Investigation of the ill-posed inverse problem in measuring near-threshold electron and positron-induced atomic inner-shell ionization cross-sections based on neural network approaches

Accurate measurement of near-threshold positron or electron impact atomic inner-shell ionization cross-sections is of significant importance both theoretically and in practical applications. Due to the complexity of preparing thin targets, difficulty in precisely measuring their thickness, and the low characteristic X-ray collection efficiency of thin targets in positron collision experiments, the thick-target method is typically employed. Solving the corresponding inner-shell ionization cross-section from the experimental yield of thick targets is an ill-posed inverse problem. Existing methods such as the direct comparison method, yield differential method, and regularization method, do not achieve ideal accuracy. Although our research group has recently developed the MC-neural network method, which significantly improves solution accuracy, it is highly time-consuming to use Monte Carlo simulations to generate neural network datasets. To address this issue, we have developed a numerical-neural network method, which uses numerical calculations to quickly generate large-scale, high-quality datasets for training convolutional neural network models to solve the inverse problem of thick target experimental cross-sections. In this study, numerical-neural network and MC-neural network methods are used to process the experimental yield data of K_ɑβ characteristic X-rays from Ti and L_ɑβγ characteristic X-rays from Ag in positron collisions with pure thick targets at energies below 10 keV. The positron-induced K-shell ionization cross-section of Ti and the L_ɑβγ characteristic X-ray production cross-section of Ag are obtained, which are compared with the experimental cross-sections obtained by the direct comparison method by other researchers and the DWBA theoretical values. The results show that the outcomes obtained by both neural network methods are in good agreement with the results from others and the DWBA theoretical values, and the numerical-neural network method is superior to the MC-neural network method in terms of solution accuracy. In addition, this study also reprocessed the experimental yield data of K_ɑβ characteristic X-rays of Zr and L_ɑβ characteristic X-rays of Au induced by electrons at energies below 27 keV, obtained by Li et al. of Sichuan University using the thick-target method. This also confirmed that the numerical-neural network method has more advantages than other methods.