Sum rules are derived for the circular dichroic response of a core line (CMXD). They relate the intensity of the CMXD signal to the ground-state expectation value of the magnetic field operators (orbital, spin, and magnetic dipole) of the valence electrons. The results obtained are discussed and tested for transition metals and rare earths. PACS numbers: 78.70.Dm, 78.20.Ls For circular dichroism in the x-ray region (CMXD), Thole et al [I] ha. ve recently derived a new magneto-optical sum rule. It shows that, to a good approximation, the intensity of the CMXD signal, integrated over a complete core-level edge of a ferromagnet (or ferrimagnet), is proportional to the ground-state expectation value of the orbital angular momentum operator L, . The derivation was carried out for electric dipole transitions in a localized model, considering a single ion in an arbitrary crystal-field symmetry and including hybridization effects.In this Letter we show that, within the same framework, another sum rule can be obtained. It relates the CMXD signal, integrated over a single partner of a spinorbit-split core-level edge, to the ground-state expectation value of the operators (L"total spin S"and magnetic di-(+~r C~' ( W' jm ) = g (+ ) c/t !i ]l+j 'm )(cjm~Cq ' (l!i )R, t = X,&+lc, t 4~,. I+'& pole [g;s; -3r";(r"; s;)l, ) that describe the magnetic field generated by the valence electrons. Our results indicate that, besides (L, ), as described in Ref.[I], CMXD spectroscopy can provide an independent determination of the ground-state expectation value of S, [2]; this has been tested using CMXD data, taken at the L23 edges of the ferromagnetic metals Fe, Co, and Ni [3]. Furthermore, valuable, site-specific information on the magnetic anisotropy of the sample can be obtained, as discussed below. We consider the electric dipole transitions of a single partner of spin-orbit-split edge, in an ion with the valence shell only partly filled. Let~+) denote any state of the ground configuration l" of the ion. The final-state configuration is represented by~+ 'jm) =~cj~l"+'(%')); here +' denotes any state of the outer shell l"+' and cj stands for a hole in a core level. The dipole matrix element is given by I/2 c j c I I x ( -) r '/ [ ] I/2Po. y -m -yqT he notation is as follows: cj and l~represent creation operators for core and valence electrons, respectively; Cq . . (]) denotes a normalized spherical harmonic; [j] =2j+ I, R,t stands for the radial matrix element of the c 1 dipole transition; and P,t =(c~~C '~~l )R,t. The total intensity of the j edge is expressed by r '4l 44'

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