Nuclear fission
Binding energy

The binding energy of the nucleus is the difference between the rest-mass energy of the nucleus and the rest-mass energy of the neutron and proton nucleons. The binding energy formula includes volume, surface and Coulomb energy terms that include empirically derived coefficients for all three, plus energy ratios of a deformed nucleus relative to a spherical form for the surface and Coulomb terms. Additional terms can be included such as symmetry, pairing, the finite range of the nuclear force, and charge distribution within the nuclei to improve the estimate. Normally binding energy is referred to and plotted as average binding energy per nucleon.

According to Lilley, "The binding energy of a nucleus B is the energy required to separate it into its constituent neutrons and protons." m (A, Z) = Z m H + N m n − B / c 2 {\displaystyle m(\mathbf {A},\mathbf {Z})=\mathbf {Z} m_{H}+\mathbf {N} m_{n}-\mathbf {B} /c^{2}} where A is mass number, Z is atomic number, m is the atomic mass of a hydrogen atom, m is the mass of a neutron, and c is the speed of light. Thus, the mass of an atom is less than the mass of its constituent protons and neutrons, assuming the average binding energy of its electrons is negligible. The binding energy B is expressed in energy units, using Einstein's mass-energy equivalence relationship. The binding energy also provides an estimate of the total energy released from fission.

The curve of binding energy is characterized by a broad maximum near mass number 60 at 8.6 MeV, then gradually decreases to 7.6 MeV at the highest mass numbers. Mass numbers higher than 238 are rare. At the lighter end of the scale, peaks are noted for helium-4, and the multiples such as beryllium-8, carbon-12, oxygen-16, neon-20 and magnesium-24. Binding energy due to the nuclear force approaches a constant value for large A, while the Coulomb acts over a larger distance so that electrical potential energy per proton grows as Z increases. Fission energy is released when a A is larger than 120 nucleus fragments. Fusion energy is released when lighter nuclei combine.

Carl Friedrich von Weizsäcker's semi-empirical mass formula may be used to express the binding energy as the sum of five terms, which are the volume energy, a surface correction, Coulomb energy, a symmetry term, and a pairing term:

B = a v A − a s A 2 / 3 − a c Z 2 A 1 / 3 − a a (N − Z) 2 A ± Δ {\displaystyle B=a_{v}\mathbf {A} -a_{s}\mathbf {A} ^{2/3}-a_{c}{\frac {\mathbf {Z} ^{2}}{\mathbf {A} ^{1/3}}}-a_{a}{\frac {(\mathbf {N} -\mathbf {Z})^{2}}{\mathbf {A} }}\pm \Delta } where the nuclear binding energy is proportional to the nuclear volume, while nucleons near the surface interact with fewer nucleons, reducing the effect of the volume term. According to Lilley, "For all naturally occurring nuclei, the surface-energy term dominates and the nucleus exists in a state of equilibrium." The negative contribution of Coulomb energy arises from the repulsive electric force of the protons. The symmetry term arises from the fact that effective forces in the nucleus are stronger for unlike neutron-proton pairs, rather than like neutron–neutron or proton–proton pairs. The pairing term arises from the fact that like nucleons form spin-zero pairs in the same spatial state. The pairing is positive if N and Z are both even, adding to the binding energy.

In fission there is a preference for fission fragments with even Z, which is called the odd–even effect on the fragments' charge distribution. This can be seen in the empirical fragment yield data for each fission product, as products with even Z have higher yield values. However, no odd–even effect is observed on fragment distribution based on their A. This result is attributed to nucleon pair breaking.

In nuclear fission events the nuclei may break into any combination of lighter nuclei, but the most common event is not fission to equal mass nuclei of about mass 120; the most common event (depending on isotope and process) is a slightly unequal fission in which one daughter nucleus has a mass of about 90 to 100 daltons and the other the remaining 130 to 140 daltons.

Stable nuclei, and unstable nuclei with very long half-lives, follow a trend of stability evident when Z is plotted against N. For lighter nuclei less than N = 20, the line has the slope N = Z, while the heavier nuclei require additional neutrons to remain stable. Nuclei that are neutron- or proton-rich have excessive binding energy for stability, and the excess energy may convert a neutron to a proton or a proton to a neutron via the weak nuclear force, a process known as beta decay.

Neutron-induced fission of U-235 emits a total energy of 207 MeV, of which about 200 MeV is recoverable, Prompt fission fragments amount to 168 MeV, which are easily stopped with a fraction of a millimeter. Prompt neutrons total 5 MeV, and this energy is recovered as heat via scattering in the reactor. However, many fission fragments are neutron-rich and decay via β emissions. According to Lilley, "The radioactive decay energy from the fission chains is the second release of energy due to fission. It is much less than the prompt energy, but it is a significant amount and is why reactors must continue to be cooled after they have been shut down and why the waste products must be handled with great care and stored safely."
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