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30 2. Theoretical background
must distinguish between the different polarizations q of the light, with q = M ′ − M. Therefore,
we choose the basis |J, M〉 including the magnetic quantum number, and the final states
|J ′ ,M ′ 〉, using the electric dipole operator P q . A summation over all light polarization q yields
the total photoemission intensity as
∑ ∑
σ JJ′ ∝ σq JJ′ =
q
q
∣
∣〈
J ′ ,M ∣ ′ ∣
∣ Pq
∣∣J,
〉∣ ∣∣∣ 2
M , (2.29)
where P q denotes the q component of the electric dipole moment P q = √ 4/3π r ·Y q l (θ,ϕ),
expressed by the spherical harmonics Y q l (θ,ϕ).
We focus on transitions with ΔM = −q = ±1 according to the angular momentum q of the
circularly polarized light. The transition probability σ JJ′ can be divided into two factors
which can be treated separately: (i) the line strength S JJ ′ for every allowed transition J → J ′ ,
and (ii) the algebra of angular momenta which includes the change of M due to q and the
dipole selection rules. Applying the Wigner-Eckart theorem yields
σ JJ′
q
(2.29)
=
=
〈 ∣ J ′ ,M ∣ ′ ∣ 〉∣ ∣ ∣∣J, ∣∣∣ 2
Pq M 〈 ∣ ∣ J ′ ∣ 〉∣ ∣ P ∣∣J ∣∣∣ 2
·
}{{}
S JJ ′
(
J 1 J
′
−M q M ′ ) 2
.
}{{}
3j-symbol
(2.30)
The 3j-symbol vanishes for q = M ′ −M = 0, which means, that the key prerequisite in order to
obtain an MCD is the transfer of the angular momentum of circularly polarized light onto the
final state M ′ . The line strength S JJ ′ shows only a non-vanishing MCD difference spectrum,
if the different sub-levels for every M are unequally populated.
The MCD spectrum is defined as
∑ (
JJ
I MCD = σ ′
q=−1 − )
σJJ′ q=+1
J ′
∝ 〈 M 〉 k B T , (2.31)
whereas 〈M〉 kB T is the Boltzmann average (at temperature T ) which is proportional to the
macroscopic magnetization |M(T )|. In this approximation, the MCD spectrum I MCD is a
measure for the macroscopic magnetization.
Example of an MCD spectrum by core-level photoemission
We discuss the MCD in Fe 2p core-level photoemission, which was initially experimentally
observed by Baumgarten et al. (1990) prior to the development of the MCD theory. 108 In Fe
2p, the spin–orbit interaction is dominant. Hence, a good approximation is the use of |j, m j 〉
spin–orbit final states as a basis. 106 Then, the MCD spectrum can be described by
∑ (
I MCD (j, m j )= σ
l ′ s ′
q=−1 − s
q=+1) ′
σl′ . (2.32)
l ′ s ′
Here, l ′ and s ′ denote the final state angular and spin momenta of the photo-ionized p 5 shell.
The sum in eq. 2.32 runs over the six spin–orbit lines as indicated in Tab. 2.1. Radial matrix