We can use the shell model to give a qualitative explanation for one more observation, although the details are beyond the scope of this text. FIGURE 42.12 shows the neutron and proton potential-energy wells of a high-$Z$ nucleus. In a nucleus with many protons, the electrostatic potential energy lifts the proton potential-energy well higher than the neutron potential-energy well. Protons and neutrons now have a different set of energy levels.

FIGURE 42.12 The proton energy levels are displaced upward in a high-$Z$ nucleus.

As a nucleus is “built,” by the addition of protons and neutrons, the proton energy well and the neutron energy well must fill to just about the same height. If there were neutrons in energy levels above vacant proton levels, the nucleus would lower its energy by using beta decay to change the neutron into a proton. Similarly, beta decay would change a proton into a neutron if there were a vacant neutron energy level beneath a filled proton level. The net result of beta decay is to keep the levels on both sides filled to just about the same energy.

Because the neutron potential-energy well starts at a lower energy, more neutron states are available than proton states. Consequently, a high-$Z$ nucleus will have more neutrons than protons. This conclusion is consistent with our observation in Figure 42.4 that $N>Z$ for heavy nuclei.