478 Introduction to Quantum PhysicsP40.38From the condition for Bragg reflection,mddλθφ==FHGIKJ222sincos.Butda=FHGIKJsin2where ais the lattice spacing.Thus, with m=1,λφφ=FHGIKJFHGIKJ=2aasincossinFIG. P40.38hphmKe2=×⋅×××=×−−−−6626 101054 0 1 60 10167 1034311910....Js2 9.11kgJmejej.Therefore, the lattice spacing is a×°=−−sin.sin.50 0218 1002181010mnm .P40.39(a)Epcmc224=+with Ehf=,ph=andmch=Csohfhc2λλC2andfcFHGIKJ2211C2(Eq. 1).(b)For a photonfc=1.The third term 1Cin Equation 1 for electrons and other massive particles shows thatthey will always have a different frequency from photons of the same wavelength .*P40.40For the massive particle, Kmc=−γ12bgand mhphmv. For the photon (which we represent as ),EK=and == = =−cfchEchKchmc12. Then the ratio is mch mvmc hvc=
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