固体物理作业题.docx
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固体物理作业题
Chapter1
Problem1.1:
Computethepackingfractionfforthebcclattice.
Problem1.2:
(a)Showthatthepackingfractionfforthediamondlatticeisπ3/16.
(b)Whatisthepackingfractionandcoorinationnumberofthehoneycomblattice?
Problem1.3:
Considerthehexagonalclosepackedlattice.(a)Showthatc=a83=1.633a.Frequentlyacrystal
structureiscalledhcpevencisnotexactlyequaltotheidealvalue.(b)Showthatthepackingfractionfor
theidealhcplatticeisπ2/6=0.7405
Problem1.4:
TheioniccompoundA+B-crystallizesintheNaClstructure.Plotthepackingfractionasafunctionofthe
ratio+−ζ=r/r.Assumethatζ<1.
Problem1.5:
Repeatthecalculationofproblem1.4fortheCsClstructure.
Problem1.6:
Usetheinformationinthetextbooktocalculatethedensities(inkgm-3)ofthefollowingsolids:
(a)
Aluminum,(b)Iron,(c)Siliconand(d)Zinc.Atomicweightsofsomecommonelementsarelistedinthe
textbook.
Problem1.7:
SrTiO3crystallizesintheperovskitestructure.Thestrontiumatomsareatthecornersofthecubewithside
a,thetitaniumatomsareatthebodycenter,whiletheoxygenatomsoccupythecubefaces.
(a)WhatistheBravaislatticetype?
2
(b)VerifythattheprimitiveunitcellcontainsoneSr,oneTiandthreeOatoms.
(c)Writedownasetofprimitivelatticevectorsandbasisvectorsfortheperovskitestructure.
Problem1.8:
TheprimitivelatticevectorsofacertainBravaislatticecanbewritten
Rnnaxnbynzvrrr
12213
(2)1
2
=1+++
Whatisthelatticetype?
Problem1.9:
IneachofthefollowingcasesindicatewhetherthestructureisaBravaislattice.Ifitis,givethreeprimitive
latticevectors.IfitisnotdescribeitasaBravaislatticewithassmallaspossiblebasis.Inallcasesthe
lengthofthesideoftheunitcubeisa.
(a)Basecenteredcubic(simplecubicwithadditionalpointsinthecentersofthehorizontalfacesofthe
cubiccell).
(b)Sidecenteredcubic(simplecubicwithadditionalpointsinthecentersoftheverticalfacesofthecubic
cell).
(c)Edgecenteredcubic(simplecubicwithadditionalpointsatthemidpointsofthelinesjoiningnearest
neighbors).
Problem1.10:
指出体心立方晶格(111)面与(100)面,(111)面与(110)的交线的晶向。
Chapter2
Problem2.1:
(i)Whatisalattice?
Expressalatticemathematically.
(ii)Whatdoyoumeanbya“basis”?
(iii)Howcanyoucombinealatticewithabasistoobtainacrystalstructure?
Problem2.2:
(i)Whatarethe3fundamentaltranslationvectors?
(ii)Showwith2dimensionalexamples,howthefundamentaltranslationvectorsmaydefine
eitheranon-primitive(conventional)unitcelloraprimitiveunitcell.
(iii)Expressmathematicallythesize(areain2dimensional&volumein3dim)ofaunitcell.
Problem2.3:
(i)WhatisaBravaislattice?
(ii)Drawthefive2-dimensionalBravaislatticesclearlyshowingthefundamentallattice
translationvectors.Whatisthedifferencebetweenacenteredrectangularlatticeanda
simplerectangularlattice?
Problem2.4:
(i)Howmany3dimensionalBravaislatticesarepresent?
(ii)Howmany3dimensionalcrystalsystemsarethere?
(iii)MakeaTablehavingthefollowingcolumns:
LatticeSystem--Bravaislattice--Diagram
ofConventionalunit--Name&Symbolunitcell--Cellcharacteristics.
Problem2.5:
(i)Whatdoyoumeanbypackingfraction?
(ii)Calculatethepackingfractioninasimplecubic,basecenteredcubicandafacecentered
cubicstructures.
(iii)Inwhichstructurearetheatomsmostcloselypacked?
(iv)Whatdoyoumeanby“coordinationnumber”inacrystalstructure?
2
(v)ExplainwithdiagramtheNaClstructureandtheCsClstructure?
Whatisthestructureof
diamond?
Problem2.6:
(i)Whatarecrystalplanes?
(ii)WhatdotheMillerIndicesrepresent?
(iii)Whatdothefollowingindicesrepresent?
(hkl),{hkl},[hkl]and?
(iv)Drawtheunitcell&thefollowingplanesinasimplecubiclattice:
(100),(ī00),(200),
(1ī1),(201),(2ī0),(122).
(v)Whatdoyoumeanbycrystallatticeinterplanarspacing(dhkl)?
(vi)Writetheformulaefordhklforaorthogonallatticeandacubiclattice.Alsowritethe
formulafortheanglebetween2planesinacubiclattice.
Problem2.7:
(i)Whatisthedensityofatoms(numberperunitarea)ona(111)planeofafcclattice?
(ii)Whatisthedensityofatoms(numberperunitarea)ona(110)planeofabcclattice?
Problem2.8:
(i)ConstructtheWigner-Seitzprimitivecellsforone-dimensionallattice.
(ii)ConstructtheWigner-Seitzprimitivecellsforsquarelatticeandforhoneycomblattice
(i.e.,hexagonallattice)(2D).
(iii)ConstructtheWigner-Seitzprimitivecellsforsimplecubic,body-centeredcubicand
face-centeredcubiclattices(3D),respectively.
Problem2.9:
(i)Constructthereciprocallatticefortwo-dimensionalrectangularlattice,squarelattice,
obliquelatticeandhexagonallattice.
(ii)Constructthereciprocallatticesforsimplecubic,body-centeredcubicandface-centered
cubiclattices.
3
Problem2.10:
(i)ConstructthefirstBrillouinzonefortwo-dimensionalrectangularlattice,squarelattice,
obliquelatticeandhexagonallattice.
(ii)ConstructthefirstBrillouinzoneforsimplecubic,body-centeredcubicand
face-centeredcubiclattices.
Problem2.11:
考虑晶格中的一个晶面hkl。
(i)证明倒格矢垂123Ghbkblb
rvvv
=++直于这个晶面。
(ii)证明晶格中两个相邻平行晶面的间距为
G
hkldr
2π()=
(iii)证明对于简单立方晶格有d=a/h2+k2+l2
Problem2.12:
证明第一布里渊区的体积为(2π)3/Vc。
其中Vc是晶格原胞的体积。
提示:
布里渊区的体积等于傅里
叶空间中的初基平行六面体的体积,同时利用矢量恒等式
(c×a)×(a×b)=(c•a×b)a
Problem2.13:
(i)ExplaininshorthowX-rayscanbediffractedbyacrystal.Aneutronbeamcanalsobe
usedinsteadofX-raystostudydiffraction.Why?
(StatedeBroglieshypothesisofmatter
wavesi.e.waveandparticleduality:
λ=h/p).
(ii)DrawaneatdiagramanddeduceBragg’sLawfordiffractionbyacrystal(2dsinθ=nλ).
Visiblelightcannotbeusedtostudydiffractionbycrystals,why?
(iii)TheBraggangleforreflectionfromthe(111)planesinAl(fcc)is19.2degreesforan
X-raywavelengthofλ=1.54Ǻ.Compute:
(i)thelengthofthecubeedgeoftheunitcell;
(ii)theinterplanardistancefortheseplanes.(Ans4.04angstromand2.33angstrom).
Problem2.14:
4
Anx-raysourceemitsanx-raylineofwavelengthl=0.154nm.Thelatticeconstantand
crystalstructuresofironandaluminumarefoundinthetableslistedinthetextbook.
(i)FindtheBraggangle(s)forreflectionsfromthe(111)planesofAl.
(ii)FindtheBraggangle(s)forreflectionsfromthe(110)planesofFe.
Problem2.15:
Theprimitivelatticevectorsofa2-dimensionaltriangularlatticeare
aai
rr=;baiaj
rrr
2
3
2
=+
whereaisthenearestneighbordistance.
(i)Findthereciprocallattice
(ii)DrawtheWignerSeitzcellandlocatethecoordinatesofitscorners.
(iii)DrawtheBrillouinzoneandlocatethecoordinatesofitscorners.
Problem2.16:
AnX-rayreflectionfromacertaincrystaloccursatanangleofincidenceof45owhenthe
crystalismaintainedat0oC.Whenitisheatedto150oCtheanglechangesby6.4minutesof
arc.Whatisthelinearthermalexpansioncoefficientofthematerial?
Chapter3
Problem3.1
OntheoriginofVanderWaalsforce
(a)GiveaqualitativeinterpretationontheoriginoftheVanderWaalsforce.
(b)GiveaquantitativeinterpretationontheoriginoftheVanderWaals.
Problem3.2
Anapproximatewayofcombiningtherepulsiveandattractiveinteractionsbetweentheatomsina
molecularcrystalistheLennard-Jonespotential
==
126
126()-4-
rrr
B
r
VrAσσ
εwhereAandBareconstantswhichdependonwhichatomormoleculeisinvolved.Itisconventionalto
prameterizethepotentialintermsofanenergyparameterεandlengthparameterσ.Table3.1liststhe
Lennard-Jonesparametersforinertgases.
Table3.1liststheLennard-Jonesparametersforinertgases
Elementσ(angstrom)ε(eV)
Ne2.740.0031
Ar3.400.0104
Kr3.650.0140
Xe3.980.0200
(a)PlottheLennard-JonespotentialandforcefortheinertgasesNe,Ar,KeandXe,respectively.
(b)DerivetheequilibriumdistancerofortheinertgasesinTable3.1.
(c)DerivethepotentialintheneighborhoodoftheequilibriumdistancerofortheinertgasesinTable3.1
(d)ComparewiththeresultfoundinTable4(C.Kittel,p.41).
Problem3.3
ThelatticeparametersofKClaregivenintable5.1
(a)CalculatetheCoulombenergybetweenaK+andaCl-ionatthenearestneighbourdistanceinunitsof
eV.
(b)AssumethattheparameterssandeofthevanderWaalsattractionbetweentheions(theterm
proportionalto1/r6intheLennard-Jonespotential)arethesameasforAr(table3.1).Calculatethe
vanderWaalsenergybetweenaK+andaCl-ionatthenearestneighbordistanceofKCl.Compare
withtheresultfoundunder(a).
Problem3.4
CalculatetheMadelungconstantforthecrystalstructureofNaClandcompareyourresultswiththose
listedintable3.2.
CrystalstructureofNaCl
Table3.2Madelungconstantforsomecrystalstructureslistedintable
Structureα
NaCl1.7476
CsCl1.7627
ZnS1.6381
Chapter4
Problem4.1
Consideralinearchainofatoms.Eachatominteractwithitsnearestneighboroneithersideviaa
Lennard-Jonespotential.Assumeparametervaluesapproximatetokrypton(table3.1)
(a)Findtheequilibriumspacingbetweentheatoms.
(b)Findthesoundvelocity.
(c)Whatisthemaximumfrequency?
Problem4.2
Theharmonicchainmodelcanbesolvedalsowhentheinteractionbetweenthemassesextendsbeyond
thenearestneighbors.Considerthecasewhenthen’thmassisconnectedtomassesn+1andn-1withthe
spacingconstantK1andtomassesn+2andn-2withthespacingco