X-ray spin form factors of rare-earth ions

The x-ray spin form factor obtained by the nonresonant magnetic diffraction where the magnetic moment in the target material is parallel to the scattering vector, being a unique microscopic approach to the aspherical density distribution of scatterers, has been theoretically investigated for the $4f$ electrons of the rare-earth ions. Examining the contribution from the lowest-order asphericity with the operator-equivalent technique, it is shown that the aspherical effect is relatively serious for the ions with small total-spin quantum numbers, such as ${\mathrm{Ce}}^{3+},$ ${\mathrm{Pr}}^{3+},$ ${\mathrm{Tm}}^{3+},$ and ${\mathrm{Yb}}^{3+},$ and it is illustrated that the relation of the expanse of the present spin form factor and the aspherical distortion of the $4f$ charge density along the moment direction is inverse between the less-than-half and more-than-half cases. It is also shown that, while the recent experiment on ${\mathrm{SmAl}}_2$ appears to support the factorization of the relevant operators into the spatial and spin parts in calculating the thermal averages, the reliability of such a treatment could be tested by studying the thermal variation of the form factors for various rare earths or, if detectable, measuring the form factor of ${\mathrm{Eu}}^{3+}.$ It is emphasized through the paper that the x-ray magnetic diffraction of this specific geometry is a limited but hopeful way to study the spatial distribution of spin polarization in magnetic materials and could be complementary to the neutron-diffraction and Compton-scattering methods.