Nucleon magnetic moment

The CODATA recommended value for the magnetic moment of the proton is The best available measurement for the value of the magnetic moment of the neutron is Here, μN is the nuclear magneton, a standard unit of meausure for the magnetic moments of nuclear components, and μB is the Bohr magneton, an alternate unit from spectroscopy, both being physical constants. In SI units, these values are and A magnetic moment is a vector quantity, and the direction of the nucleon's magnetic moment is determined by its spin. The torque on the neutron that results from an external magnetic field is towards aligning the neutron's spin vector opposite to the magnetic field vector. The nuclear magneton is the spin magnetic moment of a Dirac particle, a charged, elementary particle, with a proton's mass p, in which anomalous corrections are ignored. The magnetic moment of such a particle is parallel to its spin. The sign of the neutron's magnetic moment is that of a negatively charged particle. Similarly, that the magnetic moment of the proton, is not almost equal to indicates that it too is not an elementary particle. Although the nucleons interact with normal matter through magnetic forces, the magnetic interactions are many orders of magnitude weaker than the nuclear interactions. The influence of the neutron's magnetic moment is therefore only apparent for low energy, or slow, neutrons. The magnetic moments of the antiproton and antineutron have the same magnitudes as their antiparticles, the proton and neutron, but they have opposite sign. ==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. The proton's magnetic moment was determined by measuring the deflection of a beam of molecular hydrogen by a magnetic field. Stern won the Nobel Prize in Physics in 1943 for this discovery. Neutron The neutron was discovered in 1932, and since it had no charge, it was assumed to have no magnetic moment. Indirect evidence suggested that the neutron had a non-zero value for its magnetic moment, however, until direct measurements of the neutron's magnetic moment in 1940 resolved the issue. Values for the magnetic moment of the neutron were independently determined by R. Bacher at the University of Michigan at Ann Arbor (1933) and I. Y. Tamm and S. A. Altshuler 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 (), the result was met with skepticism. The measured values for these particles were only in rough agreement between the groups, but the Rabi group confirmed the earlier Stern measurements that the magnetic moment for the proton was unexpectedly large. Since a deuteron is composed of a proton and a neutron with aligned spins, the neutron's magnetic moment could be inferred by subtracting the deuteron and proton magnetic moments. The resulting value was not zero and had a sign opposite to that of the proton. By the late 1930s, accurate values for the magnetic moment of the neutron had been deduced by the Rabi group using measurements employing newly developed nuclear magnetic resonance techniques. Unexpected consequences The large value for the proton's magnetic moment and the inferred negative value for the neutron's magnetic moment were unexpected and could not be explained. The 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. == Nucleon gyromagnetic ratios ==

The magnetic moment of a nucleon is sometimes expressed in terms of its -factor, a dimensionless scalar. The convention defining the -factor for composite particles, such as the neutron or proton, is : \boldsymbol{\mu} = \frac{\ g\ \mu_\mathsf{N}\ }{ \hbar }\ \boldsymbol{I}\ , where    is the intrinsic magnetic moment,    is the spin angular momentum, and is the effective -factor. While the -factor is dimensionless, for composite particles it is defined relative to the nuclear magneton. For the neutron,    is so the neutron's -factor is while the proton's -factor is The gyromagnetic ratio, symbol , of a particle or system is the ratio of its magnetic moment to its spin angular momentum, or : \ \boldsymbol{\mu} = \gamma\ \boldsymbol{I} ~. For nucleons, the ratio is conventionally written in terms of the proton mass and charge, by the formula : \ \gamma = \frac{\ g\ \mu_\mathsf{N}\ }{ \hbar } = g\ \frac{ e }{\ 2\ m_\mathsf{p}\ } ~. The neutron's gyromagnetic ratio is The proton's gyromagnetic ratio is 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, such as in MRI imaging. For this reason, the quantity called "gamma bar", expressed in the unit MHz / T, is often given. The quantities and are therefore convenient. == Physical significance ==

Larmor precession When a nucleon is put into a magnetic field produced by an external source, it is subject to a torque tending to orient its magnetic moment parallel to the field (in the case of the neutron, its spin is antiparallel to the field). As with any magnet, this torque is proportional the product of the magnetic moment and the external magnetic field strength. Since the nucleons have spin angular momentum, this torque will cause them to precess with a well-defined frequency, called the Larmor frequency. It is this phenomenon that enables the measurement of nuclear properties through nuclear magnetic resonance. The Larmor frequency can be determined from the product of the gyromagnetic ratio with the magnetic field strength. Since for the neutron the sign of γn is negative, the neutron's spin angular momentum precesses counterclockwise about the direction of the external magnetic field. Proton nuclear magnetic resonance Nuclear magnetic resonance employing the magnetic moments of protons is used for nuclear magnetic resonance (NMR) spectroscopy. Since hydrogen-1 nuclei are within the molecules of many substances, NMR can determine the structure of those molecules. Determination of neutron spin The interaction of the neutron's magnetic moment with an external magnetic field was exploited to determine the spin of the neutron. In 1949, D. Hughes and M. Burgy measured neutrons reflected from a ferromagnetic mirror and found that the angular distribution of the reflections was consistent with spin . 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. They recorded the two such spin states, consistent with a spin  particle. B. Brockhouse and C. Shull won the Nobel Prize in physics in 1994 for developing these scattering techniques. Control of neutron beams by magnetism As neutrons carry no electric charge, neutron beams cannot be controlled by the conventional electromagnetic methods employed in particle accelerators. The magnetic moment of the neutron allows some control of neutrons using magnetic fields, however, including the formation of polarized neutron beams. The reflection preferentially selects particular spin states, thus polarizing the neutrons. Neutron magnetic mirrors and guides use this total internal reflection phenomenon to control beams of slow neutrons. Nuclear magnetic moments Since an atomic nucleus consists of a bound state of protons and neutrons, the magnetic moments of the nucleons contribute to the nuclear magnetic moment, or the magnetic moment for the nucleus as a whole. In this calculation, the spins of the nucleons are aligned, but their magnetic moments offset because of the neutron's negative magnetic moment. == Nature of the nucleon magnetic moments ==

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

The anomalous values for the magnetic moments of the nucleons presented a theoretical quandary for the 30 years from the time of their discovery in the early 1930s to the development of the quark model in the 1960s. 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. By this idea, the magnetic moment of the neutron was caused by the fleeting existence of the large magnetic moment of the electron in the course of these quantum-mechanical fluctuations, the value of the magnetic moment determined by the length of time the virtual electron was in existence. The theory proved to be untenable, however, when H. Bethe and R. Bacher showed that it predicted values for the magnetic moment that were either much too small or much too large, depending on speculative assumptions. Similar considerations for the electron proved to be much more successful. In quantum electrodynamics (QED), the anomalous magnetic moment of a particle stems from the small contributions of quantum mechanical fluctuations to the magnetic moment of that particle. The g-factor for a "Dirac" magnetic moment is predicted to be 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 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 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 J. Schwinger in 1948. Computed to fourth order, the QED prediction for the electron's anomalous magnetic moment agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron one of the most accurately verified predictions in the history of physics. The Yukawa interaction for nucleons was discovered in the mid-1930s, and this nuclear force is mediated by pion mesons. The Feynman diagram at right is roughly the first-order diagram, with the role of the virtual particles played by pions. As noted by A. Pais, "between late 1948 and the middle of 1949 at least six papers appeared reporting on second-order calculations of nucleon moments". These theoretical approaches were incorrect because the nucleons are composite particles with their magnetic moments arising from their elementary components, quarks. ==Quark model of nucleon magnetic moments==

In the quark model for hadrons, the neutron is composed of one up quark (charge  ) and two down quarks (charge  ) while the proton is composed of one down quark (charge  ) and two up quarks (charge  ). The magnetic moment of the nucleons can be modeled as a sum of the magnetic moments of the constituent quarks, The calculation assumes that the quarks behave like pointlike Dirac particles, each having their own magnetic moment, as computed using an expression similar to the one above for the nuclear magneton: \ \mu_\text{q} = \frac{\ e_\text{q} \hbar\ }{2 m_\text{q}}\ , where the q-subscripted variables refer to quark magnetic moment, charge, or mass. The measured value for this ratio is . 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 O. Greenberg in 1964. Furthermore, the complex system of quarks and gluons that constitute a nucleon requires a relativistic treatment. Nucleon magnetic moments have been successfully computed from first principles, requiring significant computing resources. ==See also==