Neutron magnetic moment
The neutron magnetic moment is the intrinsic magnetic dipole moment of the neutron, symbol μn. Protons and neutrons, both nucleons, comprise the nucleus of atoms, and both nucleons behave as small magnets whose strengths are measured by their magnetic moments. The neutron interacts with normal matter primarily through the nuclear force and through 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 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. For an elementary particle to have an intrinsic magnetic moment, it must have both spin and electric charge. The neutron has spin 1/2 ħ, but it has no net charge. The existence of the neutron's magnetic moment was puzzling and defied a correct 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.
The best available measurement for the value of the magnetic moment of the neutron is μn = 04272(45) μN−1.913. Here μN is the nuclear magneton, a physical constant and standard unit for the magnetic moments of nuclear components. In SI units, μn = 3647(23)×10−27 J⋅T−1−9.662. A magnetic moment is a vector quantity, and the direction of the neutron's magnetic moment is defined by its spin. The torque on the neutron resulting from an external magnetic field is towards aligning the neutron's spin vector opposite to the magnetic field vector.
where e is the elementary charge and ħ is the reduced Planck constant. The magnetic moment of this particle is parallel to its spin. Since the neutron has no charge, it should have no magnetic moment by this expression. The non-zero magnetic moment of the neutron indicates that it is not an elementary particle. The sign of the neutron's magnetic moment is that of a negatively charged particle. Similarly, the fact that the magnetic moment of the proton, μp = 2.793 μN, is not equal to 1 μN indicates that it too is not an elementary particle. Protons and neutrons are composed of quarks, and the magnetic moments of the quarks can be used to compute the magnetic moments of the nucleons.
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 magnetic moment of the electron is therefore about 1000 times larger than that of the neutron.
Soon after the neutron was discovered in 1932, indirect evidence suggested 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. The proton's magnetic moment had been determined by measuring the deflection of a beam of molecular hydrogen by a magnetic field. Stern won the Nobel Prize in 1943 for this discovery.
By 1934 groups led by Stern, now in Pittsburgh, and I. I. Rabi in New York had independently measured the magnetic moments of the proton and deuteron. The measured values for these particles were only in rough agreement between the groups, but the Rabi group confirmed the earlier Stern measurements that the magnetic moment for the proton was unexpectedly large. Since a deuteron is composed of a proton and a neutron with aligned spins, the neutron's magnetic moment could be inferred by subtracting the deuteron and proton magnetic moments. The resulting value was not zero and had sign opposite to that of the proton. Values for the magnetic moment of the neutron were also determined by R. Bacher at Ann Arbor (1933) and I.Y. Tamm and S.A. Altshuler (1934) in the Soviet Union from studies of the hyperfine structure of atomic spectra. Although Tamm and Altshuler's estimate had the correct sign and order of magnitude (μn = −0.5 μN), the result was met with skepticism. 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. The large value for the proton's magnetic moment and the inferred negative value for the neutron's magnetic moment were unexpected and could not be explained. The anomalous values for the magnetic moments of the nucleons would remain a puzzle until the quark model was developed in the 1960s.
The refinement and evolution of the Rabi measurements led to the discovery in 1939 that the deuteron also possessed an electric quadrupole moment. This electrical property of the deuteron had been interfering with the measurements by the Rabi group. The discovery meant that the physical shape of the deuteron was not symmetric, which provided valuable insight into the nature of the nuclear force binding nucleons. Rabi was awarded the Nobel Prize in 1944 for his resonance method for recording the magnetic properties of atomic nuclei.
The value for the neutron's magnetic moment was first directly measured by Luis Alvarez and Felix Bloch at Berkeley, California in 1940, using an extension of the magnetic resonance methods developed by Rabi. Alvarez and Bloch determined the magnetic moment of the neutron to be μn = −1.93(2) μN. 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.
Neutron g-factor and gyromagnetic ratio
The magnetic moment of a nucleon is sometimes expressed in terms of its g-factor, a dimensionless scalar. The convention defining the g-factor for composite particles, such as the neutron or proton, is
where μ is the intrinsic magnetic moment, I is the spin angular momentum, and g is the effective g-factor. While the g-factor is dimensionless, for composite particles it is defined relative to the natural unit of the nuclear magneton. For the neutron, I is 1/2 ħ, so the neutron's g-factor, symbol gn, is 08545(90). −3.826
For nucleons, the ratio is conventionally written in terms of the proton mass and charge, by the formula
The neutron's gyromagnetic ratio, symbol γn, is 47179(43)×108 s−1⋅T−1. −1.832 The gyromagnetic ratio is also the ratio between the observed angular frequency of Larmor precession (in rad s−1) and the strength of the magnetic field in nuclear magnetic resonance applications, such as in MRI imaging. For this reason, the value of γn is often given in units of MHz/T. The quantity γn/2π ("gamma bar") is therefore convenient, which has the value 6943(69) MHz⋅T−1. −29.164
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 (hence its spin antiparallel to the field). Like any magnet, the amount of this torque is proportional both to the magnetic moment and the external magnetic field. Since the neutron has spin angular momentum, this torque will cause the neutron to precess with a well-defined frequency, called the Larmor frequency. It is this phenomenon that enables the measurement of nuclear properties through nuclear magnetic resonance. The Larmor frequency can be determined by the product of the gyromagnetic ratio with the magnetic field strength. Since the sign of γn is negative, the neutron's spin angular momentum precesses counterclockwise about the direction of the external magnetic field.
The magnetic moment of the neutron has been exploited to probe the properties of matter using scattering or 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 Å using cold or thermal neutrons. Bertram Brockhouse and Clifford Shull won the Nobel Prize in physics in 1994 for developing these scattering techniques.
Since neutrons are neutral particles, they do not have to overcome Coulomb repulsion as they approach charged targets, as experienced by protons or alpha particles. Neutrons can deeply penetrate matter. On the other hand, without an electric charge, neutron beams cannot be controlled by the conventional electromagnetic methods employed for particle accelerators. The magnetic moment of the neutron allows some control of neutrons using magnetic fields, however, including the formation of polarized neutron beams.
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 nucleons. The deuteron has the simplest example of a nuclear magnetic moment, with measured value 0.857 µN. This value is within 3% of the sum of the moments of the proton and neutron, which gives 0.879 µN. In this calculation, the spins of the nucleons are aligned, but their magnetic moments offset because of the neutron's negative magnetic moment.
Anomalous magnetic moments and meson physics
The anomalous values for the magnetic moments of the nucleons presented a theoretical quandary for the 30 years from the time of their discovery in the early 1930s to the development of the quark model in the early 1960s. Considerable theoretical efforts were expended in trying to understand the origins of these magnetic moments, but the failures of these theories were glaring. Much of the theoretical focus was on developing a nuclear-force equivalence to the remarkably successful theory explaining the small anomalous magnetic moment of the electron.
In quantum electrodynamics (QED), the anomalous magnetic moment of a particle stems from the small contributions of quantum mechanics to the magnetic moment of that particle. The g-factor for a "Dirac" magnetic moment is predicted to be 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 a small fraction of a percent; the difference compared to the classical value is the anomalous magnetic moment. The actual g-factor for the electron is measured to be 31930436153(53). −2.002 QED results from 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 small effects of these quantum mechanical fluctuations can be theoretically computed using Feynman diagrams with loops.
The one-loop contribution to the anomalous magnetic moment of the electron, corresponding to the first order and largest correction in QED, is found by calculating the vertex function shown in the diagram on the right. The calculation was discovered by Julian Schwinger in 1948. Computed to fourth order, the QED prediction for the electron's anomalous magnetic moment agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron one of the most accurately verified predictions in the history of physics.
Compared to the electron, the anomalous magnetic moments of the nucleons are enormous. The g-factor for the proton is 5.6, and the chargeless neutron should have no magnetic moment at all. 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: μp − = + 1.00 μN1.79 μN, μn − = 0.00 μN−1.91 μN.
The Yukawa interaction for nucleons was discovered in the mid-1930s, and this nuclear force is mediated by pion mesons. In parallel with the theory for the electron, the hypothesis was that higher-order loops involving nucleons and pions may generate the anomalous magnetic moments of the nucleons. The physical picture was that the effective magnetic moment of the neutron arose from the combined contributions of the "bare" neutron, which is zero, and the cloud of "virtual" pions and photons that surround this particle as a consequence of the nuclear and electromagnetic forces. The Feynman diagram at right is roughly the first order diagram, with the role of the virtual particles played by pions. As noted by Abraham Pais, "between late 1948 and the middle of 1949 at least six papers appeared reporting on second order calculations of nucleon moments." These theories were also, as noted by Pais, "a flop" – they gave results that grossly disagreed with observation. Nevertheless, serious efforts continued along these lines for the next couple of decades, to little success. These theoretical approaches were incorrect because the nucleons are composite particles with their magnetic moments arising from their elementary components, quarks.
Quark model for nucleon magnetic moments
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). The magnetic moment of the neutron can be modeled as a sum of the magnetic moments of the constituent quarks, although this simple model belies the complexities of the Standard Model of particle physics. 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:
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.
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%. The measured value for this ratio is 89806(34). −1.459 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 in 1964.
From the nonrelativistic, quantum mechanical wavefunction for baryons composed of three quarks, a straightforward calculation gives fairly accurate estimates for the magnetic moments of neutrons, protons, and other baryons. For a neutron, the end result of this calculation is that the magnetic moment of the neutron is given by μn = 4/3 μd − 1/3 μu, where μd and μu 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 the three quarks are in a particular, dominant quantum state.
of quark model
|p||4/3 μu − 1/3 μd||2.79||2.793|
|n||4/3 μd − 1/3 μu||−1.86||−1.913|
The results of this calculation are encouraging, but the masses of the up or down quarks were assumed to be 1/3 the mass of a nucleon. The masses of the quarks are actually only about 1% that of a nucleon. The discrepancy stems from the complexity of the Standard Model for nucleons, where most of their mass originates in the gluon fields, virtual particles, and their associated energy that are essential aspects of the strong force. Furthermore, the complex system of quarks and gluons that constitute a neutron requires a relativistic treatment. Nucleon magnetic moments have been successfully computed from first principles, requiring significant computing resources.
- Bohr magneton
- Electron magnetic moment
- Proton magnetic moment
- Nuclear magnetic moment
- Anomalous magnetic moment
- Neutron diffraction
- Neutron triple-axis spectrometry
- LARMOR neutron microscope
- Beringer, J.; et al. (Particle Data Group) (2012). "Review of Particle Physics, 2013 partial update" (PDF). Phys. Rev. D 86: 010001. Bibcode:2012PhRvD..86a0001B. doi:10.1103/PhysRevD.86.010001. Retrieved May 8, 2015.
- Bjorken, J.D.; Drell, S.D. (1964). Relativistic Quantum Mechanics. New York: McGraw-Hill. pp. 241–246. ISBN 0070054932.
- Hausser, O. (1981). "Nuclear Moments". In Lerner, R. G.; Trigg, G. L. Encyclopedia of Physics. Reading, Massachusetts: Addison-Wesley Publishing Company. pp. 679–680. ISBN 0201043130.
- "CODATA values of the fundamental constants". NIST. Retrieved May 8, 2015.
- Schreckenbach, K. (2013). "Physics of the Neutron". In Stock, R. Encyclopedia of Nuclear Physics and its Applications. Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. pp. 321–354. ISBN 978-3-527-40742-2.
- Frisch, R.; Stern, O. (1933). "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.". Z. Phys. 85: 4–16. Bibcode:1933ZPhy...85....4F. doi:10.1007/bf01330773. Retrieved May 9, 2015.
- Esterman, I.; Stern, O. (1933). "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.". Z. Phys. 85: 17–24. Bibcode:1933ZPhy...85...17E. doi:10.1007/bf01330774. Retrieved May 9, 2015.
- Toennies, J. P.; Schmidt-Bocking, H.; Friedrich, B.; Lower, J. C. A. "Otto Stern (1888–1969): The founding father of experimental atomic physics". arXiv.org. pp. 1–39. Retrieved May 9, 2015.
- "The Nobel Prize in Physics 1943". Nobel Foundation. Retrieved 2015-01-30.
- Esterman, I.; Stern, O. (1934). "Magnetic moment of the deuton". Physical Review 45: 761(A109). Bibcode:1934PhRv...45..739S. doi:10.1103/PhysRev.45.739. Retrieved May 9, 2015.
- Rabi, I.I.; Kellogg, J.M.; Zacharias, J.R. (1934). "The magnetic moment of the proton". Physical Review 46: 157–163. Bibcode:1934PhRv...46..157R. doi:10.1103/physrev.46.157. Retrieved May 9, 2015.
- Rabi, I.I.; Kellogg, J.M.; Zacharias, J.R. (1934). "The magnetic moment of the deuton". Physical Review 46: 163–165. Bibcode:1934PhRv...46..163R. doi:10.1103/physrev.46.163. Retrieved May 9, 2015.
- Breit, G.; Rabi, I.I. (1934). "On the interpretation of present values of nuclear moments". Physical Review 46: 230–231. Bibcode:1934PhRv...46..230B. doi:10.1103/physrev.46.230. Retrieved May 9, 2015.
- John S. Rigden (1987). Rabi, Scientist and Citizen. New York: Basic Books, Inc. pp. 99–114. ISBN 9780674004351. Retrieved May 9, 2015.
- Bacher, R.F. (1933). "Note on the Magnetic Moment of the Nitrogen Nucleus". Physical Review 43: 1001–1002. Bibcode:1933PhRv...43.1001B. doi:10.1103/physrev.43.1001. Retrieved 2015-02-10.
- Tamm, I.Y.; Altshuler, S.A. (1934). "Magnetic Moment of the Neutron". Doklady Akad. Nauk SSSR 8: 455. Retrieved 2015-01-30.
- Sergei Vonsovsky (1975). Magnetism of Elementary Particles. Moscow: Mir Publishers. pp. 73–75.
- Kellogg, J.M.; Rabi, I.I.; Ramsey, N.F.; Zacharias, J.R. (1939). "An electrical quadrupole moment of the deuteron". Physical Review 55: 318–319. Bibcode:1939PhRv...55..318K. doi:10.1103/physrev.55.318. Retrieved May 9, 2015.
- "The Nobel Prize in Physics 1944". Nobel Foundation. Retrieved 2015-01-25.
- Alvarez, L. W; Bloch, F. (1940). "A quantitative determination of the neutron magnetic moment in absolute nuclear magnetons". Physical Review 57: 111–122. Bibcode:1940PhRv...57..111A. doi:10.1103/physrev.57.111.
- Ramsey, Norman F. (1987). "Chapter 5: The Neutron Magnetic Moment". In Trower, W. Peter. Discovering Alvarez: Selected Works of Luis W. Alvarez with Commentary by His Students and Colleagues. University of Chicago Press. pp. 30–32. ISBN 978-0226813042. Retrieved May 9, 2015.
- Povh, B.; Rith, K.; Scholz, C.; Zetsche, F. (2002). Particles and Nuclei: An Introduction to the Physical Concepts. Berlin: Springer-Verlag. pp. 74–75,259–260. ISBN 978-3-540-43823-6. Retrieved May 10, 2015.
- "CODATA values of the fundamental constants". NIST. Retrieved May 8, 2015.
- "CODATA values of the fundamental constants". NIST. Retrieved May 8, 2015.
- Jacobsen, Neil E. (2007). NMR spectroscopy explained. Hoboken, New Jersey: Wiley-Interscience. ISBN 9780471730965. Retrieved May 8, 2015.
- "CODATA values of the fundamental constants". NIST. Retrieved May 8, 2015.
- B. D. Cullity, C. D. Graham (2008). Introduction to Magnetic Materials (2 ed.). Hoboken, New Jersey: Wiley-IEEE Press. p. 103. ISBN 0-471-47741-9. Retrieved May 8, 2015.
- M. H. Levitt (2001). Spin dynamics: basics of nuclear magnetic resonance. West Sussex, England: John Wiley & Sons. pp. 25–30. ISBN 0-471-48921-2.
- S.W. Lovesey (1986). Theory of Neutron Scattering from Condensed Matter Volume 1: Nuclear Scattering. Oxford: Clarendon Press. pp. 1–30. ISBN 0198520298.
- "The Nobel Prize in Physics 1994". Nobel Foundation. Retrieved 2015-01-25.
- Oku, T.; Suzuki, J.; et al. (2007). "Highly polarized cold neutron beam obtained by using a quadrupole magnet". Physica B 397: 188–191. Bibcode:2007PhyB..397..188O. doi:10.1016/j.physb.2007.02.055.
- Arimoto, Y.; Geltenbort, S.; et al. (2012). "Demonstration of focusing by a neutron accelerator". Physical Review A 86: 023843. Bibcode:2012PhRvA..86b3843A. doi:10.1103/PhysRevA.86.023843. Retrieved May 9, 2015.
- Semat, Henry (1972). Introduction to Atomic and Nuclear Physics: 5th edition. London: Holt, Rinehart and Winston. p. 556. ISBN 978-1-4615-9701-8. Retrieved May 8, 2015.
- Pais, Abraham (1986). Inward Bound. Oxford: Oxford University Press. p. 299. ISBN 0198519974.
- See section 6.3 in Peskin, M. E.; Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Reading, Massachusetts: Perseus Books. pp. 175–198. ISBN 978-0201503975.
- "CODATA values of the fundamental constants". NIST. Retrieved May 11, 2015.
- Aoyama, T.; Hayakawa, M.; Kinoshita, T.; Nio, M. (2008). "Revised value of the eighth-order QED contribution to the anomalous magnetic moment of the electron". Physical Review D 77 (5): 053012. arXiv:0712.2607. Bibcode:2008PhRvD..77e3012A. doi:10.1103/PhysRevD.77.053012.
- Schwinger, J. (1948). "On Quantum-Electrodynamics and the Magnetic Moment of the Electron". Physical Review 73 (4): 416–417. Bibcode:1948PhRv...73..416S. doi:10.1103/PhysRev.73.416.
- See chapter 1, section 6 in deShalit, A.; Feschbach, H. (1974). Theoretical Nuclear Physics Volume I: Nuclear Structure. New York: John Wiley and Sons. p. 31. ISBN 0471203858.
- Brown, L.M.; Rechenberg, H. (1996). The Origin of the Concept of Nuclear Forces. Bristol and Philadelphia: Institute of Physics Publishing. pp. 95–312. ISBN 0750303735.
- Drell, S.; Zachariasen, F. (1961). Electromagnetic Structure of Nucleons. New York: Oxford University Press. pp. 1–130.
- Drell, S.; Pagels, H.R. (1965). "Anomalous Magnetic Moment of the Electron, Muon, and Nucleon". Physical Review 140: B397–B407. Bibcode:1965PhRv..140..397D. doi:10.1103/PhysRev.140.B397. Retrieved May 10, 2015.
- Gell, Y.; Lichtenberg, D. B. (1969). "Quark model and the magnetic moments of proton and neutron". Il Nuovo Cimento A. Series 10 61: 27–40. Bibcode:1969NCimA..61...27G. doi:10.1007/BF02760010.
- Perkins, Donald H. (1982), Introduction to High Energy Physics, Addison Wesley, Reading, Massachusetts, pp. 201–202, ISBN 0-201-05757-3
- Cho, Adiran (2 April 2010). "Mass of the Common Quark Finally Nailed Down". http://news.sciencemag.org. American Association for the Advancement of Science. Retrieved 27 September 2014.
- Greenberg, O. W. (2009), "Color charge degree of freedom in particle physics", Compendium of Quantum Physics, Springer Berlin Heidelberg, pp. 109–111, doi:10.1007/978-3-540-70626-7_32
- Beg, M.A.B.; Lee, B.W.; Pais, A. (1964). "SU(6) and electromagnetic interactions". Physical Review Letters 13: 514–517, erratum 650. Bibcode:1964PhRvL..13..514B. doi:10.1103/physrevlett.13.514.
- Sakita, B. (1964). "Electromagnetic properties of baryons in the supermultiplet scheme of elementary particles". Physical Review Letters 13: 643–646. Bibcode:1964PhRvL..13..643S. doi:10.1103/physrevlett.13.643.
- Mohr, P.J.; Taylor, B.N. and Newell, D.B. (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 6.0). The database was developed by J. Baker, M. Douma, and S. Kotochigova. (2011-06-02). National Institute of Standards and Technology, Gaithersburg, Maryland 20899. Retrieved May 9, 2015.
- Wilczek, F. (2003). "The Origin of Mass" (PDF). MIT Physics Annual: 24–35. Retrieved May 8, 2015.
- Ji, Xiangdong (1995). "A QCD Analysis of the Mass Structure of the Nucleon". Phys.Rev.Lett. 74: 1071–1074. arXiv:hep-ph/9410274. Bibcode:1995PhRvL..74.1071J. doi:10.1103/PhysRevLett.74.1071. Retrieved May 8, 2015.
- Martinelli, G.; Parisi, G.; Petronzio, R.; Rapuano, F. (1982). "The proton and neutron magnetic moments in lattice QCD". Physics Letters B 116: 434–436. Bibcode:1982PhLB..116..434M. doi:10.1016/0370-2693(82)90162-9. Retrieved May 8, 2015.
- Kincade, Kathy (2 February 2015). "Pinpointing the magnetic moments of nuclear matter". http://phys.org. Phys.org. Retrieved May 8, 2015.
- S.W. Lovesey (1986). Theory of Neutron Scattering from Condensed Matter. Oxford University Press. ISBN 0198520298.
- 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.