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Gyromagnetic ratio

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Ratio of magnetic moment to angular momentum

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Inphysics, thegyromagnetic ratio (also sometimes known as themagnetogyric ratio^[1] in other disciplines) of a particle or system is theratio of itsmagnetic moment to itsangular momentum, and it is often denoted by the symbolγ, gamma. ItsSI unit is the reciprocal second pertesla (s⁻¹⋅T⁻¹) or, equivalently, thecoulomb perkilogram (C⋅kg⁻¹).

Theg-factor of a particle is a relateddimensionless 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

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Consider anonconductive 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 {q}{2m}},

where $q {\displaystyle q}$ is its charge, and $m {\displaystyle m}$ 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 anintegration. Suppose the ring has radiusr, areaA =πr², massm, chargeq, and angular momentumL =mvr. Then the magnitude of the magnetic dipole moment is

\mu =IA={\frac {qv}{2\pi r}}\,\pi r^{2}={\frac {q}{2m}}\,mvr={\frac {q}{2m}}L.

For an isolated electron

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An isolated electron has an angular momentum and a magnetic moment resulting from itsspin. 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'sg-factor, which is denotedg_e: $\gamma _{{\text{e}}^{-}}={\frac {\mu _{{\text{e}}^{-}}}{\hbar /2}}=g_{\text{e}}{\frac {-e}{2m_{\text{e}}}}=-g_{\text{e}}{\frac {\mu _{\text{B}}}{\hbar }},$ whereμ_e⁻ is the electron's magnetic moment,ħ/2 is the angular momentum (spin) of the electron, andμ_B is theBohr magneton.

The gyromagnetic ratio due to electron spin is twice that due to the orbiting of an electron.

The electron gyromagnetic ratio is^[2]^[3]

\gamma _{{\text{e}}^{-}}

=−1.76085962784(55)×10¹¹ s⁻¹⋅T⁻¹

The ratio of the electron'sLarmor frequency to the magnetic flux density is^[4]

{\bar {\gamma }}_{{\text{e}}^{-}}={\frac {\gamma _{{\text{e}}^{-}}}{2\pi }}

=−28024.9513861(87) MHz⋅T⁻¹

The electron gyromagnetic ratioγ (and itsg-factorg_e) are in excellent agreement with theory; seePrecision tests of QED for details.^[5]

In the framework of relativistic quantum mechanics, $g_{\text{e}}=2\left(1+{\frac {\alpha }{2\pi }}+\cdots \right),$ where $\alpha$ is thefine-structure constant. Here the small corrections tog = 2 come from the quantum field theory calculations of theanomalous magnetic dipole moment. The electrong-factor is known to twelve decimal places by measuring theelectron magnetic moment in a one-electron cyclotron:^[6]

g_{\text{e}}

=2.00231930436092(36).

Gyromagnetic factor not as a consequence of relativity

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Since a gyromagnetic factor equal to 2 follows from Dirac's equation, it is a frequent misconception to think that ag-factor 2 is a consequence of relativity; it is not. The factor 2 can be obtained from the linearization of both theSchrödinger equation (known as theLévy-Leblond equation) and the relativisticKlein–Gordon equation (which is implied by theDirac equation). In both cases a 4-spinor is obtained and for both linearizations theg-factor is found to be equal to 2. Therefore, the factor 2 is aconsequence of the minimal coupling and of the fact of having the same order of derivatives for space and time.^[7]

For a nucleus

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The sign of the gyromagnetic ratio,γ, determines the sense of precession. While the magnetic moments (the black arrows) are oriented the same for both cases ofγ, the precession is in opposite directions. Spin and magnetic moment are in the same direction forγ > 0 (as for protons).

Protons, neutrons, and many nuclei carrynuclear 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. The formula is:

\gamma _{\text{n}}={\frac {e}{2m_{\text{p}}}}\,g_{\text{n}}=g_{\text{n}}\,{\frac {\mu _{\text{N}}}{\hbar }},

where $\mu _{\text{N}}$ is thenuclear magneton, and $g_{\text{n}}$ is theg-factor of the nucleon or nucleus in question. The ratio ${\frac {\gamma _{n}}{2\pi \,g_{\text{n}}}}=\mu _{\text{N}}/h=7.622\ 593\ 2188(24)$ MHz/T.^[8]

The gyromagnetic ratio of a nucleus plays a role innuclear magnetic resonance (NMR) andmagnetic resonance imaging (MRI). These procedures rely on the fact that bulk magnetization due to nuclear spinsprecess in a magnetic field at a rate called theLarmor frequency, which is simply the product of the gyromagnetic ratio with the magnetic field strength. With this phenomenon, the sign ofγ 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 as¹H and¹³C have positive gyromagnetic ratios.^[9]^[10] Approximate values for some common nuclei are given in the table below.^[11]^[12]

Nucleus	γ_n [s⁻¹⋅T⁻¹]	γ_n [MHz⋅T⁻¹]
¹H⁺	2.6752218708(11)×10⁸‍^[13]	42.577478461(18)‍^[14]
²H	4.1065×10⁷	6.536
³H	2.853508×10⁸	45.415^[15]
³He	−2.0378946078(18)×10⁸^[16]	−32.434100033(28)^[17]
⁷Li	1.03962×10⁸	16.546
¹³C	6.72828×10⁷	10.7084
¹⁴N	1.9331×10⁷	3.077
¹⁵N	−2.7116×10⁷	−4.316
¹⁷O	−3.6264×10⁷	−5.772
¹⁹F	2.51815×10⁸	40.078
²³Na	7.0761×10⁷	11.262
²⁷Al	6.9763×10⁷	11.103
²⁹Si	−5.3190×10⁷	−8.465
³¹P	1.08291×10⁸	17.235
⁵⁷Fe	8.681×10⁶	1.382
⁶³Cu	7.1118×10⁷	11.319
⁶⁷Zn	1.6767×10⁷	2.669
¹²⁹Xe	−7.3995401(2)×10⁷	−11.7767338(3)^[18]

A full list can be found in the external link section below.

Larmor precession

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Main article:Larmor precession

Any free system with a constant gyromagnetic ratio, such as a rigid system of charges, anucleus, or anelectron, when placed in an externalmagnetic field $\mathbf {B}$ (measured in teslas) that is not aligned with itsmagnetic moment, willprecess at afrequencyf (measured inhertz) that is proportional to the external field:

f={\frac {\gamma }{2\pi }}B.

For this reason, values of ${\overline {\gamma }}={\frac {\gamma }{2\pi }}$ , with the unithertz pertesla (Hz/T), are often quoted instead of $\gamma$ .

Heuristic derivation

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The derivation of this ratio is as follows: First we must prove the torque resulting from subjecting a magnetic moment $\mathbf {m}$ to a magnetic field $\mathbf {B}$ is ${\boldsymbol {\mathrm {T} }}=\mathbf {m} \times \mathbf {B} .$ 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 $m=I\pi r^{2}$ , or in the following way, imitating the moment $\mathbf {p}$ of an electric dipole: The magnetic dipole can be represented by a needle of a compass with fictitious magnetic charges $\pm q_{\text{m}}$ on the two poles and vector distance between the poles $\mathbf {d}$ under the influence of the magnetic field of earth $\mathbf {B} .$ By classical mechanics the torque on this needle is ${\boldsymbol {\mathrm {T} }}=q_{\text{m}}(\mathbf {d} \times \mathbf {B} ).$ But as previously stated $q_{\text{m}}\mathbf {d} =I\pi r^{2}{\hat {\mathbf {d} }}=\mathbf {m} ,$ so the desired formula comes up. ${\hat {\mathbf {d} }}$ 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 $\mathbf {J}$ equals the applied torque $\mathbf {T}$ :

{\frac {d\mathbf {J} }{dt}}=\mathbf {T} .

Note as an example theprecession 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 $\mathbf {J}$ and downwards⁠ $\mathbf {m}$ ⁠. Replace the gravity with a magnetic flux density⁠ $\mathbf {B}$ ⁠.

${\frac {d\mathbf {J} }{dt}}$ represents the linear velocity of the pike of the arrow $\mathbf {J}$ along a circle whose radius is $J\sin {\phi },$ where $\phi$ is the angle between $\mathbf {J}$ and the vertical. Hence the angular velocity of the rotation of the spin is

\omega =2\pi \,f={\frac {1}{J\sin {\phi }}}\left|{\frac {d\mathbf {J} }{dt}}\right|={\frac {|\mathbf {T} |}{J\sin {\phi }}}={\frac {|\mathbf {m} \times \mathbf {B} |}{J\sin {\phi }}}={\frac {m\,B\sin {\phi }}{J\sin {\phi }}}={\frac {m\,B}{J}}=\gamma \,B.

Consequently, $f={\frac {\gamma }{2\pi }}\,B,\quad {\text{q.e.d.}}$

This relationship also explains an apparent contradiction between the two equivalent terms,gyromagnetic ratio versusmagnetogyric ratio: whereas it is a ratio of a magnetic property (i.e. dipole moment) to agyric (rotational, fromGreek:γύρος, "turn") property (i.e.angular momentum), it is also a ratio between theangular precession frequency (anothergyric property) $\omega =2\pi f$ and themagnetic flux density.

The angular precession frequency has an important physical meaning: It is theangular 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.

References

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^E.R. Cohen et al. (2008).Quantities, Units and Symbols in Physical Chemistry : IUPAC Green Book. 3rd Edition, 2nd Printing. Cambridge:IUPAC & RSC Publishing.ISBN 0-85404-433-7. p. 23.Electronic version.
^"2022 CODATA Value: electron gyromagnetic ratio".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved18 May 2024.
^NIST puts a positive sign on the quantity; however, to be consistent with the formulas in this article, hereγ_e⁻ is treated as being negative, as for many references; for example,Weil & Bolton (2007).Electron Paramagnetic Resonance. Wiley. p. 578.^{[full citation needed]}
^"2022 CODATA Value: electron gyromagnetic ratio in MHz/T".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved18 May 2024.
^Knecht, Marc (12 October 2002)."The anomalous magnetic moments of the electron and the muon". In Duplantier, Bertrand; Rivasseau, Vincent (eds.).Poincaré Seminar 2002. Poincaré Seminar. Progress in Mathematical Physics. Vol. 30. Paris, FR: Birkhäuser (published 2003).ISBN 3-7643-0579-7. Archived fromthe original(PostScript) on 15 October 2005.
^"2022 CODATA Value: electron g factor".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved18 May 2024.
^Greiner, Walter (4 October 2000).Quantum Mechanics: An introduction.Springer Verlag.ISBN 9783540674580 – via Google Books.
^"Nuclear magneton in MHz/T:μ_N/h".NIST. 2022. (citingCODATA-recommended values)
^Levitt, M.H. (2008).Spin Dynamics. John Wiley & Sons Ltd.ISBN 978-0470511176.
^Palmer, Arthur G. (2007).Protein NMR Spectroscopy.Elsevier Academic Press.ISBN 978-0121644918.
^Bernstein, M.A.; King, K.F.; Zhou, X.J. (2004).Handbook of MRI Pulse Sequences. San Diego, CA: Elsevier Academic Press. p. 960.ISBN 0-12-092861-2 – via archive.org.
^Weast, R.C.; Astle, M.J., eds. (1982).Handbook of Chemistry and Physics. Boca Raton, FL:CRC Press. p. E66.ISBN 0-8493-0463-6.
^"2022 CODATA Value: proton gyromagnetic ratio".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved18 May 2024.
^"2022 CODATA Value: proton gyromagnetic ratio in MHz/T".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved18 May 2024.
^"Tritium Solid State NMR Spectroscopy at PNNL for Evaluation of Hydrogen Storage Materials"(PDF). November 2015.
^"shielded helion gyromagnetic ratio". NIST 2022. Retrieved9 July 2024.
^"shielded helion gyromagnetic ratio in MHz/T". NIST 2022. Retrieved9 July 2024.
^Makulski, Wlodzimierz (2020)."Explorations of Magnetic Properties of Noble Gases: The Past, Present, and Future".Magnetochemistry.6 (4): 65.doi:10.3390/magnetochemistry6040065.