X-ray magnetic circular dichroism

X-ray magnetic circular dichroism (XMCD) is a difference spectrum of two X-ray absorption spectra (XAS) taken in a magnetic field, one taken with left circularly polarized light, and one with right circularly polarized light.

[1] By closely analyzing the difference in the XMCD spectrum, information can be obtained on the magnetic properties of the atom, such as its spin and orbital magnetic moment.

Using XMCD magnetic moments below 10−5 μB can be observed.

[2] In the case of transition metals such as iron, cobalt, and nickel, the absorption spectra for XMCD are usually measured at the L-edge.

This corresponds to the process in the iron case: with iron, a 2p electron is excited to a 3d state by an X-ray of about 700 eV.

[3] Because the 3d electron states are the origin of the magnetic properties of the elements, the spectra contain information on the magnetic properties.

In rare-earth elements usually, the M4,5-edges are measured, corresponding to electron excitations from a 3d state to mostly 4f states.

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.

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

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]

satisfy the following conditions giving us the following selection rules for dipole transitions with circular polarised light:[4] 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:

represents linear 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:

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:

Applying the earlier approximation, we can express the L2,3-edges as:

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

is neglected, the term is commonly referred to as the effective spin

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.