Theory and Measurement of the X-Ray Satellite Reflections in Holmium Due to the Aspherical4fCharge Density

The theory and experimental measurement of the satellite reflections appearing about the normal Laue-Bragg reflections in holmium are discussed. These reflections arise from the asphericity of the $4f$ charge density as distinct from the core charge density. In holmium metal the $5d^1$ and $6s^2$ valence electrons are believed to enter the conduction bands, leaving a tripositive ion core approximated by the configuration ${}_{}{}^5{}_{}{}^{}I_8^{}{}_{}{}^{}$. The $4f$ shell lacking four electrons is screened from the crystalline environment by the filled $5s^2$ and $5p^6$ shells. Because of the unpaired electrons in the $4f$ shell, the ions have a magnetic moment which between 20 and 132°K tends to align them in a flat spiral with a propagation vector $\mathbf{\tau }$ lying along the hexagonal axis of the crystal. This tendency of the ions to order into such a spiral, coupled with the aspherical charge density, induces periodicities in the scattering of the structure containing the first three even harmonics of $\mathbf{\tau }$. These periodicities produce x-ray satellite reflections about all allowed reflections except ($00\overline{0}·L$) at ±2, ±4, and $\pm 6\mathbf{\tau }$. At a nonzero temperature the ions are disordered to the extent that they are distributed amongst states with magnetic quantum number $M$ and inner quantum number $J=8\mathrm{}$. Using the molecular-field approximation suggested by Nagamiya to describe this disorder, it was found that past observation of neutron satellite intensities of Koehler et al. were explained. The theory was then used to calculate the average x-ray scattering factor from the scattering factors for the various ionic states $M$ whose computation rests upon parameters derived by Blume et al. from recently determined Hartree-Fock wave functions. The agreement with experimentally observed values of the scattering of the first satellite pair about the ($22\overline{4}·0$) reflection discussed in the paper is excellent.