The absorption of the axion is then detected, exploiting a laser field that induces an electronic transition from \(b\) to the the conduction band of the matrix, whose edge is \(c\), or, similarly, to an atomic excited level. In order to find out which is the best combination of guest atoms and solid-matrix atoms or molecules, minimizing the spectral broadening, it is important to obtain reliable spectroscopic data for different combinations of guest atoms (alkali atoms, in this paper) and inert-gas cold matrices. It is also of fundamental importance to develop simple theoretical models to estimate the potential energy between the atoms or molecules, which can be easily used to compare different combinations of guest atoms and solid-crystal atoms or molecules. Their predictions can be then compared with available experimental data. Analysis of experimental data and theoretical predictions can also give important hints on how the guest atoms diffuse in the cold matrix in specific experimental setups and depending from the preparation of the doped cold crystal. In fact, the potential energies between the alkali atoms and the matrix atoms or molecules, as we will show, can give an indication of the inhomogeneous broadening of the alkali spectral lines, due to the atoms-matrix interaction energies, as well as indications on their spatial distribution inside the cold matrix. The linewidth of the Zeeman-shifted energy levels should be the same for the unperturbed atoms, i.e., in the absence of the magnetic field, since their electronic wavefunction is the same. We wish to mention that, at the cryogenic temperatures attainable in our proposed detection scheme based on Matrix Isolation Spectroscopy, considering a 100 GHz frequency between the two Zeeman sublevels (that corresponds to a temperature of about 5 K), and tuned to an axion mass around 0.5 meV, essentially all guest atoms are in their ground state; thus, the axion field is going to excite them from the ground state. A more detailed discussion on this point for the proposed detection scheme, as well as on the expected counting rate and signal-to-noise ratio, can be found in [23; 30], in terms of the local axion energy density, the axion decay constant and other relevant parameters. The signal-to-noise ratio depends also on the lifetime of the Zeeman sublevels; for example, in a system recently considered, assuming a level lifetime of 1 ms, axions with mass larger than 80 GHz can be searched with a signal to noise ratio of 3 and with a significant count number in a 1-hour observation time, provided the crystal is cooled below 100 mK [30]. In this paper, in Section 2 we first present experimental spectroscopic data of dilute Rubidium atoms in Argon and Neon cold matrices, with particular attention given to the lineshape of the emission spectra. Next, in Section 3 we propose a simple general theoretical model of the interaction potential energy between the alkali atoms and the inert atoms or molecules of the matrix, in terms of few data that can be found in the literature (theoretically calculated or experimental), specifically equilibrium distance, dissociation energy, van der Waals constant, and the curvature of the potential energy at the equilibrium distance. The theoretical results are then used to estimate the order of magnitude of the broadening of the alkali spectral lines due to their interaction with the atoms or molecules of the cold matrix. This expected broadening is compared with our spectral experimental results. The results can be usefully exploited also to estimate the position distribution of the Rubidium atoms in the Argon or Neon cold matrix. Finally, we consider theoretically the case of Lithium atoms in a parahydrogen cold matrix, and estimate the relevant interaction energies, and the expected inhomogeneous spectral broadening in such a case.