The nucleus of an atom consists of an incredibly dense assembly of protons and neutrons. The positively charged protons exert extremely strong electric repulsive forces on each other, so you might wonder how the nucleus keeps from exploding. During the 1930s, physicists found that protons and neutrons also exert an attractive force on each other. This force, one of the fundamental forces of nature, is called the strong force. It is the force that holds the nucleus together.

The primary characteristic of the strong force, other than its strength, is that it is a short-range force. The attractive strong force between two nucleons (a nucleon is either a proton or a neutron; the strong force does not distinguish between them) rapidly decreases to zero if they are separated by more than about 2 fm. This is in sharp contrast to the long-range nature of the electric force.

A reasonable model of the nucleus is to think of the protons and neutrons as particles in a nuclear potential well that is created by the strong force. The diameter of the potential well is equal to the diameter of the nucleus (this varies with atomic mass), and nuclear physics experiments have found that the depth of the potential well is $\approx 50\text{MeV}$.

The real potential well is three-dimensional, but let’s make a simplified model of the nucleus as a one-dimensional potential well. FIGURE 40.17 shows the potential energy of a neutron along an $x$-axis passing through the center of the nucleus. Notice that the zero of energy has been chosen such that a “free” neutron, one outside the nucleus, has $E=0$. Thus the potential energy inside the nucleus is $-50\text{MeV}$. The 8.0 fm diameter shown is appropriate for a nucleus having atomic mass number $A\approx 40$, such as argon or potassium. Lighter nuclei will be a little smaller, heavier nuclei somewhat larger. (The potential-energy diagram for a proton is similar, but is complicated a bit by the electric potential energy.)

FIGURE 40.17 There are four allowed energy levels for a neutron in this nuclear potential well.

A numerical solution of the Schrödinger equation finds the four stationary states shown in Figure 40.17. The wave functions have been omitted, but they look essentially identical to the wave functions in Figure 40.14a. The major point to note is that the allowed energies differ by several million electron volts! These are enormous energies compared to those of an electron in an atom or a semiconductor. But recall that the energies of a particle in a rigid box, $E_n=n^2{h}^2\text{/}8m{L}^2$, are proportional to $1\text{/}{L}^2$. Our previous examples, with nanometer-size boxes, found energies in the eV range. When the box size is reduced to femtometers, the energies jump up into the MeV range.

It often happens that the nuclear decay of a radioactive atom leaves a neutron in an excited state. For example, Figure 40.17 shows a neutron that has been left in the $n=3$ state by a previous radioactive decay. This neutron can now undergo a quantum jump to the $n=1$ ground state by emitting a photon with energy

and wavelength

This photon is $\approx {10}^7$ times more energetic, and its wavelength $\approx {10}^7$ times smaller, than the photons of visible light! These extremely high-energy photons are called gamma rays. Gamma-ray emission is, indeed, one of the primary processes in the decay of radioactive elements.

Our one-dimensional model cannot be expected to give accurate results for the energy levels or gamma-ray energies of any specific nucleus. Nonetheless, this model does provide a reasonable understanding of the energy-level structure in nuclei and correctly predicts that nuclei can emit photons having energies of several million electron volts. This model, when extended to three dimensions, becomes the basis for the shell model of the nucleus in which the protons and neutrons are grouped in various shells analogous to the electron shells around an atom that you remember from chemistry. You can learn more about nuclear physics and the shell model in Chapter 42.