Correction of beam hardening in X-ray radiograms

The intensity of a monochromatic X-ray beam decreases exponentially with the distance it has traveled inside a material; this behavior is commonly referred to as Beer-Lambert's law. Knowledge of the material-specific attenuation coefficient m allows us to determine the thickness of a sample from the intensity decrease the beam has experienced. However, classical X-ray tubes emit a polychromatic bremsstrahlung-spectrum. And the attenuation coefficients of all materials depend on the photon energy: photons with high energy are attenuated less than photons with low energy. In consequence, the X-ray spectrum changes while traveling through the medium; due to the relative increase in high energy photons, this effect is called beam hardening. For this varying spectrum, the Beer-Lambert law only remains valid if m is replaced by an effective attenuation coefficient meff which depends not only on the material but also on its thickness x and the details of the X-ray setup used. We present here a way to deduce meff(x) from a small number of auxiliary measurements using a phenomenological model. This model can then be used to determine an unknown material thickness or in the case of a granular media its volume fraction.