Neutron magnetic moment: Difference between revisions
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#REDIRECT [[Nucleon magnetic moment]] |
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The '''neutron magnetic moment''' is the intrinsic [[magnetic dipole moment]] of the [[neutron]], symbol ''μ''<sub>n</sub>. [[Protons]] and neutrons, both [[nucleons]], comprise the [[atomic nucleus|nucleus]] of [[atoms]], and both nucleons behave as small [[magnets]] whose strengths are measured by their magnetic moments. The neutron interacts with normal matter through either the [[nuclear force]] or its magnetic moment. 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 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 existence of the neutron's magnetic moment indicates the neutron is not an [[elementary particle]], because for an elementary particle to have an intrinsic magnetic moment, it must have both [[Spin (physics)|spin]] and [[electric charge]]. The neutron has [[Spin-½|spin 1/2 ''ħ'']], but no net charge. The existence of the neutron's magnetic moment was puzzling and defied a valid explanation until the [[quark model]] for particles was developed in the 1960s. The neutron is composed of three quarks, and the magnetic moments of these elementary particles combine to give the neutron its magnetic moment. |
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==Description== |
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[[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.]] |
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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=8 May 2015 |doi=10.1103/PhysRevD.86.010001 |bibcode=2012PhRvD..86a0001B|doi-access=free }}</ref> Here, {{mvar|μ}}<sub>N</sub> is the [[nuclear magneton]], a [[physical constant]] and standard unit for the magnetic moments of nuclear components. In [[SI units]], {{physconst|mun|symbol=yes|after=.}} A magnetic moment is a vector quantity, and the direction of the neutron's magnetic moment is determined by its spin. 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. |
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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. The nuclear magneton is |
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<math display="block">\mu_\text{N} = \frac{e \hbar}{2 m_\text{p}},</math> |
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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. Since the neutron has no charge, it should have no magnetic moment by the analagous expression. 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 [[proton magnetic moment|magnetic moment of the proton]], {{physconst|mup_muN|symbol=yes|round=3|after=,}} is not almost equal to 1 {{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. |
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Although the neutron interacts with normal matter primarily through either nuclear or magnetic forces, the magnetic interactions are about seven 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]]. The [[electron magnetic moment|magnetic moment of the electron]] is therefore about 1000 times larger than that of the neutron.<ref>{{cite web |work=NIST |title=CODATA values of the fundamental constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?muemsmunn |access-date=8 May 2015}}</ref> |
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The magnetic moments of the neutron and [[antineutron]] have the same magnitude, 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> |
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==Measurement== |
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{{see also|Discovery of the neutron}} |
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Soon after the neutron was discovered in 1932, indirect evidence suggested that the neutron had an unexpected non-zero value for its magnetic moment. Attempts to measure the neutron's magnetic moment originated with the discovery by [[Otto Stern]] in 1933 in [[Hamburg]] that the proton had an anomalously large magnetic moment.<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 |lang=de |trans-title=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 |access-date=9 May 2015 |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 |lang=de |trans-title=Magnetic Deviation of Hydrogen Molecules and the Magnetic Moment of the Proton. II |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 |access-date=9 May 2015 |bibcode=1933ZPhy...85...17E |s2cid=186232193 }}</ref> The proton's magnetic moment had been 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 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=2015-01-30 |df=dmy-all}}</ref> |
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By 1934 groups led by Stern, now in [[Pittsburgh]], and [[Isidor Isaac Rabi|I. I. Rabi]] 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}}</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">{{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><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. The resulting value was not zero and had a sign opposite to that of the proton. |
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Values for the magnetic moment of the neutron were also determined by [[Robert Bacher|R. 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 [[Ann Arbor]] (1933) and [[Igor Tamm|I. Y. Tamm]] and [[Semen Altshuler|S. 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">{{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 |pages=[https://archive.org/details/MagnetismOfElementaryParticles/page/n75 73]–75 |year=1975}}</ref> 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 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. |
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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. 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. 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> |
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The value for the neutron's magnetic moment was first directly measured by [[Luis Walter Alvarez|Luis Alvarez]] and [[Felix Bloch]] at [[Berkeley, California|Berkeley]], [[California]] in 1940.<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> 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> |
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== Neutron gyromagnetic ratio == |
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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 |
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<math display="block">\boldsymbol{\mu} = \frac{g \mu_\text{N}}{\hbar} \boldsymbol{I},</math> |
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where '''{{mvar|μ}}''' is the intrinsic magnetic moment, '''{{mvar|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 natural unit of the [[nuclear magneton]]. For the neutron, '''{{mvar|I}}''' is {{sfrac|1|2}} {{mvar|ħ}}, so the neutron's {{mvar|g}}-factor, symbol {{mvar|g}}<sub>n</sub>, is {{val|-3.82608545|(90)}}.<ref>{{cite web |publisher=NIST |title=CODATA values of the fundamental constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?gnn |access-date=2022-09-18}}</ref> |
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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 |
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<math display="block">\boldsymbol{\mu} = \gamma \boldsymbol{I}.</math> |
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For nucleons, the ratio is conventionally written in terms of the proton mass and charge, by the formula |
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: <math>\gamma = \frac{g \mu_\text{N}}{\hbar} = g \frac{e}{2m_\text{p}}.</math> |
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The neutron's gyromagnetic ratio is {{mvar|γ}}<sub>n</sub> = {{val|-1.83247171|(43)|e=8|u=rad/(s⋅[[Tesla (unit)|T]])}}.<ref>{{cite web |publisher=NIST |title=CODATA values of the fundamental constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?gamman |access-date=8 August 2019}}</ref> The gyromagnetic ratio is also the ratio between the observed angular frequency of [[Larmor precession]] (in rad/s) 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=8 May 2015}}</ref> such as in [[Magnetic resonance imaging|MRI imaging]]. For this reason, the value of {{mvar|γ}}<sub>n</sub> is often given in units of [[MHz]]/[[Tesla (unit)|T]]. The quantity {{mvar|γ}}<sub>n</sub>/(2{{mvar|π}}) = {{val|-29.1646931|(69)|u=[[MHz]]/[[Tesla (unit)|T]]}}, called "gamma bar", is therefore convenient.<ref>{{cite web |publisher=NIST |title=CODATA values of the fundamental constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?gammanbar |access-date=2022-09-17}}</ref> |
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== Physical significance == |
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[[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.]] |
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When a neutron 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 (and hence its spin 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 neutron has spin angular momentum, this torque will cause the neutron 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 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> |
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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 |title=Neutrons, Nuclei and Matter: An Exploration of the Physics of Slow Neutrons |author=J. Byrne | isbn= 978-0486482385 | year=2011 |location=Mineola, New York | publisher=Dover Publications | pages=28–31}}</ref> In 1949, Hughes and Burgy measured neutrons reflected from a ferromagnetic mirror and found that the angular distribution of the reflections was consistent with spin 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 |access-date=2016-06-26 |archive-date=2016-08-13 |archive-url=https://web.archive.org/web/20160813204055/http://physics.princeton.edu/~mcdonald/examples/EP/hughes_pr_76_1413_49.pdf |url-status=dead }}</ref> In 1954, Sherwood, Stephenson, and Bernstein employed neutrons in a [[Stern–Gerlach experiment]] 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.<ref name="Sherwood">{{cite journal |last1=Sherwood |first1=J. E. |last2=Stephenson |first2=T. E. | first3=S. | last3=Bernstein |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><ref name="Byrne"/> Until these measurements, the possibility that the neutron was a spin-3/2 particle could not have been ruled out. |
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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. Neutrons can deeply penetrate matter. 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. These methods provide information that is complementary to [[X-ray spectroscopy]]. In particular, the magnetic moment of the neutron is used to determine magnetic properties of materials at length scales of 1–100 [[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]] and [[Clifford 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> |
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Without an electric charge, [[Particle beam|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 field]]s, however,<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">{{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}}</ref> including the formation of [[Spin polarization|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.<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. |
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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. 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''<sub>N</sub>, which is within 3% of the measured value 0.857 ''µ''<sub>N</sub>. 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">{{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> |
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== Nature of the neutron's magnetic moment == |
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[[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 neutron's magnetic moment is Ampèrian.]] |
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A magnetic dipole moment can be generated by [[Magnetic moment#Two representations of the cause of the magnetic moment|two possible mechanisms]].<ref name="McDonald"> |
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{{Cite journal |
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|last1=McDonald |first1=K. T. |
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|year=2014 |
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|title=The Forces on Magnetic Dipoles |
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|journal=Joseph Henry Laboratory, Princeton University |
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|url=http://physics.princeton.edu/~mcdonald/examples/neutron.pdf |
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|access-date=18 June 2017 |
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}}</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 neutron's intrinsic magnetic moment. In 1930, [[Enrico Fermi]] showed that the magnetic moments of nuclei (including the proton) are Ampèrian.<ref name="Fermi"> |
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{{Cite journal |
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|last1=Fermi |first1=E. |
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|year= 1930 |
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|title=Uber die magnetischen Momente der Atomkerne |
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|lang=de |
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|journal=[[Z. Phys.]] |
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|volume=60 |issue=5–6 |pages=320–333 |
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|bibcode=1930ZPhy...60..320F |
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|doi=10.1007/bf01339933 |
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|s2cid=122962691 |
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}}</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 neutron, 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"> |
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{{Cite journal |
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|last1=Jackson |first1=J. D. |
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|year=1977 |
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|title=The nature of intrinsic magnetic dipole moments |
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|journal=[[CERN]] |
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|volume=77-17 |pages=1–25 |
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|url=http://physics.princeton.edu/~mcdonald/examples/EP/jackson_CERN-77-17.pdf |
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|access-date= 18 June 2017 |
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}}</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"> |
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{{Cite journal |
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|last1=Mezei |first1=F. |
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|year=1986 |
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|title=La Nouvelle Vague in Polarized Neutron Scattering |
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|journal=[[Physica (journal)|Physica]] |
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|volume=137B |issue=1 |pages=295–308 |
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|bibcode=1986PhyBC.137..295M |
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|doi=10.1016/0378-4363(86)90335-9 |
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}}</ref><ref name="Hughes"> |
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{{Cite journal |
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|last1=Hughes |first1=D. J. |
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|last2=Burgy |first2=M. T. |
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|year=1951 |
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|title=Reflection of neutrons from magnetized mirrors |
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|journal=[[Physical Review]] |
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|volume=81 |issue=4 |pages=498–506 |
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|bibcode=1951PhRv...81..498H |
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|doi=10.1103/physrev.81.498 |
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}}</ref><ref name="Shull"> |
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{{Cite journal |
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|last1=Shull |first1=C. G. |
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|last2=Wollan |first2= E. O. |
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|last3=Strauser |first3= W. A. |
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|year=1951 |
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|title=Magnetic structure of magnetite and its use in studying the neutron magnetic interaction |
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|journal=[[Physical Review]] |
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|volume=81 |issue=3 |pages=483–484 |
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|bibcode=1951PhRv...81..483S |
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|doi=10.1103/physrev.81.483 |
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}}</ref> |
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== Anomalous magnetic moments and meson physics == |
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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.<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> 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. |
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The problem of the origins of the magnetic moments of nucleons was recognized as early as 1935. [[Gian Carlo 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]] and [[Robert 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> |
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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 mechanics|quantum mechanical]] fluctuations to the [[magnetic moment]] of that particle.<ref name="Peskin"> |
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{{cite book |
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|last1=Peskin |
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|first1=M. E. |
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|last2=Schroeder |
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|first2=D. V. |
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|year=1995 |
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|title=An Introduction to Quantum Field Theory |
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|section=6.3. The Electron VertexFunction: Evaluation |
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|location=Reading, Massachusetts |
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|publisher=Perseus Books |
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|pages=[https://archive.org/details/introductiontoqu0000pesk/page/175 175–198] |
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|isbn=978-0201503975 |
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|url=https://archive.org/details/introductiontoqu0000pesk/page/175 |
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}}</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"> |
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{{Cite journal |
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|last1=Aoyama |first1=T. |
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|last2=Hayakawa |first2=M. |
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|last3=Kinoshita |first3=T. |
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|last4=Nio |first4=M. |
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|year=2008 |
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|title=Revised value of the eighth-order QED contribution to the anomalous magnetic moment of the electron |
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|journal=[[Physical Review D]] |
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|volume=77 |issue=5 |pages=053012 |
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|arxiv=0712.2607 |
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|bibcode=2008PhRvD..77e3012A |
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|doi=10.1103/PhysRevD.77.053012 |
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|s2cid=119264728 |
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}}</ref> |
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[[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).]] |
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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]] in 1948.<ref name="Peskin"/><ref> |
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{{cite journal |
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|last1=Schwinger |first1=J. |
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|year=1948 |
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|title=On Quantum-Electrodynamics and the Magnetic Moment of the Electron |
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|journal=[[Physical Review]] |
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|volume=73 |issue=4 |pages=416–417 |
|||
|bibcode=1948PhRv...73..416S |
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|doi=10.1103/PhysRev.73.416 |
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|doi-access=free |
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}}</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"/> |
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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 |
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|last1=deShalit |first1=A. |
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|last2=Feschbach |first2=H. |
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|year=1974 |
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|title=Theoretical Nuclear Physics Volume I: Nuclear Structure |
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|location=New York |
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|publisher=[[John Wiley and Sons]] |
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|page=31 |
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|isbn=978-0471203858 |
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}}</ref> |
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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="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]], "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"> |
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{{cite journal |
|||
|last1=Drell |first1=S. |
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|last2=Pagels |first2=H. R. |
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|year=1965 |
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|title=Anomalous Magnetic Moment of the Electron, Muon, and Nucleon |
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|journal=[[Physical Review]] |
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|volume=140 |issue=2B |
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|pages=B397–B407 |
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|doi=10.1103/PhysRev.140.B397 |bibcode = 1965PhRv..140..397D |osti=1444215 |
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}}</ref> These theoretical approaches were incorrect because the nucleons are composite particles with their magnetic moments arising from their elementary components, quarks. |
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==Quark model of nucleon magnetic moments== |
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In the [[quark model]] for [[hadrons]], the neutron is composed of one up quark (charge +2/3 ''e'') and two down quarks (charge −1/3 ''e'').<ref> |
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{{cite journal |
|||
|last1=Gell |first1=Y. |
|||
|last2=Lichtenberg |first2=D. B. |
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|year=1969 |
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|title=Quark model and the magnetic moments of proton and neutron |
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|journal=[[Il Nuovo Cimento A]] |
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|series=Series 10 |volume=61 |issue=1 |
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|pages=27–40 |
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|doi=10.1007/BF02760010 |
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|bibcode = 1969NCimA..61...27G |s2cid=123822660 |
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}}</ref> The magnetic moment of the neutron can be modeled as a sum of the magnetic moments of the constituent quarks,<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 this simple model belies the complexities of the [[Standard Model]] of [[particle physics]].<ref name="Mass">{{cite web |url=https://www.science.org/content/article/mass-common-quark-finally-nailed-down |title=Mass of the Common Quark Finally Nailed Down |last1=Cho |first1=Adiran |date=2 April 2010 |publisher=Science Magazine, American Association for the Advancement of Science |access-date=27 September 2014}}</ref> 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: |
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<math display="block">\mu_\text{q} = \frac{e_\text{q} \hbar}{2 m_\text{q}},</math> |
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where the q-subscripted variables refer to quark magnetic moment, charge, or mass. Simplistically, the magnetic moment of the neutron 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 the neutron. |
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In one of the early successes of the Standard Model (SU(6) theory), in 1964 Mirza A. B. Beg, [[Benjamin W. Lee]], and [[Abraham Pais]] theoretically calculated the ratio of proton-to-neutron magnetic moments to be −3/2, which agrees with the experimental value to within 3%.<ref name="Greenberg"> |
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{{cite book |
|||
|last=Greenberg |first=O. W. |
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|year=2009 |
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|chapter=Color charge degree of freedom in particle physics |
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|title=Compendium of Quantum Physics |
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|publisher=Springer Berlin Heidelberg |
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|pages=109–111 |
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|doi=10.1007/978-3-540-70626-7_32 |
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|arxiv=0805.0289 |
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|isbn=978-3-540-70622-9 |
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|s2cid=17512393 |
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}}</ref><ref name="Beg"> |
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{{cite journal |
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|last1=Beg |first1=M. A. B. |
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|last2=Lee |first2=B. W. |
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|last3=Pais |first3=A. |
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|year=1964 |
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|title=SU(6) and electromagnetic interactions |
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|journal=[[Physical Review Letters]] |
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|volume=13 |issue=16 |pages=514–517, erratum 650 |
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|doi=10.1103/physrevlett.13.514 |bibcode = 1964PhRvL..13..514B }}</ref><ref name="Sakita">{{cite journal |
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|last1=Sakita |first1=B. |
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|year=1964 |
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|title=Electromagnetic properties of baryons in the supermultiplet scheme of elementary particles |
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|journal=[[Physical Review Letters]] |
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|volume=13 |issue=21 |pages=643–646 |
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|doi=10.1103/physrevlett.13.643 |
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|bibcode = 1964PhRvL..13..643S |
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}}</ref> The measured value for this ratio is {{val|-1.45989806|(34)}}.<ref name="2010 CODATA">Mohr, P. J.; Taylor, B. N. and Newell, D. B. (2011), [http://physics.nist.gov/constants "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.</ref> A contradiction of the [[quantum mechanics|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]] in 1964.<ref name="Greenberg"/> |
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From the [[Special relativity|nonrelativistic]] quantum-mechanical [[wavefunction]] 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 {{nowrap|1=''μ''<sub>n</sub> = 4/3 ''μ''<sub>d</sub> − 1/3 ''μ''<sub>u</sub>}}, where ''μ''<sub>d</sub> and ''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. |
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{| class="wikitable" style="text-align:center;" |
|||
|- |
|||
! Baryon |
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! Magnetic moment<br />of quark model |
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! Computed<br />(<math>\mu_\text{N}</math>) |
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! Observed<br />(<math>\mu_\text{N}</math>) |
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|- |
|||
| p |
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| 4/3 ''μ''<sub>u</sub> − 1/3 ''μ''<sub>d</sub> |
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| 2.79 |
|||
| 2.793 |
|||
|- |
|||
| n |
|||
| 4/3 ''μ''<sub>d</sub> − 1/3 ''μ''<sub>u</sub> |
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| −1.86 |
|||
| −1.913 |
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|} |
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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.<ref name="Perk"/> The masses of the quarks are actually only about 1% that of a nucleon.<ref name="Mass"/> 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 |
|||
|last1=Wilczek |first1=F. |
|||
|year=2003 |
|||
|title=The Origin of Mass |
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|url=http://web.mit.edu/physics/news/physicsatmit/physicsatmit_03_wilczek_originofmass.pdf |
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|access-date=May 8, 2015 |
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|journal=MIT Physics Annual |
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|pages=24–35 |
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}}</ref> Furthermore, the complex system of quarks and gluons that constitute a neutron requires a relativistic treatment.<ref> |
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{{cite journal |
|||
|last1=Ji |first1=Xiangdong |
|||
|year=1995 |
|||
|title=A QCD Analysis of the Mass Structure of the Nucleon |
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|journal=Phys. Rev. Lett. |
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|volume=74 |issue=7 |
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|pages=1071–1074 |
|||
|doi=10.1103/PhysRevLett.74.1071 |
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|pmid=10058927 |
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|arxiv = hep-ph/9410274 |bibcode = 1995PhRvL..74.1071J |s2cid=15148740 |
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}}</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 |
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|journal=Physics Letters B |
|||
|volume=116 |issue=6 |
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|pages=434–436 |
|||
|doi=10.1016/0370-2693(82)90162-9 |
|||
|bibcode = 1982PhLB..116..434M |
|||
|url=http://cds.cern.ch/record/138281/files/198207343.pdf |
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}}</ref><ref name="MagMom"> |
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{{cite web |
|||
|url=http://phys.org/news/2015-02-magnetic-moments-nuclear.html |
|||
|title=Pinpointing the magnetic moments of nuclear matter |
|||
|last1=Kincade |first1=Kathy |
|||
|date=2 February 2015 |
|||
|publisher=Phys.org |
|||
|access-date=May 8, 2015 |
|||
}}</ref> |
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==See also== |
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* [[Neutron electric dipole moment]] |
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* [[Bohr magneton]] |
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* [[Electron magnetic moment]] |
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* [[Proton magnetic moment]] |
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* [[Nuclear magnetic moment]] |
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* [[Anomalous magnetic dipole moment]] |
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* [[Neutron diffraction]] |
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* [[Neutron triple-axis spectrometry]] |
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* [[LARMOR neutron microscope]] |
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* [[Antineutron]] |
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* [[Aharonov–Casher effect]] |
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==References== |
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{{reflist}} |
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==Bibliography== |
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* S. W. Lovesey (1986). Theory of Neutron Scattering from Condensed Matter. Oxford University Press. {{ISBN|0198520298}}. |
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* [[Donald Hill Perkins|Donald H. Perkins]] (1982). Introduction to High Energy Physics. Reading, Massachusetts: Addison Wesley, {{ISBN|0-201-05757-3}}. |
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* [[John S. Rigden]] (1987). Rabi, Scientist and Citizen. New York: Basic Books, Inc., {{ISBN|0-465-06792-1}}. |
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* [[Sergei Vonsovsky]] (1975). Magnetism of Elementary Particles. Moscow: Mir Publishers. |
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==External links== |
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* {{Commons category-inline}} |
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[[Category:Electric and magnetic fields in matter]] |
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[[Category:Magnetic moment]] |
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[[Category:Magnetism]] |
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[[Category:Magnetostatics]] |
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[[Category:Neutron|magnetic moment]] |
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[[Category:Physical quantities]] |
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Revision as of 11:30, 1 October 2022
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