186
7 Spherical Symmetry and Spins
Γ 8i T 1j |Γ 8k and describe the spin-orbit coupling coefficients for the spin-orbit levelsofa 4 T 1 state. However, since all coefficients in the table are real, turning them
around does not make any difference. Hence, we can directly use the spin-orbit
tables to obtain the Zeeman matrix. Only one of the coupling channels is seen to
link the κ and ν levels. In spherical symmetry this requires a jump of 3 spin units,
which can be bridged only by an ℓ = 3 operator. The coupling coefficients of this
channel are thus of spherical octupole parentage (ℓ = 3), while the other set is of
dipole parentage (ℓ = 1). We shall parameterize the corresponding reduced matrix
elements as J f and J p , respectively. The constant μ B /, as well as a common factor of 1/
√
15, is also absorbed into these parameters. The electronic operator will
be represented as |T i |, and the Zeeman Hamiltonian is then recast in complex form
as
H Ze = B x |T x |+B y |T y |+B z |T z |
= B z |T 0 |−
1
√
2
(B x − iB y )|T +1 |+
1
√
2
(B x + iB y )|T −1 |
(7.73)
This expression follows the convention of Eq. (7.39). The operator part is defined
by
|T 0 |=|T z |
|T +1 |=−
1
√
2
|T x |+i|T y |
(7.74)
|T −1 |=
1
√
2
|T x |−i|T y |
The elements of the interaction matrix are then given by
H ij = B z
J p Γ 8 i|T 0 Γ 8 j p + J f Γ 8 i|T 0 Γ 8 j f
−
1
√
2
(B x − iB y )
J p Γ 8 i|T +1 Γ 8 j p + J f Γ 8 i|T +1 Γ 8 j f
+
1
√
2
(B x + iB y )
J p Γ 8 i|T −1 Γ 8 j p + J f Γ 8 i|T −1 Γ 8 j f
(7.75)
The resulting interaction matrix is given in Table 7.8. Since the Zeeman interaction leads to a splitting of the levels that conserves the barycentre, the secular
equation does not contain odd powers in the energy:
aE
4 + bE
2 + c = 0
(7.76)
7 Spherical Symmetry and Spins
Γ 8i T 1j |Γ 8k and describe the spin-orbit coupling coefficients for the spin-orbit levelsofa 4 T 1 state. However, since all coefficients in the table are real, turning them
around does not make any difference. Hence, we can directly use the spin-orbit
tables to obtain the Zeeman matrix. Only one of the coupling channels is seen to
link the κ and ν levels. In spherical symmetry this requires a jump of 3 spin units,
which can be bridged only by an ℓ = 3 operator. The coupling coefficients of this
channel are thus of spherical octupole parentage (ℓ = 3), while the other set is of
dipole parentage (ℓ = 1). We shall parameterize the corresponding reduced matrix
elements as J f and J p , respectively. The constant μ B /, as well as a common factor of 1/
√
15, is also absorbed into these parameters. The electronic operator will
be represented as |T i |, and the Zeeman Hamiltonian is then recast in complex form
as
H Ze = B x |T x |+B y |T y |+B z |T z |
= B z |T 0 |−
1
√
2
(B x − iB y )|T +1 |+
1
√
2
(B x + iB y )|T −1 |
(7.73)
This expression follows the convention of Eq. (7.39). The operator part is defined
by
|T 0 |=|T z |
|T +1 |=−
1
√
2
|T x |+i|T y |
(7.74)
|T −1 |=
1
√
2
|T x |−i|T y |
The elements of the interaction matrix are then given by
H ij = B z
J p Γ 8 i|T 0 Γ 8 j p + J f Γ 8 i|T 0 Γ 8 j f
−
1
√
2
(B x − iB y )
J p Γ 8 i|T +1 Γ 8 j p + J f Γ 8 i|T +1 Γ 8 j f
+
1
√
2
(B x + iB y )
J p Γ 8 i|T −1 Γ 8 j p + J f Γ 8 i|T −1 Γ 8 j f
(7.75)
The resulting interaction matrix is given in Table 7.8. Since the Zeeman interaction leads to a splitting of the levels that conserves the barycentre, the secular
equation does not contain odd powers in the energy:
aE
4 + bE
2 + c = 0
(7.76)