6 to the unoccupied states with down-spin. This excitation may be considered as a core-hole plus d $^{9}$, since one of the quasi bound states, which are almost localized at the core-hole site, is empty. In this calculation, however, the states are not split off from the band bottom edge, indi- cating that the core-hole-dn-hole states are only weakly bound. The satellite binding energy of the calculated spectra is 1 eV smaller than that of the observed spec- tra. This discrepancy might be owing to the LDA. The Excitations of two e-h pairs would give rise to consider- able intensities in the energy range ω$_{q}$ − ϵ = 110 ∼ 120 eV. Excitations of three e-h pairs give rise to only small intensities. Table II. This could be understood as follows. Since the up-spin 3 d bands are almost fully occupied in the ground state, up-spin one-electron states constituting the final state | f$_{0}$ 〉 could be represented by a unitary transform of those constituting the ground state | g 〉 . Therefore, since the determinant is invariant under unitary transforma- tion, A$_{↑}$ ’s are close to unity. Final states | f$_{ν}$ 〉 ’s containing up -spin e-h pairs could give rise to only small intensities, since the states of the excited electrons with up-spin are almost orthogonal to occupied states with up-spin in the ground state | g 〉 , and thereby the overlap determinants would vanish. On the other hand, the final states | f$_{1}$ 〉 ’s containing one down - spin e-h pair could give rise to considerable intensities, since the corresponding one-electron wave functions con- tain the amplitudes of the unoccupied one-electron states in the ground state | g 〉 , and thereby the overlap deter- minants would not vanish. Considering various combina- tions of one e-h pair, we obtain intensities distributed in a wide range of binding energy. Figure 5: 3 s XPS spectra in ferromagnetic cobalt as a function of binding energy. (a) and (b) are for the up-spin and down- spin channels, respectively. The experimental data are taken from Ref. [10]. Figure 4: 3 s XPS spectra in ferromagnetic nickel as a function of binding energy. (a) and (b) are for the up-spin and down- spin channels, respectively. The experimental data are taken from Ref. [13]. 80 85 90 95 100 105 110 115 0.0 0.1 0.2 0.3 Intensity (arb.units) expt. total |f$_{0}$> dn |f$_{1}$> |f$_{2}$> |f$_{3}$> 80 85 90 95 100 105 110 115 Binding Energy (eV) 0.0 0.2 0.4 Intensity (arb.units) expt. total |f$_{0}$> dn |f$_{1}$> |f$_{2}$> a) up spin channel b) dn spin channel Co 3s XPS 95 100 105 110 115 120 125 0.0 0.1 0.2 Intensity (arb.units) expt. total |f$_{0}$> dn |f$_{1}$> |f$_{2}$> |f$_{3}$> 95 100 105 110 115 120 125 Binding Energy (eV) 0.0 0.1 0.2 Intensity (arb.units) expt. total |f$_{0}$> up |f$_{1}$> dn |f$_{1}$> |f$_{2}$> a) up spin channel b) dn spin channel Ni 3s XPS Figure 5(a) shows the calculated spectra for Co. The calculated spectra have a broad peak structure with the maximum intensity at the threshold around ω$_{q}$ − ϵ = 101 eV. They also have large shoulder intensities around ω$_{q}$ − ϵ = 104 eV. The former peak originates from the excitations with one and two e-h pairs. The contribution of the lowest-energy final state | f$_{0}$ 〉 (no e-h pair) is quite small due to small | A$_{↓}$ | $^{2}$. The latter shoulder originates from excitations of two and three e-h pairs, probably in- cluding the excitations from the quasi-bound states to the unoccupied states with down-spin. Figure 4(a) shows the calculated spectra for Ni in com- parison with experimental observations in the up-spin channel. The calculated spectra have maximum intensity at the threshold around ω$_{q}$ − ϵ = 110 eV and the signifi- cant satellite intensity around ω$_{q}$ − ϵ = 115 eV. Note that that the final states containing one down-spin e-h pair give rise to considerable intensities, extending over main and satellite regions. The satellite intensity corresponds to excitations of one e-h pair from the quasi-bound states