In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol γ, gamma. Its SI unit is the reciprocal second per tesla (s−1⋅T−1) or, equivalently, the coulomb per kilogram (C⋅kg−1). The g-factor of a particle is a related dimensionless value of the system, derived as the ratio of its gyromagnetic ratio to that which would be classically expected from a rigid body of which the mass and charge are distributed identically, and for which total mass and charge are the same as that of the system.
For a classical rotating body Consider a nonconductive charged body rotating about an axis of symmetry. According to the laws of classical physics, it has both a magnetic dipole moment due to the movement of charge and an angular momentum due to the movement of mass arising from its rotation. It can be shown that as long as its charge and mass densities and currents are distributed identically and rotationally symmetric, its gyromagnetic ratio is
γ = q 2 m , {\displaystyle \gamma ={\frac {q}{2m}},}
where q is its charge, and m is its mass. The derivation of this relation is as follows. It suffices to demonstrate this for an infinitesimally narrow circular ring within the body, as the general result then follows from an integration. Suppose the ring has radius r, area A = πr2, mass m, charge q, and angular momentum L = mvr. Then the magnitude of the magnetic dipole moment is
μ = I A = q v 2 π r π r 2 = q 2 m m v r = q 2 m L . {\displaystyle \mu =IA={\frac {qv}{2\pi r}}\,\pi r^{2}={\frac {q}{2m}}\,mvr={\frac {q}{2m}}L.}
For an isolated electron An isolated electron has an angular momentum and a magnetic moment resulting from its spin. While an electron's spin is sometimes visualized as a rotation of a rigid body about an axis, the magnetic moment cannot be attributed to mass distributed identically to the charge in such a model since it is close to twice what this would predict. The correcting factor needed relative to classical relation is called the electron's g-factor, which is denoted ge:
γ e − = μ e − ℏ / 2 = − g e − e 2 m e = g e μ B ℏ , {\displaystyle \gamma _{{\text{e}}^{-}}={\frac {\mu _{{\text{e}}^{-}}}{\hbar /2}}=-g_{\text{e}}{\frac {-e}{2m_{\text{e}}}}=g_{\text{e}}{\frac {\mu _{\text{B}}}{\hbar }},}
where μe− is the electron's magnetic moment, ħ/2 is the angular momentum (spin) of the electron, and μB is the Bohr magneton. The gyromagnetic ratio due to electron spin is roughly twice that due to the orbiting of an electron. The electron gyromagnetic ratio is
γ e − {\displaystyle \gamma _{{\text{e}}^{-}}} = −1.76085962784(55)×1011 s−1⋅T−1 The ratio of the electron's Larmor frequency to the magnetic flux density is
γ ¯ e − = γ e − 2 π {\displaystyle {\bar {\gamma }}_{{\text{e}}^{-}}={\frac {\gamma _{{\text{e}}^{-}}}{2\pi }}} = −28024.9513861(87) MHz⋅T−1 The electron gyromagnetic ratio γ (and its g-factor ge) are in excellent agreement with theory; see Precision tests of QED for details. In the framework of relativistic quantum mechanics,
g e = − 2 ( 1 + α 2 π + ⋯ ) , {\displaystyle g_{\text{e}}=-2\left(1+{\frac {\alpha }{2\pi }}+\cdots \right),}
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