A new magneto-optical sum rule is derived for circular magnetic dichroism in the x-ray region (CMXD). The integral of the CMXD signal over a given edge allows one to determine the ground-state expectation value of the orbital angular momentum. Applications are discussed to transition-metal and rare-earth magnetic systems.PACS numbers: 78.70.Dm, 78.20.Ls The orbital part of the magnetic moment, pL, in magnetic materials is determined by the interplay among several effects: Coulomb and spin-orbit interactions, hybridization, and crystal fields. To study these effects, the independent determination of pt and ps (the spin part) is of prime importance.In neutron scattering, the two contributions can be separated by fitting the measured form factors with a suitable model [I]. In nonresonant x-ray diffraction, as shown by a theoretical analysis [2,3], different polarization responses directly separate spin and orbital densities; however, all attempts to implement a quantitative separation experimentally have been, so far, inconclusive [4].In this Letter we show that, to a good approximation, it is possible to measure directly the ground-state expectation value of the orbital angular momentum operator L. -by core-level absorption spectroscopy. This is achieved by considering the difference between the integrated absorption intensity for right and left circularly polarized t light. This integral of the circular magnetic x-ray dichroism (CMXD) has to be taken over a complete corelevel edge of magnetically oriented ferromagnetic or ferrimagnetic materials. If the edge is spin-orbit split, the integration must be over the two components. Our results agree well with the available experimental data, such as those obtained at the L2 3 edges of ferromagnetic Ni [5], and the M45 edges of Gd + in the gadolinium iron garnet [6].The strong final-state interactions of the valence shell with the core hole normally allow one to draw only indirect conclusions about the ground state from core-level spectra. However, the importance of the integrated CMXD is that it directly measures a ground-state property.The importance of sum rules is well known in optical spectroscopy, and has often been used to derive nontrivial ground-state properties, such as the number of electrons participating in a band of optical transitions, the plasma frequency, etc. In magneto-optics, the following sum rule was derived by Smith for zero external magnetic field [7]: g(f p -f p) =-(aIL, Ia)+ 2 (aIS, (XV"V+YVrV)Ia) p & 2mc' with f,t i the oscillator strengths for left and right circularly polarized light. The second term in the right-hand side arises from the spin-orbit interaction in the valence shell; order-of-magnitude estimates, based on hydrogenic wave functions, show that it gives a very small correction to the first term. Expression (I), derived by considering E1 transitions only, is formally exact; nevertheless, it has a limited degree of application, as it implies a sum over an infinite number of transitions.
