The semi-empirical mass formula

The mass of a nucleus defined by A and Z is given by

The semi-empirical mass formula, based on the liquid drop model, considers five contributions to the binding energy:

1. The volume term . Since the nuclear force is saturated, each nucleon contributes about 16MeV to the binding of the nucleus.
2. The surface term, which gives the reduction in binding resulting from the reduced binding at the nuclear surface,
3. The Coulomb term, which represents the Coulomb repulsion of the Z(Z-1)/2 pairs of protons in the nucleus. For a spherical nucleus of radius with the charge spread evenly throughout the sphere the Coulomb energy is

   

   For a general charge distribution not too different from the above, this can be parameterised as

   

4. The asymmetry term, which accounts for the difference between proton and neutron number. If there were no Coulomb interaction between protons, we would expect, from symmetry arguments applied to a Fermi gas, to find equal numbers of protons and neutrons. In order to generate the observed neutron excess (in most nuclei) we need to shift nucleons from the `proton side' to the `neutron side' of these two Fermi gases. These neutrons can only be added above the Fermi level, so energy must be put into the system. This is the asymmetry energy which reduces the nuclear binding. Note that the system is symmetrical about N=Z; the same energy would be required to shift nucleons the other way if we require a proton excess. Thus to lowest order, we can expect the energy to vary as ; in addition, the Fermi gas energy level spacing varies as 1/A so that the asymmetry term is

   

5. An empirical term to take into account the observed pairing of nuclei:

   

   We note that of 342 (beta-) stable nuclei in the 1993 mass compilation, there are are 209 with even A, even Z; 70 with odd A, even Z; 59 with even A, odd Z and only 4 with odd A and Z ( H, Li, B, . Clearly pairing enhances stability (or binding energy). This can also be seen, for instance, in the neutron separation energies of neighbouring isotopes, etc.

The binding energy is thus

The coefficients are determined by fitting to a suitably large data set of masses (hence semi-empirical). A typical set is (all values in MeV):