10.2 Nuclear Binding Energy

Mass Defect

According to nuclear particle experiments, the total mass of a nucleus $(m_{\text{nuc}})$ is less than the sum of the masses of its constituent nucleons (protons and neutrons). The mass difference, or mass defect, is given by

where $Z{m}_p$ is the total mass of the protons, $\left(A-Z\right)m_n$ is the total mass of the neutrons, and $m_{\text{nuc}}$ is the mass of the nucleus. According to Einstein’s special theory of relativity, mass is a measure of the total energy of a system ($E=m{c}^2$). Thus, the total energy of a nucleus is less than the sum of the energies of its constituent nucleons. The formation of a nucleus from a system of isolated protons and neutrons is therefore an exothermic reaction—meaning that it releases energy. The energy emitted, or radiated, in this process is $\left(\text{Δ}m\right)c^2.$

Now imagine this process occurs in reverse. Instead of forming a nucleus, energy is put into the system to break apart the nucleus (Figure 10.6). The amount of energy required is called the total binding energy (BE), $E_{\text{b}}.$

Experimental results indicate that the binding energy for a nucleus with mass number $A>8$ is roughly proportional to the total number of nucleons in the nucleus, A. The binding energy of a magnesium nucleus (${}^{24}\text{Mg}$), for example, is approximately two times greater than for the carbon nucleus (${}^{12}\text{C}$).

Figure 10.6 The binding energy is the energy required to break a nucleus into its constituent protons and neutrons. A system of separated nucleons has a greater mass than a system of bound nucleons.

Graph of Binding Energy per Nucleon

In nuclear physics, one of the most important experimental quantities is the binding energy per nucleon (BEN), which is defined by

This quantity is the average energy required to remove an individual nucleon from a nucleus—analogous to the ionization energy of an electron in an atom. If the BEN is relatively large, the nucleus is relatively stable. BEN values are estimated from nuclear scattering experiments.

A graph of binding energy per nucleon versus mass number A is given in Figure 10.7. This graph is considered by many physicists to be one of the most important graphs in physics. Two notes are in order. First, typical BEN values range from 6–10 MeV, with an average value of about 8 MeV. In other words, it takes several million electron volts to pry a nucleon from a typical nucleus, as compared to just 13.6 eV to ionize an electron in the ground state of hydrogen. This is why nuclear force is referred to as the “strong” nuclear force.

Second, the graph rises at low A, peaks very near iron $\left(\text{Fe},A=56\right),$ and then tapers off at high A. The peak value suggests that the iron nucleus is the most stable nucleus in nature (it is also why nuclear fusion in the cores of stars ends with Fe). The reason the graph rises and tapers off has to do with competing forces in the nucleus. At low values of A, attractive nuclear forces between nucleons dominate over repulsive electrostatic forces between protons. But at high values of A, repulsive electrostatic forces between forces begin to dominate, and these forces tend to break apart the nucleus rather than hold it together.

Figure 10.7 In this graph of binding energy per nucleon for stable nuclei, the BEN is greatest for nuclei with a mass near ${}^{56}\text{Fe}$. Therefore, fusion of nuclei with mass numbers much less than that of Fe, and fission of nuclei with mass numbers greater than that of Fe, are exothermic processes.

As we will see, the BEN-versus-A graph implies that nuclei divided or combined release an enormous amount of energy. This is the basis for a wide range of phenomena, from the production of electricity at a nuclear power plant to sunlight.