Analytical-kernel B-spline unfolding of fast-neutron spectra from a proton recoil telescope: a deterministic alternative to Monte Carlo response matrices

Unfolding the neutron energy spectrum from proton-recoil data requires a detector response matrix, which has conventionally been obtained by Monte Carlo particle transport — a step that is computationally costly and injects statistical noise into the inversion. We remove that step. Retaining the B-spline expansion of the neutron spectrum introduced for organic-scintillator unfolding, we instead build the response matrix of a proton recoil telescope (PRT) from a recently derived closed-form analytical response, which expresses the equal-radius, on-axis response in terms of elliptic integrals and includes proton energy loss in the converter. The result is a fully deterministic pipeline: the response matrix follows directly from closed-form expressions, with no Monte Carlo sampling at any stage, free of statistical fluctuations and reproducible to machine precision. The inversion is regularized by a second-derivative Tikhonov penalty with automatic L-curve selection of the regularization parameter, non-negativity is enforced exactly by bounded-variable least squares, and Poisson counting statistics are accounted for through a weighted (whitened) least-squares formulation. A first independent, out-of-model validation is provided: proton-recoil spectra generated by Geant4 transport at two neutron energies — 2.45 MeV (D–D), for five radiator thicknesses between 5 and 100 μm, and 14.1 MeV (D–T) — are unfolded with the analytical matrix and recover the neutron lines to within ≤4% of the true energies, without forward and inverse operators matched by construction; a joint two-line unfolding resolves both lines simultaneously. Matched-operator synthetic benchmarks, which characterize the inversion rather than the physics of the kernel, localize well-separated peaks to better than 0.15 MeV and yield bootstrap confidence intervals; peaks overlapping within their widths are not resolved. The complete pipeline runs in seconds — orders of magnitude faster than a direct Monte Carlo construction of an equivalent response matrix — making it well-suited to iterative detector design and rapid-turnaround neutron diagnostics.