## 3 Dilute Alkali Atoms in a Solid Inert-Gas Matrix: Theoretical Analysis
### Numerical Estimates of the Atoms-Matrix Interaction Energies
As mentioned above, we need to evaluate the potential energy at the distance \(a^{Rb-Ar}=a_{Ar}/\sqrt{2}=3.72\ \text{\AA}\), assuming that the Rubidium atom replaces one Argon atom of the matrix. Since \(a^{Rb-Ar}<R_{eq}^{Rb-Ar}\), we use the Morse potential (9). The \(\alpha=\sqrt{k/(2D)}\) parameter in Equation (9) in the present case is therefore \(0.818\ \text{\AA}^{-1}\). Using all these data in (9), we immediately find  
\[V_{\text{M}}^{Rb-Ar}(r=a^{Rb-Ar}=3.72\ \text{\AA})\simeq 4.35\cdot 10^{-2}\ \text{eV}. \tag{14}\]  
In this case we have been able to estimate the potential energy using the Morse potential only, because the potential curvature \(\kappa\) is known. In Section 3.1 we proposed that \(\kappa\) could be obtained using the Buckingham-Hill potential through Equation (10). The van der Waals constant \(C_{6}\) can be found in [46, 47], for example, as \(C_{6}^{Rb-Ar}=201\ \text{eV\AA}^{6}\). From Equation (10) we thus obtain \(\kappa_{M}=\kappa^{Rb-Ar}=4.73\cdot 10^{-3}\ \text{eV\AA}^{-2}\), that is relatively close to the tabulated value shown in (13). This is an important consistency check for the analytical method presented in this section. Summing the potential (14) over the 12 nearest-neighbors of the Rubidium atom (pairwise summation), we obtain  
\[V_{tot}^{Rb-Ar}\simeq 0.52\ \text{eV}. \tag{15}\]  
Using the very recent data for the equilibrium distance and the dissociation energy in [61], \(R_{eq}^{Rb-Ar}=5.541\ \text{\AA}\), \(D^{Rb-Ar}=4.716\cdot 10^{-3}\ \text{eV}\), and the data for the curvature of the potential at the equilibrium distance communicated to us by the authors of that paper [62], \(\kappa^{Rb-Ar}=5.617\cdot 10^{-3}\ \text{eV\AA}^{-2}\), we get \(V_{\text{M}}^{Rb-Ar}(r=a^{Rb-Ar}/\sqrt{2}=3.72\ \text{\AA})\simeq 3.99\cdot 10^{-2}\ \text{eV}\), that is close to (14). Interpolation of recent ab initio numerical data for the potential energy [62] (see also [61]), yields a somehow lower value, \(\simeq\)\(3.1\cdot 10^{-2}\ \text{eV}\), in any case not far from our analytical estimate. We can thus say that, even using numerical data from different sources, we obtain essentially the same approximate estimate of \(V_{\text{M}}^{Rb-Ar}\). Even if we are mainly interested to the ground state, as discussed later on, we can use the same procedure to estimate the potential energy of an excited state of the Rubidium atom in the solid Argon matrix. Specifically, we consider the first excited state in the Rb-Ar molecular configuration \(A^{2}\Pi_{1/2}\) (corresponding to a \({}^{2}P_{1/2}\) configuration of Rubidium, and the Argon in its ground state). We use the following tabulated data: \(R_{eq}^{Rb-Ar}\)\(=3.60\) A, \(D_{(exc)}^{Rb-Ar}\)\(=3.162\cdot 10^{-3}\) eV [58]; \(C_{6~{}(exc)}^{Rb-Ar}\)\(=325.7\) eVA [6]. Because in this case the distance at which we should evaluate the potential, \(r=a^{Rb-Ar}/\sqrt{2}=3.72\) A, is very close (slightly larger, to be precise) to the equilibrium distance \(R_{eq}^{Rb-Ar}\), we can use the Buckingham-Hill potential (8), and we obtain \(V_{BH~{}(exc)}^{Rb-Ar}\)\(\simeq-3.07\cdot 10^{-2}\) eV. Since the interatomic distance now considered is close to the equilibrium distance, we must expect that the Morse potential should yield the same result too. An explicit evaluation by using Equation (9), with the potential curvature obtained from (10), gives indeed practically the same result. After pairwise sum over the 12 nearest neighbors Argon atoms, we have \(V_{tot~{}(exc)}^{Rb-Ar}\)\(\simeq-0.37\) eV. These interaction energies are of the same order of magnitude of the ground-state interaction energies evaluated above. Our main aim here is the role of these interactions in the linewidth broadening of the guest atoms in the solid matrix. Together with other effects, phonon excitations in the matrix for example, they have also other consequences, of course, for example energy shifts due to the degeneracy lifting of excited atomic levels. We will not consider these effects in the present preliminary study.