## 3 Dilute Alkali Atoms in a Solid Inert-Gas Matrix: Theoretical Analysis
### Numerical Estimates of the Atoms-Matrix Interaction Energies
We indicate with \(\Delta r\) a small displacement from the average distance \(r_{0}\), which is equal to \(r_{Rb-Ar}\) in the present Rb-Ar case, we can obtain the consequent change of the potential energy as  
\[\Delta V_{M}(\Delta r)=V_{M}(r_{0}+\Delta r)-V_{M}(r_{0})\simeq 2\alpha De^{ \alpha(R_{eq}-2r_{0})}\left(e^{\alpha r_{0}}-e^{\alpha R_{eq}}\right)\Delta r, \tag{16}\]  
and thus the relative absolute value of the change of the potential energy is  
\[\left|\frac{\Delta V_{M}(\Delta r)}{V_{M}(r_{0})}\right|\simeq\left|\frac{2 \alpha\left(e^{\alpha r_{0}}-e^{\alpha R_{eq}}\right)}{e^{\alpha R_{eq}}-2e^{ \alpha r_{0}}}\Delta r\right|, \tag{17}\]  
This relation allows us to estimate the position dispersion \(\Delta r\) of the alkali atoms in the solid matrix, once the relative dispersion of the potential energy is known. The latter can be obtained from the spectral experimental results of the previous section, specifically from the relative energy widths (2). In the case of Rb-Ar here considered, for the peak 3 of Table 1 (see also Figure 3) with an energy width of 4%, and using the numerical data in (13) and (17) yields the Argon-atoms position dispersion  
\[|\Delta r|\sim 1.7\cdot 10^{-2}\text{\AA}\, \tag{18}\]  
that means a relative dispersion of \(\sim\)0.5%. Indeed, this should be considered as an upper limit to the position dispersion around the trapping site, because other effects, not considered here, can contribute to the line broadening. Our experimental measurements, combined with our theoretical model, have thus allowed us to obtain relevant hints on how the guest atoms in our Rb-Ar experimental setup diffuse in the cold matrix, and relate the measured spectral widths to the alkali-matrix interactions. This method can be also applied also to other doping-atoms matrix, for example Rubidium atoms in a cold Neon matrix. (ii) _Rb-Ne_  
We now consider the Rb-Ne case, whose spectral measurements have been reported at the end of Section 2. The relevant numerical values for equilibrium distance, dissociation energy and van der Waals constant can be found in [46; 47; 56; 57; 61; 63]. In this case, some differences are found in the literature among the data obtained with different methods, in particular for the dissociation energy. This can yield different estimates of the interaction energy in the solid matrix, depending on the numerical data used. Solid Neon has a fcc crystal structure as solid Argon, with a reticular distance \(a_{Ne}=4.43\) A. We first use the very recent data of the equilibrium distance and dissociation energy for the Rb-Ne dimer reported in [61], \(R_{eq}^{Rb-Ne}=6.188\) A and \(D^{Rb-Ne}=7.28\cdot 10^{-4}\) eV, respectively, supplemented with the value of the potential curvature at the potential minimum [62], \(x^{Rb-Ne}=1.107\cdot 10^{-3}\) eVA\({}^{-2}\). In this case we obtain  
\[V_{\text{M}}^{Rb-Ne}(r=3.13\text{\AA})\simeq 0.13\text{\ eV}, \tag{19}\]  
which is larger than that for Rb-Ar. A similar result is obtained using the data in [57], \(R_{eq}^{Rb-Ne}=6.212\) A, \(D^{Rb-Ne}=6.07\cdot 10^{-4}\) eV, \(\kappa=1.061\cdot 10^{-3}\) eVA\({}^{-2}\), where the value of the curvature \(\kappa\) of the potential at the equilibrium position has been inferred from their data of the harmonic wavenumbers, obtaining in this case \(V_{\text{M}}^{Rb-Ne}(r=3.13\text{\AA})\simeq 0.17\) eV. This is also confirmed by other recent numerical data [62] (see also [61]), yielding \(V(r=3.13\text{\AA})=0.061\) eV: this value is fairly lower than our theoretical estimate based on the Morse potential (it is, however, larger compared to the Rb-Ar case), showing again a larger variability of the energies in the Rb-Ne case according to the analytical or numerical method used. Although our theoretically predicted values for the ground-state interaction energy for Rb-Ne are larger than for Rb-Ar, as a comparison with (14) immediately shows, while our spectra of Section 2 show sharper linewidths in the Rb-Ne case, this is not necessarily a contradiction between experimental results and theoretical analysis.