Possibility of phase-matched nonlinear interaction of x-ray waves in ideal crystals

A dynamic theory is developed for the generation of a second x-ray harmonic in an ideal crystal characterized by a quadratic nonlinearity. New types of phase matching near the Bragg diffraction direction are predicted. Directions are found of one-dimensional phase matching in x-ray frequency doubling in accordance with the Laue and Bragg schemes. Calculations are made of the dependences of the amplitudes of second-harmonic waves on the deviation from the exact Bragg reflection direction and on the thickness of a crystal.