Critical sizes for the submersion of alkali clusters into liquid helium

Alkali atoms do not stably embed in liquid helium-4 because the interatomic attractive potential is unable to overcome the short-range Pauli repulsion of the $s$ electrons and the surface tension cost of the surrounding bubble. Similarly, small alkali complexes reside on the surface of helium nanodroplets instead of inside. However, as the size of the metal cluster increases, its van der Waals attraction to the helium matrix grows faster than the repulsive energies and above a certain size it should become favorable for clusters to submerge in the liquid. Based on an evaluation of the relevant energy terms, we characterize the bubble dimensions and estimate the critical submersion sizes. The latter range from $N\sim 20$ for ${\text{Li}}_N$ and ${\text{Na}}_N$ to $N>100$ for ${\text{Rb}}_N$ in helium-4 and from $N\sim 5$ for ${\text{Li}}_N$ and ${\text{Na}}_N$ to $N\sim 20$ for ${\text{Cs}}_N$ in helium-3. These results are discussed in the context of nanodroplet pick-up experiments with alkali atoms.

Figure 1

(Color online) Model of an alkali nanocluster submerged in a liquid-helium bath, residing within a bubble.

Figure 2

(Color online) Examples of differences in the attractive cluster-helium van der Waals energy calculated by pairwise addition of the interatomic ${\text{C}}_6/r^6$ potential (dashed line) and the full expression in Eq. (6) (solid line). The energy scale is in atomic units, and $N$ is the number of atoms in the cluster particle. Top plot: sodium and bottom plot: rubidium.

Figure 3

(Color online) Appearance of cluster submersion with increasing size, for the example, of ${\text{Na}}_N$. The three dashed curves depict the three contributions to Eq. (1): van der Waals attraction [Eqs. (6, 7, 8, 9), red dots], short-range repulsion [first term in Eq. (5), green long dash], and the surface-tension energy of the cavity wall [last term in Eq. (1) with the use of Eq. (3), brown short dash]. The solid blue line is the sum of these, i.e., the total binding energy of a solvated cluster, ${\mathcal{E}}_{\text{solv}}$. The solid black line is the binding energy at the surface, ${\mathcal{E}}_{\text{surf}}$, Eqs. (10, 11, 12). Submersion corresponds to the crossing of these lines, as marked by an arrow.