Effective structure factors in many-beam X-ray diffraction – use of the second Bethe approximation

An analytical expression for the effective structure factor, F^{n, p}_h0, generally valid for each polarization direction p in the many-beam X-ray case, is derived. The second Bethe approximation is utilized with the additional assumption that certain small σ.π-type terms may be neglected. It is shown that the many-beam effects observed in both plane-wave- and integrated-intensity-type experiments may be described by the existing standard two-beam expressions provided the effective structure factor is introduced. Examples based on three- and four-beam interactions are given with focus on the absorption-independent effects. In particular, the deviation of F^{3, p}_h0F^{3, p}_0h in the general three-beam case from the corresponding two-beam value is shown to depend on the three-phase structure invariant and the deviation parameter of the coupled beam. The results which agree with full three-beam calculations may be applied to determine invariants experimentally from any type of three-beam experiment in principle. Applications to Pendellösung and mosaic-crystal experiments are briefly discussed. The accuracy of the method is evaluated from calculated dispersion surfaces in three- and four-beam examples.