250
5 Atomic Systems
given at low temperatures by
μ =
A
N
=
W
N
+
9hν D
8
−
π 4 k B T 4
5 3
D
.
(5.6.24)
More generally, μ can be written as
μ = − 0 +
9k B T 4
3
D
D /T
0
ln(1 − e
−x )x
2 dx,
(5.6.25)
with 0 representing the heat of sublimation per molecule at 0 K, defined by
0 ≡ −
W
N
−
9hν D
8
.
(5.6.26)
Such a procedure may be regarded as exact, provided that we use the experimental
value for 0 in expression (5.6.25) rather than the model expression given in
Eq. (5.6.26).
Now, to obtain an expression for the vapour pressure P of a crystal, we may use
the thermodynamic criterion for phase equilibrium to set μ crystal equal to μ gas and
then treat the vapour as an ideal gas. Hence, for sufficiently low temperatures, we
have
μ crystal = − 0 −
π 4 k B T 4
5 3
D
= −k B T ln
2πmk B T
h 2
3
2 k B T
p
= μ gas .
(5.6.27)
This equation gives for P the expression
ln P =
5
2 ln T −
0
k B T
−
π 4 T 3
5 3
D
+ ln
2πmk B
h 2
3
2
k B
.
(5.6.28)
Somewhat more generally, we would obtain the relation
ln P =
5
2 ln T −
0
k B T
+ 9
T
D
3 D /T
0
x
2 ln(1 − e
−x ) dx
+ ln
2πmk B
h 3
3
2
k B
,
(5.6.29)
for the vapour pressure above a monatomic Debye crystal.