Nuclear binding energy curve

Nuclear binding energy curve

Binding energy per nucleon of common isotopes.

In the periodic table of elements, the series of light elements from hydrogen up to sodium is observed to exhibit generally increasing binding energy per nucleon as the atomic mass increases. This increase is generated by increasing forces per nucleon in the nucleus, as each additional nucleon is attracted by all of the other nucleons, and thus more tightly bound to the whole.

The region of increasing binding energy is followed by a region of relative stability (saturation) in the sequence from magnesium through xenon. In this region, the nucleus has become large enough that nuclear forces no longer completely extend efficiently across its width. Attractive nuclear forces in this region, as atomic mass increases, are nearly balanced by repellent electromagnetic forces between protons, as atomic number increases.

Finally, in elements heavier than xenon, there is a decrease in binding energy per nucleon as atomic number increases. In this region of nuclear size, electromagnetic repulsive forces are beginning to gain against the strong nuclear force.

At the peak of binding energy, nickel-62 is the most tightly-bound nucleus (per nucleon), followed by iron-58 and iron-56.^[1] This is the approximate basic reason why iron and nickel are very common metals in planetary cores, since they are produced profusely as end products in supernovae and in the final stages of silicon burning in stars. However, it is not binding energy per defined nucleon (as defined above) which controls which exact nuclei are made, because within stars, neutrons are free to convert to protons to release even more energy, per generic nucleon, if the result is a stable nucleus with a larger fraction of protons. Thus, iron-56 has most binding energy of any group of 56 nucleons (because of its relatively larger fraction of protons), even while having less binding energy per nucleon than nickel-62, if this binding energy is computed by comparing Ni-62 with its disassembly products of 28 protons and 34 neutrons. In fact, it has been argued that photodisintegration of ⁶²Ni to form ⁵⁶Fe may be energetically possible in an extremely hot star core, due to this beta decay conversion of neutrons to protons.^[2]

It is generally believed that iron-56 is more common than nickel isotopes in the universe for mechanistic reasons, because its unstable progenitor nickel-56 is copiously made by staged build-up of 14 helium nuclei inside supernovas, where it has no time to decay to iron before being released into the interstellar medium in a matter of a few minutes as a star explodes. However, nickel-56 then decays to iron-56 within a few weeks. The gamma ray light curve of such a process has been observed to happen in type IIa supernovae, such as SN1987a. In a star, there are no good ways to create nickel-62 by alpha-addition processes, or else there would presumably be more of this highly-stable nuclide in the universe.

Measuring the binding energy

The existence of a maximum in binding energy in medium-sized nuclei is a consequence of the trade-off in the effects of two opposing forces which have different range characteristics. The attractive nuclear force (strong nuclear force), which binds protons and neutrons equally to each other, has a limited range due to a rapid exponential decrease in this force with distance. However, the repelling electromagnetic force, which acts between protons to force nuclei apart, falls off with distance much more slowly (as the inverse square of distance). For nuclei larger than about four nucleons in diameter, the additional repelling force of additional protons more than offsets any binding energy which results between further added nucleons as a result of additional strong force interactions; such nuclei become less and less tightly bound as their size increases, though most of them are still stable. Finally, nuclei containing more than 209 nucleons (larger than about 6 nucleons in diameter) are all too large to be stable, and are subject to spontaneous decay to smaller nuclei.

Nuclear fusion produces energy by combining the very lightest elements into more tightly-bound elements (such as hydrogen into helium), and nuclear fission produces energy by splitting the heaviest elements (such as uranium and plutonium) into more tightly-bound elements (such as barium and krypton). Both processes produce energy, because middle-sized nuclei are the most tightly bound of all.

As seen above in the example of deuterium, nuclear binding energies are large enough that they may be easily measured as fractional mass deficits, according to the equivalence of mass and energy. The atomic binding energy is simply the amount of energy (and mass) released, when a collection of free nucleons are joined together to form a nucleus.

Nuclear binding energy can be easily computed from the easily measurable difference in mass of a nucleus, and the sum of the masses of the number of free neutrons and protons that make up the nucleus. Once this mass difference, called the mass defect or mass deficiency, is known, Einstein's mass-energy equivalence formula E = mc² can be used to compute the binding energy of any nucleus. (As a historical note, early nuclear physicists used to refer to computing this value as a "packing fraction" calculation.)

For example, the atomic mass unit (1 u) is defined to be 1/12 of the mass of a ¹²C atom—but the atomic mass of a ¹H atom (which is a proton plus electron) is 1.007825 u, so each nucleon in ¹²C has lost, on average, about 0.8% of its mass in the form of binding energy.