hw3Solution

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online