Nuclear Binding energy

Question

The nuclear binding energy, is the energy that is needed to seperate the nucleons in a nucleus. And binding energy is also defined as the energy given out when a nucleus forms from nucleons.

Also the larger the nucleus is, the more energy is required to break it apart, so why doesn't that mean that larger nuclei are more stable? I mean Uranium has a lower binding energy per nucleon than Iron, but there are many many more nucleons in Uranium that Iron so the total binding energy is going to be much greater.

Basically I don't understand why whether an element gives out energy by fusion or fission (why the lighter element provide energy by fusion not fission and vice versa for heavy elements) depends on binding energy per nucleon and not "total" binding energy

Answer 1

Your basic nuclear reaction conserves the number of nucleons present.¹

That is important, because at a bit less than 1 GeV each the mass of the nucleons dominates the total energy of all these states.

So the only place available to get or lose energy in a reaction is by

1. Changing the flavor of nucleons. Every neutron converted to a proton gets you a neutrino and some gammas (once the positron has captured and annihilated).

2. By changing the total binding energy. Notice that getting one nucleon more bound doesn't help if another one gets less bound by a large amount. The decision to express this in terms of the average binding energy is completely arbitrary because for $N$ total nucleons (which doesn't change, remember?) $E_\text{total} = E_\text{AVG} * N$.

Another questions addresses what is so special about iron that it has the highest binding energy per nucleon.

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¹ This is more or less required by Baryon number conservation in the Standard Model, and we will ignore the need for Baryon non-conservation in most beyond the Standard Model candidate theories.

Answer 2

Nuclear physics is in the realm of quantum mechanics. To first order one can think of "one nucleon" in the collective left over strong interaction field of all the rest, as a potential. As for strong interactions nuclei are indistinguishable to first order the binding energy per nucleon makes sense. The larger this energy the harder to extract a nucleon from the potential.

The total binding energy will depend on second order effects coming in from the Pauli exclusion principle and the electromagnetic repulsion of protons, from clusterings within large nuclei etc. For example there are a large number of unstable isotopes when the electromagnetic repulsion of the protons is not balanced by the neutrality of neutrons or when there are too many neutrons. A simple curve cannot be defined the way it can with the binding energy per nucleon.

The binding energy per nucleon is a necessary condition to see whether fusion can happen, but it is not sufficient to make sure the reaction will happen as all quantum numbers have to be satisfied . In this article there is a discussion of conditions and possibilities for fusion.