mass defect

TheInfoList

OR:

Nuclear binding energy in
experimental physics Experimental physics is the category of disciplines and sub-disciplines in the field of physics that are concerned with the observation of physical Phenomenon, phenomena and experiments. Methods vary from discipline to discipline, from simple exper ...
is the minimum
energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...
that is required to disassemble the
nucleus Nucleus (plural, : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA ...
of an
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...
into its constituent protons and neutrons, known collectively as
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number, mass number (nucleon number). Until the 1960s, n ...
s. The binding energy for stable nuclei is always a positive number, as the nucleus must gain energy for the nucleons to move apart from each other. Nucleons are attracted to each other by the
strong nuclear force The strong interaction or strong force is a fundamental interaction that confines Quark, quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is calle ...
. In theoretical nuclear physics, the nuclear binding energy is considered a negative number. In this context it represents the energy of the nucleus relative to the energy of the constituent nucleons when they are infinitely far apart. Both the experimental and theoretical views are equivalent, with slightly different emphasis on what the binding energy means. The
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
of an atomic nucleus is less than the sum of the individual masses of the free
constituent Constituent or constituency may refer to: Politics * An individual voting, voter within an electoral district, state, community, or organization * Advocacy group or constituency * Constituent assembly * Constituencies of Namibia Other meanings * ...
protons and neutrons. The difference in mass can be calculated by the Einstein equation, , where ''E'' is the nuclear binding energy, ''c'' is the speed of light, and ''m'' is the difference in mass. This 'missing mass' is known as the mass defect, and represents the energy that was released when the nucleus was formed. The term "nuclear binding energy" may also refer to the energy balance in processes in which the nucleus splits into fragments composed of more than one nucleon. If new
binding energy In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
is available when light nuclei fuse (
nuclear fusion Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is mani ...
), or when heavy nuclei split (
nuclear fission Nuclear fission is a nuclear reaction, reaction in which the atomic nucleus, nucleus of an atom splits into two or more smaller atomic nucleus, nuclei. The fission process often produces gamma ray, gamma photons, and releases a very large ...
), either process can result in release of this binding energy. This energy may be made available as ''nuclear energy'' and can be used to produce electricity, as in
nuclear power Nuclear power is the use of nuclear reactions to produce electricity. Nuclear power can be obtained from nuclear fission, nuclear decay and nuclear fusion reactions. Presently, the vast majority of electricity from nuclear power is produced b ...
, or in a
nuclear weapon A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either nuclear fission, fission (fission bomb) or a combination of fission and nuclear fusion, fusion reactions (Thermonuclear weapon, thermonu ...
. When a large nucleus
splits A split (commonly referred to as splits or the splits) is a physical position in which the legs are in line with each other and extended in opposite directions. Splits are commonly performed in various athletic activities, including dance, figu ...
into pieces, excess energy is emitted as gamma rays and the kinetic energy of various ejected particles (
nuclear fission Nuclear fission is a nuclear reaction, reaction in which the atomic nucleus, nucleus of an atom splits into two or more smaller atomic nucleus, nuclei. The fission process often produces gamma ray, gamma photons, and releases a very large ...
products). These nuclear binding energies and forces are on the order of one million times greater than the electron binding energies of light atoms like hydrogen.

Introduction

Nuclear energy

An absorption or release of nuclear energy occurs in
nuclear reaction In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic phy ...
s or
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive Decay chain, disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nucl ...
; those that absorb energy are called
endothermic In thermochemistry, an endothermic process () is any thermodynamic process with an increase in the enthalpy (or internal energy ) of the system.Oxtoby, D. W; Gillis, H.P., Butler, L. J. (2015).''Principle of Modern Chemistry'', Brooks Cole. p. ...
reactions and those that release energy are
exothermic In thermodynamics, an exothermic process () is a thermodynamic process or Chemical reaction, reaction that releases energy from the system to its Environment (systems), surroundings, usually in the form of heat, but also in a form of light (e.g. ...
reactions. Energy is consumed or released because of differences in the nuclear binding energy between the incoming and outgoing products of the nuclear transmutation. The best-known classes of exothermic nuclear transmutations are
nuclear fission Nuclear fission is a nuclear reaction, reaction in which the atomic nucleus, nucleus of an atom splits into two or more smaller atomic nucleus, nuclei. The fission process often produces gamma ray, gamma photons, and releases a very large ...
and
nuclear fusion Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is mani ...
. Nuclear energy may be released by 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 fusion, when light nuclei like
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...
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 might ultimately be converted to thermal energy, emitted as heat. 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.

The nuclear force

Electrons The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...
and nuclei are kept together by
electrostatic attraction Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that quantifies the amount of force between two stationary, electric charge, electrically charged particles. The electric force between char ...
(negative attracts positive). Furthermore, electrons are sometimes shared by neighboring atoms or transferred to them (by processes of
quantum physics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
); this link between atoms is referred to as a
chemical bond A chemical bond is a lasting attraction between atoms or ions that enables the formation of Molecule, molecules and crystals. The bond may result from the Coulomb's law, electrostatic force between oppositely charged ions as in Ionic bonding, ...
and is responsible for the formation of all
chemical compounds A chemical compound is a chemical substance composed of many identical molecules (or molecular entity, molecular entities) containing atoms from more than one chemical element held together by chemical bonds. A homonuclear molecule, molecule co ...
. The
electric force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that quantifies the amount of force between two stationary, electric charge, electrically charged particles. The electric force between char ...
does not hold nuclei together, because all
protons A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Ele ...
carry a positive charge and repel each other. If two protons were touching, their repulsion force would be almost 40 Newton. Because each of the
neutrons The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be relate ...
carries total charge zero, a proton could electrically attract a neutron if the proton could induce the neutron to become electrically polarized. However, having the neutron between two protons (so their mutual repulsion decreases to 10 N) would attract the neutron only for an electric quadrupole arrangement. Higher multipoles, needed to satisfy more protons, cause weaker attraction, and quickly become implausible. After the proton and neutron magnetic moments were measured and verified, it was apparent that their magnetic forces might be 20 or 30 newtons, attractive if properly oriented. A pair of protons would do 10−13 joules of work to each other as they approach – that is, they would need to release energy of 0.5 MeV in order to stick together. On the other hand, once a pair of nucleons magnetically stick, their external fields are greatly reduced, so it is difficult for many nucleons to accumulate much magnetic energy. Therefore, another force, called the ''
nuclear force The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nucle ...
'' (or ''residual strong force'') holds the
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number, mass number (nucleon number). Until the 1960s, n ...
s of nuclei together. This force is a residuum of the
strong interaction The strong interaction or strong force is a fundamental interaction that confines Quark, quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is calle ...
, which binds quarks into nucleons at an even smaller level of distance. The fact that nuclei do not clump together (fuse) under normal conditions suggests that the nuclear force must be weaker than the electric repulsion at larger distances, but stronger at close range. 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. Unlike
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
or electrical forces, the nuclear force is effective only at very short distances. At greater distances, the electrostatic force dominates: the protons repel each other because they are positively charged, and like charges repel. For that reason, the protons forming the nuclei of ordinary
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...
—for instance, in a balloon filled with hydrogen—do not combine to form
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol (chemistry), symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert gas, inert, monatomic gas and the first in the noble gas gr ...
(a process that also would require some protons to combine with electrons and become
neutrons The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be relate ...
). They cannot get close enough for the nuclear force, which attracts them to each other, to become important. Only under conditions of extreme
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
and
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
(for example, within the core of a
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
), can such a process take place.

Physics of nuclei

There are around 94 naturally occurring elements on earth. The
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...
s of each element have a
nucleus Nucleus (plural, : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA ...
containing a specific number of
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s (always the same number for a given element), and some number of
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...
s, which is often roughly a similar number. Two atoms of the same element having different numbers of neutrons are known as
isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbe ...
s of the element. Different isotopes may have different properties – for example one might be stable and another might be unstable, and gradually undergo
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive Decay chain, disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nucl ...
to become another element. The hydrogen nucleus contains just one proton. Its isotope deuterium, or heavy hydrogen, contains a proton and a neutron. Helium contains two protons and two neutrons, and carbon, nitrogen and oxygen – six, seven and eight of each particle, respectively. However, a helium nucleus weighs less than the sum of the weights of the two heavy hydrogen nuclei which combine to make it. The same is true for carbon, nitrogen and oxygen. For example, the carbon nucleus is slightly lighter than three helium nuclei, which can combine to make a carbon nucleus. This difference is known as the mass defect.

Mass defect

Mass defect (also called "mass deficit") is the difference between the mass of an object and the sum of the masses of its constituent particles. Discovered by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
in 1905, it can be explained using his formula ''E'' = ''mc''2, which describes the equivalence of energy and mass. The decrease in mass is equal to the energy emitted in the reaction of an atom's creation divided by ''c''2. By this formula, adding energy also increases mass (both weight and inertia), whereas removing energy decreases mass. For example, a helium atom containing four nucleons has a mass about 0.8% less than the total mass of four hydrogen atoms (each containing one nucleon). The helium nucleus has four nucleons bound together, and the
binding energy In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
which holds them together is, in effect, the missing 0.8% of mass. If a combination of particles contains extra energy—for instance, in a molecule of the explosive TNT—weighing it reveals some extra mass, compared to its end products after an explosion. (The end products must be weighed after they have been stopped and cooled, however, as the extra mass must escape from the system as heat before its loss can be noticed, in theory.) On the other hand, if one must inject energy to separate a system of particles into its components, then the initial mass is less than that of the components after they are separated. In the latter case, the energy injected is "stored" as
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
, which shows as the increased mass of the components that store it. This is an example of the fact that energy of all types is seen in systems as mass, since mass and energy are equivalent, and each is a "property" of the other. 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'' = Δ''mc''2 for each nucleus, where Δ''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 lighter elements, the energy that can be released by assembling them from lighter elements decreases, and energy can be released when they fuse. This is true for nuclei lighter than
iron Iron () is a chemical element with Symbol (chemistry), symbol Fe (from la, Wikt:ferrum, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 element, group 8 of the periodic table. It is, Abundance ...
/
nickel Nickel is a chemical element with Chemical symbol, symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and Ductility, ductile transition metal. Pure nickel is chemically reactive bu ...
. For heavier nuclei, more energy is needed to bind them, and that energy may be released by breaking them up into fragments (known as
nuclear fission Nuclear fission is a nuclear reaction, reaction in which the atomic nucleus, nucleus of an atom splits into two or more smaller atomic nucleus, nuclei. The fission process often produces gamma ray, gamma photons, and releases a very large ...
). Nuclear power is generated at present by breaking up uranium nuclei in nuclear power reactors, and capturing the released energy as heat, which is converted to electricity. As a rule, very light elements can fuse comparatively easily, and very heavy elements can break up via fission very easily; elements in the middle are more stable and it is difficult to make them undergo either fusion or fission in an environment such as a laboratory. The reason the trend reverses after iron is the growing positive charge of the nuclei, which tends to force nuclei to break up. It is resisted by the strong nuclear interaction, which holds nucleons together. The electric force may be weaker than the strong nuclear force, but the strong force has a much more limited range: in an iron nucleus, each proton repels the other 25 protons, while the nuclear force only binds close neighbors. So for larger nuclei, the electrostatic forces tend to dominate and the nucleus will tend over time to break up. As nuclei grow bigger still, this disruptive effect becomes steadily more significant. By the time
polonium Polonium is a chemical element A chemical element is a species of atoms that have a given number of protons in their atomic nucleus, nuclei, including the pure Chemical substance, substance consisting only of that species. Unlike chemical ...
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.) Because of this process, nuclei with more than 94 protons are not found naturally on Earth (see
periodic table The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an Cultural i ...
). The isotopes beyond uranium (atomic number 92) with the longest half-lives are plutonium-244 (80 million years) and curium-247 (16 million years).

Nuclear reactions in the sun

The
nuclear fusion Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is mani ...
process works as follows: five billion years ago, the new Sun formed when gravity pulled together a vast cloud of
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...
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 Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association ...
proposed. Thermal energy appears as the motion of atoms and molecules: the higher the temperature of a collection of particles, 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 hydrogen nuclei to overcome their electric repulsion, and bring them into the short range of the attractive
nuclear force The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nucle ...
, nuclei began to stick together. When this began to happen, protons combined into deuterium and then helium, with some protons changing in the process to neutrons (plus positrons, positive electrons, which combine with electrons and annihilate into gamma-ray photons). This released nuclear energy now keeps up the high temperature of the Sun's core, and the heat also keeps the gas pressure high, keeping the Sun at its present size, and stopping gravity from compressing it any more. There is now a stable balance between gravity and pressure. 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 that combine small nuclei into bigger ones, typically to heat boilers, whose steam could turn turbines and produce electricity. 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. For elements that weigh more than
iron Iron () is a chemical element with Symbol (chemistry), symbol Fe (from la, Wikt:ferrum, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 element, group 8 of the periodic table. It is, Abundance ...
(a nucleus with 26 protons), the fusion process no longer releases energy. In even heavier nuclei energy is consumed, not released, by combining similarly sized nuclei. With such large nuclei, overcoming the electric repulsion (which affects all protons in the nucleus) requires more energy than 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. With the nuclei of elements heavier than
lead Lead is a chemical element with the Symbol (chemistry), symbol Pb (from the Latin ) and atomic number 82. It is a heavy metals, heavy metal that is density, denser than most common materials. Lead is Mohs scale of mineral hardness#Intermediate ...
, the electric repulsion is so strong that some of them spontaneously eject positive fragments, usually nuclei of helium that form stable (
alpha particles Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 atomic nucleus, nucleus. They are generally produced in the process of alpha decay, but may ...
). This spontaneous break-up is one of the forms of radioactivity exhibited by some nuclei. Nuclei heavier than lead (except for
bismuth Bismuth is a chemical element with the Symbol (chemistry), symbol Bi and atomic number 83. It is a post-transition metal and one of the pnictogens, with chemical properties resembling its lighter group 15 siblings arsenic and antimony. Elemental ...
,
thorium Thorium is a weakly radioactive decay, radioactive metallic chemical element with the Symbol (chemistry), symbol Th and atomic number 90. Thorium is silvery and tarnishes black when it is exposed to air, forming thorium dioxide; it is moderatel ...
, and
uranium Uranium is a chemical element with the Symbol (chemistry), symbol U and atomic number 92. It is a silvery-grey metal in the actinide series of the periodic table. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons. ...
) spontaneously break up too quickly to appear in nature as primordial elements, though they can be produced artificially or as intermediates in the
decay chain In nuclear science, the decay chain refers to a series of radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive Decay chain, disintegration, or nuclear disintegration) is the process by which an unsta ...
s of heavier elements. Generally, the heavier the nuclei are, the faster they spontaneously decay. Iron nuclei are the most stable nuclei (in particular
iron-56 Iron-56 (56Fe) is the most common isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), a ...
), 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. Alternatively, one can break up the heaviest ones—nuclei of uranium or plutonium—into smaller fragments, and that is what
nuclear reactor A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plants for electricity generation and in nuclear marine propulsion. Heat from nu ...
s do.

Nuclear binding energy

An example that illustrates nuclear binding energy is the nucleus of 12C (carbon-12), which contains 6 protons and 6 neutrons. The protons are all positively charged and repel each other, but the
nuclear force The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nucle ...
overcomes the repulsion and causes them to stick together. The nuclear force is a close-range force (it is strongly attractive at a distance of 1.0 fm and becomes extremely small beyond a distance of 2.5 fm), and virtually no effect of this force is observed outside the nucleus. The nuclear force also pulls neutrons together, or neutrons and protons. 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 that form them, and the difference—by the formula —gives the binding energy of the nucleus.

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 having a nucleus consisting of 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. 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 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. The protons of hydrogen 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 that 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. Producing helium from normal hydrogen would be practically impossible on earth because of the difficulty in creating
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two Stable isotope ratio, stable isotopes of hydrogen (the other being Hydrogen atom, protium, or hydrogen-1). The atomic nucleus, nucleus of a deuterium ato ...
. Research is being undertaken on developing a process using deuterium and
tritium Tritium ( or , ) or hydrogen-3 (symbol T or H) is a rare and radioactive isotope of hydrogen with half-life about 12 years. The nucleus of tritium (t, sometimes called a ''triton'') contains one proton and two neutrons, whereas the nucleu ...
. The earth's oceans contain a large amount of deuterium that could be used and tritium can be made in the reactor itself from
lithium Lithium (from el, λίθος, lithos, lit=stone) is a chemical element with the Symbol (chemistry), symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard temperature and pressure, standard conditions, it ...
, and furthermore the helium product does not harm the environment, so some consider nuclear fusion a good alternative to supply our energy needs. Experiments to carry out this form of fusion have so far only partially succeeded. Sufficiently hot deuterium and tritium must be 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.

The binding energy maximum and ways to approach it by decay

In the main
isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbe ...
s of light elements, 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. 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.
Iron-56 Iron-56 (56Fe) is the most common isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), a ...
(56Fe) is the most efficiently bound nucleus meaning that it has the least average mass per nucleon. However,
nickel-62 Nickel-62 is an isotope of nickel having 28 protons and 34 neutrons. It is a stable isotope, with the highest nuclear binding energy, binding energy per nucleon of any known nuclide (8.7945 MeV). It is often stated that iron-56, 56Fe is the "mos ...
is the most tightly bound nucleus in terms of binding energy per nucleon. (Nickel-62's higher binding energy does not translate to a larger mean mass loss than 56Fe, because 62Ni has a slightly higher ratio of neutrons/protons than does iron-56, and the presence of the heavier neutrons increases nickel-62's average mass per nucleon). 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 nucleons. If changing one proton into a neutron or one neutron into a proton increases the stability (lowering the mass), then this will happen through
beta decay In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar (nuclide), isobar of that ...
, meaning the nuclide will be radioactive. The two methods for this conversion are mediated by the weak force, and involve types of
beta decay In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar (nuclide), isobar of that ...
. In the simplest beta decay, neutrons are converted to protons by emitting a negative electron and an antineutrino. This is always possible outside a nucleus 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 The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 ''elementary charge, e'', a spin (physics), spin of 1/2 (the same as the electron), and the same Electron rest ...
and an electron neutrino. 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 Electron capture (K-electron capture, also K-capture, or L-electron capture, L-capture) is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron, usually from the K or L electron shells. Thi ...
, in which a proton simply electron captures one of the atom's K orbital electrons, emits a neutrino, and becomes a neutron. Among the heaviest nuclei, starting with tellurium nuclei (element 52) containing 104 or more nucleons, electric forces may be so destabilizing that entire chunks of the nucleus may be ejected, usually as
alpha particles Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 atomic nucleus, nucleus. They are generally produced in the process of alpha decay, but may ...
, which consist of two protons and two neutrons (alpha particles are fast helium nuclei). (
Beryllium-8 Beryllium-8 (8Be, Be-8) is a radionuclide A radionuclide (radioactive nuclide, radioisotope or radioactive isotope) is a nuclide that has excess nuclear energy, making it unstable. This excess energy can be used in one of three ways: emitted f ...
also decays, very quickly, into two alpha particles.) This type of decay becomes more and more probable as elements rise in atomic weight past 104. The curve of binding energy is a graph that plots the binding energy per nucleon against atomic mass. This curve has its main peak at iron and nickel and then slowly decreases again, and also a narrow isolated peak at helium, which is more stable than other low-mass nuclides. The heaviest nuclei in more than trace quantities in nature, uranium 238U, are unstable, but having a half-life of 4.5 billion years, close to the age of the Earth, they are still relatively abundant; they (and other nuclei heavier than helium) have formed in stellar evolution events like supernova explosions preceding the formation of the solar system. The most common isotope of thorium, 232Th, also undergoes alpha 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 that are also unstable, starting a chain of decays that ends in some stable isotope of lead.

Calculation of nuclear binding energy

Calculation can be employed to determine the nuclear binding energy of nuclei. The calculation involves determining the ''mass defect'', converting it into energy, and expressing the result 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. 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.
Guides

Fission and fusion

Nuclear energy is released by the splitting (fission) or merging (fusion) of the nuclei of
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...
(s). The conversion of nuclear
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
energy In physics, energy (from Ancient Greek: wikt:ἐνέργεια#Ancient_Greek, ἐνέργεια, ''enérgeia'', “activity”) is the physical quantity, quantitative physical property, property that is #Energy transfer, transferred to a phy ...
to a form of energy, which can remove some mass when the energy is removed, is consistent with the
mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
formula: :Δ''E'' = Δ''m'' ''c''2, where :Δ''E'' = energy release, :Δ''m'' =
mass defect Nuclear binding energy in experimental physics is the minimum energy that is required to disassemble the atomic nucleus, nucleus of an atom into its constituent protons and neutrons, known collectively as nucleons. The binding energy for stable n ...
, and ''c'' = the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
in a
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Ph ...
. Nuclear energy was first discovered by French physicist
Henri Becquerel Antoine Henri Becquerel (; 15 December 1852 – 25 August 1908) was a French engineer, physicist, Nobel laureate, and the first person to discover evidence of radioactivity. For work in this field he, along with Marie Skłodowska-Curie and Pier ...
in 1896, when he found that photographic plates stored in the dark near
uranium Uranium is a chemical element with the Symbol (chemistry), symbol U and atomic number 92. It is a silvery-grey metal in the actinide series of the periodic table. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons. ...
were blackened like
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 Picometre, picometers to 10 Nanometre, nanometers, corresponding to frequency, ...
plates (X-rays had recently been discovered in 1895).
Nickel-62 Nickel-62 is an isotope of nickel having 28 protons and 34 neutrons. It is a stable isotope, with the highest nuclear binding energy, binding energy per nucleon of any known nuclide (8.7945 MeV). It is often stated that iron-56, 56Fe is the "mos ...
has the highest binding energy per
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number, mass number (nucleon number). Until the 1960s, n ...
of any
isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbe ...
. If an atom of lower average binding energy per nucleon is changed into two atoms of higher average binding energy per nucleon, energy is emitted. (The average here is the weighted average.) Also, if two atoms of lower average binding energy fuse into an atom of higher average binding energy, energy is emitted. The chart shows that fusion, or combining, of
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...
nuclei to form heavier atoms releases energy, as does fission of uranium, the breaking up of a larger nucleus into smaller parts. Nuclear energy is released by three ''exoenergetic'' (or
exothermic In thermodynamics, an exothermic process () is a thermodynamic process or Chemical reaction, reaction that releases energy from the system to its Environment (systems), surroundings, usually in the form of heat, but also in a form of light (e.g. ...
) processes: *
Radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive Decay chain, disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nucl ...
, where a neutron or proton in the
radioactive Radioactive decay (also known as nuclear decay, radioactivity, radioactive Decay chain, disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nucl ...
nucleus decays spontaneously by emitting either particles,
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
(gamma rays), or both. Note that for radioactive decay, it is not strictly necessary for the binding energy to increase. What is strictly necessary is that the mass decrease. If a neutron turns into a proton and the energy of the decay is less than 0.782343 MeV, the difference between the masses of the neutron and proton multiplied by the speed of light squared, (such as
rubidium-87 Rubidium Rubidium is the chemical element with the Symbol (chemistry), symbol Rb and atomic number 37. It is a very soft, whitish-grey solid in the alkali metal group, similar to potassium and caesium. Rubidium is the first alkali metal in the ...
decaying to
strontium-87 The alkaline earth metal strontium (38Sr) has four stable, naturally occurring isotopes: 84Sr (0.56%), 86Sr (9.86%), 87Sr (7.0%) and 88Sr (82.58%). Its standard atomic weight is 87.62(1). Only 87Sr is radiogenic; it is produced by decay from the ...
), the average binding energy per nucleon will actually decrease. * 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, and some neutrons The energy producing nuclear interaction of light elements requires some clarification. Frequently, all light element energy-producing nuclear interactions are classified as fusion, however by the given definition above fusion requires that the products include a nucleus that is heavier than the reactants. Light elements can experience energy producing nuclear interactions by fusion or fission. All energy producing nuclear interactions between two Hydrogen isotopes and between hydrogen and helium-3 are fusion as the product of these interactions include a heavier nucleus. However, the energy producing nuclear interaction of a neutron with Lithium–6 produces Hydrogen-3 and Helium-4, each a lighter nucleus. By the definition above, this nuclear interaction is fission, not fusion. When fission is caused by a neutron, as in this case, it is called induced fission. Light element energy-producing nuclear interactions: ''Fusion'' :1H + 1H → 2He Q ≈ 1.44 MeV :1H + 2H → 3He Q ≈ 5.52 MeV :2H + 2H → 3H + p+ Q ≈ 4.08 MeV :2H + 2H → 3He + n Q ≈ 3.27 MeV :2H + 3H → 4He + n Q ≈ 17.53 MeV :2H + 3He → 4He + p+ Q ≈ 18.34 MeV :3He + 3He → 4He + p+ + p+ Q ≈ 12.85 MeV :3He + 6Li → 4He + 4He + p+ Q ≈ 22.36 MeV ''Fission'' :6Li + p+4He + 3He Q ≈ 4.02 MeV :6Li + 2H → 4He + 4He Q ≈ 11.18 MeV :6Li + 3He → 4He + 4He + p+ Q ≈ 0.94 MeV :7Li + p+4He + 4He Q ≈ 17.34 MeV :7Li + 2H → 4He + 4He + n Q ≈ 15.11 MeV :11B + p+4He + 4He + 4He Q ≈ 8.68 MeV

Binding energy for atoms

The binding energy of an atom (including its electrons) is not exactly the same as the binding energy of the atom's nucleus. The measured mass deficits of
isotopes Isotopes are two or more types of atoms that have the same atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the ...
are always listed as mass deficits of the neutral
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...
s of that isotope, and mostly in . As a consequence, the listed mass deficits are not a measure of the stability or binding energy of isolated nuclei, but for the whole atoms. There is a very practical reason for this, namely that it is very hard to totally
ion An ion () is an atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has n ...
ize heavy elements, i.e. strip them of all of their
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...
s. This practice is useful for other reasons, too: stripping all the electrons from a heavy unstable nucleus (thus producing a bare nucleus) changes the lifetime of the nucleus, or the nucleus of a stable neutral atom can likewise become unstable after stripping, indicating that the nucleus cannot be treated independently. Examples of this have been shown in bound-state β decay experiments performed at the GSI heavy ion accelerator. This is also evident from phenomena like
electron capture Electron capture (K-electron capture, also K-capture, or L-electron capture, L-capture) is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron, usually from the K or L electron shells. Thi ...
. Theoretically, in orbital models of heavy atoms, the electron orbits partially inside the nucleus (it does not ''orbit'' in a strict sense, but has a non-vanishing probability of being located inside the nucleus). A
nuclear decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive Decay chain, disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nucl ...
happens to the nucleus, meaning that properties ascribed to the nucleus 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 Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...
up to
sodium Sodium is a chemical element with the Symbol (chemistry), symbol Na (from Latin ''natrium'') and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 element, group 1 of the ...
is observed to exhibit generally increasing binding energy per nucleon as the
atomic mass The atomic mass (''m''a or ''m'') is the mass of an atom. Although the SI unit of mass is the kilogram (unit), kilogram (symbol: kg), atomic mass is often expressed in the non-SI unit dalton (unit), dalton (symbol: Da) – equivalently, unif ...
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. Helium-4 and oxygen-16 are particularly stable exceptions to the trend (see figure on the right). This is because they are doubly magic, meaning their protons and neutrons both fill their respective nuclear shells. The region of increasing binding energy is followed by a region of relative stability (saturation) in the sequence from about mass 30 through about mass 90. 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 The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...
increases. Finally, in the heavier elements, there is a gradual 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 Nickel-62 is an isotope of nickel having 28 protons and 34 neutrons. It is a stable isotope, with the highest nuclear binding energy, binding energy per nucleon of any known nuclide (8.7945 MeV). It is often stated that iron-56, 56Fe is the "mos ...
is the most tightly bound nucleus (per nucleon), followed by
iron-58 Naturally occurring iron Iron () is a chemical element with Symbol (chemistry), symbol Fe (from la, Wikt:ferrum, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 element, group 8 of the perio ...
and
iron-56 Iron-56 (56Fe) is the most common isotope Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), a ...
. 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
supernova A supernova is a powerful and luminous explosion of a star. It has the plural form supernovae or supernovas, and is abbreviated SN or SNe. This transient astronomical event occurs during the last stellar evolution, evolutionary stages of a mass ...
e and in the final stages of
silicon burning In astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements ...
in stars. However, it is not binding energy per defined nucleon (as defined above), which controls exactly which nuclei are made, because within stars, neutrons and protons can inter-convert to release even more energy per generic nucleon. In fact, it has been argued that
photodisintegration Photodisintegration (also called phototransmutation, or a photonuclear reaction) is a nuclear process in which an atomic nucleus absorbs a high-energy gamma ray, enters an excited state, and immediately decays by emitting a subatomic particle. The ...
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. This favors the creation of 56Fe, the nuclide with the lowest mass per nucleon. However, at high temperatures not all matter will be in the lowest energy state. This energetic maximum should also hold for ambient conditions, say and , for neutral condensed matter consisting of 56Fe atoms—however, in these conditions nuclei of atoms are inhibited from fusing into the most stable and low energy state of matter. Elements with high binding energy per nucleon, like iron and nickel, cannot undergo fission, but they can theoretically undergo fusion with hydrogen, deuterium, helium, and carbon, for instance: :Ni + C → Se Q = 5.467 MeV 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 Naturally occurring nickel Nickel is a chemical element with Chemical symbol, symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and Ductility, ductile transition metal. Pure ni ...
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 the supernova explodes. However, nickel-56 then decays to
cobalt-56 Naturally occurring cobalt Cobalt is a chemical element with the Symbol (chemistry), symbol Co and atomic number 27. As with nickel, cobalt is found in the Earth's crust only in a chemically combined form, save for small deposits found in alloy ...
within a few weeks, then this radioisotope finally decays to iron-56 with a half life of about 77.3 days. The radioactive decay-powered light curve of such a process has been observed to happen in
type II supernova A Type II supernova (plural: ''supernovae'' or ''supernovas'') results from the rapid collapse and violent explosion of a massive star. A star must have at least 8 times, but no more than 40 to 50 times, the solar mass, mass of the Sun () to ...
e, such as
SN 1987A SN 1987A was a type II supernova in the Large Magellanic Cloud, a dwarf satellite galaxy of the Milky Way. It occurred approximately from Earth and was the closest observed supernova since Kepler's Supernova. 1987A's light reached Earth on Febr ...
. 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.

Binding energy and nuclide masses

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 that have different range characteristics. The attractive nuclear force (
strong nuclear force The strong interaction or strong force is a fundamental interaction that confines Quark, quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is calle ...
), 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 that results between further added nucleons as a result of additional strong force interactions. Such nuclei become increasingly 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 Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is mani ...
produces energy by combining the very lightest elements into more tightly bound elements (such as hydrogen into
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol (chemistry), symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert gas, inert, monatomic gas and the first in the noble gas gr ...
), and
nuclear fission Nuclear fission is a nuclear reaction, reaction in which the atomic nucleus, nucleus of an atom splits into two or more smaller atomic nucleus, nuclei. The fission process often produces gamma ray, gamma photons, and releases a very large ...
produces energy by splitting the heaviest elements (such as
uranium Uranium is a chemical element with the Symbol (chemistry), symbol U and atomic number 92. It is a silvery-grey metal in the actinide series of the periodic table. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons. ...
and
plutonium Plutonium is a radioactive decay, radioactive chemical element with the Symbol (chemistry), symbol Pu and atomic number 94. It is an actinide metal of silvery-gray appearance that tarnishes when exposed to air, and forms a dull coating plutoni ...
) into more tightly bound elements (such as
barium Barium is a chemical element A chemical element is a species of atoms that have a given number of protons in their atomic nucleus, nuclei, including the pure Chemical substance, substance consisting only of that species. Unlike chemical com ...
and
krypton Krypton (from grc, κρυπτός, translit=kryptos 'the hidden one') is a chemical element with the symbol (chemistry), symbol Kr and atomic number 36. It is a colorless, odorless, tasteless noble gas that occurs in trace element, trace amount ...
). The nuclear fission of a few light elements (such as Lithium) occurs because Helium-4 is a product and a more tightly bound element than slightly heavier elements. Both processes produce energy as the sum of the masses of the products is less than the sum of the masses of the reacting nuclei. As seen above in the example of deuterium, nuclear binding energies are large enough that they may be easily measured as fractional
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a body. It was traditionally believed to be related to the physical quantity, quantity of matter in a Physical object, physical body, until the discovery of the atom and par ...
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 Nucleus (plural, : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA ...
. 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 In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
formula 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
dalton Dalton may refer to: Science * Dalton (crater), a lunar crater * Dalton (program), chemistry software * Dalton (unit) (Da), the atomic mass unit * John Dalton, chemist, physicist and meteorologist Entertainment * Dalton (Buffyverse), minor cha ...
(1 Da) is defined as 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 Da, 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 semi-empirical formula for the binding energy (''E''B) per nucleon is: $\frac = a - \frac - \frac - \frac \pm \frac$ where the coefficients are given by: $a = 14.0$; $b = 13.0$; $c = 0.585$; $d = 19.3$; $e = 33$. The first term $a$ is called the saturation contribution and ensures that the binding energy per nucleon is the same for all nuclei to a first approximation. The term $-b/A^$ 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 $-cZ^2/A^$ is the Coulomb electrostatic repulsion; this becomes more important as $Z$ increases. The symmetry correction term $-d\left(N-Z\right)^2/A^2$ 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 $\pm e/A^$ is purely empirical; it is + for even–even nuclei and − for
odd–odd nuclei In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physi ...
. When A is odd, the pairing term is identically zero.

Example values deduced from experimentally measured atom nuclide masses

The following table lists some binding energies and mass defect values.Jagdish K. Tuli, Nuclear Wallet Cards, 7th edition, April 2005, Brookhaven National Laboratory, US National Nuclear Data Center Notice also that we use 1  Da = . To calculate the binding energy we use the formula ''Z'' (''m''p + ''m''e) + ''N'' ''m''n − ''m''nuclide where ''Z'' denotes the number of protons in the nuclides and ''N'' their number of neutrons. We take , and . The letter ''A'' denotes the sum of ''Z'' 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 (''Z'' + ''N'') ''m''n − ''m''nuclide. The "''total'' binding energy per nucleon" would be this value divided by ''A''. 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 56Fe would release the most energy per nucleon, since it has a larger fraction of protons than 62Ni. However, if nuclei 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. 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. (Note that the Binding Energy given above for hydrogen-1 is the atomic binding energy, not the nuclear binding energy which would be zero.)