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Prob.set.3 - Physics 361 Problem set 3 due in class P3.1...

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Physics 361 Problem set 3 due in class October 21, 2009 P3.1 Anomalous density of states A certain two-dimensional simple-square lattice of lat- tice constant a has the dispersion relation ( ω L ( k )) 2 = c 2 L k 2 , for longitudinal vibrations and ( ω L ( k )) 2 = c 2 T k 2 for transverse vibrations for ka 1. As explained in Chapter 23, the density of mode frequencies g ( ω ) determines the temper- ature dependence of the specific heat. a) What is g ( ω ) for an N - atom crystal at frequencies ω described by the dispersion relation above? That is, what is the number of modes between ω and ω + ? Suppose that for some wave-vector ~ k 0 , the ω L ( k ) has an absolute maximum. Near this maximum ω L ( k ) = ω 0 - A ( ~ k - ~ k 0 ) 2 . You can assume ω T ( k ) always lies well below ω 0 , so that these modes don’t contribute to b). b) Find the form of g ( ω ) when ω < ω 0 , and when ω > ω 0 . P3.2 Poor-man’s localization In the alternating-spring system of Figure 22.9, there is a gap in the density of states— i.e., a frequency range where there are no normal modes.
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