Three-beam diffraction in a finite perfect crystal

By using the Laplace transform on the Takagi-Taupin equations for three coupled waves in a perfect crystal it has been possible to obtain general boundary-value Green functions for the wave fields D_o, D_h and D_g. For a crystal shaped as a parallelepiped the integrated power P_h is calculated in the kinematical limit by suitable integrations over one divergence angle and over the entrance and exit surfaces. The result, which is expressed as a function of the deviation from the Bragg condition for the third wave, is continuous through the three-beam point, and gives the expected asymmetry associated with the invariant phase of the product of the three structure factors involved. The asymptotic behaviour is the same as that obtained from pseudo-two-beam formulations based on standard plane-wave theories. In the expression for the integrated power the dimensions of the crystal, scaled to appropriate extinction lengths, occur as parameters. The movement of the reciprocal-lattice point g owing to the rotation of the crystal when P_h is to be measured is taken explicitly into account. When this movement is negligible or small, it is found that the diffracted power in the vicinity of the three-beam point shows oscillations due to a functional dependence corresponding to the Laue interference function. Both Umweganregung and Aufhellung situations are covered.