Theory of X-Ray Diffraction by a Vibrating Crystal

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Abstract
The scattering amplitude for x‐ray diffraction by a vibrating crystal is obtained. The Bragg diffracted propagators for the perfect crystal are replaced by new propagators in which the Fourier transform of the polarizability of the crystal, v(H), is multiplied by the Bessel function of zero order, J0(H·Aq), of the product of the vibrational amplitude Aq and a reciprocal lattice vector H, in a similar fashion to the correction by the Debye‐Waller factor in the thermal problem. The scattering amplitude contains both dynamical and kinematical contribution. The case of ideal thickness vibration is studied in detail in the Laue geometry. The real part of the perturbed scattering amplitude is proportional to v(H) sin (H·Aq). This theory elucidates the transition from dynamical to kinematical scattering, that is, a decrease in extinction, as a function of the vibration amplitude.