The simplest form of shell model is rather unstructured. We consider a Fermi gas of nucleons confined to a nuclear volume . Since neutrons and protons are distinguishable, we consider two independent gases; the particles do not interact and we neglect the Coulomb interaction in the case of protons; and we include a factor of two to account for the spin degeneracy (i.e. two possible spin states in each Fermi gas level). The gas is assumed to be degenerate, so that the lowest possible levels are fully occupied.
The number of nucleons that can be contained in a certain volume of phase space is obtained by dividing that volume by the volume of one state in phase space, :
These states will be filled with Z protons or N neutrons up to some maximum momentum (for Fermi momentum). Thus, for example,
Thus the Fermi momentum is
where z is the density of protons. A similar expression holds for neutrons.
For the nucleus we can take
and find a value of MeV (where is the mass of a nucleon).
Thus the kinetic energy of a nucleon at the top of the Fermi distribution (at the Fermi surface) is about 35 MeV. Such a nucleon is still bound by about 8 MeV, so the shell model potential must be about 43 MeV deep.
The average kinetic energy of a nucleon is
The total kinetic energy of the protons in the gas (i.e. the internal energy of the gas) is
Thus the energy of the nucleons in the nucleus is
We can expand this expression around the symmetrical position N=Z=A/2 to obtain a contribution to the asymmetry energy. Let t=N-Z. Then, expanding in powers of t,
This then justifies the form of the asymmetry term in the semiempirical mass formula. From this one obtains MeV; about half the usual value. (The rest is accounted for in the dependence of the potential well on t).