So, one encounters the so-called over-determined problem .
The number of channels is higher than the number of pertinent library spectra ( n > m) in gamma-ray spectroscopy analysis.
Here, N =(N 1 ,N 2 ,…,N n ) T, α =( α 1, α 2,…, α m ) T and R is the n × m response matrix. This equation can also be written in a matrix form as follows: N + ε = R α Here, N i is the recorded count in the ith channel of multichannel analyzer, ε i is the inherent error related to the ith channel of the multichannel analyzer, R i j is the true number of counts in the ith channel of the jth component and α j is the multiplier of the jth component. The basic assumption of the approach is that the detector count ( N) in any unknown spectrum can be considered as a linear combination of detector count in contributing pertinent library spectra ( R) : N i + ε i = ∑ j R i j α j It may provide some advantages, if the resolution of the detector is a limiting factor for example in case of NaI(Tl) and BGO detectors. On the other hand, entire gamma-ray spectrum including both the photopeaks and the Compton continua are taken into account during calculations in WSA. This method of gamma ray spectra analysis have some shortcomings like: need of a gamma-ray detector with relatively high energy resolution and loss of the Compton continua information due to considering only the photopeak counts for weight fraction calculations . On one hand, the amounts or weight fractions of the constituents are calculated based on the integral number of counts in the most intense peaks of the background-subtracted spectra in photopeak analysis method. Two different approaches can be used for gamma-ray spectroscopic analysis : The photopeak analysis method and the library least-squares (LLS) approach or Whole Spectrum Analysis (WSA). In gamma-ray spectroscopic analysis, following the acquisition of the gamma-ray spectra a quantitative analysis of gamma radioactive materials are carried out to determine the respective amounts of different isotopes of a sample . The results of the developed PSO algorithm show a better match with the real fractional abundances than that of MLR method. The performance of the developed PSO algorithm is compared to the multiple linear regression (MLR) method as well. To test the developed algorithm, a number of experimentally measured gamma-ray spectra related to a mixed gamma-ray source including different combinations of 60 Co, 137 Cs, 22 Na, 152 Eu and 241 Am isotopes are analyzed using information of whole spectrum. The PSO method is used for complex fitting to the response of a 3 × 3 inch NaI (Tl) scintillation detector and the fitting process is controlled by a test for significance of abundance and computation of Theil coefficient. PSO is an iterative algorithm that imitates the social behaviors observed in nature to solve complex optimization problems. In this paper a new gamma-ray spectra analysis algorithm based on Particle Swarm Optimization (PSO) is developed to identify different isotopes of a mixed gamma-ray source and determine their fractional abundances. Analysis of gamma-ray spectra is an important step for identification and quantification of radionuclides in a sample.
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