# 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 act 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 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. While the neutron has spin 1/2 ħ, it has no net charge.

## Description

Schematic diagram depicting the spin of the neutron as the black arrow and magnetic field lines associated with the neutron's negative magnetic moment. While the spin of the neutron is upward in this diagram, the magnetic field lines at the center of the dipole are downward.

The best available measurement for the value of the magnetic moment of the neutron is μn = −1.91304272(45) μN.[1] Here μN is the nuclear magneton, a physical constant and standard unit for the magnetic moments of nuclear components. In SI units, μn = −9.6623647(23)×10−27 J·T−1. 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.

The nuclear magneton is the spin magnetic moment of a Dirac particle, a charged, spin 1/2 elementary particle, with a proton's mass mp. In SI units, the nuclear magneton is

$\mu_\mathrm{N} = {{e \hbar} \over {2 m_\mathrm{p}}},$

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.[2] 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 2000 times larger than that of the nucleons.

The magnetic moment of the antineutron is the same magnitude, but is of opposite sign, as that of the neutron.

## Measurement

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.[3][4] Stern won the Nobel Prize in 1943 for this discovery using molecular beams.[5]

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.[6][7][8] While the measured values for these particles were only in rough agreement between the groups, the Rabi group confirmed the earlier Stern measurements that the magnetic moment for the proton was unexpectedly large.[9][10] 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[11] at Ann Arbor (1933) and I.Y. Tamm and S.A. Altshuler[12] (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.[9][13] 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.[10] The large value for the proton's magnetic moment and the inferred negative value for the neutron's magnetic moment were unexpected and raised many questions.[9] 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.[10][14] 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 won the Nobel Prize in 1944 for his resonance method for recording the magnetic properties of atomic nuclei.[15]

The value for the neutron's magnetic moment was first directly measured by Luis Alvarez and Felix Bloch at Berkeley, California in 1940,[16] 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 conventional formula is

$\boldsymbol{\mu} = \frac{g \mu_\mathrm{N}}{\hbar}\boldsymbol{I}$

where μ is the intrinsic magnetic moment of the nucleon, I is the nuclear spin angular momentum, and g is the effective g-factor. For the neutron, I is 1/2 ħ, so the neutron's g-factor, symbol gn, is −3.82608545(90).[17]

The gyromagnetic ratio, symbol γ, 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

$\gamma = \frac{g \mu_\mathrm{N}}{\hbar} = g\frac{e}{2m_\text{p}}$

The neutron's gyromagnetic ratio, symbol γn, is −1.83247179(43)×108 s−1·T−1.[18] 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,[19] 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 −29.1646943(69) MHz·T−1.[20]

## Physical significance

Direction of Larmor precession for a neutron. The central arrow denotes the magnetic field, the small red arrow the spin of the neutron.

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).[21] 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.[22] Bertram Brockhouse and Clifford Shull won the Nobel Prize in physics in 1994 for the development of these scattering techniques.[23]

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,[24][25] 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.

## Thirty years in the theoretical wilderness

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.[26] 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.

One-loop correction to the fermion's magnetic dipole moment. The solid lines at top and bottom represent the fermion (electron or nucleon), the wavey lines represent the photon mediating the electromagnetic force. The middle solid lines represent a virtual pair of particles, electron and positron for QED, pions for the nuclear force.

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 −2.002319304361.... 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, discovered in 1948 by Julian Schwinger, is relatively straightforward.[27][28] 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 the most accurately verified prediction 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, while 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 μN = +1.79 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 thinking was that higher-order loops involving nucleons and pions may give rise to the anomalous magnetic moments of the nucleons.[2] 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."[26] 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.[2][29] These theoretical approaches were incorrect because the nucleons are composite particles with their magnetic moments arising from their elementary components, quarks.

## Magnetic moment, quarks, and the Standard Model

Within the quark model for hadrons, such as the neutron, the neutron is composed of one up quark (charge +2/3 e) and two down quarks (charge −1/3 e).[30] The magnetic moment of the neutron can be modeled as a sum of the magnetic moments of the constituent quarks,[31] although this simple model belies the complexities of the Standard Model of particle physics.[32]

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%.[33][34][35] The measured value for this ratio is −1.45989806(34).[36] 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.[33]

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.[31] 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. 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.

Baryon Magnetic moment
of quark model
Computed
($\mu_\mathrm{N}$)
Observed
($\mu_\mathrm{N}$)
p 4/3 μu − 1/3 μd 2.79 2.793
n 4/3 μd − 1/3 μu −1.86 −1.913

While the results of this calculation are encouraging, the masses of the up or down quarks were assumed to be 1/3 the mass of a nucleon,[31] whereas the masses of these quarks are only about 1% that of a nucleon.[32] The discrepancy stems from the complexity of the Standard Model for nucleons, where most of their mass originates in the gluon fields and virtual particles that are essential aspects of the strong force.[32] Further, the complex system of quarks and gluons that constitute a neutron requires a relativistic treatment. A calculation of nucleon magnetic moments from first principles is not yet available.

## References

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4. ^ 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. doi:10.1007/bf01330774.
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