Department of Physics, University of Washington, Seattle, WA 98195-1560, USA The Royal Institute of Technology, Physics Department Frescati, Frescativägen 24, S-10405, Stockholm, SWEDEN Institute of Physics, Warsaw University of Technology, ul. Koszykowa 75, PL-00662, Warsaw, POLAND
Yongle YU, Aurel BULGAC and Piotr MAGIERSKI
February 4, 1999
The positioning of a bubble inside a many fermion system does not affect the volume, surface or curvature terms in the liquid drop expansion of the total energy. Besides possible Coulomb effects, the only other contribution to the ground state energy of such a system arises from shell effects. We show that the potential energy surface is a rather shallow function of the displacement of the bubble from the center and in most cases the preferential position of a bubble is off center. Systems with bubbles are expected to have bands of extremely low lying collective states, corresponding to various bubble displacements.
PACS numbers: 21.10.-k, 21.10.Dr, 21.60.-n,24.60.Lz
There are a number of situations when the formation of voids is favored. When a system of particles has a net charge, the Coulomb energy can be significantly lowered if a void is created [1, 2] and despite an increase in surface energy the total energy decreases. One can thus naturally expect that the appearance of bubbles will be favored in relatively heavy nuclei. This situation has been considered many times over the last 50 years in nuclear physics and lately similar ideas have been put forward for highly charged alkali metal clusters .
The formation of gas bubbles is another suggested mechanism which could lead to void(s) formation . The filling of a bubble with gas prevents it from collapsing. Various heterogeneous atomic clusters  and halo nuclei  can be thought of as some kind of bubbles as well. In these cases, the fermions reside in a rather unusual mean-field, with a very deep well near the center of the system and a very shallow and extended one at its periphery. Since the amplitude of the wave function in the semiclassical limit is proportional to the inverse square root of the local momentum, the single particle wave functions for the weakly bound states will have a small amplitude over the deep well. If the two wells have greatly different depths, the deep well will act almost like a hard wall (in most situations).
Several aspects of the physics of bubbles in Fermi systems have not
been considered so far in the literature. It is tacitly assumed that
a bubble position has to be determined according to symmetry
considerations. For a Bose system one can easily show that a bubble
has to be off-center . In the case of a Fermi system the
most favorable arrangement is not obvious . The total
energy of a many fermion system has the general form
where the first three terms represent the smooth liquid drop part of the total energy and is the pure quantum shell correction contribution, the amplitude of which grows in magnitude approximately as , see Ref. . We shall consider in this work only one type of fermions with no electric charge. In a nuclear system the Coulomb energy depends rather strongly on the actual position of the bubble, but in a very simple way. In an alkali metal cluster, as the excess charge is always localized on the surface, the Coulomb energy is essentially independent of the bubble position. The character of the shell corrections is in general strongly correlated with the existence of regular and/or chaotic motion [10, 11]. If a spherical bubble appears in a spherical system and if the bubble is positioned at the center, then for certain ``magic'' fermion numbers the shell correction energy , and hence the total energy E(N), has a very deep minimum. However, if the number of particles is not ``magic'', in order to become more stable the system will in general tend to deform. Real deformations lead to an increased surface area and liquid drop energy. On the other hand, merely shifting a bubble off-center deforms neither the bubble nor the external surface and therefore, the liquid drop part of the total energy of the system remains unchanged.
Moving the bubble off-center can often lead to a greater stability of the system due to shell correction energy effects. In recent years it was shown that in a 2-dimensional annular billiard, which is the 2-dimensional analog of spherical bubble nuclei, the motion becomes more chaotic as the bubble is moved further from the center . One might thus expect that the importance of the shell corrections diminishes when the bubble is off-center. We shall show that this is not the case however.
One can anticipate that the relative role of various periodic orbits (diameter, triangle, square etc.) is modified in unusual ways in systems with bubbles. In 3-dimensional systems the triangle and square orbits determine the main shell structure and produce the beautiful supershell phenomenon [10, 13]. A small bubble near the center will affect only diameter orbits. After being displaced sufficiently far from the center, the bubble will first touch and destroy the triangle orbits. In a 3-dimensional system only a relatively small fraction of these orbits will be destroyed. Thus one might expect that the existence of supershells will not be critically affected, but that the supershell minimum will be less pronounced. A larger bubble will simultaneously affect triangular and square orbits, and thus can have a dramatic impact on both shell and supershell structure.
The change of the total energy of a many fermion system can be computed quite accurately using the shell corrections method, once the single-particle spectrum is known as a functions of the shape of the system [9, 11]. The results presented in this Letter have been obtained using the 3d-version of the conformal mapping method described in  as applied to an infinite square well potential with Dirichlet boundary conditions. The magic numbers are hardly affected by the presence or absence of a small diffuseness . The absence of a spin-orbit interaction leads to quantitavive, but to no qualitative differences. Consequently, we expect that our results are generic.
In Fig. 1 we show the unfolded single-particle spectrum for the case
of a bubble of half the radius of the system, a=R/2, as a function
of the displacement d/R of the bubble from the center. The size of
the system is determined as usual from . The unfolded
single-particle spectrum is determined using the Weyl formula
 for the average cumulative number of states.
Figure 1: The unfolded single-particle spectrum for the case of a bubble of radius a= R/2 as a function of the bubble dispalcement d/R. Only the lowest 128 levels are shown here.
where are the actual single-particle energies of the Schrödinger equation, is the Weyl formula for the total number of states with energy smaller than e in a 3d-cavity and are the unfolded eigenvalues, which by construction leads to a spectrum with an unit average level density.
As the bubble is moved off center, the classical problem becomes more chaotic  and one can expect that the single particle spectrum would approach that of a random Hamiltonian  and that the nearest-neighbor splitting distribution would be given by the Wigner surmise. A random Hamiltonian would imply that ``magic'' particle numbers are as a rule absent. There is a large number of level crossings in Fig. 1 and one can clearly see a significant number of relatively large gaps in the spectrum. These features are definitely not characteristic of a random Hamiltonian. If the particle number is such that the Fermi level is at a relatively large gap, then the system at the corresponding ``deformation'' is very stable. This situation is very similar to the celebrated Jahn-Teller effect in molecules. A simple inspection of Fig. 1 suggests that for various particle numbers the energetically most favorable configuration can either have the bubble on- or off-center. Consequently, a ``magic'' particle number could correspond to a ``deformed'' system. In this respect this situation is a bit surprising, but not unique. It is well known that many nuclei prefer to be deformed, and there are particularly stable deformed ``magic'' nuclei or clusters [11, 13, 14, 20].
The variation of the ground state energy of an interacting N-fermion system, with respect to shape deformation or other parameters, is quite accurately given by the shell correction energy . In our case, the eigenspectrum and the shell correction energy are functions of N, R, a and d. When the particle number N is varied at constant density, we have and . There is a striking formal analogy between the energy shell correction formula and the recipe for extracting the renormalized vacuum Casimir energy in quantum field theory  or the critical Casimir energy in a binary liquid mixture near the critical demixing point . In Fig. 2 we show the contour plot of the shell correction energy for a system with a=R/2 as a function of the bubble displacement d/R versus . The overall regularity of ``mountain ridges' and ``canyons'' seem to be due the interference effects arising from two periodic orbits along the diameter passing through the centers of the two spheres. Various mountain tops and valleys form an alternating network almost orthogonal to the ``mountain ridges'' and ``canyons''. For some N's the bubble ``prefers'' to be in the center, while for other values that is the worst energy configuration. For a given particle number N the energy is an oscillating function of the displacement d and many configurations at different d value have similar energies. However, in all cases, moving the bubble all the way to the edge of the systems leads to the lowest values of . This drop in the shell correction energy as a function of d is preceeded by the highest ``mountain range''. Thus, with the exception of the alternating peaks and deep lakes for the on-center configuration, the largest variations in the shell correction energy occur when the bubble is close to the surface.
Figure 2: Contour plot of the shell correction energy for the case of a bubble of radius a = R/2 for up to N=8,000 spinless fermions. Calculations have been perfomed up to . Energy is measured in units of .
In Fig. 3 we show the unfolded single-particle spectra for a bubble with a smaller radius a=R/5. The number of level crossing is significantly smaller than in Fig. 1 and as a result, the shell correction energy contour plot has less structure, see Fig. 4, and thus a system with a smaller bubble is also significantly softer.
Pairing correlations can lead to a further softening of the potential energy surface of a system with one or more bubbles. We have seen that the energy of a system with a single bubble is an oscillating function of the bubble displacement. When the energy of the system as a function of this displacement has a minimum, the Fermi level is in a relatively large gap, where the single-particle level density is very low. When the energy has a maximum, just the opposite takes place. Pairing correlations will be significant when the Fermi level occurs in a region of high single-particle level density and it is thus natural to expect that the total energy is lowered by paring correlations at ``mountain tops'', and be less affected at ``deep valleys". All this ultimately leads to a further leveling of the potential energy surface.
Figure 3: The same as in Fig. 1 but for a= R/5.
Figure 4: The same as in Fig. 2 but for a = R/5 and for up to N=1,000 spinless fermions.
With increasing temperature the shell correction energy decreases in magnitude, but the most probable position of a bubble is still off-center. The reason in this case is however of a different nature, the ``positional'' entropy of such a system favours configurations with the bubble off-center, as a simple calculation shows, namely . Moreover, making more bubbles could lead to a further decrease of the free energy, even though the total energy might increase. Thus the problem of one or more bubbles at finite temperatures has its own intricacies.
A system with one or several bubbles should be a very soft system. The energy to move a bubble is parametrically much smaller than any other collective mode. All other familiar nuclear collective modes for example involve at least some degree of surface deformation. For this reasons, once a system with bubbles is formed, it could serve as an extremely sensitive ``measuring device'', because a weak external field can then easily perturb the positioning of the bubble(s) and produce a system with a completely different geometry. There are quite a number of systems where one can expect that the formation of bubbles is possible . Known nuclei are certainly too small and it is difficult at this time to envision a way to create nuclei as big as those predicted in Ref. . On the other hand voids, not always spherical though, can be easily conceived to exist in neutron stars . Metallic clusters with bubbles, one or more fullerenes in a liquid metal or a metallic ball placed inside a superconducting microwave resonator  in order to study the ball energetics and maybe even dynamics, are all very promising candidates.
Financial support for this research was provided by DOE.