Analytical modeling of a proton recoil telescope

Neutron spectrometry in fusion diagnostics requires precise knowledge of the Proton Recoil Telescope (PRT) response functions. We present a comprehensive analytical model, extending a lesser-known approach (published in German), that computes response functions and detection efficiencies for a wide range of geometrical configurations, including extended, off-axis, and asymmetric neutron sources. The model successfully recovers Gotoh's integral formulation for axially symmetric, thin-radiator configurations with parallel beams, validating our broader framework. We derive, for the first time, a closed-form analytical solution for the equal-radius, on-axis geometry with proton energy loss using elliptic integrals. This result eliminates the need for numerical integration of Gotoh's equations and provides immediate computational access to exact response functions. Asymptotic approximations for varying source-detector distances yield simplified expressions suitable for practical visualization and rapid system optimization. The framework incorporates proton energy loss through Bragg-Kleeman range-energy relations in flat, uniform radiator surfaces and has been validated against full numerical integration. This analytical approach enables efficient calculation of response functions for any combination of source geometry, radiator configuration, and detector arrangement, offering computational advantages that are several orders of magnitude faster than Monte Carlo simulations and making it particularly valuable for iterative experimental design optimization and real-time diagnostic applications.