固体物理习题解答Chapter-5-problems课件.ppt

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1、1.Singularity in density of states(a)The density of modes for a monatomic linear latticeThe density of modes is Considering the nearest neighbor interactions,we have the dispersion relation for a monatomic linear lattice:(b)The density of modes for an optical branchSuppose tat an optical branch has

2、the dispersion relation near K=0 in three dimension:The density of modes:For 0,the number of modes:For 0,the number of modes N=0,therefore D()=0.2.Rms thermal dilation of crystal cell(a)The root mean square thermal dilation V/VFor sodium,B 71010 erg/cm3,a 4108 cm.At room temperature T=300 K.The pote

3、ntial energy of a primitive cell associated with the dilation is(b)The root mean square thermal dilation a/aIn the Debye approximation,Then(b)One dimension latticeIn one dimension,D()=L/v.Then Whence 1 and diverge for a one-dimension lattice.The mean square strainIn one dimension,D()=L/v,and for the

4、 Debye approximation =vK.Then i.e.the mean square strain is finite.The thermal energywhereAt low temperature,xD Then The total number of modes with the wavevector less than K isIn the Debye approximation,=Kv.ThenThe density of modesFor 1d vibrations,The thermal energy5.Gruneisen constant(a)The parti

5、tion function of a phonon mode of frequency is:The free energy of a phonon mode of frequency is:(b)Consider the volume dilation.The free energy of the crystal may be written as where is the fractional volume change,and B is the bulk modulus.Assume the volume dependence of K is/=,where is a constant which is supposed independent of the mode K.Then The thermal energy:Therefore or(c)Because we suppose that is independent of K,we haveIn the Debye mode,the Debye temperature is defined asHence Therefore