Gyromagnetic Ratio
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physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the
ratio In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of its
magnetic moment In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
to its
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
, and it is often denoted by the symbol , gamma. Its SI unit is the reciprocal second per tesla (s−1⋅T−1) or, equivalently, the
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). It is defined to be equal to the electric charge delivered by a 1 ampere current in 1 second, with the elementary charge ''e'' as a defining c ...
per
kilogram The kilogram (also spelled kilogramme) is the base unit of mass in the International System of Units (SI), equal to one thousand grams. It has the unit symbol kg. The word "kilogram" is formed from the combination of the metric prefix kilo- (m ...
(C⋅kg−1). The -factor of a particle is a related
dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
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 : \gamma = \frac, 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 , area , mass , charge , and angular momentum . Then the magnitude of the magnetic dipole moment is : \mu = I A = \frac \, \pi r^2 = \frac \, m v r = \frac 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 -factor, which is denoted : \gamma_ = \frac = g_\text \frac = -g_\text \frac, where is the electron's magnetic moment, is the angular momentum (spin) of the electron, and is the
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. In SI units, the Bohr magneton is defined as \mu_\mat ...
. The gyromagnetic ratio due to electron spin is twice that due to the orbiting of an electron. The electron gyromagnetic ratio is : The ratio of the electron's Larmor frequency to the magnetic flux density is : The electron gyromagnetic ratio (and its -factor ) are in excellent agreement with theory; see '' Precision tests of QED'' for details. In the framework of relativistic quantum mechanics, g_\text = 2 \left(1 + \frac + \cdots\right), where \alpha is the
fine-structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
. Here the small corrections to come from the quantum field theory calculations of the anomalous magnetic dipole moment. The electron -factor is known to twelve decimal places by measuring the electron magnetic moment in a one-electron cyclotron: : .


Gyromagnetic factor not as a consequence of relativity

Since a gyromagnetic factor equal to 2 follows from Dirac's equation, it is a frequent misconception to think that a -factor 2 is a consequence of relativity; it is not. The factor 2 can be obtained from the linearization of both the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
(known as the Lévy-Leblond equation) and the relativistic Klein–Gordon equation (which is implied by the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
). In both cases a 4- spinor is obtained and for both linearizations the -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.


For a nucleus

Proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
s, neutrons, and many nuclei carry
nuclear spin Nuclear may refer to: Physics Relating to the nucleus of the atom: * Nuclear engineering * Nuclear physics * Nuclear power * Nuclear reactor * Nuclear weapon * Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics * Nuclear space * ...
, 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. The formula is: : \gamma_\text = \frac \, g_\text = g_\text\, \frac, where \mu_\text is the
nuclear magneton The nuclear magneton (symbol ) is a physical constant of magnetic moment, defined in SI units by: \mu_\text = and in Gaussian CGS units by: \mu_\text = where: * is the elementary charge, * is the reduced Planck constant, * is the proton ...
, and g_\text is the -factor of the nucleon or nucleus in question. The ratio \frac = \mu_\text/h = 7.622\ 593\ 2188(24) MHz/T. The gyromagnetic ratio of a nucleus plays a role in
nuclear magnetic resonance Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are disturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
(NMR) and
magnetic resonance imaging Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to generate pictures of the anatomy and the physiological processes inside the body. MRI scanners use strong magnetic fields, magnetic field gradients, and ...
(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 determines the sense (clockwise vs. counterclockwise) of precession. Within atoms and molecules some shielding occurs, with the effect that the nucleus experiences a slightly modified magnetic flux density, which changes the observed precession frequency compared to that of an isolated nucleus in the same applied magnetic field. Most common nuclei such as 1H and 13C have positive gyromagnetic ratios. Approximate values for some common nuclei are given in the table below. A full list can be found in the external link section below.


Larmor precession

Any free system with a constant gyromagnetic ratio, such as a rigid system of charges, a nucleus, or an
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
, when placed in an external
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
(measured in teslas) that is not aligned with its
magnetic moment In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
, will precess at a
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
(measured in
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in ter ...
) that is proportional to the external field: : f = \frac B . For this reason, values of , with the unit
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in ter ...
per tesla (Hz/T), are often quoted instead of .


Heuristic derivation

The derivation of this ratio is as follows: First we must prove the torque resulting from subjecting a magnetic moment \mathbf to a magnetic field \mathbf is \boldsymbol = \mathbf \times \mathbf. 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\pi r^2, or in the following way, imitating the moment of an electric dipole: The magnetic dipole can be represented by a needle of a compass with fictitious magnetic charges \pm q_\text on the two poles and vector distance between the poles \mathbf under the influence of the magnetic field of earth \mathbf. By classical mechanics the torque on this needle is \boldsymbol = q_\text(\mathbf \times \mathbf). But as previously stated q_\text \mathbf = I\pi r^2 \hat = \mathbf, so the desired formula comes up. \hat is the unit distance vector. The spinning electron model here is analogous to a gyroscope. For any rotating body the rate of change of the angular momentum \mathbf equals the applied torque \mathbf: : \frac = \mathbf. Note as an example the
precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...
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 place of a gyroscope, imagine a sphere spinning around the axis with its centre on the pivot of the gyroscope, and along the axis of the gyroscope two oppositely directed vectors both originated in the centre of the sphere, upwards \mathbf and downwards . Replace the gravity with a magnetic flux density . \frac represents the linear velocity of the pike of the arrow \mathbf along a circle whose radius is J\sin, where \phi is the angle between \mathbf and the vertical. Hence the angular velocity of the rotation of the spin is : \omega = 2\pi \,f = \frac \left, \frac\ = \frac = \frac = \frac = \frac = \gamma\, B. Consequently, f = \frac\,B,\quad \text 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 , "turn") property (i.e.
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
), it is also a ratio between the angular precession frequency (another ''gyric'' property) and the magnetic flux density. The angular precession frequency has an important physical meaning: It is the angular cyclotron frequency, the resonance frequency of an ionized plasma being under the influence of a static finite magnetic field, when we superimpose a high frequency electromagnetic field.


See also

* Charge-to-mass ratio * Chemical shift * Landé -factor * Larmor equation * Proton gyromagnetic ratio


References


External links


Konstantin's gyromagnetic ratio table
A full list of all known gyromagnetic ratios. {{DEFAULTSORT:Gyromagnetic Ratio Atomic physics Nuclear magnetic resonance Ratios