10 Measuring Atomic Magnetic Moments in Magnetic Nanostructures …
245
L z = 2
l ν (l ν + 1)
l c (l c + 1) − l ν (l ν + 1) − 2
edge
μ ↑↑ (ω) − μ ↑↓ (ω)
dω
edge
μ ↑↑ (ω) + μ ↑↓ (ω) + μ 0
dω
n h (10.1)
Here, μ ↑↑ (ω) and μ ↑↓ (ω) are the X-ray absorption spectra measured with the
photon angular momentum parallel and antiparallel with the applied (saturating)
magnetic field, respectively, and μ 0 (ω) is the average of the two, or, alternatively,
the absorption spectrum measured with linearly polarised light. The term n h is the
number of holes per atom at the Fermi level, which for a solid state system, will be a
non-integer number. For example, in the case of Fe, this is normally taken to be 3.39
holes per atom [10].
The integrals in this case are over the entire absorption edge. For absorption by a
transition metal L edge, i.e. 2p–3d transitions, l c = 1 and l v , = 2 and using
μ 0 (ω) =
1
2
μ ↑↑ (ω) + μ ↑↓ (ω)
(10.2)
(8.1) becomes:
L z =
4
3
L 2 +L s
μ ↑↑ (ω) − μ ↑↓ (ω)
dω
L 2 +L s
μ ↑↑ (ω) + μ ↑↓ (ω)
dω
n h
(10.3)
Thus, the orbital moment is proportional to the total area in the dichroism
spectrum.
The projection of the spin moment along the photon spin is given by [7]:
l ν (l ν + 1) − 2 − l c (l c + 1)
3l c
S z
+
l ν (l ν + 1)[l ν (l ν + 1) + 2l c (l c + 1) + 4] − 3(l c − 1)
2
(l c + 2)
2
6l ν l c (l ν + 1)
T z
=
j +
μ ↑↑ (ω) − μ ↑↓ (ω)
dω −
l c +1
l c
j −
μ ↑↑ (ω) − μ ↑↓ (ω)
dω
edge
μ ↑↑ (ω) + μ ↑↓ (ω) + μ 0
dω
n h (10.4)
where < T Z > is the expectation value of the magnetic dipole operator along the
photon spin and is a measure of the anisotropy of the spin distribution, which XMCD
is sensitive to since the photon polarisation samples a directional cut through the
atomic electron density. This term, discussed in more detail below, is small relative
to and for bulk samples has often been assumed to be zero but its significance
increases in nanoscale particles and it must be taken into consideration. Note that
the integrals in the numerator of (4) are taken over individual j components of the L
edge.
In the case of transition metal L edges (4) becomes.
245
L z = 2
l ν (l ν + 1)
l c (l c + 1) − l ν (l ν + 1) − 2
edge
μ ↑↑ (ω) − μ ↑↓ (ω)
dω
edge
μ ↑↑ (ω) + μ ↑↓ (ω) + μ 0
dω
n h (10.1)
Here, μ ↑↑ (ω) and μ ↑↓ (ω) are the X-ray absorption spectra measured with the
photon angular momentum parallel and antiparallel with the applied (saturating)
magnetic field, respectively, and μ 0 (ω) is the average of the two, or, alternatively,
the absorption spectrum measured with linearly polarised light. The term n h is the
number of holes per atom at the Fermi level, which for a solid state system, will be a
non-integer number. For example, in the case of Fe, this is normally taken to be 3.39
holes per atom [10].
The integrals in this case are over the entire absorption edge. For absorption by a
transition metal L edge, i.e. 2p–3d transitions, l c = 1 and l v , = 2 and using
μ 0 (ω) =
1
2
μ ↑↑ (ω) + μ ↑↓ (ω)
(10.2)
(8.1) becomes:
L z =
4
3
L 2 +L s
μ ↑↑ (ω) − μ ↑↓ (ω)
dω
L 2 +L s
μ ↑↑ (ω) + μ ↑↓ (ω)
dω
n h
(10.3)
Thus, the orbital moment is proportional to the total area in the dichroism
spectrum.
The projection of the spin moment along the photon spin is given by [7]:
l ν (l ν + 1) − 2 − l c (l c + 1)
3l c
S z
+
l ν (l ν + 1)[l ν (l ν + 1) + 2l c (l c + 1) + 4] − 3(l c − 1)
2
(l c + 2)
2
6l ν l c (l ν + 1)
T z
=
j +
μ ↑↑ (ω) − μ ↑↓ (ω)
dω −
l c +1
l c
j −
μ ↑↑ (ω) − μ ↑↓ (ω)
dω
edge
μ ↑↑ (ω) + μ ↑↓ (ω) + μ 0
dω
n h (10.4)
where < T Z > is the expectation value of the magnetic dipole operator along the
photon spin and is a measure of the anisotropy of the spin distribution, which XMCD
is sensitive to since the photon polarisation samples a directional cut through the
atomic electron density. This term, discussed in more detail below, is small relative
to
increases in nanoscale particles and it must be taken into consideration. Note that
the integrals in the numerator of (4) are taken over individual j components of the L
edge.
In the case of transition metal L edges (4) becomes.