Environmental Engineering Reference
In-Depth Information
The specific heat
C
may be derived in a simple way, within our current
spin-wave approximation, by noting that the excitation spectrum is the
same as that for a non-interacting Bose system, so that the entropy
is fully determined by the statistics of independent bosons of energies
E
q
(
T
):
S
=
k
B
q
(1 +
n
q
)ln(1+
n
q
)
n
q
ln
n
q
,
−
(5
.
3
.
2)
and hence
C
=
T∂S/∂T
=
k
B
T
q
(
dn
q
/dT
)ln
{
(1 +
n
q
)
/n
q
},
or, with
n
q
=
e
βE
q
(
T
)
1
−
1
,
−
C
=
q
E
q
(
T
)
dn
q
/dT
=
β
q
(5
.
3
.
3)
n
q
(1 +
n
q
)
E
q
(
T
)
E
q
(
T
)
/T
∂E
q
(
T
)
/∂T
,
−
as in (3.4.17).
The first derivative of
F
with respect to the angles
θ
and
φ
can be
obtained in two ways. The first is to introduce
S
, as given by (5.3.2)
into (5.3.1), so that
∂F
∂θ
=
∂U
E
q
(
T
)
∂n
q
∂θ
∂θ
−
q
m
q
,b
q
∂U
∂m
q
+
q
=
∂U
∂θ
∂m
q
∂θ
∂U
∂b
q
∂b
q
∂θ
−
E
q
(
T
)
∂n
q
∂θ
+
m
q
,b
q
=
∂U
∂θ
,
(5
.
3
.
4)
=
J A
q
(
T
)and
∂U/∂b
q
=
J B
q
(
T
),
as it can be shown that
∂U/∂m
q
when
U
=
, and hence that each term in the sum over
q
in the second line of (5.3.4) vanishes, when (5.2.32) is used. This result
is only valid to second order in 1
/J
. However, a result of general validity
is
H
0
+
H
1
+
H
2
∂F/∂θ
=
∂
/∂θ
,
H
(5
.
3
.
5)
asdiscussedinSection2.1,inconnectionwitheqn(2.1.5). Thetwodif-
ferent expressions for
∂F/∂θ
, and corresponding expressions for
∂F/∂φ
,
agree if
H
2
, i.e. to second
order in 1
/J
. However, the results obtained up to now are based on the
H
in (5.3.5) is approximated by
H
0
+
H
1
+
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