# Waves in plasmas

Last updated

In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electrons and a single species of positive ions, but it may also contain multiple ion species including negative ions as well as neutral particles. Due to its electrical conductivity, a plasma couples to electric and magnetic fields. This complex of particles and fields supports a wide variety of wave phenomena.

Plasma is one of the four fundamental states of matter, and was first described by chemist Irving Langmuir in the 1920s. It consists of a gas of ions, atoms which have some of their orbital electrons removed, and free electrons. Plasma can be artificially generated by heating or subjecting a neutral gas to a strong electromagnetic field to the point where an ionized gaseous substance becomes increasingly electrically conductive, and long-range electromagnetic fields dominate the behaviour of the matter.

In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress, or external force. Fluids are a phase of matter and include liquids, gases and plasmas. They are substances with zero shear modulus, or, in simpler terms, substances which cannot resist any shear force applied to them.

The electron is a subatomic particle, symbol
e
or
β
, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ. Being fermions, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

## Terminology and classification

Waves in plasmas can be classified as electromagnetic or electrostatic according to whether or not there is an oscillating magnetic field. Applying Faraday's law of induction to plane waves, we find ${\displaystyle \mathbf {k} \times {\tilde {\mathbf {E} }}=\omega {\tilde {\mathbf {B} }}}$, implying that an electrostatic wave must be purely longitudinal. An electromagnetic wave, in contrast, must have a transverse component, but may also be partially longitudinal.

Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.

In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant over any plane that is perpendicular to a fixed direction in space.

Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the direction of propagation of the wave. Mechanical longitudinal waves are also called compressional or compression waves, because they produce compression and rarefaction when traveling through a medium, and pressure waves, because they produce increases and decreases in pressure.

Waves can be further classified by the oscillating species. In most plasmas of interest, the electron temperature is comparable to or larger than the ion temperature. This fact, coupled with the much smaller mass of the electron, implies that the electrons move much faster than the ions. An electron mode depends on the mass of the electrons, but the ions may be assumed to be infinitely massive, i.e. stationary. An ion mode depends on the ion mass, but the electrons are assumed to be massless and to redistribute themselves instantaneously according to the Boltzmann relation. Only rarely, e.g. in the lower hybrid oscillation, will a mode depend on both the electron and the ion mass.

Temperature is a statistical quantity. The formal definition is T = dU/dS, the change in internal energy with respect to entropy, holding volume and particle number constant. A practical definition comes from the fact that the atoms, molecules, or whatever particles in a system have an average kinetic energy. The average means to average over the kinetic energy of all the particles in a system.

In a plasma, the Boltzmann relation describes the number density of an isothermal charged particle fluid when the thermal and the electrostatic forces acting on the fluid have reached equilibrium.

In plasma physics, a lower hybrid oscillation is a longitudinal oscillation of ions and electrons in a magnetized plasma. The direction of propagation must be very nearly perpendicular to the stationary magnetic field, within about me/mi radians. Otherwise the electrons can move along the field lines fast enough to shield the oscillations in potential. The frequency of oscillation is

The various modes can also be classified according to whether they propagate in an unmagnetized plasma or parallel, perpendicular, or oblique to the stationary magnetic field. Finally, for perpendicular electromagnetic electron waves, the perturbed electric field can be parallel or perpendicular to the stationary magnetic field.

Summary of elementary plasma waves
EM characteroscillating speciesconditions dispersion relation name
electrostaticelectrons${\displaystyle {\vec {B}}_{0}=0\ {\rm {or}}\ {\vec {k}}\|{\vec {B}}_{0}}$${\displaystyle \omega ^{2}=\omega _{p}^{2}+3k^{2}v_{th}^{2}}$ plasma oscillation (or Langmuir wave)
${\displaystyle {\vec {k}}\perp {\vec {B}}_{0}}$${\displaystyle \omega ^{2}=\omega _{p}^{2}+\omega _{c}^{2}=\omega _{h}^{2}}$ upper hybrid oscillation
ions${\displaystyle {\vec {B}}_{0}=0\ {\rm {or}}\ {\vec {k}}\|{\vec {B}}_{0}}$${\displaystyle \omega ^{2}=k^{2}v_{s}^{2}=k^{2}{\frac {\gamma _{e}KT_{e}+\gamma _{i}KT_{i}}{M}}}$ ion acoustic wave
${\displaystyle {\vec {k}}\perp {\vec {B}}_{0}}$ (nearly)${\displaystyle \omega ^{2}=\Omega _{c}^{2}+k^{2}v_{s}^{2}}$ electrostatic ion cyclotron wave
${\displaystyle {\vec {k}}\perp {\vec {B}}_{0}}$ (exactly)${\displaystyle \omega ^{2}=[(\Omega _{c}\omega _{c})^{-1}+\omega _{i}^{-2}]^{-1}}$ lower hybrid oscillation
electromagneticelectrons${\displaystyle {\vec {B}}_{0}=0}$${\displaystyle \omega ^{2}=\omega _{p}^{2}+k^{2}c^{2}}$ light wave
${\displaystyle {\vec {k}}\perp {\vec {B}}_{0},\ {\vec {E}}_{1}\|{\vec {B}}_{0}}$${\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}}{\omega ^{2}}}}$ O wave
${\displaystyle {\vec {k}}\perp {\vec {B}}_{0},\ {\vec {E}}_{1}\perp {\vec {B}}_{0}}$${\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}}{\omega ^{2}}}\,{\frac {\omega ^{2}-\omega _{p}^{2}}{\omega ^{2}-\omega _{h}^{2}}}}$ X wave
${\displaystyle {\vec {k}}\|{\vec {B}}_{0}}$ (right circ. pol.)${\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}/\omega ^{2}}{1-(\omega _{c}/\omega )}}}$ R wave (whistler mode)
${\displaystyle {\vec {k}}\|{\vec {B}}_{0}}$ (left circ. pol.)${\displaystyle {\frac {c^{2}k^{2}}{\omega ^{2}}}=1-{\frac {\omega _{p}^{2}/\omega ^{2}}{1+(\omega _{c}/\omega )}}}$ L wave
ions${\displaystyle {\vec {B}}_{0}=0}$ none
${\displaystyle {\vec {k}}\|{\vec {B}}_{0}}$${\displaystyle \omega ^{2}=k^{2}v_{A}^{2}}$ Alfvén wave
${\displaystyle {\vec {k}}\perp {\vec {B}}_{0}}$${\displaystyle {\frac {\omega ^{2}}{k^{2}}}=c^{2}\,{\frac {v_{s}^{2}+v_{A}^{2}}{c^{2}+v_{A}^{2}}}}$ magnetosonic wave

${\displaystyle \omega }$ - wave frequency, ${\displaystyle k}$ - wave number, ${\displaystyle c}$ - speed of light, ${\displaystyle \omega _{p}}$ - plasma frequency, ${\displaystyle \omega _{i}}$ - ion plasma frequency, ${\displaystyle \omega _{c}}$ - electron gyrofrequency, ${\displaystyle \Omega _{c}}$ - proton gyrofrequency, ${\displaystyle \omega _{h}}$ - upper hybrid frequency, ${\displaystyle v_{s}}$ - plasma "sound" speed, ${\displaystyle v_{A}}$ - plasma Alfvén speed

The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299792458 metres per second. It is exact because by international agreement a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity, c is the upper limit for the speed at which conventional matter and information can travel. Though this speed is most commonly associated with light, it is also the speed at which all massless particles and field perturbations travel in vacuum, including electromagnetic radiation and gravitational waves. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. Particles with nonzero rest mass can approach c, but can never actually reach it. In the special and general theories of relativity, c interrelates space and time, and also appears in the famous equation of mass–energy equivalence E = mc2.

Plasma parameters define various characteristics of a plasma, an electrically conductive collection of charged particles that responds collectively to electromagnetic forces. Plasma typically takes the form of neutral gas-like clouds or charged ion beams, but may also include dust and grains. The behaviour of such particle systems can be studied statistically.

In plasma physics, an upper hybrid oscillation is a mode of oscillation of a magnetized plasma. It consists of a longitudinal motion of the electrons perpendicular to the magnetic field with the dispersion relation

## Related Research Articles

In electromagnetism, there are two kinds of dipoles:

Electron cyclotron resonance is a phenomenon observed in plasma physics, condensed matter physics, and accelerator physics. It happens when the frequency of incident radiation coincides with the natural frequency of rotation of electrons in magnetic fields. A free electron in a static and uniform magnetic field will move in a circle due to the Lorentz force. The circular motion may be superimposed with a uniform axial motion, resulting in a helix, or with a uniform motion perpendicular to the field resulting in a cycloid. The angular frequency of this cyclotron motion for a given magnetic field strength B is given by

A Penning trap is a device for the storage of charged particles using a homogeneous axial magnetic field and an inhomogeneous quadrupole electric field. This kind of trap is particularly well suited to precision measurements of properties of ions and stable subatomic particles. Geonium atoms have been created and studied this way, to measure the electron magnetic moment. Recently these traps have been used in the physical realization of quantum computation and quantum information processing by trapping qubits. Penning traps are used in many laboratories worldwide, including CERN, to store antimatter like antiprotons.

Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is just the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequency do not change as a result of the scattering. This limit is valid as long as the photon energy is much smaller than the mass energy of the particle: , or equivalently, if the wavelength of the light is much greater than the Compton wavelength of the particle.

Plasma oscillations, also known as Langmuir waves, are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability in the dielectric function of a free electron gas. The frequency only depends weakly on the wavelength of the oscillation. The quasiparticle resulting from the quantization of these oscillations is the plasmon.

In physics, the motion of an electrically charged particle such as an electron or ion in a plasma in a magnetic field 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.

In plasma physics, an electrostatic ion cyclotron wave is a longitudinal oscillation of the ions in a magnetized plasma, propagating nearly perpendicular to the magnetic field. The angle between the direction of propagation and the direction perpendicular to the magnetic field must be greater than about the square root of the mass ratio,

The two-stream instability is a very common instability in plasma physics. It can be induced by an energetic particle stream injected in a plasma, or setting a current along the plasma so different species can have different drift velocities. The energy from the particles can lead to plasma wave excitation.

In plasma physics, an electromagnetic electron wave is a wave in a plasma which has a magnetic field component and in which primarily the electrons oscillate.

The gyroradius is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field. In SI units, the gyroradius is given by

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 oscillating electric field tuned to this resonance to add kinetic energy to charged particles.

In plasma physics, the Hasegawa–Mima equation, named after Akira Hasegawa and Kunioki Mima, is an equation that describes a certain regime of plasma, where the time scales are very fast, and the distance scale in the direction of the magnetic field is long. In particular the equation is useful for describing turbulence in some tokamaks. The equation was introduced in Hasegawa and Mima's paper submitted in 1977 to Physics of Fluids, where they compared it to the results of the ATC tokamak.

The Appleton–Hartree equation, sometimes also referred to as the Appleton–Lassen equation is a mathematical expression that describes the refractive index for electromagnetic wave propagation in a cold magnetized plasma. The Appleton–Hartree equation was developed independently by several different scientists, including Edward Victor Appleton, Douglas Hartree and German radio physicist H. K. Lassen. Lassen's work, completed two years prior to Appleton and five years prior to Hartree, included a more thorough treatment of collisional plasma; but, published only in German, it has not been widely read in the English speaking world of radio physics. Further, regarding the derivation by Appleton, it was noted in the historical study by Gilmore that Wilhelm Altar first calculated the dispersion relation in 1926.

Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by virtual particles, particles that exist for only a short time determined by the uncertainty principle. The virtual particles, also known as force carriers, are bosons, with different bosons associated with each force.

Plasmonic nanoparticles are particles whose electron density can couple with electromagnetic radiation of wavelengths that are far larger than the particle due to the nature of the dielectric-metal interface between the medium and the particles: unlike in a pure metal where there is a maximum limit on what size wavelength can be effectively coupled based on the material size.

The Farley–Buneman instability, or FB instability, is a microscopic plasma instability named after Donald T. Farley and Oscar Buneman. It is similar to the ionospheric Rayleigh-Taylor instability.

## References

Thomas Howard Stix was an American physicist. Stix performed seminal work in plasma physics, and wrote the first mathematical treatment of the field in 1962's The Theory of Plasma Waves.