Environmental Engineering Reference
In-Depth Information
1
a
>
=
|
1
>
/
√
2
i
, the only non-zero matrix elements of
and
|
+1
>
−|−
1
a
>
=
J
(
J
+1)
/
2,
plus their Hermitian conjugates. In Pr, the matrix element of
J
ζ
is a
J
are
<
1
a
|
J
ζ
|
1
s
>
=
i
and
<
0
|
J
ξ
|
1
s
>
=
<
0
|
J
η
|
factor of
√
10 smaller than the other matrix elements. This means that
the transformation of the (
J
= 4) ion of Pr to an effective
J
=1system
introduces a scaling of the two-ion couplings
J
ξξ
(
q
)and
J
ηη
(
q
)bya
factor of 10, compared to
J
ζζ
(
q
), and the latter may therefore be ne-
glected to a first approximation. Hence the (
J
=1)
XY
-model is an
appropriate low-temperature description of the hexagonal ions in Pr.
The RPA theory of the
XY
-model, in the singlet-doublet case, is
nearly identical to that developed above for the Ising model in a trans-
verse field. One difference is that
n
0
+
n
1
+
n
2
=
n
0
+2
n
1
=1,instead
of
n
0
+
n
1
= 1, but since this condition has not been used explicitly (the
population of any additional higher-lying levels is neglected), it may be
considered as accounted for. The other modification of the above results
is that there are now two components of
χ
(
q
,ω
) which are important:
χ
xx
(
ω
)=
χ
yy
(
ω
) are given by the same expression as
χ
αα
(
ω
)ineqn
(7.1.3) (with
M
α
=1when
J
= 1), whereas
χ
xy
(
ω
)
0 (the (
xyz
)-axes
are assumed to coincide with the (
ξηζ
)-axes). This means that, for a
Bravais lattice, there are two poles at positive energies in the RPA sus-
ceptibility (7.1.2) at each
q
-vector. As long as
≡
J
xy
(
q
) = 0, one of the
modes describes a time variation of
J
x
alone, and the other
J
y
alone,
and their dispersion relations are both given by eqn (7
.
1
.
4
b
), with
α
set
equal to
x
or
y
. It is interesting to compare this result with the spin-
wave case. Although the magnetic response is there also determined by
a2
2 matrix equation, it only leads to one (spin-wave) pole at posi-
tive energies, independently of whether the two-ion coupling is isotropic.
The cancellation of one of the poles is due to the specific properties of
χ
xy
(
ω
) in (5.1.3), produced by the molecular field (or the broken time-
reversal symmetry) in the ordered phase. In the case considered above,
the two modes may of course be degenerate, but only if
×
J
xx
(
q
)isequal
to
J
yy
(
q
). In an hcp system, such a degeneracy is bound to occur, by
symmetry, if
q
is parallel to the
c
-axis. If the degeneracy is lifted by
anisotropic two-ion couplings, which is possible in any other direction in
q
-space, the
x
-and
y
-modes mix unless
q
is parallel to a
b
-axis. The va-
lidity of the results derived above is not restricted to the situation where
the doublet lies above the singlet. If the
XY
-model is taken literally,
all the results apply equally well if ∆, and hence also
n
01
, is negative.
However, if the
z
-components are coupled to some extent, as in Pr, the
importance of this interaction is much reduced at low temperature if ∆
is positive. In this case the
zz
-response, which is purely elastic,
χ
zz
(
ω
)=2
βn
1
δ
ω
0
χ
zz
(
q
,ω
)
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