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The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the
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 mas ...
s and neutrons of
atom Every atom is composed of a 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, liquid, gas, and ...
s. Neutrons and protons, both nucleons, are affected by the nuclear force almost identically. Since protons have charge +1 ''e'', they experience an electric force that tends to push them apart, but at short range the attractive nuclear force is strong enough to overcome the electromagnetic force. The nuclear force binds nucleons into atomic nuclei. The nuclear force is powerfully attractive between nucleons at distances of about 0.8  femtometre (fm, or 0.8×10−15
metre The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pr ...
), but it rapidly decreases to insignificance at distances beyond about 2.5 fm. At distances less than 0.7 fm, the nuclear force becomes repulsive. This repulsion is responsible for the size of nuclei, since nucleons can come no closer than the force allows. (The size of an atom, measured in
angstrom The angstromEntry "angstrom" in the Oxford online dictionary. Retrieved on 2019-03-02 from https://en.oxforddictionaries.com/definition/angstrom.Entry "angstrom" in the Merriam-Webster online dictionary. Retrieved on 2019-03-02 from https://www.m ...
s (Å, or 10−10 m), is five orders of magnitude larger). The nuclear force is not simple, though, as it depends on the nucleon spins, has a tensor component, and may depend on the relative momentum of the nucleons. The nuclear force has an essential role in storing energy that is used 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 ...
and
nuclear weapons A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission (fission bomb) or a combination of fission and fusion reactions (thermonuclear bomb), producing a nuclear explosion. Both bo ...
. Work (energy) is required to bring charged protons together against their electric repulsion. This energy is stored when the protons and neutrons are bound together by the nuclear force to form a nucleus. The mass of a nucleus is less than the sum total of the individual masses of the protons and neutrons. The difference in masses is known as the mass defect, which can be expressed as an energy equivalent. Energy is released when a heavy nucleus breaks apart into two or more lighter nuclei. This energy is the electromagnetic potential energy that is released when the nuclear force no longer holds the charged nuclear fragments together. A quantitative description of the nuclear force relies on equations that are partly
empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
. These equations model the internucleon potential energies, or potentials. (Generally, forces within a system of particles can be more simply modeled by describing the system's potential energy; the negative gradient of a
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
is equal to the vector force.) The constants for the equations are phenomenological, that is, determined by fitting the equations to experimental data. The internucleon potentials attempt to describe the properties of nucleon–nucleon interaction. Once determined, any given potential can be used in, e.g., the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
to determine the quantum mechanical properties of the nucleon system. The discovery of the neutron in 1932 revealed that atomic nuclei were made of protons and neutrons, held together by an attractive force. By 1935 the nuclear force was conceived to be transmitted by particles called mesons. This theoretical development included a description of the
Yukawa potential In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential named after the Japanese physicist Hideki Yukawa. The potential is of the form: :V_\text(r)= -g^2\frac, where is a ...
, an early example of a nuclear potential.
Pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
s, fulfilling the prediction, were discovered experimentally in 1947. By the 1970s, the quark model had been developed, by which the mesons and nucleons were viewed as composed of quarks and gluons. By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force.

# Description

While the nuclear force is usually associated with nucleons, more generally this force is felt between
hadron In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the el ...
s, or particles composed of
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
s. At small separations between nucleons (less than ~ 0.7 fm between their centers, depending upon spin alignment) the force becomes repulsive, which keeps the nucleons at a certain average separation. For identical nucleons (such as two neutrons or two protons) this repulsion arises from the Pauli exclusion force. A Pauli repulsion also occurs between quarks of the same
flavour Flavor or flavour is either the sensory perception of taste or smell, or a flavoring in food that produces such perception. Flavor or flavour may also refer to: Science *Flavors (programming language), an early object-oriented extension to Lis ...
from different nucleons (a proton and a neutron).

## Field strength

At distances larger than 0.7 fm the force becomes attractive between spin-aligned nucleons, becoming maximal at a center–center distance of about 0.9 fm. Beyond this distance the force drops exponentially, until beyond about 2.0 fm separation, the force is negligible. Nucleons have a radius of about 0.8 fm. At short distances (less than 1.7 fm or so), the attractive nuclear force is stronger than the repulsive Coulomb force between protons; it thus overcomes the repulsion of protons within the nucleus. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. The nuclear force has a spin-dependent component. The force is stronger for particles with their spins aligned than for those with their spins anti-aligned. If two particles are the same, such as two neutrons or two protons, the force is not enough to bind the particles, since the spin vectors of two particles of the same type must point in opposite directions when the particles are near each other and are (save for spin) in the same quantum state. This requirement for
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s stems from the Pauli exclusion principle. For fermion particles of different types, such as a proton and neutron, particles may be close to each other and have aligned spins without violating the Pauli exclusion principle, and the nuclear force may bind them (in this case, into a deuteron), since the nuclear force is much stronger for spin-aligned particles. But if the particles' spins are anti-aligned the nuclear force is too weak to bind them, even if they are of different types. The nuclear force also has a tensor component which depends on the interaction between the nucleon spins and the angular momentum of the nucleons, leading to deformation from a simple spherical shape.

## Nuclear binding

To disassemble a nucleus into unbound protons and neutrons requires work against the nuclear force. Conversely, energy is released when a nucleus is created from free nucleons or other nuclei: the
nuclear binding energy Nuclear binding energy in experimental physics is the minimum energy that is required to disassemble the nucleus of an atom into its constituent protons and neutrons, known collectively as nucleons. The binding energy for stable nuclei is always ...
. Because of
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 physici ...
(i.e. Einstein's formula ), releasing this energy causes the mass of the nucleus to be lower than the total mass of the individual nucleons, leading to the so-called "mass defect". The nuclear force is nearly independent of whether the nucleons are neutrons or protons. This property is called ''charge independence''. The force depends on whether the
spins The spins (as in having "the spins")Diane Marie Leiva. ''The Florida State University College of Education''Women's Voices on College Drinking: The First-Year College Experience"/ref> is an adverse reaction of intoxication that causes a state of v ...
of the nucleons are parallel or antiparallel, as it has a non-central or ''
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
'' component. This part of the force does not conserve orbital
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sys ...
, which under the action of central forces is conserved. The symmetry resulting in the strong force, proposed by Werner Heisenberg, is that protons and neutrons are identical in every respect, other than their charge. This is not completely true, because neutrons are a tiny bit heavier, but it is an approximate symmetry. Protons and neutrons are therefore viewed as the same particle, but with different
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
quantum numbers; conventionally, the proton is ''isospin up,'' while the neutron is ''isospin down''. The strong force is invariant under SU(2) isospin transformations, just as other interactions between particles are invariant under SU(2) transformations of intrinsic spin. In other words, both isospin and intrinsic spin transformations are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word iso ...
to the SU(2) symmetry group. There are only strong attractions when the total isospin of the set of interacting particles is 0, which is confirmed by experiment. Our understanding of the nuclear force is obtained by scattering experiments and the binding energy of light nuclei. The nuclear force occurs by the exchange of virtual light mesons, such as the virtual
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
s, as well as two types of virtual mesons with spin ( vector mesons), the
rho meson Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the s ...
s and the omega mesons. The vector mesons account for the spin-dependence of the nuclear force in this "virtual meson" picture. The nuclear force is distinct from what historically was known as the weak nuclear force. The weak interaction is one of the four
fundamental interactions In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravity, gravitational and ...
, and plays a role in processes such as
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 of that nuclide. For exam ...
. The weak force plays no role in the interaction of nucleons, though it is responsible for the decay of neutrons to protons and vice versa.

# History

The nuclear force has been at the heart of
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 physics, which studies th ...
ever since the field was born in 1932 with the discovery of the neutron by James Chadwick. The traditional goal of nuclear physics is to understand the properties of atomic nuclei in terms of the "bare" interaction between pairs of nucleons, or nucleon–nucleon forces (NN forces). Within months after the discovery of the neutron, Werner Heisenberg and
Dmitri Ivanenko Dmitri Dmitrievich Ivanenko (russian: Дми́трий Дми́триевич Иване́нко; July 29, 1904 – December 30, 1994) was a Ukrainian theoretical physicist who made great contributions to the physical science of the twentieth cen ...
had proposed proton–neutron models for the nucleus. Heisenberg approached the description of protons and neutrons in the nucleus through quantum mechanics, an approach that was not at all obvious at the time. Heisenberg's theory for protons and neutrons in the nucleus was a "major step toward understanding the nucleus as a quantum mechanical system". Heisenberg introduced the first theory of nuclear exchange forces that bind the nucleons. He considered protons and neutrons to be different quantum states of the same particle, i.e., nucleons distinguished by the value of their nuclear
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
quantum numbers. One of the earliest models for the nucleus was the liquid-drop model developed in the 1930s. One property of nuclei is that the average binding energy per nucleon is approximately the same for all stable nuclei, which is similar to a liquid drop. The liquid-drop model treated the nucleus as a drop of incompressible nuclear fluid, with nucleons behaving like molecules in a liquid. The model was first proposed by
George Gamow George Gamow (March 4, 1904 – August 19, 1968), born Georgiy Antonovich Gamov ( uk, Георгій Антонович Гамов, russian: Георгий Антонович Гамов), was a Russian-born Soviet and American polymath, theore ...
and then developed by
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
, Werner Heisenberg, and
Carl Friedrich von Weizsäcker Carl Friedrich Freiherr von Weizsäcker (; 28 June 1912 – 28 April 2007) was a German physicist and philosopher. He was the longest-living member of the team which performed nuclear research in Germany during the Second World War, under W ...
. This crude model did not explain all the properties of the nucleus, but it did explain the spherical shape of most nuclei. The model also gave good predictions for the binding energy of nuclei. In 1934, Hideki Yukawa made the earliest attempt to explain the nature of the nuclear force. According to his theory, massive
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer sp ...
s ( mesons) mediate the interaction between two nucleons. In light of
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
(QCD)—and, by extension, the Standard Model—meson theory is no longer perceived as fundamental. But the meson-exchange concept (where
hadron In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the el ...
s are treated as elementary particles) continues to represent the best working model for a quantitative ''NN'' potential. The Yukawa potential (also called a screened Coulomb potential) is a potential of the form : $V_\text\left(r\right) = -g^2 \frac,$ where ''g'' is a magnitude scaling constant, i.e., the amplitude of potential, $\mu$ is the Yukawa particle mass, ''r'' is the radial distance to the particle. The potential is monotone increasing, implying that the force is always attractive. The constants are determined empirically. The Yukawa potential depends only on the distance ''r'' between particles, hence it models a central force. Throughout the 1930s a group at
Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...
led by I. I. Rabi developed magnetic-resonance techniques to determine the magnetic moments of nuclei. These measurements led to the discovery in 1939 that the deuteron also possessed an electric quadrupole moment. This electrical property of the deuteron had been interfering with the measurements by the Rabi group. The deuteron, composed of a proton and a neutron, is one of the simplest nuclear systems. The discovery meant that the physical shape of the deuteron was not symmetric, which provided valuable insight into the nature of the nuclear force binding nucleons. In particular, the result showed that the nuclear force was not a central force, but had a tensor character.
Hans Bethe Hans Albrecht Bethe (; July 2, 1906 – March 6, 2005) was a German-American theoretical physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics, and solid-state physics, and who won the 1967 Nobel Prize ...
identified the discovery of the deuteron's quadrupole moment as one of the important events during the formative years of nuclear physics. Historically, the task of describing the nuclear force phenomenologically was formidable. The first semi-empirical quantitative models came in the mid-1950s, such as the Woods–Saxon potential (1954). There was substantial progress in experiment and theory related to the nuclear force in the 1960s and 1970s. One influential model was the Reid potential (1968) : $V_\text\left(r\right) = -10.463 \frac - 1650.6 \frac + 6484.2 \frac,$ where $\mu = 0.7~\text^,$ and where the potential is given in units of
MeV In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacuum ...
. In recent years, experimenters have concentrated on the subtleties of the nuclear force, such as its charge dependence, the precise value of the π''NN'' coupling constant, improved phase-shift analysis, high-precision ''NN'' data, high-precision ''NN'' potentials, ''NN'' scattering at intermediate and high energies, and attempts to derive the nuclear force from QCD.

# The nuclear force as a residual of the strong force

The nuclear force is a residual effect of the more fundamental strong force, or strong interaction. The strong interaction is the attractive force that binds the elementary particles called
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
s together to form the nucleons (protons and neutrons) themselves. This more powerful force, one of the
fundamental forces In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electr ...
of nature, is mediated by particles called
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind ...
s. Gluons hold quarks together through color charge which is analogous to electric charge, but far stronger. Quarks, gluons, and their dynamics are mostly confined within nucleons, but residual influences extend slightly beyond nucleon boundaries to give rise to the nuclear force. The nuclear forces arising between nucleons are analogous to the forces in chemistry between neutral atoms or molecules called London dispersion forces. Such forces between atoms are much weaker than the attractive electrical forces that hold the atoms themselves together (i.e., that bind electrons to the nucleus), and their range between atoms is shorter, because they arise from small separation of charges inside the neutral atom. Similarly, even though nucleons are made of quarks in combinations which cancel most gluon forces (they are "color neutral"), some combinations of quarks and gluons nevertheless leak away from nucleons, in the form of short-range nuclear force fields that extend from one nucleon to another nearby nucleon. These nuclear forces are very weak compared to direct gluon forces ("color forces" or strong forces) inside nucleons, and the nuclear forces extend only over a few nuclear diameters, falling exponentially with distance. Nevertheless, they are strong enough to bind neutrons and protons over short distances, and overcome the electrical repulsion between protons in the nucleus. Sometimes, the nuclear force is called the residual strong force, in contrast to the strong interactions which arise from QCD. This phrasing arose during the 1970s when QCD was being established. Before that time, the ''strong nuclear force'' referred to the inter-nucleon potential. After the verification of the quark model, ''strong interaction'' has come to mean QCD.

# Nucleon–nucleon potentials

Two-nucleon systems such as the deuteron, the nucleus of a deuterium atom, as well as proton–proton or neutron–proton scattering are ideal for studying the ''NN'' force. Such systems can be described by attributing a ''
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
'' (such as the
Yukawa potential In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential named after the Japanese physicist Hideki Yukawa. The potential is of the form: :V_\text(r)= -g^2\frac, where is a ...
) to the nucleons and using the potentials in a
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
. The form of the potential is derived phenomenologically (by measurement), although for the long-range interaction, meson-exchange theories help to construct the potential. The parameters of the potential are determined by fitting to
experimental data Experimental data in science and engineering is data produced by a measurement, test method, experimental design or quasi-experimental design. In clinical research any data produced are the result of a clinical trial. Experimental data may be qual ...
such as the deuteron binding energy or ''NN'' elastic scattering cross sections (or, equivalently in this context, so-called ''NN'' phase shifts). The most widely used ''NN'' potentials are the Paris potential, the Argonne AV18 potential, the CD-Bonn potential, and the Nijmegen potentials. A more recent approach is to develop effective field theories for a consistent description of nucleon–nucleon and three-nucleon forces. Quantum hadrodynamics is an effective field theory of the nuclear force, comparable to QCD for color interactions and QED for electromagnetic interactions. Additionally, chiral symmetry breaking can be analyzed in terms of an effective field theory (called chiral perturbation theory) which allows perturbative calculations of the interactions between nucleons with pions as exchange particles.

## From nucleons to nuclei

The ultimate goal of
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 physics, which studies th ...
would be to describe all nuclear interactions from the basic interactions between nucleons. This is called the ''microscopic'' or ''ab initio'' approach of nuclear physics. There are two major obstacles to overcome: *Calculations in many-body systems are difficult (because of multi-particle interactions) and require advanced computation techniques. *There is evidence that three-nucleon forces (and possibly higher multi-particle interactions) play a significant role. This means that three-nucleon potentials must be included into the model. This is an active area of research with ongoing advances in computational techniques leading to better first-principles calculations of the nuclear shell structure. Two- and three-nucleon potentials have been implemented for nuclides up to '' A'' = 12.

## Nuclear potentials

A successful way of describing nuclear interactions is to construct one potential for the whole nucleus instead of considering all its nucleon components. This is called the ''macroscopic'' approach. For example, scattering of neutrons from nuclei can be described by considering a plane wave in the potential of the nucleus, which comprises a real part and an imaginary part. This model is often called the optical model since it resembles the case of light scattered by an opaque glass sphere. Nuclear potentials can be ''local'' or ''global'': local potentials are limited to a narrow energy range and/or a narrow nuclear mass range, while global potentials, which have more parameters and are usually less accurate, are functions of the energy and the nuclear mass and can therefore be used in a wider range of applications.

* *
Nuclear binding energy Nuclear binding energy in experimental physics is the minimum energy that is required to disassemble the nucleus of an atom into its constituent protons and neutrons, known collectively as nucleons. The binding energy for stable nuclei is always ...

# Bibliography

* Gerald Edward Brown and A. D. Jackson (1976). ''The Nucleon–Nucleon Interaction''. Amsterdam North-Holland Publishing. . * R. Machleidt and I. Slaus
"The nucleon–nucleon interaction"
''J. Phys. G'' 27 (May 2001) R69. . (Topical review.) * E. A. Nersesov (1990). ''Fundamentals of atomic and nuclear physics''. Moscow: Mir Publishers. . *