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 + dω ? Suppose that for some wave-vector ~ k0 , the ω L ( k ) has an absolute maximum. Near this maximum ω L ( k ) = ω0-A ( ~ k-~ k0 ) 2 . You can assume ω T ( k ) always lies well below ω0 , so that these modes don’t contribute to b). b) Find the form of g ( ω ) when ω < ∼ ω0 , and when ω > ∼ ω0 . P3.2 Poor-man’s localization
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This note was uploaded on 03/02/2010 for the course PHYSICS 336 taught by Professor Pucker during the Spring '10 term at King's College London.