Using critical numbers in the problem of identifying an unknown substance

The problem of identifying the chemical composition of a homogeneous medium is considered based on the available approximate data on the values of the attenuation coefficient of X-ray radiation in this medium for a finite set of energies. It is assumed that the medium is one of the substances included in a predetermined finite set $M$. For each substance of the set $M$, the exact values of the attenuation coefficient are known for all energies from a predetermined finite set of energies $E_1,…,E_N$ from the interval ${[0.001 – 20]}$ MeV. The data on the attenuation coefficients of the medium are known with a given error. The concept of a critical number for a pair of substances and energy is introduced. At the first stage of solving the problem, the set of all critical numbers for all possible pairs of substances from the set $M$ and all the energies used is found. After that, the characteristics of the set of all critical numbers that are necessary for solving the problem are studied. Next, for each energy $E_k$, a set of substances close in value of the attenuation coefficient to the substance of the unknown medium is constructed. Then, from the resulting family of sets, the substance most suitable as a solution to the identification problem is selected. The formulation of the problem is considered justified in the problems of customs control and in medical tomography. The issue of the uniqueness of the solution to the identification problem is considered. Examples of calculations for a set of specific substances are given using some results of previously performed studies.