Spin magnetic moment

In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. For example, the electron is an elementary spin-1/2 fermion. Quantum electrodynamics gives the most accurate prediction of the anomalous magnetic moment of the electron. 'Spin' is a non-classical property of elementary particles, since classically the 'spin angular momentum' of a material object is really just the total orbital angular momenta of the object's constituents about the rotation axis. Elementary particles are conceived as concepts which have no axis to 'spin' around (see wave–particle duality). In general, a magnetic moment can be defined in terms of an electric current and the area enclosed by the current loop. Since angular momentum corresponds to rotational motion, the magnetic moment can be related to the orbital angular momentum of the charge carriers in the constituting current. However, in magnetic materials, the atomic and molecular dipoles have magnetic moments not just because of their quantized orbital angular momentum, but, due to the spin of elementary particles constituting them (electrons, and the quarks in the protons and neutrons of the atomic nuclei). A particle may have a spin magnetic moment without having an electric charge. For example, the neutron is electrically neutral but has a non-zero magnetic moment because of its internal quark structure. We can calculate the observable spin magnetic moment, a vector, μ→S, for a sub-atomic particle with charge q, mass m, and spin angular momentum (also a vector), S→, via: where γ ≡ g q 2 m {displaystyle gamma equiv g{frac {q}{2m}}} is the gyromagnetic ratio, g is a dimensionless number, called the g-factor, q is the charge, and m is the mass. The g-factor depends on the particle: it is g = −2.0023 for the electron, g = 5.586 for the proton, and g = −3.826 for the neutron. The proton and neutron are composed of quarks, which have a non-zero charge and a spin of ħ/2, and this must be taken into account when calculating their g-factors. Even though the neutron has a charge q = 0, its quarks give it a magnetic moment. The proton and electron's spin magnetic moments can be calculated by setting q = +e and q = −e, respectively, where e is the elementary charge. The intrinsic electron magnetic dipole moment is approximately equal to the Bohr magneton μB because g ≈ −2 and the electron's spin is also ħ/2: Equation (1) is therefore normally written as Just like the total spin angular momentum cannot be measured, neither can the total spin magnetic moment be measured. Equations (1), (2), (3) give the physical observable, that component of the magnetic moment measured along an axis, relative to or along the applied field direction. Assuming a Cartesian coordinate system, conventionally, the z-axis is chosen but the observable values of the component of spin angular momentum along all three axes are each ±ħ/2. However, in order to obtain the magnitude of the total spin angular momentum, S→ be replaced by its eigenvalue, √s(s + 1), where s is the spin quantum number. In turn, calculation of the magnitude of the total spin magnetic moment requires that (3) be replaced by: