Line intensities and selection rules
The line intensities and selection rules of XMCD can be understood by considering the transition matrix elements of an atomic state
excited by circularly polarised light. [4] [5] Here
is the principal,
the angular momentum and
the magnetic quantum numbers. The polarisation vector of left and right circular polarised light can be rewritten in terms of spherical harmonics
leading to an expression for the transition matrix element
which can be simplified using the 3-j symbol:

The radial part is referred to as the line strength while the angular one contains symmetries from which selection rules can be deduced. Rewriting the product of three spherical harmonics with the 3-j symbol finally leads to: [4]
The 3-j symbols are not zero only if
satisfy the following conditions giving us the following selection rules for dipole transitions with circular polarised light: [4]


Derivation of sum rules for 3d and 4f systems
We will derive the XMCD sum rules from their original sources, as presented in works by Carra, Thole, Koenig, Sette, Altarelli, van der Laan, and Wang. [6] [7] [8] The following equations can be used to derive the actual magnetic moments associated with the states:

We employ the following approximation:

where
represents linear polarization,
right circular polarization, and
left circular polarization. This distinction is crucial, as experiments at beamlines typically utilize either left and right circular polarization or switch the field direction while maintaining the same circular polarization, or a combination of both.
The sum rules, as presented in the aforementioned references, are:

Here,
denotes the magnetic dipole tensor, c and l represent the initial and final orbital respectively (s,p,d,f,... = 0,1,2,3,...). The edges integrated within the measured signal are described by
, and n signifies the number of electrons in the final shell.
The magnetic orbital moment
, using the same sign conventions, can be expressed as:

For moment calculations, we use c=1 and l=2 for L2,3-edges, and c=2 and l=3 for M4,5-edges. Applying the earlier approximation, we can express the L2,3-edges as:

For 3d transitions,
is calculated as:

For 4f rare earth metals (M4,5-edges), using c=2 and l=3:

The calculation of
for 4f transitions is as follows:

When
is neglected, the term is commonly referred to as the effective spin
. By disregarding
and calculating the effective spin moment
, it becomes apparent that both the non-magnetic XAS component
and the number of electrons in the shell n appear in both equations. This allows for the calculation of the orbital to effective spin moment ratio using only the XMCD spectra.