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Symmetry, Group Theory, and the Physical Properties of Crystals (Paperback, 2010)
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Symmetry, Group Theory, and the Physical Properties of Crystals (Paperback, 2010)
Series: Lecture Notes in Physics, 824
Expected to ship within 10 - 15 working days
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Whydowelookatsomethingsandthinktheyarebeautifulwhileotherthingsdo
notappearestheticallypleasingtous?Thisisaquestionthathasalwaysinterested
mankind.
Oneanswerisgivenbythefollowingquotationfromanearlypresidentof
theCollegeofNewJersey(nowPrincetonUniversity):
"Beautyisfoundinimmaterialthingslikeproportionoruniformity...calledbyvariousnamesofregularity,order,uniformity,symmetry,
1 proportion,harmony,etc. "...JonathanEdwards
Symmetrynotonlyprovidesthenaturalharmonythatmakessomethingappear
beautifultous,butalsoisofgreatvaluetosciencebecauseitdictatesthephysical
traitsofmanyobjects.
Natureitselfseemstolovebeautysinceatomstendtoself? assemble into
shapes with speci?c symmetry and crystals grow in geometric
lattices. Inmanycases,ifweknowthesymmetryofsomethingwecanpredict
someofitsimportantpropertieswithouthavingtoresorttoexperimentationor
complicatedcalculations. One area where the concept of symmetry
plays an important role is that of crystalline solids. Crystals, by
their very nature, exhibit speci?c symmetries.
Crystallinematerialshavemanyimportantapplicationsindevicesbasedontheir
electronic,optical,thermal,magnetic,andmechanicalproperties.
Solidstatep-
sicistsandchemists,aswellasmaterialscientistsandengineers,havedeveloped
rigorousquantumtheoreticalmodelstodescribethesepropertiesandsophisticated
measurementtechniquestoverifythesemodels.
Manytimes,however,inscreeningmaterialsforanewapplicationitisuseful
to be able to quickly and easily determine if a speci?c material
will have the
appropriatepropertieswithoutmakingdetailedcalculationsorexperiments.
This canbedonebyanalyzingthesymmetrypropertiesofthematerial.
Themathema-
calformalismthathasbeendevelopedtoaccomplishthisiscalledgrouptheory.
Thesymmetrypropertiesofacrystalcanbedescribedbyagroupofmathematical
1 J.
Edwards,WorksofJonathanEdwards(BannerofTruthTrust,Edinburgh,1979) v
vi Preface operations.
Thenusingsimplegrouptheoryprocedures,thephysicalpropertiesof
thecrystalcanbedetermined.
Duringthe45yearsIhavebeeninvolvedinteachingandresearchinvarious
areasofsolidstatephysics,Ihavemadeextensiveuseoftheconceptsofgroup
theory.
YetIhavebeensurprisedathowlittleemphasisthistopicreceivesinany
formaleducationalcurriculum.
Generally,astudentstudyingsolidstatephysicsor
chemistrywillbeexposedtocrystalstructuresearlyinthesemesterandthenhave
nofurtherexposuretocrystalsymmetryuntilsomespecialtopicsuchasnonlinear
opticsisdiscussed.
Thisbookfocusesonthesymmetryofcrystalsandthedescr-
tionofthissymmetrythroughtheuseofgrouptheory.
Althoughspeci?cexamples are provided of using this formalism to
determine both the microscopic and
macroscopicpropertiesofmaterials,theemphasisisonthecomprehensive,per-
sivenatureofsymmetryinallareasofsolidstatescience.
Theintentofthebookistobeareferencesourceforthosedoingresearchor
teachinginsolidstatescienceandengineering,oratextforaspecialtycoursein
grouptheoryappliedtothepropertiesofcrystals. Tucson,AZ RichardC.
Powell June2010 Contents 1 SymmetryinSolids...1 1. 1 Symmetry...1
1. 2 CrystalStructures...4 1. 3 SymmetryinReciprocalSpace...15 1. 4
Problems...24 References...24 2 GroupTheory...25 2. 1
BasicConceptsofGroupTheory...27 2. 2 CharacterTables...31 2. 3
GroupTheoryExamples...40 2. 3. 1 C PointGroup...40 3v 2. 3. 2 O
PointGroup...45 h 2. 4 GroupTheoryinQuantumMechanics...47 2. 5
Problems...52 References...53 3 TensorPropertiesofCrystals...55 3.
1 First-RankMatterTensors...5 7 3. 2 Second-RankMatterTensors...62
3. 3 Third-RankMatterTensors...68 3. 4
Fourth-RankMatterTensors...73 3. 5 Problems...77 References...77 4
SymmetryPropertiesofPointDefectsinSolids...79 4. 1
EnergyLevelsofFreeIons...79 4. 2 CrystalFieldSymmetry...85 4. 3
EnergyLevelsofIonsinCrystals...87 vii viii Contents 4. 4
Example:d?Electrons...95 4. 5 Example:f-Electrons...100 4. 6
Problems...104 References...104 5
SymmetryandtheOpticalPropertiesofCrystals ...
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