Gyromagnetic ratio

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 radian per second per tesla (rad⋅s−1⋅T−1) or, equivalently, the coulomb per kilogram (C⋅kg−1). 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 radian per second per tesla (rad⋅s−1⋅T−1) or, equivalently, the coulomb per kilogram (C⋅kg−1). The term 'gyromagnetic ratio' is often used as a synonym for a different but closely related quantity, the g-factor. The g-factor, unlike the gyromagnetic ratio, is dimensionless. For more on the g-factor, see below, or see the article g-factor. Any free system with a constant gyromagnetic ratio, such as a rigid system of charges, a nucleus, or an electron, when placed in an external magnetic field B (measured in teslas) that is not aligned with its magnetic moment, will precess at a frequency f (measured in hertz), that is proportional to the external field: For this reason, values of γ/(2π), in units of hertz per tesla (Hz/T), are often quoted instead of γ. The derivation of this relation is as follows: First we must prove that the torque resulting from subjecting a magnetic moment m ¯ {displaystyle {overline {m}}} to a magnetic field B ¯ {displaystyle {overline {B}}} is T ¯ = m ¯ × B ¯ {displaystyle {overline {mathrm {T} }}={overline {m}} imes {overline {B}}} . The identity of the functional form of the stationary electric and magnetic fields has led to defining the magnitude of the magnetic dipole moment equally well as m = I π r 2 {displaystyle m=Ipi r^{2}} , or in the following way, imitating the moment p of an electric dipole: The magnetic dipole can be represented by a needle of a compass with fictitious magnetic charges ± q m {displaystyle pm q_{m}} on the two poles and vector distance between the poles l ¯ {displaystyle {overline {l}}} under the influence of the magnetic field of earth B ¯ {displaystyle {overline {B}}} . By classical mechanics the torque on this needle is T ¯ = l ¯ × B ¯ ⋅ q m = q m ⋅ l ¯ × B ¯ . {displaystyle {overline {mathrm {T} }}={overline {l}} imes {overline {B}}cdot q_{m}=q_{m}cdot {overline {l}} imes {overline {B}}.} But as previously stated q m ⋅ l ¯ = I π r 2 = m ¯ , {displaystyle q_{m}cdot {overline {l}}=Ipi r^{2}={overline {m}},} so the desired formula comes up. The model of the spinning electron we use in the derivation has an evident analogy with a gyroscope. For any rotating body the rate of change of the angular momentum J ¯ {displaystyle {overline {J}}} equals the applied torque T ¯ {displaystyle {overline {T}}} : Note as an example the precession of a gyroscope. The earth's gravitational attraction applies a force or torque to the gyroscope in the vertical direction, and the angular momentum vector along the axis of the gyroscope rotates slowly about a vertical line through the pivot. In the place of the gyroscope imagine a sphere spinning around the axis and with its center on the pivot of the gyroscope, and along the axis of the gyroscope two oppositely directed vectors both originated in the center of the sphere, upwards J ¯ {displaystyle {overline {J}}} and downwards m ¯ . {displaystyle {overline {m}}.} Replace the gravity with a magnetic flux density B. Consequently, f = γ 2 π B q . e . d . {displaystyle f={frac {gamma }{2pi }}Bquad q.e.d.} This relationship also explains an apparent contradiction between the two equivalent terms, gyromagnetic ratio versus magnetogyric ratio: whereas it is a ratio of a magnetic property (i.e. dipole moment) to a gyric (rotational, from Greek: γύρος, 'turn') property (i.e. angular momentum), it is also, at the same time, a ratio between the angular precession frequency (another gyric property) ω = 2πf and the magnetic field.