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 …