Quantum theory of X-ray reflection and scattering

Summary

When X-rays fall upon a crystal, the characteristic vibrations of the crystal lattice may be excited thereby, in much the same way as in the phenomenon of the scattering of light in crystals with diminished frequencey, the excitation being a quantum mechanical effect. From the equations for the conservation of energy and momentum, the geometrical relations entering in this effect are deduced theoretically for the two cases in which the lattice vibrations fall within (1) the acoustic range of frequency and (2) the optical range. In the first case, the incident X-rays are scattered in directions falling within, a cone having the incident ray as axis and with a semi-vertical angle 2 sin⁻¹ λ 2γ ^* whereγ ^* is the minimum acoustical wave-length. In the second case, we have a quantum-mechanical reflection of the X-rays with diminished frequency in a direction which generally follows the geometric formula 2d sin 1/2(θ+ϕ)=nλ where θ and ϕ are the glancing angles of incidence and reflection on the crystal spacings. For crystals with specially rigid bindings, the alternative fomulad sin (θ+ϕ)=nλ cosϕ is indicated as being more appropriate. In either case, the intensity of the reflection should fall off rapidly as ϕ and ϕ diverge.