Mathematical modeling of gamma field intensity for semi-infinite soil using the <i>q</i> -exponential attenuation law

Gamma spectrometry of radioactive media and objects addresses various issues, including mineral exploration, environmental monitoring, medicine, security, agriculture, anti-terrorist protection and the assessment of the consequences of both man-made and natural disasters. Despite the extensive experimental and theoretical research on gamma radiation processes, mathematical modeling of gamma fields remains a relevant topic. In particular, these studies concern the analytical calculations of gamma ray intensity in a medium that reduces the gamma radiation flux in accordance with a modified attenuation law. The revised attenuation law is described by the Tsallis q-exponential function representing a power generalization of the classical exponential function and, as shown in this research, can provide a more flexible tool for fitting experimental data. The parameter q serves mainly for measuring the deviation from exponential behavior, but is also related to quantifying the system’s nonadditivity, i.e., a property that is inherent in complex systems with long-range interactions and memory effects. Using this law, we modify the classical problem of evaluating the intensity of a gamma field generated by a semi-infinite volume (half-space). This problem mathematically leads us to a triple improper integral which as shown can be expressed in terms of generalized incomplete beta-functions under different auxiliary conditions. Technically, we derive the gamma ray intensity at a point located in the air above the radioactive soil that is assumed to consist of horizontal homogeneous layers. The intensity dependences on the distance to a soil surface are evaluated at different parameter values q and fixed typical values of soil densities and attenuation coefficients. These findings can be of interest to specialists in modeling gamma and similar fields, as well as for enhancing remote monitoring techniques for gamma fields.