4 X-ray Dichroisms in Spherical Tensor and Green’s Function Formalism
91
τ 1 ,τ 2 ,τ 3 ,τ 4
Y l 1 ,m 1 (θ, φ)σ 1 Y l 2 ,m 2 (θ
, φ
)σ 2 |
∞
k=0
k
m=−k
Y
∗
k,m (θ
, φ
)Y k,m (θ, φ)|
Y l 3 ,m 3 (θ, φ)σ 3 Y l 4 ,m 4 (θ
, φ
)σ 4
=
∞
k=0
k
m=−k
τ 1 ,τ 2 ,τ 3 ,τ 4
Y l 1 ,m 1 (θ, φ)σ 1 |Y k,m (θ, φ)|Y l 3 ,m 3 (θ, φ)σ 3
Y l 2 ,m 2 (θ
, φ
)σ 2 |Y
∗
k,m (θ
, φ
)|Y l 4 ,m 4 (θ
, φ
)σ 4 .
(4.14)
We have in (4.14) integrals involving three spherical harmonics which are given
by the Gaunt coefficients. This can be used to restrict the values of the summation over k and m. The Gaunt coefficients are different from zero in the first integral only for m = m 1 − m 3 . Similarly, the second integral is different from zero for
m = m 4 − m 2 . Hence, in combination, one concludes that the total M z is conserved
(m 1 + m 2 = m 3 + m 4 ) for the integrals. Furthermore, the values of k are restricted
to values of k ≤ min(|l 1 + l 4 |, |l 2 + l 3 |). This simplifies the angular part.
Let us now investigate the radial part. The general expression for a scattering
event involving four different shells is expressed in (4.15)
τ 1 ,τ 2 ,τ 3 ,τ 4
R n 1 ,l 1 (r )R n 2 ,l 2 (r
)|
min(|l 1 +l 4 |,|l 2 +l 3 |)
k=0
r
k
<
r
k+1
>
|R n 3 ,l 3 (r )R n 4 ,l 4 (r
) .
(4.15)
In the case of Coulomb interaction in a single shell, n 1 = n 2 = n 3 = n 4 and
l 1 = l 2 = l 3 = l 4 and hence one obtains the expression
F
(k)
=
2l
k=0
R n,l (r )R n,l (r
)|
r
k
<
r
k+1
>
|R n,l (r )R n,l (r
) .
(4.16)
For a 3d
n configuration, 2l = 4 yielding values of k = 0, 2, 4 and one has to evaluate
three Slater integrals F
( k) for such a configuration. For an excited state with a core
hole, like the excited state of a L absorption edge with a configuration 2 p
5 3d
n+1 ,
it is necessary to take into consideration the Coulomb interaction between the 2 p
and the 3d shells. There are two possible cases for these scattering events (see also
Chap. 2):
1. Direct interaction with n 1 = n 3 and n 2 = n 4 . This means that each electron scatters in its shell. The matrix elements read
F
(k)
=
min(|2l 1 ,2l 2 |)
k=0
R n 1 ,l 1 (r )R n 2 ,l 2 (r
)|
r
k
<
r
k+1
>
|R n 1 ,l 1 (r )R n 2 ,l 2 (r
) .
(4.17)
2. Exchange interaction with n 1 = n 4 and n 2 = n 3 . This means that electrons
exchange shells. The matrix elements read
91
τ 1 ,τ 2 ,τ 3 ,τ 4
Y l 1 ,m 1 (θ, φ)σ 1 Y l 2 ,m 2 (θ
, φ
)σ 2 |
∞
k=0
k
m=−k
Y
∗
k,m (θ
, φ
)Y k,m (θ, φ)|
Y l 3 ,m 3 (θ, φ)σ 3 Y l 4 ,m 4 (θ
, φ
)σ 4
=
∞
k=0
k
m=−k
τ 1 ,τ 2 ,τ 3 ,τ 4
Y l 1 ,m 1 (θ, φ)σ 1 |Y k,m (θ, φ)|Y l 3 ,m 3 (θ, φ)σ 3
Y l 2 ,m 2 (θ
, φ
)σ 2 |Y
∗
k,m (θ
, φ
)|Y l 4 ,m 4 (θ
, φ
)σ 4 .
(4.14)
We have in (4.14) integrals involving three spherical harmonics which are given
by the Gaunt coefficients. This can be used to restrict the values of the summation over k and m. The Gaunt coefficients are different from zero in the first integral only for m = m 1 − m 3 . Similarly, the second integral is different from zero for
m = m 4 − m 2 . Hence, in combination, one concludes that the total M z is conserved
(m 1 + m 2 = m 3 + m 4 ) for the integrals. Furthermore, the values of k are restricted
to values of k ≤ min(|l 1 + l 4 |, |l 2 + l 3 |). This simplifies the angular part.
Let us now investigate the radial part. The general expression for a scattering
event involving four different shells is expressed in (4.15)
τ 1 ,τ 2 ,τ 3 ,τ 4
R n 1 ,l 1 (r )R n 2 ,l 2 (r
)|
min(|l 1 +l 4 |,|l 2 +l 3 |)
k=0
r
k
<
r
k+1
>
|R n 3 ,l 3 (r )R n 4 ,l 4 (r
) .
(4.15)
In the case of Coulomb interaction in a single shell, n 1 = n 2 = n 3 = n 4 and
l 1 = l 2 = l 3 = l 4 and hence one obtains the expression
F
(k)
=
2l
k=0
R n,l (r )R n,l (r
)|
r
k
<
r
k+1
>
|R n,l (r )R n,l (r
) .
(4.16)
For a 3d
n configuration, 2l = 4 yielding values of k = 0, 2, 4 and one has to evaluate
three Slater integrals F
( k) for such a configuration. For an excited state with a core
hole, like the excited state of a L absorption edge with a configuration 2 p
5 3d
n+1 ,
it is necessary to take into consideration the Coulomb interaction between the 2 p
and the 3d shells. There are two possible cases for these scattering events (see also
Chap. 2):
1. Direct interaction with n 1 = n 3 and n 2 = n 4 . This means that each electron scatters in its shell. The matrix elements read
F
(k)
=
min(|2l 1 ,2l 2 |)
k=0
R n 1 ,l 1 (r )R n 2 ,l 2 (r
)|
r
k
<
r
k+1
>
|R n 1 ,l 1 (r )R n 2 ,l 2 (r
) .
(4.17)
2. Exchange interaction with n 1 = n 4 and n 2 = n 3 . This means that electrons
exchange shells. The matrix elements read