Omar_Elementary Solid State Physics_Lecture2

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1、Chapter2Howdowedeterminecrystalstructure?Bystudyingthediffractionpatternofabeamofradiationincidentoncrystal.Typesofradiationused:X-rays,neutrons,electrons.Treatmentofthesethreetypesisverysimilar:Needtoexamineonlyoneindetail(X-rays)through:•simplederivationofBragg’slaw−→mirroratomicplanes;•sophistica

2、tedderivationofBragg’slaw−→basedonscatteringtheory.2.2GenerationandAbsorptionofX-raysX-raysareEMwaveswithwavelengthsλ∼1˚A=0.1nm=⇒λofsameorderofmagnitudeaslatticeconstants.hc−34E=hν=;h=6.6×10Js;ν=c/λ=⇒λ6.6×10−34×3×1082×10−25J104eVE=≈≈λλ(m)λ(˚A)since:−19−101eV=1.6×10J;1˚A=10m.Electronsemittedbycathod

3、eareacceleratedbyalargepotentialactingacrossthetable.Whenelectronsimpingeontarget(seeFig.2.1):•continuousspectrumisduetodeclerationbynuclearcharges;•discretespectrumduetoexcitations/reemmisionsofmolecules.Maxfrequencyofcontinuousspectrumisrelatedtoacceleratingpotential:eV=hν0⇓12.3λ0=˚A;Visinkilovol

4、tsVUponpassagethroughmedium,beamisattenuated:−αxI=I0eα=extinctionorattenuationcoefficient;Attenuationisduetoabsorptionandscatteringofthebeambyatomsinthemedium.Bragg’sLawBeamisreflectedwhenincidentonsurfaceofacrystal.ReflectionoccursonlyforcertainanglesofincidenceasillustratedinFig.2.2.Model:Crystalisre

5、presentedbyasetofparallelplanes,correspondingtotheatomicplanes.Theyactasmirrorsoftheincidentrays.Reflectedraysinterfereatthedetector.Constructiveinterferenceif:∆=PathDifference=2dsinθ=nλ,n=1,2,3..2dsinθ=nλBragg’sLawn=1isthefirstorderreflection;n=2isthesecondorderreflection.Intensitydecreaseswiththeorder

6、n.Willconsideronlyn=1here.2Moreappropriatetothinkofreflectionasadiffractionbecauseinterferenceisessential.Since:λcanbedeterminedindependently,andθcanbemeasured:Wecanuseexpression2dsinθ=nλtocalculateinterplanardistanced.•Sincesinθ<1=⇒λ<2d=⇒visiblelightcannotbeused.•Ifcrystalrotated,anewdiffractedbeamma

7、yappearcorrespondingtoanewsetofplanes.Modelusedaboveisoversimplifiedbecausescatteringiscausedbydiscreteelec-tronsratherthanbyasetofmirrorplanes.Willnowconsider:•amoresophisticatedmodel,inwhichthediffractedbea