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. 57 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 .