Module thickness 

Transmission of a sample whose areal density varies over the pixels a count sums.

A region’s pixels see the areal density x·n, x distributed with mean 1 and relative variance v over the beam that reaches them. Their summed counts transmit the mean ⟨exp(−x·D)⟩, D = Σ n σ the optical depth at the mean density. For x Gamma distributed the mean is

T(D, v) = (1 + v·D)^(−1/v) = exp(−D · ln(1 + v·D)/(v·D))

exact for that distribution only. At v = 0 it is exp(−D).

Functions

interacting_per_depth

q = (1 − T)/D, the fraction of the beam that interacts per unit optical depth, with ∂q/∂D and ∂q/∂v, at optical depth depth (D, 0 or more) and relative variance variance (v, 0 or more): Σ n σ_x · q is the chance that a neutron’s first collision in the sample is of kind x. q is 1 at D = 0.

uneven_transmission

The mean transmission T, −∂T/∂D and ∂T/∂v at optical depth depth (D, 0 or more) and relative variance variance (v, 0 or more) of the areal density.