Compton and Rayleigh double scattering of unpolarized radiation

Analytical expressions describing double-scattering intensities of the Compton and Rayleigh effects (Compton-Compton, Compton-Rayleigh, Rayleigh-Compton, and Rayleigh-Rayleigh contributions), are deduced in the framework of the transport theory for an infinitely thick sample irradiated with collimated monochromatic radiation. An orders-of-interaction solution of the integro-differential Boltzmann equation for unpolarized photons is used to separate the multiple-order terms. Interaction kernels for coherent and incoherent scattering include atomic form factors describing the effect of the electronic distribution in multielectron atoms. The total attenuation coefficient of the target takes into account, besides the mentioned scattering processes, the photoelectric effect, important in the x-ray regime. First-order Compton and Rayleigh interactions give monochromatic peaks according to the theoretical model that neglects electron motion. In contrast, Compton-Compton, Compton-Rayleigh, and Rayleigh-Compton contribute asymmetric continuous spectra whose wavelength breadths are the DuMond width and 2${\lambda }_{\mathit{C}}$ (${\lambda }_{\mathit{C}}$ is the Compton wavelength), respectively. Single- and double-scattering intensities of the Rayleigh and Compton effects are computed for pure and composite materials as a function of the excitation energy and the angular orientations of the incident and take-off beams. Since absorption in the target is considered, computations can be straightforwardly compared with experimental data and with realistic Monte Carlo simulations. The agreement is good for low excitation energies because the second-order term remains dominant in multiple scattering and bremsstrahlung emission is weaker. However, for higher excitation energies the probability of higher orders of multiple scattering increases, and they cannot be neglected. Although analytical calculations are performed up to the second order in this work, a Monte Carlo simulation is used to show the importance of higher orders in light elements.