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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 relat ...
, the motion of an electrically charged particle such as 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 no kno ...
or
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conv ...
in a plasma in a
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 ...
can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.


Gyration

If the magnetic field is uniform and all other forces are absent, then the
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 ...
will cause a particle to undergo a constant acceleration perpendicular to both the particle velocity and the magnetic field. This does not affect particle motion parallel to the magnetic field, but results in circular motion at constant speed in the plane perpendicular to the magnetic field. This circular motion is known as the gyromotion. For a particle with mass m and charge q moving in a magnetic field with strength B, it has a frequency, called 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 ...
, of :\omega_ = \frac . \,\! For a speed perpendicular to the magnetic field of v_, the radius of the orbit, called the
gyroradius The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field. In SI units, the non-relativistic gyroradius is given by :r_ ...
or Larmor radius, is :\rho_ =\frac . \,\!


Parallel motion

Since the magnetic Lorentz force is always perpendicular to the magnetic field, it has no influence (to lowest order) on the parallel motion. In a uniform field with no additional forces, a charged particle will gyrate around the magnetic field according to the perpendicular component of its velocity and drift parallel to the field according to its initial parallel velocity, resulting in a
helical Helical may refer to: * Helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is for ...
orbit. If there is a force with a parallel component, the particle and its guiding center will be correspondingly accelerated. If the field has a parallel gradient, a particle with a finite Larmor radius will also experience a force in the direction away from the larger magnetic field. This effect is known as the
magnetic mirror A magnetic mirror, known as a magnetic trap (магнитный захват) in Russia and briefly as a pyrotron in the US, is a type of magnetic confinement device used in fusion power to trap high temperature plasma using magnetic fields. Th ...
. While it is closely related to guiding center drifts in its physics and mathematics, it is nevertheless considered to be distinct from them.


General force drifts

Generally speaking, when there is a force on the particles perpendicular to the magnetic field, then they drift in a direction perpendicular to both the force and the field. If \boldsymbol is the force on one particle, then the drift velocity is :\boldsymbol_f = \frac \frac. These drifts, in contrast to the mirror effect and the non-uniform ''B'' drifts, do not depend on finite Larmor radius, but are also present in cold plasmas. This may seem counterintuitive. If a particle is stationary when a force is turned on, where does the motion perpendicular to the force come from and why doesn't the force produce a motion parallel to itself? The answer is the interaction with the magnetic field. The force initially results in an acceleration parallel to itself, but the magnetic field deflects the resulting motion in the drift direction. Once the particle is moving in the drift direction, the magnetic field deflects it back against the external force, so that the average acceleration in the direction of the force is zero. There is, however, a one-time displacement in the direction of the force equal to (''f''/''m'')ωc−2, which should be considered a consequence of the polarization drift (see below) while the force is being turned on. The resulting motion is a
cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another cur ...
. More generally, the superposition of a gyration and a uniform perpendicular drift is a
trochoid In geometry, a trochoid () is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the ...
. All drifts may be considered special cases of the force drift, although this is not always the most useful way to think about them. The obvious cases are electric and gravitational forces. The grad-B drift can be considered to result from the force on a magnetic dipole in a field gradient. The curvature, inertia, and polarisation drifts result from treating the acceleration of the particle as
fictitious force A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. It is related to Newton's second law of motion, which tre ...
s. The diamagnetic drift can be derived from the force due to a pressure gradient. Finally, other forces such as radiation pressure and collisions also result in drifts.


Gravitational field

A simple example of a force drift is a plasma in a gravitational field, e.g. the
ionosphere The ionosphere () is the ionized part of the upper atmosphere of Earth, from about to above sea level, a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays an im ...
. The drift velocity is :\boldsymbol_g = \frac \frac Because of the mass dependence, the gravitational drift for the electrons can normally be ignored. The dependence on the charge of the particle implies that the drift direction is opposite for ions as for electrons, resulting in a current. In a fluid picture, it is this current crossed with the magnetic field that provides that force counteracting the applied force.


Electric field

This drift, often called the \boldsymbol\times\boldsymbol (''E''-cross-''B'') drift, is a special case because the electric force on a particle depends on its charge (as opposed, for example, to the gravitational force considered above). As a result, ions (of whatever mass and charge) and electrons both move in the same direction at the same speed, so there is no net current (assuming quasineutrality of the plasma). In the context of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
, in the frame moving with this velocity, the electric field vanishes. The value of the drift velocity is given by :\boldsymbol_E = \frac


Nonuniform E

If the electric field is not uniform, the above formula is modified to read :\boldsymbol_E = \left( 1 + \frac\rho_^2\nabla^2 \right) \frac


Nonuniform B

Guiding center drifts may also result not only from external forces but also from non-uniformities in the magnetic field. It is convenient to express these drifts in terms of the parallel and perpendicular kinetic energies :K_\, = \fracmv_\, ^2 :K_\perp = \fracmv_\perp^2 In that case, the explicit mass dependence is eliminated. If the ions and electrons have similar temperatures, then they also have similar, though oppositely directed, drift velocities.


Grad-B drift

When a particle moves into a larger magnetic field, the curvature of its orbit becomes tighter, transforming the otherwise circular orbit into a
cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another cur ...
. The drift velocity is :\boldsymbol_ = \frac \frac


Curvature drift

In order for a charged particle to follow a curved field line, it needs a drift velocity out of the plane of curvature to provide the necessary
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 ...
. This velocity is :\boldsymbol_= \frac\frac where \boldsymbol_ is the
radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
pointing outwards, away from the center of the
circular arc Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...
which best approximates the curve at that point. :\boldsymbol_ = \frac\, \hat\times\frac, where \hat=\boldsymbol/B is the unit vector in the direction of the magnetic field. This drift can be decomposed into the sum of the curvature drift and the term :\frac\, \hat\times\left frac + (\boldsymbol_E\cdot\nabla\hat) \right In the important limit of stationary magnetic field and weak electric field, the inertial drift is dominated by the curvature drift term.


Curved vacuum drift

In the limit of small plasma pressure,
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
provide a relationship between gradient and curvature that allows the corresponding drifts to be combined as follows :\boldsymbol_R + \boldsymbol_ = \frac\frac For a species in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
, 2K_\, +K_\perp can be replaced by 2k_BT (k_BT/2 for K_\, and k_BT for K_\perp). The expression for the grad-B drift above can be rewritten for the case when \nabla B is due to the curvature. This is most easily done by realizing that in a vacuum, Ampere's Law is \nabla\times\boldsymbol = 0 . In cylindrical coordinates chosen such that the azimuthal direction is parallel to the magnetic field and the radial direction is parallel to the gradient of the field, this becomes :\nabla\times\boldsymbol = \frac \frac \left( r B_\theta \right) \hat = 0 Since r B_\theta is a constant, this implies that : \nabla B = - B \frac and the grad-B drift velocity can be written :\boldsymbol_ = -\frac \frac


Polarization drift

A time-varying electric field also results in a drift given by :\boldsymbol_p = \frac\frac Obviously this drift is different from the others in that it cannot continue indefinitely. Normally an oscillatory electric field results in a polarization drift oscillating 90 degrees out of phase. Because of the mass dependence, this effect is also called the inertia drift. Normally the polarization drift can be neglected for electrons because of their relatively small mass.


Diamagnetic drift

The diamagnetic drift is not actually a guiding center drift. A pressure gradient does not cause any single particle to drift. Nevertheless, the fluid velocity is defined by counting the particles moving through a reference area, and a pressure gradient results in more particles in one direction than in the other. The net velocity of the fluid is given by :\boldsymbol_D = -\frac


Drift Currents

With the important exception of the \boldsymbol\times\boldsymbol drift, the drift velocities of differently charged particles will be different. This difference in velocities results in a current, while the mass dependence of the drift velocity can result in chemical separation.


See also

*
List of plasma (physics) articles This is a list of plasma physics topics. A * Ablation * Abradable coating * Abraham–Lorentz force * Absorption band * Accretion disk * Active galactic nucleus * Adiabatic invariant * ADITYA (tokamak) * Aeronomy * Afterglow plasma * A ...


References


T.G. Northrop, The guiding center approximation to charged particle motion, Annals of Physics 15, p.79-101, 1961H.J. de Blank, Guiding center motion, Fusion Science and Technology / Volume 61 / Number 2T / February 2012 / Pages 61-68
''Cosmic Plasma'' (1981), Hannes Alfvén {{Cite book , last=Sulem , first=P.L. , date=2005 , title=Introduction to Guiding center theory, url=https://books.google.com/books?id=By2G4s-PaM8C&q=sulem+2006+guiding+center+theory&pg=PA109, journal= Fields Institute Communications, volume=46 , pages=109–149 , access-date=22 October 2014, isbn=9780821837238 Plasma physics