9814579203.pdf

A Group Theoretical Approach

Crystal Symmetry, Lattice Vibrations, and Optical Spectroscopy of Solids

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N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I

World Scientific

Baldassare Di BartoloBoston College, USA

Richard C. PowellUniversity of Arizona, USA

A Group Theoretical Approach

Crystal Symmetry, Lattice Vibrations, and Optical Spectroscopy of Solids

9052_9789814579209_tp.indd 2 22/4/14 9:05 am

Published by

World Scientific Publishing Co. Pte. Ltd.

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Library of Congress Cataloging-in-Publication DataDi Bartolo, Baldassare, author. Crystal symmetry, lattice vibrations and optical spectroscopy of solids : a group theoretical approach / Baldassare Di Bartolo (Boston College, USA) & Richard C. Powell (University of Arizona, USA). pages cm Includes bibliographical references and index. ISBN 978-9814579209 (hardcover : alk. paper) 1. Lattice dynamics. 2. Phonons. 3. Symmetry (Physics) 4. Optical spectroscopy. 5. Solid state physics. I. Powell, Richard C. (Richard Conger), 1939 author. II. Title. QC176.8.L3D53 2014 530.4'11--dc23 2013047941

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April 9, 2014 11:57 9in x 6in Crystal Symmetry, Lattice Vibrations and. . . b1701-fm

To Rita and Gwen

v


Preface

The eld of solid state physics consists of many dierent topics, eachaddressing a specic physical property of the material, e.g. electrical,mechanical, optical, or magnetic properties. However, there are twofundamental aspects of a solid that aect all of its physical properties.The rst is the microscopic arrangement of the atomic constituentsof the material. This can be amorphous leading to a glass mate-rial or have a specic order leading to a crystalline material. Thebasic structure of crystals and glasses is a major factor in determin-ing all of their physical properties. The second is the motion of theatomic constituents about their equilibrium positions. These atomicvibrations modulate the physical properties of the material. Thisbook focuses on these two fundamental aspects of solids and thus isappropriate to use as a textbook for an introductory class in solidstate physics within a curriculum which includes additional classeson specic properties of solids. It assumes the student has taken aone year course in quantum mechanics.

The major mathematical tool used to describe the symmetryproperties of crystal lattices and their thermal vibrations is grouptheory. The techniques of group theory have important applicationsin almost every area of physics. This book covers the principles ofgroup theory starting at a fundamental level and providing specicexamples of how to use group theory in studying crystalline solids. Itassumes no prior knowledge of the subject. Because of the extensivetreatment of the topic, it would be possible to use this book as atextbook for a one semester course on group theory.

vii


viii Preface

The book is divided into three parts:

I. The rst part focuses on the symmetry of crystal structures. TheHamiltonian of a crystalline solid is introduced and the adiabaticapproximation is described. The importance of symmetry for thisHamiltonian is discussed. The basic concepts of group theory arereviewed and applied to the treatment of spatial symmetries andquantum mechanical processes. Dierent types of crystal latticesare described and the experimental measurement of crystal lat-tices by x-ray diraction is discussed.

II. The second part of the book deals with the lattice vibrations ofcrystals (i.e. phonons). The Hamiltonian of the crystal developedin Part I is modied to describe phonons and dierent typesof phonon modes are discussed. The thermodynamics of latticevibrations is described as well as the experimental measurementsof lattice vibrations by neutron scattering and the eects of ther-mal vibrations on x-ray diraction. Again symmetry and grouptheory play important roles in describing dierent types of latticevibrations.

III. In the third part of the book the optical spectroscopy of crys-tals is described with a special emphasis on the eects of latticesymmetry and thermal vibrations on the spectroscopic proper-ties. The energy levels of a solid are in general determined by theelectrons on its atomic constituents or by its lattice vibrations.These energy levels can be either delocalized throughout thematerial or localized at a specic point in the lattice. Transitionsbetween energy levels can produce spectral lines throughout theultraviolet, visible, and infrared regions of the spectra. Measure-ment techniques such as absorption, emission, and scattering oflight are used to determine the properties of these energy levels.This section describes in detail the important case of electronictransitions between the energy levels of an impurity ion in acrystal with special emphasis on the eects of thermal vibrationson these transitions. In addition, the measurement of phononenergies through infrared absorption and inelastic light scatter-ing is described. The important application of these concepts tosolid state lasers is summarized in the nal chapter. This ties


Preface ix

together how crystal symmetry, lattice vibrations, and opticalspectroscopic properties determine the operational parametersof a laser.

This book provides an example of the importance of symme-try and group theory techniques in describing fundamental physi-cal properties of solids. As well as its potential use as a textbook,it should be of interest to researchers studying lattice vibrations ofcrystals or spectroscopists whose results are aected by lattice vibra-tions. It should be noted that there are other thermal eects in solids(such as phonon transport properties) that are not treated here sincethey are not closely related to spectral data and have been treatedextensively elsewhere.

Much of the basic material in this book was originally publishedin 1976 by John Wiley & Sons under the title Phonons and Reso-nances in Solids which is now out of print. We found the book tobe a useful text for advanced courses in solid state physics but feltthat the organization of the book was somewhat cumbersome andthe connection between the various chapters was not transparent.Therefore we have revised the book by reorganizing the chapters andadding new material. In addition, numerous typographical errors inthe original book have been corrected and the title has been changedto reect the new structure of the text. These changes should signif-icantly increase the usefulness of the book as a basic text on grouptheory and solid state physics.

Baldassare Di Bartolo Richard C. PowellChestnut Hill, Massachusetts Tucson, ArizonaAugust 2013


Contents

Preface vii

Part I. Symmetry of Crystals 1

Chapter 1. Introduction 3

1.1 The Hamiltonian of a Crystalline Solid . . . . 31.2 The Adiabatic Approximation . . . . . . . . . 61.3 The Role of Symmetry . . . . . . . . . . . . . 71.4 The Symmetries of the Hamiltonian . . . . . 9Reference . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 2. Concepts of Group Theory 13

2.1 Properties of a Group . . . . . . . . . . . . . 132.2 Subgroups, Cosets, and Classes . . . . . . . . 152.3 Theory of Representations . . . . . . . . . . . 172.4 Orthogonality Relations . . . . . . . . . . . . 202.5 Characters of a Matrix Representation . . . . 232.6 Reduction of a Reducible Representation . . . 252.7 Basis Functions for Irreducible

Representations . . . . . . . . . . . . . . . . . 272.8 Direct Product Representations . . . . . . . . 292.9 The Fundamental Theorem for Functions

Transforming Irreducibly . . . . . . . . . . . . 312.10 Product Groups and Their

Representations . . . . . . . . . . . . . . . . . 342.11 Connection of Quantum Mechanics

with Group Theory . . . . . . . . . . . . . . . 35

xi


xii Contents

Chapter 3. Crystal Symmetries 41

3.1 Unit Cells and Space Lattices . . . . . . . . . 413.2 Miller Indices . . . . . . . . . . . . . . . . . . 443.3 The Crystal Systems . . . . . . . . . . . . . . 47

3.3.1 The Four Two-Dimensional CrystalSystems . . . . . . . . . . . . . . . . . 48

3.3.2 The Seven Three-Dimensional CrystalSystems . . . . . . . . . . . . . . . . . 48

3.4 The Bravais Lattices . . . . . . . . . . . . . . 503.4.1 The Five Bravais Lattices

in Two Dimensions . . . . . . . . . . . 503.4.2 The Fourteen Bravais Lattices

in Three Dimensions . . . . . . . . . . 53

Chapter 4. Group Theoretical Treatment of CrystalSymmetries 57

4.1 Space Groups . . . . . . . . . . . . . . . . . . 574.2 The Crystallographic Point Groups . . . . . . 60

4.2.1 Two-Dimensional CrystallographicPoint Groups . . . . . . . . . . . . . . 60

4.2.2 Three-Dimensional CrystallographicPoint Groups . . . . . . . . . . . . . . 61

4.2.3 Site Groups . . . . . . . . . . . . . . . 654.3 The Invariant Subgroup of Primitive

Translations: Bravais Lattices . . . . . . . . . 684.4 The Compatibility of Rotational

and Translational Symmetriesand Its Relevance to Space Groups . . . . . . 69

4.5 The Irreducible Representations of a Groupof Primitive Translations Brillouin Zones . . . 72

4.6 The Irreducible Representations of SpaceGroups . . . . . . . . . . . . . . . . . . . . . . 774.6.1 Eects of Translational

Symmetry . . . . . . . . . . . . . . . . 784.6.2 Eects of Rotational Symmetry . . . . 79


Contents xiii

4.6.3 General Properties of the IrreducibleRepresentations . . . . . . . . . . . . . 80

4.6.4 Small Representations for DierentPoints of the Brillouin Zone . . . . . . 83

4.7 Example I. Symmorphic Group C14v . . . . . . 864.8 Example II. Nonsymmorphic Group C24v . . . 106References . . . . . . . . . . . . . . . . . . . . . . . . 127

Chapter 5. Scattering of X-Rays by Crystals 129

5.1 Introduction . . . . . . . . . . . . . . . . . . . 1295.2 Scattering from a Single Electron . . . . . . . 1305.3 Scattering from a Single Atom . . . . . . . . 1335.4 Scattering from the Atoms in the Unit Cell

of a Crystal . . . . . . . . . . . . . . . . . . . 1365.5 Scattering from a Crystal . . . . . . . . . . . 1375.6 Interpretation of Laue Equations

in Reciprocal Space . . . . . . . . . . . . . . . 1415.7 Methods of X-Ray Diraction . . . . . . . . . 143

5.7.1 The Laue Method (see Fig. 5.8a) . . . 1435.7.2 The Bragg Method

(see Fig. 5.8b) . . . . . . . . . . . . . 1445.7.3 The Debye-Scherrer Method

(see Fig. 5.8c) . . . . . . . . . . . . . . 145References . . . . . . . . . . . . . . . . . . . . . . . . 145

Part II. Lattice Vibrations of Crystals 147

Chapter 6. Lattice Vibrations of Crystals 149

6.1 The Innite Linear Crystal . . . . . . . . . . 1496.2 The Finite Linear Crystal . . . . . . . . . . . 1546.3 Normal Modes of Vibration of a Linear

Crystal . . . . . . . . . . . . . . . . . . . . . . 1576.4 Linear Crystal with a Basis . . . . . . . . . . 1676.5 Lattice Vibrations in Three Dimensions . . . 173

6.5.1 The Equations of Motion . . . . . . . 173


xiv Contents

6.5.2 Allowed Values of k. Density

of Phonon Modes . . . . . . . . . . . . 1766.5.3 Normal Modes of Vibration . . . . . . 1796.5.4 Energy Levels . . . . . . . . . . . . . . 1846.5.5 Particular Modes of Vibration . . . . 1886.5.6 Spectrum of Lattice Vibrations . . . . 190

6.6 Group Theory and Lattice Vibrations . . . . 1916.6.1 Properties of the Normal

Coordinates . . . . . . . . . . . . . . . 1916.6.2 The Frequency Eigenvalues

and the Polarization Vectors . . . . . 1936.6.3 Additional Degeneracies Not Due

to Spacelike Symmetries . . . . . . . . 1976.6.4 Time-Reversal Degeneracy . . . . . . 198

6.7 Group-Theoretical Analysis of the LatticeVibrations of a Linear Crystal . . . . . . . . . 2026.7.1 Case of One Atom Per Unit Cell . . . 2026.7.2 Case of Two Atoms

Per Unit Cell . . . . . . . . . . . . . . 2066.8 Group-Theoretical Analysis of the Lattice

Vibrations of a Three-DimensionalCrystal . . . . . . . . . . . . . . . . . . . . . . 209

6.9 Example I. Lattice Vibrationsof a Two-Dimensional Crystalwith Symmetry C14v . . . . . . . . . . . . . . 212

6.10 Example II. Lattice Vibrationsof a Two-Dimensional Crystalwith Symmetry C24v . . . . . . . . . . . . . . 221

References . . . . . . . . . . . . . . . . . . . . . . . . 232

Chapter 7. Thermodynamics of Lattice Vibrations 233

7.1 Thermodynamics of Specic Heats . . . . . . 2337.2 The Classical Theory of the Specic Heats

of Solids . . . . . . . . . . . . . . . . . . . . . 2357.3 The Einstein Theory of Specic Heat . . . . . 2367.4 The Debye Theory of Specic Heat . . . . . . 240


Contents xv

7.4.1 The Specic Heatof a Linear Crystal . . . . . . . . . . . 240

7.4.2 The Debye Theory Appliedto a Linear Crystal . . . . . . . . . . . 243

7.4.3 The Debye Theory Appliedto a Three-Dimensional Crystal . . . . 244

7.5 Temperature Dependence of the Amplitudeof Vibrations Solids. The Lindemann Lawof Melting . . . . . . . . . . . . . . . . . . . . 252

References . . . . . . . . . . . . . . . . . . . . . . . . 255

Chapter 8. Eect of Lattice Vibrations on X-rayScattering and Neutron Scattering 257

8.1 Eect of Lattice Vibrations on the Intensityof the Scattered Radiation . . . . . . . . . . . 2578.1.1 The Intensity of the Scattered

Radiation . . . . . . . . . . . . . . . . 2578.1.2 The Eect of Lattice Vibrations:

Einstein Model . . . . . . . . . . . . . 2588.1.3 The Eect of Lattice Vibrations:

Normal Mode Treatment . . . . . . . 2618.2 Theory of Neutron Scattering . . . . . . . . . 2678.3 Elastic Neutron Scattering . . . . . . . . . . . 2748.4 Inelastic Neutron Scattering . . . . . . . . . . 2788.5 Application of Neutron Scattering

to the Study of Lattice Vibrations . . . . . . 283References . . . . . . . . . . . . . . . . . . . . . . . . 286

Part III. Optical Spectroscopy of Crystals 289

Chapter 9. Interaction of Radiation with Matter 291

9.1 The Classical Radiative Field . . . . . . . . . 2929.2 The Quantum Theory

of the Radiative Field . . . . . . . . . . . . . 3019.3 The Hamiltonian of a Charged Particle

in an Electromagnetic Field . . . . . . . . . . 303


xvi Contents

9.4 The Interaction Between a ChargedParticle and a Radiative Field . . . . . . . . . 305

9.5 First-Order Processes. Absorptionand Emission of Radiation . . . . . . . . . . . 308

9.6 Second-Order Processes . . . . . . . . . . . . 3159.6.1 Matrix Element Due to H1 . . . . . . 3179.6.2 Matrix Elements Due to H2 . . . . . . 3189.6.3 Eective Matrix Element . . . . . . . 3199.6.4 Transition Rates of Scattering

Processes . . . . . . . . . . . . . . . . 323References . . . . . . . . . . . . . . . . . . . . . . . . 325

Chapter 10. Optical Spectra of Impurities in Solids I 327

10.1 Impurities in Crystals . . . . . . . . . . . . . 32810.2 Review of the Theory of Small Vibrations

(Classical) . . . . . . . . . . . . . . . . . . . . 32810.3 Harmonic and Anharmonic Relaxation . . . . 33710.4 Review of the Theory of Small Vibrations

(Quantum Mechanical) . . . . . . . . . . . . . 34110.5 The Eect of Impurities on Lattice

Vibrations . . . . . . . . . . . . . . . . . . . . 34610.6 The Franck-Condon Principle . . . . . . . . . 35210.7 Absorption and Emission in Crystals . . . . . 35910.8 Purely Electronic (Zero-Phonon)

Transitions . . . . . . . . . . . . . . . . . . . 36210.9 Characteristics of the Zero-Phonon Lines . . . 37010.10 Phonon-Assisted Transitions . . . . . . . . . . 37210.11 Radiative Transitions in the Presence

of Localized Vibrations . . . . . . . . . . . . . 38010.12 Classication of Vibronic Spectra . . . . . . . 390References . . . . . . . . . . . . . . . . . . . . . . . . 391

Chapter 11. Optical Spectra of Impurities in Solids II 393

11.1 Summary of Previous Results . . . . . . . . . 39311.2 Deviations from the Franck-Condon

Approximation . . . . . . . . . . . . . . . . . 397


Contents xvii

11.3 Deviations from the Adiabatic Approximation.Radiationless Transitions . . . . . . . . . . . . 410

11.4 A Simple Model for Laser Crystals: AnEective Hamiltonian . . . . . . . . . . . . . 413

11.5 Radiative, Vibronic, and RadiationlessTransitions of Magnetic Impurities . . . . . . 416

11.6 Selection Rules for Vibronic Transitions . . . 42611.7 Eect of Temperature on the Position and

Shape of a Purely Electronic Line . . . . . . . 42711.7.1 Thermal Line Shift . . . . . . . . . . . 42811.7.2 Thermal Broadening of Sharp Lines . 429

References . . . . . . . . . . . . . . . . . . . . . . . . 433

Chapter 12. Interaction of Light with Lattice Vibrations:Infrared Absorption and InelasticLight Scattering 435

12.1 General Characteristics of Infrared Absorptionby Crystals . . . . . . . . . . . . . . . . . . . 436

12.2 Infrared Transitions in a Molecular System . 43612.3 Momentum and Energy Conservation

in Infrared Absorption . . . . . . . . . . . . . 43812.4 Quantum Theory of Infrared Absorption . . . 44112.5 Reststrahl (One-Phonon) Absorption . . . . . 45112.6 Two-Phonon Absorption . . . . . . . . . . . . 45312.7 Selection Rules for Infrared Absorption . . . . 45712.8 The Eect of Impurities on Infrared

Absorption Spectra . . . . . . . . . . . . . . . 45912.9 Infrared Absorption in Homopolar Crystals . 46012.10 General Characteristics of Raman Scattering

from Crystals . . . . . . . . . . . . . . . . . . 46812.11 Theory of Raman Scattering . . . . . . . . . . 47112.12 Transition Polarizability . . . . . . . . . . . . 47512.13 Energy Scattered in Raman Scattering

Experiments . . . . . . . . . . . . . . . . . . . 48012.14 Selection Rules for Raman Scattering . . . . . 48312.15 The Eect of Impurities on Raman Scattering 48512.16 Brillouin Scattering . . . . . . . . . . . . . . . 486References . . . . . . . . . . . . . . . . . . . . . . . . 490

May 26, 2014 14:47 9in x 6in Crystal Symmetry, Lattice Vibrations and. . . b1701-fm

xviii Contents

Chapter 13. Lattice Vibrations and Lasers 491

13.1 Nonradiative Transitions . . . . . . . . . . . . 49413.2 Single Wavelength Lasers . . . . . . . . . . . 498

13.2.1 Optical Transitions in Rare Earth IonLasers . . . . . . . . . . . . . . . . . . 499

13.2.2 Radiationless Decay Processes in RareEarth Ion Lasers . . . . . . . . . . . . 502

13.3 Multiple Wavelength Lasers . . . . . . . . . . 505References . . . . . . . . . . . . . . . . . . . . . . . . 509

Subject Index 511

April 8, 2014 15:56 9in x 6in Crystal Symmetry, Lattice Vibrations and. . . b1701-ch01

Part I

Symmetry of Crystals

The rst part of this book deals with the symmetry of an orderedarray of atoms. Group Theory is introduced starting from funda-mental principles and is applied to the study of the symmetry ofcrystalline solids. The symmetry of solids is intimately related totheir physical properties and a study of their symmetry is essentialfor an understanding of their behaviour.

1


Chapter 1

Introduction

This chapter sets the stage for the entire book by treating theHamiltonian of a crystalline solid, introducing the adiabatic approx-imation, and considering the consequences of this approximation onthe role played by symmetry.

1.1. The Hamiltonian of a Crystalline Solid

A crystalline solid is an ordered array of atoms bound together. TheHamiltonian of such a system that includes n electrons and N nucleiis given by

H =ni=1

p2i

2m+

N=1

P2

2M+ V(r

i,R

), (1.1.1)

where m is the mass of the electron, M is the mass of the thnucleus, r

iis the position coordinate of the ith electron, p

iis the

linear momentum of the ith electron, R

is the position coordinate ofthe th nucleus, and P

is the linear momentum of the th nucleus.

Also

V(ri

,R

) = Vee + Vnn + Vne, (1.1.2)

where

Vee =12

ni=1

i =j

nj=1

e2

|ri r

j| ,

3


4 Introduction

Vnn =12

N=1

=

N=1

e2ZZ|R

R | ,

Vne = ni=1

N=1

e2Z|R

ri| ,

The Schrodinger equation of the system is given by

H(ri

,R

) = E(ri

,R

), (1.1.3)

where the Hamiltonian operator H is expressed as follows:

H = 2

2m

ni=1

2i 2

2

N=1

2M

+ V(ri

,R

). (1.1.4)

Therefore the eigenfunctions and the eigenvalues of the system aregiven by

2

2m

ni=1

2i2

2

N=1

1M

2+ V = E. (1.1.5)

In order to solve the above Schrodinger equation we seek solutionsof the type

(ri

,R

) = (R

)(ri

,R

). (1.1.6)

Using (1.1.6) in (1.1.5) we obtain, dropping the subscripts i and ,

2

2m(R

)

ni=1

2i(r,R)2

2

N=1

2M

(R)(r

,R)

+V(r,R)(R)(r

,R) = E(R

)(r

,R). (1.1.7)

But

2(R)(r,R) = (R)2(r

,R) +(r

,R)2(R)

+ 2(R) (r,R). (1.1.8)


1.1. The Hamiltonian of a Crystalline Solid 5

Then

N=1

2

M(R) (r,R)

N=1

2

2M(R

)2(r,R)

(r,R)

N=1

2

2M2(R) (R)

ni=1

2

2m2i (r,R)

+V(r,R)(R

)(r

,R) = E(R

)(r

,R). (1.1.9)

We will assume thatN=1

2

M

[ +

122

]

N=1

2

2M2

.(1.1.10)

We shall return to the validity of this assumption later. Because of(1.1.10), (1.1.9) becomes

2

2

N=1

1M

2+[

2

2m

ni=1

2i + V(r,R)] = E.

(1.1.11)

The operator in the square brackets on the left-hand side of (1.1.11)represents the Hamiltonian for the system if the nuclei are assumedto be xed in space. We call this Hamiltonian He:

He = 2

2m

ni=1

2i + V(r,R). (1.1.12)

The corresponding eigenfunctions and eigenvalues are given by theSchrodinger equation,

2

2m

ni=1

2i (r,R) + V(r,R)(r,R) = (R)(r,R). (1.1.13)

Replacing (1.1.13) in (1.1.11) we obtain

2

2

N=1

1M

2(R) + (R)(R) = E(R). (1.1.14)


6 Introduction

The solution of the Schrodinger equation (1.1.3) reduces then to thesolution of the two equations (1.1.13) and (1.1.14); we now rewritethis solution with the proper subscripts as follows:

2

2m

ni=1

2ik(r,R) + V(r,R)k(r,R) = k(R)k(r,R), (1.1.15)

2

2

N=1

1M

2kl(R) + k(R)kl(R) = Eklkl(R). (1.1.16)

A stationary state of the system will be represented by the eigen-function

n(r,R) = k(r

,R)kl(R

), (1.1.17)

where k(r,R) and kl(R

) are eigenfunctions of the Hamiltonians

He = 2

2m

ni=1

2i + V(r,R), (1.1.18)

and

Hv = 2

2

N=1

2M

+ k(R), (1.1.19)

respectively.

1.2. The Adiabatic Approximation

Let us consider now in detail equations (1.1.15) and (1.1.16). Equa-tion (1.1.15) is an eigenvalue equation whose eigenfunctions rep-resent the motion of the electrons in the crystal when the nucleiare kept xed in space. The energy eigenvalues of (1.1.15) dependparametrically on the nuclear coordinates. Equation (1.1.16) is aneigenvalue equation whose eigenfunctions represent the motion ofthe nuclei in the crystal. In the Hamiltonian (1.1.19) for nuclearmotion, the energy k(R

), which is a function of the nuclear coordi-

nates regarded as parameters, plays the role of the potential energyfor nuclear motion. This potential energy k(R

) is an eigenvalue

of (1.1.15) and, as such, depends on the quantum number k. The


1.3. The Role of Symmetry 7

eigenfunction (R) also depends on k; however, k does not play the

role of a quantum number for (R) even if it is used as one of its

subscripts. Therefore the functions kl(R) and kl(R

) with k = kare not mutually orthogonal.

We now return to the approximation (1.1.10) and discuss itsphysical meaning. The implication of (1.1.10) is that the function(r

,R), which represents the motion of the electrons, is a function

that varies slowly with the nuclear coordinates, so that |(r,R)|is much smaller than |(R)|. In pictorial terms we may say thatthis is the case since the electrons, having much smaller masses thanthe nuclei, go through their orbits many times before the nuclei haveshifted from their equilibrium position by any considerable distance.

In the light of this fact the approximation (1.1.10), which hasallowed us to express the eigenstates of the system in the productform (1.1.6), is called the adiabatic approximation. It is also calledsometimes the BornOppenheimer approximation.1

The implications of the adiabatic approximation are far reach-ing. We are now in the position of treating the electrons and thenuclei independently; we will, however, keep in mind that we areallowed to do so only within the limits of validity of the adiabaticapproximation.

1.3. The Role of Symmetry

The quantum-mechanical treatment of a physical system implies gen-erally the solution of a Schrodinger equation. This solution gives theenergy eigenvalues and the eigenfunctions of the Hamiltonian. Ingeneral the eigenfunctions are degenerate; that is, several of themcorrespond to the same energy eigenvalue. The degeneracy and thetransformation properties of the eigenfunctions are closely related tothe symmetry properties of the Hamiltonian; indeed, both degener-acy and transformation properties can be derived from the knowledgeof symmetries.

Before considering these symmetries for the case of a crys-talline solid, it is worthwhile to review some basic concepts regard-ing coordinate transformations. In particular we are concerned withthose transformations that leave the distance between two points


8 Introduction

unchanged. The most general transformation of this type can beexpressed by the symbol {R|t

} and involves a rotational operation R

followed by a translation t. A position vector x

, when acted upon by

{R|t}, becomes

x = R

x+ t

, (1.3.1)

with

x1 = R11x1 + R12x2 + R13x3 + t1,x2 = R21x1 + R22x2 + R23x3 + t2,x3 = R31x1 + R32x2 + R33x3 + t3.

(1.3.2)

R

is a 3 3 real orthogonal matrix: if its determinant is +1, therotation is called proper; if its determinant is 1, the rotation iscalled improper.

A pure rotation is indicated by {R|0} and pure translation by

{E|t}. The identity operation is represented by {E|0

}.

If two operations {R1|t} and {R2|t

} act in succession upon a

vector x, the result is

x = R1

x+ t

, (1.3.3)

and

x = R

2x + t

= R

2(R1

x+ t

) + t

= R2

R1

x+ R

2t + t

. (1.3.4)

Therefore, the product of the two operations {R2|t} and {R1|t

}

is given by

{R2|t}{R1|t

} = {R2R1|R2t

+ t

}. (1.3.5)

The inverse of an operation {R|t} is given by

{R|t}1 = {R1| R1t

}. (1.3.6)

In fact, applying (1.3.5) we nd

{R1| R1t}{R|t

} = {R1R|R1t

R1t

}

= {E|0}. (1.3.7)


1.4. The Symmetries of the Hamiltonian 9

1.4. The Symmetries of the Hamiltonian

Let us consider rst the exact Hamiltonian (1.1.4) given by

H = ni=1

2

2m2i

N=1

2

2M2

+12

ni=1

i =j

nj=1

e2

|ri r

j| +

12

N=1

=

N=1

e2ZZ|R

R |

ni=1

N=1

e2Z|R

ri| . (1.4.1)

It is clear that if we translate each coordinate by an amount d, that

is, if we replace the electronic and nuclear coordinates as follows,

R = R

+ d,

ri = ri

+ d,

(1.4.2)

the Hamiltonian does not change. This invariance can be eliminatedfrom our consideration if we replace the 3(n + N) coordinates bymeans of the three coordinates of the center of mass of the systemand 3(n + N 1) coordinates depending only on the relative vecto-rial distances between particles. In the new coordinate system theHamiltonian can be separated into the sum

H = 2

2M2 + H, (1.4.3)

where M is the total mass of the crystal and the operator 2 isdened with respect to the coordinates of the center of mass. In(1.4.3) the rst term is the kinetic energy of the center of mass andthe second term is the Hamiltonian of the motion relative to thecenter of mass. H depends only on the relative coordinates and istherefore translationally invariant.

The eigenfunctions of the Hamiltonian H can be written as

= eikU

, (1.4.4)


10 Introduction

and the energy is given by

E = E +2

2M|k|2, (1.4.5)

where U

is the coordinate of the center of mass, is an eigenfunctionof H with energy E, and is an eigenfunction of H with energy E,and k

is an arbitrary vector. The problem is then restricted to nding

the energy eigenvalues E and H and the eigenfunctions of H.Since the rotational symmetry properties of H are the same as

those of H, we can now restrict ourselves to the consideration of therotational symmetries of H.

Let us consider now the eect of a real orthogonal transfor-mation R

on the Hamiltonian H. This transformation does not

change the distances between particles, and therefore the terms inthe Hamiltonian representing Coulomb interactions remain invariant.The Laplacian operator 2 also remains invariant:

2 =3

i=1

2

x2i=i,jk

RijRik

xj

xk

=jk

jk

xj

xk= 2. (1.4.6)

Therefore the exact Hamiltonian H is invariant under all spatialtransformations, which include translations, rotations (proper andimproper), and combinations of the two. It is evident that underthese nonrestrictive conditions no relevant information on the eigen-functions and eigenvalues of H can be gained by the consideration ofthe symmetry properties of H.

The situation becomes completely dierent when we consider theHamiltonian

He = 2

2m

ni=1

2i + V(r,R). (1.4.7)

The Hamiltonian He represents the motion of the n electrons inthe eld of the nuclei, which are assumed to be at xed positions.Considering the symmetry properties of He, the nuclear coordinatesare not to be operated upon, because they appear only as parameters


1.4. The Symmetries of the Hamiltonian 11

in the Hamiltonian; we can only perform symmetry operations on theelectronic coordinates. Let us perform the most general symmetryoperation {R|t

} on the electronic coordinates. If we call

ri = {R |t}ri, (1.4.8)

then the Hamiltonian He in the new coordinates is given by

2

2m

ni=1

2i +12

ni=1

i =j

nj=1

e2

|ri r

j| +

12

N=1

=

N=1

e2ZZ|R

R |

ni=1

N=1

e2Z|R

ri|. (1.4.9)

The only term that has changed is the last sum. But, if the operation{R|t} is such that

|R

ri| = |R ri|, (1.4.10)where R

indicates the position of a nucleus identical to the nucleus,

then the last sum in (1.4.9) will go into itself. This fact implies that{R|t} must be such that

{R|t}1R

= R

, (1.4.11)

or

{R|t}R

= R

, (1.4.12)

for all identical nuclei. Therefore the Hamiltonian He is invariantunder all those operations on the electronic coordinates which, whenused on the nuclear coordinate parameters, send identical nuclei intoone another.

The Hamiltonian Hv represents the nuclear motion. The eigen-values of He are used as the potential in which the nuclei experiencetheir motion; this Hamiltonian depends only on the nuclear coor-dinates. The symmetry operations are now to be performed on thenuclear coordinates, and it is clear that Hv is invariant under allthose operations which send identical nuclei into one another.

We have gone through this long discussion to show how symmetryconsiderations are relevant to the quantum-mechanical treatment of a


12 Introduction

crystalline solid. We have found that in the adiabatic approximationthe relevant symmetries are those related to the nuclear coordinates.We learn how to consider these symmetries in a formal way in thenext three chapters.

Reference1. M. Born and J. R. Oppenheimer, Ann. Phys. 84, 457 (1927).


Chapter 2

Concepts of Group Theory

Group theory studies the algebraic structures known as groups. Thischapter denes a group and presents its basic properties, most impor-tant among them the representations. It then examines the connec-tions between group theory and quantum mechanics.

2.1. Properties of a Group

An ensemble of elements forms a group if the elements have the fol-lowing properties:

a. Closure: the product of two elements is an element of theensemble.

b. The ensemble contains the element E (identity).c. The multiplication of elements is associative: A(BC) = (AB)C.d. Every element has a reciprocal element contained in the ensemble:

Given an element A, the ensemble contains an element B such thatAB = BA = E.

In particular, the elements of a group may be symmetry opera-tions that transform a geometrical body of a certain shape into itself.

Example. Equilateral triangle (see Fig. 2.1). The operations thattransform the equilateral triangle into itself are the following:

1. E: identity,2. C3: 120 clockwise rotation about the axis Z,3. C23: 240

clockwise rotation about the axis Z,4. 1: reection through the plane containing the axis Z and axis 11,

13


14 Concepts of Group Theory

Fig. 2.1. Equilateral triangle with related axes. The axis Z is perpendicular tothe X and Y axes.

5. 2: reection through the plane containing the axis Z and axis 22,6. 3: reection through the plane containing the axis Z and axis 33.

The group of these operations is called C3v.The number of elements in a group is called the order of the

group.The elements of a group can be arranged in a multiplication table.

In the example of the equilateral triangle the table is given by

E C3 C23 1 2 3

E E C3 C23 1 2 3

C3 C3 C23 E 3 1 2

C23 C23 E C3 2 3 1

1 1 2 3 E C3 C23

2 2 3 1 C23 E C3

3 3 1 2 C3 C23 E

In general, if A and B are two elements of a group, then AB =BA. If AB = BA, A and B being any two elements of the group, thegroup is said to be Abelian.


2.2. Subgroups, Cosets, and Classes 15

2.2. Subgroups, Cosets, and Classes

Given a group G of order g, any subset of elements of G that is itselfa group is called a subgroup. The group C3v then has the followingsubgroups:

{E}, {E, 1}, {E, 2}, {E, 3}, {E,C3,C23}. (2.2.1)

Let us consider now the subgroup {E, 1} of C3v. We can multiplyall the elements of {E, 1} by all the other elements of C3v not in{E, 1}. Multiplying from the left, we obtain the following left cosets:

C3{E, 1} = {C3, 3}C23{E, 1} = {C23, 2}2{E, 1} = {2,C23}3{E, 1} = {3,C3}

(2.2.2)

Multiplying from the right, we obtain the right cosets:

{E, 1}C3 = {C3, 2}{E, 1}C23 = {C23, 3}{E, 1}2 = {2,C3}{E, 1}3 = {3,C23}

(2.2.3)

For each case we have two distinct cosets.We notice here that the group C3v can be expressed in terms of

the subgroup {E, 1} and the corresponding left or right cosets:

C3v: {E, 1}+ C3{E, 1}+ C23{E, 1}= {E, 1}+ {C3, 3}+ {C23, 2}, (2.2.4)

C3v: {E, 1}+ {E, 1}C3 + {E, 1}C23= {E, 1}+ {C3, 2}+ {C23, 3}. (2.2.5)

This property is of general nature, in the sense that a group maybe decomposed into right or left cosets with respect to a denitesubgroup.



If A and B are two elements of a group, the element

C = B1AB (2.2.6)

is called conjugate of A with respect to B.If a group G contains a certain subgroup H such that, for any

element X in G

X1HX = H, (2.2.7)

then the subgroup is said to be invariant or self-conjugate. A sub-group of order g/2 of a group G of order g is an invariant subgroupof G. The proof of this last property is left to the reader.

The right and left cosets of a group with respect to an invariantsubgroup are the same, because HX = XH.

Given a certain group G with an invariant subgroup H, the collec-tion of elements that consists of H and of all the cosets with respectto H forms a group that is called the factor group and is indicatedby G/H. In the case of the group C3v an invariant subgroup is givenby H = {E,C3,C23}. The factor group C3v/H is then given by

{E,C3,C23}, {1, 2, 3}and is a group of order 2.

A collection of elements conjugate to each other form a class.

Example. Group C3v. The elements conjugate of C3 and C23 canbe found from the following operations:

EC3E = C3 EC23E = C23

C23C3C3 = C3 C23C

23C3 = C

23

C3C3C23 = C3 C3C23C

23 = C

23

1C31 = C23 1C231 = C3

2C32 = C23 2C232 = C3

3C33 = C23 3C233 = C3

Therefore the elements C3 and C23 belong to the same class.It can be shown easily that the identity E forms a class by itself

(a general property) and that the elements 1, 2, and 3 also forma class.


2.3. Theory of Representations 17

The group C3v consists therefore of the following three classes:

C 1 : E

C2 : C3,C23C3 : 1, 2, 3.

Any group can be decomposed into non-overlapping classes.The product of two classes can be expressed in general as a linear

combination of all the classes of the group:

CiCj =hi

=1

hjk=1

RiRjk =

rm=1

cijmCm (2.2.8)

where

hi = number of elements in Cihj = number of elements in CjRi = generic element of CiRjk = generic element of Cjr = number of classes in the group

cijm = positive integer or zero.

In the case of group C3v, for example,

C2C3 = (C3,C23) (1, 2, 3)= C31,C32,C33,C231,C

232,C

233

= 3, 1, 2, 2, 3, 1 = 2C3. (2.2.9)

If a group is Abelian, then

B1AB = AB1B = A, (2.2.10)

B and A being any two elements of the group. Therefore in an Abeliangroup every element forms a class by itself.

2.3. Theory of Representations

Given a group G, any set of elements that multiply according tothe multiplication table of G is said to be a representation of G. Ifthe elements of the representation are matrices the representation



is called a matrix representation. A restriction on these matrices isthat they must be nonsingular; that is, each matrix must possesan inverse. A necessary and sucient condition for a matrix to benonsingular is that its determinant must not vanish.

Example. Three matrix representations of the group C3v are givenin the following table.

C3v E C3 C23

1 1 1 12 1 1 1

3

1 00 1

0BB@12

3

2

3

212

1CCA

0BB@12

3

23

212

1CCA

C3v 1 2 3

1 1 1 12 1 1 1

3

1 00 1

! 0BB@1

23

2

3

212

1CCA

0BB@

1

2

3

23

212

1CCA

It is easy to see that the matrices multiply as the elements theyrepresent. For example C3 1 = 3 corresponds to

1

2

32

32

12

(1 00 1

)=

12

32

32

12

Let us assume that

a,b, c,d, e, f, . . . (2.3.1)

are the matrices of a representation of a group. Let us transform eachmatrix by using a certain matrix

in a similarity transformation as


2.3. Theory of Representations 19

follows:

a =

1a

, b

=

1b

, c

=

1c

. (2.3.2)

The new matrices a,b, c, . . . form a representation of the group.

In fact, if

ab= d

, (2.3.3)

then

1a

b=

1a

1b

=

1d

, (2.3.4)

that is

ab = d

. (2.3.5)

A representation with matrices a,b, c,d, e, f, . . . is said to be

reducible if it is possible to nd a similarity transformations whichtransform each matrix of as follows:

a =

1a

=

a1 0 0 0

0 a2 0 0

0 0 a3 0

0 0 0 a4

. (2.3.6)

The matrices a1 are square matrices. Correspondingly the matrices

b, c, . . . in their reduced form are blocked out in the same way. Since

ab = d

, (2.3.7)

it is also true that

a1b

1 = d

1

a2b

2 = d

2

a3b

3 = d

3

(2.3.8)

Therefore the matrices

a1,b

1, c

1,d

1, e

1, f1

, . . . (2.3.9)



form a representation of the group. Other representations aregiven by

a2,b

2, c

2,d

2, e

2, f2

, . . . (2.3.10)

a3,b

3, c

3,d

3, e

3, f3

, . . . (2.3.11)

If we indicate the representations (2.3.9), (2.3.10), (2.3.11), . . . by1,2,3, . . . , then the reduction of the matrix representation isindicated as follows:

= 1 + 2 + 3 + (2.3.12)

Given a matrix representation , if it is not possible to nd asimilarity transformation that reduces it, the representation is saidto be irreducible.

Two irreducible representations that dier only by a similaritytransformation are said to be equivalent.

The representations of the group C3v given in the example areinequivalent and irreducible; this statement is given here withoutproof, but will be checked later.

2.4. Orthogonality Relations

We now examine some important relations that apply to irreducibleinequivalent matrix representations.

We consider in particular the representations of the group C3vreported in Section 2.3. If we call R the generic symmetry operationwe nd

R

1(R)111(R)11 = 1 + 1 + 1 + 1 + 1 + 1 = 6

R

2(R)112(R)11 = 1 + 1 + 1 + 1 + 1 + 1 = 6

R

3(R)113(R)11 = 1 +14+

14+ 1 +

14+

14

= 3

R

3(R)123(R)12 =34+

34+

34+

34

= 3


2.4. Orthogonality Relations 21

R

3(R)213(R)21 =34+

34+

34+

34

= 3

R

3(R)223(R)22 = 1 +14+

14+ 1 +

14+

14

= 3

Also, we ndR

1(R)112(R)11 = 1 + 1 + 1 1 1 1 = 0R

1(R)113(R)11 = 1 12 12+ 1 1

2 1

2= 0

R

2(R)113(R)22 = 1 12 12 1 + 1

2+

12

= 0

R

3(R)113(R)22 = 1 +14+

14 1 1

4 1

4= 0

In summary we obtain

R

(R)ik(R)j = 0, if = ,

or i = ,or k = j,

(2.4.1)

and R

(R)ik(R)ik =gn

, (2.4.2)

where g is the order of the group and n is the dimension of therepresentation .

The relations (2.4.1) and (2.4.2) reect a general property of thematrix elements of irreducible inequivalent representations that canbe expressed as follows:

R

(R1)ki(R)j = ikjgn

. (2.4.3)

The generic element of a matrix may be a complex number.



These relations are called orthogonality relations and apply toinequivalent irreducible representations. If the matrices are unitarythe relation above can be expressed as follows:

R

(R)ik(R)j = ikjgn

. (2.4.4)

The g numbers

(R1)ij,(R2)ij,(R3)ij, . . . (2.4.5)

may be considered to be the components of a g-dimensional vectororthogonal to any other vector of the same type, obtained by a dier-ent choice of the subscripts i, j, or . If the dimensions of the repre-sentations are n,n,n, . . . then the total number of these vectors is

n2 + n2 + n

2 + (2.4.6)

This number must be equal to g, because it is possible to constructonly g orthogonal vectors in a g-dimensional space:

n2 + n2 + n

2 + = g, (2.4.7)

orr

=1

n2 = g, (2.4.8)

where r is the number of irreducible inequivalent representations. Inthe example of the group C3v,

n1 = 1n2 = 1n3 = 2

and the number of elements in the group is

3=1

n2 = 1 + 1 + 4 = 6.

Example. If g = 4, since we always have a one-dimensional repre-sentation with all matrices equal to 1, it must also be

n2 = 12 + 12 + 12 + 12 = 4.


2.5. Characters of a Matrix Representation 23

Therefore if g = 4, the group has four one-dimensional irreduciblerepresentations.

If g = 10

n2 = 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12 + 12,

or

12 + 12 + 12 + 12 + 12 + 12 + 22,

or

12 + 12 + 22 + 22,

or

12 + 32.

That is, the group has either ten one-dimensional or six one-dimensional and one two-dimensional or two one-dimensionaland two two-dimensional or one one-dimensional and one three-dimensional irreducible representations.

2.5. Characters of a Matrix Representation

Given a certain matrix representation, the traces of the matrices arecalled characters of the representation. The character of a represen-tation corresponding to the generic operation R is given by

(R) =m

(R)mm. (2.5.1)

The characters of the irreducible representations of the group C3vcan be easily derived by looking at the table of these representationsin Section 2.3:

C3v E C3 C23 1 2 3

1 1 1 1 1 1 12 1 1 1 1 1 13 2 1 1 0 0 0



We notice here that the character corresponding to the oper-ation E (identity) is equal to the dimension of the representation,namely

(E) = n. (2.5.2)

The character of a matrix is unchanged by a similarity transforma-tion. In fact, if the character of a matrix A is given by

A =j

Ajj, (2.5.3)

and if

B = C1AC, (2.5.4)

then

B =i

Bii =i

j

k

C1ij AjkCki

=j

k

(i

C1ij Cki

)Ajk =

j

k

jkAjk

=j

Ajj = A. (2.5.5)

From (2.5.4) and (2.5.5) we can deduce that if two operations belongto the same class, the corresponding matrices for a given representa-tion have the same character.

Also, two nonequivalent, irreducible representations have dier-ent character systems and two irreducible representations with thesame character system are equivalent.

Let us consider again the orthogonality relations (2.4.4) and letus put i = k, = j. We obtain

R

(R)ii(R) = ign

. (2.5.6)


2.6. Reduction of a Reducible Representation 25

Summing over i from 1 to n and over form 1 to n

R

[ni=1

(R)ii

n=1

(R)

]

=R

(R)(R) =gn

ni=1

n=1

i = g.

(2.5.7)

Therefore we nd the following orthogonality relation for thecharacters:

R

(R)(R) = g , (2.5.8)

and we can say that the characters of the irreducible representationsof a group of order g form a set of orthogonal g-dimensional vectors.

An important property can be derived from (2.5.8). Since for acertain representation the characters corresponding to operations ofthe same class are equal, we may write (2.5.8) as

r=1

(R)(R)g = g, (2.5.9)

orr

=1

[(R)

gg

] [(R)

gg

]= , (2.5.10)

where g is the number of elements in the th class, R is the genericoperation in the th class, and r is the number of classes. The nor-malized characters (R)

g/g are the components of a set of

orthogonal vectors in an r-dimensional space. Since we can constructonly r such vectors, we see that the number of irreducible inequiv-alent representations of a group must be equal to the number ofclasses.

2.6. Reduction of a Reducible Representation

Any reducible representation can be reduced in terms of itsirreducible representations by a proper similarity transformation.



This transformation leaves the character of the representationunchanged. Therefore, if we designate by (R) and j(R) the char-acter of the reducible representation and of the generic irreduciblerepresentation, respectively, we can write

(R) =r

j=1

cjj(R), (2.6.1)

where r is the number of irreducible representations and cj is thenumber of times the jth irreducible representation is contained inthe reducible representation.

Let us multiply both members of (2.6.1) by i(R) and sumover R:

R

(R)i(R) =R

j

cjj(R)i(R)

=j

cjR

j(R)i(R)

=j

cjgij

= cig, (2.6.2)

or

ci =1g

R

(R)i(R). (2.6.3)

It is also true thatR

|(R)|2 =R

(R)(R)

=R

[

i

cii(R)]

j

cjj(R)

=i

j

cicjR

i(R)j(R) = gi

c2i , (2.6.4)

where we have made use of (2.5.8).


2.7. Basis Functions for Irreducible Representations 27

Example. Group C3v. A reducible representation of this grouphas the following character system:

E C3 C23 1 2 3

5 1 1 1 1 1

The representation will reduce as

= c11 + c22 + c33,

where

c1 =16

R

(R)1(R) =16[5 1 1 + 1 + 1 + 1] = 1,

c2 =16

R

(R)2(R) =16[5 1 1 1 1 1] = 0,

c3 =16

R

(R)3(R) =16[10 + 1 + 1] = 2.

Therefore

= 1 + 23.

Also R

|(R)|2 = 30 = gi

c2i .

2.7. Basis Functions for Irreducible Representations

Let us consider an irreducible representation of dimension nand let

u1,u2, . . . ,u

n (2.7.1)

be a set of n functions. These functions are said to form a basis forthe representation if when applying an operation R of the group



to a function ui of the set we obtain

Rui =nj=1

(R)jiuj . (2.7.2)

If a set of functions form a basis for a representation it is alsosaid that they transform irreducibly according to . In particular itis said that ui transforms according to the i-th column of the matrixrepresentation .

Example. Group C3v. The representation 3 of the group C3v isgiven by

E C3 C23

1 00 1

0BB@12

3

2

3

212

1CCA

0BB@12

3

23

212

1CCA

1 2 3

1 00 1

! 0BB@1

23

2

3

212

1CCA

0BB@

1

2

3

2

3

212

1CCA

A set of functions that form a basis for this representation isgiven by x and y. Applying (2.7.2) to these two functions, we obtain,for example, in the case R = C3

C3x = 12x32

y,

C3y =32

x 12y.

The operation R, being in general a symmetry operation, may beconsidered to be linear and unitary. Therefore

(Rui ,Ruj ) = (u

i ,u

j ), (2.7.3)


2.8. Direct Product Representations 29

where the parentheses indicate the inner product of two functions.Let us assume that the functions of the set (2.7.1) are orthonormal:

(ui ,uj ) = ij. (2.7.4)

Then, since

Rui =n

=1

(R)iu , (2.7.5)

Ruj =nm=1

(R)mjum, (2.7.6)

(2.7.3) becomes

(Rui ,Ruj ) =

n=1

nm=1

(R)i(R)mj(u ,u

m)

=n

=1

nm=1

(R)i(R)mjm

=n

=1

(R)i(R)j = ij, (2.7.7)

or

+ (R)

(R) = 1. (2.7.8)

Thus we can say:

If the functions forming a basis for an irreducible representation ofa group of linear unitary operations are orthonormal, the matricesrepresenting the operations are unitary.

2.8. Direct Product Representations

Let us assume that

u1,u2, . . . ,u

n

v1, v2, . . . , v

n

(2.8.1)



are two sets of functions that form the basis for two representationsof the same group and , respectively. The products ui v

k form

a set of functions that is called the direct product of the two sets offunctions. The direct product set forms a basis for a representationthat has dimension n n and is in general reducible.

If R is an operation of the group, then

Rui =nj=1

(R)jiuj ,

Rvk =n

=1

(R)kv , (2.8.2)

and

R(ui vk) =

nj=1

n=1

(R)ji(R)kuj v

=j

(R)j,ikuj v . (2.8.3)

The character of the direct product representation is equal to theproduct of the characters of the individual representations:

(R) =j

(R)j,j

=nj=1

n=1

(R)jj(R) = (R)(R). (2.8.4)

A direct product representation can, in general, be reduced in termsof the irreducible representation of the group.

Example. Group C3v. We obtain

11 = 1; 22 = 1

33 = 1 + 2 + 3; 12 = 2

13 = 3; 23 = 3


2.9. The Fundamental Theorem for Functions Transforming Irreducibly 31

2.9. The Fundamental Theorem for FunctionsTransforming Irreducibly

A set of n functions ui are said to form a basis for an irreduciblerepresentation or to transform irreducibly according to represen-tation of a symmetry group G if

Rui =nj=1

(R)jiuj , (2.9.1)

where R is any operation of G. The above relation can be rewrittenin the following form:

R(u1,u2, . . . ,u

n)

= (u1,u2, . . . ,u

n)

(R)11(R)12 . . .(R)1n(R)21(R)22 . . .(R)2n(R)n1(R)n2 . . .(R)nn

, (2.9.2)

or, more concisely,

R u = u

(R). (2.9.3)

In (2.9.1) the i-th function ui is said to transform according tothe ith column of the representation. We assume that the basisfunctions are orthonormal, and therefore the representations are uni-tary.

A set of n operators Qi are said to transform irreducibly accord-ing to a representation of a symmetry group G if

RQi R1 =

j

Qj (R)ji, (2.9.4)

where R is any operation of G.The case of an invariant operator is a particular one in which the

operator transforms according to the identity representation 1:

RQ(x)R1 = Q(x). (2.9.5)

Let us consider a set of orthonormal functions ui transform-ing irreducibly according to a representation and a set oforthonormal functions vj transforming irreducibly according to a



representation . The product functions ui vj form a basis for the

representation that is, in general, reducible: =

g. (2.9.6)

The representation is brought into a reduced from by aunitary matrix T

:

T+[

(R)

(R)]T

=

(R). (2.9.7)

The generic element of T

may be indicated by T,qij,n where i and jindividuate two basis functions, respectively, in the and man-ifold, is the generic irreducible representation in (2.9.6), and nindividuates a basis function in the manifold. Finally, since maybe contained more than once in (g may be greater than 1),the index q distinguishes among the dierent representations.

We may consider the function

w,qn =i,j

T,qij,n ui v

j . (2.9.8)

Then

ui vj =

nq

[T,q

ij,n ] w

,qn , (2.9.9)

where [T,qij,n ] is the generic element of T

+.

Let us consider now an integral of the type

(ui ,Qvj ), (2.9.10)

where Q is an invariant operator. If the representations and are unitary, besides being irreducible, then

(ui ,Qvj ) = (Ru

i ,RQv

j )

= (Rui ,QRvj ) =

1g

R

(Rui ,QRvj )

=1g

R

k

m

(R)ki(R)mj(uk,Qv

m)

=1nij

k

(uk,Qvk). (2.9.11)


2.9. The Fundamental Theorem for Functions Transforming Irreducibly 33

We see that the matrix element is dierent from zero if = ,i = j and also is independent of the subscript i. A particular caseis given by

(ui , vj ) = 0, unless = , i = j, (2.9.12)

as expected.We may now enunciate the rst part of the Fundamental

Theorem:

I. Matrix elements of invariant operators taken between two givenfunctions vanish unless the two functions transform according to thesame column of the same irreducible representation. If it does notvanish, the matrix element is independent of the column according towhich the two functions transform.

Let us consider, further, an integral of the following type:

(fn,Qi v

j ), (2.9.13)

where Qi indicates the i component of a set of operators that trans-form irreducibly according to the representation. Following (2.9.8)we may write

Qi vj =

nq

[T,q

ij,n ] w

qn , (2.9.14)

where the wqn function transforms irreducibly according to the nth

column of the representation.The integral (2.9.13) is then given by

(fn,Qi v

j ) =

gq=1

cq[T,qij,n ]

, (2.9.15)

where the coecient cq are independent of the indices i, j, and n anddepend only on the operators in question and on the indices , , ,and q.

If g = 1, then

(fn ,Qi v

j ) = [T

,ij,n ]

const, (2.9.16)where the constant is independent of the subscripts.



If Pi is the i component of another set of operators that transformirreducibly according to we get also

(fn ,Pi v

j ) = [T

,ij,n ]

const, (2.9.17)

the constant being dierent from the one in (2.9.16), but still inde-pendent of subscripts. From (2.9.16) and (2.9.17) we get

(fn ,Qi v

j ) = const (fn ,Pi vj ),

the constant being independent of the subscript.We can now enunciate the second part of the Fundamental

Theorem:

II. Matrix elements of operators Qi transforming irreducibly accord-ing to a representation , taken between two functions f

n and v

j

that transform irreducibly according to columns n and j, respectively,of representation and , vanish unless the representation contains the representation . If is contained q times in ,then the value of the matrix element is uniquely determined by thesymmetry property of f n ,Qi , and v

j apart from q constants. These

constants do not depend on the subscripts n, i, and j.

2.10. Product Groups and Their Representations

Suppose we are given two groups,

g = e, a,b, c, . . .

G = E,A,B,C, . . .(2.10.1)

and suppose that all products between the elements of the two groupsare commutative:

rR = Rr, (2.10.2)

where r is any element of g and R is any element of G.If the groups g and G are, respectively, of order m and M and if

(r) and

(R) are the matrices representing g and G, respectively, it

is possible to order the (mM)2 numbers (r)ij (R)k in such a


2.11. Connection of Quantum Mechanics with Group Theory 35

way that they form a mM mM matrix. For example, the elementappearing in the ik-th row and j-th column may be

(r)ij (R)k. (2.10.3)This matrix is called the product group matrix.

The character (rR) of (r)

(R) is given by

(rR) = (r)(R). (2.10.4)

All the irreducible representations of g G can be obtained byforming the direct product representations that have as componentsthe respective irreducible representations of the component groups.

If g has s irreducible representations and G has S irreduciblerepresentations, g G has sS irreducible representations.

2.11. Connection of Quantum Mechanicswith Group Theory

Let us consider a quantum-mechanical system that is represented bythe eigenfunctions of the equation

H = E. (2.11.1)

Let us also assume that the Hamiltonian is invariant under certainoperations

RH(x)R1 = H(x

). (2.11.2)

It is easily seen that these operations form a group.

1. The operation that does not change the change the coordinatescorresponds to the identity element of the group.

2. If R and S are two generic operations that leave H invariant, thatis, if

RHR1 = H,SHS1 = H,

(2.11.3)

then

(SR)H(SR)1 = (SR)H(R1S1) = S(RHR1)S1

= SHS1 = H. (2.11.4)



3. The associative law also holds.4. Moreover, from (2.11.2), multiplying by R1 on the left and R on

the right we obtain

H = R1HR,

that is, the inverse of any operation R that leaves H invariant isalso an operation that leaves H invariant.

Therefore the operations that leave the Hamiltonian invariant forma group. We call this group the group of the Schrodinger equationand designate it by the symbol G.

Let us now apply an operation R of G to both sides of (2.11.1):

RH = ER. (2.11.5)

Because of the invariance of H,

RH = HR, (2.11.6)

and

H(R) = E(R). (2.11.7)

R is a solution of the eigenvalue equation belonging to the eigen-value E.

If E is not degenerate, it must be

R = c, (2.11.8)

with c a constant.If E is an m-fold degenerate eigenvalue, then

R =mi=1

ai i, (2.11.9)

where i are the degenerate eigenfunctions corresponding to theeigenvalue E. If we apply another operation S of G to we get

S =mj=1

bjj. (2.11.10)



If we apply the operation T = SR to we obtain

(SR) =mj=1

cjj. (2.11.11)

Since the operation SR consists of operation R followed by operationS, we have also from (2.11.9)

SR = S(R) =mi=1

aiSi

=mi=1

aimj=1

bjij

=mi=1

mj=1

aibjij. (2.11.12)

Then from (2.11.11) and (2.11.12)

cj =mi=1

bji ai. (2.11.13)

The matrix of the coecients cj is the product of the matrix of thecoecients bji and of the matrix of the coecients ai:

C

= BA

. (2.11.14)

We note that if the eigenfunctions are orthonormal these matricesare unitary. The above relation mirrors the relation

T = SR. (2.11.15)

By applying all the operations of G to we nd a matrix forevery element of the group. These matrices form a representation ofthe group G of the Schrodinger equation and the set of degenerateeigenfunctions ( = 1, 2, . . . ,m) are said to form a basis for arepresentation of the group G.

The representation is irreducible. In fact, if it were not so, itcould be possible to form subsets of linear combinations of the functions

1,2, . . . ;

i,

i+1, . . . ;

j,

j+1, . . .

m, (2.11.16)



such that each operation of G transforms a function of a subset intoa linear combination of functions of the same subset. But, if this werethe case, the eigenvalues corresponding to the dierent subsets couldbe dierent, contrary to the assumption that all the m eigenfunctions have the same eigenvalue.

Therefore we can state the following:

Eigenfunctions belonging to the same eigenvalue of the energy of asystem form a basis for an irreducible representation of the group ofoperations that leave the Hamiltonian invariant. The dimensionof this irreducible representation is equal to the degree of degeneracyof the eigenvalue.

Example. Let us assume that the system under consideration hasa Hamiltonian that is left invariant by all the symmetry operationsof the group C3v.

An eigenfunction representing the system is of one of the follow-ing types:

a. An eigenfunction that forms a basis for the representation 1 ofC3v; that is, it remains unchanged if acted upon by any operationof C3v.

b. An eigenfunction that forms a basis for the irreducible represen-tation 2 of C3v; that is, it remains unchanged if acted uponby the operations E,C3,C23, but changes sign if subjected to theoperations 1, 2, and 3 of C3v.

c. One of two functions that form a basis for the representation 3of C3v and are bound by the relation

R(12) = (12)

(3(R)11 3(R)123(R)21 3(R)22

), (2.11.17)

or

R1 = 3(R)111 + 3(R)212,

R2 = 3(R)121 + 3(R)222,(2.11.18)

where R is the generic operation of C3v.



The rule just enunciated has the following exceptions:

1. Accidental Degeneracy

The eigenvalues of two eigenfunctions may be equal by fortuitouscoincidence. The eigenfunctions do not belong to the same set ofbasis function for a representation of the group of the Schrodingerequation. Examples of systems that present this type of degeneracyare the following:

a. An atomic system under the action of a magnetic eld. It mayhappen that two Zeeman levels corresponding to two dierentorientations of the magnetic moment of the atom in the magneticeld may have the same energy in correspondence to a certainvalue of the magnetic eld. If we change the intensity of the mag-netic eld this degeneracy is lifted.

b. Atomic hydrogen presents degeneracy between s,p,d, f, . . . levelswith the same n number. If the Coulomb eld is replaced by a non-Coulombic central eld (as in hydrogen-like atoms, for example,Na), the degeneracy is lifted.

2. Excess Degeneracy

This type of degeneracy derives from the fact that the group of theSchrodinger equation under consideration does not contain all theoperation that leave the Hamiltonian invariant. An example of thistype of degeneracy is the Kramers degeneracy due to time-reversalsymmetry.

In summary, we can say, that, apart from the two cases describedabove,

1. All the possible eigenfunctions of a Hamiltonian must form a basisfor some irreducible representation of the group of the operationsthat leave the Hamiltonian invariant.

2. By knowing the irreducible representations of the group of opera-tions that leave the Hamiltonian invariant, we know the possibledegrees of degeneracy of the eigenfunctions and the transforma-tion properties of the eigenfunctions under the operations of thegroup.


Chapter 3

Crystal Symmetries

This chapter deals with the basic concepts underlying the symmetryof an ordered array of atoms. The notions of crystal systems andBravais lattices are introduced.

3.1. Unit Cells and Space Lattices

A crystal is a periodic array of atoms whose basic repeating unit iscalled unit cell. The smallest unit cell is called primitive unit cell.

The three linearly independent vectors a, b, and c

that dene the

primitive unit cell are called basic primitive translations. The arrayof points generated by the vectors

Tn = n1a

+ n2b

+ n3c

, (3.1.1)

where n1,n2, and n3 are integer numbers, is called a lattice.In a crystal lattice, each lattice site has the same surroundings

and is at a corner of a primitive unit cell. Therefore the primitiveunit cell contains one lattice site. Unit cells other than primitive aresometimes used; these cells contain more than one lattice site.

Every site that occurs on the surface or edge or corner of a unitcell must also occur at equivalent sites on the same surface or edgeor corner, respectively, of every other unit cell (see Fig. 3.1).

Any point in or on the unit cell may be indicated by three coor-dinates with respect to the axes a

, b

, and c

. In order to identify

a position that repeats itself at equivalent locations, it is sucientto give the coordinate of one location. In general, the coordinates

41


42 Crystal Symmetries

Fig. 3.1. Equivalent sites on a (a) surface, (b) edge, and (c) corner of a unit cell.

involving as many zeros as possible are used. In Fig. 3.1 the equiva-lent positions are identied by the following coordinates:

a:12

,12

, 0

b:12

, 0, 0

c: 0, 0, 0

Atomic positions in the unit cell may or may not coincide withlattice sites. In general, only the inequivalent atomic sites in the unitcell are given. The unit cell of Fig. 3.2 has one atomic site at (0, 0, 0).The unit cell of Fig. 3.3 has two atomic sites at (12 , 0, 0) and (0,

12 , 0).

The unit cell of Fig. 3.4 has three atomic sites at (12 ,12 , 0), (

12 , 0,

12),

and (0, 12 ,12).


3.1. Unit Cells and Space Lattices 43

Fig. 3.2. Unit cell with one atomic site at 000.

Fig. 3.3. Unit cell with two atomic sites at 1200 and 0 1

20.

Fig. 3.4. Unit cell with three atomic sites at 12

120, 1

20 12, 0 1

212.

There are, in general, several ways of positioning and choosingthe unit cell with respect to the atomic sites. However, it should beclear that the lattice associated with a certain crystal structure isunique.



3.2. Miller Indices

A specic direction in a crystal may be identied by consideringa vector T

n parallel to this direction and giving its three compo-

nents in the basic directions of the unit cell. The vector is so chosenthat the three components n1,n2,n3 are the lowest possible inte-gers dening the direction. These three numbers are called Millerindices of the direction and are presented within square bracketsas follows: [n1n2n3]. In Fig. 3.5 the directions s

and t

have indices

[121] and [111], respectively. The directions of the basic vectorsa, b, and c

are designated by [100], [010], and [001], respectively. If

any of the three indices is negative a minus sign is put on top of thenegative index. The directions opposite to s

in Fig. 3.5 is indicated

by [121].A plane in a crystal is identied by three Miller indices that

are determined from the intersection of the plane with the threecoordinate axes. The intercepts are expressed in terms of the basicvectors a

, b, and c

. We illustrate the procedure to be followed in order

to nd the Miller indices of a plane with the following example.

Fig. 3.5. Directions in a crystal: s= [121], t

[111].


3.2. Miller Indices 45

Example. Let us assume that a plane has the following intercepts,with the axes a

, b, and c

, respectively: 2, 1, 3. We proceed as follows:

1. We take the reciprocal of the intercepts:

12

, 1,13

2. We given the above fractions the minimum common denominator:

36

,66

,26

3. We take the numerators of the fractions above as indices:

3, 6, 2

4. We enclose the indices in parentheses:

(362)

If the point of interception of a plane has a negative coordinate,the correspondent Miller index is given with a negative sign on topof it. Notice that two planes (h k l) and (h k l) are parallel.

If a plane is parallel to one of the axes, the point of intersec-tion between the plane and the axis is at innity; the correspondingMiller index is, in this case, zero. In particular we notice the followingplanes:

Plane Miller indices

ab

(001)

bc

(100)

ac

(010)

In a cubic lattice a plane (h k l) and a direction [h k l] aremutually perpendicular.

Also, in a cubic lattice the directions of fourfold rotation

[100], [010], [001], [100], [010], [001]



are considered equivalent and collectively designated by 100. Simi-larly, the directions of threefold rotation

[111], [111], [111], [111], [111], [111], [111], [111]

are equivalent and are designated by 111. The six twofold rotationaxes are of the form 110.

In a cubic lattice the six faces are considered to be equivalentand collectively designated by {100}. The six dodecahedral planesare designated by {110} and the eight octahedral planes by {111}.

Consider now in Fig. 3.6 the unit cell of a hexagonal latticedened by the basic vectors a

1, a

2, and c

. The six planes

(100), (010), (110), (100), (010), (110)

Fig. 3.6. Unit cell of a hexagonal lattice.


3.3. The Crystal Systems 47

are equivalent in the hexagonal lattice, but do not have similar Millerindices. This inconvenience is obviated by using a four-index nota-tion, referred to the four basic vectors a

1, a

2, a

3, and c

. In this

notation the six planes become

(1010), (0110), (1100), (1010), (0110), (1100)

A consequence of choosing the three basic vectors a1, a

2, and a

3 with

120 between each other is the fact that in a four-index notation thesum of the rst three indices is always zero. Therefore one index canalways be determined if the other three are known and the redundantindex may be replaced by a dot. For example, plane (1120) may beindicated by (11.0).

If a direction is given in terms of the unit cell dened by thevectors a

1, a

2, and c

in Fig. 3.6 by means of the indices [MNQ], it

is possible to express it in terms of the unit cell dened by the fourvectors a

1, a

2, a

3, and c

by means of four indices [m n p q]. These

indices can be obtained by using the following relations

m =13(2MN),

n =13(2NM),

p = (m + n),q = Q,

(3.2.1)

and reducing the new indices to the lowest integers. Thus, if [MNQ] =[110], then

m =13

, n =13

,

p = 23

, q = 0,

and [m n p q] = [1120].

3.3. The Crystal Systems

The unit cells of a crystal structure can take only a certain numberof distinct shapes. This number is four for two-dimensional crystals



Table 3.1. The four two-dimensional crystal systems.

Axial Angular Characteristic

System relationship relationshipa symmetryb C

Oblique a = b = 90 No symmetry C2Rectangular a = b = 90 1 C2vSquare a = b = 90 1 C4vHexagonal a = b = 120 1 C6vaThe convention for the angle is

b , 2-fold symmetry axis, 4-fold symmetry axis, 6-fold symmetry axiscThe meaning of the notation used in this column is explained in Chapter 4.

and seven for four-dimensional crystals. Correspondingly there arefour crystal systems in two dimensions and seven crystal systems inthree dimensions.

3.3.1. The Four Two-Dimensional Crystal Systems

The crystal systems in two dimensions are reported together with theaxial and angular relationships of their unit cells in Table 3.1. Eachsystem has also a characteristic symmetry. The square lattice has,for example, fourfold rotational symmetry; that is, it will reproduceitself four times in a complete 360 rotation.

In general, a gure with an n-fold axis reproduces itself every360/n degrees in a rotation about such an axis.

3.3.2. The Seven Three-Dimensional Crystal Systems

The crystal systems in three dimensions are reported in Table 3.2together with the axial and angular relationships of their unit cells.

The only rotation axes that can occur in crystals are 1-, 2-, 3-,4-, and 6-fold. In a rotation about an n 1 fold axis, a body withsuch an axis reproduces itself every 360/n degrees.


3.3. The Crystal Systems 49

Table 3.2. The seven three-dimensional crystal systems.

Axial Angular Characteristic

System relationship relationshipa symmetryb C

Triclinic a = b = c = = No symmetry S2Monoclinic a = b = c = = 90 = 1 in b direction C2hOrthorhombic a = b = c = = = 90 3 mutually D2hTetragonal a = b = c = = = 90 1 in c direction D4hCubic a = b = c = = = 90 4 to octahedral

planesOh

3 to cube facesTrigonal a = b = c = = < 120 = 90 1 D3dHexagonal a = b = c = = 90 = 120 1 D6haThe convention for the angles is

b , two-fold symmetry axis, three-fold symmetry axis, four-fold symmetry axis, six-fold symmetry axis

cThe meaning of the notation used in this column is explained in Chapter 4.

Each system has a characteristic symmetry. Given a certain crys-tal the characteristic symmetry can be used to individuate the systemto which the structure belongs.

For the cubic systems, either the presence of the four 3-fold axes(perpendicular to the octahedral {111} planes) or the presence ofthe three fourfold axes (perpendicular to the cube faces {100} aresucient to identify the crystal system.

The characteristic symmetries of trigonal, tetragonal, andhexagonal systems are one 3-fold, one 4-fold and one 6-fold axis,respectively.

The presence of three mutually orthogonal 2-fold axes is sucientto identify the orthorhombic system.



The monoclinic system is identiable by the presence of one 2-fold axis in the direction b.

Finally the triclinic system, having a unit cell with the threebasis vectors and the three angles dierent from each other, has nosymmetry.

We notice that, in general, a crystal structure has more sym-metry elements than the ones strictly necessary to identify it. Forexample, the cubic system has also 2-fold axes perpendicular to thedodecahedral planes {110}.

3.4. The Bravais Lattices

A crystal structure is represented by an array of points called a lat-tice. Each lattice site is at a corner of a primitive cell and has sur-roundings identical to those of any other site. We have shown thatthere are four distinct shapes of primitive cells (and crystal systems)in two dimensions and seven distinct shapes of primitive cells (andcrystal system) in three dimensions. The corresponding lattices arecalled primitive Bravais lattices.

There are other lattices that have additional sites not at thecorners of their unit cells but with surroundings identical to thoseat the corner. These lattices are called nonprimitive Bravais lattices.There is one nonprimitive Bravais lattice in two dimensions and thereare seven nonprimitive Bravais lattices in three dimensions.

Therefore we have a total of ve Bravais lattices in two dimen-sions and fourteen Bravais lattices in three dimensions.

3.4.1. The Five Bravais Lattices in Two Dimensions

The ve Bravais lattices in two dimensions are given in Table 3.3 andFig. 3.7. Of these, four (the oblique, the rectangular, the square andthe hexagonal) are primitive. The internally centered rectangular isthe only nonprimitive Bravais lattice in two dimensions.

We shall illustrate now the reason for the presence of the non-primitive Bravais lattices. The following considerations apply to boththe two-dimensional and the three-dimensional cases.

Any crystal can be represented by one of the primitive Bravaislattices; that is, it may have a primitive cell in one of the four (seven)


3.4. The Bravais Lattices 51

Table 3.3. The ve Bravais latticesin two dimensions.

System Bravais latticea

Oblique PRectangular P, ISquare PHexagonal P

aP, primitive; I, internally centered.

Fig. 3.7. The ve Bravais lattices in two dimensions: (a) oblique P, (b) rectan-gular P, (c) rectangular I, (d) square P, (e) hexagonal P.



Fig. 3.8. Unit cells of (a) square lattices, (b) rectangular lattice.

crystal systems. Therefore, a primitive unit cell can always be drawnfor any lattice, including a nonprimitive Bravais lattice. This primi-tive cell contains the smallest possible number of atomic sites.

We illustrate these considerations with an example. Consider thesquare lattice in Fig. 3.8(a) with a nonprimitive unit cell of basis vec-tors a

and b

in the form of an internally centered square. A primitive

cell can be found for this lattice; this is the cell with basis vectorsa

and b

. This primitive cell belongs to the square crystal system.The Bravais unit cell is always chosen as the smallest unit cell

that presents the symmetry of the structure. Therefore, since theprimitive unit cell (with basis vectors a

and b

) belongs to thesame crystal system of and is smaller than the unit cell with basisvector a

and b

, the square P is the Bravais lattice of the structure in

Fig. 3.8(a) and the internally centered square is not a Bravais lattice.Consider now the rectangular I lattice in Fig. 3.8(b). A primitive

cell can also be found for this lattice; this is the cell with basis vectors


3.4. The Bravais Lattices 53

a

and b

. This cell, however, does not belong to the rectangular butto the oblique system, and therefore it cannot be the Bravais latticefor this structure. Therefore the internally centered rectangle is aBravais lattice.

3.4.2. The Fourteen Bravais Lattices in ThreeDimensions

The fourteen Bravais lattices in three dimensions are given inTable 3.4 and Figs. 3.93.15. Of these fourteen lattices seven areprimitive and seven are nonprimitive.

The primitive Bravais lattices have one lattice site at 000.

Table 3.4. The fourteen Bravais latticesin three dimensions.

System Bravais latticea

Triclinic PMonoclinic P, COrthorhombic P, C, F, ITetragonal P, ICubic P, F, ITrigonal R(P)Hexagonal P

aP, primitive; C, two faces centered; F,all faces centered; I, internally centered.

Fig. 3.9. Triclinic Bravais lattice. Vector lengths are indi

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