The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) is the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of the circular motion of a
charged particle
In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...
in the presence of a uniform
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 ...
. In
SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
, the non-relativistic gyroradius is given by
:
where
is the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
of the particle,
is the component of the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
perpendicular to the direction of the magnetic field,
is the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
of the particle, and
is the strength of the magnetic field.
The
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of this circular motion is known as the gyrofrequency, or
cyclotron frequency
Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillati ...
, and can be expressed as
:
in units of
radians
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 c ...
/second.
Variants
It is often useful to give the gyrofrequency a sign with the definition
:
or express it 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 on ...
with
:
.
For electrons, this
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 ...
can be reduced to
:
.
In cgs-units the gyroradius
:
and the corresponding gyrofrequency
:
include a factor
, that is the velocity of light, because the magnetic field is expressed in units
.
Relativistic case
For relativistic particles the classical equation needs to be interpreted in terms of particle momentum
:
:
where
is the
Lorentz factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
. This equation is correct also in the non-relativistic case.
For calculations in
accelerator and
astroparticle
Cosmic rays are high-energy particles or clusters of particles (primarily represented by protons or atomic nuclei) that move through space at nearly the speed of light. They originate from the Sun, from outside of the Solar System in our own ...
physics, the formula for the gyroradius can be rearranged to give
:
,
where
is the speed of light,
is the unit of
Giga
Giga ( or ) is a unit prefix in the metric system denoting a factor of a short-scale billion or long-scale milliard (109 or ). It has the symbol G.
''Giga'' is derived from the Greek word (''gígas''), meaning "giant". The ''Oxford English Dic ...
-
electronVolts
In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacuum. ...
, and
is the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
.
Derivation
If the charged particle is moving, then it will experience a
Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
given by
:
,
where
is the velocity
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
and
is the magnetic field vector.
Notice that the direction of the force is given by the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of the velocity and magnetic field. Thus, the Lorentz force will always act perpendicular to the direction of motion, causing the particle to
gyrate, or move in a circle. The radius of this circle,
, can be determined by equating the magnitude of the Lorentz force to the
centripetal force
A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
as
:
.
Rearranging, the gyroradius can be expressed as
:
.
Thus, the gyroradius is
directly proportional
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constant ...
to the particle mass and perpendicular velocity, while it is inversely proportional to the particle electric charge and the magnetic field strength. The time it takes the particle to complete one revolution, called the
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
, can be calculated to be
:
.
Since the period is the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of the frequency we have found
:
and therefore
:
.
See also
*
Beam rigidity In accelerator physics, rigidity is the effect of particular magnetic fields on the motion of the charged particles.
It is a measure of the momentum of the particle, and it refers to the fact that a higher momentum particle will have a higher resis ...
*
Cyclotron
A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Janu ...
*
Magnetosphere particle motion
The ions and electrons of a plasma interacting with the Earth's magnetic field generally follow its magnetic field lines. These represent the force that a north magnetic pole would experience at any given point. (Denser lines indicate a strong ...
*
Gyrokinetics
Gyrokinetics is a theoretical framework to study plasma behavior on perpendicular spatial scales comparable to the gyroradius and frequencies much lower than the particle cyclotron frequencies.
These particular scales have been experimentally sho ...
References
{{Reflist
Plasma physics
Accelerator physics
Kinematic properties