Spin-Orbit Coupling and Double
Groups
C734b 2008
Spin-orbit coupling and double
groups
1
Spin-orbit term in H induces coupling of orbital and spin angular momenta to give
total angular momentum:
r r r
J = L+S
-splits Russell-Saunders multiplets into their components labeled by the J quantum number.
- recall:
⎡ (2 L + 1)α ⎤
sin ⎢
⎥⎦
2
χ (DL (α )) = ⎣
α
⎛ ⎞
sin ⎜ ⎟
⎝2⎠
- A deeper analysis shows that this result is related to commutation relations for L operators.
⎡ (2 J + 1)α ⎤
sin ⎢
⎥⎦
2
⎣
⇒ χ (DJ (α )) =
⎛α ⎞
sin ⎜ ⎟
⎝2⎠
Since S and J obey the same relations:
Spin-orbit coupling and double
groups
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1
Since L ≡ integer and S ≡ integer or ½-integer
Means J can be an integer or ½-integer
no problem problem
Consider a proper rotation through an angle α + 2π
⎡ (2 J + 1)α
⎤
sin ⎢
+ (2 J + 1)π ⎥
2
⎣
⎦
χ (DJ (α + 2π )) =
⎡ (α + 2π ) ⎤
sin ⎢
⎥⎦
2
⎣
Q sin ( A + B ) = sin ( A) cos(B ) + cos( A)sin (B )
⎡ (2 J + 1)α ⎤
sin ⎢
⎥⎦ cos[(2 J + 1)π ]
2
⎣
=
⎛α ⎞
− sin ⎜ ⎟
⎝2⎠
= (− 1) χ [DJ (α )]
2J
Spin-orbit coupling and double
groups
∴
3
J ≡ integer χ [DJ (α + 2π )] = χ [DJ (α )] as expected
However, when J ≡ ½-integer (odd # of electrons)
χ [DJ (α + 2π )] = − χ [DJ (α )]
This behaviour arises because state functions are spinors (orbital x spin function)
and not vectors.
Need a new operator:
Adding
E
E = R(2π , n ) ≠ E = R(0, n )
to the group G = {R} gives a double group G
with elements {R}+ E {R}
Spin-orbit coupling and double
groups
4
2
Spin-orbit coupling and double
groups
5
What about Improper Rotations?
i ∈ G, iψ j = ±ψ j
±
±
and χ (D J (IR )) = ± χ (D J (R ))
If
If
IR here means improper rotation
i ∉ G, χ (DJ (IR )) = χ (DJ (R ))
These rules hold for J integer or ½-integer, and so for L and S also.
Spin-orbit coupling and double
groups
6
3
Note:
G
contains 2 x # elements of G (hence the name), but not necessarily 2 x # classes
The number of new classes in G
are given by Opechewski’
Opechewski’s Rules
1.) C2n = EC 2n and C2n are in the same class iff there is another C2 axis with a rotation axis
perpendicular to n
2.) Cn = EC n are always in different classes if n ≠ 2
k
−k
3.) for n > 2, Cnk and Cn-k are in the same class, and so are Cn and Cn
Spin-orbit coupling and double
groups
Let
7
⎡ (2 J + 1)α ⎤
sin ⎢
⎥⎦
2
χ J (α ) = ⎣
= χ J [R(α , n )]
⎛α ⎞
sin ⎜ ⎟
⎝2⎠
χ J (α + 2π ) = (− 1)2 J χ J (α ) = χ [R (α , n )]
Equations work for integer and ½-integer J-values
Note: the characters of the new classes Ck of G are for integer J the same as those
of the classes Ck of G, but for ½-integer J have the same magnitude but opposite sign.
The new representations for G by ½-integer J are called spinor representations
Spin-orbit coupling and double
groups
8
4
Labels for DJ representations
1.) Bethe’s notation: Γj where j ≡ the number of integer values necessary to label all
the spinor represntations.
2.) Mulliken-Herzberg notation: IRs are labelled E, G, H, … according to their
dimensionality 2, 4, 6, with a subscripts J which corresponds to the representation DJ in
which the IR first occurs
Example:
O
Spin-orbit coupling and double
groups
9
Spin-orbit coupling and double
groups
10
5
All new representations generated by half-integer J are at least 2-fold degenerate
in any electrostatic field ≡ Kramer’
Kramer’s Theorem
Further splittings are possible in a magnetic field.
For inversion symmetry:
1.) Bethe’s notation:
even (gerade)
Γ j±
odd (ungerade)
2.) Mulliken-Herzberg notation: use superscripts g and u.
Spin-orbit coupling and double
groups
11
Weak Crystal Fields
Means weaker than
Ĥ Lr ⋅Sr .
Therefore a weak crystal field acts on the components of the Russell-Saunders multiplets.
Depending on their degeneracy these components may undergo further splittings in a
weak crystal field.
Example: In O the following splittings:
D 3 = Γ8
2
D 5 = Γ 7 ⊕ Γ8
2
Selected spin-orbit components
D 7 = Γ 6 ⊕ Γ 7 ⊕ Γ8
2
D 9 = Γ 6 ⊕ 2Γ8
2
{Γj that given in Table}
Spin-orbit coupling and double
groups
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6
Question:
Examine the effect of spin-orbit coupling on the states that result
from an intermediate-field of O symmetry on the Russell-Saunders multiplet 4F.
Correlate these states with those produced by a weak crystal field of O symmetry on
the components produced by spin-orbit coupling on the 4F multiplet.
Answer:
4F
implies L =3 in an intermediate field
⇒ A 2 ⊕ T1 ⊕ T2 = Γ 2 ⊕ Γ 4 ⊕ Γ 5
To examine the effect of spin-orbit coupling on the intermediate field use ψ = ϕiχj where
ϕi forms a basis for Γi and χj forms a basis for Γj. This means ψ = ϕiχj forms a basis for the
direct product Γ i ⊗ Γ j
Q 2S + 1 = 4 ⇒ S =
3
2
∴ Γ j ≡ D 3 ≡ Γ8
2
Spin-orbit coupling and double
groups
13
∴ Γ8 ⊗ Γ 2 = Γ8
Γ8 ⊗ Γ 4 = Γ 6 ⊕ Γ 7 ⊕ 2Γ8
Γ8 ⊗ Γ 5 = Γ 6 ⊕ Γ 7 ⊕ 2Γ8
Final correlation diagram on the next slide:
Spin-orbit coupling and double
groups
14
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Spin-orbit coupling and double
groups
15
8