Environmental Engineering Reference
In-Depth Information
The singlet-triplet model, relevant in the case of cubic symmetry
and, with some modifications, also for the cubic ions in Pr, introduces
one new feature; each component of the single-ion susceptibility includes
a mixture of an elastic and an inelastic response. In surroundings with
cubic symmetry,
χ
o
(
ω
) is proportional to the unit tensor, and the diag-
onal component is
2
n
01
M
1
∆
∆
2
χ
o
(
ω
)=
(
hω
)
2
+2
βn
1
M
2
δ
ω
0
,
(7
.
1
.
11)
−
where now
n
0
+3
n
1
= 1. This result follows from the circumstance
that
J
x
, for instance, has a matrix element between the singlet state
and one of the triplet states, and a matrix element between the two
other triplet states, the numerical values of which are denoted by
M
1
and
M
2
respectively. In the Γ
1
−
Γ
4
case with
J
= 4, corresponding to
Pr,
M
1
=
20
/
3and
M
2
=1
/
2. The inelastic
χ
(
q
,ω
=0)isequivalent
to (7.1.4) for the singlet-singlet system, but with
M
α
replaced by
M
1
.
Because of the elastic contribution, the critical condition
R
(
T
N
)=1is
now determined from
R
(
T
)=
2
n
01
M
1
+2
β
∆
n
1
M
2
J
αα
(
Q
)
∆
.
(7
.
1
.
12)
The inelastic neutron-scattering spectrum is also determined by eqn
(7.1.4) with
M
α
=
M
1
and
α
=
x
,
y
,or
z
, when the off-diagonal cou-
pling is neglected. The only difference is that there may now be three
different branches, depending on the polarization. In addition to the in-
elastic excitations, the spectrum also includes a diffusive, elastic mode.
In order to determine its contribution to the scattering function,
δ
ω
0
in
(7.1.11) may be replaced by
δ
2
/
δ
2
(
hω
)
2
, and if the limit
δ
−
→
0is
taken at the end, the result is found to be:
χ
o
(0)
χ
o
(
ω
−
→
0)
αα
d
S
(
q
,ω
≈
0) =
δ
(
hω
)
β
{
1
−
χ
o
(
ω
→
0)
J
αα
(
q
)
}{
1
−
χ
o
(0)
J
αα
(
q
)
}
∆
E
q
2
χ
αα
(
q
,
0)
χ
o
(0)
=2
n
1
M
2
δ
(
hω
)
.
(7
.
1
.
13)
The two-ion coupling is assumed to be diagonal, and
χ
o
(
ω
0) is
the static susceptibility without the elastic contribution. The scatter-
ing function at
q
=
Q
, integrated over small energies, diverges when
T
approaches
T
N
, as it also does in the singlet-singlet system. In the
latter case, and in the singlet-doublet system, the divergence is related
to the softening of the inelastic mode (
E
Q
→
→
T
N
), as
in eqn (7.1.5). In the singlet-triplet system, it is the intensity of the
0when
T
→
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