4 X-ray Dichroisms in Spherical Tensor and Green’s Function Formalism
105
linearly polarized light is an effect referred to as linear dichroism as discussed previously.
Octahedral Crystal Field with an Exchange Field z
Another interesting system to investigate is a magnetic 3d
9 ion where the crystal
field is O h with an exchange field aligned along the z-axis. Hence, the z-axis is
inequivalent to the x- and y-axes due to the exchange field. The conductivity tensor
of such a system is shown in Fig. 4.6. The exchange field is aligned along a high
symmetry direction in this example which preserves the C 4 rotation symmetry of
the system and consequently preserves the symmetry of the conductivity tensor. All
off-diagonal elements are zero. Note that the off-diagonal elements are zero because
we chose to calculate the tensor using the symmetry adapted transition operators.
Three fundamental spectra come into play and are plotted in Fig. 4.7:
• R(0, 0) which gives the isotropic cross section,
• R(1, 0) which has a polarization dependence of the form
1
2
i
∗
x y − i x
∗
y
,
• R(2, 0) which has a polarization dependence of the form
1
6
2| z |
2
− | x |
2
− | y |
2
.
Nearly no angular dependence can be observed by rotating the incident linear
polarization vector in the x − y-, x − z-, and y − z-planes (see Fig. 4.8a, b, and c).
This is consistent with the fact that the fundamental spectrum R(2, 0) responsible
for the angular dependence is nearly zero [R(2, 0) is about two orders of magnitude
smaller than the other two fundamental spectra in this system]. The difference in the
absorption cross section of linear polarized light in a magnetic system is an effect
referred to as XMLD [24]. The magnitude of the XMLD effect for this system can
be seen in Fig. 4.9. Note that the magnitude of the XMLD effect in Fe 3 O 4 is ∼1%
of the XAS signal, which could be reliably measured on existing beamlines [25].
A strong dichroism is observed when circularly polarized light is used as in the
case for Fig. 4.10a. Here the incident polarization vector is either left or right polarized
about the z-axis leading to a difference in the absorption. This is an effect referred to
as X-ray magnetic circular dichroism (XMCD) [26]. It can be seen from the expression of the polarization part of the cross-section that if the incident wave vector is
aligned perpendicular to the exchange field, for example, for = [0, −
i
√
2
,
1
√
2
], and
= [0, −
i
√
2
, −
1
√
2
], no XMCD effect is observed. This is shown in Fig. 4.10b and c.
This dichroism can be used to quantify the ground state spin and orbital moments of
the system as given by the sum rules [27].
Octahedral Crystal Field with an Exchange Field [210]
As a last example, we consider a system in C 1 symmetry. Consider aligning the
exchange field along a low symmetry direction, e.g., [210]. The exchange field now
completely breaks the symmetry of the system and the conductivity tensor has offdiagonal elements (bottom of Fig. 4.6). Contrary to the previous case (where the
exchange field was aligned to the z-axis), it is now not possible to find a rotated
105
linearly polarized light is an effect referred to as linear dichroism as discussed previously.
Octahedral Crystal Field with an Exchange Field z
Another interesting system to investigate is a magnetic 3d
9 ion where the crystal
field is O h with an exchange field aligned along the z-axis. Hence, the z-axis is
inequivalent to the x- and y-axes due to the exchange field. The conductivity tensor
of such a system is shown in Fig. 4.6. The exchange field is aligned along a high
symmetry direction in this example which preserves the C 4 rotation symmetry of
the system and consequently preserves the symmetry of the conductivity tensor. All
off-diagonal elements are zero. Note that the off-diagonal elements are zero because
we chose to calculate the tensor using the symmetry adapted transition operators.
Three fundamental spectra come into play and are plotted in Fig. 4.7:
• R(0, 0) which gives the isotropic cross section,
• R(1, 0) which has a polarization dependence of the form
1
2
i
∗
x y − i x
∗
y
,
• R(2, 0) which has a polarization dependence of the form
1
6
2| z |
2
− | x |
2
− | y |
2
.
Nearly no angular dependence can be observed by rotating the incident linear
polarization vector in the x − y-, x − z-, and y − z-planes (see Fig. 4.8a, b, and c).
This is consistent with the fact that the fundamental spectrum R(2, 0) responsible
for the angular dependence is nearly zero [R(2, 0) is about two orders of magnitude
smaller than the other two fundamental spectra in this system]. The difference in the
absorption cross section of linear polarized light in a magnetic system is an effect
referred to as XMLD [24]. The magnitude of the XMLD effect for this system can
be seen in Fig. 4.9. Note that the magnitude of the XMLD effect in Fe 3 O 4 is ∼1%
of the XAS signal, which could be reliably measured on existing beamlines [25].
A strong dichroism is observed when circularly polarized light is used as in the
case for Fig. 4.10a. Here the incident polarization vector is either left or right polarized
about the z-axis leading to a difference in the absorption. This is an effect referred to
as X-ray magnetic circular dichroism (XMCD) [26]. It can be seen from the expression of the polarization part of the cross-section that if the incident wave vector is
aligned perpendicular to the exchange field, for example, for = [0, −
i
√
2
,
1
√
2
], and
= [0, −
i
√
2
, −
1
√
2
], no XMCD effect is observed. This is shown in Fig. 4.10b and c.
This dichroism can be used to quantify the ground state spin and orbital moments of
the system as given by the sum rules [27].
Octahedral Crystal Field with an Exchange Field [210]
As a last example, we consider a system in C 1 symmetry. Consider aligning the
exchange field along a low symmetry direction, e.g., [210]. The exchange field now
completely breaks the symmetry of the system and the conductivity tensor has offdiagonal elements (bottom of Fig. 4.6). Contrary to the previous case (where the
exchange field was aligned to the z-axis), it is now not possible to find a rotated