Spin (physics)

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Magnetic field lines around a magnetostatic dipole; the magnetic dipole itself is in the center and is seen from the side.

This article is about spin in quantum mechanics. For rotation in classical mechanics, see angular momentum. For other uses, see spin (disambiguation).

In particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles, composite particles (hadrons), and atomic nuclei.[notes 1]

All elementary particles of a given kind have the same spin quantum number, which is an important part of a particle's quantum state. The spin of electrons, when combined with the spin-statistics theorem, results in the Pauli exclusion principle, which in turn grounds the periodic table of chemical elements. The spin direction (also called spin for short) of a particle is an important intrinsic degree of freedom.

Wolfgang Pauli was the first to propose the concept of spin, but he did not name it. In 1925, Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit suggested a physical interpretation of particles spinning around their own axis. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part thereof.

The head-on collision of a quark (the red ball) from one proton (the orange ball) with a gluon (the green ball) from another proton with opposite spin, spin is represented by the blue arrows circling the protons and the quark. The blue question marks circling the gluon represents the question: Are gluons polarized? The particles ejected from the collision are a shower of quarks and one photon of light (the purple ball).

Spin quantum number

As the name suggests, spin was originally conceived as the rotation of a particle around some axis. This picture is correct in so far as spins obey the same mathematical laws as do quantized angular momenta. On the other hand, spins have some peculiar properties that distinguish them from orbital angular momenta:

Spin quantum numbers may take on half-integer values;
The spin of a charged particle is associated with a magnetic dipole moment with a g-factor differing from 1. This is incompatible with classical physics, assuming that the charge and mass of the particle are distributed evenly in spheres of equal radius.

Elementary particles

Elementary particles are particles for which there is no known way of dividing them into smaller units. Theoretical and experimental studies have shown that the spin possessed by such particles cannot be explained by postulating that they are made up of even smaller particles rotating about a common center of mass (see classical electron radius); as far as can be determined, these elementary particles are true point particles. The spin of an elementary particle is a truly intrinsic physical property, akin to the particle's electric charge and rest mass.

Let the spin quantum number s be n/2, where n can be any non-negative integer. Hence the allowed values of s are 0, 1/2, 1, 3/2, 2, etc. The value of s for an elementary particle depends only on the type of particle, and cannot be altered in any known way (in contrast to the spin direction described below). The spin angular momentum S of any physical system is quantized. The allowed values of S are:

$S = \hbar \, \sqrt{s (s+1)},$

where $\hbar$ is the reduced Planck's constant. In contrast, orbital angular momentum can only take on integer quantum numbers.

All known matter is ultimately composed of elementary particles called fermions, and all elementary fermions have s=1/2. Examples of fermions are the electron and positron, the quarks making up protons and neutrons, and the neutrinos. Elementary particles emit and receive one or more particles called bosons. This boson exchange gives rise to the three fundamental interactions ("forces") of the Standard model of particle physics; hence bosons are also called force carriers. These bosons have s=1. The best understood boson is the photon. Electromagnetism is the force that results when charged particles exchange photons.

Theory predicts the existence of two bosons whose s differs from 1. The force carrier for gravity is the hypothetical graviton; theory suggests that it has s=2. The Higgs mechanism predicts that elementary particles acquire nonzero rest mass by exchanging hypothetical Higgs bosons with an all-pervasive Higgs field. Theory predicts that the Higgs boson has s=0. If so, it would be the only elementary particle for which this is the case.

Composite particles

The spin of composite particles, such as protons, neutrons, and atomic nuclei is usually understood to mean the total angular momentum, which is the sum of the spins and orbital angular momenta of the constituent particles. Such a composite spin is subject to the same quantization condition as any other angular momentum.

Composite particles are often referred to as having a definite spin, just like elementary particles; for example, the proton is a spin-1/2 particle. This is understood to refer to the spin of the lowest-energy internal state of the composite particle (i.e., a given spin and orbital configuration of the constituents).[1]

It is not always easy to deduce the spin of a composite particle from first principles; for example, even though we know that the proton is a spin-1/2 particle, the question of how this spin is distributed among the three internal valence quarks and the surrounding sea quarks and gluons is an active area of research.

Delta baryons, which decay into protons and neutrons, have spin 3/2. All the three quarks inside a Δ particle have their spin axis pointing in the same direction, unlike the nearly identical proton and neutron (called "nucleons") in which the intrinsic spin of one of the three constituent quarks is always opposite the spin of the other two. This difference in spin alignment is the only quantum number distinction between the Δ⁺ and Δ⁰ and ordinary nucleons.

Atoms and molecules

The spin of atoms and molecules is the sum of the spins of unpaired electrons. It is responsible for paramagnetism.

The spin-statistics theorem

The spin of a particle has crucial consequences for its properties in statistical mechanics. Particles with half-integer spin obey Fermi-Dirac statistics, and are known as fermions. They are required to occupy antisymmetric quantum states (see the article on identical particles.) This property forbids fermions from sharing quantum states – a restriction known as the Pauli exclusion principle. Particles with integer spin, on the other hand, obey Bose-Einstein statistics, and are known as bosons. These particles occupy "symmetric states", and can therefore share quantum states. The proof of this is known as the spin-statistics theorem, which relies on both quantum mechanics and the theory of special relativity. In fact, "the connection between spin and statistics is one of the most important applications of the special relativity theory".[2]

Magnetic moments

Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern–Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves.

The intrinsic magnetic moment μ of an elementary particle with charge q, mass m, and spin angular momentum S, is

$\mu = g \, \frac{q}{2m}\, S$

where the dimensionless quantity g is called the g-factor. For exclusively orbital rotations it would be 1 (assuming that the mass and the charge occupy spheres of equal radius).

The electron, being a charged elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g-factor, which has been experimentally determined to have the value −2.002 319 304 3622(15), with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation.[3] The value of 2 arises from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties, and the correction of 0.002 319 304… arises from the electron's interaction with the surrounding electromagnetic field, including its own field.[4] Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions.

Neutrinos are both elementary and electrically neutral. The minimally extended Standard Model that takes into account finite neutrino masses predicts neutrino magnetic moments of:[5][6][7]

$\mu_{\nu}\approx 3\times 10^{-19}\mu_{B}\frac{m_{\nu}}{\text{eV}}$

where the $μ ν$ are the neutrino magnetic moments, $m ν$ are the neutrino masses, and $μ B$ is the Bohr magneton. New physics above the electroweak scale could, however, lead to significantly higher neutrino magnetic moments. It can be shown in a model independent way that neutrino magnetic moments larger than about 10⁻¹⁴ $μ B$ are unnatural, because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses cannot exceed about 1 eV, these radiative corrections must then be assumed to be fine tuned to cancel out to a large degree.[8]

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