固体物理作业题Word下载.docx

固体物理作业题Word下载.docx

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Problem1.7:

SrTiO3crystallizesintheperovskitestructure.Thestrontiumatomsareatthecornersofthecubewithside

a,thetitaniumatomsareatthebodycenter,whiletheoxygenatomsoccupythecubefaces.

（a）WhatistheBravaislatticetype?

2

（b）VerifythattheprimitiveunitcellcontainsoneSr,oneTiandthreeOatoms.

（c）Writedownasetofprimitivelatticevectorsandbasisvectorsfortheperovskitestructure.

Problem1.8:

TheprimitivelatticevectorsofacertainBravaislatticecanbewritten

Rnnaxnbynzvrrr

12213

（2）1

=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?

（v）ExplainwithdiagramtheNaClstructureandtheCsClstructure?

Whatisthestructureof

diamond?

Problem2.6:

（i）Whatarecrystalplanes?

（ii）WhatdotheMillerIndicesrepresent?

（iii）Whatdothefollowingindicesrepresent?

（hkl）,{hkl},[hkl]and<

hkl>

?

（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,

（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

=+

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-

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