Mathematical Crystallography
Worked examples in the
Geometry of Crystals
Fellow of Darwin College, Cambridge
Mathematical Crystallography
DOWNLOAD PDF
Mathematical Crystallography
Preface
First Edition
A large part of crystallography deals with the way in which atoms are arranged in single crystals.
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 crystallography.
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, updated July 2008
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
APPENDIX 2: TRANSFORMATION TEXTURE . . . . . . . . . . . . . . . . . . . . . . . .96
APPENDIX 3: TOPOLOGY OF GRAIN DEFORMATION . . . . . . . . . . . . . . . . . . . .100
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Worked examples in the
Geometry of Crystals
Fellow of Darwin College, Cambridge
Mathematical Crystallography
DOWNLOAD PDF
Mathematical Crystallography
Preface
First Edition
A large part of crystallography deals with the way in which atoms are arranged in single crystals.
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 crystallography.
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, updated July 2008
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
APPENDIX 2: TRANSFORMATION TEXTURE . . . . . . . . . . . . . . . . . . . . . . . .96
APPENDIX 3: TOPOLOGY OF GRAIN DEFORMATION . . . . . . . . . . . . . . . . . . . .100
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
