Improvement of the Geometric-Progression Formula for Approximating Gamma-Ray Buildup Factors Up to Depths of 100 mfp

The geometric-progression (G-P) formula can accurately reproduce buildup factor data up to depths of 40 mean free paths (mfp) within a few percent. This formula was improved to apply to depths up to 100 mfp, using the buildup factor data of Shimizu et al. (2004) calculated with the Invariant Embedding method.

The behavior of the K parameter as a function of distance was examined, and a new formula was introduced from the depth of X_m ([approximately]40 mfp). The fitting parameters were determined using a minimizing procedure of the maximum fractional deviation (MMD). Within some sets of parameters determined by the MMD fit, one set of parameters was selected that realized the interpolation of the buildup factor with regard to energy, using interpolated G-P parameters. Consequently, discrete buildup factor data were converted to continuous data with regard to both energy and distance.