100
H. Elnaggar et al.
σ (1, 1) = −4παω × Im
−1
2
√
2
∗
x z − x
∗
z + i y
∗
z − i
∗
y z
(4.31)
I |rC
∗
1,0 G
+ rC 1,1 |I + +I |rC
∗
1,−1 G
+ rC 1,0 |I
,
(4.32)
σ (1, −1) = −4παω × Im
1
2
√
2
∗
x z − x
∗
z + i
∗
y z − i y
∗
z
×
I |rC
∗
1,0 G
+ rC 1,−1 |I + +I |rC
∗
1,1 G
+ rC 1,0 |I
. (4.33)
One notices from (4.30), (4.32) and (4.33) that the spectra of a = 1 are not active if
any of these two cases are satisfied:
1. If linearly polarized light is used for the measurement.
2. If all off-diagonal elements are zero and the diagonal elements are equal.
Another conclusion that can be drawn from the recoupling is the necessity to perform
XAS measurements using both linearly and circularly polarized light to probe these
fundamental spectra.
4.2.1.3 Term a = 2
The term a = 2 consists of five components, namely, σ (2, 0), σ (2, 1), σ (2, −1),
σ (2, 2), and σ (2, −2). These are given below as follows:
σ (2, 0) = −4παω × Im
1
6
2| z |
2
− | x |
2
− | y |
2
2I |rC
∗
1,0 G
+ rC 1,0 |I
−−I |rC
∗
1,−1 G
+ rC 1,−1 |I − −I |rC
∗
1,1 G
+ rC 1,1 |I
,
(4.34)
σ (2, 1) = −4παω × Im
1
2
√
2
x
∗
z +
∗
x z − i y
∗
z − i
∗
y z
×
I |rC
∗
1,−1 G
+ rC 1,0 |I − −I |rC
∗
1,0 G
+ rC 1,1 |I
,
(4.35)
σ (2, −1) = −4παω × Im
1
2
√
2
x
∗
z +
∗
x z + i
∗
y z + i y
∗
z
×
I |rC
∗
1,0 G
+ rC 1,−1 |I − −I |rC
∗
1,1 G
+ rC 1,0 |I
, (4.36)