184
7 Spherical Symmetry and Spins
7.7 Application: Spin Hamiltonian for the Octahedral Quartet
State
The octahedral double group contains a four-dimensional spin representation, which
is commonly denoted as the Γ 8 quartet, or U ′ in Griffith’s notation. The direct square
of this irrep is given by
Γ 8 × Γ 8 =[A 2 + 2T 1 + T 2 ]+{A 1 + E + T 2 }
(7.67)
According to the time-reversal selection rules, time-odd interactions with a magnetic
field will be based on the symmetrized square. The spin-operator of the Zeeman
Hamiltonian transforms as T 1g , which is indeed included in the symmetrized square.
The present case is, however, special since the T 1g irrep occurs twice in the product.
The multiplicity separation cannot be achieved on the basis of symmetrization since
both T 1g irreps appear in the symmetrized part. One way to distinguish the two
products is through the subduction process from spherical symmetry. The addition
rules of angular momenta give rise to
3/2 × 3/2 =[p + f ]+{s + d}
(7.68)
On these grounds the two T 1g interactions can be distinguished on the basis of a
different spherical parentage corresponding to p or f coupling. We shall return to
this point in a moment. For a systematic treatment of this problem, we start by setting up a suitable function space. The spherical S = 3/2 spin-quartet level subduces
directly the octahedral Γ 8 . We can thus use the quartet spin functions as symmetry
bases. The components of S = 3/2 can be obtained by a fully symmetrized product
of the basic spinor:
|3/2 + 3/2=α 1 α 2 α 3
|3/2 + 1/2=
1
√
3
(α 1 α 2 β 3 + α 1 β 2 α 3 + β 1 α 2 α 3 )
|3/2 − 1/2=
1
√
3
(α 1 β 2 β 3 + β 1 α 2 β 3 + β 1 β 2 α 3 )
|3/2 − 3/2=β 1 β 2 β 3
(7.69)
In this expression the fundamental {|α, |β} spinor resembles a quark state: three
quarks are coupled together to form the quartet result. The use of the quartet spin
bases does not mean that our Γ 8 really corresponds to a quartet spin. It only means
that we can introduce a fictitious spin operator, ˜
S, which acts on the Γ 8 components
in the same way as the real spin momentum would act on the components of a
spin quartet. The transformations of the Γ 8 spinor under the elements of the group
O ∗ may be obtained by combining the transformation matrices for the fundamental
(|α|β) spins with the quartet coupling scheme in Eq. (7.69). In the group O ∗ ,this
irrep is denoted as Γ 6 . In Table 7.7 the results are shown for two generators of the
octahedral group. These matrices can be taken as the canonical basis relationships
7 Spherical Symmetry and Spins
7.7 Application: Spin Hamiltonian for the Octahedral Quartet
State
The octahedral double group contains a four-dimensional spin representation, which
is commonly denoted as the Γ 8 quartet, or U ′ in Griffith’s notation. The direct square
of this irrep is given by
Γ 8 × Γ 8 =[A 2 + 2T 1 + T 2 ]+{A 1 + E + T 2 }
(7.67)
According to the time-reversal selection rules, time-odd interactions with a magnetic
field will be based on the symmetrized square. The spin-operator of the Zeeman
Hamiltonian transforms as T 1g , which is indeed included in the symmetrized square.
The present case is, however, special since the T 1g irrep occurs twice in the product.
The multiplicity separation cannot be achieved on the basis of symmetrization since
both T 1g irreps appear in the symmetrized part. One way to distinguish the two
products is through the subduction process from spherical symmetry. The addition
rules of angular momenta give rise to
3/2 × 3/2 =[p + f ]+{s + d}
(7.68)
On these grounds the two T 1g interactions can be distinguished on the basis of a
different spherical parentage corresponding to p or f coupling. We shall return to
this point in a moment. For a systematic treatment of this problem, we start by setting up a suitable function space. The spherical S = 3/2 spin-quartet level subduces
directly the octahedral Γ 8 . We can thus use the quartet spin functions as symmetry
bases. The components of S = 3/2 can be obtained by a fully symmetrized product
of the basic spinor:
|3/2 + 3/2=α 1 α 2 α 3
|3/2 + 1/2=
1
√
3
(α 1 α 2 β 3 + α 1 β 2 α 3 + β 1 α 2 α 3 )
|3/2 − 1/2=
1
√
3
(α 1 β 2 β 3 + β 1 α 2 β 3 + β 1 β 2 α 3 )
|3/2 − 3/2=β 1 β 2 β 3
(7.69)
In this expression the fundamental {|α, |β} spinor resembles a quark state: three
quarks are coupled together to form the quartet result. The use of the quartet spin
bases does not mean that our Γ 8 really corresponds to a quartet spin. It only means
that we can introduce a fictitious spin operator, ˜
S, which acts on the Γ 8 components
in the same way as the real spin momentum would act on the components of a
spin quartet. The transformations of the Γ 8 spinor under the elements of the group
O ∗ may be obtained by combining the transformation matrices for the fundamental
(|α|β) spins with the quartet coupling scheme in Eq. (7.69). In the group O ∗ ,this
irrep is denoted as Γ 6 . In Table 7.7 the results are shown for two generators of the
octahedral group. These matrices can be taken as the canonical basis relationships