International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 6.4, p. 610

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.
References
Darwin, C. G. (1922). The reflexion of Xrays from imperfect crystals. Philos. Mag. 43, 800–829.Hamilton, W. C. (1957). The effect of crystal shape and setting on secondary extinction. Acta Cryst. 10, 629–634.