Environmental Engineering Reference
In-Depth Information
Heisenberg exchange. The mean temperature (
θ
+2
θ
⊥
)
/
3 is determined
by
(
0
) which, from (2.1.13), is the algebraic sum of the isotropic two-
ion interactions, and this temperature may be measured directly with a
polycrystalline sample. The two basal-plane components are found to
be equal. This is not just due to the assumption of high temperatures,
but is generally valid as long as there is no ordered moment in the basal-
plane. In this case, the
c
-axis is a three-fold symmetry axis, or effectively
a six-fold axis, due to the symmetry of the basal-plane anisotropy
B
6
in
the Hamiltonian. The susceptibility is a second-rank tensor, according
to (2.1.9), and it cannot therefore vary under rotation about a three- or
six-fold axis.
J
2.1.2 The mean-field approximation
The high-temperature expansion may be extended to higher order in
β
,
but the calculations rapidly become more complex, so we shall instead
adopt another approach, the mean-field approximation. In this method,
the correlated fluctuations of the moments around their equilibrium val-
ues are neglected. In order to introduce
J
i
into the Hamiltonian, we
utilize the identity
J
i
·
J
j
=(
J
i
−
J
i
)
·
(
J
j
−
J
j
)+
J
i
·
J
j
+
J
j
·
J
i
−
J
i
·
J
j
.
The MF approximation then consists in neglecting the first term on
the right-hand side, which is associated with two-site fluctuations, since
i
=
j
. The Hamiltonian (2.1.1) is then effectively decoupled into a sum
of
N
independent terms for the single sites;
H
i
H
MF
(
i
), where
−
J
i
·
h
i
−
J
i
−
2
J
i
·
H
MF
(
i
)=
H
cf
(
i
)
J
(
ij
)
J
j
,
(2
.
1
.
16)
j
in the presence of an external magnetic field
h
i
=
gµ
B
H
i
. Introducing
the effective field
=
h
i
+
j
h
eff
i
J
(
ij
)
J
j
,
(2
.
1
.
17
a
)
we may write the MF Hamiltonian
+
2
J
i
·
−
J
i
·
h
eff
(
h
eff
i
H
MF
(
i
)=
H
cf
(
i
)
−
h
i
)
.
(2
.
1
.
17
b
)
i
Self-consistent solutions of the MF equations may sometimes be obtained
analytically, but numerical methods may be used more generally, pro-
vided that the periodicity of the magnetic structure is commensurable
with that of the lattice. For an assumed distribution of
, the effec-
tive field and hence the MF Hamiltonian for the
i
th site is calculated.
J
j
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