Point Groups, Space Groups, Crystals, Molecules 1st Edition by R Mirman ISBN 9813105364 9789813105362

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ISBN 10: 9813105364  
ISBN 13: 9789813105362
Author: R Mirman

The study of point groups, space groups, crystals, and molecules lies at the intersection of several important branches of physics and chemistry, particularly solid-state physics, crystallography, and molecular symmetry. These concepts provide a unified way of understanding the symmetry and properties of both molecules and solid materials at a fundamental level. They are crucial for describing the behavior of molecules in various environments and for predicting the properties of crystalline materials.

Point Groups, Space Groups, Crystals, Molecules 1st Table of contents:

Chapter I Transformations With a Point Fixed: Point Groups

I.1 SYMMETRIES OF SPACE AND OF OBJECTS

I.2 WHY STRUCTURES OF CRYSTALS AND MOLECULES ARE LIMTTED

I.2.a What are the essential properties of these objects?

I.2.b There are only a few categories of crystals

I.3 LATTICES

I.3.a Lattices and crystals

I.3.b Unit cells

I.4 THE ROTATION GROUP AND ITS FINITE SUBGROUPS

I.4.a The finite subgroups of the two-dimensional rotation group

I.4.b The finite subgroups for three-dimensions

I.4.c Regular polyhedra

I.4.d How the rotation group determines its finite subgroups

I.5 HOW TRANSLATIONS LIMIT ROTATION GROUPS OF CRYSTALS

I.5.a Translations limit the trace of the rotation matrix

I.5.b Discreteness is essential

I.5.c Complex numbers, rotations and translations

I.5.d Why are rotational symmetries of lattices limited?

I.6 POINT GROUPS

I.6.a The symmetry operations on crystals and molecules

I.6.b Naming and describing the point groups

I.6.c Point groups with reflections

I.6.d There are thus 32 point groups

I.6.e Why are these all the point groups?

I.6.f The molecular point groups

I.6.g Objects invariant under the point groups

I.7 STRUCTURE OF POINT GROUPS

I.7.a Point groups as semi-direct products

I.7.b Classes of the point groups

I.8 DOUBLE GROUPS

I.8.a Definition of double groups

I.8.b Construction of double groups

I.8.c Double group classes

I.8.d The single group is a factor group

I.9 SIMPUCITY AND SYMMETRY IN A COMPLEX ENVIRONMENT

Chapter II Crystal Structures and Bravais Lattices

II.1 CRYSTALS AND LATTICES

II.2 LATTICES AND CRYSTAL SYSTEMS

II.2.a Definition of a lattice

II.2.b Unit cells are parallelepipeds

II.2.c What is a Bravais lattice?

II.2.d The holohedry groups and their subgroups

II.3 LATTICES IN TWO DIMENSIONS

II.3.a The two-dimensional lattices

II.3.b Why we consider all these lattices, and all distinct

II.3.c Why are points added to unit cells?

II.3.d Where can points be put simultaneously in two dimensions?

II.3.e The symmetry groups of two-dimensional lattices

II.3.f Finding the two-dimensional Bravais lattices analytically

II.3.g What makes two lattices distinct?

II.4 THE SEVEN THREE-DIMENSIONAL CRYSTAL SYSTEMS

II.4.a Labeling axes and faces

II.4.b Classification of lattices by their sides and angles

II.4.c Description of the seven systems

II.4.d Why the systems have the names they do

II.4.e The restrictions on projections of lattices from the point groups

II.4.f What determines these systems?

II.4.g Where can axes be placed?

II.4.h What symmetry elements must a lattice contain?

II.5 PRIMITIVE AND NON-PRIMITIVE BRAVAIS LATTICES

II.5.a The conditions on added points in three dimensions

II.5.b Thus there are limits on lattices

II.5.c Finding the Bravais lattices analytically

II.6 DESCRIPTIONS OF THE FOURTEEN LATTICES

II.6.a Triclinic, monoclinic, orthorhombic and tetragonal systems

II.6.b The cubic system

II.6.c The trigonal and hexagonal systems

II.6.d The rhombohedral unit cell

II.7 THE SEVEN CRYSTAL SYSTEMS AND THEIR SYMMETRY GROUPS

II.7.a Systems, groups and lattices

II.7.b Derivation of the Bravais lattices from their symmetry

II.7.c Seven lattice symmetry groups and fourteen lattices

II.7.d Decreasing the symmetry

II.7.e Dilatations relating lattices

II.7.f Crystals and lattices

II.8 THE SYMMETRIES OF CUBIC AND HEXAGONAL LATTICES

II.8.a The rotational symmetries of the cube

II.8.b The symmetry group as a subgroup of symmetric groups

II.8.c The representation matrices of the symmetry group

II.8.d Adjunction of the inversion

II.8.e Generalizations of the cube

II.8.f The symmetries of the hexagonal lattice

Chapter III Space Groups

III.1 GROUPS WITH DISCRETE TRANSLATIONS

III.2 SPACE GROUPS: DEFINITIONS AND NOTATIONS

III.2.a Denoting space group operations

III.2.b The affine group

III.2.c Types of space groups

III.2.d Translations are invariant

III.2.e Semi-direct products

III.3 SYMMORPHIC SPACE GROUPS

III.3.a A symmorphic group in two-dimensions

III.3.b Glide planes

III.4 NONSYMMORPHIC SPACE GROUPS

III.4.a Screw axes

III.4.b The nonprimitive glide reflection

III.4.c Why are these the only non-primitive operations?

III.4.d The factor group of a space group gives a point group

III.4.e How nonprimitive elements are restricted

III.4.f Example of a two-dimensional nonsymmorphic group

III.5 THE DIAMOND STRUCTURE

III.5.a The point group of the diamond space group

III.5.b Where are the nonprimitive elements placed?

III.5.c Why is the space group of diamond so rich?

III.5.d Spinel

III.6 INHOMOGENEOUS ROTATION GROUPS AND NONSYMMORPHIC GROUPS

III.6.a Are nonsymmorphic space groups semi-direct products?

III.6.b Why are there nonsymmorphic groups?

III.7 GEOMETRIC AND ARITHMETIC EQUIVALENCE OF CRYSTAL CLASSES

III.8 THE SPACE GROUPS IN ONE AND TWO DIMENSIONS

III.8.a The symmetry groups of linear objects (frieze groups)

III.8.b Space groups with two translations

III.9 THE THREE-DIMENSIONAL SPACE GROUPS

III.9.a The symmorphic space groups

III.9.b Enantiomorphic space groups

III.9.c Deriving the space groups by enumeration

Chapter IV Representations of Translation Groups

IV.1 REPRESENTATIONS AND THE ROLE THEY PLAY

IV.2 REPRESENTATIONS OF TRANSLATIONS

IV.2.a The reciprocal space

IV.2.b Lattice points and vectors

IV.2.c Brillouin zones

IV.2.d The representation basis states of the translations

IV.2.e Bloch’s theorem

IV.2.f The translation irreducible representations

IV.3 THE WIGNER-SEITZ CELLS

IV.3.a The centered rectangular lattice — what determines its cell?

IV.3.b The space-filling parallelohedra

IV.3.c Dual figures

IV.3.d Inscription in, and circumscription about, spheres

IV.3.e Illustrations of parallelohedra

IV.3.f Parallelohedra give lattices

IV.3.g Determining parallelohedra from their properties

IV.3.h The number of sides of parallelohedra

IV.4 LATTICES CAN HAVE SEVERAL WIGNER-SEITZ CELLS

IV.4.a Emphasizing the importance of understanding what is geometry, what is convention

IV.4.b The centered tetragonal lattice

IV.4.c Implications of multiple Wigner-Seitz cells

IV.5 WIGNER-SEITZ CELLS FOR THE LATTICES

IV.5.a The cubic system

IV.5.b The tetragonal system

IV.5.c The orthorhombic system

IV.5.d The other systems

IV.5.e Why are there five categories of Wigner-Seitz cells?

IV.5.f Obtaining three-dimensional cells from their crosssections

IV.5.g Lattices as spaces and representations of their translation groups

IV.6 BRILLOUIN ZONES FOR THE LATTICES

IV.6.a The cube and its relatives

IV.6.b The Brillouin zones of the tetragonal lattices

IV.6.c The other systems

Chapter V Representations: Point Groups and Projective

V.1 FORMULATING REPRESENTATIONS

V.2 REPRESENTATIONS OF THE CRYSTALLOGRAPHIC POINT GROUPS

V.2.a The characters of point groups

V.2.b Representations and characters of cylic groups

V.2.c Dihedral group representations and characters

V.2.d Computation of characters

V.2.e The table of characters

V.2.f Representations of the tetrahedral and octahedral groups

V.2.g Adjunction of reflections and the inversion

V.3 REPRESENTATIONS OF DOUBLE GROUPS

V.3.a The representations of the double cyclic groups

V.3.b Representations of D2 *

V.3.c Representations of other double groups

V.4 PROJECTIVE REPRESENTATIONS

V.4.a Definition of projective representation

V.4.b Factor systems

V.4.c Some groups with projective representations

V.5 CENTRAL EXTENSIONS

V.5.a Nonuniqueness of central extensions

V.5.b Central extensions and factor systems

V.5.c Forming a central extension using a specific factor system

Chapter VI Induced Representations

VI.1 INDUCING AND SUBDUCING TO FIND REPRESENTATIONS

VI.2 SUBDUCED REPRESENTATIONS

VI.2.a Conjugate representations

VI.2.b Orbit of a representation

VI.2.c Little groups

VI.2.d The multiplicity of an orbit is representation independent

VI.2.e There is but one orbit in the decomposition

VI.2.f The set of subduced representations of a normal subgroup — Clifford’s theorem

VI.3 INDUCED REPRESENTATIONS

VI.3.a Representation finding using induction

VI.3.b Definition of induced representation

VI.3.c The matrices give a representation of the group

VI.4 EXAMPLES OF INDUCED REPRESENTATIONS

VI.4.a Representations of S3 as examples of induction

VI.4.b Induced representations of D4

VI.4.c The representations of tetrahedral group T (A4)

VI.4.d Induced representations of octahedral group O

VI.4.e Inducing from a nonnormal subgroup

VI.4.f Representations of S4 from S3

VI.5 PROPERTIES OF INDUCED REPRESENTATIONS

Vl.5.a Basis functions of induced representations

VI.5.b Conjugate representations, little groups, and orbits

VI.5.c Conjugate subgroups

VI.5.d The characters of the induced representation

VI.5.e The Frobenius reciprocity theorem

VI.6 IRREDUCIBILITY AND COMPLETENESS FOR ARBITRARY SUBGROUPS

VI.7 INDUCING FROM A NORMAL SUBGROUP

VI.7.a Proof of irreducibility of the induced representation

VI.7.b Irreducibility of representations induced from normal subgroup

VI.7.c Allowable representations of the little group

VI.7.d All representations are given by the allowable representations

VI7.e Obtaining only nonequivalent representations

VI.7.f Finding the induced representation using the little group

VI.7.g The matrices of the induced representations

VI.7.h Induced representations of direct product groups

VI.7.i Semi-direct product groups

Chapter VII Representations of Space Groups

VII.1 FINDING THE REPRESENTATIONS

VII.2 CONCEPTS FOR THE REPRESENTATIONS

VII.2.a The type of space group representations studied here

VII.2.b Induced representations in the terminology of space groups

VII.2.c Orbits

VII.2.d The star of a vector

VII.2.e Classification of positions of Brillouin zones

VII.2.f The cosets formed from the translations

VII.2.g little groups and reciprocal lattice vectors

VII.2.h Little co-groups and little groups

VII.3 REPRESENTATIONS OF LITTLE GROUPS AND LITTLE CO-GROUPS

VII.3.a Small representations, and why they help

VII.3.b The central extensions of the little co-group

VII.3.c Symmorphic space groups

VII.4 INDUCING REPRESENTATIONS OF NONSYMMORPHIC SPACE GROUPS

VII.4.a Representations, allowable and not

VII.4.b Inducing representations of symmorphic space groups

VII.4.c Representations of nonsymmorphic groups

VII.4.d The allowable representations of the little group

VII.5 WHAT THE PROCEDURE IS, AND WHAT IT MEANS

VII.5.a The procedure in essence

VII.5.b What operators are diagonal?

VII.5.c The meaning of space group representations

VII.5.d What is the dimension of the space group representation?

VII.5.e Representations can contain more than one momentum magnitude value

VII.6 THE SQUARE AS AN EXAMPLE

VII.6.a The reciprocal lattice vectors of the square

VII.6.b General vectors and vectors giving symmetry

VII.6.c The stars of the points of the square

VII.6.d The square and the rectangle

VII.6.e Little groups of general, and of special, vectors

VII.6.f Representations of the space group of the square

VII.6.g The square with nonsymmorphic glides

VII.6.h A nonsymmorphic group of the rectangle

VII.6.i What determines the space group and its representations?

VII.7 THE CUBIC AND DIAMOND STRUCTURES

VII.7.a The representations of the factors for Fm3m

VII.7.b The representations of the factors for Fd3m

Chapter VIII Spin and Time Reversal

VIII.1 MORE COMPUCATED CRYSTALS

VIII.2 TIME REVERSAL

VIII.2.a Antilinear and antiunitary operators

VIII.2.b The general form of an antilinear operator

VIII.2.c The general form of the time reversal operator

VIII.2.d Kramer’s Theorem

VIII.3 COMPLEX CONJUGATE REPRESENTATIONS

VIII.3.a When are conjugate representations equivalent?

VIII.3.b Operators mixing representations and their conjugates

VIII.3.c Classification of groups under complex conjugation

VIII.3.d Equivalent sets can be distinguishable

VIII.4 COLOR GROUPS

VIII.4.a The conditions on color groups

VIII.4.b Point groups with time reversal

VIII.4.c Magnetic groups

VIII.4.d An example of a magnetic group

VIII.4.e Construction of a magnetic group

VIII.4.f The orthorhombic magnetic crystal

VIII.4.g What determines which groups are different?

VIII.5 MAGNETIC BRAVAIS LATTICES

VIII.5.a Properties of the magnetic Bravais lattices

VIII.5.b Two-dimensional magnetic Bravais lattices

VIII.5.c Three-dimensional magnetic Bravais lattices

VIII.6 MAGNETIC SPACE GROUPS

VIII.6.a Types of colored space groups

VIII.6.b A space group obtained from the primitive cubic lattice

VIII.6.c Implications and extensions

VIII.7 REPRESENTATIONS OF GROUPS WITH ANTILINEAR OPERATORS

VIII.7.a Corepresentations

VIII.7.b Multiplication rules for corepresentations

VIII.7.c Representation Spaces

VIII.7.d Constructing the corepresentations

VIII.7.e Transformations of the corepresentations

VIII.7.f Reducibility of corepresentations

VIII.7.g The types of corepresentations

VIII.7.h The case of inequivalent subgroup representations, ∆ and ∆+

VIII.7.i Equivalent subgroup representations

VIII.7.j A simple corepresentation

VIII.7.k Which representations are which?

VIII.7.l The Herring test

VIII.7.m Are corepresentations representations?

VIII.8 SPIN AND COREPRESENTATIONS

VIII.8.a Integral spin

VIII.8.b Half-integer spin

VIII.8.c Classifying the corepresentations

VIII.9 APPUCAHON OF COREPRESENTATIONS TO MAGNETIC SPACE GROUPS

VIII.9.a The translation operators

VIII.9.b Corepresentations of the magnetic space groups

VIII.9.c Projective corepresentations of magnetic groups

VIII.9.d Time reversal invariance and gray space groups

VIII.9.e The physical meaning of the representations

VIII.10 REPRESENTATIONS OF DOUBLE SPACE GROUPS

Chapter IX Tensors, Groups and Crystals

IX.1 MACROSCOPIC PHYSICAL PROPERTIES OF CRYSTALS

IX.2 TENSORS FOR THE ROTATION GROUPS

IX.2.a The number of representations symbolized by a tensor

IX.2.b Pseudo-tensors

IX.2.c Tensors relating vectors

IX.2.d Required symmetry in indices

IX.3 TENSORS AND SYMMETRY

IX.3.a Foundations of the tensor analysis of crystal properties

IX.3.b Equilibrium and non-equilibrium properties

IX.3.c Magnetic tensors and the effect of time reversal

IX.3.d The number of independent components

IX.3.e The meaning of the number of independent components

IX.3.f Requirements imposed by symmetry elements on tensors

IX.3.g How group theory provides information about tensors

IX.4 RANK-1 TENSORS — ELECTRIC AND MAGNETIC DIPOLE MOMENTS

IX.5 SECOND RANK TENSORS

IX.5.a Thermal conductivity

IX.5.b Thermal expansion

IX.5.c Stress and strain

IX.5.d Second-rank tensors for group C3V

IX.5.e Nonzero components for different crystal systems

IX.6 THIRD-RANK TENSORS

IX.6.a Piezoelectricity

IX.6.b The Hall effect

IX.6.c Optical activity

IX.6.d Groups that can, and cannot, have these tensor properties

IX.7 FOURTH RANK TENSORS

IX.7.a Relating stress and strain

IX.7.b Photoelasticity

IX.8 THE EFFECT OF IRREDUCIBILITY ON THE PHYSICS OF TENSORS

IX.9 THE USEFULNESS OF TENSORS IN ANALYZING CRYSTALS

Chapter X Groups, Vibrations, Normal Modes

X.1 WHAT GROUPS TELL US ABOUT MOLECULES AND CRYSTALS

X.2 VIBRATIONAL STATES AND SYMMETRY

X.2.a Normal modes

X.2.b The simple harmonic oscillator

X.2.c Group theory of the one-dimensional simple harmonic oscillator

X.2.d The linear triatomic molecule

X.3 WHY, AND HOW, GROUP THEORY IS RELEVANT TO VIBRATIONS

X.3.a Symmetry and normal coordinates

X.3.b Finding normal coordinates

X.4 CHARACTERS AND COUNTING

X.5 EXAMPLES OF APPUCATION OF SYMMETRY TO VIBRATIONS

X.5.a Group theory, symmetry and the vibrations of water

X.5.b Ammonia

X.5.c How vibrational spectra depend on the molecule

X.5.d Breaking of symmetry

X.6 MULTIPLE EXCITATIONS

X.7 TRANSITIONS AND SELECTION RULES

X.7.a Transitions due to electric dipole moments

X.7.b Polarizability and the Raman effect

X.8 HOW REASONABLE ARE THE APPROXIMATIONS?

X.9 HOW DIFFERENT GROUPS GIVE DIFFERENT VIBRATIONAL SPECTRA

Chapter XI Bands, Bonding, and Phase Transitions

XI.1 WHY IS SYMMETRY RELEVANT?

XI.1.a Different types of objects can be studied separately

XI.1.b Degeneracy, necessary and accidental

XI.1.c What can we know about objects in crystals?

XI.1.d Symmetry varies; how is it useful?

XI.1.e Selection rules, perhaps not exact, but still productive

XI.2 ELECTRON STATES IN CRYSTALS

XI.2.a Labels for states are necessary

XI.2.b Bands, and why they are

XI.2.c What group theory tells about electron bands

XI.2.d Symmetry reduction

XI.2.e The equation governing the objects

XI.2.f Translational symmetry and bands

XI.2.g Energy eigenvalues

XI.2.h Energies and statefunctions for cubic lattices

XI.2.i Energy bands for nonsymmorphic groups

XL2.j The close-packed hexagonal structure

XI.2.k What do we learn from this group-theoretical analysis?

XI.3 THE EFFECT OF TIME REVERSAL

XI.3.a The energy surfaces must have inversion symmetry

XL3.b Degeneracy due to time reversal

XI.3.c Degeneracy at general points

XI.4 LATTICE VIBRATIONS

XI.4.a The dynamical matrix

XI.4.b Transitions in the simple cubic lattice

XI.5 ATOMS IN CRYSTALS AND ENERGY LEVEL SPLITTING

XI.5.a Level splitting in crystals

XI.5.b The octahedral group as an example

XI.6 SPIN-ORBIT COUPLING

XI.6.a Spin-orbit coupling and removal of degeneracy

XI.6.b Cubic symmetry

XI.7 MOLECULAR ORBITALS

XI.7.a Relating molecular orbitals and atomic orbitals

XI.7.b The types of orbitals we consider

XI.7.c Benzene

XI.7.d Bonding and antibonding states

XI.8 CHANGE OF PHASE

XI.8.a First and second order phase transitions

XI.8.b Limitations on the analysis

XI.8.c Specifying the crystal thermodynamics

XI.8.d Equilibrium

XI.8.e How symmetry change is found

XI.8.f The order parameter

XI.8.g Expansion of the density

XI.8.h Physically irreducible representations

XI.8.i Symmetry restrictions on the expansion coefficients

XI.8.j Implications of the requirement that Φ be a minimum

XI.8.k Conditions from necessity of terms being zero

XI.8.l Halving the symmetry always allows a transition

XI.8.m Active and passive representations

XI.9 CLASSICAL VIEWS, QUANTUM VIEWS, REALITY

Appendix A Symbols and definitions

Appendix B The Point Groups

Appendix C Objects Invariant Under the Point Groups

Appendix D Two-Dimensional Space Groups

Appendix E Point Group Character Tables

E.1 DENOTING THE REPRESENTATIONS

E.2 THE CHARACTER TABLES

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