Spontaneous spherical symmetry breaking in atomic confinement

Abstract

The effect of spontaneous breaking of initial SO(3) symmetry is shown to be possible for an H-like atom in the ground state, when it is confined in a spherical box under general boundary conditions of "not going out" through the box surface (i.e. third kind or Robin's ones), for a wide range of physically reasonable values of system parameters. The reason is that such boundary conditions could yield a large magnitude of electronic wavefunction in some sector of the box boundary, what in turn promotes atomic displacement from the box center towards this part of the boundary, and so the underlying SO(3) symmetry spontaneously breaks. The emerging Goldstone modes, coinciding with rotations around the box center, restore the symmetry by spreading the atom over a spherical shell localized at some distances from the box center. Atomic confinement inside the cavity proceeds dynamically -due to the boundary condition the deformation of electronic wavefunction near the boundary works as a spring, that returns the atomic nuclei back into the box volume.

Atomic H over plane: Effective potential and level reconstruction

“…In a cavity with finite R, such situation also takes place when the equilibrium position of H shifts from the center to the border, that happens whenever λ < q with q being the nucleus charge. [34,35] In ℜ 3 / 2 the spherical symmetry restores only for infinite distances between the atom and plane and only in the case, when the atomic electron is localized in the nucleus vicinity, where it falls into the eigenstates of the free atom. But this is not the general case.…”

“…[25][26][27][28][29][30][31][32][33] Moreover, such conditions are able to take into account the interaction of confined particles with medium, surrounding the cavity or demarcating the half-space. [29][30][31][32]34,35] It would be worth to note that the term "not going through," used here, underlines that these conditions do not necessarily originate from the actual confinement of particles inside the given volume, rather they may be caused by a significantly wider number of reasons, as it takes place, in particular, in the Wigner-Seitz model of an alkali metal, [4,5] where the valence electron state is principally delocalized. The latter circumstance turns out to be quite important, because in some cases the cavities, where a particle or an atom could reside, form a lattice, similar to that of an alkali metal, like certain interstitial sites of a metal supercell, for example, next-to-nearest octahedral positions of palladium face-centered-cubic lattice.…”

“…[36] In this case a particle (or a valence electron, provided that the whole lattice of cavities is occupied by identical atoms) finds itself in a periodic potential of a cubic lattice, and so the description of its ground state could be based on the first principles of the Wigner-Seitz model. [31,32,34,35] Another approach to atomic confinement by means of "not going through" condition has been considered recently by Burrows and Cohen [37] in terms of the electronic flux, which is assumed to vanish on some notional bounding surface of arbitrary shape.…”

“…with growing R the equilibrium position of the atom in the center of cavity ceases to be stable and it shifts to the border. [34,35] When the curvature of the border becomes much larger than a B , then the problem of an H over plane with Robin boundary condition for the electronic WF appears in a natural way. In this problem there are two most important questions.…”

“…The first one is under which conditions imposed on the surface interaction the effective atomic potential, treated as a function of the distance h to the plane, reveals a minimum for finite and nonzero h. A preliminary, rather rough estimate for this effect has been considered in Refs. [34,35]. The second concerns the additional "power-like" levels, which appear always under Robin boundary condition with attractive surface interaction and become the lowest ones, when the attraction is sufficiently strong.…”