`Debye frequencyIn general3N =0G() d.So in the Debye model 3N = whence D = cs In summary,   9N 2  3  D G() =   0 for  &lt; D for  &gt; D33V 2 2 c3 sD 2 d =03 3V D 2 c3 3 2 s6 2 N . VDebye model energy and heat capacityThe total energy isE(T, V, N ) =0E()G() dwhere E() is the expected energy in a single harmonic oscillator mode of frequency . From the &quot;simple harmonic oscillator&quot; 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 0The first part of this integral is 9N ¯ h 3 2 DD 3 d =04 9N ¯ D h 9 = N ¯ D . h 3 2 D 4 8The second part isD 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 hwhere 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 x3x 0ett3 dt. -1For 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 TV,N3N 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 xdx dT3N kB D(x) + 3N kBt3 dt + -1 0 x t3 dt + t 0 e -1 3x 3D(x) - x e -1 etx3 x3 x3 ex - 1 3 x3 x3 ex - 1h ¯ D kB T 2 x - T -`

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