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. 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.

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. 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.

Line intensities and selection rules

The line intensities and selection rules of XMCD can be understood by considering the transition matrix elements excited by circularly polarised light. the angular momentum the magnetic quantum numbers. The polarisation vector of left and right circular polarised light can be rewritten in terms of spherical harmonicsleading 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:The 3-j symbols satisfy the following conditions giving us the following selection rules for dipole transitions with circular polarised light:

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. The following equations can be used to derive the actual magnetic moments associated with the states:

We employ the following approximation:

represents linear polarization 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:

denotes the magnetic dipole tensor, c and l represent the initial and final orbital respectively (s,p,d,f,..., and n signifies the number of electrons in the final shell.

, 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:

is calculated as:

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

for 4f transitions is as follows:

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.

See also

• EMCD

• Faraday effect

• Magnetic circular dichroism

• Magnetic field

• Transition metals

References