Gyromagnetic ratio
Its SI unit is the radian per second per tesla (rad⋅s−1⋅T−1) or, equivalently, the coulomb per kilogram (C⋅kg−1).
[citation needed] The term "gyromagnetic ratio" is often used[2] as a synonym for a different but closely related quantity, the g-factor.
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 density and flow [clarification needed] are distributed identically and rotationally symmetric, its gyromagnetic ratio is where
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 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 literal rotation about an axis, it cannot be attributed to mass distributed identically to the charge.
The above classical relation does not hold, giving the wrong result by the absolute value of the electron's g-factor, which is denoted ge:
Here the small corrections to the relativistic result g = 2 come from the quantum field theory calculations of the anomalous magnetic dipole moment.
The electron g-factor and γ are in excellent agreement with theory; see Precision tests of QED for details.
[7] Since a gyromagnetic factor equal to 2 follows from Dirac's equation, it is a frequent misconception to think that a g-factor 2 is a consequence of relativity; it is not.
In both cases a 4-spinor is obtained and for both linearizations the g-factor is found to be equal to 2.
Therefore, the factor 2 is a consequence of the minimal coupling and of the fact of having the same order of derivatives for space and time.
[8] Physical spin-1/2 particles which cannot be described by the linear gauged Dirac equation satisfy the gauged Klein–Gordon equation extended by the g e/4 σμν Fμν term according to,[9] Here, 1/2σμν and Fμν stand for the Lorentz group generators in the Dirac space, and the electromagnetic tensor respectively, while Aμ is the electromagnetic four-potential.
This particle has been shown to be characterized by g = −+2/3 and consequently to behave as a truly quadratic fermion.
Protons, neutrons, and many nuclei carry nuclear spin, which gives rise to a gyromagnetic ratio as above.
The ratio is conventionally written in terms of the proton mass and charge, even for neutrons and for other nuclei, for the sake of simplicity and consistency.
[10] The gyromagnetic ratio of a nucleus plays a role in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI).
These procedures rely on the fact that bulk magnetization due to nuclear spins precess in a magnetic field at a rate called the Larmor frequency, which is simply the product of the gyromagnetic ratio with the magnetic field strength.
With this phenomenon, the sign of γ determines the sense (clockwise vs counterclockwise) of precession.
[13][14] 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) 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 ratio is as follows: First we must prove the torque resulting from subjecting a magnetic moment
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
, 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
For any rotating body the rate of change of the angular momentum
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.
represents the linear velocity of the pike of the arrow
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.
