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 [3].
The formation of gas bubbles is another suggested mechanism which could lead to void(s) formation [4]. The filling of a bubble with gas prevents it from collapsing. Various heterogeneous atomic clusters [5] and halo nuclei [6] 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 [7]. In the case of a Fermi system the
most favorable arrangement is not obvious [8]. 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. [9]. 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 [12]. 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 [8] 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 [14]. 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
[15] 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 [12] and one can expect that the single particle spectrum would approach that of a random Hamiltonian [16] and that the nearest-neighbor splitting distribution would be given by the Wigner surmise[17]. 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
[11]. 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 [18]
or the critical Casimir energy in a binary liquid mixture near the
critical demixing point [19]. 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 [8]. 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. [2]. On the other hand voids, not always spherical though, can be easily conceived to exist in neutron stars [21]. Metallic clusters with bubbles, one or more fullerenes in a liquid metal or a metallic ball placed inside a superconducting microwave resonator [22] 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.