4.4 Nuclear mass formula

There is more structure in Fig. 4.1 than just a simple linear dependence on $A$. A naive analysis suggests that the following terms should play a rôle:

1. Bulk energy: This is the term studied above, and saturation implies that the energy is proportional to $B_{\text{bulk}}=\alpha A$.
2. Surface energy: Nucleons at the surface of the nuclear sphere have less neighbours, and should feel less attraction. Since the surface area goes with $R^2$, we find $B_{\text{surface}}=-\beta A$.
3. Pauli or symmetry energy: nucleons are fermions (will be discussed later). That means that they cannot occupy the same states, thus reducing the binding. This is found to be proportional to $B_{\text{symm}}=-\gamma {\left(N∕2-Z∕2\right)}^2∕A^2$.
4. Coulomb energy: protons are charges and they repel. The average distance between is related to the radius of the nucleus, the number of interaction is roughly $Z^2$ (or $Z\left(Z-1\right)$). We have to include the term $B_{\text{Coul}}=-𝜖Z^2∕A$.

Taking all this together we fit the formula

to all know nuclear binding energies with $A\ge 16$ (the formula is not so good for light nuclei). The fit results are given in table 4.1.

In Fig. 4.3 we show how well this fit works. There remains a certain amount of structure, see below, as well as a strong difference between neighbouring nuclei. This is due to the superfluid nature of nuclear material: nucleons of opposite momenta tend to anti-align their spins, thus gaining energy. The solution is to add a pairing term to the binding energy,

The results including this term are significantly better, even though all other parameters remain at the same position, see Table 4.2. Taking all this together we fit the formula