …

7

3hkl

311, 622, …

11, 44, …

8

3h2k0

320, 640, …

13, 52, …

9

3h2kl

321, 642, …

14, 56, …

10

4hk0

410, 820, …

17, 68, …

11

3h3kl

331, 662, …

19, 76, …

12

4h2kl

421, 842, …

21, 84, …

13

3h3k2l

332, 664, …

22, 88, …

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6 Study of Diffraction Results of Some Cubic Crystals

Table 6.3 PXRD data of CsCl along with Po α-Polonium, a = 3.345 Å

CsCl crystal, a = 4.126 Å, λ = 1.5045 Å

Intensity, I

Intensity, I

d (Å)

2θ°

(100)

100

39

4.126

21.50

(110)

91

100

2.918

30.70

(111)

38

12

2.382

37.80

(200)

21

13

2.063

43.90

(210)

62

12

1.845

49.40

(211)

50

22

1.684

54.50

(220)

18

05

1.459

63.80

(221)

42

05

1.375

68.20

(310)

32

06

1.305

72.40

(311)

32

02

1.244

76.50

(222)

11

01

1.191

80.60

(hkl) plane

so that we can generalize the same for other such cases. The reason cited in literature about the absence of 100 reflection in bcc are: 1. That in the body centred cubic lattice, the (100) planes are interleaved by an equivalent set of planes at the halfway position, the angle/position where the Braggs’ condition would give the second order 100 reflection or the first order 200 reflection. These interleaved planes will give a reflection exactly out of phase with that from the primary planes, which will exactly cancel the signal. Therefore, there is no signal from (n00) planes with odd values of n. 2. That if the wave scattered from the middle plane (Fig. 6.8a) of bcc Molybdenum is out of phase with the ones scattered from the top and bottom planes, i.e. if the top and bottom rays (1 and 3) are in phase (path difference of λ) then the ray (2) will be exactly out of phase with the rays 1 and 3 (path difference of λ/2). Further, the scattering power of each plane is identical, as they contain same number of atoms (one atom per unit area, in this case) [1, 2].

Fig. 6.8 BCC molybdenum

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157

Fig. 6.9 a bcc α-Fe; b (100) planes; c (200) planes

3. Elsewhere, the same thing is given in slightly different form. Here it simply says that the beam cancels because the body center atoms scatter exactly 180° out of phase when the plane is lying in between 100 planes as shown in Fig. 6.8b, while another situation expresses that strong reflection occurs because all atoms lie on 200 planes which scatter in phase (Fig. 6.8c). 4. Consider the case of α-Fe which has a bcc lattice shown in Fig. 6.9a. Like S.N.1, it says that the reflection from the (100) planes in (b) has zero intensity and is systematically absent. This is because, at the Braggs’ angle for these planes, the body center atoms which lie midway between the two adjacent (100) planes diffract X-rays exactly 180° out-of-phase relative to the corner atoms which lie on the (100) planes. Averaged over the whole crystal, there are equal numbers of corner and body centre atoms and the beams diffracted by each cancel completely [3, 4]. From the above stated reasons, two pertinent questions arise. (i) When the interleaved planes give a reflection exactly out of phase with that from the primary planes, and exactly cancels the signal, then why the signal from (n00) planes with only odd values of n should disappear (instead, even and odd all should disappear). (ii) It is the use of an arbitrary word, ‘if’ the top and bottom rays are in phase (path difference of λ) then the middle ray will be exactly out of phase with the 1 and 3 rays (path difference of λ/2) and exactly cancels the signal. Now, let us look at the problem by taking into account the Fig. 5.4d for (200) plane, the Fig. 5.4f for (222) plane, and the above mentioned fundamental aspects together. According to Fig. 5.4d, as soon as an identical atom is placed at the body center of a simple cubic unit cell to make it a bcc structure, the (100) planes lose their independent identity because they automatically become the member of (200) family of planes. This supports the criteria mentioned in Sect. 6.3 that ‘the first XRD peak should be due to diffraction from the closest packed planes in the given crystal lattice, and (100) plane is definitely not the closest packed plane for a bcc crystal structure. Actually, the planes (110) are found to represent the closest packed in bcc (as they are also widely spaced) and hence appear as first diffracting planes.

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6 Study of Diffraction Results of Some Cubic Crystals

Fig. 6.10 Powder diffraction pattern of bcc α-Fe

Therefore, the actual reason of the absence of (100) plane from bcc pattern is not due the cancellation by equal and opposite phases of waves but because of its merging with the (200) family as shown in Figs. 6.8c and 6.9c, where all the three planes in the unit cell are parallel, identical in composition and hence they diffract in phase to give rise a (200) peak as evident from Fig. 6.10 for bcc iron. A similar argument will explain the absence of (111) plane from bcc pattern, as it becomes a part of (222) family of planes. In a similar manner, other low index planes, such as (210), (221), (311), etc. will merge with their respective higher order family of planes and hence are found to be missing from bcc patterns. This particular feature, in a way, appears as ‘crystallographic illusion’. Case III: Face Centered Cubic Crystal In this case also, let us first check the reasons cited in literature and then analyse the case in view of the Fig. 5.4g for (200) plane and Fig. 5.4h for (220) plane, respectively and the above mentioned fundamental aspects. 1. The reflection from (200) plane is exactly out of phase with (100) reflection, as a result, a destructive interference occurs and no (100) reflection is observed. 2. Furthermore, in order for a reflection to be observed for fcc, the hkl indices must be all odd or all even [1]. Again, let us take into account the Fig. 5.4g for (200) plane, the Fig. 5.4h for (220) plane, and the above mentioned fundamental aspects together. Since, the nature of the first problem is identical to that of bcc, therefore, as per the Fig. 5.4g, h, the answer is also identical as mentioned above in case of bcc. On the other hand, the second reason stated above is nothing but a part of the so called extinction rules, whose basis itself needs a review. In fact, the ambiguity associated with this is to be removed. As far as the fcc structure is concerned, the sequence of diffracting Braggs’ planes obtained in Table 5.2 matches very well with the respective patterns observed in metals (viz. copper, nickel, aluminium, etc.) and MX crystals (viz. CsBr, NaCl, etc.).

6.5 Analytical Study of Diffraction Results of Some Elemental Cubic Crystals

159

Fig. 6.11 Powder diffraction pattern of Cu

Here, let us consider the case of simulated powder diffraction pattern of copper shown in Fig. 6.11 as a representative case for elemental crystalline solid. In Fig. 6.3, we observed that (111) plane is the most widely spaced, so it is the densest plane among the fcc planes and hence expected to give an intense first peak in the diffraction pattern. From the experimentally observed intensity pattern of Cu (Fig. 6.11) it is evident, this turns out to be correct [5]. Rest of the peaks in the pattern are also on the expected line. Now, let us consider the compositional aspect of fcc unit cell. Here, while introducing the face centered atoms in a simple cubic unit cell, a new plane is created exactly halfway between the top and bottom planes. Similarly, both the top and bottom planes acquire one additional atom, each at its center, as shown in Fig. 6.12a. It is also important to note that due to face centering, all the three parallel planes show identical atomic density, it is 2 atoms per unit area. Therefore, like bcc, the (100) planes of fcc also lose their independent identity and automatically become the member of (200) family of planes according to Fig. 5.4g, as also shown in Fig. 6.12b. The face centering action also introduces a new plane exactly halfway between the two (110) planes (not shown here). As a result of this, they lose their identity and become the member of 220 family of planes as shown in Fig. 5.4h. All (220) planes are parallel, equidistant and have identical atomic density (satisfying the above mentioned criteria), and hence will diffract in phase. The set of (111) planes remain unaffected from the centering action (except that they have become the most dense planes) as shown in Figs. 5.4i and 6.12c, and hence give rise their independent reflections. Because this plane corresponds to the smallest reciprocal lattice vector, it provides the first diffraction peak in the pattern (Fig. 6.11). On a similar ground as in bcc and fcc, based on Figs. 5.4 and 6.12, the planes (100) and (110) become the members of their respective next higher order planes, i.e. the planes (200) and (220) in DC. Accordingly, the first DC peak is from the closest packed low index (111) planes, and the next peak is from the (220) planes, respectively. Further, as a result of the introduction of atoms at (¼, ¼, ¼) and other equivalent positions in the unit cell, a new set of planes (400) appear, which are parallel to both (200) and (100) planes and are located exactly halfway between the

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6 Study of Diffraction Results of Some Cubic Crystals

Fig. 6.12 a (100) planes, b (200) planes, c (111) planes

Fig. 6.13 a (100), (200), (400) planes, b (111) planes, c DC structure

two (200) planes and one quarter and three quarter between two (100) planes. The next two reflections in the diffraction pattern will come from the set of planes (311) and (400), respectively. The (001) plane projection DC unit cell is shown in Fig. 6.14. Finally, it is necessary to mention that the above discussed analytical results will be applicable to other crystal systems also.

6.6 Analytical Study of Diffraction Results of Some MX-Cubic Crystals Keeping in view the above mentioned fundamental principles, we intend to discuss about the results obtained for some important members of MX cubic crystals such as CsCl, CsBr, NaCl, etc. in this section to see the behaviour of their peak patterns (and only qualitatively about their intensity) to draw some suitable conclusions based on them to understand the subject better. In these cases, the elemental composition of two atoms in a basis can vary, but their arrangement and orientation remain the same. A simple example of two-element

6.6 Analytical Study of Diffraction Results of Some MX-Cubic Crystals

161

Fig. 6.14 (001) plane projection DC unit cell

lattice is CsCl. It is treated as a simple cubic lattice with a basis that has one Cs+ ion at 000 (the origin) and one Cl− ion at (1/2)a (1/2)b (1/2)c (the center). Another example of a different composition but the same arrangement and orientation is NaCl. It can be treated as having alternate Na+ and Cl− ions at the lattice points of a simple cubic structure. In this case, for each ion of one type there are six nearest neighbors of oppositely charged ions. However, a different view is that NaCl can be thought of as the interweaving of an FCC lattice of Na+ ions and Cl− ions, respectively. In such a situation, the basis consists of one Cl− ion at 000 (the origin) and one Na+ at (1/2)a (1/2)b and (1/2)c (the center). Case I: CsCl Crystal Structure CsCl is an important MX material which crystallizes in a simple cubic structure. As per Dent Glasser [4], its axial parameter is found to be, a = 4.126 Å. Looking at its single unit cell, it appears to have a bcc structure. However, if we represent it in the form of two interpenetrating unit cells as shown in Fig. 6.15, the clarity about its simple cubic nature is better. The unit cell contains one formula unit of CsCl, with the eight corner chlorine ions each being shared by eight unit cells. In ionic structures like this, individual molecule is indistinguishable because each ion is surrounded by the ions of opposite charge. Since, cesium chloride consists of equal numbers of cesium and chlorine ions, placed at the points of a bcc lattice so that each ion has eight nearest neighbours of the other kind. Ratio of their ionic radii are given by rCs + 1.70 = 0.939 = rCl − 1.81 Since, the resulting radius ratio lies between 0.732 < 0.939 < 1.0, therefore, cesium chloride prefers the configuration of a simple cubic structure.

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6 Study of Diffraction Results of Some Cubic Crystals

Fig. 6.15 CsCl structure. a A basis containing two atoms, b two interpenetrating unit cells

Accordingly, in principle its powder pattern is expected to contain all the planes with integral values of h, k, and l indices which are possible for simple cubic crystal, the PXRD data of CsCl [4] in fact confirm this (Table 6.3). Structure Factor (F) 1. It is the resultant wave scattered by all atoms of the unit cell. 2. It is independent of the shape and size of the unit cell; but is dependent on the position of the atoms/ions etc. within the cell. Fhkl = f Cs + f Cl = an even integer = f Cs − f Cl = an odd integer where, the intensity of a reflection is I ∝ |Fhkl |2 Therefore, for: Even indices (200), (400), (220), etc. ⇒ |Fhkl |2 = ( f Cs + f Cl )2 . Odd indices (100), (111), (300), etc. ⇒ |Fhkl |2 = ( f Cs − f Cl )2 . Case II: CsBr Crystal Structure CsBr is another important MX material which crystallises in a body centered cubic structure, where the atomic masses of Cs and Br are 133 and 80 amu, respectively.

6.6 Analytical Study of Diffraction Results of Some MX-Cubic Crystals

163

Fig. 6.16 CsBr crystal structure with two different ions per unit cell

Its axial parameter is found to be equal to 4.366 Å. Like CsCl, CsBr can also be represented in the form of two interpenetrating unit cells, where Cs+ ion is at the body centre and Br– ions at the corners as shown in Fig. 6.16. The corresponding ionic radii are 1.88 and 1.82 Å, respectively. Here, Cs+ ion is at the body centre and Br– ions are at the corners, therefore the relationship is √ 3a = 2(r+ + r− ) = 2(1.88 + 1.82) = 7.4 Å or 7.4 a = √ = 4.27 Å 3 Therefore, the packing fraction is given by )] ] [( 3 [ 4 r+ + r−3 × 3.14 (1.88)3 + (1.82)3 3 = P.F = a3 (4.27)3 4.186[6.645 + 6.038] = = 0.6815 = 68% 77.854 4 π 3

This corresponds to the efficiency of a bcc structure. The powder XRD pattern also provides the same result.

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6 Study of Diffraction Results of Some Cubic Crystals

Case III: NaCl Crystal Structure NaCl is well-known to crystallize as an fcc structure and hence should in principle follow its rule, as far as the peak positions in the diffraction patterns are concerned. Accordingly, the planes (100) and (110), respectively become the members of (200) and (220) family of planes. However, the NaCl structure can also be considered to be made of two interlocking CCP arrays of Na+ and Cl− ion layers, where the close packed layers of Cl− ions lie parallel to the body diagonal layers of (111) planes in the unit cell and located exactly halfway between the two close packed layers of Na+ ions as shown in Fig. 6.17a. On the other hand, Na+ and Cl− ion layers are shown in Fig. 6.17b as if they form a set of (222) planes, such that the d-spacing between the two different types of ion layers remains identical, i.e. ) ) ( ( d111 o f N a + − N a + ion layer s = d111 o f Cl − − Cl − ion layer s = 3.25 Å ∗ Since both ion layers have the same d-spacing, they will have same d⭢111 value and therefore they will diffract at the same position. However, because of compositional difference between the two sets of layers, the reflection from a Cl− close packed layer will be exactly out of phase with the close packed Na+ layer [diffraction of (111) Cl− layer planes in NaCl crystal is shown in Fig. 6.18]. Since a Cl atom has 18 electrons, it scatters X-rays more than a Na atom with 10 electrons, the scattered waves will partially cancel each other and hence the peak intensity corresponding to (111) planes is expected to be weak, which is very clear from Fig. 6.19. However, when the two close packed layers are treated identical as found in metal crystals such as Cu, Co, etc., they diffract in phase and the peak intensity becomes high as we saw earlier for copper in Fig. 6.11. The (200) and (220) planes look similar in composition (Fig. 6.20), because they contain the close packed layers of both Na+ and Cl− ions together. Since, in each case the set of planes is parallel, equidistant and identical in composition, the X-rays

Fig. 6.17 (111) layer planes of a Cl− (Na+ planes lying in between), b both Cl− and Na+

6.6 Analytical Study of Diffraction Results of Some MX-Cubic Crystals

165

Fig. 6.18 Diffraction of (111) Cl− layer planes in NaCl crystal Fig. 6.19 NaCl powder intensity pattern

will reinforce each other and diffract in phase as per the above mentioned criteria [diffracting planes of both (200) and (220) in NaCl crystal are also shown in Fig. 6.21], resulting in high intensity peaks as evident from Fig. 6.19. Unlike (200) and (220) planes, the (222) planes contain the separate close-packed layers of Na+ and Cl− ions such as in Fig. 6.17b, one each alternately and a set of empty parallel layers between them, the scattered waves will partially cancel each other and hence the resulting peak intensity from the (222) planes is expected to be weak as compared to (200) and (220) planes, as evident from Fig. 6.19. Structure Factor and Intensity

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6 Study of Diffraction Results of Some Cubic Crystals

Fig. 6.20 a The (200) and b the (220) planes of atoms in NaCl

Fig. 6.21 Diffraction of a (200) layer planes, b (220) layer planes in NaCl crystal

Fhkl = 4( f N a + f Cl ) = an even integer = 4( f N a − f Cl ) = an odd integer where, the intensity of a reflection is I ∝ |Fhkl |2 Therefore, when hkl are all even, the reflections are called primary reflections (200), (220), (222), (420), etc. ⇒ |Fhkl |2 = 16( f N a + f Cl )2 When hkl are all odd, the reflections are called super lattice reflections. (111), (311), (331), (333), etc. ⇒ |Fhkl |2 = 16( f N a − f Cl )2 When hkl are mixed, there will be no reflections.

6.7 Summary

167

|Fhkl |2 = 0

6.7 Summary 1. Diffraction (of X-rays, electrons, neutrons, etc.) requires two important conditions to be met: Coherent waves (with wavelength λ), and (b) Crystalline array of atoms (or molecules) with interatomic spacing of the order of ~ λ. 2. Different kinds of waves could be: (a) electromagnetic waves (light waves, X-rays…), (b) matter waves (electrons, neutrons…) or (c) mechanical waves (sound waves, waves on water surface…). 3. Diffraction is treated as a special case of constructive and destructive interference, where the constructive interference is the result of synchronized diffracted waves that add together to increase its intensity and the destructive interference is the result of diffracted waves which cancel each other out to decrease its intensity. The processes of constructive and destructive interference are illustrated in Fig. 6.1. 4. The Braggs’ law predicts the angle and d-spacing at which the peaks will be diffracted, but not their intensities. In a diffraction pattern, the Peak position provides the information about crystal structure or d-spacing and some other information in a crystalline phase. 5. Diffraction intensities are influenced by the atomic number (Z) of the atoms in the structure, shape and size of the specimen, and other factors related to the machine. The peak intensities are used to determine the positions of atoms located in the unit cell. 6. Because each mineral material is different from one another (either in terms of its chemistry or the geometric pattern of its atomic arrangement), therefore each PXRD pattern is a fingerprint. Intuitively, we might expect that a plane of atoms with larger ‘Z’ value will give a more intense reflection, because the heavier the atom, the better at scattering of X-rays. The physical basis for the diffraction of neutron beam and electron beam is the same as that for the diffraction of X-rays, the only difference is in the mechanism of scattering. 7. Structures with lighter elements (having smaller ‘Z’ values) can be studied using neutron diffraction. Since the neutrons are scattered by the nucleus of an atom, therefore their scattering vary less from element to element, whereas X-rays are scattered by the electron cloud, and the lighter elements barely re-emit them. 8. Electrons are charged particles and interact strongly with all atoms, so an electron beam of low energy, such as a few eV would be completely absorbed by the specimen. Therefore, to make the electron beam to penetrate into the specimen, it is necessary that a beam of very high energy (~50 keV–1 meV) is used. Similarly, because of the less penetrating power of an electron beam, a thin (~100–1000 nm) specimen is preferred.

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6 Study of Diffraction Results of Some Cubic Crystals

9. X-ray scattering from a single molecule is very weak and extremely difficult to detect above the noise level. On the other hand, in a crystal in which, a huge number of molecules is arranged in an identical orientation, the scattered waves can add up and raise the signal to a measurable level. Therefore, a crystal acts as an amplifier. 10. In direct space lattice, we come across the term infinite lattice points invariably. However, the diffraction of crystals always provide us only a limited number of reciprocal lattice points (reflections) and diffraction peaks. This raises an automatic question on associating the term ‘infinite lattice’ with crystals. 11. We know that all kinds of diffraction work are in reciprocal space and the diffraction patterns providing different sets of Braggs’ planes actually represent the zone boundaries of different Brillouin zones. 12. According to the atomic density criterion, the most close packed planes in simple cubic crystal is the set of equivalent planes {100}, where h2 + k2 + l2 = 1. Similarly, in body-centered cubic crystal it is {110}, where h2 +k2 +l2 = 2; and in face-centered cubic crystal it is {111}, where h2 + k2 + l2 = 3, respectively will represent the first XRD peak in each case. 13. From the study carried out for four cubic crystal systems, we can conclude that irrespective of a single crystal diffraction or a powder diffraction, the number of diffraction peaks (N) in the XRD spectrum of different cubic crystal systems decreases with the increase in number of atoms in the unit cell, progressively from sc to DC. In other words, this can be written in terms of a decrease in the number of diffraction peaks in cubic crystal systems in the order, Diamond cubic < face centered cubic < body centered cubic < simple cubic

References 1. Cullity, B.D., Stock, S.R.: Elements of X-Ray Diffraction. Prentice Hall, USA (2001) 2. West, A.R.: Solid State Chemistry and its Application, 2nd edn. John Wiley and Sons Ltd, New York (2014) 3. Dent Glasser, L.S.: Crystallography and Its Application. Van Nostrand Reinhold Company Limited, New York (1977) 4. De Graef, M., McHenry, M.E.: Structure of Materials: An Introduction to Crystallography, Diffraction, and Symmetry. Cambridge University Press, Cambridge (2007) 5. Pecharsky, V.K., Zavalij, P.Y.: Fundamental of Powder Diffraction and Structural Characterization of Materials. Springer, Berlin (2005)

Chapter 7

Possibility of Translational Symmetry (if Any) in Crystals

7.1 Introduction From Chaps. 3, 4, 5, 6, we have come across many evidences which clearly indicate that due to spherically symmetric nature of the Wigner–Seitz cells, the Brillouin zones (both of them are ideal representatives of mirror combination scheme) and the diffraction patterns giving rise to various sets of Braggs’ planes in the form of zone boundaries of Brillouin zones, forbid the presence of translational symmetry of any kind (either microscopic or macroscopic) in crystals, because they all possess defined origin. In fact, such spherically symmetric systems allow only the symmetries like rotation, mirror, inversion, and their compatible combinations. However, for the sake of argument and to the satisfaction of readers, in this chapter we intend to explore and analyse the possibility (if any) of the existence of translational symmetries (particularly the microscopic symmetries such as screw axes and glide planes) in crystals and whether there exists any experimental evidence and/or theoretical justification in their favour. In the process, we shall also discuss the role of the lattice centring (if any) in the creation of screw axes, glide planes and the so called ‘systematic absences’ in cubic crystal system. The results obtained from the same can be generalized to understand the situation in other crystal systems.

7.2 A Brief Historical Background of Crystallographic Developments In order to know the actual reasons behind the inclusion of microscopic symmetries, such as screw axes and glide planes to create a new class of symmetries, the space group symmetries and presume them to represent the special arrangements of various crystal structures in three dimensional system, we must look back into the historical

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. A. Wahab, Mirror Symmetry, Springer Series in Solid-State Sciences 200, https://doi.org/10.1007/978-981-99-8361-2_7

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7 Possibility of Translational Symmetry (if Any) in Crystals

background and see the chronological developments of crystallography as a subject that took place especially during Bravais, Sohncke and Fedorov era. Literature survey provides us enough materials related to the chronological development of crystallography as a subject, prominent among them are provided by [1–3]. However, the developments that took place during those days as mentioned above were largely mathematical and group theoretical in nature, and practically there is hardly any evidence left with us about actual illustrations of atomic/molecular arrangements or any other experimental observations used in the derivation of space groups applicable to crystals. Taking this fact into account, here we provide a brief description of the developments, which actually led to the inclusion of microscopic symmetries (such as the screw axes and glide planes) in support of the assumption of associating infinite symmetry system to crystals. The group theoretical development of the space lattice was initiated by Hessel (1830) and Gadolin (1867) with the derivation of finite crystal classes (also known as the point groups). They deduced the 32 crystal classes before many of them were illustrated by minerals; of these 32 point groups, Gadolin found only 20 examples in nature. The laws governing the finite point group symmetries were then extended to cover the symmetries of infinite crystal lattices (so far this has gone unchallenged except an extremely mild objection in some standard texts about associating the term ‘infinite’ with crystals). Based on this, Frankenheim (1801–1869), Bravias (1811– 1863), and Sohncke (1842–1898) introduced the infinite symmetries of lattices to explain the structural arrangements in crystals. In the middle of nineteenth century Frankenheim and Bravais further developed the concept of the crystal lattice and enumerated 14 ‘frameworks’ in the form of space lattices, which form the basis of the modern structural crystallography. Further, literature survey [4, 5] suggests that in order to investigate some results obtained by pure mathematicians, Leonhard Sohncke first introduced the concepts of screw axes and glide planes in crystallography in 1879 by modifying the Bravais concept of space lattice from ‘identical environment and similar orientation of lattice points’ to ‘identical environment but without necessarily similar orientation’. He then proposed 65 structural arrangements (and called them the space groups) obtained by reducing the number from the work of Camille Jordan (1869), who originally derived 174 possible point groups where he considered both continuous and discrete groups of ‘proper’ motions of sets of points while exploring the possible combinations of rotations, screw rotations and translations, however, without discussing the applications of his work to crystallography. Inspired from the Sohncke’s proposal (without assessing the consequences of associating infinite symmetries to crystals), both Fedorov and Schoenflies (1853–1928) extended their works independently by taking into account the improper rotations (i.e. rotoreflection and rotoinversion) also and theoretically derived a total of 230 possible structural arrangements in three dimensional space, they called them the space groups. They essentially tell us that every crystal in nature or that we grow in our laboratory, and every crystalline material that we fabricate, must have one of these 230 space symmetries. William Barlow (1845–1934) also derived the same number of space groups, but he delayed publishing his list (1894). Using a method based on Sohncke’s, but

7.3 Translational Symmetries/Translational Periodicity in Crystals

171

instead of systems of points he considered geometric patterns of orientated motifs. To pursue his investigations, Barlow used pairs of gloves to model his results. However, Barlow’s contribution to the theory of crystal structure outweighed his derivation of the space groups. In this connection, one important fact which cannot be denied is that the majority of the space groups (out of 230) yet to find a crystal structure even after a long time (nearing 200 years).

7.3 Translational Symmetries/Translational Periodicity in Crystals In the present day crystallography, the translational symmetry is considered to be the most fundamental symmetry and all other symmetries such as reflection, rotation, inversion and their compatible combinations are supposed to comply with the translation for their validity. However, the recent study carried out related to symmetry, provides us enough evidence to suggest that mirror symmetry is the only fundamental symmetry in crystals and all other symmetries such as rotation, inversion, rotoreflection, rotoinversion and translational periodicity can be easily derived from the same [6]. This can be proved on the basis of the evidences obtained from different experiments discussed in Chaps. 3 and 4. To understand them better, let us consider the following examples: 1. Using the fundamental principle of parallel mirrors with the object lattice (or an atom) in between them, infinite images of the same could be obtained in the form of a one-dimensional lattice (or crystal). This simple experiment suggests that actual translation of the lattice point or the atom is, neither required nor possible to get the (infinite) lattice points (or crystal), rather it is the property of the combination of parallel mirrors and the lattice point (or the atom) between them. 2. Using the same basic principle as above, the formation of 2-D and 3-D lattices (or crystal structure) can be understood in terms of their Wigner–Seitz unit cells and Brillouin zones, which possess several pairs of parallel lines/planes (in the form of mirrors) in 2-D/3-D, respectively with an atom at their center. Different pairs of parallel mirror lines/planes of the given Wigner–Seitz unit cell/Brillouin zone in turn produce the desired infinite 2-D/3-D direct/reciprocal lattices exhibiting perfect translational periodicity. 3. The intimate connection between the Braggs’ planes and the zone boundaries discussed in Chaps. 4 and 5 strongly suggests that the diffractions of X-rays, electrons or neutrons recognize only the W–S/B–Z cells and not the conventional unit cells of the space lattice for diffraction purpose. The diffraction geometry first of all selects an atom from the closest packed plane of the given crystal structure to be at the center (acting as defined origin) and then completes the diffraction

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7 Possibility of Translational Symmetry (if Any) in Crystals

process by selecting the first nearest neighbor planes of atoms of smallest reciprocal lattice vector to construct the zone boundaries of the first Brillouin zone around the central atom on satisfying the Braggs’ condition. The diffraction geometry then selects the next nearest neighbor planes of atoms with increasing reciprocal lattice vectors (or next less close packed planes with less interplanar spacing in direct lattice) one by one, each time and completes the construction of higher zones (as per the Braggs’ condition), in spherically symmetric manner. This is how the diffraction seems to be taking place in crystals. 4. The distribution of Braggs’ planes in different Brillouin zones (as exhibited in a diffraction pattern) is spherically symmetric around the central atom acting as defined origin. The spherically symmetric nature of diffraction patterns in reciprocal space allows the mirror, rotational and inversion symmetries while forbids the presence of translational symmetry of any kind (either macroscopic or microscopic) in crystals [7]. This is an extremely important result and must be taken into account during the crystal structure work. The spherically symmetric distribution of Braggs’ planes (indexed one quadrant only) for the first few low order Brillouin zones obtained from 2-D square lattice is shown in Fig. 5.8. Similar distributions can be shown in case of other 2-D and 3-D lattices.

7.4 Earlier Proposed Systematic Absences in Crystals The following three different types of systematic absences have been proposed earlier in crystalline solids. They are: 1. Systematic absences in crystals due to lattice centering (Table 7.1) 2. Systematic absences in crystals due to screw axes (Table 7.2) 3. Systematic absences in crystals due to glide planes (Table 7.3) Table 7.1 Proposed systematic absences in crystals due to lattice centering Lattice type

Symbol

Set of reflections

Conditions for possible reflections

Primitive

P

(hkl)

None

Centered on A face

A

k + l = 2n

Centered on B face

B

h + l = 2n

Centered on C face

C

h + k = 2n

Centered on all faces

F

h, k, l all odd or all even

Body centered

I

h + k + l = 2n

Rhombohedral, obverse

R

−h + k + l = 3n

Rhombohedral, reverse

R

h − k + l = 3n

7.5 Role of Lattice Centering (if Any) in Systematic Absences

173

Table 7.2 Proposed systematic absences in crystals due to screw axes Screw axes Type

Orientation

Set of reflections (hkl)

Condition for possible reflection

Translation*

Non-hexagonal crystals 21 , 42

[100]

a/2

h00

h = 2n

21 , 42

[010]

b/2

0k0

k = 2n

21 , 42

[001]

c/2

00l

l = 2n

21

[110]

a/2 + b/2

hh0

h = 2n

41 , 43

[100]

a/4

h00

h = 4n

41 , 43

[010]

b/4

0k0

k = 4n

41 , 43

[001]

c/4

00l

l = 4n

Hexagonal crystals 63

[0001]

c/2

000l

l = 2n

31 , 32 , 62 , 64

[0001]

c/3

000l

l = 3n

61 , 65

[0001]

c/6

000l

l = 6n

*

‘o’ subscripts omitted

7.5 Role of Lattice Centering (if Any) in Systematic Absences From literature, we know that certain sets of Braggs’ planes are not exhibited by the diffraction patterns in centered lattices. They can be determined by the socalled extinction rules. For three cubic cases (i.e. primitive, body centered and face centered), they are like: P: No extinction (absence), I: h + k + l /= 2n are extinct (absent), and, F: h, k, l mixed with respect to evenness or oddness are extinct (absent). Now, in order to understand this aspect, let us consider the case of either bcc or fcc of cubic crystal system. There are actually two ways of looking at this problem: Case I: First Option If a bcc crystal lattice is considered in its Wigner–Seitz form (also a primitive lattice, which exhibits the complete symmetry of the original lattice) as considered by Ashcroft and Mermin (Indian edition) [7], then the Braggs’ reflections will be expected to occur when the change in wave vector k⭢ is a vector of its reciprocal lattice, i.e. fcc crystal lattice form (the Brillouin zone of bcc, also a primitive), of side 4 π /a. This matches with the result obtained from interplanar spacing calculations provided in Table 5.2. This suggests that the direct lattice must be considered in primitive or Wigner–Seitz form to avoid the ‘extinction’ issue in crystals completely. Similarly, for fcc lattice.

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7 Possibility of Translational Symmetry (if Any) in Crystals

Table 7.3 Proposed systematic absences in crystals due to glide planes Set of reflections (hkl)

Condition for possible reflection**

b/2

0kl

k = 2n

c/2

0kl

l = 2n

a

a/2

h0l

h = 2n

c

c/2

h0l

l = 2n

(001)

a

a/2

hk0

h = 2n

(001)

b

b/2

hk0

k = 2n

(110)

c

c/2

hhl

l = 2n

(1100)

c

c/2

hh2hl

l = 2n

(1120)

c

c/2

hh0l

l = 2n

Glide plane Orientation

Symbol

Translation*

(100)

b

(100)

c

(010) (010)

Axial glides

Diagonal glides (100)

n

b/2 + c/2

0kl

k + l = 2n

(010)

n

a/2 + c/2

h0l

h + l = 2n

(001)

n

a/2 + b/2

hk0

h + k = 2n

(110)

n

a/2 + b/2 + c/2

hhl

h + l = 2n

Diamond glides (100)

d

b/4 + c/4

0kl

k + l = 4n

(010)

d

a/4 + c/4

h0l

h + l = 4n

(001)

d

a/4 + b/4

hk0

h + k = 4n

(110)

d

a/4 + b/4 + c/4

hhl

2h + l = 4n

* The ‘0’ subscripts are omitted from a0 , b0 and c0 in this column, this being a common practice ** The indices of reflection cited in this column represent sets of lattice planes that are (a) perpendicular to the glide plane and (b) doubled or quadrupled in number by the glide plane translation

Case II: Second Option If the bcc crystal lattice is considered as a simple cubic crystal lattice with a basis containing two atoms (this is a common practice followed for all centered lattices): one at the origin (000) and the other at its body center (1/2 1/2 1/2), this will provide its reciprocal lattice also to be a simple cubic crystal lattice, with its side as 2π/a. Also, there will now be a structure factor F(hkl) associated with each Braggs’ reflection, which is given by F(hkl) = f [1 + ex pπi (h + k + l)] = f [1 + 1] = 2 f, i f (h + k + l)is an even integer and I ∝ |F(hkl)| = 4 f 2 2

7.5 Role of Lattice Centering (if Any) in Systematic Absences

175

Similarly, F(hkl) = f [1 − 1] = 0, i f (h + k + l)is an odd integer. and I ∝ |F(hkl)|2 = 0 where, f is the atomic scattering factor or atomic form factor. According to Ashcroft and Mermin [8], the zero structure factor suggests that those points in the simple cubic reciprocal lattice, the sum of whose coordinates with respect to the cubic primitive vectors are odd (corresponding to white balls), will have no Braggs’ reflections associated with them. When these white balls are removed, the simple cubic reciprocal lattice gets converted into the face centered cubic crystal lattice (Fig. 7.1) that we would have had if we had treated the bcc crystal lattice in its primitive (or its W–S) form rather than as a simple cubic crystal lattice with a basis. Similar treatment can be carried out for fcc and other non-primitive crystal lattices. However, according to Braggs’ formulation and the concepts of mirror combination scheme based on the principles of Wigner–Seitz and Brillouin-zone, diffraction of crystals actually starts from the widest set of crystal planes of atoms having the least set of reciprocal lattice vector (as d* = 1/d) and continue with the gradually decreasing width of planes and not by selected atoms as considered above by Ashcroft and Mermin and in other texts. Further, since each crystal lattice (e.g. sc, bcc and fcc) has its own widest spacing plane and other subsequent planes, which give rise to the Fig. 7.1 Process of conversion of simple cubic into fcc lattice

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7 Possibility of Translational Symmetry (if Any) in Crystals

diffraction peaks according to the scheme of the concerned structure. Therefore, the assumption/consideration of bcc crystal lattice as a simple cubic crystal lattice with a basis containing two atoms and similarly fcc crystal lattice as a simple cubic crystal lattice with a basis containing four atoms (or in general considering any centered lattice in such a manner) is not permissible in crystal diffraction. Therefore, each of the 16 crystal lattices is independent and needs to be considered independently in its primitive form or its Wigner–Seitz form. The above discussion suggests that if we treat the given centered crystal lattice in its primitive (or its W–S form), then the corresponding Brillouin zone (is a set of selected Braggs’ planes) obtained from each diffraction peak (provided by the Powder pattern) is sufficient to establish its crystal structure (without needing the structure factor information for the same). This also in turn suggests that diffraction (of X-rays, electrons or neutrons) recognizes only the W–S and B–Z forms of the crystal lattice. A particular set of crystal planes, prima facie appears to be absent from the diffraction patterns and also appears to follow the conditions of systematic absences arising due to lattice centering provided in Table 7.1. However, as shown in Fig. 5.4 that the lower order planes are not absent but they actually merge with their next higher order parallel planes (because of identical interplanar spacing due to the presence of additional atomic planes in bcc and fcc) and together they give rise to the diffraction peaks of appreciable intensity, as they are identical and hence diffract in phase. Therefore, the consideration of systematic absences due to the so called lattice centering in cubic crystals is not consistent with the facts. A similar result is expected in centered lattices of other crystal systems. This phenomenon can be treated as a crystallographic illusion.

7.6 Role of Screw Axes (if Any) in Systematic Absences As discussed above, it is because of the fact that the Wigner–Seitz cells, the Brillouin zones and the diffraction patterns possess defined origin, they cannot have translational symmetry of any kind (either microscopic or macroscopic). Therefore, the presence of either screw axes or glide planes in crystals is completely ruled out and hence the question of space groups containing microscopic symmetries (screw axes and glide planes) in them does not arise in principle. However, just for the sake of argument and to provide some other satisfactory reasons in support of the justification, we propose to analyse them in brief. The screw axes and glide planes are known as microscopic symmetries and assumed to be present in all crystals (which are well ordered otherwise), which according to literature could be seen only through a beam of X-rays/electrons/ neutrons [9], while their concepts are definitely borrowed from purely classical and macroscopic examples, such as climbers, spiral stairs, image of the footsteps and so on as shown in Fig. 7.2. This consideration is same as if we wish to solve the relativistic/quantum mechanical problems by using the formulae of classical mechanics

7.6 Role of Screw Axes (if Any) in Systematic Absences

177

Fig. 7.2 Macroscopic examples of screw axis and glide plane

in physics. In fact, it is because of the influence of macroscopic phenomenon and the dominance of design representations in early days of crystallography (particularly during Sohncke’s period), the 2-D space lattices were also named wallpaper groups. However, this is also a fact that no diffraction pattern has exhibited in any form, the presence of screw axis or glide plane in it so far. This is certainly contrary to the claim [9] that they can be viewed under X-rays, etc., this obviously raises the pertinent question of their existence in crystals (which are well ordered otherwise). As per the literature, when a ‘non-primitive translation’ combines with rotation to produce screw axes, and with mirror to produce glide planes—the symmetry elements of a crystal structure do not all pass through one point as found in point groups [10]. But the screw axes and glide planes formed in this way are supposed to be arranged at regular intervals and consistent with the crystal lattice in space. Further, the resulting symmetry operations in a space group can be combined only in certain consistent ways in order to satisfy group theory. This suggests that the systematic absences which are supposed to be arising due to screw axes and glide planes also remain compatible with the crystal lattice (irrespective of primitive or centered). It means that the initial (low index) Braggs’ planes which prima facie appear to be missing as a result of lattice centering (considered above) represent the planes whose proper fractions (non-primitive) correspond to the distances where the screw axes and/or glide planes are supposed to be effective. In the light of this, let us try to understand the role of screw axes and glide planes in the creation of systematic absences (if any) in cubic crystal system, and hence in other crystal systems.

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7 Possibility of Translational Symmetry (if Any) in Crystals

A screw axis is defined as an axis about which a proper rotation (θ = 2π/n) combines with a ‘non-primitive translation’ parallel to it, transforms an array of atoms to self-coincidence. The permissible screw axes are: 21 , 31 , 32 , 41 , 42 , 43 , 61 , 62 , 63 , 64 and 65 as provided in Table 7.2. They are symbolically represented by n m (where m and n are integers with m < n), n representing the rotation (2, 3, 4, 6) and the subscript m represents the pitch of the screw. Therefore, m/n represents the translation for each rotation around the axis. According to the classification of screw axes based on the number of threads or pitch of the screw, the following suggestions were made [11, 12]. 1. The screws of enantiomorphic pairs such as 31 and 32 are the same except that they are right handed and left handed, respectively. Similarly, 41 and 43 , 61 and 65 and 62 and 64 . The screw axes 21 , 42 and 63 have neither right nor left sense, and are called neutral. 2. Any screw axis with m threads is supposed to have m-fold proper rotation axis. However, observations suggest that each of the following screw axes 21 , 31 , 32 (≡ 3−1 ), 41 , 43 (≡ 4−1 ), 61 and 65 (≡ 6−1 ) has only onefold; each of the following axes 42 , 62 and 64 (≡ 6−2 ) has twofold; and 63 has threefold proper rotations, respectively. Therefore according to above suggestions, if the space group contains the screw axis 21 or the enantiomorphic pairs 31 /32 (≡ 62 /64 ), 41 /43 or 61 /65 , then the respective translation component reproduces a plane, each of which is normal to the axis at half, one third, one fourth, or one sixth of the unit cell spacing (in direct space) as shown in Figs. 7.3a and 7.4a (figures are not to the scale). In reciprocal space, their respective positions are twice, thrice, 4 times and 6 times the plane responsible for first order reflection/diffraction and hence they correspond to second, third, fourth and sixth order plane, respectively as shown in Figs. 7.3b and 7.4b.

7.7 Role of Glide Planes (if Any) in Systematic Absences Like screw axis, a glide plane is defined as the plane across which a mirror plane combine with a ‘non-primitive translation’ parallel to it, transforms an array of atoms to self-coincidence (the order in which the two operations are carried out is unimportant). The permissible glide planes are: axial glide, diagonal glide or diamond glide in different crystal structures including cubic as provided in Table 7.3. In direct lattice, the axial and diagonal glides occur at half the axial and diagonal distances of the unit cell, respectively. Similarly, the diamond glide occurs at one-fourth the face or body diagonal distance as can be shown in Fig. 7.3a. In reciprocal space, they can be represented respectively at twice and four times away from the first order (normal) reflection, implying that the reflections due to glide planes if any, will coincide with the second and fourth order of normal reflection (Fig. 7.3b).

7.7 Role of Glide Planes (if Any) in Systematic Absences

179

Fig. 7.3 Location of screw axes and glide planes for orthogonal system in a direct space, b reciprocal space

Further, from above interplanar d-spacing calculations (Sect. 5.5) and diffraction results (Sects. 5.7, 6.5 and 6.6), we come to know that the planes required to form the first Brillouin zone in three cubic crystal systems are: 1. {100} for sc. 2. {110} for bcc, where {100} planes are supposedly absent as they lose their identity when merge with higher order reflections. 3. {111} and {200} combined for fcc, where {100} and {110} planes are supposedly absent as they lose their identity when merge with their corresponding higher order reflections, respectively. 4. Further, we also know that the next higher order planes corresponding to {100}, {110} and {111} planes are: {200}, {300},… etc.; and {220}, {330},… etc. and {222}, {333},… etc. planes, respectively. They will automatically correspond to the appropriate screw axis/glide plane in reciprocal space (according to its type) as can be seen from the set of reflections corresponding to different non-primitive screw axes/glide orientations provided in Tables 7.2 and 7.3. From the above discussion, we find that like lattice centering, the screw axes and glide planes also create no systematic absences as such in any crystal lattice (centered or primitive). They only appear to be so due to merging of the low index planes with the next higher order parallel planes. Therefore, the consideration of systematic absences due to the so called lattice centering, screw axes and glide planes in diffraction of crystals is not consistent with the facts and appears to be like a phenomenon called ‘crystallographic illusion’ (similar to other types of illusions in physics).

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7 Possibility of Translational Symmetry (if Any) in Crystals

Fig. 7.4 Location of screw axes and glide planes in a direct space, b reciprocal space

7.8 Reality of Enantiomorphous and Some Other Pairs in Crystals Literature survey [12, 13] suggests that the following 11 enantiomorphous pairs and 4 other pairs as provided in Table 7.4 have been found to exhibit the same intensity sequence and systematic absences by each pair. However, no suitable explanations

7.8 Reality of Enantiomorphous and Some Other Pairs in Crystals

181

of the same have come from anyone (from the crystallographic community), so far. It appears to be one of the long pending issues in crystallography. In order to understand clearly the enantiomorphous pairs of space groups (or crystals), let us consider first the case of rhombohedral pair 31 /32 . Let they represent . the rhombohedral close packed (RCP) crystal structures with sequences ABC..A… in . cyclic orientation having screw axis 31 and ACB..A… in anti-cyclic orientation having screw axis 32 as shown in Fig. 7.5. As per the criterion for diffraction (discussed above), if they possess identical layer separation (d-spacing), they will give rise their diffraction at the same point (because of the identical reciprocal lattice vector) and hence they are supposed to represent the same crystal structure. Therefore considering the two orientations given in Fig. 7.5 and making simple calculation based on their geometry we obtain identical interlayer spacing (separation between any two consecutive layers). For example [14], (√ ) (√ ) 2 8 h(AB) = h(AC) = √ a = √ R 3 3 where a = 2R, and R is the radius of the sphere (atom). As the separation in two orientations is found to be identical and d* (= 1/d), they are bound to diffract at the same point and hence they will provide the identical diffraction pattern according to the criterion for diffraction. That means, both the orientations represent the same crystal structure. This matches with the result obtained from a recent study on genesis of rhombohedral structures made by Wahab and Wahab [14] that 3R exhibits only one structure with the lattice parameters a = b = c and α = β = γ = 60◦ . In a similar manner, other enantiomorphic pairs can also be shown to represent identical crystal structure. As far as the two other pairs are concerned, they belong to a body centered orthorhombic and cubic crystal systems, respectively. Let us take the case of bcc pair I23/I21 3 first. As we have already observed above that the first B–Z of bcc consists of 12 {110} planes (where {100} planes appear to be absent from their normal sites). Here, for the sake of an argument, the plane corresponding to a 21 screw axis will lie at half the axial separation in real space as shown in (Figs. 7.3a, 7.4a). Therefore, in reciprocal space (it will be at twice the normal distance), the same will coincide with the {200} planes, which provides the same result as obtained for the normal bcc structure with the point group I23 [where (100) planes merge with (200) planes] as shown in Figs. 7.3b, 7.4b. Accordingly, the two sequences are bound to exhibit the Table 7.4 Reported enantiomorphous and four other pairs Enantiomorphous pairs

Other pairs

31 /32 , 31 21/32 21, 31 12/32 12, 41 /43 , 41 22/43 22, 41 21 2/43 21 2, 61 /65 , 62 /64 , 61 22/65 22, 62 22/64 22, 41 32/43 32

I222/I21 21 21 , I23/I21 3 C31m/C3m1, C31m/C3m1

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7 Possibility of Translational Symmetry (if Any) in Crystals

Fig. 7.5 Rhombohedral (3R) crystal structure in a cyclic and b anticyclic orientation

identical XRD pattern, hence they should represent identical crystal structure. The orthorhombic pair I222/I21 21 21 can be explained in a similar manner.

7.9 Important Observations Against Translational Symmetries in Crystals The following important observations suggest that the inclusion of translational symmetries (either microscopic or macroscopic) is not found fit in crystalline solids: 1. A pair of parallel mirrors with an object between them is able to produce infinite images in the form of a one dimensional crystal lattice with perfect translational periodicity. Similarly, the Wigner–Seitz cells and Brillouin zones respectively generate direct and reciprocal lattices on the basis of mirror combination scheme both in 2-D and 3-D lattices. These unit cells are primitive, centro-symmetric and bounded by spherically symmetric zone boundaries. They suggest that the resulting lattices have translational periodicity and contain no translational symmetry. 2. A Brillouin-zone is a representation of spherically symmetric distribution of zone boundaries of a diffraction peak. In a diffraction pattern, each Brillouin zone for a given peak has a defined origin in it, which permits the rotational, mirror and

7.10 Some Obvious Reasons for Reduction in the Number of Space Groups

3.

4.

5.

6. 7.

8.

183

inversion symmetries but forbids any translational symmetry (either microscopic or macroscopic). As a result, the possibility of existence of glide planes and screw axes in crystals is ruled out. No screw axes are found to coincide with the corresponding principal axes, they are simply parallel to it and pass only through low symmetry sites in all cases. Therefore when the screw component is removed, they do not match with the symmetry of the corresponding principal axis, it is against its basic assumption. According to the classification [11] of screw axes based on the number of threads (pitch), the screw axes 21 , 31 , 32 , 41 , 43 , 61 and 65 contain only onefold, 42 , 62 and 64 contain twofold and 63 contains a threefold of pure rotation, respectively. Enanntiomorphic pairs 31 /32 , 31 21/32 21, 31 12/32 12, 41 /43 , 41 22/43 22, 41 21 2/43 21 2, 61 /65 , 62 /64 , 61 22/65 22, 62 22/64 22, 41 32/43 32 are found to represent identical crystal structure. Inconsistencies and contradictions are found to be very common in the space groups proposed by Fedorov, Schoenflies and Barlow. Some space groups may be mathematically legitimate, but chemically impossible, and the crystal structures of organic compounds so far determined belong to a rather very restricted number of space groups [15]. Majority of the 230 space groups proposed about 200 years ago, yet to find a crystal structure.

7.10 Some Obvious Reasons for Reduction in the Number of Space Groups The number of space groups, 230 derived/proposed by Fedorov, Shoenflies and Barlow is believed to be fixed and unchangeable (no less and no more according to literature) in crystallography. However, during the course of study of mirror as the only fundamental symmetry [6], we came across a number of important aspects (some of them are mentioned below), according to which this number (230) is bound to reduce (even if the translational symmetry is taken into account for the sake of an argument). They are: 1. According to mirror combination scheme, the triclinic crystal system contains the point groups’ m, 1 and 1 (representing the minimum form of ‘mirror’, ‘proper rotation’ and ‘inversion’ symmetries, respectively), while as per earlier classification the point group ‘m’ belong to monoclinic crystal system, a reshuffle of the same is inevitable. Similarly, the point groups ‘mm2’ and ‘2/m’ need to be interchanged their places between monoclinic and orthorhombic (Table 2.9) for correct representation. As a result of these changes, the effective number of space groups is bound to decrease. 2. The enantiomorphic pairs of space groups actually represent the same structure and hence 11 such pairs should be counted simply 11. This further reduces the total strength of the space groups.

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7 Possibility of Translational Symmetry (if Any) in Crystals

3. In the two other pairs I222/I21 21 21 and I23/I21 3, each pair represents identical structure. This also causes the reduction in the total number of space groups. Hence, the resulting number of space groups will be significantly less than 230 even if the translational symmetry is taken into account.

7.11 Some Obvious Inconsistencies/Contradictions About Screw Axes/Glide Planes Inconsistencies and contradictions are found to be very common in the 230 space groups derived by Fedorov, Schoenflies and Barlow based on translational symmetries like screw axes and glide planes (they will also contribute in the reduction of 230 space groups). For example, only some of the very obvious cases are pointed out below: 1. Consider the space groups P222, P2221 and P21 21 2 connected to the point group 222 of the orthorhombic crystal system. In matrix notation, the symmetry multiplication of the point group 222 can be expressed as [16]: ⎛

⎞⎛ ⎞ ⎛ ⎞ −1 0 0 −1 0 0 1 0 0 2[001] × 2[010] = ⎝ 0 −1 0 ⎠⎝ 0 1 0 ⎠ = ⎝ 0 −1 0 ⎠ = 2[100] 0 0 1 0 0 −1 0 0 −1 This implies that when a twofold rotation along z-axis is combined with another twofold rotation along y-axis, a new twofold rotation along x-axis is generated. This is well established, as known to us from symmetry combinations. However, then it is clear that the space groups P2221 and P21 21 2, mentioned above contradict this multiplication scheme and hence they are not possible on the basis of symmetry considerations. Similar is the case with the space groups C222 and C2221 . 2. Consider the space groups P422, P41 22, P42 22 and P43 22 connected to the point group 422 of the tetragonal crystal system. In matrix notation, the symmetry multiplication of the point group 422 can be expressed as: ⎛

⎞⎛ ⎞ ⎛ ⎞ 0 −1 0 −1 0 0 0 −1 0 ] [ 4[001] × 2[010] = ⎝ 1 0 0 ⎠⎝ 0 1 0 ⎠ = ⎝ −1 0 0 ⎠ = 2 110 0 0 1 0 0 −1 0 0 −1 This implies that when a fourfold rotation along z-axis is combined with a twofold rotation along y-axis, a new twofold rotation along the face diagonal is generated. This is well established, as known to us from symmetry combinations. However, then it is clear that the other space groups P41 22, P42 22 and P43 22, mentioned above contradict this multiplication scheme and hence they are not possible. Similar is the case with the space groups, I41 22.

7.11 Some Obvious Inconsistencies/Contradictions About Screw Axes/Glide …

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3. Consider the space groups P622, P61 22, P62 22, P63 22, P64 22 and P65 22 connected to the point group 622 of the hexagonal crystal system. In matrix form, the symmetry multiplication of the point group 622 can be expressed as: ⎛

⎞⎛ ⎞ ⎛ ⎞ 1 −1 0 −1 0 0 0 −1 0 ] [ 6[001] × 2[010] = ⎝ 1 0 0 ⎠⎝ −1 1 0 ⎠ = ⎝ −1 0 0 ⎠ = 2 110 0 0 1 0 0 −1 0 0 −1 This implies that when a sixfold rotation along z-axis is combined with a twofold rotation along y-axis, a new twofold rotation along the face diagonal is generated. This is well established, as known to us from symmetry combinations. However, then it is clear that the other space groups, P61 22, P62 22, P63 22, P64 22 and P65 22, contradict this multiplication scheme and hence they are not possible. 4. Consider the space groups Pmm2, Pcc2 and Pnn2 connected to the point group mm2 of the orthorhombic crystal system as per earlier classifications. However, from the present classification (Table 2.9) based on symmetry, it belongs to monoclinic crystal system. In matrix form, the symmetry multiplication of the point group mm2 can be expressed as: ⎛

⎞⎛ −1 0 0 1 0 m[100] × m[010] = ⎝ 0 1 0 ⎠⎝ 0 −1 0 01 0 0

⎞ ⎛ ⎞ 0 −1 0 0 0 ⎠ = ⎝ 0 −1 0 ⎠ = 2[001] 1 0 0 1

This implies that when a mirror plane perpendicular to x-axis is combined with another mirror perpendicular to y-axis, a twofold rotation along z-axis is generated (or in general, when two mirror planes are held perpendicular to each other, a twofold axis is created along the line of their intersection as in Fig. 2.3). This is well established, as known to us from symmetry combinations. However, then it is clear that the space groups Pcc2 and Pnn2, mentioned above contradict this multiplication scheme (because cc and nn glide planes contain additional translational terms in them and hence cannot give rise to the same product) and hence they are not possible. Similar is the case with the space groups Ccc2 and Fdd2, they are also not correct. 5. Finally, let us consider the space groups P43m, P43n, F43c and I43d of cubic crystal system connected to the point group 43m . In matrix form, the first point group P43m can be expressed as: ⎛

⎞⎛ ⎞ ⎛ ⎞ 0 1 0 001 1 0 0 4[001] × 3[111] = ⎝ −1 0 0 ⎠⎝ 1 0 0 ⎠ = ⎝ 0 0 −1 ⎠ = m[011] 0 0 −1 010 0 −1 0 This implies that when a 4-fold rotation along z-axis is combined with a threefold rotation along body diagonal of the cube, a mirror plane perpendicular to [011] axis of the cube is generated. This is well established, as known to us from

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symmetry combinations. However, then it is clear that the space groups P43n, F43c and I43d containing the n, c and d glides (which contain the additional translational term) do not fit into the above equation and are not possible. Another example to know about the involvement of glide planes and screw axes in crystals is to discuss the case of well-known hexagonal close packed structures 2H and 4H exhibited by a large number of Periodic Table elements as their space groups (P63 /mmc) and (P63 mc), which contain both screw axis and glide plane. However, as far as the HCP structure (2H) with its stacking sequence AB/A… is concerned, neither Bravais himself nor the literature ever accepted it, an independent lattice. But in our recent study of packing of identical atoms [17], we discovered it to be a fundamental lattice by constructing its actual unit cell, primitive unit cell, Wigner–Seitz unit cell and Brillouin zone, supported by mathematical derivation of its lattice parameters, unit cell volume, reciprocal lattice parameters and the volume of the reciprocal unit cell. We further discussed its symmetries and also determined its space group to be P6m2 and not P63 /mmc as reported in literature. The same space group of this structure was also determined by [18] using a plane projection (2-D) approach. Neither Jaswon and Rose nor we found any evidence of presence of screw axis or glide plane in this structure. In fact, close packing of identical atoms cannot have a 63 rotation axis, but can have only 3, 3 and 6 rotation axes (Tables 2. 9 and 3.2). A similar exercise carried out for (4H) structure with stacking sequence ABCB/A… provides the same result.

7.12 Summary 1. The crystallographic developments that took place during early days of Bravais, Sohncke and Fedorov era were largely mathematical/group theoretical in nature and had little concern with the actual illustrations of atomic and molecular arrangements or any experimental observation in the derivation of 230 space groups. 2. The group theoretical development of the space lattice was initiated by Hessel (1830) and Gadolin (1867) with the derivation of 32 finite crystal classes (also known as point groups); of these 32 point groups, Gadolin found only 20 examples in nature. 3. The laws governing the point group symmetries dealing with finite symmetries were actually extended to cover the symmetries of crystal lattices. Frankenheim (1801–1869), Bravias (1811–1863), and Sohncke (1842–1898) then introduced the concept of infinite symmetries of lattices to crystals, simply based on conjectures. 4. In the middle of nineteenth century, Frankenheim and Bravais developed the concept of crystal lattice and enumerated the 14 ‘frameworks’ in the form space lattices that formed the basis of the modern structural crystallography.

7.12 Summary

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5. Leonhard Sohncke was the first to introduce the concepts of screw axes in crystallography in 1879 by modifying the Bravias concept of space lattice from ‘identical environment and similar orientation of lattice points’ to ‘identical environment but without necessarily similar orientation’. He then obtained 65 space groups by curtailing the number from the original work of Jordon (174 point groups), keeping only those that were relevant from crystallographic point of view. 6. Inspired by the Sohncke’s work, both Fedorov and Schoenflies (1853–1928) extended their works independently by taking into account the improper rotations (i.e. rotoreflection and rotoinversion) and theoretically derived a total of 230 possible structural arrangements in three dimensional space, and called them the space groups. 7. William Barlow (1845–1934) also derived the same number of space groups, but he delayed publishing his list (1894). Using a method based on Sohncke’s, instead of systems of points, he considered geometric patterns of orientated motifs, a pair of gloves to model his results. 8. The study suggests that spherically symmetric nature of the Wigner–Seitz unit cells, the Brillouin zones and the diffraction patterns/crystal structural data, crystals cannot possess translational symmetry of any kind (either macroscopic or microscopic). In fact, crystals possess the translational periodicity. 9. The assumption/consideration of bcc crystal lattice as a simple cubic crystal lattice with a basis containing two atoms and similarly fcc crystal lattice as a simple cubic crystal lattice with a basis containing four atoms (or in general considering any centered lattice in such a manner) is not permissible in crystal diffraction. Therefore, each of the 16 crystal lattices is independent and they need to be considered independently in its primitive or Wigner–Seitz form. 10. Introduction of the concepts of microscopic symmetries such as screw axes and glide planes in crystals are definitely borrowed from purely classical and macroscopic examples, such as climbers, spiral stairs, image of the footsteps and so on, as shown in Fig. 7.2. Explaining the microscopic system with the help of macroscopic examples does not appear to be logical. 11. There are many important practical observations which suggest that the inclusion of translational symmetries (either macroscopic or microscopic) is not found fit in crystalline solids. 12. The number of space groups, 230 derived by Fedorov, Shoenflies and Barlow is believed to be fixed and unchangeable (no less and no more according to literature) in crystallography. However, there are some obvious reasons due to which this number is bound to decrease. 13. Inconsistencies and contradictions are found to be very common in the space groups derived by Fedorov, Schoenflies and Barlow based on translational symmetries like screw axes and glide planes. 14. In the absence of any experimental evidence or theoretical justification in favour of screw axes and/or glide planes, their considerations appear to be simply on conjectures.

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References 1. De Graef, M., McHenry, M.E.: Structure of Materials: An Introduction to Crystallography, Diffraction, and Symmetry. Cambridge University Press, Cambridge (2007) 2. Schuh, C.P.: Mineralogy and Crystallography: On the History of These Sciences From Beginnings Through 1919, Tucson, Arizona (2007) 3. Forman, P.: The discovery of the diffraction of X-rays by crystals; a critique of the myths. Arch. Hist. Exact Sci. 6(1), 38–71 (1969) 4. Philips, F.C.: An Introduction to Crystallography, ELBS and Longman Group Limited, 4th edn. The University Press, Glasgow (1971) 5. Hammond, C.: The Basics of Crystallography and Diffraction, IUCr, 2nd edn. Oxford University Press, Oxford (2001) 6. Wahab,M. A.: The Mirror: Mother of all Symmetries in Crystals, Adv. Sci. Eng. Med. 12, 289–313 (2020) 7. Leslie, A.: Crystals, Symmetry and Space Groups. MRC Laboratory of Molecular Biology (LMB) Crystallography Course, Cambridge (2013) 8. Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Harcourt Asia PTE Ltd., Singapore (2001) 9. Chatterjee, S.K.: Crystallography and the World of Symmetry. Springer, Berlin Heidelberg (2008) 10. Clegg, W.: X Ray Crystallography, Oxford University Primers, 2nd edn. Oxford University Press, Oxford (2015) 11. Burger, M.J.: Elementary Crystallography: An Introduction to the Fundamental Geometrical Features of Crystals. John Wiley and Sons Inc., New York (1963) 12. Zolotoyabko, E.: Basic Concept of Crystallography: An Outline from Crystal Symmetry), 1st edn. Wiley-VCH, New York (2011) 13. Burger, M.J.: The crystallographic symmetries determinable by X-ray diffraction. PNAS 36(5), 324 (1950) 14. Wahab, M.A., Wahab, K.M.: Genesis of rhombohedral structures and mode of polytype transformations in close packing of identical atoms. Mater. Focus 7(2), 321 (2018) 15. Gavezzotti, A., Flack, H.: Crystal Packing, International Union of Crystallography, 2 Abbey Square, Chester CH1 2HU, U.K. (2005) 16. Wahab, M.A.: Essentials of Crystallography, 2nd edn. Narosa Publishing House, New Delhi (2014) 17. Wahab, M. A.., Wahab, K.M.: Resolution of Ambiguities and the Discovery of Two New Space Lattices, ISST J. Appl. Phys. 6, 1, 1 (2015) 18. Jaswon, M.A., Rose, M. A.: Crystal Symmetry: Theory of Colour Crystallography, Ellis Horwood Limited, U.K., p. 78, (1983)

Chapter 8

Resolution of Existing Discrepancies, Ambiguities and Confusions

8.1 Introduction Literature survey suggests that certain serious discrepancies, confusions and ambiguities have been persisting ever since the beginning of crystallography as a subject. Many of them are found to exist in the present day crystallography as well. In this chapter, we are going to discuss about the modes of resolutions of some of them, in brief. Considerable confusion, complexities and ambiguities are found to exist in literature as far as the representations of trigonal, hexagonal and rhombohedral lattices are concerned. A comprehensive study of the same, clearly suggests the necessity of HCP and RCP as the two new and independent lattices in 3-D, and hence a brief description about them is provided in this chapter. An immediate consequence of this, is the consideration of 16 space lattices in crystals instead of 14. Another related confusion about the space lattice remains to be answered is the discrepancy between the number of lattices (16) and the Laue groups (11), this confusion remains even if the number of lattices is 14, because as per the fundamental crystallography, each space lattice contains a center of symmetry, and hence in principle all of them should have been the member of the Laue group. Also some discrepancies are found in the allocation of point groups to 2-D (oblique and rectangular) and 3-D (triclinic, monoclinic, orthorhombic, trigonal and hexagonal) lattices. The discrepancy is also found to persist in the reported and the actual symmorphic space groups. Resolution of the issues mentioned above could become possible only with the help of the recent study of symmetries based on mirror combination scheme and using the concept of minimum symmetry form (MSF). The mirror combination scheme is based on the widely known and extremely important concepts, the Wigner–Seitz cells and the Brillouin zones as they are widely used to understand the structure and properties of crystal materials in solid state physics, crystallography and other subjects of science and technology.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. A. Wahab, Mirror Symmetry, Springer Series in Solid-State Sciences 200, https://doi.org/10.1007/978-981-99-8361-2_8

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8 Resolution of Existing Discrepancies, Ambiguities and Confusions

8.2 Necessity of HCP and RCP Lattices We know that according to the present nomenclature, simple hexagonal (SH) and hexagonal close packing (HCP) have been put together to represent the hexagonal crystal system, with the only difference that SH as primitive with a basis containing one atom and HCP the primitive with a basis containing two atoms (in 2-D) as shown in Fig. 8.1. However, this consideration has been the cause of considerable confusion, complexities and ambiguities ever since the crystallography has come up as an independent subject. Keeping this in view, a comprehensive study of SH and HCP units based on the model of identical atoms (spheres) was carried out for the first time by constructing their unit cells (e.g. the conventional, primitive, Wigner–Seitz and Brillouin zone) for comparison. This study was further supported by comprehensive crystallographic calculations. Constructions of SH, HCP and RCP unit cells have been discussed in Sect. 3.6. Here, we provide only the summary of results obtained from various geometrical constructions and the crystallographic calculations [1] to highlight the necessity to consider HCP and RCP as independent lattices. 1. A simple hexagonal lattice (SH) is quite well known and defined as a = b = c = 2R, α = β = 90° and γ = 120° (when the balls of the two close packed layers just touch each other), while a hexagonal close packed (HCP) lattice is defined for the first time with a = b = c = 2R and α = 60◦ , β = γ = 120◦ (when the balls of the second close packed layer is displaced to B or C voids w.r.t the balls of the first close packed layer). 2. One SH unit is made up of three parallelepiped units, each with a = b = c = 2R, α = β = 90° and γ = 120°, while one HCP unit is made up of three rhombohedral close packed (RCP) units, each with a = b = c = 2R, α = β = γ = 60°. That is, the primitive unit of SH remains SH and non-close packed, while the primitive unit of HCP is RCP, which remains close packed. The volume of the primitive

Fig. 8.1 Plane projection of SH and HCP unit cells

8.3 Discrepancy in the Representation of RCP and CCP

191

rhombohedron is given by VR =

VH C P 3

where three rhombohedral units completely fill the HCP unit cell (Fig. 3.5c, d). 3. The axis of the simple hexagonal unit is the same as the principal axis, i.e. [along] [001] direction, while the axis of the cyclic AB unit of HCP is tilted along 113 direction, the angle of tilt with respect to vertical [001] axis is given by (√ ) 2 cos−1 √ = 35.26 3 4. Both, the constructions as well as calculations of Wigner–Seitz units and the Brillouin zones corresponding to the unit cells of SH and HCP suggest that the resulting shape for SH remains simple hexagonal, while for HCP they are rhombic dodecahedron, the (W–S) and truncated octahedron, the (B–Z), respectively. 5. A trigonal unit is found to be associated with SH throughout while the rhombohedral unit is associated with HCP only. Thus a trigonal unit remains non-close packed, while the RCP unit remains close packed. This implies that: a trigonal unit ≇ the RCP unit Thus equating a trigonal unit and a rhombohedral unit as found in common text books is not justified. Similarly, SH (a non-close packed) unit and HCP (a close packed) unit are entirely different and need to be treated separately. All results indicate that HCP and RCP need to be represented as independent lattices. 6. Both the HCP and RCP lattices are called Wahab lattices, and the 16 lattices together as Bravais–Wahab or (B–W) lattices or simply space lattices as before.

8.3 Discrepancy in the Representation of RCP and CCP After constructing the HCP unit cell as in Fig. 3.5, if the spheres are placed in the third and fourth layers respectively in C and A voids without changing the direction of packing, we obtain rhombohedral close packing (RCP) and the so called cubic close packing (CCP) unit cells as shown in Fig. 8.2. The RCP unit cell is obtained by joining the first nearest neighbour atoms (shown by blue lines) while the CCP unit is the outcome of joining the second nearest atoms (shown by red lines), however, the two unit cells are represented by the same ABC/A … sequence. In fact, the CCP unit is nothing but the second order RCP unit, therefore only the RCP unit should be used to represent the close packed structure with the ABC/A … sequence instead of CCP, to remove any confusion arising due to CCP and FCC representations for the same structure (where RCP and FCC are certainly different structures as no

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Fig. 8.2 RCP and CCP unit cells, where CCP is simply the second order RCP

face centered cubic require to be represented by ABC/A … sequence). It appears logical because of the fact that RCP exhibits a large number different structural modifications (called polytypes) like HCP, while the so called CCP (or even FCC) does not. Accordingly, the vertical manner of close packing of atomic layers should be represented fundamentally in two extreme forms, i.e. either HCP or RCP (and not CCP) because both HCP and RCP forms are complementary to each other and both exhibit polytypism independently [2].

8.4 Discrepancy in the Number of Space Lattices 14 or 16 We know that FCC, HCP and RCP all represent close packed structures (however, their conventional unit cells have different shapes). In mono atomic system, they have the same coordination number (12), their Wigner–Seitz cells bound to have the same shape, but oriented differently due to different ways of packing of atoms in their unit cells and hence they represent different crystal structures. For example, different ways of packing of identical atoms in three layers of HCP and RCP are shown in Fig. 8.3. Thus on the basis of geometrical consideration, there can be only 14 shapes. This is correct according to Bravais classification, where the three different orientations belonging to FCC, HCP and RCP in close packing of identical atoms correspond to a single W–S/B–Z shape. However, due to three separate orientations, one each for FCC, HCP and RCP, there exist three different crystal lattices (or crystal structures). Therefore, 14 W–S/B–Z shapes, actually represent 16 space lattices. These 16 space lattices (of course cannot be explained on the basis of conventional lattices) as has

8.5 Ambiguity in 16 Space Lattices and 11 Laue Groups

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Fig. 8.3 Different ways of packing of three close packed layers in HCP and RCP

been demonstrated by Wahab and Wahab (2015) after deriving the lattice parameters of HCP and RCP unit cells, and constructing their Wigner–Seitz cells and Brillouin zones for the first time in the history of crystallographic developments.

8.5 Ambiguity in 16 Space Lattices and 11 Laue Groups From fundamental crystallography we know that each space lattice has a center of symmetry. Accordingly, for 16 space lattices there should in principle be 16 Laue groups (it means, even for 14 space lattices, there should have been 14 Laue groups). The ambiguity existing between the number of space lattices (16) and the Laue groups (11) remained unattended and unresolved mystery so far in the history of crystallography (the ambiguity remains even if we consider 14 space lattices). However, it is the discovery of HCP and RCP lattices together with the mirror combination scheme that has made it possible to provide a clear answer to this longstanding problem. It is due to the addition of HCP and RCP lattices into the fold of crystal systems, four center of symmetries (two point groups in the form of 3, 3m from each crystal system) are added to 11 centrosymmetric point groups, to make the total Laue groups to be 15. Further, according to mirror combination scheme, the point group 2/m had to be shifted from monoclinic to orthorhombic crystal system. As a result, the monoclinic crystal system is left without any centrosymmetric partner, which is in contradiction with the exhibition of center of symmetry by each lattice and the diffraction of crystals. In order to remove this contradiction the point group (1) has to be added to the monoclinic crystal system. This makes the total (or actual) Laue groups to be 16, however some of them being identical, the number of ‘unique Laue groups’ remains to be 11.

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8.6 Discrepancies in the Allocation of Point Groups to 2-D and 3-D Lattices Here, we consider only those cases which have come to light very recently in the study of symmetry based on mirror combination scheme and the concept of minimum symmetry form (Wahab 2020), according to which a few significant changes in the allotment of point groups from earlier classifications have become inevitable. The allocation of point groups in 2-D and 3-D crystal systems made earlier, allows us to point out the following discrepancies: Case I: Allocation of Symmetries (m and 2) In Chap. 2, we discussed about the concept of minimum symmetry form (MSF) of each symmetry category and found that: MSF of mirror and rotation operations in a 2-D system are: m and 1, and similarly, MSF of mirror, rotation and inversion operations in a 3-D system are: m, 1 and 1. Accordingly, the point groups’ m and 1, the two MSF members should be associated with each 2-D lattice. Similarly, the point groups’ m, 1 and 1, the three MSF members should be associated with each 3-D lattice, respectively. Based on the above considerations, the point groups’ m and 1, the two MSF members should be associated with the oblique crystal system (in 2-D), which actually represents the minimum symmetry crystal system. However, according to earlier classifications the point groups 2 and m have been found wrongly associated with the oblique and rectangular crystal systems, respectively. Therefore, they need to be interchanged their places for correct representation. In a similar manner for correct representation, the point groups’ m, 1 and 1, the three MSF members (and not only the two MSF members 1 and 1) should be associated with triclinic crystal system (in 3-D), which actually represents the minimum symmetry crystal system. Case II: Allocation of Symmetries (mm2 and 2/m) The illustrative example (Fig. 2.19a) shown in Chap. 2 clearly explains that two intersecting mirrors at 90° can only produce twofold symmetry along the axis of intersection of two mirrors and hence the point group ‘mm2’ and not ‘2/m’ (which can be produced only when a third horizontal mirror is placed perpendicular to the above pair of mirrors, shown in Fig. 2.19b). Therefore the two point groups, mm2 and 2/m need to be interchanged their places for correct representation (Table 2.8).

8.8 Confusion Over Centro-symmetric Nature of Diffraction Patterns

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8.7 Resolution of Symmorphic Space Groups In order to determine the symmorphic space groups in a given crystal system, it is necessary to know the number of lattices and the number of point groups associated with that crystal system. Thus taking into account the 14 space lattices and the point groups associated to each of the seven crystal systems (according to earlier classification), only 61 symmorphic space groups could be obtained as mentioned by Ashcroft and Mermin [3], Dove [4], etc. However, based on the corrected crystallographic data (provided in Tables 2.8 and 2.9 with no change in 1-D and the addition of two newly found lattices in the form of HCP and RCP in 3-D), the results of new calculation for 1-D, 2-D and 3-D have been summarized in Table 3.4. The HCP contains 7 point groups and the RCP 5, and together they contribute 12, the number that is required in 3-D to make the total symmorphic space groups to be 61 + 12 = 73. This, in turn, justifies the inclusion of HCP and RCP as the two new and independent 3-D lattices. The difference of one (i.e. 73 from arithmetic groups and 74 from present calculation) that occurs may be due to the inevitable reshuffle of point groups in low symmetry crystal systems as discussed in Chap. 2.

8.8 Confusion Over Centro-symmetric Nature of Diffraction Patterns Crystallographers have been trying hard to understand and provide a satisfactory explanation for the observation of center of symmetry exhibited by all crystals (irrespective of centro-symmetric or not themselves) in their diffraction patterns. However, this can be understood by considering the actual diffraction process. From simple observation of diffraction pattern and the related crystallographic data of a crystalline or polycrystalline material, we find that the first reflection/ diffraction peak is always due to the set of planes with largest d-spacing (in direct lattice), and for the subsequent reflections or diffraction peaks, the d-spacing of the set of planes decrease gradually. In other words, the first reflection or diffraction peak always corresponds to the least Braggs’ angle, which is found to increase gradually for subsequent peak positions. This implies that diffraction process first of all selects an atom from the closest packed plane of the given crystal structure to be at the center (acting as defined origin) and then completes the diffraction process by selecting the first nearest neighbour planes of atoms of smallest reciprocal lattice vector to construct the zone boundaries of the first Brillouin zone around the central atom after satisfying the Braggs’ condition. The diffraction process then selects the next nearest neighbour planes of atoms with increasing reciprocal lattice vectors one by one, each time and completes the construction of higher Brillouin zones (as per the Braggs’ condition), in spherically symmetric manner as shown earlier in Figs. 4.21, 4.22 (in 2-D), and Figs. 4.23, 4.24 (in 3-D), respectively. The creation of successive

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Brillouin zones around the central atom by the planes of increasing reciprocal lattice vectors takes place in a similar manner in 2-D and 3-D. The exhibition of center of symmetry in diffraction experiments by all crystals (whether centro-symmetric or not) is only due to the presence of an atom at the center (acting as a defined origin) of all Brillouin zones, which is a natural outcome of the diffraction process. This phenomenon is known to be due to Friedel’s law. This historic result definitely supports the idea of space lattice to be represented by W–S/B–Z, instead of the conventional lattices as far as the diffraction of crystals is concerned. This seems to be quite logical because all important concepts of solid state physics such as motion of electrons/phonons through periodic potential of the lattice, origin of the band gap in solid state, relationship between Fermi surface and B–Z, etc. can be understood only in terms of Brillouin zone and not in terms of conventional unit cell of a given lattice.

8.9 Confusion Over Translational Symmetries in Crystals We have discussed about this aspect in detail in Chap. 7. However, the consideration of screw axes and glide planes as a part of the translational symmetry is not found fit for crystalline solids because of the following important observations: 1. A pair of parallel mirrors with an object atom between them is able to produce its infinite images in the form of a one dimensional crystal lattice with perfect translational periodicity. Similarly, the Wigner–Seitz cells and Brillouin zones respectively in 2-D and 3-D able to generate both direct and the reciprocal crystal lattices on the basis of mirror combination scheme. The corresponding unit cells are primitive, centro-symmetric and bounded by spherically symmetric zone boundaries. These suggest that the resulting lattices have translational periodicity and not translational symmetry. 2. Brillouin-zones are nothing but the representation of spherically symmetric distribution of zone boundaries corresponding to diffraction peaks. The Brillouin zones and the diffraction patterns have a single defined origin in them and hence cannot have any translational symmetry [5]. As a result, the possibility of existence of glide planes and screw axes in crystals is ruled out. 3. According to the classification of screw axes [6] based on the number of threads (pitch), the screw axes 21 , 31 , 32 , 41 , 43 , 61 and 65 contain only onefold, 42 , 62 and 64 contain twofold and 63 contains a threefold of pure rotation, respectively. Now a very important question arises that how can they regain a rotational symmetry which they do not possess? 4. No screw axes are found to coincide with the principal axis after removal of screw component from them. They are simply parallel and found to pass only through the low symmetry sites in all cases. Therefore when the screw component is removed, they do not match with the symmetry of the principal axis, it is against its basic assumption.

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8.10 Summary 1. FCC, HCP and RCP all represent close packed structures. In mono atomic system, they have the same coordination number (12), their Wigner–Seitz cells bound to have the same shape but oriented differently due to different ways of packing of atoms in their conventional unit cells and hence they represent different crystal structures. As a result of this, 14 different geometrical shapes represent 16 space lattices. 2. A trigonal unit is found to be associated with SH throughout while the rhombohedral unit is associated with HCP only. Thus a trigonal unit remains non-close packed, while an RCP unit remains close packed. This implies that: a trigonal unit ≇ the RCP unit

3.

4.

5.

6.

7.

Thus equating a trigonal unit and a rhombohedral unit as found in common text books is not justified. Similarly, SH (a non-close packed) unit and HCP (a close packed) unit are entirely different and need to be treated separately. All results indicate that HCP and RCP need to be represented as independent lattices. The point groups’ m and 1 are the two MSF (minimum symmetry form) members in 2-D crystal systems. Therefore, they should be associated with the oblique crystal system (in 2-D) instead of the point groups 2 and 1. In a similar manner, the point groups’ m, 1 and 1 are the three MSF members (and not only the two, 1 and 1) should be associated with triclinic crystal system (in 3-D), for correct representation. The illustrative example (Fig. 2.19a) clearly explains that two intersecting mirrors at 90° can produce the point group ‘mm2’ and not ‘2/m’, which can be produced only when a third horizontal mirror is placed perpendicular to the above pair of mirrors, shown in Fig. 2.19b. Therefore the two point groups, 2/m and mm2 have been wrongly associated with monoclinic and orthorhombic crystal systems, they need to be interchanged their places for correct representation (Table 2.8). Since there are 16 space lattices (where each is of them is centrosymmetric), therefore in principle there should be 16 ‘actual’ Laue groups and not 11 as found in literature. However, some of them being identical, the number of ‘unique Laue groups’ remains to be 11. The exhibition of center of symmetry in diffraction experiments by all crystals (whether centro-symmetric or not) is only due to the presence of an atom at the center of Wigner–Seitz cells, Brillouin zones and the diffraction patterns/data, each has a defined origin. Therefore, the centro symmetry is a natural outcome of the diffraction process. This phenomenon is known to be due to Friedel’s law. The consideration of screw axes and glide planes as a part of the translational symmetry is not found fit for crystalline solids because there are many important observations going against this idea.

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References 1. Wahab, M.A., Wahab, K.M.: Resolution of Ambiguities and the Discovery of Two New Space Lattices, ISST J. Appl. Phys. 6 1, 1 (2015) 2. Verma, A.R., Krishna, P.: Polymorphism and Polytypism in Crystals. Wiley, New York (1966) 3. Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Harcourt Asia PTE Ltd., Singapore (2001) 4. Dove, M. T.: Structure and Dynamics (an atomic view of materials), Oxford Master Series in Condensed Matter Physics. Oxford University Press, Oxford (2003) 5. Leslie, A.: Crystals Symmetry and Space Groups. MRC Laboratory of Molecular Biology (LMB) Crystallography Course, Cambridge (2013) 6. Burger, M. J.: Elementary Crystallography: An introduction to the fundamental geometrical features of crystals, John Wiley and Sons Inc., Revised Printing Edition, USA (1963

Chapter 9

Fundamental Crystallography

9.1 Introduction Historical developments suggest that the subject crystallography began with a systematic study of macroscopic crystal forms, and the term ‘crystal’ was traditionally defined in terms of the arrangement and symmetry of the building blocks associated with different crystal forms. At some stage during the development of the subject, the mathematical treatment of geometry and symmetry became the most important aspect in the description of crystals, and it was during this period, the process of nomenclature, such as crystal lattices and crystal systems was took place. Later on, using the mathematical theory of crystal symmetry based on the concept of crystal lattice, Bravais formalized them in 1848, and the crystallographic community accepted the same as final. Further, with the advent of the X-rays (and other forms of) diffraction, focus shifted to the study of atomic arrangements in crystalline materials, and the definition of a crystal became a region of matter within which the atoms, ions or molecules were found to be arranged in periodic manner in all three-dimensions. This orderly arrangement in a crystalline material is now known as the ‘crystal structure’.

9.2 Crystals and Different Polyhedral Shapes The early atomic theories did not provide any useful hypothesis about the forms of crystals. However in 1611, Johannes Kepler published the title ‘Strena Seu de Nive Sexangula’, generally considered to be the first treatise on geometrical crystallography. In it, he raised the question why snowflakes always have six corners, never three, four, five, or seven. Although he could not formulate any explanation as to why this phenomenon occurred. However, he tried to explain it by considering the water crystals as made of fundamental spherical particles. Based on this, he analyzed how

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 M. A. Wahab, Mirror Symmetry, Springer Series in Solid-State Sciences 200, https://doi.org/10.1007/978-981-99-8361-2_9

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these tiny spheres could be stacked together to form the larger crystals, which eventually led him to describe the nature of packing in simple hexagonal and three cubic (sc, bcc and fcc) crystals. From these, he also deduced the forms of the polyhedron that could arise by compressing some of these arrangements of spheres. Robert Hooke was the first scientist to consider the tiny spheres used in modelling, to be the ultimate particles of matter equivalent to atoms. He was also among the first researchers to consider the structure of crystals as a general problem. In his famous book ‘Micrographia’ published in 1665, he emphasized that all crystalline forms could be built from a few basic packing arrangements of spherical atoms. He then went on to describe and illustrate how various polygonal and polyhedral figures could be constructed. In 1668, the first detailed geometrical theory of crystal structure was published by Domenico Guglielmini in his book ‘Riflessioni Filosofiche Dedotte dalle Figure de’ Sali’. In it, he described several experiments related to the precipitation of crystals from various saline solutions. Careful study of the resulting specimens led him to conclude that only four primary crystal shapes existed, and that all other geometries observed to arise from various internal combinations of these four basic shapes and their associated salts. The four salts that possessed these forms were common salt (the cube), salt of vitriol (the rhombohedron), nitre (the hexagonal prism), and alum (the octahedron). The author further emphasized that these four basic shapes were built from collections of very small, invisible pieces of the named salt. In 1705, he extended his original ideas by developing them mathematically based on his observations of the physical crystals. In this work, he also anticipated the theories of Bergman and Haüy by recognizing that there is a tiny crystal molecule that formed the nucleus inside the crystal. It is the crystals with their plane faces, sharp edges, pointed vertices and the varied colours, which attracted the interest of mineralogists/crystallographers from early times. Indeed, these properties also formed the first definition of a crystal; since most other naturally occurring substances exhibit rounded or curved outlines. Further, the external shape of a crystal was believed to be the manifestation of its internal structural regularity. Ideally speaking, it comprises a three-dimensional, regular stacking of identical building blocks. For an example, Fig. 9.1 illustrates the crystal form of an octahedron obtained by a three-dimensional stacking of minute and identical cubical blocks, as envisaged by the French mineralogist Renée Just Haüy (1743–1822), sometimes referred to as Abbé Haüy.

9.3 Crystal Polyhedra and the Concept of Lattice Based on the above fundamental studies and other similar isolated time to time crystallography related developments, suggest that mineralogists had acquired the knowledge of crystals of different solid materials in the form of different polyhedral shapes with passage of time. Some of the important regular polyhedral shapes (also

9.3 Crystal Polyhedra and the Concept of Lattice

201

Fig. 9.1 Crystal form of an octahedron obtained from minute cubical blocks

known as Platonic solids) are provided in Table 9.1. Also, some other important and fundamental information related to polyhedra are [1]: 1. In 1752, the Swiss mathematician Leonhard Euler published his discovery which states that for any three-dimensional polyhedron, the number of faces (F), edges (E) and vertices (V) are related by the formula: F − E + V = 2. 2. In 1768, Carl Linne in third volume of his ‘Systema Naturae’, applied morphological approach, favouring it over other physical or chemical examinations while classifying crystals. Relying on the external form of the crystals, he grouped together those that are cubic, hexagonal, octahedral, and rhombic dodecahedral. His approach of sorting crystals by external form had tremendous impact on emerging crystallographers and mineralogists of that time. 3. Towards the end of eighteenth century, two innovative French mathematicians, Claude Navier and Augustin Cauchy simultaneously with Seeber proposed solids as arrays of mathematical points extending symmetrically through 3-dimensional space. But at the time, these researches were largely unrecognized by most crystallographers. However, the idea of a space filling lattice inside a crystal was proposed by some other researchers. Based on the above, it is not wrong to say that almost all required information, such as the idea of various polyhedral shapes, their symmetries, concept of lattice, etc. were known to Bravais from other sources. He then very rightly made use of the available information to his mathematical theory of crystal symmetry based on the concept of crystal lattice and proposed/confirmed the existence of 1 linear (1-D) lattice, 5 plane (2-D) lattices and 14 space (3-D) lattices in the year 1848.

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Table 9.1 Characteristics of regular polyhedron Name

Polygon Vertices (V) Edges (E) Faces (F) Euler equation Image F−E+V=2

Tetrahedron

3

4

6

4

2

Cube

4

8

12

6

2

Octahedron

3

6

12

8

2

Dodecahedron 5

20

30

12

2

Icosahedron

12

30

20

2

3

A simple representation relating the crystal structure with the lattice and the associated basis was also put forward during the time of Bravais Crystal Structure = Lattice + Basis

(9.1)

Crystallography continued to develop, and in the process 2 (1-D), 10 (2-D) and 32 (3-D) point groups (also called crystal classes) were obtained using non-translational symmetry operations like rotation, mirror, inversion and their compatible combinations. Similarly, 230 space groups were derived using the combination of microscopic translations, such as screw axes and glide planes with point group symmetries. However, the space group related developments are found to have some inherited shortcomings as pointed out in earlier chapters. Equation 9.1 is of fundamental nature and explains that the crystals are built up of regular arrangements of atoms in one, two and three dimensions; these arrangements are represented by a motif or a repeat unit, called the unit cell. This unit cell is the smallest volume unit which represents the crystal in terms of its structure and symmetries. When such a unit cell is repeated large number of times (theoretically infinity without producing overlaps or gaps) in three dimensions, generates the entire crystal. In simplest (monoatomic) crystals, the unit cell contains a single atom, as in copper, silver, gold, iron, aluminium, and other metal crystals.

9.3 Crystal Polyhedra and the Concept of Lattice

203

Fig. 9.2 a Various choices of primitive unit cells, b the choice of a centered unit cell

In reality, the assumption of indefinite repetition of the identical structural unit cell in space to construct a crystal, simplifies our problem drastically. This means that the consideration of entire crystal lattice is not necessary because the whole crystal lattice is represented by only one unit cell. However, since the lattice is supposed to be infinite in extent, so there are infinite number of choices of the unit cells (some are shown in Fig. 9.2). Under this situation, our objective is to select the unit cell in a best possible manner. In order to select such a unit cell, the following criteria are to be adopted. 1. If possible, select the smallest unit cell, as this requires least amount of information to describe the contents of the unit cell. 2. If possible, a cell with 90° angles (and in some cases 120°) should be selected so that visualization and geometrical calculations are easier. 3. Select a unit cell which reflects both structure and symmetry of the lattice, it represents. In reality, this is the most important criterion for selecting the correct unit cell. The crystal lattices are assumed to be infinite (while in reality a crystal has finite size), since the order of magnitude of the unit cell parameters a, b and c of a crystal is just a few angstroms (1 Å = 10−1 nm), whereas the average size crystal with dimensions of the order of a micro-meter will contain several thousand such unit cells distributed along three crystal dimensions. However, the concept of infinite lattice is simply an assumption and needs to be used with caution to understand crystal structure.

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9.4 Lattice, Basis and the Crystal Structures A space lattice is defined as an infinite three-dimensional array of points in space where each point in the lattice has identical surrounding of points. In other words, it can be defined as the set of points in space such that the surrounding of one point is identical with the surrounding of all others. Crystal lattices are classified on the basis of their symmetries [2, 3]. Likewise, the basis (or the motif) can be defined as a group of one or more number of atoms, ions or molecules, located in a specific manner with respect to each other and associated with each and every lattice points. Similarly, a unit cell is defined as an ‘imaginary’ parallel sided region of a structure enclosing space, from which the entire crystal can be constructed by purely translational displacements. It is either primitive or non-primitive (in conventional forms of representation) and can be understood as: 1. It is the smallest possible ‘structural’ unit that is repeated, three-dimensionally. 2. It contains a complete description of the structure as a whole. 3. Complete crystal structure can be generated by the repeated stacking of adjacent unit cells, face to face, throughout three dimensional space. 4. It is the crystallographic analogue of an ‘atom’. Keeping the molecular and crystal structures and their geometrical shapes in view, the lattice representations can be classified into the following four possible categories (Fig. 9.3): (i) Point lattice (zero dimensional lattice) (ii) Linear lattice (one dimensional lattice)

Fig. 9.3 Four different types of lattice representations

9.4 Lattice, Basis and the Crystal Structures

205

Fig. 9.4 a A sphere, b corresponding lattice point

(iii) Plane lattice (two dimensional lattice) (iv) Space lattice (three dimensional lattice) 1. Zero Dimensional Lattice From geometrical point of view, a lattice point is known to have no (or zero) dimension. Accordingly, a lattice point is an imaginary concept. On the other hand, an atom, an ion or a molecule (such as fullerene C60 ) containing large number of identical atoms associated with a single lattice point is real, but from geometrical point of view it remains a zero dimensional object. An isolated object such as a spherical ball (Fig. 9.4) which has infinite symmetry elements in the form of infinite mirror planes and infinite rotational axes all passing through its center. In fact, a sphere is the only object in the geometrical universe which possess perfect symmetry. This is the reason, a sphere is called the ‘holy grail’ of symmetry. 2. One Dimensional Lattice A collection of atoms/ions in a row is a one dimensional array where the arrangement of atoms/ions is perfectly periodic with the periodicity say ‘a’ (Fig. 9.5a). The corresponding lattice points is the representation of a linear lattice (Fig. 9.5b). The characteristic feature of a linear lattice is that the environment around one point is identical with the environment around any other point in the lattice. In a hard-sphere model, the shortest distance between two like atoms is one atomic diameter. We can easily define Eq. 9.1, and understand the crystal structure in terms of a linear lattice with a basis (the motif) in one dimensional representation as:

Fig. 9.5 a Linear array of number 7, b corresponding linear lattice

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Fig. 9.6 Two dimensional array of: a objects, b points; a plane lattice

3. Two Dimensional Lattice ⭢ to the entire lattice array due When we add another non-collinear translation ‘b’ a ’, a two-dimensional array of objects is obtained (Fig. 9.6a). The to the translation ‘⭢ corresponding collection of two-dimensional points shown in Fig. 9.6b is called a plane lattice. In principle, there are infinite number of possible plane lattices as there are no ⭢ A 2–D restrictions on the length and direction of the primitive vectors a⭢ and b. lattice with arbitrary vectors and angles is known as an oblique lattice and belongs to a minimum symmetry system. However, choosing the lattice parameters (i.e. the axial lengths a, b, and angle between them) with care, other possible lattices can be constructed which possess 2-, 3-, 4- and sixfold pure rotations along with higher order mirror symmetries. In order to get such lattices, we need to impose restrictions on the primitive vectors and angles between them. Each distinct set of restrictions will lead to a distinct type of lattice, which can be characterised by the set of symmetry operations that transform the lattice into itself. In 2-D, there are four such sets of restrictions giving rise to four special lattices. Since, the rectangular system has two lattices (primitive and non-primitive, one each), in all there are five 2-D lattices, one general (or oblique) type and four special types. We can easily define Eq. 9.1 and understand the crystal structure in terms of a square lattice (where a = b, γ = 90°) with a basis (the motif) in a 2-D representation, as shown in Fig. 9.7.

9.4 Lattice, Basis and the Crystal Structures

207

Fig. 9.7 a A simple square lattice, b basis with two atoms, and c crystal structure

Three Dimensional Lattice When a third non-coplanar translation ‘⭢ c’ is added to the entire plane pattern due to ⭢ a three dimensional array of objects is obtained (Fig. 9.8a). a ’ and ‘b’, translations ‘⭢ The corresponding collection of three-dimensional points shown in Fig. 9.8b is called a space lattice. This lattice is nothing but a 3-D assembly of mathematical points in space. The characteristic feature of a space lattice is that the environment around one point is identical with the environment around any other point in the lattice (same as in 1- and 2-D systems).

Fig. 9.8 a 3-D array of number 7, b corresponding 3-D lattice

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Adopting the similar procedure as in 2-D, all 3-D lattices can be obtained. As mentioned above, Bravais proposed 14 lattices for 3-D system, including one general (called triclinic) and 13 special, based on their symmetry axes.

9.5 Crystal Dimension and Related Symmetries Symmetry is one of the most important properties used in the identification of crystalline substances. Further, crystal shape (dimension) and the associated symmetries are inherently related to each other. If the orientation of the crystal (or an object) is brought into coincidence after applying some symmetry operation(s), then the crystal (the object) is said to possess the symmetry with respect to that operation(s). Therefore, the symmetry of a given crystal can be determined by the reference of three different types of symmetry elements, i.e. planes of symmetry, symmetry axes, and the centre of symmetry as discussed below. Using a cubic unit cell, these symmetries are illustrated in Fig. 9.9. Mirror (or Plane of) Symmetry A plane of symmetry (also called the ‘mirror plane’ or ‘symmetry plane’) is an imaginary plane, which divides the crystal into two halves, they are mirror images of each other, shown for cubic crystal in Fig. 9.9. A crystal can have one or more planes of symmetry and denoted as σ. A symmetry plane parallel with the principal axis is called vertical (σv ) mirror and the one perpendicular to it, horizontal (σh ) mirror. Other type of symmetry plane exists such that a vertical symmetry plane additionally bisects the angle between the two twofold rotation axes perpendicular to the principal axis, such a plane is called dihedral (σd ) mirror, there are also simple diagonal planes. Besides, a symmetry plane can also be identified by its Cartesian orientation, e.g., (xy), (xz), or (yz). Similarly in 2-D, the mirror lines are found to exist in oblique, rectangular, triangular, square and hexagonal lattices as shown in Fig. 9.10. Rotational (or Axial) Symmetry The axis of symmetry is an imaginary line about which a crystal is rotated through 360°/n until it assumes a congruent position; where n may be 1, 2, 3, 4, or 6, depending on the number of times the congruent position is repeated. In other words, if an object is brought into a position of self-congruence through the smallest non-zero rotation angle (θ), then the axis is said to have the n-fold rotational symmetry, where n=

360◦ θ

These correspond to onefold (monad), twofold (diad), threefold (triad), fourfold (tetrad), and sixfold (hexad) axes, respectively. For example, if an object is rotated about an axis that repeats itself after every 90° of rotation, then it is said to have a fourfold rotational symmetry. Therefore, the axis

9.5 Crystal Dimension and Related Symmetries

Fig. 9.9 Representation of different symmetries of a cube

Fig. 9.10 Illustrations showing mirror lines dividing the objects into two halves

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Fig. 9.11 Proper rotational symmetries in 3-D

along which the rotation is performed, is an element of symmetry and is referred to as a rotation axis. The following types of rotational symmetry axes are possible in crystals (Fig. 9.11). Onefold Rotation Axis: An object that requires a rotation of complete 360° in order to restore it to its original appearance, has no rotational symmetry. Since it repeats itself only one time after every 360° rotation, it is said to have only onefold axis of rotational symmetry. Every object, irrespective of symmetric or asymmetric will possess this. Twofold Rotation Axis: If an object appears identical after a rotation of 180°, that is the identical appearance repeated twice in a 360° rotation, then it is said to have a twofold rotation axis (360/180 = 2). Note that in these examples the axes we are referring to are imaginary lines that extend perpendicular to the page. A filled oval shape represents the point where the twofold rotation axis intersects the page. Threefold Rotation Axis: Objects that repeat themselves upon rotation of 120° are said to have a threefold rotational symmetry (360/120 = 3), and the identical appearance is repeated 3 times in a 360° rotation. A filled triangle is used to symbolize the location of threefold rotation axis. Fourfold Rotation Axis: If an identical appearance of an object is repeated after 90° of rotation, it will repeat 4 times in a 360° rotation, as mentioned above. A filled square is used to symbolize the location of fourfold axis of rotational symmetry. Sixfold Rotation Axis: A rotation of 60° about an axis causes the object to repeat itself, then it has sixfold axis of rotational symmetry (360/60 = 6). A filled hexagon is used as the symbol for a sixfold rotation axis. All five proper rotational symmetries in 3-D are shown in Fig. 9.11. 4. Centre of Symmetry (or Inversion Center, I) Centre of symmetry (or inversion center, i) is the symmetry of a crystal (or unit cell) with respect to a point situated at its center. Actually, the center of symmetry is an act of inversion about this central point of the unit cell. Therefore, a crystal is said to have a centre of symmetry when like (or similar) faces and edges are arranged

9.6 Miller Indices to Represent Points, Directions and Planes in Crystals

211

Fig. 9.12 Triclinic unit cell exhibiting center of symmetry

in corresponding positions on opposite sides of the central point. If a line is drawn through the central point, the points of similar character are present equidistantly on both sides of this point as shown in Fig. 9.12 for a least symmetric triclinic lattice. Therefore, one can define that when similar faces or edges occur in parallel pairs, on opposite sides of a crystal, then the crystal is said to possess a centre of symmetry. The occurrence of parallel pairs of faces or edges is a must for any crystal (or any unit cell, i.e. conventional, W–S or B–Z) to possess a center of symmetry. Similarly, for all other lattices. Identity Symmetry This symmetry is abbreviated as E (or 1), from the German ‘Einheit’ meaning unity. This symmetry element simply makes no change in the crystal/unit cell; every unit cell/molecule has this symmetry element. Actually, it is analogous to multiplying by one (or unity). This is a very useful symmetry in group theory. Using these ‘elements of symmetry’, crystallographers recognized 32 crystal classes (also called the point groups) and they classified them into seven crystal systems based on symmetry, earlier. The cubic crystals belong to the highest symmetry system, while the triclinic crystals to the lowest symmetry system.

9.6 Miller Indices to Represent Points, Directions and Planes in Crystals A simple inspection of any well-developed crystal, suggests the desirability of a method to describe the geometrical features of a crystal appearing in the form of its faces, edges and corners. As a three-dimensional body, it may be assigned to three non-coplanar, right-handed, ‘reference axes’ designated by x, y and z, as illustrated in Fig. 9.13. The inter-axial angles are symbolized as α between y and z, β between z and x, and γ between x and y. The crystallographic reference axes are set always along (or in close relation to) the symmetry directions of the crystal, but they are not

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Fig. 9.13 Indices of crystal lattice sites and the direction [q r s]

always orthogonal: two lines are orthogonal if they are mutually perpendicular, that is there is no component of any one on to another [2, 4]. Point (or Site) Indices The position of any lattice site relative to a chosen origin is defined by three of its coordinates x, y, z as shown in Fig. 9.13. These coordinates are in general expressed as; x = qa, y = rb, and z = sc, where a, b and c are the lattice parameters and q, r and s are integers. However, if the lattice parameters are used as the units of lengths along the respective axes then the lattice coordinates can be obtained simply in the form of numbers q, r and s. These numbers are then termed as point (or site) indices and are written either in the form [[q r s]] or simply q r s without brackets. For a negative index, the minus sign is written above the index. As an example, for a point or site with coordinates x = − 2a, y = 1b and z = − 3c, the point (or site) indices are written as [[213]] or simply 213. While labelling different points (or sites) of a unit cell, we follow the same procedure as we do for listing the points in any Cartesian coordinate system. For the purpose, let us consider a point P as in Fig. 9.14a and start labelling from the origin of the unit cell, from where we travel a distance qa along x-direction, rb along ydirection, and sc along z-direction, respectively without any disconnection to reach the point P. To obtain the point or site indices q, r and s in a given unit cell, we adopt the following procedure: 1. Start from the origin. 2. In order to count the number of lattice constants (the axial lengths), we need to move in the x-, y-, and z-directions to reach the given point. 3. Write the point as q r s with or without square brackets. Do not convert the coordinates to reduced integers. We can verify that the point 1/2 1/2 1/2 in the bcc structure is not the same as the point 111. Process of finding the location for a given point or site indices in the unit cell can be done in a similar way. For example, in a bcc unit cell shown in Fig. 9.14b, the

9.6 Miller Indices to Represent Points, Directions and Planes in Crystals

213

Fig. 9.14 Crystallographic point coordinates a of a point P, b positions of atoms in bcc

path to be followed from the point number 1 to reach the point number 9, is 0a in the x-direction, 1a in both y-, and z-directions, respectively. Thus the point coordinates of the point number 9 are 011. On the other hand, the point coordinates for the point number 5 are 1/2 1/2 1/2. Similarly, the indices for all other points can be obtained. In a simple cube, the site indices of all the eight corners are shown in Fig. 9.15a, while the site indices of three principal points and the corresponding directions are shown in Fig. 9.15b. Similarly, the site indices in bcc and fcc are shown in Fig. 9.16. Indices of Direction A crystallographic direction is defined as a line or vector between two points. To draw a given direction with [uvw] indices, we adopt the following procedure: 1. Choose a point as the origin in a (cubic) unit cell for the given direction(s). If any of the index u, v, or w is negative, shift the origin to another point by moving in

Fig. 9.15 a Point (or site) indices, b site indices and principal directions in a cube

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Fig. 9.16 Point (or site) indices in a bcc, and b fcc

2.

3. 4. 5.

the positive direction on that axis, the tail of the direction vector will be at this point. To determine the co-ordinates of two points that lie on the given direction, we start from the chosen origin (tail of the vector) and move a distance ua in xdirection, vb in y-direction, and wc in z-direction without any disconnection to reach the final point. Subtract the co-ordinates of the tail point from those of the head point and express the same in terms of the unit cell dimensions. If necessary, multiply or divide these numbers by a common factor to reduce them to the smallest integers. The three indices are enclosed in square brackets as [uvw]. A negative integer is represented with a bar over the number. Draw a line from the chosen origin to the final point. Add an arrow head at the point. For a better clarity, let us consider the following examples.

Example 1: Let us consider some simple directions, such as [110], [011] and [101] shown in Fig. 9.17a, and slightly more complicated directions, such as [111], [111] and [111] shown in Fig. 9.17b and try to understand how to draw them. Simpler directions such as [110], [011] and [101] shown in Fig. 9.17a are easy to draw, just select a crystallographic point as the origin and move along the x-, y-, or z-directions, the required distances. However, for slightly more complicated directions containing negative signs, such as [111] and [111] shown in Fig. 9.17b, it is more convenient to shift the origin towards the point containing negative indices as stated in the procedure. Draw and label the final point. To check that the direction of shifted origin is properly made, subtract the point coordinates of the arrow tail from those of its head. For [111] direction, we have

9.6 Miller Indices to Represent Points, Directions and Planes in Crystals

215

Fig. 9.17 Crystallographic directions a [110], [011] and [101], b [111], [111] and [111]

u = 1 − 0 = 1, v = 1 − 0 = 1 and w = 0 − 1 = −1 Similarly for other direction [111], u = 0 − 1 = −1, v = 1 − 0 = 1 and w = 0 − 1 = −1 The resulting indices of directions [uvw] are: [111] and [111]. Miller Indices of Planes Miller indices of planes are obtained by adopting the following procedure: 1. If the given plane passes through the origin of co-ordinate system, the origin must be shifted to another corner of unit cell. 2. Identify the points at which the given plane intercepts the x-, y-, z-axes and express the intercepts in terms of the lattice parameters. Determine the intercepts of the plane along x-, y- and z-axes in terms of lattice constant (the axial lengths). 3. Divide the intercepts by appropriate unit translations. 4. Take reciprocals of these numbers. Intercept for a plane parallel to an axis is to be taken as infinity with reciprocal equal to zero. 5. If fractions result, multiply each of them by smallest common divisor. 6. Put the resulting integers in parenthesis, i.e. (hkl) to get the required indices of that and all other parallel planes. Example 2: Consider the example of some simple planes, such as (100), (010), (001), (110) and (111) shown in Fig. 9.18. They are easy to draw by selecting again the 000 crystallographic point as the origin. However, for slightly more complicated planes containing negative indices, such as (110) and (111) shown in the lower figure, we need to shift the origin towards the axis containing the minus (−) sign. This point will act as the new origin from where the intercepts are to be found.

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Fig. 9.18 Crystallographic planes (100), (010), (001), (110), (110), (111) and (111)

Next, let us consider a plane (hkl) which passes through the origin, for example as shown in Fig. 9.19a. For such cases, do the following:

Fig. 9.19 a Plane that passes through the origin. b The plane shifted + 1 lattice constant in the z-direction

9.7 Representation of Directions and Planes in Cubic Crystal System

217

1. Shift this plane to one lattice constant (the axial length) along the axis such that it no longer passes through the origin (Fig. 9.19b). This new plane is crystallographic equivalent and parallel to the first plane. 2. Now, find the intercepts of this plane along x-, y-, and z-axes. If the plane doesn’t intercept an axis (then it is parallel to that axis), call its intercept as infinity. Write down these values by taking care of the negative intercepts. 3. Take the reciprocal of these intercepts. The value of 1/∞ is zero. 4. If necessary, multiply or divide by a common number to put the indices in the most reduced integer form. 5. Enclose the indices in parentheses without commas (hkl).

9.7 Representation of Directions and Planes in Cubic Crystal System (a) Representation of Directions Directions of known indices can be represented in a (for example in cubic) unit cell by using the following procedure: 1. Divide the given indices of direction by a number such that the resulting indices become ≤ 1, they represent the coordinates of the lattice site nearest to the origin in the given direction and lie within the unit cell. 2. If a lattice site nearest to the origin contains fractional coordinates, remove the fractions by multiplying them with suitable number. 3. Mark the length of the position vector along the respective coordinate axes without disconnection. Join the origin with the end point to get the required [uvw] or [hkl] direction. Example 1: Draw the [112], [121], [121] and [110] directions in a cubic unit cell. Because of the presence of the digit 2 in first three directions, their position coordinates are obtained by dividing each of them by 2, while the last one remains as such. Doing so, the position coordinates of the atoms nearest from the origin along four directions are: [[ ]] [[ ]] [[ ]] 11 1 1 1 1 1 , 1 , −1 and [[110]] 22 2 2 2 2 Therefore, to determine the locations of the position coordinates of the first, second and fourth directions, we need to move half the unit along x and y directions and one unit along z direction; half the unit along x and z directions and one unit along y direction and one unit each along x and y directions, respectively. On the other hand, to determine the locations of the position coordinates of the third direction, we need to move half the unit along x and z directions and one unit along −y direction, respectively. These directions are shown in Fig. 9.20.

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Fig. 9.20 Representation of [112], [121], [121] and [110] directions in a cubic unit cell

Fig. 9.21 Representation of [121] direction in a cubic unit cell

However, there is an alternative method to represent the third direction. Because of minus sign with digit 2, we need to move two steps in the positive y-direction to shift the origin to locate the ‘tail’ of the direction at this point. Now to determine the position coordinates of the third direction, we need to move one unit each in the positive x- and z-directions, and 2 units in the negative y-direction, respectively as shown in Fig. 9.21. (b) Representation of Planes Planes of known Miller indices can be represented in a cubic unit cell by using the following procedure: 1. Take the reciprocal of given Miller indices. They will represent the intercepts in terms of axial units. 2. If any of the Miller index is negative, shift the origin by moving it along that axis by one unit. 3. Mark the length of the intercepts on the respective coordinate axes, each one starting from the origin. Join their end points; the resulting sketch will represent the required (hkl) plane. Example 2 Represent the (210) plane in a cubic unit cell.

9.8 Crystal Parameters in 2-D and 3-D Systems

219

Fig. 9.22 Representation of (210) plane in a cubic unit cell

In order to draw the plane inside the cubic unit cell, first take reciprocals of the indices to obtain their intercepts, they are: 1 −1 1 1 , = 1, = ∞ = 2 1 0 −2 Since the x-intercept contains a negative sign, and we wish to draw the plane within the unit cell, so we need to move the origin to one unit in the positive xdirection to [[100]] point. Then we move −1/2 unit to locate the x-intercept and + 1 unit to locate y-intercept. The required plane is parallel to the z-axis as shown in Fig. 9.22.

9.8 Crystal Parameters in 2-D and 3-D Systems Based on the lattice parameters a, b and γ, application of different restrictions and verification of the centering conditions, we finally obtain five types of 2-D lattices. A summarized form of five 2-D lattices is provided in Table 9.2. In a similar manner, considering the lattice parameters a, b, c, α, β and γ, and applying different restrictions and centering conditions, according to Bravais we have 14 different space lattices and 7 crystal systems in three dimensions. However, two new lattices were found in a recent study of close packing of identical atoms [5], which makes the total space lattices to be 16, they are then classified into 8 crystal systems. A summarized form of 16 3-D lattices is provided in Table 9.3.

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Table 9.2 A summary of five 2-D lattices Axes and angles

Point group

Oblique

a /= b, γ /= 90°

m, 1

Square

a = b, γ = 90°

4, 4mm

Hexagonal

a = b, γ = 120°

3, 3m 6, 6mm

Rectangular primitive

a /= b, γ /= 90°

2, mm2

Centered rectangular

a /= b, γ /= 90°

2, mm2

Lattice

Unit cell

9.9 Crystal Systems and Axial Systems The crystal systems were initially introduced by Christian Samuel Weiss (1780– 1856) when he translated the text books written by Haüys. They were based on the analysis of the alignment of particularly striking directions of the crystals, the axes ‘around which everything was equally distributed’. Accordingly, the crystal systems constitute a symmetry-related classification of crystals by means of crystallographic axes of coordinates. Weiss used the axes to clearly denote the position of all crystal faces or planes within the lattice for the first time by the ratios between their intercepts on the axes, called the Weiss indices. However, nowadays we mainly use the lowest integral common multiples of the reciprocal intercepts, called the Miller indices. In a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case both the crystal and lattice systems have the same name. However, a set of five point groups are found to be assigned to three lattice systems, to trigonal, rhombohedral and HCP, and two other point groups to trigonal and HCP, they all exhibit three fold proper rotational symmetry. In total there are eight crystal systems: they are triclinic, monoclinic, orthorhombic, rhombohedral (also called RCP), tetragonal, trigonal/HCP, hexagonal, and cubic.

9.10 Summary

221

Table 9.3 A summary of crystal systems and related information S. No

Crystal system

Lattice type

Unit cell parameters

Associated point group symmetries

1

Triclinic

Primitive (P)

a /= b /= c α /= β /= γ

m, 1,1

2

Monoclinic

Primitive (P) Base-centered (c)

a /= b /= c α = β = 90° /= γ

m, mm2, (1)

3

Orthorhombic

Primitive (P) Base-centered (c) Body-centered (I) Face-centered (F)

a /= b /= c α = β = γ = 90°

222, 2/m, mmm

4

Rhombohedral (RCP)

Primitive (P)

a=b=c α = β = γ = 60°

3, 3, 32, 3m, 3m

5

Tetragonal

Primitive (P) Body-centered (I)

a = b /= c α = β = γ = 90°

4, 4, 4/m, 422, 4mm, 42m, 4/mmm

6

Trigonal

Primitive (P)

a=b=c 3, 3, 32, 3m, 3m, α = β = γ /= 90° and < 6, 62m 120° a = b = c = 2R α = 60°, β = γ = 120°

HCP 7

Hexagonal

Primitive (P)

a = b /= c α = β = 90° and γ = 120°

6, 6/m, 622, 4mm, 6/mmm

8

Cubic

Primitive (P) Body-centered (I) Face-centered (F)

a=b=c α = β = γ = 90°

23, m3, 432, 43m, m3m

* R radius of the sphere, (1) not counted towards the total

Generally speaking, based on the axial systems and angle between them, two different axial systems are in use: (a) the orthogonal axial system, and (b) the crystallographic (or hexagonal) axial system. They are provided in Figs. 9.23, 9.24. However, triclinic crystal system does not fit in either of the two axial systems, while the monoclinic crystal system fits only partially. Table 9.4 provides a general classification of crystal system/s belonging to different axial systems, especially for the purpose of crystal structure determination.

9.10 Summary 1. According to Leonhard Euler, in any 3-dimensional polyhedron, the number of faces (F), edges (E) and vertices (V) are related according to the formula F − E + V =2

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Fig. 9.23 Orthogonal axial systems, a right handed and b left handed Fig. 9.24 Hexagonal axial system

Table 9.4 Relationship between axial system and crystal system S. No

Axial system

Angular relations

Crystal system

1

General

α /= β /= γ /= 90◦

Triclinic

90◦

2

Semi-orthogonal

α = γ, β /=

3

Orthogonal

α = β = γ = 90◦

4

Hexagonal

α=β=

90◦ , γ

Monoclinic =

Orthorhombic, Tetragonal, Cubic 120◦

Trigonal/HCP, RCP, Hexagonal

2. A simple representation relating the crystal structure with the lattice and the associated basis is given by Crystal Structure = Lattice + Basis 3. In order to select a suitable unit cell in the given lattice, the following criteria should be adopted. (a) If possible, select the smallest unit cell, as this requires least amount of information to describe the contents of the unit cell. (b) If possible, a cell with 90° angles (and in some cases 120°) should be selected so that visualization and geometrical calculations are easier.

9.10 Summary

223

(c) Select a unit cell which reflects both structure and symmetry of the lattice, it represents. In reality, this is the most important criterion for selecting the correct unit cell. 4. Symmetry of a given crystal can be determined by the reference of three different types of symmetry elements, i.e. planes of (or mirror) symmetry, symmetry (rotation) axes, and the centre of symmetry. 5. To obtain the point or site indices q, r and s in a given unit cell, we adopt the following procedure: (a) Start from the origin. (b) In order to count the number of lattice constants (the axial lengths), we need to move in the x-, y-, and z-directions to reach the given point. (c) Write the point as q r s with or without square brackets. Do not convert the fractional coordinates to reduced integers. 6. Directions of known Miller indices can be represented in a cubic unit cell by using the following procedure: (a) Divide the given Miller indices by a number such that the resulting indices become ≤ 1, they represent the coordinates of the lattice site nearest to the origin in the given direction and lie within the unit cell. (b) If a lattice site nearest to the origin contains fractional coordinates, remove the fraction by multiplying them with suitable number. (c) Mark the length of the position vector along the respective coordinate axes without disconnection. Join the origin with the end point to get the required [uvw] or [hkl] direction. 7. Planes of known Miller indices can be represented in a cubic unit cell by using the following procedure: (a) Take the reciprocal of the given Miller indices. They will represent the intercepts in terms of axial units. (b) If a Miller index is negative, shift the origin by moving it in the positive direction along that axis by one unit. (c) Mark the length of the intercepts on the respective coordinate axes, each one starting from the origin. Join their end points; the resulting sketch will represent the required (hkl) plane. 8. Based on the lattice parameters a, b and γ, application of different restrictions and verification of the centering conditions, we obtain five types of 2-D lattices. 9. Similarly, based on the lattice parameters a, b, c, α, β and γ and applying different restrictions, centering conditions and including two new found lattices, we obtain 16 lattices, which are classified into 8 crystal systems in three dimensions. 10. Based on the axial systems and angle between them, two different axial systems are in use: (a) the orthogonal axial system and (b) the crystallographic (or hexagonal) axial system.

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References 1. Schuh, C. P.: Mineralogy and Crystallography: On the History of These Sciences From Beginnings Through 1919, Tucson, Arizona (2007) 2. Wahab, M.A.: Solid State Physics: Structure and Properties of Materials, 3rd ed. Narosa Publishing House, New Delhi (2015) 3. Wahab, M.A.: Essentials of Crystallography, 2nd ed. Narosa Publishing House, New Delhi (2014) 4. Wahab, M.A.: Numerical problems in Crystallography, eBook, Springer Nature, Singapore (2021) 5. Wahab, M.A., Wahab, K.M.: Resolution of ambiguities and the discovery of two new space lattices. ISST J. Appl. Phys. 6(1), 1 (2015)