en>D.Lazard: Fixing typo raised by BracketBot

2014-01-28T14:50:23Z

Fixing typo raised by BracketBot

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In [[physics]], '''Larmor precession''' (named after [[Joseph Larmor]]) is the [[precession]] of the [[magnetic moment]]s of [[electron]]s, [[muons]], all [[leptons]] with [[magnetic moment]]s which are [[quantum effects]] of [[particle spin]], [[atomic nucleus|atomic nuclei]], and [[atom]]s about an external [[magnetic field]]. The magnetic field exerts a [[torque]] on the magnetic moment,
:<math>\vec{\Gamma} =
\vec{\mu}\times\vec{B}=
\gamma\vec{J}\times\vec{B}</math>

where <math>\vec{\Gamma}</math> is the torque, <math>\vec{\mu}</math> is the magnetic dipole moment, <math>\vec{J}</math> is the [[angular momentum]] vector, <math>\vec{B}</math> is the external magnetic field, <math>\times</math> symbolizes the [[cross product]], and <math>\ \gamma</math> is the [[gyromagnetic ratio]] which gives the proportionality constant between the magnetic moment and the angular momentum.

==Larmor frequency==
The angular momentum vector <math>\vec{J}</math> precesses about the external field axis with an [[angular frequency]] known as the '''Larmor frequency''',

:<math>\omega = -\gamma B</math>

where <math>\omega</math> is the [[angular frequency]],<ref>Spin Dynamics, Malcolm H. Levitt, Wiley, 2001</ref> <math>\gamma=\frac{-e g}{2m}</math> is the [[gyromagnetic ratio]], and <math>B</math> is the magnitude of the magnetic field<ref>{{cite book | isbn = 978-0-521-57572-0 | url = http://books.google.com/?id=1J2hzvX2Xh8C&pg=PA192&lpg=PA192&dq=Larmor's+Theorem | page = 192 | author = Louis N. Hand and Janet D. Finch. | year = 1998 | publisher = Cambridge University Press | location = Cambridge, England | title = Analytical Mechanics}}</ref> and <math>g</math> is the [[g-factor (physics)|g-factor]] (normally 1, except in quantum physics).

Simplified, this becomes:

<math>\omega = \frac{egB}{2m}</math>

where <math>\omega</math> is the Larmor frequency, m is mass, e is charge, and B is applied field. For a given nucleus, the g-factor includes the effects of the spin of the nucleons as well as their orbital angular momentum and the coupling between the two. Because the nucleus is so complicated, g factors are very difficult to calculate, but they have been measured to high precision for most nuclei. Each nuclear isotope has a unique Larmor frequency for [[NMR spectroscopy]], which is tabulated [http://www-lcs.ensicaen.fr/pyPulsar/index.php/List_of_NMR_isotopes here].

==Including Thomas precession==
The above equation is the one that is used in most applications. However, a full treatment must include the effects of [[Thomas precession]], yielding the equation (in [[CGS units]]):
:<math>\omega_s = \frac{geB}{2mc} + (1-\gamma)\frac{eB}{mc\gamma}</math>
where <math>\gamma</math> is the relativistic [[gamma factor]] (not to be confused with the gyromagnetic ratio above). Notably, for the electron g is very close to 2 (2.002..), so if one sets g=2, one arrives at
:<math>\omega_{s(g=2)} = \frac{eB}{mc\gamma}</math>

==Bargmann–Michel–Telegdi equation==

The spin precession of an electron in an external electromagnetic field is described by the Bargmann–Michel–Telegdi (BMT) equation <ref>V. Bargmann, [[Louis Michel|L. Michel]], and V. L. Telegdi, ''Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field'', Phys. Rev. Lett. 2, 435 (1959).</ref>
:<math>\frac{da^{\tau}}{ds} = \frac{e}{m} u^{\tau}u_{\sigma}F^{\sigma \lambda}a_{\lambda}
+ 2\mu (F^{\tau \lambda} - u^{\tau} u_{\sigma} F^{\sigma \lambda})a_{\lambda},</math>
where <math>a^{\tau}</math>, <math>e</math>, <math>m</math>, and <math>\mu</math> are polarization four-vector, charge, mass, and magnetic moment, <math>u^{\tau}</math> is four-velocity of electron, <math>a^{\tau}a_{\tau} = -u^{\tau}u_{\tau} = -1</math>, <math>u^{\tau} a_{\tau}=0</math>, and <math>F^{\tau \sigma}</math> is electromagnetic field-strength tensor. Using equations of motion,
:<math>m\frac{du^{\tau}}{ds} = e F^{\tau \sigma}u_{\sigma},</math>
one can rewrite the first term in the right side of the BMT equation as <math>(- u^{\tau}w^{\lambda} + u^{\lambda}w^{\tau})a_{\lambda}</math>, where <math>w^{\tau} = du^{\tau}/ds</math> is four-acceleration. This term describes [[Fermi–Walker transport]] and leads to [[Thomas precession]]. The second term is associated with Larmor precession.

When electromagnetic fields are uniform in space or when gradient forces like <math>\nabla({\boldsymbol\mu}\cdot{\boldsymbol B})</math> can be neglected, the particle's translational motion is described by
:<math>{du^\alpha\over d\tau}={e\over m}F^{\alpha\beta}u_\beta\;.</math>
The BMT equation is then written as <ref>Jackson, J. D., ''Classical Electrodynamics'', 3rd edition, Wiley, 1999, p. 563.</ref>
:<math>{\;\,dS^\alpha\over d\tau}={e\over m}\bigg[{g\over2}F^{\alpha\beta}S_\beta+\left({g\over2}-1\right)u^\alpha\left(S_\lambda F^{\lambda\mu}U_\mu\right)\bigg]\;,</math>

The Beam-Optical version of the Thomas-BMT, from the ''Quantum Theory of Charged-Particle Beam Optics'', applicable in accelerator optics
<ref>M. Conte, [http://scholar.google.com/citations?user=mp7XSDAAAAAJ&hl=en R. Jagannathan], [http://scholar.google.com/citations?user=hZvL5eYAAAAJ&hl S. A. Khan] and M. Pusterla, Beam optics of the Dirac particle with anomalous magnetic moment, Particle Accelerators, 56, 99-126 (1996); (Preprint: IMSc/96/03/07, INFN/AE-96/08).</ref>
<ref>[http://inspirehep.net/author/S.A.Khan.5/ Khan, S. A.] (1997). [http://www.imsc.res.in/xmlui/handle/123456789/75?show=full Quantum Theory of Charged-Particle Beam Optics], ''Ph.D Thesis'', [[University of Madras]], [[Chennai]], [[India]]. (complete thesis available from [http://www.imsc.res.in/xmlui/ Dspace of IMSc Library], [[Institute of Mathematical Sciences, Chennai|The Institute of Mathematical Sciences]], where the doctoral research was done).</ref>

==Applications==
A 1935 paper published by [[Lev Landau]] and [[Evgeny Lifshitz]] predicted the existence of [[ferromagnetic resonance]] of the Larmor precession, which was independently verified in experiments by J. H. E. Griffiths (UK) and E. K. Zavoiskij (USSR) in 1946.

Larmor precession is important in [[nuclear magnetic resonance]], [[electron paramagnetic resonance]] and [[muon spin spectroscopy|muon spin resonance]]. It is also important for the alignment of [[cosmic dust]] grains, which is a cause of the [[Polarization (waves)|polarization]] of [[starlight]].

To calculate the spin of a particle in a magnetic field, one must also take into account [[Thomas precession]].

==Precession direction==
The precession direction is determined by the [[Chirality_(physics)|chirality]] of the particle.

==See also==
* [[Rabi cycle]]
* [[LARMOR neutron microscope]]

==Notes==
{{reflist}}

== External links ==
* [http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/larmor.html Georgia State University HyperPhysics page on Larmor Frequency]
* [http://bio.groups.et.byu.net/LarmourFreqCal.phtml Larmor Frequency Calculator]

[[Category:Precession]]
[[Category:Electromagnetism]]
[[Category:Atomic physics]]

Complex dimension - Revision history

en>D.Lazard: Fixing typo raised by BracketBot