nucleon magnetic moment
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The nucleon magnetic moments are the intrinsic
magnetic dipole moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnet ...
s of 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 mass ...
and
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 ...
, symbols ''μ''p and ''μ''n.
Protons 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 m ...
and neutrons, both
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 (nucleon number). Until the 1960s, nucleons were ...
s, comprise the
nucleus Nucleus ( : 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 Nucl ...
of
atoms 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 both nucleons behave as small
magnets A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nickel ...
whose strengths are measured by their magnetic moments. The nucleons interact with normal matter through either 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 their magnetic moments, with the charged proton also interacting by the
Coulomb force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
. The proton's magnetic moment, surprisingly large, was directly measured in 1933, while the neutron was determined to have a magnetic moment by indirect methods in the mid 1930s. Luis Alvarez and
Felix Bloch Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss-American physicist and Nobel physics laureate who worked mainly in the U.S. He and Edward Mills Purcell were awarded the 1952 Nobel Prize for Physics for "their development of new ...
made the first accurate, direct measurement of the neutron's magnetic moment in 1940. The proton's magnetic moment is exploited to make measurements of molecules by
proton nuclear magnetic resonance Proton nuclear magnetic resonance (proton NMR, hydrogen-1 NMR, or 1H NMR) is the application of nuclear magnetic resonance in NMR spectroscopy with respect to hydrogen-1 nuclei within the molecules of a substance, in order to determine the struct ...
. The neutron's magnetic moment is exploited to probe the atomic structure of materials using scattering methods and to manipulate the properties of neutron beams in particle accelerators. The existence of the neutron's magnetic moment and the large value for the proton magnetic moment indicate the nucleons are not
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ...
s. For an elementary particle to have an intrinsic magnetic moment, it must have both
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
and
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
. The nucleons have
spin-1/2 In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one full ...
 ''ħ'', but the neutron has no net charge. Their magnetic moments were puzzling and defied a valid explanation until the
quark model In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Ei ...
for
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 ...
particles was developed in the 1960s. The nucleons are composed of three quarks, and the magnetic moments of these elementary particles combine to give the nucleons their magnetic moments.


Description

The
CODATA The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the International ...
recommended value for the magnetic moment of the proton is   = . The best available measurement for the value of the magnetic moment of the neutron is Here, N is the
nuclear magneton The nuclear magneton (symbol ''μ'') is a physical constant of magnetic moment, defined in SI units by: :\mu_\text = and in Gaussian CGS units by: :\mu_\text = where: :''e'' is the elementary charge, :''ħ'' is the reduced Planck constant ...
, a
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant ...
and standard unit for the magnetic moments of nuclear components. In
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
, these values are and A magnetic moment is a vector quantity, and the direction of the nucleon's magnetic moment is determined by its spin. The
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
on the neutron that results from an external
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
is towards aligning the neutron's spin vector opposite to the magnetic field vector. The nuclear magneton is the
spin magnetic moment In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. For example, the electron is an elementary spin-1/2 fermion. Quantum electrodynamics gives the ...
of a
Dirac particle In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its Dirac equation#Covariant form and relativistic invariance, free form, or including Dirac equation#Comparison with the ...
, a charged, spin-1/2 elementary particle, with a proton's mass p, in which anomalous corrections are ignored. The nuclear magneton is \mu_\text = \frac, where is the
elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...
, and is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
. The magnetic moment of such a particle is parallel to its spin. Since the neutron has no charge, it should have no magnetic moment by the analogous expression. The non-zero magnetic moment of the neutron thus indicates that it is not an elementary particle. The sign of the neutron's magnetic moment is that of a negatively charged particle. Similarly, that the magnetic moment of the proton, is not almost equal to 1 N indicates that it too is not an elementary particle. Protons and neutrons are composed of
quarks 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 common ...
, and the magnetic moments of the quarks can be used to compute the magnetic moments of the nucleons. Although the nucleons interact with normal matter through magnetic forces, the magnetic interactions are many orders of magnitude weaker than the nuclear interactions. The influence of the neutron's magnetic moment is therefore only apparent for low energy, or slow, neutrons. Because the value for the magnetic moment is inversely proportional to particle mass, the nuclear magneton is about 1/2000 as large as the
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. The Bohr magneton, in SI units is defined as \mu_\m ...
. The magnetic moment of the electron is therefore about 1000 times larger than that of the nucleons. The magnetic moments of the
antiproton The antiproton, , (pronounced ''p-bar'') is the antiparticle of the proton. Antiprotons are stable, but they are typically short-lived, since any collision with a proton will cause both particles to be annihilated in a burst of energy. The exis ...
and
antineutron The antineutron is the antiparticle of the neutron with symbol . It differs from the neutron only in that some of its properties have equal magnitude but opposite sign. It has the same mass as the neutron, and no net electric charge, but has ...
have the same magnitudes as their antiparticles, the proton and neutron, but they have opposite sign.


Measurement


Proton

The anomalously large magnetic moment of the proton was discovered in 1933 by O. Stern at the
University of Hamburg The University of Hamburg (german: link=no, Universität Hamburg, also referred to as UHH) is a public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('' Allgemeines Vo ...
. The proton's magnetic moment was determined by measuring the deflection of a beam of molecular hydrogen by a magnetic field. Stern won the Nobel Prize in 1943 for this discovery.


Neutron

The neutron was discovered in 1932, and since it had no charge, it was assumed to have no magnetic moment. Indirect evidence suggested that the neutron had a non-zero value for its magnetic moment, however, until direct measurements of the neutron's magnetic moment in 1940 resolved the issue. Values for the magnetic moment of the neutron were independently determined by R. Bacher at the
University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...
at
Ann Arbor Anne, alternatively spelled Ann, is a form of the Latin female given name Anna. This in turn is a representation of the Hebrew Hannah, which means 'favour' or 'grace'. Related names include Annie. Anne is sometimes used as a male name in the ...
(1933) and I. Y. Tamm and S. A. Altshuler in the
Soviet Union The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nationa ...
(1934) from studies of the hyperfine structure of atomic spectra. Although Tamm and Altshuler's estimate had the correct sign and order of magnitude (), the result was met with skepticism. By 1934 groups led by Stern, now at the
Carnegie Institute of Technology Carnegie Mellon University (CMU) is a private research university in Pittsburgh, Pennsylvania. One of its predecessors was established in 1900 by Andrew Carnegie as the Carnegie Technical Schools; it became the Carnegie Institute of Technolog ...
in
Pittsburgh Pittsburgh ( ) is a city in the Commonwealth (U.S. state), Commonwealth of Pennsylvania, United States, and the county seat of Allegheny County, Pennsylvania, Allegheny County. It is the most populous city in both Allegheny County and Wester ...
, and I. I. Rabi 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 ...
in New York had independently measured the magnetic moments of the proton and
deuteron Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one n ...
. The measured values for these particles were only in rough agreement between the groups, but the Rabi group confirmed the earlier Stern measurements that the magnetic moment for the proton was unexpectedly large. Since a deuteron is composed of a proton and a neutron with aligned spins, the neutron's magnetic moment could be inferred by subtracting the deuteron and proton magnetic moments. The resulting value was not zero and had a sign opposite to that of the proton. By the late 1930s, accurate values for the magnetic moment of the neutron had been deduced by the Rabi group using measurements employing newly developed
nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
techniques. The value for the neutron's magnetic moment was first directly measured by L. Alvarez and F. Bloch at the
University of California The University of California (UC) is a public land-grant research university system in the U.S. state of California. The system is composed of the campuses at Berkeley, Davis, Irvine, Los Angeles, Merced, Riverside, San Diego, San Franci ...
at Berkeley in 1940. Using an extension of the magnetic resonance methods developed by Rabi, Alvarez and Bloch determined the magnetic moment of the neutron to be . By directly measuring the magnetic moment of free neutrons, or individual neutrons free of the nucleus, Alvarez and Bloch resolved all doubts and ambiguities about this anomalous property of neutrons.


Unexpected consequences

The large value for the proton's magnetic moment and the inferred negative value for the neutron's magnetic moment were unexpected and could not be explained. The unexpected values for the magnetic moments of the nucleons would remain a puzzle until the
quark model In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Ei ...
was developed in the 1960s. The refinement and evolution of the Rabi 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 discovery meant that the physical shape of the deuteron was not symmetric, which provided valuable insight into the nature of 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 ...
binding nucleons. Rabi was awarded the Nobel Prize in 1944 for his resonance method for recording the magnetic properties of atomic nuclei.


Nucleon gyromagnetic ratios

The magnetic moment of a nucleon is sometimes expressed in terms of its -factor, a dimensionless scalar. The convention defining the -factor for composite particles, such as the neutron or proton, is \boldsymbol = \frac \boldsymbol, where is the intrinsic magnetic moment, is the spin
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 syst ...
, and is the effective -factor. While the -factor is dimensionless, for composite particles it is defined relative to the natural unit of the
nuclear magneton The nuclear magneton (symbol ''μ'') is a physical constant of magnetic moment, defined in SI units by: :\mu_\text = and in Gaussian CGS units by: :\mu_\text = where: :''e'' is the elementary charge, :''ħ'' is the reduced Planck constant ...
. For the neutron, is  , so the neutron's -factor, symbol n, is . The gyromagnetic ratio, symbol , of a particle or system is the
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of its magnetic moment to its spin angular momentum, or \boldsymbol = \gamma \boldsymbol. For nucleons, the ratio is conventionally written in terms of the proton mass and charge, by the formula : \gamma = \frac = g \frac. The proton's gyromagnetic ratio is p =  rad⋅s−1T−1. The neutron's gyromagnetic ratio is n = . The gyromagnetic ratio is also the ratio between the observed angular frequency of
Larmor precession In physics, Larmor precession (named after Joseph Larmor) is the precession of the magnetic moment of an object about an external magnetic field. The phenomenon is conceptually similar to the precession of a tilted classical gyroscope in an extern ...
(in rad/s) and the strength of the magnetic field in nuclear magnetic resonance applications, such as in MRI imaging. For this reason, the value of is often given in units of
MHz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one he ...
/ T. The quantities p/(2) =  MHz⋅ T−1 and n/(2) = , called "gamma bar", are therefore convenient.


Physical significance


Larmor precession

When a nucleon is put into a magnetic field produced by an external source, it is subject to a torque tending to orient its magnetic moment parallel to the field (in the case of the neutron, its spin is antiparallel to the field). As with any magnet, this torque is proportional the product of the magnetic moment and the external magnetic field strength. Since the nucleons have spin angular momentum, this torque will cause them to
precess Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
with a well-defined frequency, called the Larmor frequency. It is this phenomenon that enables the measurement of nuclear properties through nuclear magnetic resonance. The Larmor frequency can be determined from the product of the gyromagnetic ratio with the magnetic field strength. Since for the neutron the sign of ''γ''n is negative, the neutron's spin angular momentum precesses counterclockwise about the direction of the external magnetic field.


Proton nuclear magnetic resonance

Nuclear magnetic resonance employing the magnetic moments of protons is used for
Nuclear Magnetic Resonance (NMR) spectroscopy Nuclear magnetic resonance spectroscopy, most commonly known as NMR spectroscopy or magnetic resonance spectroscopy (MRS), is a spectroscopic technique to observe local magnetic fields around atomic nuclei. The sample is placed in a magnetic fie ...
. Since
hydrogen-1 Hydrogen (1H) has three naturally occurring isotopes, sometimes denoted , , and . and are stable, while has a half-life of years. Heavier isotopes also exist, all of which are synthetic and have a half-life of less than one zeptosecond (10 ...
nuclei are within the molecules of many substances, NMR can determine the structure of those molecules.


Determination of neutron spin

The interaction of the neutron's magnetic moment with an external magnetic field was exploited to determine the spin of the neutron. In 1949, D. Hughes and M. Burgy measured neutrons reflected from a ferromagnetic mirror and found that the angular distribution of the reflections was consistent with spin 1/2. In 1954, J. Sherwood, T. Stephenson, and S. Bernstein employed neutrons in a
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throug ...
that used a magnetic field to separate the neutron spin states. They recorded the two such spin states, consistent with a spin-1/2 particle. Until these measurements, the possibility that the neutron was a spin-3/2 particle could not have been ruled out.


Neutrons used to probe material properties

Since neutrons are neutral particles, they do not have to overcome
Coulomb repulsion Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
as they approach charged targets, unlike protons and
alpha particle 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 nucleus. They are generally produced in the process of alpha decay, but may also be pr ...
s. Neutrons can deeply penetrate matter. The magnetic moment of the neutron has therefore been exploited to probe the properties of matter using scattering or diffraction techniques. These methods provide information that is complementary to
X-ray spectroscopy X-ray spectroscopy is a general term for several spectroscopic techniques for characterization of materials by using x-ray radiation. Characteristic X-ray spectroscopy When an electron from the inner shell of an atom is excited by the energy o ...
. In particular, the magnetic moment of the neutron is used to determine magnetic properties of materials at length scales of 1–100  Å using cold or thermal neutrons. B. Brockhouse and C. Shull won the
Nobel Prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
in physics in 1994 for developing these scattering techniques.


Control of neutron beams by magnetism

Without an electric charge, neutron beams cannot be controlled by the conventional electromagnetic methods employed for particle accelerators. The magnetic moment of the neutron allows some control of neutrons using magnetic fields, however, including the formation of polarized neutron beams. One technique employs the fact that cold neutrons will reflect from some magnetic materials at great efficiency when scattered at small grazing angles. The reflection preferentially selects particular spin states, thus polarizing the neutrons. Neutron magnetic mirrors and guides use this
total internal reflection Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflect ...
phenomenon to control beams of slow neutrons.


Nuclear magnetic moments

Since an atomic nucleus consists of a bound state of protons and neutrons, the magnetic moments of the nucleons contribute to the
nuclear magnetic moment The nuclear magnetic moment is the magnetic moment of an atomic nucleus and arises from the spin of the protons and neutrons. It is mainly a magnetic dipole moment; the quadrupole moment does cause some small shifts in the hyperfine structure as ...
, or the magnetic moment for the nucleus as a whole. The nuclear magnetic moment also includes contributions from the orbital motion of the charged protons. The deuteron, consisting of a proton and a neutron, has the simplest example of a nuclear magnetic moment. The sum of the proton and neutron magnetic moments gives 0.879 ''µ''N, which is within 3% of the measured value 0.857 ''µ''N. In this calculation, the spins of the nucleons are aligned, but their magnetic moments offset because of the neutron's negative magnetic moment.


Nature of the nucleon magnetic moments

A magnetic dipole moment can be generated by two possible mechanisms. One way is by a small loop of electric current, called an "Ampèrian" magnetic dipole. Another way is by a pair of magnetic monopoles of opposite magnetic charge, bound together in some way, called a "Gilbertian" magnetic dipole. Elementary magnetic monopoles remain hypothetical and unobserved, however. Throughout the 1930s and 1940s it was not readily apparent which of these two mechanisms caused the nucleon intrinsic magnetic moments. In 1930, Enrico Fermi showed that the magnetic moments of nuclei (including the proton) are Ampèrian. The two kinds of magnetic moments experience different forces in a magnetic field. Based on Fermi's arguments, the intrinsic magnetic moments of elementary particles, including the nucleons, have been shown to be Ampèrian. The arguments are based on basic electromagnetism, elementary
quantum mechanics Quantum mechanics is a fundamental 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 quantum chemistr ...
, and the hyperfine structure of atomic s-state energy levels. In the case of the neutron, the theoretical possibilities were resolved by laboratory measurements of the scattering of slow neutrons from ferromagnetic materials in 1951.


Anomalous magnetic moments and meson physics

The anomalous values for the magnetic moments of the nucleons presented a theoretical quandary for the 30 years from the time of their discovery in the early 1930s to the development of the quark model in the 1960s. Considerable theoretical efforts were expended in trying to understand the origins of these magnetic moments, but the failures of these theories were glaring. Much of the theoretical focus was on developing a nuclear-force equivalence to the remarkably successful theory explaining the small anomalous magnetic moment of the electron. The problem of the origins of the magnetic moments of nucleons was recognized as early as 1935. G. C. Wick suggested that the magnetic moments could be caused by the quantum-mechanical fluctuations of these particles in accordance with Fermi's 1934 theory of beta decay. By this theory, a neutron is partly, regularly and briefly, disassociated into a proton, an electron, and a neutrino as a natural consequence 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 of that nuclide. For ...
. By this idea, the magnetic moment of the neutron was caused by the fleeting existence of the large magnetic moment of the electron in the course of these quantum-mechanical fluctuations, the value of the magnetic moment determined by the length of time the virtual electron was in existence. The theory proved to be untenable, however, when H. Bethe and R. Bacher showed that it predicted values for the magnetic moment that were either much too small or much too large, depending on speculative assumptions. Similar considerations for the electron proved to be much more successful. In
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED), the
anomalous magnetic moment In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. (The ''magnetic moment'', also called '' ...
of a particle stems from the small contributions of quantum mechanical fluctuations to the
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...
of that particle. The g-factor for a "Dirac"
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...
is predicted to be for a negatively charged, spin-1/2 particle. For particles such as the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
, this "classical" result differs from the observed value by around 0.1%; the difference compared to the classical value is the anomalous magnetic moment. The ''g''-factor for the electron is measured to be QED is the theory of the mediation of the electromagnetic force by photons. The physical picture is that the ''effective'' magnetic moment of the electron results from the contributions of the "bare" electron, which is the Dirac particle, and the cloud of "virtual", short-lived electron–positron pairs and photons that surround this particle as a consequence of QED. The effects of these quantum mechanical fluctuations can be computed theoretically using
Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s with loops. The one-loop contribution to the anomalous magnetic moment of the electron, corresponding to the first-order and largest correction in QED, is found by calculating the vertex function shown in the diagram on the right. The calculation was discovered by J. Schwinger in 1948. Computed to fourth order, the QED prediction for the electron's anomalous magnetic moment agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron one of the most accurately verified predictions in the history of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. Compared to the electron, the anomalous magnetic moments of the nucleons are enormous. The g-factor for the proton is 5.6, and the chargeless neutron, which should have no magnetic moment at all, has a g-factor of −3.8. Note, however, that the anomalous magnetic moments of the nucleons, that is, their magnetic moments with the expected Dirac particle magnetic moments subtracted, are roughly equal but of opposite sign: , but . The
Yukawa interaction In particle physics, Yukawa's interaction or Yukawa coupling, named after Hideki Yukawa, is an interaction between particles according to the Yukawa potential. Specifically, it is a scalar field (or pseudoscalar field) and a Dirac field of the ...
for nucleons was discovered in the mid-1930s, and this nuclear force is mediated by
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 ...
meson In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...
s. In parallel with the theory for the electron, the hypothesis was that higher-order loops involving nucleons and pions may generate the anomalous magnetic moments of the nucleons. The physical picture was that the ''effective'' magnetic moment of the neutron arose from the combined contributions of the "bare" neutron, which is zero, and the cloud of "virtual" pions and photons that surround this particle as a consequence of the nuclear and electromagnetic forces. The Feynman diagram at right is roughly the first-order diagram, with the role of the virtual particles played by pions. As noted by A. Pais, "between late 1948 and the middle of 1949 at least six papers appeared reporting on second-order calculations of nucleon moments". These theories were also, as noted by Pais, "a flop" they gave results that grossly disagreed with observation. Nevertheless, serious efforts continued along these lines for the next couple of decades, to little success. These theoretical approaches were incorrect because the nucleons are composite particles with their magnetic moments arising from their elementary components, quarks.


Quark model of nucleon magnetic moments

In the
quark model In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Ei ...
for
hadrons 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 ele ...
, the neutron is composed of one up quark (charge +2/3 ''e'') and two down quarks (charge −1/3 ''e'') while the proton is composed of one down quark (charge −1/3 ''e'') and two up quarks (charge +2/3 ''e''). The magnetic moment of the nucleons can be modeled as a sum of the magnetic moments of the constituent quarks, although this simple model belies the complexities of the Standard Model of
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
. The calculation assumes that the quarks behave like pointlike Dirac particles, each having their own magnetic moment, as computed using an expression similar to the one above for the nuclear magneton: \mu_\text = \frac, where the q-subscripted variables refer to quark magnetic moment, charge, or mass. Simplistically, the magnetic moment of a nucleon can be viewed as resulting from the vector sum of the three quark magnetic moments, plus the orbital magnetic moments caused by the movement of the three charged quarks within it. In one of the early successes of the Standard Model (SU(6) theory), in 1964 M. Beg, B. Lee, and A. Pais theoretically calculated the ratio of proton-to-neutron magnetic moments to be −3/2, which agrees with the experimental value to within 3%. The measured value for this ratio is .Mohr, P. J.; Taylor, B. N. and Newell, D. B. (2011)
"The 2010 CODATA Recommended Values of the Fundamental Physical Constants"
(Web Version 6.0). The database was developed by J. Baker, M. Douma, and S. Kotochigova. (2011-06-02). National Institute of Standards and Technology, Gaithersburg, Maryland 20899. Retrieved May 9, 2015.
A contradiction of the quantum mechanical basis of this calculation with the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
led to the discovery of the
color charge Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). The "color charge" of quarks and gluons is completely unrelated to the everyday meanings of colo ...
for quarks by O. Greenberg in 1964. From the
nonrelativistic The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
quantum-mechanical
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
for
baryon In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classif ...
s composed of three quarks, a straightforward calculation gives fairly accurate estimates for the magnetic moments of neutrons, protons, and other baryons. For a neutron, the magnetic moment is given by , where ''μ''d and ''μ''u are the magnetic moments for the down and up quarks respectively. This result combines the intrinsic magnetic moments of the quarks with their orbital magnetic moments and assumes that the three quarks are in a particular, dominant quantum state. The results of this calculation are encouraging, but the masses of the up or down quarks were assumed to be 1/3 the mass of a nucleon. The masses of the quarks are actually only about 1% that of a nucleon. The discrepancy stems from the complexity of the Standard Model for nucleons, where most of their mass originates in the gluon fields, virtual particles, and their associated energy that are essential aspects of the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
. Furthermore, the complex system of quarks and gluons that constitute a nucleon requires a relativistic treatment. Nucleon magnetic moments have been successfully computed from
first principle In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
s, requiring significant computing resources.


See also

* Neutron triple-axis spectrometry * LARMOR neutron microscope * Neutron electric dipole moment *
Aharonov–Casher effect The Aharonov–Casher effect is a quantum mechanical phenomenon predicted in 1984 by Yakir Aharonov and Aharon Casher, in which a traveling magnetic dipole is affected by an electric field. It is dual to the Aharonov–Bohm effect, in which the ...


References


Bibliography

* S. W. Lovesey (1986). Theory of Neutron Scattering from Condensed Matter. Oxford University Press. . * Donald H. Perkins (1982). Introduction to High Energy Physics. Reading, Massachusetts: Addison Wesley, . * John S. Rigden (1987). Rabi, Scientist and Citizen. New York: Basic Books, Inc., . * Sergei Vonsovsky (1975). Magnetism of Elementary Particles. Moscow: Mir Publishers.


External links

* {{Commons category-inline Electric and magnetic fields in matter Magnetic moment Magnetism Magnetostatics
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagne ...
Physical quantities