Combined Effects of Magnetic Structure and Local Crystal Fields in Anisotropic X-ray Anomalous Scattering

The X-ray extinction conditions are studied for the cases when both magnetic interaction and crystalline fields cause the anisotropy of X-ray scattering amplitudes near the absorption edge. It is demonstrated that the simultaneous existence of those two anisotropies, referred to as the combined anisotropy, can result in excitation of additional Bragg reflections otherwise forbidden by extinction rules. To show this, the structure amplitudes of reflections are computed and two Laue (L) groups are compared: one, L_m, which corresponds to the anomalous X-ray scattering in the presence of magnetic structure and another one, L_q, which takes into account the anisotropic atomic environment. Both groups can contain additional reflections and differ from the Laue group, L_p, associated with the usual potential X-ray scattering. The cross of these groups is considered and it is shown that several extinctions, typical for the L_p group, still remain in L_q and L_m. Nevertheless, the new reflections may appear instead of those extinctions when the combined anisotropy is taken into account. In this case, the diffraction pattern is characterized by the L_c group. Then, the transformation of L_q into L_c under the phase transition is studied. It is shown that the additional reflections, not typical for L_m, can appear in the diffraction pattern below the magnetic ordering temperature T_M. Two cases are discussed: (i) L_q above T_M contains the same reflections as L_c and (ii) L_q contains the extinctions corresponding to the additional reflections in L_c. Several examples are considered for real magnetic crystals.