Spin-polarized energy-band structure, conduction-electron polarization, spin densities, and the neutron magnetic form factor of ferromagnetic gadolinium

Conduction-electron polarization, spin densities, and neutron magnetic scattering in ferromagnetic Gd metal were studied using the spin-polarized augmented-plane-wave (APW) method in a warped-muffin-tin-potential formulation. The spin-up and spin-down bands were found to be very similar in shape to the bands from a paramagnetic calculation, with the exchange splitting proportional to the amount of $d$ character in the bands. It was also found that the conduction-electron spin density determined from the APW wave functions is of mostly $d$ character. This dominance of the $d$-like wave functions for the spin-dependent interactions is explained by (i) the much greater overlap of the $4f$ states with the $d$-like wave functions as compared to the $s\ensuremath{-}p$ wave functions; (ii) the nearly complete $d$ character of the bands in the region of the Fermi surface. The magnetic form factor was calculated from the conduction-electron spin density and compared with the recent neutron magnetic - form - factor measurement of Moon, Koehler, Cable, and Child. The calculated spin density was found to have the same shape as the "diffuse" density derived by Moon et al. (including a negative but much smaller in magnitude spin density at the $c$ site in the unit cell). After the inclusion of core - polarization effects we conclude that large nonspherical contributions with ${Y}_{33}\ensuremath{-}{Y}_{3\ensuremath{-}3}$, and ${Y}_{40}$ angular dependence are needed to explain the experimental results.

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