A beam of X-rays travels through a slab of material with linear attenuation coefficient

PHYSICS · Subjective · Fermi University Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems · Monte Carlo and lattice simulation of physical systems

Computational Physics, Ch Drawing numbers: random walks and Monte Carlo

Problem

A beam of X-rays travels through a slab of material with linear attenuation coefficient μ=0.15 mm−1\mu = 0.15\,\mathrm{mm^{-1}}. The distance xx a photon travels before its first interaction has probability density P(x)=μe−μxP(x) = \mu e^{-\mu x} for x≥0x \ge 0. (a) Derive the cumulative distribution function F(x)F(x). (b) Derive the inverse-transform rule x=F−1(u)x = F^{-1}(u) for drawing an interaction distance from a uniform draw u∈[0,1)u \in [0,1). (c) A particular photon corresponds to the drawn value u=0.60u = 0.60. Find its sampled path length, to three significant figures. (d) State the mean interaction distance predicted by this density, and say in words why a single draw such as the one in part (c) need not equal it.

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