4.5.1: Fission and Fusion Energy

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As noted, in atomic nuclei the constituent nucleons, protons and neutrons, are held together by an attractive interaction called the "strong force". If we want to remove a nucleon from the nucleus, we need to do some work against the strong force. This work is equal to the energy with which the nucleon was "bound" to the nucleus, so we call it the binding energy. If we measure the mass of the nucleus before removing the nucleon, and after removing it, we finf that the mass of the original nucleus is lower than the sum of masses of the nucleus after the operation plus the mass of the removed nucleon. How comes? Well, from the famous Einstein's relation between mass and energy, \(E=m c^{2}\). Note that in the process of removing the nucleus we did work \(W\), so we added energy \(E=W\) to the nucleus + nucleon system. If we convert the work done to mass, \(\Delta m=W / c^{2}\), then we obtain the difference:

\[\Delta m=\left[\left(M_{\text {after removal }}+M_{\text {nucleon }}\right)-M_{\text {original }}\right]=\frac{W}{c^{2}} \]

If we had a magic tool enabling us to remove nucleons one by one, and measuring the work done at each step, and then added up the work done at each step, we would get the total binding energy \(E_{\text {tbe }}\) for the original nucleus. As we say,

\[ E_{\text {tbe }}=\sum_{i=1}^{N-1} W_{i} \]

where \(W i\) is the work we did in each \(i^{\text {h }}\) step of removing nuclei (Note that if there are \(N\) nucleons, we need only \(N\) - 1 steps to completely separate them from one another).

Figure \(\PageIndex{1}\): : Binding energy per nucleon versus the A number (from Wikimedia Commons).

So, by the same token as before, we may conclude:

\[ \Delta M=\sum m_{i}-M_{\text {original }}=\frac{E_{\text {tbe }}}{c^{2}} \]

where the sum is over the masses of all constituent nuclei. The \(\Delta M\) is called "the mass defect" of the given nucleus. The mass defect of all stable nuclei and important radioisotopes are listed in many publications and Web documents - but seldom in mass units, usually as mass defect per nucleon \((\Delta M / N)\) and converted to energy units by multiplying by \(c^{2}-\) then its called "Binding Energy" and given in the units of MeV.

A graph displaying the binding energy per nucleon is shown in Fig. \(\PageIndex{1}\).