Physics-based basis functions for low-dimensional representation of the refractive index in the high energy limit

The relationship between the refractive index decrement, $\delta$, and the real part of the atomic form factor, $f^{\mathrm{\prime }}$, is used to derive a simple polynomial functional form for $\delta (E)$ far from the K-edge of the element. The functional form, motivated by the underlying physics, follows an infinite power sum, with most of the energy dependence captured by a single term, $1\mathrm{/}{E}^2$. The derived functional form shows excellent agreement with theoretical and experimentally recorded values. This work helps reduce the dimensionality of the refractive index across the energy range of x-ray radiation for efficient forward modeling and formulation of a well-posed inverse problem in propagation-based polychromatic phase-contrast computed tomography.