Larmor precession
Encyclopedia
In physics
, Larmor precession (named after Joseph Larmor
) is the precession
of the magnetic moment
s of electron
s, atomic nuclei
, and atom
s about an external magnetic field
. The magnetic field exerts a torque
on the magnetic moment,
where
is the torque, 
is the magnetic dipole moment, 
is the angular momentum
vector,
is the external magnetic field, 
symbolizes the cross product
, and
is the gyromagnetic ratio which gives the proportionality constant between the magnetic moment and the angular momentum.
precesses about the external field axis with an angular frequency
known as the Larmor frequency,
where
is the angular frequency
,
is the gyromagnetic ratio, and 
is the magnitude of the magnetic field and 
is the g-factor
(normally 1, except for in quantum physics).
Simplified, this becomes:
where
is the Larmor frequency, m is mass, e is charge, and B is applied field.
Each isotope has a unique Larmor frequency for NMR spectroscopy
, which is tabulated here.
where
, 
, 
, and 
are polarization four-vector, charge, mass, and magnetic moment, 
is four-velocity of electron, 
, 
, and 
is electromagnetic field-strength tensor. Using equations of motion,
one can rewrite the first term in the right side of the BMT equation as
, where 
is four-acceleration. This term describes Fermi-Walker transport
and leads to Thomas precession
. The second term is associated with Larmor precession.
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 resonance
.
To calculate the spin of a particle in a magnetic field, one must also take into account Thomas precession
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, Larmor precession (named after Joseph Larmor
Joseph Larmor
Sir Joseph Larmor , a physicist and mathematician who made innovations in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter...
) is the precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
of the magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
s of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s, atomic nuclei
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
, and atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s about an external magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
. The magnetic field exerts a torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
on the magnetic moment,
where
is the torque, is the magnetic dipole moment, is the angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
vector,
is the external magnetic field, symbolizes the cross productCross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
, and
is the gyromagnetic ratio which gives the proportionality constant between the magnetic moment and the angular momentum.Larmor frequency
The angular momentum vector precesses about the external field axis with an angular frequencyAngular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
known as the Larmor frequency,
where
is the angular frequencyAngular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
,
is the gyromagnetic ratio, and is the magnitude of the magnetic field and is the g-factorG-factor
A g-factor is a dimensionless quantity which characterizes the magnetic moment and gyromagnetic ratio of a particle or nucleus...
(normally 1, except for in quantum physics).
Simplified, this becomes:
where
is the Larmor frequency, m is mass, e is charge, and B is applied field.Each isotope has a unique Larmor frequency for NMR spectroscopy
NMR spectroscopy
Nuclear magnetic resonance spectroscopy, most commonly known as NMR spectroscopy, is a research technique that exploits the magnetic properties of certain atomic nuclei to determine physical and chemical properties of atoms or the molecules in which they are contained...
, which is tabulated here.
Bargmann-Michel-Telegdi equation
The spin precession of an electron in an external electromagnetic field is described by the Bargmann-Michel-Telegdi (BMT) equationwhere
, , , and are polarization four-vector, charge, mass, and magnetic moment, is four-velocity of electron, , , and is electromagnetic field-strength tensor. Using equations of motion,one can rewrite the first term in the right side of the BMT equation as
, where is four-acceleration. This term describes Fermi-Walker transportFermi-Walker transport
Fermi-Walker transport is a process in general relativity used to define a coordinate system or reference frame such that all curvature in the frame is due to the presence of mass/energy density and not to arbitrary spin or rotation of the frame....
and leads to Thomas precession
Thomas precession
In physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular...
. The second term is associated with Larmor precession.
Applications
A 1935 paper published by Lev LandauLev Landau
Lev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
and Evgeny Lifshitz
Evgeny Lifshitz
Evgeny Mikhailovich Lifshitz was a leading Soviet physicist of Jewish origin and the brother of physicist Ilya Mikhailovich Lifshitz. Lifshitz is well known in general relativity for coauthoring the BKL conjecture concerning the nature of a generic curvature...
predicted the existence of ferromagnetic resonance
Ferromagnetic resonance
Ferromagnetic resonance, or FMR, is a spectroscopic technique to probe the magnetization of ferromagnetic materials. It is a standard tool for probing spin waves and spin dynamics...
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
Nuclear magnetic resonance
Nuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...
, electron paramagnetic resonance
Electron paramagnetic resonance
Electron paramagnetic resonance or electron spin resonance spectroscopyis a technique for studying chemical species that have one or more unpaired electrons, such as organic and inorganic free radicals or inorganic complexes possessing a transition metal ion...
and muon spin resonance
Muon spin spectroscopy
Muon spin spectroscopy is an experimental technique based on the implantation of spin-polarized muons in matter and on the detection of the influence of the atomic, molecular or crystalline surroundings on their spin motion...
.
To calculate the spin of a particle in a magnetic field, one must also take into account Thomas precession
Thomas precession
In physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular...
.