nucleon

In physics, a nucleon is a collective name for two baryons: the neutron and the proton. They are constituents of the atomic nucleus and until the 1960s were thought to be elementary particles. In those days their interactions (now called internucleon interactions) defined strong interactions. Now they are known to be composite particles, made of quarks. Understanding the nucleons' properties is one of the major goals of quantum chromodynamics, the modern theory of strong interactions.

The proton is the lightest baryon and its stability is a measure of baryon number conservation. The proton's lifetime thus puts strong constraints on speculative theories which try to extend the Standard Model of particle physics. The neutron decays into a proton through the weak interaction. The two are members of an isospin doublet (I = ¹⁄₂).

The proton

With spin ¹⁄₂ and positive parity (often shortened to J^P = ¹⁄₂⁺), a charge of +1 e, and rest mass of 938 MeV/c², the proton is the nucleus of the hydrogen-1 atom (¹H). It has a magnetic moment, denoted μ_p, of 2.79 nuclear magnetons (μ_N). The predicted electric dipole moment is zero, consistent with the experimentally found upper bound value which is less than 5.4×10⁻²⁶ e·m, for example, from NMR measurements.

A proton is made up of three quarks (two up quarks and one down quark), held together by the strong force, which is mediated by gluons. In some speculative grand unified theories it may decay with a half-life equal to that of the Universe. The half-life for this decay has been thus limited to be greater than 2.1×10²⁹ years. The charge radius is measured mainly through elastic electron-proton scattering and is 0.870 fm. For specific decay modes, into leptons or antileptons and a meson, the bound state is predicted to be longer lived than 10³² years. The proton is therefore taken to be a stable particle, and the baryon number is thus assumed to be conserved.

The neutron

The neutron has no net electrical charge; it does however have a (dipolar) magnetic moment μ_n , spin, parity of ¹⁄₂⁺, and a rest mass of 940 MeV/c². Like the proton, a neutron is made up of three fractional-charge quarks (in this case one up quark of charge +²⁄₃ e and two down quarks of charge −¹⁄₃ e, whose total charge is zero), that are being held together by the strong nuclear force. It decays weakly through the process

n⁰ → p⁺ + e⁻ + ν_e

The most precise measurements of its decay lifetime are mainly from traps of various kinds and in beams. The lifetime of a free neutron outside the nucleus is 885.7±0.8 s (about 15 minutes).

Its magnetic moment is −1.91 μ_N. Both time reversal and parity invariance of the strong interactions implies that the neutron's electric dipole moment must be zero; the current observational bound is that it is less than 6.3×10⁻²⁶ e·m. The mean-square charge radius, related to the scattering length, measured in low energy electron-neutron scattering for the neutron is −0.116 fm².

Violation of the conservation of baryon number (B) may give rise to oscillations between the neutron and antineutron, through processes which change B by two units. Using free neutrons from nuclear reactors, as well as neutrons bound inside nuclei, the mean time for these transitions is found to be greater than 1.3×10⁸ s. The much poorer bound, as compared to protons, is related to the difficulty of the observations.

A limit on electric charge non-conservation comes from the observed lack of the decay

n⁰ → p⁺ + ν_e + ν_e

The observations which limit the branching fraction of the neutron in this decay channel to less than 8×10⁻²⁷ are all done looking for appropriate decays of nuclei (A → A and Z → Z + 1).

Antinucleons

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CPT-symmetry puts strong constraints on the relative properties of particles and antiparticles and, therefore, is open to stringent tests. For example, the absolute value of the proton and antiproton charges have to be equal. This equality has been tested to one part in 10⁸. The equality of their masses is also tested to 10⁻⁸. By holding antiprotons in a Penning trap, the equality of the charge to mass ratio of the proton has been tested to 90×10⁻¹². The magnetic moment of the antiproton is found to be equal and opposite to that of the proton. For the neutron-antineutron system, the masses are equal to within a factor of 9×10⁻⁵.

Quark model classification

In the quark model with SU(2) flavour, the two nucleons are part of the ground state doublet. The proton has quark content of uud, and the neutron, udd. In SU(3) flavour, they are part of the ground state octet (8) of spin ¹⁄₂ baryons, known as the Eightfold way. The other members of this octet are the hyperons strange isotriplet Σ⁺, Σ⁰, Σ⁻, the Λ and the strange isodoublet Ξ⁰, Ξ⁻. One can extend this multiplet in SU(4) flavour (with the inclusion of the charm quark) to the ground state 20-plet, or to SU(6) flavour (with the inclusion of the top and bottom quarks) to the ground state 56-plet.

The article on isospin provides an explicit expression for the nucleon wave functions in terms of the quark flavour eigenstates.

Models of the nucleon

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Although it is known that the nucleon is made from three quarks, as of 2006^[update], it is not known how to solve the equations of motion for quantum chromodynamics. Thus, the study of the low-energy properties of the nucleon are performed by means of models. The only first-principles approach available is to attempt to solve the equations of QCD numerically, using lattice QCD. This requires complicated algorithms and very powerful supercomputers. However, several analytic models also exist:

The Skyrmion models the nucleon as a topological soliton in a non-linear SU(2) pion field. The topological stability of the Skyrmion is interpreted as the conservation of baryon number, that is, the non-decay of the nucleon. The local topological winding number density is identified with the local baryon number density of the nucleon. With the pion isospin vector field oriented in the shape of a hedgehog, the model is readily solvable, and is thus sometimes called the hedgehog model. The hedgehog model is able to predict low-energy parameters, such as the nucleon mass, radius and axial coupling constant, to approximately 30% of experimental values.

The MIT bag model confines three non-interacting quarks to a spherical cavity, with the boundary condition that the quark vector current vanish on the boundary. The non-interacting treatment of the quarks is justified by appealing to the idea of asymptotic freedom, whereas the hard boundary condition is justified by quark confinement. Mathematically, the model vaguely resembles that of a radar cavity, with solutions to the Dirac equation standing in for solutions to the Maxwell equations and the vanishing vector current boundary condition standing for the conducting metal walls of the radar cavity. If the radius of the bag is set to the radius of the nucleon, the bag model predicts a nucleon mass that is within 30% of the actual mass. An important failure of the basic bag model is its failure to provide a pion-mediated interaction.

The chiral bag model merges the MIT bag model and the Skyrmion model. In this model, a hole is punched out of the middle of the Skyrmion, and replaced with a bag model. The boundary condition is provided by the requirement of continuity of the axial vector current across the bag boundary. Very curiously, the missing part of the topological winding number (the baryon number) of the hole punched into the Skyrmion is exactly made up by the non-zero vacuum expectation value (or spectral asymmetry) of the quark fields inside the bag. As of 2006^[update], this remarkable trade-off between topology and the spectrum of an operator does not have any grounding or explanation in the mathematical theory of Hilbert spaces and their relationship to geometry. Several other properties of the chiral bag are notable: it provides a better fit to the low energy nucleon properties, to within 5–10%, and these are almost completely independent of the chiral bag radius (as long as the radius is less than the nucleon radius). This independence of radius is referred to as the Cheshire Cat principle, after the fading to a smile of Lewis Carroll's Cheshire Cat. It is expected that a first-principles solution of the equations of QCD will demonstrate a similar duality of quark-pion descriptions.

See also

• Hadrons
• Electroweak interaction

References

• Gerald Edward Brown and A. D. Jackson, The Nucleon-Nucleon Interaction, (1976) North-Holland Publishing, Amsterdam ISBN 0-7204-0335-9
• Linas Vepstas, A.D. Jackson, A.S. Goldhaber, Two-phase models of baryons and the chiral Casimir effect, Physics Letters B140 (1984) p. 280-284.
• Linas Vepstas, A.D. Jackson, Justifying the Chiral Bag, Physics Reports, 187 (1990) p. 109-143.
• Particle data group listing on the proton
• Particle data group listing on the neutron