Motivated by the desire to explore a form of strange matter that could be produced in an accelerator, people began to study very small strangelets of order A 10 - 100. Models of bulk quark matter are inappropriate for such small systems, even with the inclusion of finite scale modifications, as a simple Fermi gas model breaks down (although Fermi levels are sill qualitatively useful). Models of these systems involving QCD are quite limited and unfulfilling. For example, they do not display shell closures, and are particularly sensitive to the choice of renormalization scale. Gluon exchange effects are also very difficult to calculate for a system of more than a few baryons.
Instead, I will examine a fairly simple model of a gas of non interacting fermions in a bag. Again I am not interested in realistic predictions, but simply in getting a feel for what general characteristics such small accelerator produced strangelets might exhibit. This model only takes account of the kinetic energy of the quarks and Pauli exclusion, and requires color confinement. It produces surprisingly conceptually rich results.
The model begins by filling energy levels in a bag one quark at a time, minimizing the energy each time with respect to flavor. The bag radius is adjusted to balance a constant external pressure, B. (B is chosen by fits to models of bulk strange matter.) A zero point energy is included. The energy per baryon (in the bulk), the strange quark mass, and the baryon number of the strangelet are left as free parameters. The quarks are non-interacting, and perturbative QCD corrections are ignored. Ignoring coulomb interactions is also acceptable, since Z will typically be very small for these small A systems.
At this point there are several options. Some models fix the strangeness to baryon number ratio, which is clearly not of interest when searching for the most stable state. Some models use expansions of the density of states in order to describe average properties of the system. This has the effect of smoothing out potentially interesting features. I will describe a model by Gilson and Jaffe where the exact density of states is computed. This approach reveals a shell structure analogous to nuclear structure: near a shell closure, stability is greatly enhanced. As a result, S for the most stable configuration is an erratic function of A, and metastable configurations arise for parameters which would be unstable for bulk strange matter.
The procedure for this model is to put in the model constraints and parameters from the bulk in order to obtain a first guess for the bag radius. From this it is possible to numerically solve for the eigenvalues, which in turn provide a better value for the radius. The radius and eigenvalues are adjusted iteratively until the energy is minimized.
The model consists of light, relativistic quarks in a small bag, so the Dirac equation is applied:
with the boundary condition
This boundary condition is equivalent to requiring that no probability flux leaves the bag. For the non relativistic case, however, the wave function must explicitly go to zero at the boundary. Since the strange quarks are heavier, and therefore less relativistic, it is natural to expect that they would be concentrated more toward the center of the strangelet.
Solving the Dirac equation yields the eigenfunctions and transcendental equations for the eigenvalues. Equilibrium will require the Fermi energy to be the same for each species of quark, such that the change in the total energy from the addition of one quark (the chemical potential) will be the same for each;
Integration of the volume term of the density of states provides an expression for the number density of each quark species,
We can now use parameters and expressions appropriate for the bulk ( ) limit in order to obtain a first guess for the radius of the bag. From fits to bulk strange matter a value is chosen for B, the external vacuum pressure or bag parameter. The baryon number and total energy are then
where V is the volume of the bag. This gives a total energy per baryon, again in bulk, of .
These equations can be combined to find and approximate expression for the radius of the bag, ignoring any surface effects, as a function of A, , and , the free parameters of the model. The energy levels are computed numerically from the transcendental eigenvalue equations, and the radius is adjusted accordingly. This continues until the total energy is minimized. The outputs of the model are values for , S, and R.