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Debye frequency

In general

3N =

0

G() d.

So in the Debye model 3N = whence D = cs In summary, 9N 2 3 D G() = 0 for < D for > D

3

3V 2 2 c3 s

D

2 d =

0

3 3V D 2 c3 3 2 s

6 2 N . V

Debye model energy and heat capacity

The total energy is

E(T, V, N ) =

0

E()G() d

where E() is the expected energy in a single harmonic oscillator mode of frequency . From the "simple harmonic oscillator" problem, this is E() = h ¯ 1 1 + ¯ h -1 2 e .

So, in the Debye model, E(T, V, N ) = = 9N D 1 1 h ¯ + h 2 d 3 D 0 2 e¯ - 1 D 9N 3 3 d. h + ¯ 3 ¯ h -1 D 2 e 0

The first part of this integral is 9N ¯ h 3 2 D

D

3 d =

0

4 9N ¯ D h 9 = N ¯ D . h 3 2 D 4 8

The second part is

D 9N 3 h d 3 ¯ ¯ - 1 h D e 0 x t3 9N ¯ h = dt 3 t-1 ¯ (¯ D ) 0 e h h 9N kB T x t3 = dt t x3 0 e -1 = 3N kB T D(x),

[[. . . set t = ¯ and x = x(T ) = ¯ D . . . ]] h h

where we have used the definition D(x) In summary, E(T, V, N ) = 9 N ¯ D + 3N kB T D(x) h 8 where x = x(T ) = h ¯ D . kB T 3 x3

x 0

et

t3 dt. -1

For the heat capacity, CV (T, V, N ) = = = = = or, in summary, CV (T, V, N ) = 3N kB 4D(x) - 3x ex - 1 where x = x(T ) = h ¯ D . kB T E T

V,N

3N kB D(x) + 3N kB T D (x) 3N kB D(x) + 3N kB T - 9 x4 9 3N kB D(x) + 3N kB T - 4 x

dx dT

3N kB D(x) + 3N kB

t3 dt + -1 0 x t3 dt + t 0 e -1 3x 3D(x) - x e -1 et

x

3 x3 x3 ex - 1 3 x3 x3 ex - 1

h ¯ D kB T 2 x - T -

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