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useful for systems with larger
J
, if the higher-lying levels are not coupled
to the ground state, and are so sparsely populated that their influence is
negligible. According to
Kramers' theorem
, the states are at least dou-
bly degenerate in the absence of an external magnetic field, if 2
J
is odd.
In order to construct simple models with relevant level-schemes, we may
consider a
singlet-singlet
or a
singlet-triplet
configuration, instead of
systems with
J
=
2
or
J
=
2
. These models may show some unphysical
features, but these do not normally obscure the essential behaviour.
The simplest level scheme is that of the singlet-singlet model. This
may be realized conceptually by lifting the degeneracy of the two states
with
J
=
2
with a magnetic field, and then allowing only one of the
components of
J
perpendicular to the field to interact with the neigh-
bouring ions. This is the so-called
Ising model in a transverse field
.
Assuming the coupled components to be along the
α
-axis, we need only
calculate the
αα
-component of
χ
o
(
ω
). The lower of the two levels, at
the energy
E
0
, is denoted by
1
|
0
>
, and the other at
E
1
by
|
1
>
.The
single-ion population factors are
n
0
and
n
1
respectively, and the use of
eqn (3.5.20) then yields
2
n
01
M
α
∆
∆
2
χ
αα
(
ω
)=
(
hω
)
2
,
(7
.
1
.
3)
−
where
M
α
=
is the numerical value of the matrix element of
J
α
between the two states, while the two other (elastic) matrix elements
are assumed to be zero. ∆ =
E
1
−
|
<
0
|
J
α
|
1
>
|
E
0
is the energy difference, and
n
01
n
1
is the difference in population between the two states.
From eqn (7.1.2), we have immediately, since only
=
n
0
−
J
αα
(
q
) is non-zero,
2
n
01
M
α
∆
E
q
−
χ
αα
(
q
,ω
)=
(
hω
)
2
,
(7
.
1
.
4
a
)
where the
dispersion relation
is
E
q
=
∆
∆
2
n
01
M
α
J
αα
(
q
)
1
/
2
.
(7
.
1
.
4
b
)
These excitations are actually spin waves in this case of extreme axial
anisotropy, but they have all the characteristics of crystal-field excita-
tions. The energies are centred around ∆, the energy-splitting between
the two levels, and the bandwidth of the excitation energies, due to
the two-ion interaction, is proportional to the square of the matrix el-
ement, and to the population difference, between them. These factors
also determine the neutron-scattering intensities which, from (3.2.18)
and (4.2.3), are proportional to
−
n
01
M
α
∆
E
q
δ
hω
E
q
−
δ
hω
+
E
q
1
αα
d
S
(
q
,ω
)=
−
1
−
e
−βhω
(7
.
1
.
5)
n
0
δ
hω
E
q
+
n
1
δ
hω
+
E
q
.
∆
E
q
M
α
−
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