As an example, we’ll consider the energy levels of low-$Z$ nuclei $(Z<8)$. Because these nuclei have so few protons, we can use a reasonable approximation that neglects the electrostatic potential energy due to proton-proton repulsion and considers only the much larger nuclear potential energy. In that case, the proton and neutron potential-energy wells and energy levels are the same.

FIGURE 42.10 shows the three lowest energy levels and the maximum number of nucleons that the Pauli principle allows in each. Energy values vary from nucleus to nucleus, but the spacing between these levels is several MeV. It’s customary to draw the proton and neutron potential-energy diagrams and energy levels back to back. Notice that the radial axis for the proton potential-energy well points to the right, while the radial axis for the neutron potential-energy well points to the left.

FIGURE 42.10 The three lowest energy levels of a low-$Z$ nucleus. The neutron energy levels are on the left, the proton energy levels on the right.

Let’s apply this model to the $A=12$ isobar. Recall that an isobar is a series of nuclei with the same total number of neutrons and protons. FIGURE 42.11 shows the energy-level diagrams of ${}^{12}\text{B}\text{,}{}^{12}\text{C}$, and ${}^{12}\text{N}$. Look first at ${}^{12}\text{C}$, a nucleus with six protons and six neutrons. You can see that exactly six protons are allowed in the $n=1$ and $n=2$ energy levels. Likewise for the six neutrons. Thus ${}^{12}\text{C}$ has a closed $n=2$ proton shell and a closed $n=2$ neutron shell.

FIGURE 42.11 The $A=12$ isobar has to place 12 nucleons in the lowest available energy levels.

${}^{12}\text{N}$ has seven protons and five neutrons. The sixth proton fills the $n=2$ proton shell, so the seventh proton has to go into the $n=3$ energy level. The $n=2$ neutron shell has one vacancy because there are only five neutrons. ${}^{12}\text{B}$ is just the opposite, with the seventh neutron in the $n=3$ energy level. You can see from the diagrams that the ${}^{12}\text{B}$ and ${}^{12}\text{N}$ nuclei have significantly more energy—by several MeV—than ${}^{12}\text{C}$.

In atoms, electrons in higher energy levels decay to lower energy levels by emitting a photon as the electron undergoes a quantum jump. That can’t happen here because the higher-energy nucleon in ${}^{12}\text{B}$ is a neutron whereas the vacant lower energy level is that of a proton. But an analogous process could occur if a neutron could somehow turn into a proton. And that’s exactly what happens! We’ll explore the details in Section 42.6, but both ${}^{12}\text{B}$ and ${}^{12}\text{N}$ decay into ${}^{12}\text{C}$ in the process known as beta decay.

${}^{12}\text{C}$ is just one of three low-$Z$ nuclei in which both the proton and neutron shells are full. The other two are ${}^4\text{He}$ (filling both $n=1$ shells with $Z=2\text{,}N=2)$ and ${}^{16}\text{O}$ (filling both $n=3$ shells with $Z=8\text{,}N=8$). If the analogy with closed electron shells is valid, these nuclei should be more tightly bound than nuclei with neighboring values of $A$. And indeed, we’ve already noted that the curve of binding energy (Figure 42.6) has peaks at $A=4\text{,}12$, and 16. The shell model of the nucleus satisfactorily explains these peaks. Unfortunately, the shell model quickly becomes much more complex as we go beyond $n=3$. Heavier nuclei do have closed shells, but there’s no evidence for them in the curve of binding energy.