Set of supplementary notes.

3.7. LATTICE SPECIFIC HEAT 53
The internal energy is
∫
U =
dωD(ω)n(ω)h¯ω (3.40)
For the Einstein model
and the heat capacity
C V =
( ) ∂U
∂T
V
U E =
Nh¯ω o
e h¯ωo/k BT
− 1
(3.41)
( ) 2 h¯ωo e h¯ωo/k BT
= Nk b
k B T (e h¯ωo/k BT
− 1) . (3.42)
2
At low temperatures, this grows as exp−h¯ω o /k B T and is very small, but it saturates at a value
of Nk B (the Dulong and Petit law) above the characteristic temperature θ E = h¯ω o /k B . 6
At low temperature, the contribution of optical modes is small, and the Debye spectrum is
appropriate. This gives
∫ ωD
U D = dω V ω2 h¯ω
2π 2 v 3 e h¯ω/k BT
− 1 . (3.43)
0
Power counting shows that the internal energy then scales with temperature as T 4 and the
specific heat as T 3 at low temperatures. The explicit formula can be obtained as
C V = 9Nk B
( T
θ D
) 3 ∫ θD /T
0
x4 e x
dx
(e x − 1) , (3.44)
2
where the Debye temperature is θ D = h¯ω/k B . We have multiplied by 3 to account for the three
acoustic branches.
6 This is per branch of the spectrum, so gets multiplied by 3 in three dimensions