Handbook of Crystallography

Transcription

1 Handbook of Crystallography

2 A.G. Jackson Handbook of Crystallography For Electron Microscopists and Others With 114 Figures Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona

3 A.G. Jackson, Ph.D. Universal Energy Systems, Inc. Dayton, OH USA and Department of Mechanical Engineering and Materials Science Wright State University Dayton, OH USA Library of Congress Cataloging-in-Publication Data Jackson, A. G. (Allen G.) Handbook of crystallography: for electron microscopists and others I A.G. Jackson. p. cm. Includes bibliographical references and index. ISBN-13: Crystallography-Handbooks, manuals, etc. I. Title. QD908.J dc Printed on acid-free paper 1991 Springer-Verlag New York Inc. Sof'tcover reprint of the hardcover 18t edition 1991 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the lfade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Camera-ready copy prepared by the author ISBN-13: DOl: / e-isbn-13 :

4 Dedicated to Gene, whose dream was cut short, to Alice, who found new dreams, to Drew, who faced death and chose life.

5 Preface This book resulted from a series of frustrations. Analytical electron microscopy requires exactly what its name implies: quantitative information to conduct an analysis. The frustrations arose when I started hunting for specific forms of equations in a form understandable to a non-crystahographer, for definitions of subtle concepts related to crystallography, for intelligible interpretations of space group symbols and their significance. What I frequently discovered was that such information was buried in a giant tome and couched in terms familiar to crystallographers but not to electron microscopists in general, or it was located in an old reference not available in my library, or it was found in an out-of-print book, or it was in a Russian book no longer available, etc. So to minimize the frustrations, I started a notebook containing the details, particularly after I had found forms of equations useful for quick calculations or equations in a form useful for proving, doing, or extending calculations found in a reference. The resulting notebook grew to a respectable size, requiring some organizing of the contents. Finally, the size became large enough, and has proven useful enough, to produce the notebook as a book. The readers to whom this book is aimed are those microscopists, and others, who are not crystallographers, but who need details about crystal structures in order to interpret images and diffraction effects and patterns present in the electron microscope. Hence, this book is practically oriented. No attempt has been made to make equations mathematically rigorous beyond the rigor needed to accomplish a calculation. Every attempt has been made to insure the mathematical correctness of the equations, because I found correcting mistakes was a major frustration when using an expression from a reference. This book is not a handbook of electron microscopy. It is a handbook to be used when doing electron microscopy for which crystallographic information plays a fundamental role in the interpretation of the data and images generated. As a minimum my hope is that the book will be a helpful starting reference source for microscopists, since full treatments of the subjects presented would take considerably more pages than I have devoted to them. The book is divided into 13 chapters. Although the division is somewhat arbitrary, I have grouped items and subjects where possible. Chapter 1 contains mathematics useful in understanding structures, lattices, and the like. Chapter 2 contains specifics for several common crystal systems, bringing together several definitions to be found in the literature. The purpose here is to highlight the fact that different authors use different definitions, which sometimes are not equivalent or are so specialized that their use beyond the one application leaves a reader stranded. Chapter 3 is a brief overview of diffraction pattern analysis. Chapter 4 covers packing and stacking, one of those subjects that lends itself to misinterpretation because of loose use of symbols, particularly the A, B, C site symbols. Chapter 5 describes the seven crystal systems in detail. This chapter contains the equations for each system that one may need to do simple calculations or to refresh one's memory vii

6 viii Preface about axes used. Chapter 6 covers transformations of crystal system axes in a practical way. Chapter 7 deals with slip systems. Chapter 8 includes discussions of various projections found to be useful in transmission electron microscopy (TEM). Chapter 9 covers the essentials of symmetry and the wondrous symbolism used in this powerful representation of lattices and structures. Chapter 10 is a brief discussion of convergent beam electron diffraction, especially the use of higher order Laue zone (holz) rings and the parities of the rings. Chapter 11 contains miscellaneous tables that I have found of some value. Chapter 12 is an introduction to icosahedral symmetry and diffraction patterns. The last chapt'er contains the essentials about dislocations and has tables useful for simple contrast analysis. Although the list of references is far from complete, they include an those I have found to contain data useful for my needs and in extending my understanding of the fundamental principles underlying crystallography and structures. Of those listed, the following are recommended as good places to go when questions arise: Barrett, C. S. and T. B Massalski, Structure of Metals, Pergamon Press, New York, Brown, P. J. and J. B. Forsyth, The Crystal Structure of Solids, Edward Arnold Ltd, London, Deer, W. A, R. A Howie, and J. Zussman, An Introduction to the Rock Forming Minerals, Longman Group, Ltd., London, 1980 [available through Halsted Press, Division of John WHey & Sons, Inc., New York]. PearsQn, W. B., The Crystal Chemistry and Physics of Metals and Alloys, Wiley- Interscience, New York, Vainstein, B. K., A A Chernov and L. A Shuvalov (editors), Modern Crystallography, Volumes I-IV, Springer-Verlag, New York, It is a pleasure to acknowledge the help and assistance of a number of people. My thanks to Judy Paine for hunting for books in the library when I had given up, to Jeffrey Robbins for his encouragement, to Tom Broderick for helpful discussions concerning slip, to Karen Teal for her encouragement, to Materials Laboratory, and to Systems Research Laboratories for permission to publish, to Dr. C. Suryanarayana for helpful suggestions concerning icosahedral symmetry and comments on the contents, and to my students who have offered suggestions and pointed out errors. I am particularly grateful to the University of Dayton Department of Materials Science for the opportunities to teach from 1984 to I thank Dr. Dan Miracle for permission to use in Chapter 10 the extended form of the table on diffraction symbols and point groups prior to publication. My thanks also to the production and editorial staff at Springer-Verlag for their constructive suggestions and help. The following have given permissions to use examples and figures from publications: The Royal Society of London, John Wiley and Sons, ASM International, and the American Physical Society. Preparation of any book is inevitably a family affair. Hence, my thanks to my spouse, Marti, who played a major role by her expressions of interest and her help in reviewing the early manuscripts, to my daughter, MicheHe, for proofing several sections for me, and to my son, Mike, for giving up his tennis table for months. Any errors present are my responsibility. There are subjects of importance that I have not included, such as matrix representations of symmetry operations, disciinations, and ceramic structures related to minerals. My apologies for such

7 Preface ix omissions, but the references provided are good starting points for finding more information on these subjects. I will appreciate any comments, thoughts, and suggestions readers may have, complimentary or otherwise. Preparation of the manuscript was accomplished using the Microsoft Word program on the Macintosh SE. Equations were prepared using the mathematics typesetting codes in Word. Figures were prepared using CricketDraw. The original text was printed on Laserprint paper using a Laserprinter II NT. Dayton, Ohio January, 1991 A G.Jackson

8 Contents Preface..._..._......_..._. vii List of Tables... xv 1. Definitions and Mathematics Definitions Vector Operations Dot and cross products Basis Vectors Miller Indices (plane Indices) Direction Indices Permutation of Indices The Direct Lattice Basis vectors of the direct lattice Differentiation of the lattice from the structure Fundamentals of the Reciprocal Lattice Defining Vectors for Various Crystal Systems Face-Centered Cubic Conversion of primitive cell to FCC Other possibilities Body-Centered Cubic Hexagonal Close Packed End-Centered (Base-Centered) Simple Cubic Diffraction Pattern Analysis Introduction Errors in Measurements Measurements on the emulsion Camera constant equation approximation Distortions in patterns Statistics and error Analysis of Patterns Introduction Indexing conventions Methods Ring patterns Spot patterns Structure Factor Ratio Tables xi

9 xii Contents 4. Packing Fraction and Stacking Sequences Packing Fraction Simple cubic Face-centered cubic Body-centered cubic Hexagonal close packed Stacking Sequences Simple cubic Body-centered cubic Face-centered cubic Hexagonal close packed Interstitial Positions Face-centered cubic Body-centered cubic Hexagonal close packed Detailed Equations for Various Crystal Systems General Equations Applicable to Any System Cubic System Tetragonal System Orthorhombic System Monoclinic System (b axis unique) Triclinic System Trigonal System Basis vectors Various equations Hexagonal System Conversion Formulas Introduction BCC to Orthorhombic hcp to Orthorhombic BCC to hcp FCT to BCT and BCT to FCT Transformation FCT to BCT BCT to FCT Monoclinic Nonprimitive to Primitive Transformation Rhombohedral to HCP Some Orientation Relationships Some Ordered Structures Slip Systems Face-Centered Cubic Body-Centered Cubic Hexagonal Close Packed Miscellaneous Definitions... 86

10 Contents xiii 8. Projections..._..."..._ Introduction Direct Lattice Projections Reciprocal Space Projections The Stereographic Projection Grid Projections Structurt; Symbols. _... m... _ Crystal Designations Introduction Equivalent points... ~ Strukturbericht Symbols Pearson Symbols Symmetry Symbols Operational definitions Macroscopic symmetry elements Space group symbols International symbols...,..., Schoenflies symbol International or Hermann-Mauguin symbol Jagodzinski-Wyckoff notation B. Ramsdell notation... lob Zhdanov notation... lob 10. Convergent Beam Electron DitTraction... _ Introduction Problems with Obtaining and Interpreting the Patterns Instrument and specimen related problems Recording the pattern Interpretation Zero and Higher Order Laue Zones Lattice Parameter Along the Zone Axis Higher Order Laue Zone Lines in Diffraction Discs Symmetry Identification Some definitions Point and space group determination Thickness Measurement Using Higher Order Laue Zone Lines in a Diffraction Disc Indexing Holz Patterns Construction ofthe Holz Pattern and Identification of Planes in the Holz Ring Rings in Convergent Beam Diffraction Interpretation of hcp CBED Ring Patterns Miscellaneous Tables and Data... _... m Mendeleev Number and Chemical Scale Machlin Classification of Some Intermetallics Schlafli Symbols Fourier Series and Transforms

11 xiv Contents Introduction Fourier series Fourier transforms Crystal structures and Fourier transforms Icosahedral Structures and Paiterns....." Definitions Golden mean Icosahedron Axes Simple Projection Examples Diffraction Pattems Dislocations Definitions Image Contrast of Dislocations Analysis of Burger's Vector Thompson Tetrahedron for Face Centered Cubic Partials Twins References Index

12 List of Tables Table 3.1. Some Values of Structure Factors I F Table 3.2. Ratio Tables for FACE-CENTERED CUBIC Lattice Table 3.3. Ratio Tables for BODY-CENTERED CUBIC Lattice Table 304. Ratio Tables for Simple Cubic Lattice Table 3.5. Plane Ratios for the Tetragonal System for (hkd) Planes Table 3.6. Ratio Tables for Hexagonal Close Packed Lattice for I = G Table 4.1. Unit Cell Parameters for SC, FCC,. BCC (after Kittel, [1967, p. 18, table 3]) Table 5.1. Trigonal Indices in Rhombohedral and Hexagonal Bases Table 5.2. Indices for Various Hexagonal Indexing Systems Table 6.1. Special Transformations for BCC, hcp, and Orthorhombic Lattices.. 75 Table 6.2. Disordered to Ordered Transformations Table 7.1. Number of Independent Slip Systems for hcp Table 7.2. Number of Independent Slip Systems for FCC, BCC, and hcp Table 7.3. Burgers Vectors of Dislocations in hcp Structures Table 9.1a. Characteristics of The Seven Crystal Systems Table 9.1b. Characteristics of the Crystal Systems Table 9.2. Glide Elements Table 9.3. Definitions of the Strukturbericht Symbols Table 904. Definitions of the Pearson Symbols Table 9.5. The 32 Point Groups and Their Symbols Table 9.6. Various Symbols Used for Point Groups Table 9.7. Strukturbericht Symbols, Archtypes, Pearson Symbol and Space Group Arranged by Strukturbericht Symbol Table 9.8. Strukturbericht Symbols, Archtypes, Pearson Symbol and Space Group Arranged by Archtype Table 9.9. Strukturbericht Symbols, Archtypes, Pearson Symbol and Space Group Arranged by Pearson Symbol Table Strukturbericht Symbols, Archtypes, Pearson Symbol and Space Group Arranged by Space Group Table Sums of a Zolz and a Folz Vector (Ul) to Produce g Vectors in the FCC Folz Table Values of h,k, and I for which h2 + k is near G2a2 = (for a = 0.3 nm. H = 1/3Vz = nm-1; G2 = nm-2) Table Experimentally Observed and Calculated Values of H for Various hcp Planes oftib Table loa. Expected Values of Hexp in Terms of Htheor for Various hcp Planes in (4,4) and (3, 3) Notation xv

13 xvi List of Tables Table Forms for H for the Seven Crystal Systems ; Table Diffraction Groups, Zones, Point Groups and Crystal Systems Arranged by Diffraction Group Table Diffraction Groups, Zones, Point Groups and Crystal Systems Arranged by Point Group Table Diffraction Groups, Zones, Point Groups and Crystal Systems Arranged by Crystal System Table Diffraction Groups, Zones, Point Groups and Crystal Systems Arranged by Zone Table Matrix Relating the Diffraction Groups (vertical) to the Point Groups (horizontal) Table Symmetry Symbols for CBED Discs for the Point Groups Table Expressions for u* and for H/u* Used in the Calculation of Translation Vectors Table Mendeleev Number M and Chemical Scale (X) Table Lattice Parameters of Some Intermetallic Compounds by Strukturbericht Symbol and Machlin Classification Table Some Common Fourier Transforms Table A Listing of Peaks for the Icosahedral Quasicrystal Having Inlll<25 and TI.L < Table Stair-rod Reactions with Values of b2 for Shockley Partials Table Major Thompson Tetrahedron Vectors and Associated Directions Table Burgers Vectors of Stable Dislocations in FCC, BCC, hcp, Diamond Cubic and NaCl Table gob for FCC Lattice Table FCC Imperfect Dislocations; Values of gob for Various Planes and for b = 1/6< 110> Table FCC Imperfect Dislocations.; Values of gob for Various Planes and for b = 1/3<111> Table FCC Imperfect Dislocations; Values of gob for Various Planes and for b = 1/6<112> Table gob Values for BCC Perfect Dislocations Table Details of Various Vectors Used in hcp Vector Notation for Dislocations Table Hcp Perfect Dislocations; Values of gob for Various Planes and for b = 1/3<1123> and 1/3{0003] Table Hcp Perfect Dislocations; Values of gob for Various Planes and for b = 1/3<II20>