固体物理习题解答：Chapter 4 problems

1.Monatomiclinearlattice(a)ThetotalenergyofthewaveisthesumovertheenergyofallatomsForaharmonicoscillator(b)Thetime-averagetotalenergyperatomSubstituteus=u

cos(tsKa)intotheexpressionThetotalenergyaverageovertimewhereNisthenumberoftheatoms.Thetime-averagetotalenergyperatomFromthedispersionrelationwehaveThus2.ContinuumwaveequationWehavetheequationofthemotionInthelongwavelengthlimit,>>a,thedifferenceofthedisplacementsofnearestatomsisverysmall.Henceus(t)couldbetreatedasacontinuousfunctionu(x,t).ThenwehaveThesolutionofthisequationiswiththedispersionrelationInthelongwavelengthlimit,Ka<<1,whereThereforetheequationofmotionreducestothecontinuumwaveequation3.BasisoftwounlikeatomsFromtheequationofmotionsWehavethesolutions(1)(2)SubstituteEq(2)intoEq(1)(3)(4)AttheBrillouinzoneboundaryK=Kmax=/a,wehavei.e.ori.e.thesetwolatticesactasifdecoupled:onelatticeremainsatrestwhiletheotherlatticemoves4.KohnanomalyConsideringtheinteractionsbetweenpnearestplanes,wehavethedispersionrelationSupposingwehavewithAandk0areconstantsandprunsoverallintegers,ThenWhenK=k0i.e. isinfinitewhenK=k0.Thusaplotof2

vs

K(or

vs

K)haveaverticaltangentatK=k0:thereisakinkatk0inthephonondispersionrelation(K).5.DiatomicchainWehavetheequationofmotionsThesolutionsareSubstitutethesolutionsintoequationofmotionsThehomogenouslinearequationshaveasolutiononlyifthedeterminantofthecoefficientsoftheunknownu,vvanishes.thedispersionrelationisorDiscussions:(1)K=0(2)K=/aObviously,theacousticbranchindicatestheinteractionsbetweenmoleculeswhiletheopticalbranchshowstheinteractionsinsidethemolecules.6.AtomicvibrationsinmetalrRSupposetherestoringforceisduetotheelectricchargewithinthesphereoftheradiusrcenteredtotheequilibriumposition.Asshowninthefigure,theelectricfieldinaspherewiththechargehomogenousdistributedisTheequationofmotioni.e.(a)ThefrequencyofasingleionoscillationForaharmonicoscillationwhere(b)EstimatethevalueofthisfrequencyforsodiumSodiumhasabccstructurewithlatticeconstanta=4.225Å(p.23).ThenR=31/2a/4=1.83Å=1.83E8cmM

23Mp3.84E23g4.8E10/[3.43E23(1.83E8)3]1/2

3.3E13s1(c)EstimatethevelocityofthesoundinmetalSupposethedispersionrelationis=vgKformetal,wherevgisconstant.Inestimation,wetakeK=/a=/4.225E81E8cm-1vg=/K3E13/1E8=3E5cm/s7.Softphononmodes(a)ForceconstantoftheCoulombinteractionTheCoulombbetweenatomsandatoms+pisThenwhere(b)Thedispersionrelation(c)Discussions(1)AtthefirstBrillouinzoneboundary,Ka=2isnegativewhen