Beam hardening: analytical considerations of the effective attenuation coefficient of X-ray tomography

Polychromatic x-ray beams traveling though material are prone to beam hardening, i.e., the high energy part of the incident spectrum gets over represented when traveling farther into the material. This study discusses the concept of a mean attenuation coefficient in a formal way. The total energy fluence is one-to-one related to the traveled distance in case of a polychromatic beam moving through a given, inhomogeneous material. On the basis of this one-to-one relation, it is useful to define a mean attenuation coefficient and study its decrease with depth. Our results are based on a novel parametrization of the energy dependence of the attenuation coefficient that allows for closed form evaluation of certain spectral integrals. This approach underpins the ad hoc semianalytical expressions given in the literature. An analytical model for the average attenuation coefficient is proposed that uses a simple fit of the attenuation coefficient as a function of the photon energy as input. It is shown that a simple extension of this model gives a rather good description of beam hardening for x-rays traveling through water.