Extended mathematical model of the inverse problem of nuclear gamma-resonance. Reliability and informative of application

Research the properties of iron-based solid solutions by Mössbauer spectroscopy has the problem of interpreting the results of processing experimental data within the traditional mathematical model. Since the disordered solid solutions, for example, as a result of mechanical activation, are consisted of a set of the different local atomic configurations, the corresponding Mössbauer spectra contain a large number of the shifted relative to each other spectral components with close values of the hyperfine interaction parameters. The magnitude and sign of these shift are determined by many factors: the quantitative distribution of atoms of each type in the coordination spheres, the symmetry of their distribution relative to the quantization axis, the possible local shift relative to the average statistical position in the crystallographic structure, etc. In the mathematical model, as a rule, it's not possible to taken into account all these effects of the shift by analytically.
The proposed extended mathematical model for describing the Mössbauer spectra of solid solutions makes it possible to take into account the shifts in the spectral components by using Gaussian normal distribution function, as a function of statistical set of local distortions. The width of the Gaussian distribution makes it possible to estimate the degree of local distortions of the crystal lattice that arise due to differences in the sizes of atoms of the mixed components, local distortions of the structure and symmetry of the environment of the resonant atom.
The inverse problem of nuclear gamma-resonance is formulated by the Fredholm integral equation of the first kind and is an ill-posed problem with a priori constraints on the desired solution. The introduction of two Gaussian functions with a priori unknown linewidths into the kernel of the integral equation leads to the problem of solving the equation by classical methods. Algorithm for obtaining a reliable solution based on the Tikhonov regularization method with correction of the parameters of the kernel of the integral equation is proposed in this paper. On the examples of the study of real objects, the reliability and informative application of the extended mathematical model of the inverse problem of nuclear gamma-resonance is proved.