## 3 Dilute Alkali Atoms in a Solid Inert-Gas Matrix: Theoretical Analysis
### Numerical Estimates of the Atoms-Matrix Interaction Energies
In fact, also other aspects should be taken into account for a direct comparison of the linewidths. Firstly, our theoretical estimates involve only the ground state, while the spectral linewidths of Section 2 include also the effects of the excited level, and consequent rearrangement of the wavefunction distribution and possible phonon emission. A wavefunction rearrangement is present also in the considered case of transition to the crystal conduction band, but we expect its effect to be much smaller with respect to the transition to a higher excited level, which involves a larger size of the final discrete state and thus a larger perturbation of the crystal. Thus, we guess that our ground-state linewidth estimates could be directly compared to future measurements of transitions of the doping atom's electron from the ground state (also if Zeeman shifted) to the crystal conduction band. Secondly, we cannot exclude that the position distribution of the guest atoms in the solid matrix is smaller in Neon compared to Argon. Indeed, the two solid matrices have different physical properties, related to the interactions between the matrix atoms, for example the condensation temperature, \(T_{cond}=82\) K for Ar and \(T_{cond}=20\) K for Ne, and the reticular distance: thus, nontrivial differences in the way the guest atoms arrange themselves in the matrix are not unexpected. We plan to discuss these points in a future work. In any case, we believe that the case of Neon deserves a more precise analysis from the theoretical side, also because the alkali-RG atoms distance in the Neon matrix is smaller than in the Argon matrix, in particular in comparison with the corresponding equilibrium distance (\(R_{eq}^{Rb-Ar}=5.498\) A, \(r=3.72\) A for Rb-Ar; \(R_{eq}^{Rb-Ne}=6.19\) A, \(r=3.13\) A for Rb-Ne). At shorter distances, the potential energy is very steep, as, consistently, the plots of the potential energies in [61] clearly show. All this makes the details of the model more critical: even a small variation of the relevant parameters can yield a consistent variation of the estimated potential energy. 3. _Li-(p-H\({}_{2}\))_  
Finally, we consider theoretically the Li-(p-H\({}_{2}\)) case; that is, dilute Lithium atoms diffused in a solid parahydrogen matrix. We will obtain a strong indication that this system shows a quite smaller guest-atom-matrix interaction energy. The p-H\({}_{2}\) molecule has nuclear spin \(I=0\) and, at low temperatures, an even total angular momentum \(J\), with a spherical symmetry yielding vanishing permanent multipolar momenta [64]. Due to the spherical symmetry of the parahydrogen molecule, considerations on the potentials similar to the rare gases, discussed above, can be applied. Solid parahydrogen has a Hexagonal closed packed (Hcp) structure, with a quite large intermolecular distance compared to the relevant atomic or molecular dimensions. We assume that the alkali atom is placed at the hexagon center, substituting one of the H\({}_{2}\) molecules, as shown in Figure 6; due to the large intermolecular distance of the molecules in the p-H\({}_{2}\) matrix, its presence should not significantly change the structure of the crystal. Furthermore, the known large compressibility of parahydrogen solids allows us to easily accommodate the alkali-atom impurity [65]. The reticular distance for the parahydrogen crystal is \(a\simeq 3.78\) A [2,66]. Figure 6: The alkali atom at the center of the hexagonal structure of the solid p-H\({}_{2}\) matrix. The alkali atom (orange circle) is assumed to substitute one of the molecules of the solid matrix of p-H\({}_{2}\) (blue circles). The relevant tabulated data for the dissociation energy \(D\) and the equilibrium distance \(R_{eq}\) can be found in [52], while the van der Waals constant \(C_{6}\) can be found in [51]; the tabulated values are \(D=3.90\cdot 10^{-3}\) eV, \(R_{eq}=4.76\) A, \(C_{6}=49.4\) eV A\({}^{6}\). From (7) we then get \(R_{Rep}\simeq 1.52\) A.