14.8 Binding energy

The binding energy of a nucleus is the energy that would be needed to take it apart into its individual protons and neutrons. Binding energy explains the overall trends in nuclear reactions.

Figure 14.2: Binding energy per nucleon. [pdf]

As explained in the previous section, the binding energy $E_{\rm {B}}$ can be found from the mass of the nucleus. The specific binding energy is defined as the binding energy per nucleon, $E_{\rm {B}}$$\raisebox{.5pt}{$/$}$$A$. Figure 14.2 shows the specific binding energy of the nuclei with known masses. The highest specific binding energy is 8.8 MeV, and occurs for $\fourIdx{62}{28}{}{}{\rm {Ni}}$ nickel. Nickel has 28 protons, a magic number. However, non­magic $\fourIdx{58}{26}{}{}{\rm {Fe}}$ and $\fourIdx{56}{26}{}{}{\rm {Fe}}$ are right on its heels.

Nuclei can therefore lower their total energy by evolving towards the nickel-iron region. Light nuclei can “fusion” together into heavier ones to do so. Heavy nuclei can emit alpha particles or fission, fall apart in smaller pieces.

Figure 14.2 also shows that the binding energy of most nuclei is roughly 8 MeV per nucleon. However, the very light nuclei are an exception; they tend to have a quite small binding energy per nucleon. In a light nucleus, each nucleon only experi­ences attraction from a small number of other nucleons. For example, deuterium only has a binding energy of 1.1 MeV per nucleon.

The big exception to the exception is the doubly magic $\fourIdx{4}{2}{}{}{\rm {He}}$ nucleus, the alpha particle. It has a stunning 7.07 MeV binding energy per nucleon, exceeding its immediate neighbors by far.

The $\fourIdx{8}{4}{}{}{\rm {Be}}$ beryllium nucleus is not bad either, also with 7.07 MeV per nucleon, almost exactly as high as $\fourIdx{4}{2}{}{}{\rm {He}}$, though admittedly that is achieved using eight nucleons instead of only four. But clearly, $\fourIdx{8}{4}{}{}{\rm {Be}}$ is a lot more tightly bound than its immediate neighbors.

It is therefore ironic that while various of those neighbors are stable, the much more tightly bound $\fourIdx{8}{4}{}{}{\rm {Be}}$ is not. It falls apart in about 67 as (i.e. 67 10$\POW9,{-18}$ s), a tragic consequence of being able to come neatly apart into two alpha particles that are just a tiny bit more tightly bound. It is the only alpha decay among the light nuclei. It is an exception to the rule that light nuclei prefer to fusion into heavier ones.

But despite its immeasurably short half-life, do not think that $\fourIdx{8}{4}{}{}{\rm {Be}}$ is not important. Without it there would be no life on earth. Because of the absence of stable inter­mediaries, the Big Bang produced no elements heavier than beryllium, (and only trace amounts of that) including no carbon. As Hoyle pointed out, the carbon of life is formed in the inter­ior of aging stars when $\fourIdx{8}{4}{}{}{\rm {Be}}$ captures a third alpha particle, to produce $\fourIdx{12}{6}{}{}{\rm {C}}$, which is stable. This is called the “triple alpha process.” Under the extreme conditions in the inter­ior of collapsing stars, given time this process produces significant amounts of carbon despite the extremely short half-life of $\fourIdx{8}{4}{}{}{\rm {Be}}$. The process is far too slow to have occurred in the Big Bang, however.

For $\fourIdx{12}{6}{}{6}{\rm {C}}$ carbon, the superior number of nucleons has become big enough to overcome the doubly magic advantage of the three corre­sponding alpha particles. Carbon-12’s binding energy is 7.68 MeV per nucleon, greater than that of alpha particles.