Nuclear binding energy

Nuclear binding energy is the energy required to split a nucleus of an atom into its component parts. The component parts are neutrons and protons, which are collectively called nucleons. If the binding energy for the products is higher when light nuclei fuse, or when heavy nuclei split, either of these processes will result in a release of the extra binding energy, and this energy is referred to as nuclear energy or, more loosely, nuclear power.

The mass of an atom's nucleus is always less than the sum of the individual masses of the constituent protons and neutrons. This notable difference is a measure of the nuclear binding energy, which is a result of forces that hold the nucleus together. Because these forces result in the removal of energy when the nucleus is formed, and this energy has mass, mass is removed from the total mass of the original particles, and the mass is missing in the resulting nucleus. This missing mass is known as the mass defect, and represents the energy released when the nucleus is formed. When the nucleus splits into pieces, this energy may be emitted as photons (gamma rays) or as the mass or kinetic energy of a number of different ejected particles. Total mass is conserved throughout the process, and during each nuclear transmutation, the "mass defect" mass is relocated to, or carried away by, other particles which are no longer a part of the original nucleus.

The nuclear binding energies and forces are on the order of a million times greater than the electron binding energies of light atoms like hydrogen.^[1]

The mass defect of a nucleus represents the mass of the energy of binding of the nucleus, and is the difference between the mass of a nucleus and the sum of the masses of the nucleons of which it is composed. Determining the relevant nuclear binding energy encompasses three steps of calculation, which involves the creation of mass defect by removing the mass as released energy.^[2]

Contents 1 Introduction 1.1 Nuclear energy 1.2 Nuclear and chemical energies 1.3 The nuclear force 1.4 Physics of nuclei 1.4.1 Mass defect 1.4.2 Solar binding energy 1.4.3 Combining nuclei 1.5 Nuclear binding energy 1.5.1 Nuclear fusion 1.5.2 The binding energy maximum and ways to approach it by decay 2 Determining nuclear binding energy 2.1 Conversion of mass defect into energy 3 Fission and fusion 4 Binding energy for atoms 5 Nuclear binding energy curve 5.1 Measuring the binding energy 5.2 Semiempirical formula for nuclear binding energy 6 Example values deduced from experimentally measured atom nuclide masses 7 References

Introduction

Nuclear binding energy is explained by the basic principles involved in nuclear physics.

Nuclear energy

An absorption or release of nuclear energy occurs in nuclear reactions or radioactive decay; the former are called endothermic reactions and the latter exothermic reactions. Energy is consumed or liberated because of differences in the nuclear binding energy between the incoming and outgoing products of the nuclear transmutation.^[3]

The best-known classes of exothermic nuclear transmutations are fission and fusion. Nuclear energy may be liberated by atomic fission, when heavy atomic nuclei like uranium and plutonium are broken apart into lighter nuclei. The energy from fission is used to generate electric power in hundreds of locations worldwide. Nuclear energy is also released during atomic fusion, when light nuclei like hydrogen are combined to form heavier nuclei such as helium. The Sun and other stars use nuclear fusion to generate thermal energy which is later radiated from the surface, a type of stellar nucleosynthesis. In any exothermic nuclear process, nuclear mass may ultimately be converted to thermal energy, which is given off as heat and in doing so, carries away the mass with it.

In order to quantify the energy released or absorbed in any nuclear transmutation, one must know the nuclear binding energies of the nuclear components involved in the transmutation.

Nuclear and chemical energies

Nuclear energy is typically hundreds of thousands or millions of times greater than chemical energy or approximately 1% of the mass energy from the Einstein mass formula. The mass of a proton is :

The chemical energy of the hydrogen atom is the separation energy of an electron from a proton. It is given by the Rydberg constant from the Bohr theory of the hydrogen atom :

.

The relative change of mass is the hydrogen chemical energy divided by its mass, here the proton mass :

It is so small as to be unmeasurable directly by weighing, but can be calculated, using the Einstein formula, from the measured chemical energy. Nuclear energy is usually "explained" by a hypothetical "strong force". However, it has been shown ^[4] that it may be obtained by a similar formula with values intermediate between the Einstein mass and the Rydberg constant :

This value is not far from the deuteron binding energy, , which is also the neutron-proton separation energy. The relative change in mass is :

Knowing the formulas characterizing the nuclear and chemical energies, one obtains their ratio :

hundreds of thousands

The two preceding calculated values are comparable with the one million and one per cent ratios evaluated from experimental binding energies.

The symbols used are

Mass energy of the proton

Nuclear energy

Chemical energy

Proton mass

Electron mass :

Fine structure constant :

The nuclear force

Electrons and nuclei are kept together by electric attraction (negative attracts positive). Furthermore, electrons are sometimes shared by neighboring atoms or transferred to them (by processes of quantum physics), and this link between atoms creates many chemical compounds.^[5]

The force of electric attraction does not hold nuclei together, because all protons carry a positive charge and repel each other. Thus, Electric forces do not hold nuclei together, because they act in the opposite direction. It has been established that binding neutrons to nuclei clearly requires a non-electrical attraction.^[5]

Therefore another force, called the nuclear force, holds nuclei together. The nuclear force must be stronger than the electric repulsion at short distances, but weaker far away, or else different nuclei might tend to clump together. Therefore it has short-range characteristics. An analogy to the nuclear force is the force between two small magnets: magnets are very difficult to separate when stuck together, but once pulled a short distance apart, the force between them drops almost to zero.^[5]

Unlike gravity or electrical forces, the nuclear force is effective only at very short distances. At greater distances, the protons repel each other because they are positively charged, and charges of the same kind repel. For that reason, the protons forming the nuclei of ordinary hydrogen; for instance, in a balloon filled with hydrogen; do not combine to form helium (a process which also would require some to combine with electrons and become neutrons). They cannot get close enough for the nuclear force, which attracts them to each other, to become important. Only under conditions of extreme pressure and temperature (for example, within the core of a star), can such a process take place.^[6]

Physics of nuclei

The nuclei of atoms are found in many different sizes. In hydrogen they contain just one proton, in heavy hydrogen a proton and a neutron; in helium, two protons and two neutrons, and in carbon, nitrogen and oxygen - six, seven and eight of each particle, respectively. A helium nucleus weighs less than the sum of the weights of its components. The same phenomenon is found for carbon, nitrogen and oxygen. For example, the carbon nucleus is slightly lighter than three helium nuclei which can be combined to make a carbon nucleus. This is an illustration of the mass defect.

Mass defect

The fundamental reason for the "mass defect" is Albert Einstein's famous formula E = mc², expressing the equivalence of energy and mass. By this formula, adding energy also increases mass (both weight and inertia), whereas removing energy decreases mass.

If a combination of particles contains extra energy - for instance, in a molecule of the explosive TNT - weighing it will reveal some extra mass, compared to its end products after an explosion. On the other hand, if one must inject energy to separate it into its components, the initial weight will be less than that of the components.

The latter scenario is the case with nuclei such as helium: to break them up into protons and neutrons, one must inject energy. On the other hand, if a process existed going in the opposite direction, by which hydrogen atoms could be combined to form helium, then energy would be released. The energy can be computed using E = mc² for each nucleus, where m is the difference between the mass of the helium nucleus and the mass of four protons (plus two electrons, absorbed to create the neutrons of helium).

For elements heavier than oxygen, the energy which can be released by assembling them from lighter elements decreases, up to iron. For nuclei heavier than iron, one actually releases energy by breaking them up into 2 fragments. That is how energy is extracted by breaking up uranium nuclei in nuclear power reactors.

The reason the trend reverses after iron is the growing positive charge of the nuclei. The electric force may be weaker than the nuclear force, but its range is greater: in an iron nucleus, each proton repels 25 other protons, while the nuclear force only binds close neighbors.

As nuclei grow bigger still, this disruptive effect becomes steadily more significant. By the time polonium is reached (84 protons), nuclei can no longer accommodate their large positive charge, but emit their excess protons quite rapidly in the process of alpha radioactivity—the emission of helium nuclei, each containing two protons and two neutrons. (Helium nuclei are an especially stable combination.) This process becomes so rapid that nuclei with more than 94 protons are not found naturally on Earth.

Solar binding energy

The nuclear fusion process works as follows: five billion years ago, the new sun having formed when gravity pulled together a vast cloud of gas and dust, from which the Earth and other planets also arose. The gravitational pull released energy and heated the early Sun, much in the way Helmholtz proposed.

Heat is the motion of atoms and molecules: the higher the temperature, the greater is their velocity and the more violent are their collisions. When the temperature at the center of the newly-formed Sun became great enough for collisions between nuclei to overcome their electric repulsion, nuclei began to stick together and protons were combined into helium, with some protons changing in the process to neutrons (plus positrons, positive electrons, which combine with electrons and are destroyed). This released nuclear energy and kept up the high temperature of the Sun's core, and the heat also kept the gas pressure high, keeping the Sun puffed up and stopping gravity from pulling it together any more.

That is the "nuclear fusion" process which still takes place inside the Sun. Different nuclear reactions may predominate at different stages of the Sun's existence, including the proton-proton reaction and the carbon-nitrogen cycle which involves heavier nuclei, but whose final product is still the combination of protons to form helium.

A branch of physics, the study of controlled nuclear fusion, has tried since the 1950s to derive useful power from nuclear fusion reactions which combine small nuclei into bigger ones, typically to heat boilers, whose steam could turn turbines and produce electricity. Unfortunately, no earthly laboratory can match one feature of the solar powerhouse; the great mass of the Sun, whose weight keeps the hot plasma compressed and confines the nuclear furnace to the Sun's core. Instead, physicists use strong magnetic fields to confine the plasma, and for fuel they use heavy forms of hydrogen, which burn more easily. Magnetic traps can be rather unstable, and any plasma hot enough and dense enough to undergo nuclear fusion tends to slip out of them after a short time. Even with ingenious tricks, the confinement in most cases lasts only a small fraction of a second.

Combining nuclei

Small nuclei that are larger than hydrogen can combine into bigger ones and release energy, but in combining such nuclei, the amount of energy released is much smaller compared to hydrogen fusion. The reason is that while the overall process releases energy from letting the nuclear attraction do its work, energy must first be injected to force together positively charged protons, which also repel each other with their electric charge.^[6]

For elements that weigh more than iron (a nucleus with 26 protons), the fusion process no longer releases energy. In even heavier nuclei energy is consumed, not released, by combining similar sized nuclei. With such large nuclei, overcoming the electric repulsion (which affects all protons in the nucleus) requires more energy than what is released by the nuclear attraction (which is effective mainly between close neighbors). Conversely, energy could actually be released by breaking apart nuclei heavier than iron.^[6]

With the nuclei of elements heavier than lead, the electric repulsion is so strong that some of them spontaneously eject positive fragments, usually nuclei of helium which form very stable combinations (alpha particles). This spontaneous break-up is one of the forms of radioactivity behavior exhibited by some nuclei.^[6]

Nuclei heavier than uranium spontaneously break up too quickly to be found in nature, although they can be produced artificially. Generally, the heavier the nuclei are, the faster they spontaneously decay.^[6]

Iron nuclei (in particular isotope 56) are the most stable nuclei, and the best sources of energy are therefore nuclei whose weights are as far removed from iron as possible. One can combine the lightest ones—nuclei of hydrogen (protons)--to form nuclei of helium, and that is how the Sun generates its energy. Or else one can break up the heaviest ones—nuclei of uranium—into smaller fragments, and that is what nuclear power reactors do.^[6]

Nuclear binding energy

An example that illustrates nuclear binding energy is carbon nucleus of ¹²C, which contains 6 protons and 6 neutrons. The protons are all positively charged and repel each other, but the nuclear force overcomes the repulsion and causes them to stick together. The nuclear force is a close-range force, and virtually no effect of this force is observed outside the nucleus. The nuclear force has a strong dependence on distance—it is a short range force. The nuclear force also pulls neutrons together, or neutrons and protons.^[7]

The energy of the nucleus is negative with regard to the energy of the particles pulled apart to infinite distance (just like the gravitational energy of planets of the solar system), because energy must be utilized to split a nucleus into its individual protons and neutrons. Mass spectrometers have measured the masses of nuclei, which are always less than the sum of the masses of protons and neutrons which form them, and the difference—by the formula E = mc² --gives the binding energy of the nucleus.^[7]

Nuclear fusion

The binding energy of helium is the energy source of the Sun and of most stars. The sun is composed of 74 percent hydrogen (measured by mass), an element whose nucleus is a single proton. Energy is released in the sun when 4 protons combine into a helium nucleus, a process in which two of them are also converted to neutrons.^[7]

The conversion of protons to neutrons is the result of another nuclear force, known as the weak (nuclear) force. The weak force, like the strong force, has a short range, but is much weaker than the strong force. The weak force tries to make the number of neutrons and protons into the most energetically stable configuration. For nuclei containing less than 40 particles, these numbers are usually about equal. Protons and neutrons are closely related and are sometimes collectively known as nucleons. As the number of particles increases toward a maximum of about 209, the number of neutrons to maintain stability begins to outstrip the number of protons, until the ratio of neutrons to protons is about three to two.^[7]

The protons of hydrogen will combine to helium only if they have enough velocity to overcome each other's mutual repulsion sufficiently to get within range of the strong nuclear attraction. This means the fusion only occurs within a very hot gas. Hydrogen hot enough for combining to helium requires an enormous pressure to keep it confined, but suitable conditions exist in the central regions of the Sun, where such pressure is provided by the enormous weight of the layers above the core, pressed inwards by the Sun's strong gravity. The process of combining protons to form helium is an example of nuclear fusion.^[7]

The earth's oceans contain a large amount of hydrogen that could theoretically be used for fusion, and helium byproduct of fusion does not harm the environment, so some consider nuclear fusion to be a good alternative to supply humanities energy needs. Experiments to generate electricity from fusion have so far have only partially succeeded. Sufficiently hot hydrogen must be ionized and confined. One technique is to use very strong magnetic fields, because charged particles (like those trapped in the Earth's radiation belt) are guided by magnetic field lines. Fusion experiments also rely on heavy hydrogen which fuses more easily, and gas densities can be moderate. But even with these techniques far more net energy is consumed by the fusion experiments than is yielded by the process.^[7]

The binding energy maximum and ways to approach it by decay

In the main isotopes of light nuclei, such as carbon, nitrogen and oxygen, the most stable combination of neutrons and of protons are when the numbers are equal (this continues to element 20, calcium). However, in heavier nuclei, the disruptive energy of protons increases, since they are confined to a tiny volume and repel each other. The energy of the strong force holding the nucleus together also increases, but at a slower rate, as if inside the nucleus, only nucleons close to each other are tightly bound, not ones more widely separated.^[7]

The net binding energy of a nucleus is that of the nuclear attraction, minus the disruptive energy of the electric force. As nuclei get heavier than helium, their net binding energy per nucleon (deduced from the difference in mass between the nucleus and the sum of masses of component nucleons) grows more and more slowly, reaching its peak at iron. As nucleons are added, the total nuclear binding energy always increases—but the total disruptive energy of electric forces (positive protons repelling other protons) also increases, and past iron, the second increase outweighs the first. ⁵⁶Fe is the most efficiently bound nucleus.^[7]

To reduce the disruptive energy, the weak interaction allows the number of neutrons to exceed that of protons—for instance, the main isotope of iron has 26 protons and 30 neutrons. Isotopes also exist where the number of neutrons differs from the most stable number for that number of protons. If the ratio of protons is too far from stability, nucleons may spontaneously change from proton to neutron, or neutron to proton.

The two methods for this conversion are mediated by the weak force, and involve types of beta decay. In the simplest beta decay, neutrons are converted to protons by emitting a negative electron and an antineutrino. This is always possible because neutrons are more massive than protons by an equivalent of about 2.5 electrons. In the opposite process, which only happens within a nucleus, and not to free particles, a proton may become a neutron by ejecting a positron. This is permitted if enough energy is available between parent and daughter nuclides to do this (the required energy difference is equal to 1.022 MeV, which is the mass of 2 electrons). If the mass difference between parent and daughter is less than this, a proton-rich nucleus may still convert protons to neutrons by the process of electron capture, in which a proton simply captures one of the atom's K orbital electrons, emits a neutrino, and becomes a neutron.^[7]

Among the heaviest nuclei, starting with tellurium nuclei (element 52) containing 106 or more nucleons, electric forces may be so destabilizing that entire chunks of the nucleus may be ejected, usually as alpha particles which consist of 2 protons and 2 neutrons (alpha particles are fast helium nuclei). Alpha particles are extremely stable. This type of decay becomes more and more probable as elements rise in atomic weight past 106.

The curve of binding energy is a graph which plots the binding energy per nucleon against atomic mass. This curve has its main peak at iron and then slowly decreases again, and also a narrow isolated peak at helium, which as noted is very stable. The heaviest nuclei in nature, uranium ²³⁸U, are unstable, but having a lifetime of 4.5 billion years, close to the age of the Earth, they are still relatively abundant; they (and other nuclei heavier than iron) may have formed in a supernova explosion ^[8] preceding the formation of the solar system. The most common isotope of thorium, ²³²Th, also undergoes α particle emission, and its half-life (time over which half a number of atoms decays) is even longer, by several times. In each of these, radioactive decay produces daughter isotopes which are also unstable, starting a chain of decays which ends in some stable isotope of lead.^[7]

Determining nuclear binding energy

Calculation can be employed to determine the nuclear binding energy of nuclei. The calculation uses three general steps:^[2]

1. Determining the Mass defect
2. Conversion of mass defect into energy
3. Expressing nuclear binding energy as energy per mole of atoms, or as energy per nucleon.

Conversion of mass defect into energy

Mass defect is defined as the difference between the mass of a nucleus, and the sum of the masses of the nucleons of which it is composed. The mass defect is determined by calculating three quantities.^[2] These are: the actual mass of the nucleus, the composition of the nucleus (number of protons and of neutrons), and the masses of a proton and of a neutron. This is then followed by converting the mass defect into energy. This quantity is the nuclear binding energy, however it must be expressed as energy per mole of atoms or as energy per nucleon.^[2]

Fission and fusion

Nuclear energy is released by the splitting (fission) or merging together (fusion) of the nuclei of atom(s). The conversion of nuclear mass-energy to a form of energy which can remove some mass when the energy is removed, is consistent with the mass-energy equivalence formula ΔE = Δmc², in which ΔE = energy release, Δm = mass defect, and c = the speed of light in a vacuum (a physical constant). When this equation is used in this way, the mass "changes" only because it is removed from the system, not because it is converted or destroyed (the removed binding energy retains and accounts for the missing mass, which is a conserved quantity).

Nuclear energy was first discovered by French physicist Henri Becquerel in 1896, when he found that photographic plates stored in the dark near uranium were blackened like X-ray plates (X-rays had recently been discovered in 1895).^[9]

Nuclear chemistry can be used as a form of alchemy to turn lead into gold or change any atom to any other atom (although this may require many intermediate steps).^[8] Radionuclide (radioisotope) production often involves irradiation of another isotope (or more precisely a nuclide), with alpha particles, beta particles, or gamma rays. Nickel-62 has the highest binding energy per nucleon of any isotope. If an atom of lower average binding energy is changed into an atom of higher average binding energy, energy is given off. The chart shows that fusion of hydrogen, the combination to form heavier atoms, releases energy, as does fission of uranium, the breaking up of a larger nucleus into smaller parts. Stability varies between isotopes: the isotope U-235 is much less stable than the more common U-238.

Nuclear energy is released by three exoenergetic (or exothermic) processes:

• Radioactive decay, where a neutron or proton in the radioactive nucleus decays spontaneously by emitting either particles, electromagnetic radiation (gamma rays), neutrinos (or all of them)
• Fusion, two atomic nuclei fuse together to form a heavier nucleus
• Fission, the breaking of a heavy nucleus into two (or more rarely three) lighter nuclei

Binding energy for atoms

The binding energy of an atom (including its electrons) is not the same as the binding energy of the atom's nucleus. The measured mass deficits of isotopes are always listed as mass deficits of the neutral atoms of that isotope, and mostly in MeV. As a consequence, the listed mass deficits are not a measure for the stability or binding energy of isolated nuclei, but for the whole atoms. This has very practical reasons, because it is very hard to totally ionize heavy elements, i.e. strip them of all of their electrons.

This practice is useful for other reasons, too: Stripping all the electrons from a heavy unstable nucleus (thus producing a bare nucleus) will change the lifetime of the nucleus, indicating that the nucleus cannot be treated independently (Experiments at the heavy ion accelerator GSI). This is also evident from phenomena like electron capture. Theoretically, in orbital models of heavy atoms, the electron orbits partially inside the nucleus (it doesn't orbit in a strict sense, but has a non-vanishing probability of being located inside the nucleus).

A nuclear decay happens to the nucleus, meaning that properties ascribed to the nucleus will change in the event. In the field of physics the concept of "mass deficit" as a measure for "binding energy" means "mass deficit of the neutral atom" (not just the nucleus) and is a measure for stability of the whole atom.

Nuclear binding energy curve

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 other nearby 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 the 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 overcome the strong nuclear force attraction.

At the peak of binding energy, nickel-62 is the most tightly bound nucleus (per nucleon), followed by iron-58 and iron-56.^[10] 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 the 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.^[11]

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 fact that the maximum binding energy is found 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 computed from the 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. 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.

Semiempirical formula for nuclear binding energy

For a nucleus with A nucleons, including Z protons and N neutrons, a semiemipirical formula for the binding energy (B.E.) per nucleon (A) is:

where the binding energy is in MeV for the following numerical values of the constants: ; ; ; ; .

The first term is called the saturation contribution and ensures that the B.E. per nucleon is the same for all nuclei to a first approximation. The term is a surface tension effect and is proportional to the number of nucleons that are situated on the nuclear surface; it is largest for light nuclei. The term is the Coulomb electrostatic repulsion; this becomes more important as increases. The symmetry correction term takes into account the fact that in the absence of other effects the most stable arrangement has equal numbers of protons and neutrons; this is because the n-p interaction in a nucleus is stronger than either the n-n or p-p interaction. The pairing term is purely empirical; it is + for even-even nuclei and - for odd-odd nuclei.

Example values deduced from experimentally measured atom nuclide masses

The following table lists some binding energies and mass defect values.^[12] Notice also that we use 1 u = 1 a.m.u = 931.494028(±0.000023) MeV. To calculate the "binding energy" we use the formula P*(m_p+m_e) + N * m_n - m_nuclide where P denotes the number of protons of the nuclides and N its number of neutrons. We take m_p = 938.2723 Mev, m_e = 0.5110 MeV and m_n = 939.5656 MeV. The letter A denotes the sum of P and N (number of nucleons in the nuclide). If we assume the reference nucleon has the mass of a neutron (so that all "total" binding energies calculated are maximal) we could define the total binding energy as the difference from the mass of the nucleus, and the mass of a collection of A free neutrons. In other words, it would be [(P+N)* m_n] - m_nuclide. The "total binding energy per nucleon" would be this value divided by A.

Most strongly bound nuclides atoms

nuclide	P	N	mass excess	total mass	total mass / A	total binding energy / A	mass defect	binding energy	binding energy / A
56Fe	26	30	-60.6054 MeV	55.934937 u	0.9988382 u	9.1538 MeV	0.528479 u	492.275 MeV	8.7906 MeV
58Fe	26	32	-62.1534 MeV	57.933276 u	0.9988496 u	9.1432 MeV	0.547471 u	509.966 MeV	8.7925 MeV
60Ni	28	32	-64.4721 MeV	59.930786 u	0.9988464 u	9.1462 MeV	0.565612 u	526.864 MeV	8.7811 MeV
62Ni	28	34	-66.7461 MeV	61.928345 u	0.9988443 u	9.1481 MeV	0.585383 u	545.281 MeV	8.7948 MeV

⁵⁶Fe has the lowest nucleon-specific mass of the four nuclides listed in this table, but this does not imply it is the strongest bound atom per hadron, unless the choice of beginning hadrons is completely free. Iron releases the largest energy if any 56 nucleons are allowed to build a nuclide—changing one to another if necessary, The highest binding energy per hadron, with the hadrons starting as the same number of protons Z and total nucleons A as in the bound nucleus, is ⁶²Ni. Thus, the true absolute value of the total binding energy of a nucleus depends on what we are allowed to construct the nucleus out of. If all nuclei of mass number A were to be allowed to be constructed of A neutrons, then Fe-56 would release the most energy per nucleon, since it has a larger fraction of protons than Ni-62. However, if nucleons are required to be constructed of only the same number of protons and neutrons that they contain, then nickel-62 is the most tightly bound nucleus, per nucleon.

Some light nuclides resp. atoms

nuclide	P	N	mass excess	total mass	total mass / A	total binding energy / A	mass defect	binding energy	binding energy / A
n	0	1	8.0716 MeV	1.008665 u	1.008665 u	0.0000 MeV	0 u	0 MeV	0 MeV
1H	1	0	7.2890 MeV	1.007825 u	1.007825 u	0.7826 MeV	0.0000000146 u	0.0000136 MeV	13.6 eV
2H	1	1	13.13572 MeV	2.014102 u	1.007051 u	1.50346 MeV	0.002388 u	2.22452 MeV	1.11226 MeV
3H	1	2	14.9498 MeV	3.016049 u	1.005350 u	3.08815 MeV	0.0091058 u	8.4820 MeV	2.8273 MeV
3He	2	1	14.9312 MeV	3.016029 u	1.005343 u	3.09433 MeV	0.0082857 u	7.7181 MeV	2.5727 MeV

In the table above it can be seen that the decay of a neutron, as well as the transformation of tritium into helium-3, releases energy; hence, it manifests a stronger bound new state when measured against the mass of an equal number of neutrons (and also a lighter state per number of total hadrons). Such reactions are not driven by changes in binding energies as calculated from previously fixed N and Z numbers of neutrons and protons, but rather in decreases in the total mass of the nuclide/per nucleon, with the reaction.

References