Module exposure 

Multi-shot exposure: the energy density deposited by a list of shots, from a fitted Gaussian-sum point-spread function (GaussianPsf).

The model

A shot is a beam landing at (x, y) with a weight (the number of primaries, or any relative dwell) and an optional spot shape and beam blur. The deposited energy density at a point is the sum over shots of the shot’s weight times the PSF, E f(r) (eV/m²), centred on the shot, where E is GaussianPsf::total_ev, the energy one primary deposits in the film. The PSF is the sum of Gaussians documented in crate::tally::psf:

f(r) = 1/(π Σc) Σ_t (c_t / w_t²) exp(-r²/w_t²)

with (w_t, c_t) = (α, 1), (β, η) and, for the triple form, (γ, ν).

Beam blur. A Gaussian beam profile of width b in the same convention, exp(-r²/b²)/(π b²), is convolved with the PSF. The convolution of two normalised 2D Gaussians is the normalised Gaussian whose w² is the sum (the variances of independent Gaussians add; here w² = 2σ², so the w² add too):

∫ exp(-|r-s|²/w²)/(π w²) · exp(-s²/b²)/(π b²) d²s = exp(-r²/(w²+b²))/(π (w²+b²))

This is textbook (the convolution theorem for Gaussians; it follows from completing the square in the exponent), so a blurred point shot needs no numerical convolution: every term’s w_t² becomes w_t² + b², and the weights c_t and the normalisation are unchanged. The blur of a shot is b² = b_global² + b_shot².

Shaped spots. A uniform disc or rectangle is a fixed sub-sampling into point shots: the bounding box of the spot is cut into n × n equal cells (n is Exposure::subsamples, default DEFAULT_SUBSAMPLES); each cell centre inside the spot becomes a point shot with an equal share of the weight (the shares sum to the shot’s weight). The rectangle is the midpoint rule on the exact shape, so its error is O(h²) in the cell pitch h relative to the narrowest Gaussian width sqrt(w_t² + b²); it is accurate when h is below that width. A disc has the same quadrature error and, in addition, the boundary of the sampled cells approximates the circle to O(h) in area; raise subsamples for a spot that is large compared with the PSF’s α.

Determinism

The dose at one point sums the point sources in list order (the shots in the order given, each shot’s sub-samples in row-major order). A grid parallelises over grid points only, so each point’s sum is the same fixed sequence of floating-point operations: the result is bit-identical at any thread count.

Accuracy

The dose is only as good as the PSF it uses. A fit that does not describe the simulated profile (see the reduced χ² of PsfFit) gives a dose that does not describe the simulated exposure either; the fit’s validation is tracked separately (issue #151). Superposition also assumes the film is laterally uniform.

Direct Monte Carlo exposure

direct_dose evaluates the same shot list by transport instead of by superposition, which makes no assumption of a laterally uniform film. For each shot the electron engine (Transport::run_beam) runs primaries from the shot position with the shot’s spot shape, and the energy deposited in the film slab depth_lo <= x < depth_hi is tallied per lateral cell of the evaluation Grid (one voxel per grid point, the point at its centre, through VoxelTallyGrid). The shot’s weight scales the per-primary deposit, so the energy density at a point is

D(p) = Σ_shots w_s · (E_s(p) / N_s) / (dx · dy)

with E_s(p) the energy deposited in the cell of p by the N_s primaries of shot s. This is the cell average of the density that Exposure::dose_at gives at the cell centre; the two agree when the dose varies little over a cell (choose dx, dy below the narrowest PSF width).

Frame. Exposure coordinates (x, y) are offsets from the geometry’s entry point (the centre of the front face of a voxel grid), as for BeamSpread::landing_m: exposure x is the geometry’s y and exposure y its z. Depth is the geometry’s x.

Spots. A point shot is a Gaussian beam of the blur width (b in exp(-r²/b²), i.e. σ = b/√2 per axis); a disc or rectangle is the uniform profile of the same shape. A shaped spot with a nonzero blur is rejected: the engine’s profiles do not convolve a shape with a Gaussian.

Statistical error. The run is made in R independent replicas (DirectConfig::replicas), each with its own seeds and histories_per_shot / R primaries per shot. The reported dose at a point is the mean of the R replica doses and its standard error sqrt(s² / R), with s² the sample variance of the replica doses (Bessel-corrected). The error is statistical only (Monte Carlo noise); the cross-section tables’ own uncertainty is not included.

Determinism. Replica r and shot s run with seed point_seed(point_seed(seed, r), s) (a SplitMix64 mix, see crate::tally::psf_scan), runs are made one after the other, each is bit-identical at any thread count, and the sums over shots and replicas are taken in index order, so the result does not depend on the thread count.

Comparison with the convolution. compare_with_convolution puts the two results side by side per point (the convolution averaged over the same cell, as the tally measures a cell average) and flags the points where they differ by more than a stated number of combined standard errors, sqrt(σ_direct² + σ_conv²). The superposition has no sampling error of its own; σ_conv is the stated relative uncertainty of the fitted PSF times the convolution dose (0 if none is given).

Structs

AreaDose

The dose far inside a large uniform exposure, split by PSF term.

DirectConfig

Settings of direct_dose.

DirectDose

The energy density of a shot list by transport, with its statistical error, on a Grid.

DoseGrid

The dose on a Grid, eV/m², row-major.

Exposure

A list of shots on a PSF, ready to evaluate.

Grid

A regular grid of points: point (i, j) is at (x0 + i dx, y0 + j dy), stored row-major (j * nx + i).

PointComparison

One evaluation point of compare_with_convolution.

Shot

One shot.

Enums

ExposureError

Errors of building or reading an exposure.

SpotShape

The shape of a shot’s spot.

Constants

DEFAULT_SUBSAMPLES

Default sub-sampling per axis of a shaped spot.

Functions

compare_with_convolution

The direct and convolution doses side by side at every point of direct.grid; a point is flagged when the two differ by more than tolerance_sigmas combined standard errors (see the module docs). conv_rel_err is the relative uncertainty assigned to the convolution.

comparison_csv

CSV of a comparison: x_m,y_m,convolution_ev_per_m2,direct_ev_per_m2, combined_std_err_ev_per_m2,deviation_sigmas,flagged.

direct_dose

The energy density of shots on grid by electron transport; see the module docs (“Direct Monte Carlo exposure”) for the model, frame and error. global_blur_m is the exposure’s beam blur b.

parse_shots_csv

Reads a shot list.

uniform_area_dose

The area dose of a uniform exposure of weight_per_m2 primaries per m² over an area much larger than the PSF’s longest width, with the forward-only part reported next to the total.