Physics 361Problem set 3due in class October 21, 2009P3.1 Anomalous density of statesA certain two-dimensional simple-square lattice of lat-tice constantahas the dispersion relation (ωL(k))2=c2Lk2, for longitudinal vibrations and (ωL(k))2=c2Tk2for transverse vibrations forka1.As explained in Chapter 23, the density of mode frequenciesg(ω) determines the temper-ature dependence of the specific heat.a) What isg(ω) for anN-atom crystal at frequenciesωdescribed by the dispersionrelation above? That is, what is the number of modes betweenωandω+dω?Suppose that for some wave-vector~k0, theωL(k) has an absolute maximum.Near thismaximumωL(k) =ω0-A(~k-~k0)2. You can assumeωT(k) always lies well belowω0, sothat these modes don’t contribute to b).b) Find the form ofg(ω) whenω<∼ω0, and whenω>∼ω0.P3.2 Poor-man’s localizationIn the alternating-spring system of Figure 22.9, there is agap in the density of states—i.e.,a frequency range wherethere are no normal modes.
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