7.7 Application: Spin Hamiltonian for the Octahedral Quartet State
187
Table 7.8 Spin Hamiltonian matrix for the octahedral quartet irrep. A common factor of 1/
√
15
is absorbed into the J -parameters
B z
|Γ 8 κ| Γ 8 λ| Γ 8 μ| Γ 8 ν
Γ 8 κ|
(3J p − J f )
Γ 8 λ|
(J p + 3J f )
Γ 8 μ|− (J p + 3J f )
Γ 8 ν|− (3J p − J f )
B x − iB y
|Γ 8 κ| Γ 8 λ| Γ 8 μ| Γ 8 ν
Γ 8 κ|
√
3(J p +
1
2 J f )
Γ 8 λ|
(2J p −
3
2 J f )
Γ 8 μ|
√
3(J p +
1
2 J f )
Γ 8 ν|−
5
2 J f
B x + iB y
|Γ 8 κ| Γ 8 λ| Γ 8 μ| Γ 8 ν
Γ 8 κ|
−
5
2 J f
Γ 8 λ|
√
3(J p +
1
2 J f )
Γ 8 μ|
(2J p −
3
2 J f )
Γ 8 ν|
√
3(J p +
1
2 J f )
The parameters are identified as
a = 1
b =−10
B
2
x + B
2
y + B
2
z
J
2
p + J
2
f
c =
B
4
x + B
4
y + B
4
z
9J
4
p + 48J
3
p J f + 46J
2
p J
2
f − 48J p J
3
f + 9J
4
f
+
B
2
x B
2
y + B
2
x B
2
z + B
2
y B
2
z
×
18J
4
p − 144J
3
p J f + 32J
2
p J
2
f + 24J p J
3
f + 63J
4
f
(7.77)
The parameter b in this expression is isotropic, i.e., it does not depend on the orientation of the magnetic field in the octahedron. On the other hand, the parameter c
contains an anisotropic contribution. It depends on the orientation of the magnetic
field in the octahedron, but symmetry-equivalent orientations must, of course, yield
the same splitting. This means that c is certainly an octahedral invariant and may
thus be written as the sum of the familiar scalar L = 0 and hexadecapolar L = 4
cubic invariants that we derived in Sect. 7.2. We thus write
c = c 1
B
2
x + B
2
y + B
2
z
2
+ c 2
B
4
x + B
4
y + B
4
z − 3B
2
x B
2
y − 3B
2
x B
2
z − 3B
2
y B
2
z
(7.78)