**1. Introduction**

There are many methods for evaluating the thermal fields in radiation-matter interaction, but most of them require a complex mathematical handling [1–4]. This chapter presents a direct and powerful mathematical approach to compute the thermal field for electron beam-material and laser-sample interaction. The solving procedure is based on applying the integral trans‐ form technique which was developed in the 1960s, by the Russian School of Theoretical Physics [5]. As an example, the integral transform technique is used in [3] to solve the heat equation for a sample exposed to an infrared laser beam in order to find the solution for the absorption coefficient, which is then checked experimentally. It should be pointed out straightforwardly

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that the heat equation has the same form in the case of irradiation with a laser beam or an electron beam, at sufficiently large beam intensities [6, 7]. There is, however, a disadvantage in this model as it cannot take into account simultaneously the variation with temperature of several thermal parameters involved in the interaction like, for example, the thermal conduc‐ tivity or thermal diffusivity. In consequence, the model should be regarded as a first approx‐ imation of the thermal field. The main advantage is that the solution is a series which converges rapidly. It is important to note that the integral transform technique, as it will be shown in the next sections, belongs to the "family" of Eigen functions and Eigen values-based methods.
