Mirror Symmetry: The Mother of all Crystal Symmetries (Springer Series in Solid-State Sciences, 200)

Preface
Contents
About the Author
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
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
	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
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
4 Mirror Combination Scheme in Reciprocal Lattice
	4.1 Introduction
	4.2 The Reciprocal Lattice
	4.3 Construction of Reciprocal Lattice (First Approach)
	4.4 Construction of Reciprocal Lattice (Second Approach)
	4.5 Construction of Reciprocal Lattice According to Ewald
	4.6 Interpretation of Braggs’ Equation
	4.7 Diffraction Geometry and Ewald Sphere
	4.8 Vector Form of Braggs’ Equation from Ewald Construction
	4.9 The Ewald Sphere and the Limiting Sphere
	4.10 Construction of Brillouin Zones in 1-D, 2-D and 3-D Lattices
	4.11 Determination of Higher Order Brillouin Zones in Cubic Crystal System
	4.12 Results from Comparison of Unit Cell Data of Cubic Crystal System
	4.13 Summary
	References
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
	5.5 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
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
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
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