4 -8 -6 -4 -2 0 2 4 Energy (eV) 4 2 0 2 4 DOS (states/eV) c site 3d up spin dn spin (a) fcc Ni -8 -6 -4 -2 0 2 4 Energy (eV) 4 2 0 2 4 DOS (states/eV) c site 3d up spin dn spin (b) -8 -6 -4 -2 0 2 4 Energy (eV) 4 2 0 2 4 DOS (states/eV) c site 3d up spin dn spin (c) Figure 2: Calculated d -DOS at the core-hole site in the super- cell system in ferromagnetic nickel; (a) d -DOS with no core- hole, (b) d -DOS when the 3 s up-spin electron is removed, (c) d -DOS when the 3 s down-spin electron is removed. The Fe metal takes a bcc structure. Figures 2 and 3 show DOS’s projected onto the states with d symmetry ( d -DOS) at the core-hole site for Ni and Co, respectively. The corresponding DOS’s for Fe are shown in Fig. 3 in Ref. 19. In these calculations, six k -points are picked up in the irreducible Brillouin zone for super-cell sys- tems. The DOS’s calculated with no core-hole are essen- tially the same as those reported by Moruzzi, Janak and Williams. 25 Table I lists the screening electron number Δ n$_{dσ}$ in the d -symmetric states with spin σ , that is, the difference of the occupied electron number between in the presence and in the absence of the core hole inside the muffin-tin sphere. On the basis of these one-electron states, we calcu- late the 3 s XPS spectra, by following the procedure de- scribe in Sec. II. Figures 4, 5 and 6 are the spectra thus calculated as a function of the binding energy ω$_{q}$ − ϵ for Ni, Co, and Fe, respectively, in comparison with the experiments. 10,11,13 The spectral shape in Fig. 6 for Fe is slightly different from our previous result (Fig. 4 in Ref. 19), since the present calculation takes full account of ex- citations up to three e-h pairs in comparison with only up to two e-h pairs in Ref. 19. The spectra are strikingly different between the up-spin channel and the down-spin channel and strongly depend on elements, in good agree- ment with the experiments. In the following, we explain the origin of these behaviors in relation to one-electron states screening the core hole. Table I: Screening electron number in the d -symmetric states inside the muffin-tin sphere at the 3 s core-hole site. The radii of the muffin-tin spheres are 2 . 0 Bohr. 3s hole spin Δ n $^{d}$↑ Δ n $^{d}$↓ Δ n $^{d}$↑ + Δ n $^{d}$↓ Fe up -1.44 2.38 0.94 dn 0.47 0.48 0.95 Co up -0.52 1.55 1.03 dn 0.24 0.75 0.99 Ni up 0.07 0.87 0.94 dn 0.26 0.65 0.91