## 3 Dilute Alkali Atoms in a Solid Inert-Gas Matrix: Theoretical Analysis  
### The Theoretical Model  
We now consider some effects of a solid matrix of inert atoms or molecules at cryogenic temperatures on embedded dilute alkali atoms. In this section we estimate the potential energy of a guest alkali atom in the solid matrix, starting from tabulated values of the relevant parameters, obtained experimentally or theoretically calculated. Our aim here is to obtain estimates of the order of magnitude of the potential, rather than precise values for specific cases, using simple expressions of the dependence of the potentials from the distance, depending from a small number of parameters, and that can be easily adapted to different atom-matrix combinations. We base our analysis on three different analytical models for the potential energy curve of the doping alkali atom and one atom or molecule of the matrix; specifically, we will consider Argon, Neon, and parahydrogen as matrices. The models considered and compared are the Morse potential [36,37], the Buckingham-Hill potential [36,38,39], and the model used by [PERSON] et al for H\({}_{2}\)-H\({}_{2}\) or D\({}_{2}\)-H\({}_{2}\) interactions [40], and then used also for the Li-H\({}_{2}\) potential [41,42]). Our analysis will refer to only a specific peak of the triplet structure of the spectrum, in particular the non-shifted one as described in the previous section. Evaluating these potentials through simple analytical expressions is an important first step for estimating the contribution to the inhomogeneous linewidth of the energy levels of the dilute guest atoms in the matrix due to their interaction energy with the matrix. In fact, the actual potential energy depends on the position of the atoms in the solid crystal, which can be (slightly) different from atom to atom, yielding different energy-level shifts; this eventually yields a relevant contribution to the resulting inhomogeneous width of the spectral lines. In the case of our interest, the Buckingham-Hill and the Morse potentials can give a reliable prediction only in different distance ranges: large distances for the Buckingham-Hill potential, and short distances for the Morse potential. More sophisticated models, such as the one proposed by [PERSON] et al., have been used in the literature for the Li-H\({}_{2}\) potential as well as for the H\({}_{2}\)-H\({}_{2}\) potential [40,41], but they involve a higher number of parameters; we will consider this model for the Li-H\({}_{2}\) potential at the end of this subsection. As we will now discuss in detail, both of them can be in general necessary in order to evaluate the potential felt by the alkali atom in the solid matrix using a small number  
Figure 5: Blue dots are the spectrum of the Rb-Ne crystal, observing the light output at \(\lambda_{emission}\)= 1270 nm as a function of the excitation wavelength. The orange curve is the Lorentzian \(L(\lambda)\) fit of the data. of known and tabulated values of the relevant parameters for the potential. It can be also helpful to use an appropriate combination of both models in order to obtain an approximated analytical expression of the actual potential, assuming that the alkali atom substitutes one of the atoms or molecules of the solid matrix. This potential, together with our experimental spectroscopical data of Section 2, will be then exploited in Section 3.2 to discuss the consequent contribution to the inhomogeneous broadening of the alkali-atom spectral lines (in particular, of the ground state, involved as initial state of the proposed transition scheme for the axion detection discussed in the Introduction. We first consider the case of the doped crystal experimentally analyzed in the previous section, that is dilute Rubidium atoms in a Argon or Neon solid matrix, comparing the theoretical results with our experimental data reported in Section 2.