Hw3Solution - Physics 361 P3.1 Anomalous density of states Problem set 3 due in class A certain two-dimensional simple-square lattice of lattice

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Problem set 3 due in class October 21, 2009 P3.1 Anomalous density of states A certain two-dimensional simple-square lattice of lattice 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 temperature 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 . Solution a) The density-of-states integral in two dimensions is 1 (volume of crystal) X k Z BZ d 2 k/ (2 π ) 2 For a linear dispersion ω = ck , the density of states is g ( ω ) = Z 2 π k dk (2 π ) 2 δ ( ω - c | k | ) Using the substitution x = ck this gives g ( ω ) = 1 2 πc 2 Z x dx δ ( ω - x ) = ω 2 πc 2 . There is one transverse and one longitudinal mode, so the total density of states (per unit volume) is g ( ω ) = ω 2 π ( 1 /c L 2 + 1 /c T 2 ) . b) g ( ω ) = Z d 2 k/ (2 π ) 2 δ ( ω - ω 0 - A ( ~ k - ~ k 0 ) 2 ) Defining ( ~ k - ~ k 0 ) as ~x , we can find the contribution from ~ k near ~ k 0 . g ( ω ) = Z 1 2 (2 π x dx ) / (2 π ) 2 δ ( ω - ω 0 - Ax 2 ) = π (2 π ) 2 1 A Z d ( Ax 2 ) δ ( ω - ω 0 - Ax 2 ) . If ω < ω 0 the integral is 1, so that g ( ω ) = (4 πA ) - 1 If ω > ω 0 , there are no states (in this region of ~ k ) so g ( ω ) = 0. Thus the density of states changes discontinuously at ω 0 from (4 πA ) - 1 below to zero above. The specific heat c v must thus have a negative
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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.

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Hw3Solution - Physics 361 P3.1 Anomalous density of states Problem set 3 due in class A certain two-dimensional simple-square lattice of lattice

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