## 3 Dilute Alkali Atoms in a Solid Inert-Gas Matrix: Theoretical Analysis
### Numerical Estimates of the Atoms-Matrix Interaction Energies
Since the radius of the Lithium atom, \(\simeq 1.45\) A, is quite smaller than the alkali-parahydrogen distance in the solid matrix, \(a^{Li-p(H_{2})}=3.78\) A, our pairwise-summation of the potentials between nearest-neighbors pairs, as well as neglecting higher multipolar dispersion interactions, should be reasonably good approximations. Since the Li-H\({}_{2}\) distance in the solid matrix is smaller than the equilibrium distance, 4.76 A, we expect that the Morse potential should give a better approximation to the true potential than the Buckingham-Hill potential. With the numerical values above, taking the second derivative of the Buckingham-Hill potential (8) and (Equation (10)), and evaluating it for the distance \(r=\simeq 3.78\)A between the alkali atom and any of the nearest-neighbors parahydrogen molecules (see Figure 6), we obtain \(|\kappa|\simeq 4.34\cdot 10^{-3}\) eVA\({}^{-2}\), and then \(\alpha=\sqrt{|k|/(2D)}\simeq 0.746\) A\({}^{-1}\), to be used in the Morse potential (9). We thus obtain  
\[D^{Rb-Ar} = 4.45\cdot 10^{-3}\ \text{eV},\]0  
Taking into account that each Lithium atom has six nearest neighbors parahydrogen molecules of the solid matrix, and assuming a pairwise summation, we obtain the total interaction energy of each Lithium atom in the parahydrogen solid matrix  
\[D^{Rb-Ar} = 4.45\cdot 10^{-3}\ \text{eV},\]1  
We can also calculate the Li-(p-H\({}_{2}\)) interaction energy using the potential (11), with the following parameters given in [41]; in atomic units, for the \(X\Sigma\) configuration (ground state), they are: \(\alpha=-2.976\), \(\beta=0.1623\), \(\gamma=0.05033\), \(r_{m}=9.856\), \(C_{6}=85.00\), \(C_{8}=4050\), \(C_{10}=0\)[41]. We obtain  
\[D^{Rb-Ar} = 4.45\cdot 10^{-3}\ \text{eV},\]2  
This value is about one half of that predicted from the Morse potential, and so it is even more favorable for the feasibility of our proposed detection scheme. In fact, after summation over the six nearest-neighbors parahydrogen molecules, it yields a total potential energy of the order of one meV. Comparing (20) and (22) with (14) and (19), we see that the interaction energy in the present case is rather smaller than for both Rb-Ar and Rb-Ne. On the basis of our previous considerations, this should give a rather smaller energy width of the Lithium ground state in the matrix, at least for the contribution from its interaction with the lattice molecules, and assuming a similar dispersion of the positions of the alkali atoms in the cold solid matrix. Some experimental spectroscopic data on the Li-(p-H\({}_{2}\)) system can be found in [67], while theoretical aspects relevant for the absorption spectra are given in [41]. In [67], the spectral linewidth of the Lithium atoms in the p-H\({}_{2}\) cold matrix are of the order of 100 meV, but they cannot be directly compared with the ground-state interaction energies here evaluated. In fact, spectral lines, as already mentioned for the Ar and Ne cases, involve also other effects, such as, for example, the width of the excited level, energy spitting of degenerate excited levels due to crystal field, as well as phonon excitation in the crystal, that could be relevant since parahydrogen is a quantum crystal. We however expect that these effects play a less relevant role in our case, when we consider the transition between the alkali-atom Zeeman sublevel \(b\) (reached after the assumed axion absorption) to the crystal conduction band, whose edge in Figure 1 is indicated by \(c\). The reason is that in this case, the atomic wavefunction of the sublevels \(a\) and \(b\) is the same, and there are not other discrete levels involved in the transition. Finally, taking into account the theoretical results for Li-(p-H\({}_{2}\)) of this section, we expect that the Lithium-parahydrogen system could be a very promising physical system for the proposed axion detecting scheme, exploiting the transitions in Figure 1, with \(c\) indicating the edge of the conduction band of the cold crystal.