Environmental Engineering Reference
In-Depth Information
the population factors of the new eigenstates). The condition that
H
MF
should be diagonal in the new basis requires that
1
n
01
R
0
cos 2
θ
=
,
(7
.
1
.
8
b
)
which only has a solution if
n
01
R
0
≥
1, in accordance with the critical
condition
R
(
T
C
) = 1. The MF susceptibility is
2
n
01
M
α
∆cos2
θ
(∆
/
cos 2
θ
)
2
χ
αα
(
ω
)=
n
01
)
M
α
sin
2
2
θδ
ω
0
,
(7
.
1
.
9)
(
hω
)
2
+
β
(
n
0
+
n
1
−
−
revealing that there are now two kinds of excitation. The first is a
continuation of the paramagnetic inelastic branch, with the dispersion
relation
cos 2
θ
∆
(
q
)cos
2
2
θ
,
∆
E
q
=
2
n
01
M
α
J
cos 2
θ
−
(7
.
1
.
10)
which is again positive at all wave-vectors, consistent with the stabil-
ity of the ordered phase.
E
Q
therefore vanishes when
T
approaches
T
C
from above or below, and this kind of second-order phase transition is
frequently known as a
soft-mode transition
. In addition to the inelastic
mode, there appears a diffusive mode which, within the RPA, is purely
elastic. The diffusive mode, but not the inelastic branch, has a parallel
in the spectrum of the longitudinal fluctuations of a Heisenberg ferro-
magnet, described by eqn (3.5.27), since the spectrum analysed here is
longitudinal relative to the polarization of the spontaneously ordered
moment.
The behaviour discussed above is typical for a system where the
crystal-field ground state is a singlet. The most characteristic feature
of such a system is that the two-ion coupling must exceed a certain
threshold value, relative to the crystal-field splitting, in order to force
the system into a magnetically-ordered state at low temperatures. In
this case, the condition is that the ratio
R
0
must be greater than one.
The general (MF) condition is that
χ
αα
(0)
J
αα
(
Q
)
>
1, for at least one
of the
α
-components, where
χ
αα
(0) is the paramagnetic susceptibility
at zero temperature. This condition is a consequence of the fact that
the single-ion susceptibility remains finite in the zero-temperature limit,
ifthegroundstateisnon-degenerate. If the ground state is degenerate,
on the other hand, one or more components of the static single-ion sus-
ceptibility contains an elastic contribution proportional to 1
/k
B
T
,and
its divergence in the
T
= 0 limit results in an ordering of the moments,
within the MF approximation, no matter how weak the two-ion cou-
pling. Fluctuations not included in the MF theory modify the critical
Search WWH ::
Custom Search