Mathematical analysis of the Smakula–Dekster formula for the age determination of anthropogenic sediments

The features of application of the thermoluminescence method based on the Smakula–Dekster formula for the age determination of anthropogenic deposits have been analysed. The thermoluminescence method is widely used for the solution of geological, archaeological, and geochronological problems, in particular, the age and correlation determination of various Quaternary deposits from different sections. A new mathematical approach for anthropogenic deposits dating applying the thermoluminescence method and a physical and mathematical model of the processes of accumulation of radiation centres under natural conditions is proposed. It is noted that during the burial the fixation of accumulated age energy occurs in the rock under the influence of natural radioactive elements. During the laboratory heating process, the accumulated age energy is emitted, which allows us to determine the age of the rock. Depending on the structure and characteristics of the rock, we obtain spectra that differ in intensity. It is noted that the intensity of the peaks of thermoluminescence graphs depends on the energy accumulated by the sample during the burial process. Different types of deposits have varying thermoluminescence graphs that differ in height, width, and area. In this work, we have described in detail the solution of the first-order differential equation with separated variables, which describes the relationship between the energy flux radiated by the sample and the intensity of the given energy of the heater. By solving this differential equation, a formula was obtained for the radiation coefficient, which is a component of the Smakula–Dekster formula. It is noted that the Smakula–Dekster formula can be applied for the calculation of the concentration of radiation centres. A directly proportional relationship was established between the concentration of radiation centres and the age of the sample. It is emphasised that the greater the concentration of radiation centres, the greater the age of the sample. By applying this physical and mathematical model to the thermoluminescence method, it is possible to determine which of the studied deposits are older and which are younger. It is noted that the proposed physical and mathematical approach to the thermoluminescence method is effective for the age determination and revealing the sequence of sediment accumulation.