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{{Short description|In physics, proton and neutron magnetism}}
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{{Use mdy dates|date=December 2022}}

The '''nucleon magnetic moments''' are the intrinsic [[magnetic dipole moment]]s of the [[proton]] and [[neutron]], symbols ''μ''<sub>p</sub> and ''μ''<sub>n</sub>. The [[atomic nucleus|nucleus]] of an [[atom]] comprises protons and neutrons, both [[nucleon]]s that behave as small [[magnets]]. Their magnetic strengths are measured by their magnetic moments.  The nucleons interact with normal matter through either the [[nuclear force]] or their magnetic moments, with the charged proton also interacting by the [[Coulomb force]].

The proton's magnetic moment was directly measured in 1933 by [[Otto Stern]] team in [[University of Hamburg]]. While the neutron was determined to have a magnetic moment by indirect methods in the mid-1930s,  [[Luis Walter Alvarez|Luis Alvarez]] and [[Felix Bloch]] 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]].  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 that nucleons are not [[elementary particle]]s. For an elementary particle to have an intrinsic magnetic moment, it must have both [[Spin (physics)|spin]] and [[electric charge]]. The nucleons have [[spin-1/2|spin ''ħ''/2]], but the neutron has no net charge. Their magnetic moments were puzzling and defied a valid explanation until the [[quark model]] for [[hadron]] 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 ==
[[File:Neutron spin dipole field.jpg|thumbnail|right|Schematic diagram depicting the spin of the neutron as the black arrow and magnetic field lines associated with the neutron's magnetic moment. The spin of the neutron is upward in this diagram, but the magnetic field lines at the center of the dipole are downward.]]
The [[Committee on Data for Science and Technology|CODATA]] recommended value for the magnetic moment of the proton is {{physconst|mup/muN|symbol=yes|after=,}} or {{physconst|mup/muB|symbol=yes|after=.}}  The best available measurement for the value of the magnetic moment of the neutron is {{physconst|mun/muN|symbol=yes|after=.}}<ref name="PDG">{{cite journal |last1=Beringer |first1=J. |collaboration=Particle Data Group |year=2012 |title=Review of Particle Physics, 2013 partial update |journal=Phys. Rev. D |url=http://pdg.lbl.gov/2013/listings/rpp2013-list-n.pdf  |volume=86 |issue=1 |pages=010001 |access-date=May 8, 2015 |doi=10.1103/PhysRevD.86.010001 |bibcode=2012PhRvD..86a0001B|doi-access=free }}</ref> Here, ''μ''<sub>N</sub> is the [[nuclear magneton]], a standard unit for the magnetic moments of nuclear components, and ''μ''<sub>B</sub> is the [[Bohr magneton]], both being [[physical constant]]s. In [[SI units]], these values are {{physconst|mup|symbol=yes}} and {{physconst|mun|symbol=yes|after=.}} A magnetic moment is a vector quantity, and the direction of the nucleon's magnetic moment is determined by its spin.<ref name="Vonsovsky">{{cite book |first=Sergei |last=Vonsovsky |author-link=Sergei Vonsovsky |title=Magnetism of Elementary Particles |url=https://archive.org/details/MagnetismOfElementaryParticles |location=Moscow |publisher=Mir Publishers |year=1975}}</ref>{{rp|p=73}} The [[torque]] on the neutron that results from an external [[magnetic field]] is towards aligning the neutron's spin vector opposite to the magnetic field vector.<ref name="Shankar">{{cite book |first=R. |last=Shankar |author-link=Ramamurti Shankar |year=1994 |title=Principles of Quantum Mechanics |edition=2nd |url=https://link.springer.com/book/10.1007/978-1-4757-0576-8 |url-access=registration |pages=676 |publisher=[[Kluwer Academic]]/[[Plenum Press]] |doi=10.1007/978-1-4757-0576-8 |isbn=978-1-4757-0576-8}}</ref>{{rp|p=385}}

The nuclear magneton is the [[spin magnetic moment]] of a [[Dirac equation|Dirac particle]], a charged, spin-1/2 elementary particle, with a proton's mass {{mvar|m}}<sub>p</sub>, in which [[Anomalous magnetic dipole moment|anomalous corrections]] are ignored.<ref name="Shankar"/>{{rp|p=389}} The nuclear magneton is
<math display="block">\mu_\text{N} = \frac{e \hbar}{2 m_\text{p}},</math>
where {{mvar|e}} is the [[elementary charge]], and {{mvar|ħ}} is the [[reduced Planck constant]].<ref name="BD">{{cite book |last1=Bjorken |first1=J. D. |last2=Drell |first2=S. D. |title=Relativistic Quantum Mechanics |url=https://archive.org/details/relativisticquan0000bjor |url-access=registration |location=New York |publisher=McGraw-Hill |pages=[https://archive.org/details/relativisticquan0000bjor/page/241 241]–246 |year=1964 |isbn=978-0070054936}}</ref> The magnetic moment of such a particle is parallel to its spin.<ref name="Shankar"/>{{rp|p=389}} Since the neutron has no charge, it should have no magnetic moment by the analogous expression.<ref name="Shankar"/>{{rp|p=391}} The non-zero magnetic moment of the neutron thus indicates that it is not an elementary particle.<ref name="Hausser">{{cite book |last=Hausser |first=O. |editor1-first=R. G. |editor1-last=Lerner |editor1-link=Rita G. Lerner |editor2-first=G. L. |editor2-last=Trigg |title=Encyclopedia of Physics |location=Reading, Massachusetts |publisher=Addison-Wesley Publishing Company |year=1981 |pages=679–680 |chapter=Nuclear Moments |isbn=978-0201043136}}</ref> The sign of the neutron's magnetic moment is that of a negatively charged particle. Similarly, that the magnetic moment of the proton, {{physconst|mup/muN|symbol=yes|round=3|ref=no}} is not almost equal to 1&nbsp;{{mvar|μ}}<sub>N</sub> indicates that it too is not an elementary particle.<ref name="BD"/> Protons and neutrons are composed of [[quark model|quarks]], and the magnetic moments of the quarks can be used to compute the magnetic moments of the nucleons.<ref name="Perk">{{cite book |author1-last = Perkins |author1-first = Donald H. |title = Introduction to High Energy Physics |url  = https://archive.org/details/introductiontohi0000perk |url-access = registration |pages=[https://archive.org/details/introductiontohi0000perk/page/201 201–202] |publisher = Addison Wesley |location=Reading, Massachusetts |date = 1982 |isbn = 978-0-201-05757-7}}</ref>

Although the nucleons interact with normal matter through magnetic forces, the magnetic interactions are many orders of magnitude weaker than the nuclear interactions.<ref name="Snow">{{cite journal |last1=Snow |first1=M. |year=2013 |title=Exotic physics with slow neutrons |journal=Physics Today |url=https://physicstoday.scitation.org/doi/full/10.1063/PT.3.1918 |volume=66 |issue=3 |pages=50–55 |access-date=December 11, 2015 |doi=10.1063/PT.3.1918 |bibcode=2013PhT....66c..50S |url-access=subscription }}</ref> The influence of the neutron's magnetic moment is therefore only apparent for low energy, or slow, neutrons.<ref name="Snow"/> 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]]. The [[electron magnetic moment|magnetic moment of the electron]] is therefore about 1000&nbsp;times larger than that of the nucleons.<ref>{{cite web |publisher=NIST |title=CODATA values of the fundamental constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?muemsmunn |access-date=May 8, 2015}}</ref>

The magnetic moments of the [[antiproton]] and  [[antineutron]] have the same magnitudes as their antiparticles,
the proton and neutron, but they have opposite sign.<ref name="KS-RS">{{cite book |last=Schreckenbach |first=K. |editor1-first=R. |editor1-last=Stock |title=Encyclopedia of Nuclear Physics and its Applications |location=Weinheim, Germany |publisher=Wiley-VCH Verlag GmbH & Co. |year=2013 |pages=321–354 |chapter=Physics of the Neutron |isbn=978-3-527-40742-2}}</ref>

==Measurement==
===Proton===
The magnetic moment of the proton was discovered in 1933 by [[Otto Stern]], [[Otto Robert Frisch]] and [[Immanuel Estermann]] at the [[University of Hamburg]].<ref name="FS">{{cite journal |last1=Frisch|first1=R. |last2=Stern|first2=O.|year=1933 |title=Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. I / Magnetic Deviation of Hydrogen Molecules and the Magnetic Moment of the Proton. I | journal=Z. Phys. |url= http://web.ihep.su/owa/dbserv/hw.part2?s_c=FRISCH+1933 | volume=85 |issue=1–2 |pages=4–16 |doi=10.1007/bf01330773 |bibcode = 1933ZPhy...85....4F |s2cid=120793548 }}</ref><ref name="ES1">{{cite journal |last1=Esterman |first1=I. |last2=Stern|first2=O.|year=1933 |title=Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. II / Magnetic Deviation of Hydrogen Molecules and the Magnetic Moment of the Proton. I | journal=Z. Phys. |url= http://web.ihep.su/owa/dbserv/hw.part2?s_c=FRISCH+1933 | volume=85 |issue=1–2 |pages=17–24 |doi=10.1007/bf01330774|bibcode = 1933ZPhy...85...17E |s2cid=186232193 }}</ref><ref>{{Cite journal |last=Ramsey |first=N. F. |date=1988-06-01 |title=Molecular beams: our legacy from Otto Stern |url=https://doi.org/10.1007/BF01384845 |journal=Zeitschrift für Physik D Atoms, Molecules and Clusters |language=en |volume=10 |issue=2 |pages=121–125 |doi=10.1007/BF01384845 |bibcode=1988ZPhyD..10..121R |s2cid=120812185 |issn=1431-5866|url-access=subscription }}</ref>  The proton's magnetic moment was determined by measuring the deflection of a beam of molecular hydrogen by a magnetic field.<ref name="MolBeam">{{cite journal |last1=Toennies |first1=J. P. |last2=Schmidt-Bocking |first2=H. |last3=Friedrich |first3=B. |last4=Lower |first4=J. C. A. |title=Otto Stern (1888–1969): The founding father of experimental atomic physics |arxiv=1109.4864 |doi=10.1002/andp.201100228 |volume=523 |issue=12 |journal=Annalen der Physik |pages=1045–1070 |bibcode=2011AnP...523.1045T |year=2011|s2cid=119204397 }}</ref>  Stern won the [[Nobel Prize in Physics]] in 1943 for this discovery.<ref name="NP-1943">{{cite web | title=The Nobel Prize in Physics 1943 | publisher=Nobel Foundation | url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1943/| access-date=January 30, 2015}}</ref>

===Neutron===
{{see also|Discovery of the neutron}}
The neutron was discovered in 1932,<ref name="Neutron">{{cite journal |last=Chadwick |first=James |year=1932 |title= Existence of a Neutron |journal= [[Proceedings of the Royal Society A]] |volume=136 |issue=830 |pages=692–708 | doi=10.1098/rspa.1932.0112 | bibcode =1932RSPSA.136..692C |doi-access=free }}</ref> 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,<ref name="Breit">{{cite journal |last1=Breit |first1=G. |last2=Rabi |first2=I. I. |year=1934 |title=On the interpretation of present values of nuclear moments |journal=Physical Review |volume=46 |issue=3 |pages=230–231 |doi=10.1103/physrev.46.230 |bibcode=1934PhRv...46..230B}}</ref> however, until direct measurements of the neutron's magnetic moment in 1940 resolved the issue.<ref name="Alvarez">{{cite journal |last1=Alvarez |first1=L. W. |last2=Bloch |first2=F. |year=1940 |title=A quantitative determination of the neutron magnetic moment in absolute nuclear magnetons |journal=[[Physical Review]] |volume=57 |issue=2 |pages=111–122 |doi=10.1103/physrev.57.111 |bibcode=1940PhRv...57..111A}}</ref>

Values for the magnetic moment of the neutron were independently determined by [[Robert Bacher|R.&nbsp;Bacher]]<ref name="RB">{{cite journal |last1=Bacher |first1=R. F.|year=1933 |title=Note on the Magnetic Moment of the Nitrogen Nucleus |journal=Physical Review |volume=43 |issue=12 |pages=1001–1002 |doi=10.1103/physrev.43.1001 |url=https://authors.library.caltech.edu/51310/1/PhysRev.43.1001.pdf |bibcode=1933PhRv...43.1001B}}</ref> at the [[University of Michigan]] at [[Ann Arbor]] (1933) and [[Igor Tamm|I.&nbsp;Y. Tamm]] and [[Semen Altshuler|S.&nbsp;A. Altshuler]]<ref name="AT">{{cite journal |last1=Tamm |first1=I. Y. |last2=Altshuler |first2=S. A. |year=1934 |title=Magnetic moment of the neutron |journal=Doklady Akademii Nauk SSSR |url=http://web.ihep.su/owa/dbserv/hw.part2?s_c=ALTSHULER+1934 |access-date=2015-01-30 |df=dmy-all |volume=8 |pages=455}}</ref> in the [[Soviet Union]] (1934) from studies of the hyperfine structure of atomic spectra. Although Tamm and Altshuler's estimate had the correct sign and order of magnitude ({{nowrap|{{mvar|μ}}<sub>n</sub> {{=}} {{val|-0.5|u={{mvar|μ}}<sub>N</sub>}}}}), the result was met with skepticism.<ref name="Breit"/><ref name="Vonsovsky"/>{{rp|73–75}}

By 1934 groups led by Stern, now at the [[Carnegie Institute of Technology]] in [[Pittsburgh]], and [[Isidor Isaac Rabi|I.&nbsp;I. Rabi]] at [[Columbia University]] in [[New York City|New York]] had independently measured the magnetic moments of the proton and [[deuteron]].<ref name="ES2">{{cite journal |last1=Esterman |first1=I. |last2=Stern |first2=O. |year=1934 |title=Magnetic moment of the deuton |journal=Physical Review |url=http://web.ihep.su/owa/dbserv/hw.part2?s_c=ESTERMAN+1934 |volume=45 |issue=10 |pages=761(A109) |access-date=9 May 2015 |doi=10.1103/PhysRev.45.739 |bibcode=1934PhRv...45..739S|url-access=subscription }}</ref><ref name="RKZ1">{{cite journal |last1=Rabi |first1=I. I. |last2=Kellogg |first2=J. M. |last3=Zacharias |first3=J. R. |year=1934 |title=The magnetic moment of the proton |journal=Physical Review |volume=46 |issue=3 |pages=157–163 |doi=10.1103/physrev.46.157 |bibcode=1934PhRv...46..157R}}</ref><ref name="RKZ2">{{cite journal |last1=Rabi |first1=I. I. |last2=Kellogg |first2=J. M. |last3=Zacharias |first3=J. R. |year=1934 |title=The magnetic moment of the {{not a typo|deuton}} <!-- WARNING: "deuton" is correct; it was one variety of spelling for this particle at the time. --> |journal=Physical Review |volume=46 |issue=3 |pages=163–165 |doi=10.1103/physrev.46.163 |bibcode=1934PhRv...46..163R}}</ref> 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.<ref name="Breit"/><ref name="Rigden">{{cite book |first=John S. |last=Rigden |author-link=John S. Rigden |title=Rabi, Scientist and Citizen |location=New York |publisher=Basic Books, Inc. |pages=99–114 |year=1987 |url=https://books.google.com/books?id=Qgv9Xjv8_LYC&q=rabi+kellogg+zacharias+magnetic+moment+neutron&pg=PA106 |isbn=9780674004351 |access-date=9 May 2015}}</ref> 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.<ref name="Physicsworld">{{cite web |author=J. Rigden |title=Isidor Isaac Rabi: walking the path of God | publisher=[[Physics World]] |date=November 1, 1999| url=https://physicsworld.com/a/isidor-isaac-rabi-walking-the-path-of-god/ | access-date=December 11, 2022}}</ref> 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]] techniques.<ref name="Rigden"/>

The value for the neutron's magnetic moment was first directly measured by [[Luis Walter Alvarez|L. Alvarez]] and [[Felix Bloch|F. Bloch]] at the [[University of California, Berkeley|University of California]] at [[Berkeley, California|Berkeley]] in 1940.<ref name="Alvarez"/> Using an extension of the magnetic resonance methods developed by Rabi, Alvarez and Bloch determined the magnetic moment of the neutron to be {{nowrap|{{mvar|μ}}<sub>n</sub> {{=}} {{val|-1.93|(2)|u={{mvar|μ}}<sub>N</sub>}}}}. 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.<ref name="Trower">{{cite book |last=Ramsey |first=Norman F. |editor-first=W. Peter |editor-last=Trower |title=Discovering Alvarez: Selected works of Luis W. Alvarez with commentary by his students and colleagues |publisher=University of Chicago Press |year=1987 |pages=[https://archive.org/details/discoveringalvar0000alva/page/30 30]–32 |chapter=Chapter 5: The Neutron Magnetic Moment |chapter-url=https://books.google.com/books?id=imidr-iFYCwC&q=alvarez+and+bloch+magnetic+moment&pg=PA31 |isbn=978-0226813042 |access-date=9 May 2015 |url=https://archive.org/details/discoveringalvar0000alva |url-access=registration }}</ref>

===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.<ref name="Breit"/> The unexpected values for the magnetic moments of the nucleons would remain a puzzle until the [[quark model]] was developed in the 1960s.<ref name="Pais">{{cite book |last=Pais |first=Abraham |date=1986 |title=Inward Bound |url=https://archive.org/details/inwardboundofmat00pais_0 |url-access=registration |location=Oxford |publisher=Oxford University Press |page=[https://archive.org/details/inwardboundofmat00pais_0/page/299 299] |isbn=978-0198519973 }}</ref>

The refinement and evolution of the Rabi measurements led to the discovery in 1939 that the deuteron also possessed an [[quadrupole|electric quadrupole moment]].<ref name="Rigden"/><ref name="KRRZ">{{cite journal |last1=Kellogg |first1=J. M. |last2=Rabi |first2=I. I. |last3=Ramsey |first3=N. F. |last4=Zacharias |first4=J. R. |year=1939 |title=An electrical quadrupole moment of the deuteron |journal=Physical Review |volume=55 |issue=3 |pages=318–319 |doi=10.1103/physrev.55.318 |bibcode=1939PhRv...55..318K}}</ref> This electrical property of the deuteron had been interfering with the measurements by the Rabi group.<ref name="Rigden"/> 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.<ref name="Rigden"/> Rabi was awarded the Nobel Prize in 1944 for his resonance method for recording the magnetic properties of atomic nuclei.<ref name="NP-1944">{{cite web |title=The Nobel Prize in Physics 1944 |publisher=Nobel Foundation |url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1944/ |access-date=2015-01-25 |df=dmy-all}}</ref>

== Nucleon gyromagnetic ratios ==
The magnetic moment of a nucleon is sometimes expressed in terms of its [[g-factor (physics)|{{mvar|g}}-factor]], a dimensionless scalar. The convention defining the {{mvar|g}}-factor for composite particles, such as the neutron or proton, is
<math display="block">\boldsymbol{\mu} = \frac{g \mu_\text{N}}{\hbar} \boldsymbol{I},</math>
where {{math|'''''μ'''''}} is the intrinsic magnetic moment, {{math|'''''I'''''}} is the spin [[angular momentum]], and {{mvar|g}} is the effective {{mvar|g}}-factor.<ref name="Povh">{{cite book |last1=Povh |first1=B. |last2=Rith |first2=K. |last3=Scholz |first3=C. |last4=Zetsche |first4=F. |title=Particles and Nuclei: An Introduction to the Physical Concepts |location=Berlin |publisher=Springer-Verlag |pages=74–75, 259–260 |year=2002 |isbn=978-3-540-43823-6 |url=https://books.google.com/books?id=HC_qCAAAQBAJ&pg=PA389 |access-date=10 May 2015}}</ref> While the {{mvar|g}}-factor is dimensionless, for composite particles it is defined relative to the [[nuclear magneton]]. For the neutron, {{math|'''''I'''''}} is {{nowrap|{{sfrac|1|2}} {{mvar|ħ}}}}, so the neutron's {{mvar|g}}-factor is {{physconst|gn|symbol=yes|after=,}} while the proton's g-factor is {{physconst|gp|symbol=yes|after=.}}

The [[gyromagnetic ratio]], symbol {{mvar|[[Gamma|γ]]}}, of a particle or system is the [[ratio]] of its magnetic moment to its spin angular momentum, or
<math display="block">\boldsymbol{\mu} = \gamma \boldsymbol{I}.</math>

For nucleons, the ratio is conventionally written in terms of the proton mass and charge, by the formula
: <math>\gamma = \frac{g \mu_\text{N}}{\hbar} = g \frac{e}{2m_\text{p}}.</math>

The neutron's gyromagnetic ratio is {{physconst|gamman|symbol=yes|after=.}}  The proton's gyromagnetic ratio is {{physconst|gammap|symbol=yes|after=.}}  The gyromagnetic ratio is also the ratio between the observed angular frequency of [[Larmor precession]] and the strength of the magnetic field in nuclear magnetic resonance applications,<ref>{{cite book |title=NMR spectroscopy explained |last=Jacobsen |first=Neil E. |location=Hoboken, New Jersey |publisher=Wiley-Interscience |year=2007 |isbn=9780471730965 |url=http://books.scholarsportal.info/viewdoc.html?id=/ebooks/ebooks2/wiley/2011-12-13/1/9780470173350 |access-date=May 8, 2015}}</ref> such as in [[Magnetic resonance imaging|MRI imaging]]. For this reason, the quantity ''γ''/2''π'' with the unit [[MHz]]/[[Tesla (unit)|T]] is often given. The quantities {{mvar|γ}}<sub>n</sub>/(2{{mvar|π}}) = {{val|-29.1646931|(69)|u=[[MHz]]/[[Tesla (unit)|T]]}}<ref>{{cite web |publisher=NIST |title=CODATA values of the fundamental constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?gammanbar |access-date=September 17, 2022}}</ref> and {{mvar|γ}}<sub>p</sub>/(2{{mvar|π}}) = {{val|42.5774806|(10)}}&nbsp;MHz⋅[[Tesla (unit)|T]]<sup>−1</sup>,<ref>{{cite web|publisher=NIST|title=CODATA values of the fundamental constants|url=http://physics.nist.gov/cgi-bin/cuu/Value?gammapbar|access-date=December 10, 2022}}</ref>   called "gamma bar", are therefore convenient.<ref name="MRI_CRC">{{cite book |first1=E. |last1=Berry | first2=A. J. |last2=Bulpitt |title=Fundamentals of MRI: An Interactive Learning Approach |location=Boca Raton, Florida |publisher=[[CRC Press]] |pages=320 |year=2008 |url=https://books.google.com/books?id=oLPNBQAAQBAJ&dq=%22gamma+bar%22+gyromagnetic+ratio&pg=PA47 |isbn=9781584889021 |access-date=December 12, 2022}}</ref>

== Physical significance ==
[[File:Larmour Precession B spin negativeQ.jpg|thumb|upright|Direction of Larmor precession for a neutron. The central arrow denotes the magnetic field, the small red arrow the spin of the neutron.]]

===Larmor precession===
{{Main|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).<ref name=Graham>{{ cite book |title=Introduction to Magnetic Materials |author=B. D. Cullity |author2=C. D. Graham |url=https://books.google.com/books?id=ixAe4qIGEmwC&pg=PA103 |page=103 |isbn=978-0-471-47741-9 |year=2008 |location=Hoboken, New Jersey | publisher=[[Wiley-IEEE Press]]|edition=2nd| access-date=May 8, 2015}}</ref> 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 [[Precession|precess]] with a well-defined frequency, called the [[Larmor precession|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''<sub>n</sub> is negative, the neutron's spin angular momentum precesses counterclockwise about the direction of the external magnetic field.<ref name="Levitt">{{cite book | title=Spin dynamics: basics of nuclear magnetic resonance | author=M. H. Levitt | isbn=978-0-471-48921-4 | year=2001 | location=West Sussex, England | publisher=John Wiley & Sons | pages=[https://archive.org/details/isbn_9780471489221/page/n52 25]–30 | url=https://archive.org/details/isbn_9780471489221 | url-access=registration }}</ref>

===Proton nuclear magnetic resonance===
{{Main|Proton nuclear magnetic resonance}}
Nuclear magnetic resonance employing the magnetic moments of protons is used for [[Nuclear magnetic resonance spectroscopy|nuclear magnetic resonance (NMR) spectroscopy]].<ref name="Balci">{{cite book |first=M. |last=Balci  |title=Basic <sup>1</sup>H- and <sup>13</sup>C-NMR Spectroscopy |location=Amsterdam |edition=1st |publisher=Elsevier |pages=1–7 |year=2005 |url=https://www.elsevier.com/books/basic-1h-and-13c-nmr-spectroscopy/balci/978-0-444-51811-8 |isbn=978-0444518118 |access-date=December 12, 2022}}</ref>  Since [[hydrogen-1]] [[Atomic nucleus|nuclei]] are within the [[molecules]] of many substances, NMR can determine the structure of those molecules.<ref name="Silverstein">{{ cite book |title= Spectrometric Identification of Organic Compounds|author=R. M. Silverstein |author2=F. X. Webster |author3 = D. J. Kiemle |author4 = D. L. Bryce |pages=126–163 |isbn=978-0-470-61637-6 | url=https://www.wiley.com/en-ie/Spectrometric+Identification+of+Organic+Compounds,+8th+Edition-p-9780470616376 |year=2014 |location=Hoboken, New Jersey | publisher=[[Wiley (publisher)|Wiley]]|edition=8th| access-date=December 10, 2022}}</ref>

===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.<ref name=Byrne>{{cite book | author=Byrne, J. | year=2011 |title=Neutrons, Nuclei, and Matter: An exploration of the physics of slow neutrons | isbn= 978-0486482385 | location=Mineola, NY | publisher=Dover Publications | pages=28–31}}</ref> 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&nbsp;{{sfrac|1|2}}.<ref>{{cite journal |last1=Hughes |first1=D.J. |last2=Burgy |first2=M.T. |year=1949 |title=Reflection and polarization of neutrons by magnetized mirrors |journal=Phys. Rev. |volume=76 |issue=9 |pages=1413–1414 |doi=10.1103/PhysRev.76.1413 |bibcode=1949PhRv...76.1413H |url=http://physics.princeton.edu/~mcdonald/examples/EP/hughes_pr_76_1413_49.pdf |url-status=dead |access-date=2016-06-26 |archive-url=https://web.archive.org/web/20160813204055/http://physics.princeton.edu/~mcdonald/examples/EP/hughes_pr_76_1413_49.pdf |archive-date=2016-08-13 }}</ref> In 1954, J. Sherwood, T. Stephenson, and S. Bernstein employed neutrons in a [[Stern–Gerlach experiment]] that used a magnetic field to separate the neutron spin states.<ref name=Sherwood>{{cite journal |last1=Sherwood |first1=J.E. |last2=Stephenson |first2=T.E. | last3=Bernstein | first3=S. |year=1954 |title=Stern–Gerlach experiment on polarized neutrons |journal=Phys. Rev. |volume= 96 |issue=6 |pages=1546–1548 |doi=10.1103/PhysRev.96.1546 |bibcode =1954PhRv...96.1546S }}</ref> They recorded the two such spin states, consistent with a spin&nbsp;{{sfrac|1|2}} particle.<ref name=Sherwood/><ref name=Byrne/> Until these measurements, the possibility that the neutron was a spin&nbsp;{{sfrac|3|2}} particle could not have been ruled out.<ref name=Byrne/>

===Neutrons used to probe material properties===
Since neutrons are neutral particles, they do not have to overcome [[Coulomb's law|Coulomb repulsion]] as they approach charged targets, unlike protons and [[alpha particle]]s.<ref name="Snow"/> Neutrons can deeply penetrate matter.<ref name="Snow"/> The magnetic moment of the neutron has therefore been exploited to probe the properties of matter using [[inelastic neutron scattering|scattering]] or [[neutron diffraction|diffraction]] techniques.<ref name="Snow"/> These methods provide information that is complementary to [[X-ray spectroscopy]].<ref name="Snow"/><ref name="Silverstein"/> In particular, the magnetic moment of the neutron is used to determine magnetic properties of materials at length scales of 1–100&nbsp;[[Angstrom|Å]] using [[Neutron temperature|cold or thermal]] neutrons.<ref name="Lovesey">{{ cite book |title=Theory of Neutron Scattering from Condensed Matter |volume=1: Nuclear Scattering | author=S. W. Lovesey  | isbn=978-0198520290 | year=1986 | location=Oxford | publisher=Clarendon Press | pages=1–30}}</ref> [[Bertram Brockhouse|B. Brockhouse]] and [[Clifford Shull|C. Shull]] won the [[Nobel Prize]] in physics in 1994 for developing these scattering techniques.<ref name="NP-1994">{{cite web | title=The Nobel Prize in Physics 1994 | publisher=Nobel Foundation | url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1994/press.html | access-date=2015-01-25}}</ref>

===Control of neutron beams by magnetism===
As neutrons carry no electric charge, [[Particle beam|neutron beams]] cannot be controlled by the conventional electromagnetic methods employed in [[particle accelerators]].<ref name="Ari">{{cite journal |last1=Arimoto |first1=Y. |last2=Geltenbort |first2=S. |year=2012 |title=Demonstration of focusing by a neutron accelerator |journal=[[Physical Review A]] |volume=86 |issue=2 |pages=023843 |url=http://www.rri.kyoto-u.ac.jp/news-en/4964 |doi=10.1103/PhysRevA.86.023843 |access-date=May 9, 2015 |bibcode = 2012PhRvA..86b3843A |display-authors=etal|url-access=subscription }}</ref> The magnetic moment of the neutron allows some control of neutrons using magnetic fields, however, including the formation of [[Spin polarization|polarized]] neutron beams.<ref name="Oku">{{cite journal |last1=Oku |first1=T. |last2=Suzuki |first2=J.|year=2007 |title=Highly polarized cold neutron beam obtained by using a quadrupole magnet |journal=[[Physica B]] |volume=397 |issue=1–2 |pages=188–191 |doi=10.1016/j.physb.2007.02.055 |bibcode = 2007PhyB..397..188O |display-authors=etal}}</ref><ref name="Ari"/> One technique employs the fact that cold neutrons will reflect from some magnetic materials at great efficiency when scattered at small grazing angles.<ref name="FA">{{cite book |last1=Fernandez-Alonso |first1=Felix |last2=Price |first2=David |title=Neutron Scattering Fundamentals |location=Amsterdam |publisher= Academic Press |year=2013 |pages=103 |url= https://books.google.com/books?id=LdlAAAAAQBAJ&q=neutron+optics+mirror+magnetic&pg=PA103 |isbn=978-0-12-398374-9 |access-date=June 30, 2016 }}</ref> The reflection preferentially selects particular spin states, thus polarizing the neutrons. [[Neutron supermirror|Neutron magnetic mirrors]] and guides use this [[total internal reflection]] phenomenon to control beams of slow neutrons.<ref>{{cite web |last1=Chupp |first1=T |title=Neutron Optics and Polarization |url=https://www.ncnr.nist.gov/summerschool/ss09/pdf/Chupp_FP09.pdf |accessdate=16 April 2019}}</ref>

===Nuclear magnetic moments===
{{Main|Nuclear magnetic moment}}
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]], or the magnetic moment for the nucleus as a whole.<ref name="Byrne"/> The nuclear magnetic moment also includes contributions from the orbital motion of the charged protons.<ref name="Byrne"/> The deuteron, consisting of a proton and a neutron, has the simplest example of a nuclear magnetic moment.<ref name="Byrne"/> The sum of the proton and neutron magnetic moments gives 0.879&nbsp;''µ''<sub>N</sub>, which is within 3% of the measured value 0.857&nbsp;''µ''<sub>N</sub>.<ref name="Semat">{{cite book |last=Semat |first=Henry |title=Introduction to Atomic and Nuclear Physics |edition=5th |location=London |publisher=Holt, Rinehart and Winston |year=1972 |pages=556 |url= https://books.google.com/books?id=WJvTBwAAQBAJ&q=deuterium+magnetic+dipole+moment+sum+of+proton+and+neutron&pg=PA556 |isbn=978-1-4615-9701-8 |access-date=May 8, 2015 }}</ref>  In this calculation, the spins of the nucleons are aligned, but their magnetic moments offset because of the neutron's negative magnetic moment.<ref name="Semat"/>

== Nature of the nucleon magnetic moments ==
[[File:MagneticMoments.png|thumb|upright|A magnetic dipole moment can be created by either a current loop (top; Ampèrian) or by two magnetic monopoles (bottom; Gilbertian). The nucleon magnetic moments are Ampèrian.]]
A magnetic dipole moment can be generated by [[Magnetic moment#Models|two possible mechanisms]].<ref name="McDonald">
{{Cite journal
 |last1=McDonald |first1=K. T.
 |year=2014
 |title=The Forces on Magnetic Dipoles
 |journal=Joseph Henry Laboratory, Princeton University
 |url=http://physics.princeton.edu/~mcdonald/examples/neutron.pdf
 |access-date=18 June 2017
}}</ref> 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.<ref name="Fermi">
{{Cite journal
 |last1=Fermi |first1=E.
 |year= 1930
 |title=Uber die magnetischen Momente der Atomkerne
 |language=de
 |journal=[[Z. Phys.]]
 |volume=60 |issue=5–6 |pages=320–333
 |bibcode=1930ZPhy...60..320F
 |doi=10.1007/bf01339933
 |s2cid=122962691
}}</ref> 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]], and the [[hyperfine structure]] of atomic s-state energy levels.<ref name="Jackson">
{{Cite journal
 |last1=Jackson |first1=J. D.
 |year=1977
 |title=The nature of intrinsic magnetic dipole moments
 |journal=[[CERN]]
 |volume=77-17 |pages=1–25
 |url=http://physics.princeton.edu/~mcdonald/examples/EP/jackson_CERN-77-17.pdf
 |access-date= 18 June 2017
}}</ref> 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.<ref name="McDonald"/><ref name="Mezei">
{{Cite journal
 |last1=Mezei |first1=F.
 |year=1986
 |title=La Nouvelle Vague in Polarized Neutron Scattering
 |journal=[[Physica (journal)|Physica]]
 |volume=137B |issue=1 |pages=295–308
 |bibcode=1986PhyBC.137..295M
 |doi=10.1016/0378-4363(86)90335-9
}}</ref><ref name="Hughes">
{{Cite journal
 |last1=Hughes |first1=D. J.
 |last2=Burgy |first2=M. T.
 |year=1951
 |title=Reflection of neutrons from magnetized mirrors
 |journal=[[Physical Review]]
 |volume=81 |issue=4 |pages=498–506
 |bibcode=1951PhRv...81..498H
 |doi=10.1103/physrev.81.498
}}</ref><ref name="Shull">
{{Cite journal
 |last1=Shull |first1=C. G.
 |last2=Wollan |first2= E. O.
 |last3=Strauser |first3= W. A.
 |year=1951
 |title=Magnetic structure of magnetite and its use in studying the neutron magnetic interaction
 |journal=[[Physical Review]]
 |volume=81 |issue=3 |pages=483–484
 |bibcode=1951PhRv...81..483S
 |doi=10.1103/physrev.81.483
}}</ref>

== Anomalous magnetic moments and meson physics ==
{{See also|Anomalous magnetic dipole moment}}

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.<ref name="Pais"/> Considerable theoretical efforts were expended in trying to understand the origins of these magnetic moments, but the failures of these theories were glaring.<ref name="Pais"/> 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.<ref name="Pais"/>

The problem of the origins of the magnetic moments of nucleons was recognized as early as 1935. [[Gian Carlo Wick|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.<ref name="BrownRechenberg">{{cite book |last1=Brown |first1=L. M. |last2=Rechenberg |first2=H. |author-link2=Helmut Rechenberg |url=https://archive.org/details/originofconcepto0000brow |title=The Origin of the Concept of Nuclear Forces |publisher=Institute of Physics Publishing |year=1996 |isbn=978-0750303736 |location=Bristol and Philadelphia |pages=[https://archive.org/details/originofconcepto0000brow/page/95 95–312] |url-access=registration}}</ref> 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]].<ref name="Wick">{{Cite journal |last1=Wick |first1=G. C. |year=1935 |title=Teoria dei raggi beta e momento magnetico del protone |journal=Rend. R. Accad. Lincei |volume=21 |pages=170–175 }}</ref> 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.<ref name="Amaldi">{{cite book |last=Amaldi |first=E. |editor1-first=G. |editor1-last=Battimelli |editor2-first=G. |editor2-last=Paoloni |title=20th Century Physics: Essays and Recollections: a Selection of Historical Writings by Edoardo Amaldi |location=Singapore |publisher=World Scientific Publishing Company |year=1998 |pages=128–139 |chapter=Gian Carlo Wick during the 1930s |isbn=978-9810223694}}</ref> The theory proved to be untenable, however, when [[Hans Bethe|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.<ref name="BrownRechenberg"/><ref name="BetheBible">{{Cite journal |last1=Bethe |first1=H. A. |last2=Bacher |first2=R. F. | year=1936 |title=Nuclear Physics A. Stationary states of nuclei |journal=[[Reviews of Modern Physics]] |volume=8 |issue=5 |pages=82–229 |bibcode=1936RvMP....8...82B |doi=10.1103/RevModPhys.8.82 |url=https://authors.library.caltech.edu/51288/1/RevModPhys.8.82.pdf }}</ref>

Similar considerations for the electron proved to be much more successful. In [[quantum electrodynamics]] (QED), the [[anomalous magnetic moment]] of a particle stems from the small contributions of quantum mechanical fluctuations to the [[magnetic moment]] of that particle.<ref name="Peskin">
{{cite book
 |last1=Peskin
 |first1=M. E.
 |last2=Schroeder
 |first2=D. V.
 |year=1995
 |title=An Introduction to Quantum Field Theory
 |section=6.3. The Electron VertexFunction: Evaluation
 |location=Reading, Massachusetts
 |publisher=Perseus Books
 |pages=[https://archive.org/details/introductiontoqu0000pesk/page/175 175–198]
 |isbn=978-0201503975
 |url=https://archive.org/details/introductiontoqu0000pesk/page/175
}}</ref> The g-factor for a "Dirac" [[magnetic moment]] is predicted to be {{nowrap|1=''g'' = −2}} for a negatively charged, spin-1/2 particle. For particles such as the [[electron]], 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 {{physconst|ge|after=.}} 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]]s with loops.<ref name="Aoyama">
{{Cite journal
 |last1=Aoyama |first1=T.
 |last2=Hayakawa |first2=M.
 |last3=Kinoshita |first3=T.
 |last4=Nio |first4=M.
 |year=2008
 |title=Revised value of the eighth-order QED contribution to the anomalous magnetic moment of the electron
 |journal=[[Physical Review D]]
 |volume=77 |issue=5 |pages=053012
 |arxiv=0712.2607
 |bibcode=2008PhRvD..77e3012A
 |doi=10.1103/PhysRevD.77.053012
 |s2cid=119264728
}}</ref>

[[Image:Vertex correction.svg|thumb|right|One-loop correction to a fermion's magnetic dipole moment. The solid lines at top and bottom represent the fermion (electron or nucleon), the wavy lines represent the particle mediating the force (photons for QED, mesons for nuclear force). The middle solid lines represent a virtual pair of particles (electron and positron for QED, pions for the nuclear force).]]

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 [[Julian Schwinger|J. Schwinger]] in 1948.<ref name="Peskin"/><ref>
{{cite journal
 |last1=Schwinger |first1=J.
 |year=1948
 |title=On Quantum-Electrodynamics and the Magnetic Moment of the Electron
 |journal=[[Physical Review]]
 |volume=73 |issue=4 |pages=416–417
 |bibcode=1948PhRv...73..416S
 |doi=10.1103/PhysRev.73.416
 |doi-access=free
}}</ref> 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]].<ref name="Peskin"/>

Compared to the electron, the anomalous magnetic moments of the nucleons are enormous.<ref name="Hausser"/> 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: {{nowrap|1=''μ''<sub>p</sub> − {{val|1.00|u=''μ''<sub>N</sub>}} = +{{val|1.79|u=''μ''<sub>N</sub>}}}}, but {{nowrap|1=''μ''<sub>n</sub> − {{val|0.00|u=''μ''<sub>N</sub>}} = {{val|-1.91|u=''μ''<sub>N</sub>}}}}.<ref>See chapter 1, section 6 in {{cite book
 |last1=deShalit |first1=A.
 |last2=Feschbach |first2=H.
 |year=1974
 |title=Theoretical Nuclear Physics Volume I: Nuclear Structure
 |location=New York
 |publisher=[[John Wiley and Sons]]
 |page=31
 |isbn=978-0471203858
}}</ref>

The [[Yukawa interaction]] for nucleons was discovered in the mid-1930s, and this nuclear force is mediated by [[pion]] [[meson]]s.<ref name="BrownRechenberg"/> 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.<ref name="BD"/> 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.<ref name="Vonsovsky"/>{{rp|75–80}}<ref name="DZ">{{cite book |last1=Drell |first1=S. |last2=Zachariasen |first2=F. |date=1961 |title=Electromagnetic Structure of Nucleons |url=https://archive.org/details/electromagnetics002214mbp |pages=[https://archive.org/details/electromagnetics002214mbp/page/n12 1]–130 |location=New York |publisher=Oxford University Press }}</ref> The Feynman diagram at right is roughly the first-order diagram, with the role of the virtual particles played by pions. As noted by [[Abraham Pais|A. Pais]], "between late 1948 and the middle of 1949 at least six papers appeared reporting on second-order calculations of nucleon moments".<ref name="Pais"/> These theories were also, as noted by Pais, "a flop"{{snd}} they gave results that grossly disagreed with observation. Nevertheless, serious efforts continued along these lines for the next couple of decades, to little success.<ref name="BD"/><ref name="DZ"/><ref name="DrellPagels">
{{cite journal
 |last1=Drell |first1=S.
 |last2=Pagels |first2=H. R.
 |year=1965
 |title=Anomalous Magnetic Moment of the Electron, Muon, and Nucleon
 |journal=[[Physical Review]]
 |volume=140 |issue=2B
 |pages=B397–B407
 |doi=10.1103/PhysRev.140.B397 |bibcode = 1965PhRv..140..397D |osti=1444215
|url=https://www.slac.stanford.edu/cgi-bin/getdoc/slac-pub-0102.pdf
 }}</ref> These theoretical approaches were incorrect because the nucleons are composite particles with their magnetic moments arising from their elementary components, quarks.<ref name="Pais"/>

==Quark model of nucleon magnetic moments==

In the [[quark model]] for [[hadrons]], the neutron is composed of one up quark (charge {{sfrac|+| 2 |3}}&nbsp;{{mvar|e}}) and two down quarks (charge {{sfrac|−| 1 |3}}&nbsp;{{mvar|e}}) while the proton is composed of one down quark (charge {{sfrac|−| 1 |3}}&nbsp;{{mvar|e}}) and two up quarks (charge {{sfrac|+| 2 |3}}&nbsp;{{mvar|e}}).<ref>
{{cite journal
 |last1=Gell |first1=Y.
 |last2=Lichtenberg |first2=D. B.
 |year=1969
 |title=Quark model and the magnetic moments of proton and neutron
 |journal=[[Il Nuovo Cimento A]]
 |series=Series 10 |volume=61 |issue=1
 |pages=27–40
 |doi=10.1007/BF02760010
 |bibcode = 1969NCimA..61...27G |s2cid=123822660
}}
</ref>
The magnetic moment of the nucleons can be modeled as a sum of the magnetic moments of the constituent quarks,<ref name=Perk/> although this simple model belies the complexities of the [[Standard Model of Particle Physics]].<ref name=Mass>
{{cite news
 |last1=Cho |first1=Adiran
 |date=2 April 2010
 |title=Mass of the common quark finally nailed down
 |journal=[[Science (journal)|Science]]
 |publisher=[[American Association for the Advancement of Science]]
 |url=https://www.science.org/content/article/mass-common-quark-finally-nailed-down
 |access-date=27 September 2014
}}
</ref>
The calculation assumes that the quarks behave like pointlike [[Dirac spinor|Dirac particles]], each having their own magnetic moment, as computed using an expression similar to the one above for the nuclear magneton:
<math display="block">\ \mu_\text{q} = \frac{\ e_\text{q} \hbar\ }{2 m_\text{q}}\ ,</math>
where the q-subscripted variables refer to quark magnetic moment, charge, or mass.<ref name=Perk/> 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.<ref name=Perk/>

In one of the early successes of the Standard Model (SU(6) theory), in 1964 M. Beg, [[Benjamin W. Lee|B. Lee]], and A. Pais theoretically calculated the ratio of proton-to-neutron magnetic moments to be {{sfrac|−|3| 2 }}, which agrees with the experimental value to within 3%.<ref name=Greenberg>
{{cite book
 |last=Greenberg |first=O.W.
 |year=2009
 |chapter=Color charge degree of freedom in particle physics
 |title=Compendium of Quantum Physics
 |publisher=Springer Berlin Heidelberg
 |pages=109–111 
 |doi=10.1007/978-3-540-70626-7_32
 |arxiv=0805.0289  |s2cid=17512393
 |isbn=978-3-540-70622-9
}}
</ref><ref name=Beg>
{{cite journal
 |last1=Beg |first1=M.A.B.
 |last2=Lee |first2=B.W.
 |last3=Pais |first3=A.
 |year=1964
 |title=SU(6) and electromagnetic interactions
 |journal=[[Physical Review Letters]]
 |volume=13 |issue=16  |pages=514–517, erratum&nbsp;650
 |doi=10.1103/physrevlett.13.514 |bibcode = 1964PhRvL..13..514B
}}
</ref><ref name=Sakita>
{{cite journal
 |last1=Sakita |first1=B. 
 |year=1964
 |title=Electromagnetic properties of baryons in the supermultiplet scheme of elementary particles
 |journal=[[Physical Review Letters]]
 |volume=13  |issue=21  |pages=643–646
 |doi=10.1103/physrevlett.13.643
 |bibcode = 1964PhRvL..13..643S
}}
</ref>
The measured value for this ratio is {{val|-1.45989806|(34)}}.<ref name="2010 CODATA">{{cite report |editor1=Mohr, P.J. |editor2=Taylor, B.N. |editor3=Newell, D.B. |date=2011-06-02 |title=The 2010 CODATA Recommended Values of the Fundamental Physical Constants |id=Web Version&nbsp;6.0 |publisher=[[National Institute of Standards and Technology]] |place=Gaithersburg, MD |url=http://physics.nist.gov/constants |access-date=9 May 2015}} The database was developed by J. Baker, M. Douma, and [[Svetlana Kotochigova|S. Kotochigova]].</ref> A contradiction of the quantum mechanical basis of this calculation with the [[Pauli exclusion principle]] led to the discovery of the [[color charge]] for quarks by [[Oscar W. Greenberg|O. Greenberg]] in 1964.<ref name=Greenberg/>

From the [[Special relativity|nonrelativistic]] quantum-mechanical [[wave function]] for [[baryon]]s composed of three quarks, a straightforward calculation gives fairly accurate estimates for the magnetic moments of neutrons, protons, and other baryons.<ref name=Perk/> For a neutron, the magnetic moment is given by {{nobr|{{math|''μ''<sub>n</sub> {{=}} {{sfrac| 4 |3}} ''μ''<sub>d</sub> − {{sfrac| 1 |3}} ''μ''<sub>u</sub>}} ,}} where {{mvar|μ}}<sub>d</sub> and {{mvar|μ}}<sub>u</sub> 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.<ref name=Perk/>

{| class="wikitable" style="text-align:center;"
|-
! Baryon
! Magnetic moment<br/>of quark model
! Computed<br/>(<math>\mu_\text{N}</math>)
! Observed<br/>(<math>\mu_\text{N}</math>)
|-
| p
| {{sfrac| 4 |3}} {{mvar|μ}}<sub>u</sub> − {{sfrac| 1 |3}} {{mvar|μ}}<sub>d</sub>
| 2.79
| 2.793
|-
| n
| {{sfrac| 4 |3}} {{mvar|μ}}<sub>d</sub> − {{sfrac| 1 |3}} {{mvar|μ}}<sub>u</sub>
| −1.86
| −1.913
|}

The results of this calculation are encouraging, but the masses of the up or down quarks were assumed to be {{sfrac| 1 |3}} the mass of a nucleon.<ref name=Perk/> 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]].<ref name=Mass/><ref name=Wilczek>
{{cite journal
 |last=Wilczek  |first=F.
 |year=2003
 |title=The Origin of Mass
 |journal=MIT Physics Annual
 |pages=24–35
 |url=https://physics.mit.edu/wp-content/uploads/2021/01/physicsatmit_03_wilczek_originofmass.pdf
 |access-date=May 8, 2015
}}
</ref>
Furthermore, the complex system of quarks and gluons that constitute a nucleon requires a relativistic treatment.<ref>
{{cite journal
 |last=Ji  |first=Xiangdong
 |year=1995
 |title=A QCD analysis of the mass structure of the nucleon
 |journal=Phys. Rev. Lett.
 |volume=74 |issue=7 |pages=1071–1074
 |doi=10.1103/PhysRevLett.74.1071
 |pmid=10058927 |s2cid=15148740
 |arxiv = hep-ph/9410274
 |bibcode = 1995PhRvL..74.1071J
}}
</ref>
Nucleon magnetic moments have been successfully computed from [[first principle]]s, requiring significant computing resources.<ref>
{{cite journal
 |last1=Martinelli |first1=G.       |last2=Parisi  |first2=G.
 |last3=Petronzio  |first3=R.       |last4=Rapuano |first4=F.
 |year=1982
 |title=The proton and neutron magnetic moments in lattice QCD
 |journal=Physics Letters B
 |volume=116 |issue=6 |pages=434–436
 |doi=10.1016/0370-2693(82)90162-9
 |bibcode = 1982PhLB..116..434M
 |url=https://cds.cern.ch/record/138281/files/198207343.pdf
 |via=cern.ch
}}
</ref><ref name=MagMom>
{{cite news
 |last=Kincade |first=Kathy
 |date=2 February 2015
 |title=Pinpointing the magnetic moments of nuclear matter
 |website=Phys.org
 |url=http://phys.org/news/2015-02-magnetic-moments-nuclear.html
 |access-date=May 8, 2015
}}
</ref>

==See also==
* [[Aharonov–Casher effect]]
* [[LARMOR neutron microscope]]
* [[Neutron electric dipole moment]]
* [[Neutron triple-axis spectrometry]]

==References==
{{reflist}}

==Bibliography==
* S. W. Lovesey (1986). Theory of Neutron Scattering from Condensed Matter. Oxford University Press. {{ISBN|0198520298}}.
* [[Donald Hill Perkins|Donald H. Perkins]] (1982). Introduction to High Energy Physics. Reading, Massachusetts: Addison Wesley, {{ISBN|0-201-05757-3}}.
* [[John S. Rigden]] (1987). Rabi, Scientist and Citizen. New York: Basic Books, Inc., {{ISBN|0-465-06792-1}}.
* [[Sergei Vonsovsky]] (1975). Magnetism of Elementary Particles. Moscow: Mir Publishers.

==External links==
* {{Commons category-inline}}

[[Category:Electric and magnetic fields in matter]]
[[Category:Magnetic moment]]
[[Category:Magnetism]]
[[Category:Magnetostatics]]
[[Category:Proton|magnetic moment]]
[[Category:Neutron|magnetic moment]]
[[Category:Physical quantities]]