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. Worked examples in the Geometry of Crystals Second edition H. K. D. H. Bhadeshia Professor of Physical Metallurgy University of Cambridge Fellow of Darwin College, Cambridge i Book 377 ISBN 0 904357 94 5 First edition published in 1987 by The Institute of Metals The Institute of Metals 1 Carlton House Terrace and North American Publications Center London SW1Y 5DB Old Post Road Brookfield, VT 05036, USA c© THE INSTITUTE OF METALS 1987 ALL RIGHTS RESERVED British Library Cataloguing in Publication Data Bhadeshia, H. K. D. H. Worked examples in the geometry of crystals. 1. Crystallography, Mathematical —– Problems, exercises, etc. I. Title 548’.1 QD911 ISBN 0–904357–94–5 COVER ILLUSTRATION shows a net–like sub–grain boundary in annealed bainite, ×150, 000. Photograph by courtesy of J. R. Yang Compiled from original typesetting and illustrations provided by the author SECOND EDITION Published electronically with permission from the Institute of Materials 1 Carlton House Terrace London SW1Y 5DB ii Preface First Edition A large part of crystallography deals with the way in which atoms are arranged in single crys- tals. On the other hand, a knowledge of the relationships between crystals in a polycrystalline material can be fascinating from the point of view of materials science. It is this aspect of crystallography which is the subject of this monograph. The monograph is aimed at both undergraduates and graduate students and assumes only an elementary knowledge of crystal- lography. Although use is made of vector and matrix algebra, readers not familiar with these methods should not be at a disadvantage after studying appendix 1. In fact, the mathematics necessary for a good grasp of the subject is not very advanced but the concepts involved can be difficult to absorb. It is for this reason that the book is based on worked examples, which are intended to make the ideas less abstract. Due to its wide–ranging applications, the subject has developed with many different schemes for notation and this can be confusing to the novice. The extended notation used throughout this text was introduced first by Mackenzie and Bowles; I believe that this is a clear and unambiguous scheme which is particularly powerful in distinguishing between representations of deformations and axis transformations. The monograph begins with an introduction to the range of topics that can be handled using the concepts developed in detail in later chapters. The introduction also serves to familiarise the reader with the notation used. The other chapters cover orientation relationships, aspects of deformation, martensitic transformations and interfaces. In preparing this book, I have benefited from the support of Professors R. W. K. Honeycombe, Professor D. Hull, Dr F. B. Pickering and Professor J. Wood. I am especially grateful to Professor J. W. Christian and Professor J. F. Knott for their detailed comments on the text, and to many students who have over the years helped clarify my understanding of the subject. It is a pleasure to acknowledge the unfailing support of my family. April 1986 Second Edition I am delighted to be able to publish this revised edition in electronic form for free access. It is a pleasure to acknowledge valuable comments by Steven Vercammen. January 2001 iii Contents INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Definition of a Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Co-ordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The reciprocal basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Homogeneous deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 ORIENTATION RELATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Cementite in Steels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Relations between FCC and BCC crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Orientation relations between grains of identical structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 The metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 More about the vector cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 SLIP, TWINNING AND OTHER INVARIANT-PLANE STRAINS . . . . . . . . . . . . . . . . . . . . . . 25 Deformation twins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 The concept of a Correspondence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Stepped interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Stretch and rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Conjugate of an invariant-plane strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 MARTENSITIC TRANSFORMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 The diffusionless nature of martensitic transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 The interface between the parent and product phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Orientation relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 The shape deformation due to martensitic transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55 The phenomenological theory of martensite crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 INTERFACES IN CRYSTALLINE SOLIDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Symmetrical tilt boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 The interface between alpha and beta brass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 Coincidence site lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Multitude of axis-angle pair representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79 The O-lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Secondary dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 The DSC lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Some difficulties associated with interface theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 APPENDIX 1: VECTORS AND MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 iv 1 Introduction Crystallographic analysis, as applied in materials science, can be classified into two main subjects; the first of these has been established ever since it was realised that metals have a crystalline character, and is concerned with the clear description and classification of atomic arrangements. X–ray and electron diffraction methods combined with other structure sensitive physical techniques have been utilised to study the crystalline state, and the information obtained has long formed the basis of investigations on the role of the discrete lattice in influencing the behaviour of commonly used engineering materials. The second aspect, which is the subject of this monograph, is more recent and took off in earnest when it was noticed that accurate experimental data on martensitic transformations showed many apparent inconsistencies. Matrix methods were used in resolving these difficulties, and led to the formulation of the phenomenological theory of martensite1,2. Similar methods have since widely been applied in metallurgy; the nature of shape changes accompanying displacive transformations and the interpretation of interface structure are two examples. Despite the apparent diversity of applications, there is a common theme in the various theories, and it is this which makes it possible to cover a variety of topics in this monograph. Throughout this monograph, every attempt has been made to keep the mathematical content to a minimum and in as simple a form as the subject allows; the student need only have an elementary appreciation of matrices and of vector algebra. Appendix 1 provides a brief revision of these aspects, together with references to some standard texts available for further consultation. The purpose of this introductory chapter is to indicate the range of topics that can be tackled using the crystallographic methods, while at the same time familiarising the reader with vital notation; many of the concepts introduced are covered in more detail in the chapters that follow. It is planned to introduce the subject with reference to the martensite transformation in steels, which not only provides a good example of the application of crystallographic methods, but which is a transformation of major practical importance. At temperatures between 1185 K and 1655 K, pure iron exists as a face–centred cubic (FCC) arrangement of iron atoms. Unlike other FCC metals, lowering the temperature leads to the formation of a body–centred cubic (BCC) allotrope of iron. This change in crystal structure can occur in at least two different ways. Given sufficient atomic mobility, the FCC lattice can undergo complete reconstruction into the BCC form, with considerable unco–ordinated diffu- sive mixing–up of atoms at the transformation interface. On the other hand, if the FCC phase is rapidly cooled to a very low temperature, well below 1185 K, there may not be enough time or atomic mobility to facilitate diffusional transformation. The driving force for transformation 1 (a) BAIN STRAIN (c) Body-centered tetragonal austenite (d) Body-centered cubic martensite a a a1 2 3 b3 b1 b2 u u (b) INTRODUCTION nevertheless increases with undercooling below 1185 K, and the diffusionless formation of BCC martensite eventually occurs, by a displacive or “shear” mechanism, involving the systematic and co–ordinated transfer of atoms across the interface. The formation of this BCC martensite is indicated by a very special change in the shape of the austenite (γ) crystal, a change of shape which is beyond that expected just on the basis of a volume change effect. The nature of this shape change will be discussed later in the text, but for the present it is taken to imply that the transformation from austenite to ferrite occurs by some kind of a deformation of the austenite lattice. It was E. C. Bain 3 who in 1924 introduced the concept that the structural change from austenite to martensite might occur by a homogeneous deformation of the austenite lattice, by some kind of an upsetting process, the so–called Bain Strain. Definition of a Basis Before attempting to deduce the Bain Strain, we must establish a method of describing the austenite lattice. Fig. 1a shows the FCC unit cell of austenite, with a vector u drawn along the cube diagonal. To specify the direction and magnitude of this vector, and to relate it to other vectors, it is necessary to have a reference set of co–ordinates. A convenient reference frame would be formed by the three right–handed orthogonal vectors a 1 , a 2 and a 3 , which lie along the unit cell edges, each of magnitude aγ , the lattice parameter of the austenite. The term orthogonal implies a set of mutually perpendicular vectors, each of which can be of arbitrary magnitude; if these vectors are mutually perpendicular and of unit magnitude, they are called orthonormal. Fig. 1: (a) Conventional FCC unit cell. (b) Relation between FCC and BCT cells of austenite. (c) BCT cell of austenite. (d) Bain Strain deforming the BCT austenite lattice into a BCC martensite lattice. 2 The set of vectors ai (i = 1, 2, 3) are called the basis vectors, and the basis itself may be identified by a basis symbol, ‘A’ in this instance. The vector u can then be written as a linear combination of the basis vectors: u = u 1 a 1 + u 2 a 2 + u 3 a 3 , where u 1 , u 2 and u 3 are its components, when u is referred to the basis A. These components can conveniently be written as a single–row matrix (u 1 u 2 u 3 ) or as a single–column matrix:   u 1 u 2 u 3   This column representation can conveniently be written using square brackets as: [u 1 u 2 u 3 ]. It follows from this that the matrix representation of the vector u (Fig. 1a), with respect to the basis A is (u; A) = (u 1 u 2 u 3 ) = (1 1 1) where u is represented as a row vector. u can alternatively be represented as a column vector [A;u] = [u 1 u 2 u 3 ] = [1 1 1] The row matrix (u;A) is the transpose of the column matrix [A;u], and vice versa. The positioning of the basis symbol in each representation is important, as will be seen later. The notation, which is due to Mackenzie and Bowles2, is particularly good in avoiding confusion between bases. Co–ordinate Transformations From Fig. 1a, it is evident that the choice of basis vectors ai is arbitrary though convenient; Fig. 1b illustrates an alternative basis, a body–centred tetragonal (BCT) unit cell describing the same austenite lattice. We label this as basis ‘B’, consisting of basis vectors b 1 , b 2 and b 3 which define the BCT unit cell. It is obvious that [B;u] = [0 2 1], compared with [A;u] = [1 1 1]. The following vector equations illustrate the relationships between the basis vectors of A and those of B (Fig. 1): a 1 = 1b 1 + 1b 2 + 0b 3 a 2 = 1b 1 + 1b 2 + 0b 3 a 3 = 0b 1 + 0b 2 + 1b 3 These equations can also be presented in matrix form as follows: (a 1 a 2 a 3 ) = (b 1 b 2 b 3 )×   1 1 0 1 1 0 0 0 1   (1) This 3×3 matrix representing the co–ordinate transformation is denoted (B J A) and transforms the components of vectors referred to the A basis to those referred to the B basis. The first column of (B J A) represents the components of the basis vector a 1 , with respect to the basis B, and so on. 3 0 1 0 1 0 0 1 0 0 0 1 0A B B A 45 °°°° INTRODUCTION The components of a vector u can now be transformed between bases using the matrix (B J A) as follows: [B;u] = (B J A)[A;u] (2a) Notice the juxtapositioning of like basis symbols. If (A J’ B) is the transpose of (B J A), then equation 2a can be rewritten as (u; B) = (u; A)(A J′ B) (2b) Writing (A J B) as the inverse of (B J A), we obtain: [A;u] = (A J B)[B;u] (2c) and (u; A) = (u; B)(B J′ A) (2d) It has been emphasised that each column of (B J A) represents the components of a basis vector of A with respect to the basis B (i.e. a 1 = J 11 b 1 + J 21 b 2 + J 31 b 3 etc.). This procedure is also adopted in (for example) Refs. 4,5. Some texts use the convention that each row of (B J A) serves this function (i.e. a 1 = J 11 b 1 + J 12 b 2 + J 13 b 3 etc.). There are others where a mixture of both methods is used – the reader should be aware of this problem. Example 1: Co–ordinate transformations Two adjacent grains of austenite are represented by bases ‘A’ and ‘B’ respectively. The base vectors ai of A and bi of B respectively define the FCC unit cells of the austenite grains concerned. The lattice parameter of the austenite is aγ so that |ai| = |bi| = aγ . The grains are orientated such that [0 0 1]A‖ [0 0 1]B , and [1 0 0]B makes an angle of 45◦ with both [1 0 0]A and [0 1 0]A. Prove that if u is a vector such that its components in crystal A are given by [A;u] = [ √ 2 2 √ 2 0], then in the basis B, [B;u] = [3 1 0]. Show that the magnitude of u (i.e. |u|) does not depend on the choice of the basis. Fig. 2: Diagram illustrating the relation between the bases A and B. Referring to Fig. 2, and recalling that the matrix (B J A) consists of three columns, each column being the components of one of the basis vectors of A, with respect to B, we have [B;a 1 ] = [ cos 45 − sin 45 0] [B;a 2 ] = [ sin 45 cos 45 0] [B;a 3 ] = [ 0 0 1] and (B J A) =   cos 45 sin 45 0 − sin 45 cos 45 0 0 0 1   4 From equation 2a, [B;u] = (B J A)[A;u], and on substituting for [A;u] = [ √ 2 2 √ 2 0], we get [B;u] = [3 1 0]. Both the bases A and B are orthogonal so that the magnitude of u can be obtained using the Pythagoras theorem. Hence, choosing components referred to the basis B, we get: |u|2 = (3|b 1 |)2 + (|b 2 |)2 = 10a2γ With respect to basis A, |u|2 = ( √ 2|a 1 |)2 + (2 √ 2|a 2 |)2 = 10a2γ Hence, |u| is invariant to the co–ordinate transformation. This is a general result, since a vector is a physical entity, whose magnitude and direction clearly cannot depend on the choice of a reference frame, a choice which is after all, arbitrary. We note that the components of (B J A) are the cosines of angles between bi and aj and that (A J′ B) = (A J B) −1 ; a matrix with these properties is described as orthogonal (see appendix). An orthogonal matrix represents an axis transformation between like orthogonal bases. The Reciprocal Basis The reciprocal lattice that is so familiar to crystallographers also constitutes a special co- ordinate system, designed originally to simplify the study of diffraction phenomena. If we consider a lattice, represented by a basis symbol A and an arbitrary set of basis vectors a 1 , a 2 and a 3 , then the corresponding reciprocal basis A∗ has basis vectors a∗ 1 , a∗ 2 and a∗ 3 , defined by the following equations: a∗ 1 = (a 2 ∧ a 3 )/(a 1 .a 2 ∧ a 3 ) (3a) a∗ 2 = (a 3 ∧ a 1 )/(a 1 .a 2 ∧ a 3 ) (3b) a∗ 3 = (a 1 ∧ a 2 )/(a 1 .a 2 ∧ a 3 ) (3c) In equation 3a, the term (a 1 .a 2 ∧a 3 ) represents the volume of the unit cell formed by ai, while the magnitude of the vector (a 2 ∧a 3 ) represents the area