Analysis of the Numerical Methods for the Unfolding of Beta Spectra Obtained by Integral Detectors

Abstract

The beta spectrum, obtained by solid state or scintillation spectrometers, cannot be considered the "true" spectrum, due to the resolution and backscattering characteristics of the detectors. The "true" shape can be obtained from the experimental one by solving a Fredholm's integral equation of the first kind, having as kernel, the response function of the spectrometer. From the mathematical point of view this is an "incorrectly posed problem". A critical analysis on some usual numerical methods is presented, under the assumption of uniqueness of the solution. The replacement of the equation with a linear algebraic system is obtained by some quadrature formulas. Despite the formula used, the system is essentially illconditioned. The most currently used methods for linear system solutions are examined. The splitting of the response matrix in the product of the two matrices (resolution and back-scattering), as suggested by D. D. Slavinskas et al., appears a very useful tool. A block direct method, due to the block tridiagonal form of the resolution matrix, and a block iterative method are applied; direct inversion is suitable for the triangular matrix of the backscattering. The standard error on the unfolded spectrum, due to the standard error on the experimental spectrum, and to the response function parameter uncertainties is examined.