Iterative Chi-Square Minimization for Gamma Ray Spectrometry: Quantifying K, U and Th concentrations in Geological Samples

Abstract

Gamma ray spectrometry is widely employed for quantifying naturally occurring radioactive elements such as potassium (K), uranium (U) and thorium (Th) in geological materials, however, conventional energy window based analysis (EWBA) is constrained by spectral overlap and simplified treatment of counting uncertainties. In this study, an iterative chi-square minimization (ICSM) algorithm with adaptive uncertainty refinement is implemented in Python to perform full spectrum analysis of NaI(Tl) gamma ray spectra over the energy range 0.4–2.9 MeV. Standard spectrum from International Atomic Energy Agency (IAEA) Standard Reference Materials (SRMs) and a calibrated 5² x 4² NaI(Tl) detector are used to determine isotopic concentrations and their uncertainties in a sample spectrum. Elemental response functions derived from reference standards are used to model measured spectra as linear combinations of K, U, and Th contributions, while channel wise uncertainties are iteratively updated based on the expected (fitted) spectrum.

The method is applied to geological samples and to three in-house certified reference materials (CRMs) analyzed as unknowns. For the samples, ICSM consistently yields lower uncertainties than EWBA for identical counting times, with typical uncertainty reductions of approximately 1.3–1.8 x for K, 2–3 x for U, and 1.5–2.5 x for Th. The in-house standards, having concentrations independently determined using a high purity germanium (HPGe) detector, provide an external validation of quantitative performance. ICSM derived concentrations exhibit closer agreement with HPGe reference values and substantially reduced uncertainties compared with EWBA. Residual analysis and reduced chi-square values indicate statistically consistent fits across all analyzed spectra. These results demonstrate that iterative chi-square minimization with adaptive uncertainty refinement offers a precise and statistically robust alternative to window based methods for quantitative gamma ray spectrometric analysis of geological materials.