Environmental Engineering Reference
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where the
kinematic correction
, of the order Φ
2
S
+1
, due to the limited
number of single-spin states, which is neglected in this expression, is
unimportant when
S
≥
1. Utilizing the Hartree-Fock decoupling once
S
i
S
j
(
i
=
j
)
S
z
2
S
2
S
z
more to write
−
2
Φ, we find the internal
energy to be
(
0
)
S
2
+
q
−
1
2
U
=
H
=
N
J
E
q
n
q
(
0
)
S
(
S
+1)+
q
(3
.
4
.
15)
−
2
E
q
(
n
q
+
2
=
N
J
)
.
The second form, expressing the effect of the zero-point motion, is de-
rived using
(
ii
)=
N
q
J
0.
The thermodynamic properties of the Heisenberg ferromagnet are
determined by (3.4.10), (3.4.14), and (3.4.15), which are all valid at low
temperatures. In a cubic crystal, the energy dispersion
E
q
is isotropic
and proportional to
q
2
in the long wavelength limit, and (3.4.14) then
predicts that the magnetization
J
(
q
)
≡
S
z
decreases from its saturation value
as
T
3
/
2
. The specific heat is also found to be proportional to
T
3
/
2
.The
thermodynamic quantities have a very different temperature dependence
from the exponential behaviour (3
.
4
.
5
b
) found in the MF approxima-
tion. This is due to the presence of elementary excitations, which are
easily excited thermally in the long wavelength limit, since
E
q
→
0
when
q
→
0
in the RPA. These normal modes, which are described
as
spin waves
, behave in most aspects (disregarding the kinematic ef-
fects) as non-conserved Bose-particles, and they are therefore also called
magnons
.
We shall not present a detailed discussion of the low-temperature
properties of the Heisenberg ferromagnet. Further details may be found
in, for instance, Marshall and Lovesey (1971), and a quite complete
treatment is given by Tahir-Kheli (1976). The RPA model is correct at
T
=0where
S
z
=
S
, but as soon as the temperature is increased, the
magnons start to interact with each other, giving rise to finite lifetimes,
and the temperature dependence of the excitation energies is modified
(or
renormalized
). The temperature dependence of
E
q
=
E
q
(
T
)isre-
sponsible for the leading order 'dynamic' corrections to
S
z
and to the
heat capacity. A more accurate calculation, which we will present in
Section 5.2, adds an extra term to the dispersion:
N
1
S
z
E
q
=
{J
−J
}
k
{J
−J
}
n
k
,
(3
.
4
.
16)
(
0
)
(
q
)
+
(
k
)
(
k
+
q
)
from which the heat capacity of this non-interacting Bose-gas can be
determined as
C
=
∂U/∂T
=
q
E
q
dn
q
/dT.
(3
.
4
.
17)
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