Section 6.4.4. Radiation flow

The radiation flow is governed by the Hamilton–Darwin (H–D) equations (Darwin, 1922; Hamilton, 1957). These equations are Here, is the radiation current density (m⁻² s⁻¹) in the incident (i = initial) beam, is the current density in the diffracted (f = final) beam. The distances and are measured along the incident and diffracted beams, respectively. The coupling constant σ is the cross section per unit volume for scattering into a single Bragg reflection, while τ, which is always negative, is the cross section per unit volume for removal of radiation from the beams by any process. In what follows, it will be assumed that absorption is the only significant process, and τ is given by τ = −(σ + μ), where μ is the linear absorption coefficient (absorption cross section per unit volume). This assumption may not be true for neutron diffraction, in which incoherent scattering may have a significant role in removing radiation. In those cases, τ should include the incoherent scattering cross section per unit volume.

The H–D equations have analytical solutions in the Laue case (2θ = 0) and the Bragg case (2θ = π). The solutions at the exit surface are, respectively, and with . The path length of the diffracted beam through the crystal is D. The current density at the entrance surface is

To find formulae for the integrated intensity, it is necessary to express σ in terms of crystallographic quantities.