Quantum Statistics
223
=
3
3
3
4
2 1
3
m
t
l
V
v
v
v


π
+




(7.59)
Since each mode is associated with an average energy given by Eq. (7.55),
the total thermal energy is
E =
2
3
3
/
0
2 1
4
1
m
v
hv kT
t
l
hv
V
v d v
v
v
e


π
+


−


∫
(7.60)
which in terms of Eq. (7.59) can be written as
E =
2
0
3
/
0
9
1
m
v
hv kT
m
N
hv
v dv
v
e
−
∫
(7.61)
Defining x = hv/kT and θ = h
v
m/k where θ is called the Debye temperature,
the energy is
E =
3 /
3
0
0
9
1
T
x
T
x
N kT
dx
e
θ
 
 
θ
−
 
∫
(7.62)
The molar specific heat C v is given by ∂E/∂T for N 0 = N Avo.
In the limit T → ∞, θ/T → 0, the integral is
1
3
(θ/T), so that
C v = 3R, for T
θ << 1
(7.63)
which is the classical limit [Eq. (7.53)]. For T → 0, T
θ → ∞, one can use
3
0
1
x
x dx
e
∞
−
∫
=
4
15
π
(7.64)
to get
E =
4
3
0
3
( / ) ,
5
N kT T
T
θ
π
θ
>> 1
(7.65)
and
C v =
4
3
12
( / ) ,
5
R T
T
θ
π
θ
<< 1
(7.66)
The model predicts that the specific heat at low temperatures is proportional
to T
3
, in agreement with the experimental observation.
The behaviour of C v at other temperatures has to be evaluated numerically
from the expression