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physics Physics is the natural science that studies matter, its 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 which ...
, 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 is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
to its
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, and it is often denoted by the symbol , gamma. Its SI unit is the
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
per second per tesla (rad⋅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). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
per
kilogram The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially. ...
(C⋅kg−1). The term "gyromagnetic ratio" is often used as a synonym for a ''different'' but closely related quantity, the -factor. The -factor only differs from the gyromagnetic ratio in being dimensionless.


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 density and flow are distributed identically and rotationally symmetric, its gyromagnetic ratio is : \gamma = \frac where is its charge and 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 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 -factor, which is denoted : \gamma_\mathrm = \frac \, , g_\mathrm, = \frac \, , where 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. The Bohr magneton, in SI units is defined as \mu_\m ...
. The gyromagnetic ratio due to electron spin is twice that due to the orbiting of an electron. In the framework of relativistic quantum mechanics, g_\mathrm = -2 \left(1 + \frac + \cdots\right)~, where \alpha is the fine-structure constant. Here the small corrections to the relativistic result 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: g_\mathrm = -2.002\,319\,304\,362\,56(35). The electron gyromagnetic ratio is Note that
NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
puts a positive sign on the quantity; however, to be consistent with the formulas in this article, a negative sign is put on here. Indeed, many references say that for an electron; for example, Also note that the units of radians are added for clarity.
\gamma_\mathrm = \mathrm \frac = \mathrm . The electron -factor and are in excellent agreement with theory; see '' Precision tests of QED'' for details.


Gyromagnetic factor not as a consequence of relativity

Since a gyromagnetic factor equal to 2 follows from the 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 linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
and the relativistic Klein–Gordon equation (which leads to Dirac's). 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. Physical spin particles which can not be described by the linear gauged Dirac equation satisfy the gauged Klein–Gordon equation extended by the term according to, : \left , \left( \partial^\mu \, u + i\, e\, A^\mu \right)\, \left( \partial_\mu + i\, e\, A_\mu \right) + g \, \frac \, \sigma^ \, F_ + m^2 \,\right\; \psi \; = \; 0 ~, \quad g \ne 2 ~. Here, and stand for the Lorentz group generators in the Dirac space, and the
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. ...
respectively, while is the electromagnetic four-potential. An example for such a particle, is the spin companion to spin in the representation space of the Lorentz group. This particle has been shown to be characterized by and consequently to behave as a truly quadratic fermion.


For a nucleus

Proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
s, neutrons, and many nuclei carry
nuclear spin In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe ...
, 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_ = \frac \, g_ = g_\, \frac~, where \mu_\mathrm is the nuclear magneton, and g_ is the -factor of the nucleon or nucleus in question. The ratio of \,\frac\, , equal to \mu_\mathrm/h, is 7.622593285(47) MHz/T. The gyromagnetic ratio of a nucleus plays a role in nuclear magnetic resonance (NMR) and
magnetic resonance imaging Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio wave ...
(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. Most common nuclei such as 1H and 13C have positive gyromagnetic ratios. Approximate values for some common nuclei are given in the table below.


Larmor precession

Any free system with a constant gyromagnetic ratio, such as a rigid system of charges, a
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: * Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
, or an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
, when placed in an external
magnetic field A magnetic field is a vector 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 to its own velocity and to ...
(measured in teslas) that is not aligned with its
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
, will precess at a
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
(measured in
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
), that is proportional to the external field: :f=\fracB. For this reason, values of , in units of
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
per tesla (Hz/T), are often quoted instead of .


Heuristic derivation

The derivation of this relation is as follows: First we must prove that 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_ 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_ (\mathbf\times\mathbf) \, . But as previously stated \, q_\mathbf=I\pi r^2\hat = \mathbf \, , so the desired formula comes up. \hat is the unit distance vector. 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 \, \mathbf \, equals the applied torque \mathbf: :\frac=\mathbf~. 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 \mathbf and downwards \mathbf. Replace the gravity with a magnetic flux density \, \mathbf ~. \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 el, γύρος, "turn") property (i.e.
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
), it is also, ''at the same time'', a ratio between the angular precession frequency (another ''gyric'' property) and the
magnetic field A magnetic field is a vector 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 to its own velocity and to ...
. 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

{{DEFAULTSORT:Gyromagnetic Ratio Atomic physics Nuclear magnetic resonance Ratios