188
7 Spherical Symmetry and Spins
Fig. 7.5 Possible isotropic Zeeman splittings of the octahedral Γ 8 spinor irrep
The coefficients in this equation are identified as
c 1 = 9J
4
p + 34J
2
p J
2
f − 24J p J
3
f + 18J
4
f
c 2 = J f (4J p + 2J f )
2 (3J p − 9/4J f )
(7.79)
The c 2 coefficient is of special interest since it controls the only octahedral term
in the linear Zeeman effect. If this coefficient vanishes, the splitting will be completely isotropic and does not depend on the orientation of the magnetic field in the
octahedron. There are three possible isotropies [13].
• For J f = 0, the spin operator is strictly dipolar, and the Zeeman Hamiltonian will
induce a regular splitting of the quartet level, which is proportional to the spin
quantum number, M S . This case is illustrated in the left-hand panel of Fig. 7.5.
Such cases will arise for an octahedral 4 A 1 state and also for the quartet spin-orbit
level of a 2 T 1 state.
• For 4J p + 2J f = 0, the matrix splits into two separate 2 × 2 blocks, which
have the same eigenvalues. The splitting pattern is thus as in the central panel
of Fig. 7.5. Such a case can occur for a 2 E state. The orbital part of this state has
no angular momentum, since the corresponding operator is not included in the
direct square: T 1 /
∈ E × E. As a result, the magnetic moment of such a state is
due only to the doublet spin part. Such a state behaves as a pseudo-doublet.
• Finally, for 3J p − 9/4J f = 0, the splitting again resembles the Zeeman splitting
of a regular spherical quartet, but the spin labels are interchanged, as compared
with the standard quartet splitting: the ±3/2 levels become the inner levels of
the split manifold, while the ±1/2 levels form the outer levels. This inverted
quartet behavior is observed for 2 A 2 and 2 T 2 states. The orbital part in these
states reverses the assignment of the fictitious spin levels, as can be seen, for
instance, from the A 2 × Γ 8 coupling table in Appendix G.
7 Spherical Symmetry and Spins
Fig. 7.5 Possible isotropic Zeeman splittings of the octahedral Γ 8 spinor irrep
The coefficients in this equation are identified as
c 1 = 9J
4
p + 34J
2
p J
2
f − 24J p J
3
f + 18J
4
f
c 2 = J f (4J p + 2J f )
2 (3J p − 9/4J f )
(7.79)
The c 2 coefficient is of special interest since it controls the only octahedral term
in the linear Zeeman effect. If this coefficient vanishes, the splitting will be completely isotropic and does not depend on the orientation of the magnetic field in the
octahedron. There are three possible isotropies [13].
• For J f = 0, the spin operator is strictly dipolar, and the Zeeman Hamiltonian will
induce a regular splitting of the quartet level, which is proportional to the spin
quantum number, M S . This case is illustrated in the left-hand panel of Fig. 7.5.
Such cases will arise for an octahedral 4 A 1 state and also for the quartet spin-orbit
level of a 2 T 1 state.
• For 4J p + 2J f = 0, the matrix splits into two separate 2 × 2 blocks, which
have the same eigenvalues. The splitting pattern is thus as in the central panel
of Fig. 7.5. Such a case can occur for a 2 E state. The orbital part of this state has
no angular momentum, since the corresponding operator is not included in the
direct square: T 1 /
∈ E × E. As a result, the magnetic moment of such a state is
due only to the doublet spin part. Such a state behaves as a pseudo-doublet.
• Finally, for 3J p − 9/4J f = 0, the splitting again resembles the Zeeman splitting
of a regular spherical quartet, but the spin labels are interchanged, as compared
with the standard quartet splitting: the ±3/2 levels become the inner levels of
the split manifold, while the ±1/2 levels form the outer levels. This inverted
quartet behavior is observed for 2 A 2 and 2 T 2 states. The orbital part in these
states reverses the assignment of the fictitious spin levels, as can be seen, for
instance, from the A 2 × Γ 8 coupling table in Appendix G.