Quasi-elastic resonant x-ray scattering

Abstract

In the fast collision approximation, the scattering amplitude operator of the quasi-elastic scattering is expressed as the summation of multipole moment operators $M\sp{(k)}(l\sb{i},s\sb{i})$ of the valence shell involved in the resonance$\sp1$ with distinct polarization factors. Each multipole moment operator is expressed as the sum of an orbital moment operator and two spin-orbital moment operators with unique coefficients. The explicit form of these coefficients is obtained and the numerical values are calculated. For the transitions to continuous bands, the explicit forms of $M\sp{(k)}(l\sb{i},s\sb{i})$ are extended from electric dipole transitions to any electric multipole transitions. Within the manifolds of good total L and good total S, the $k\sp{\rm th}$ rank multipole moment operator $M\sp{(k)}(l\sb{i},s\sb{i})$ can be expressed in terms of the $k\sp{\rm th}$ rank spin-orbital moments $M\sp{(k)}({\bf L,S})$ of the total L- and total ${\bf S}$-operators of the valence shell involved in the resonance. Furthermore, within the manifolds of good total J, $M\sp{(k)}(l\sb{i},s\sb{i})$ can be further simplified in terms of the spherical tensor operators of the total J of the resonance valence shell. For Hund's rule ground states, the corresponding proportionality coefficients for both cases were obtained. For rare earths, we obtained the thermal expectation value of $M\sp{(k)}(l\sb{i},s\sb{i})$ at T = 0 for coherent elastic scattering. These results are inconsistent with Hamrick's single electron method$\sp2$ for the second half of the rare earth series. For the first half of the rare earth series, we showed that the single electron method is an approximation of our theory. In spiral antiferromagnets, such as holmium, the magnetic sensitivity results in a series of magnetic satellites distributed at each side of Bragg peak. This behavior can be understood on the basis of the XRES electric multipole transition theory we developed. As temperature increases, the higher order harmonics decrease more rapidly than the lower order harmonics, which can be qualitatively explained by mean-field theory. Just above the Neel temperature, there is weak magnetic scattering which can be understood as the short range moment-moment correlations of different spin-orbital multipole moment operators. ftn $\sp1$J. Luo, J. P. Hannon, G. T. Trammell, Phys. Rev. Lett., 71 287 (1993). $\sp2$M. Hamrick, M.A. Thesis, Physics Department, Rice University, 1991.