4 X-ray Dichroisms in Spherical Tensor and Green’s Function Formalism
123
x
y
є
θ
k
x
y
є
θ
k
z
(a)
(b)
Fig. 4.16 Angular dependence of XAS for a quadrupole transition in a 3d 9 O h ion with 10D q =
1.1 eV. a The wave vector (k) is aligned with [100]. The angular dependence is computed by
rotating the polarization vector () about [100] with θ = 0 o for [001] as depicted in the sketch
on the right. b k is aligned with [
1
√
2
1
√
2
0]. The angular dependence is computed by rotating about
[
1
√
2
1
√
2
0] with θ = 0 o for [001] as depicted in the sketch on the right
Octahedral Crystal Field with Exchange Field z
Consider a magnetic 3d
9 ion where the crystal field is O h with an exchange field
aligned along the z-axis. Seven fundamental spectra come into play, namely, R(0, 0),
R(1, 0), R(2, 0), R(3, 0), R(4, 0), R(4, −4), and R(4, 4). We have shown previously
that for O h symmetry, when k is aligned parallel to the [100] direction, and is rotated
in the z − y-plane, angular dependence is observed (see Fig. 4.16a). Repeating the
same calculation with an exchange field aligned along the z-axis leads to an angular
dependent XAS as shown in Fig. 4.18a. The exchange field reduces the symmetry
along the z-axis. The effects of rotating the incident linear polarization in the z − yplane on the fundamental cross sections are shown in Fig. 4.19. Only the terms
σ (2, 0), σ (4, 0), σ (4, 4), and σ (4, −4) are non-zero and exhibit a twofold angular
123
x
y
є
θ
k
x
y
є
θ
k
z
(a)
(b)
Fig. 4.16 Angular dependence of XAS for a quadrupole transition in a 3d 9 O h ion with 10D q =
1.1 eV. a The wave vector (k) is aligned with [100]. The angular dependence is computed by
rotating the polarization vector () about [100] with θ = 0 o for [001] as depicted in the sketch
on the right. b k is aligned with [
1
√
2
1
√
2
0]. The angular dependence is computed by rotating about
[
1
√
2
1
√
2
0] with θ = 0 o for [001] as depicted in the sketch on the right
Octahedral Crystal Field with Exchange Field z
Consider a magnetic 3d
9 ion where the crystal field is O h with an exchange field
aligned along the z-axis. Seven fundamental spectra come into play, namely, R(0, 0),
R(1, 0), R(2, 0), R(3, 0), R(4, 0), R(4, −4), and R(4, 4). We have shown previously
that for O h symmetry, when k is aligned parallel to the [100] direction, and is rotated
in the z − y-plane, angular dependence is observed (see Fig. 4.16a). Repeating the
same calculation with an exchange field aligned along the z-axis leads to an angular
dependent XAS as shown in Fig. 4.18a. The exchange field reduces the symmetry
along the z-axis. The effects of rotating the incident linear polarization in the z − yplane on the fundamental cross sections are shown in Fig. 4.19. Only the terms
σ (2, 0), σ (4, 0), σ (4, 4), and σ (4, −4) are non-zero and exhibit a twofold angular