Plasma (physics)

A plasma is a gas of mobile charged particles whose collective electromagnetic response is important to its dynamics. It commonly contains ions and electrons, and may also contain a neutral component: complete ionization is not required. A multi-species fluid description has current density $\mathbf{J}={\sum }_s{q}_s{n}_s{\mathbf{u}}_s$. The relative motion of the species can matter even when a single-fluid magnetohydrodynamics approximation describes the large-scale dynamics.

Plasma density must specify what is being counted or weighed. For species $s$, its number density $n_s$ counts particles per unit volume. Electron density $n_e$, ion density $n_i$ and total mass density $\rho ={\sum }_s{m}_s{n}_s$ need not be interchangeable. In a quasineutral mixture with ion charges $Z_{i}e$,For singly ionized hydrogen, $n_e\simeq n_i$, while the ions provide nearly all the mass. Adding heavier ions changes the relation between electron density and mass density.

An electron plasma frequency measurement offers a density diagnostic. In the cold, unmagnetized approximation with effectively fixed ions, displacement of electrons creates an electric restoring force, givingThe first formula uses SI units; the numerical version expresses electron density in particles per cubic centimetre.^[ref]

If an identified electron plasma oscillation has frequency $2000\mathrm{Hz}$, this model givesThis is an inference from an identified wave mode, not a particle counter reading. Magnetization, wave propagation, multiple species and instrument effects can change which spectral feature is observed, so interpreting a frequency requires the relevant plasma wave model.

Plasma turbulence is irregular, interacting multiscale motion of plasma particles and electromagnetic fields. At large scales a magnetohydrodynamics model can describe coupled velocity and magnetic fluctuations; near ion and electron scales their separate kinetic responses become important. A broadband spectrum alone is insufficient to establish how energy moves between scales.

A useful measurement combines spacecraft magnetometer data with plasma velocity and density measurements. Mixed third-order statistics of velocity and magnetic-field increments can test an energy-cascade relation. Ulysses observations found the predicted approximately linear scale dependence over an inertial range, providing evidence of transfer between scales rather than merely a noisy signal.^[ref]

For an idealized constant-density model, write the Elsässer fields $z^{\pm }=u\pm B/\sqrt{{\mu }_0\rho }$. Their nonlinear coupling shows why magnetic and velocity fluctuations should be measured together. Interpreting a spacecraft time lag as a spatial separation additionally requires a flow-advection approximation. At small scales, explaining plasma heating requires identifying how fluctuating field energy reaches particle distributions, beyond observing a spectral slope.

A plasma wave is a collective disturbance of charged particles and electromagnetic fields in a plasma. Restoring forces can involve charge separation, magnetic tension or pressure, producing different branches rather than one universal wave speed.

For a cold unmagnetized electron fluid with effectively stationary neutralizing ions, electrostatic density oscillations have electron plasma frequency ${\omega }_{pe}=\sqrt{n_e{e}^2/({ϵ}_0{m}_e)}$. Displacing electrons creates an electric field that pulls them back; inertia produces oscillation. Measuring that frequency therefore constrains electron density. Thermal motion, magnetic fields and collisions modify the dispersion and damping, and must be included when the cold-fluid assumptions fail.

Ionization equilibrium means that the rates populating and depopulating ionization stages balance under a specified physical model. For a neutral species photoionized at rate $\Gamma$ per atom and singly charged ions recombining with coefficient $\alpha$, a simple balance is $n_0\Gamma =n_e{n}_{+}\alpha$.

This kinetic balance need not be local thermodynamic equilibrium. Collisional ionization and radiative recombination can balance in a thin plasma that does not obey a Saha population law. Moreover, a rapidly cooling or flowing gas may retain a nonequilibrium ionization state if its ionization or recombination times exceed the evolution time. A temperature alone is therefore insufficient to declare ionization equilibrium.

A toroidal magnetic-confinement system combining an externally imposed toroidal field with a poloidal field supplied largely by toroidal plasma current. The resulting helical field-line winding supports nested magnetic flux surfaces and improves orbit confinement compared with a purely toroidal field. Confinement remains limited by orbit effects, collisions and instabilities.

A measure of toroidal field-line turns per poloidal turn. In the circular large-aspect-ratio approximation, geometry gives $q_s\simeq r{B}_{\varphi }/(R_0{B}_{\theta })$. With Ampère's law, $B_{\theta }(a)\simeq {\mu }_0{I}_p/(2\pi a)$, so the edge value relates the field-line pitch to the toroidal plasma current. This geometrical estimate alone does not establish stability for a complete current profile.

Collisional absorption of electromagnetic wave energy by charged particles scattering from other charges. In a weakly collisional unmagnetized Drude model, with $X={\omega }_p^2/{\omega }^2<1$, the intensity absorption coefficient is ${\alpha }_I\simeq ({\nu }_{ei}/c)X/\sqrt{1-X}$. The corresponding electric-field amplitude decays at half that rate.

The electron density at which the cold unmagnetized electron plasma frequency equals an applied electromagnetic frequency. In the collisionless linear model, transverse radiation below this cutoff does not propagate in a uniform bulk plasma.

The logarithm of the ratio of effective maximum to minimum impact parameter in weakly coupled Coulomb collision transport. Screening and short-distance scattering determine the cutoffs; it encodes many small-angle encounters.

The importance of electrostatic interaction energy compared with thermal energy. For mean separation $a$, a common measure is $\Gamma =e^2/(4\pi {ϵ}_{0}a{k}_{B}T)$; the weakly coupled regime has $\Gamma \ll 1$.

The number of electrons in a Debye sphere, here $N_D=(4\pi /3)n_e{\lambda }_D^3$. A large value characterizes weak fluctuations and weak coupling in the ordinary collective-screening regime.

Stochastic heating increases random particle kinetic energy through decorrelating electromagnetic kicks. A simple one-dimensional model makes the normalization explicit. Let $f(v,t)$ be a velocity distribution with number density $n=\int fdv$, and suppose constant-coefficient velocity-space diffusion givesThe coefficient has dimensions of squared velocity per time, and a freely diffusing component has variance $2Dt$. For sufficiently decaying tails, two applications of integration by parts yieldThe first velocity moment is unchanged, so this is growth of random motion rather than acceleration of the mean flow. In three independent isotropic components with coefficient $D$ in each, the heating rate per volume is instead $3mnD$. More generally each diagonal component contributes its own diffusion coefficient.

This idealization describes externally supplied, rapidly decorrelating forcing; its energy budget is not that of an isolated particle population. Screening, resonant self-consistent fields, collisions and velocity-dependent diffusion can change the result. In other plasma settings, stochastic heating also describes irregular orbits driven by spatial structure or overlapping resonances; that mechanism must be specified separately.

Transfer of fluctuation energy to particles selected by a wave-particle resonance. For a weakly damped Langmuir wave, particles near its phase velocity absorb net energy from a stable distribution by Landau damping. Under weak random forcing, a smaller damping rate produces a larger stationary wave intensity; a very long waiting time may be required before that stationary balance holds.

A drift-scale plasma instability fed by an ion temperature gradient. In common electrostatic limits, Boltzmann electron response, ion temperature evolution, magnetic drifts and polarization drift determine the dynamics. Unfavorable magnetic field-line curvature can drive a toroidal branch; straight-field branches can involve parallel compression. The threshold depends on the equilibrium gradients and geometry.

A plasma wave dominated by a longitudinal electric field. The leading dispersion relation is $\mathbf{k}\cdot \mathbf{ϵ}\cdot \mathbf{k}=0$. In a cold magnetized plasma, this becomes $k_{\perp }^{2}S+k_{\parallel }^{2}P=0$. An exactly longitudinal finite-wavenumber solution must also satisfy the full Maxwell equations; the oblique electrostatic condition is usually an asymptotic large-refractive index approximation.

For an isotropic completely degenerate nonrelativistic Fermi gas in classical-trajectory collisionless kinetic theory, the parallel marginal distribution is $F(u)=3n(v_F^2-u^2)1_{|u|<v_F}/(4v_F^3)$. Integrating $F^{\prime }(u)/(u-\omega /k)$ yields the electron dielectric function in the title for fixed ions. Its real roots outside the occupied interval have no Landau damping. Quantum recoil, collisions and finite-temperature tails require an enlarged model.

The short-wavelength root of the degenerate-plasma dielectric response, approaching the edge of the occupied parallel velocity interval from outside. Writing $s=1+\delta$ makes its logarithmic singularity give the exponentially small positive correction in the title. It is the high-wavenumber limit of the charged Langmuir wave branch in this approximation; neutral zero sound has a different interaction and long-wavelength limit.

Rapid parallel thermal electron streaming can approximately balance electrostatic and isothermal pressure forces, giving $n_e\propto e^{e\varphi /T_e}$ and its linear form in the title. This assumes a suitable connection along the magnetic field, $|\omega |\ll k_{\parallel }{v}_{\mathrm{th},e}$, and an approximately electrostatic branch. A spatially constant potential on a closed line requires its own particle-number constraint.

A plasma whose linear wave response retains the equilibrium magnetic field and particle inertia but neglects pressure forces, thermal dispersion and collisions. Its applicability requires $k{v}_{\mathrm{th},s}\ll |\omega |$ and separate care near cyclotron resonances.

A limiting point of a plasma wave branch with divergent refractive index. At finite coefficients of the magnetized cold-plasma dielectric tensor, the generic condition is $S{\sin }^2\theta +P{\cos }^2\theta =0$. Poles of that magnetized cold-plasma dielectric tensor require a separate limiting analysis. Collisions, thermal dispersion or mode conversion can regularize a singular cold prediction.

A point on a plasma wave branch where the local wavenumber tends to zero at finite frequency. The mode may change between propagation and evanescence. For a finite magnetized cold-plasma dielectric tensor, candidate cutoffs satisfy $PRL=0$; special propagation angles require checking which factor belongs to the chosen branch.

For propagation perpendicular to the equilibrium magnetic field, this plasma wave has its electric field perpendicular to that magnetic field and generally both parallel and perpendicular to its wavevector. Its dispersion relation is $n^2=RL/S$, using the magnetized cold-plasma dielectric tensor conventions $R=S-g$, $L=S+g$.

For propagation perpendicular to an equilibrium magnetic field, an ordinary plasma wave has its electric field parallel to that magnetic field, so the particle response involves no magnetic deflection. With immobile ions, the dispersion relation is ${\omega }^2={\omega }_{pe}^2+c^2{k}^2$. It is a transverse electromagnetic wave, despite its parallel particle motion.

With the same conventions as the cold-plasma right-handed wave, this transverse electromagnetic wave has $E_y=-i{E}_x$ and refractive index $n^2=S+g$. With immobile ions, $L=1-{\omega }_{pe}^2/[\omega (\omega +{\Omega }_e)]$.

For propagation along $+{\mathbf{B}}_0$, use $e^{ikz-i\omega t}$ and ${ϵ}_{xy}=ig$. This transverse electromagnetic wave has $E_y=i{E}_x$ and refractive index $n^2=S-g$. Its real electric field rotates in the sense of electron gyration. With immobile ions and positive electron cyclotron frequency magnitude ${\Omega }_e$, $R=1-{\omega }_{pe}^2/[\omega (\omega -{\Omega }_e)]$.

For $e^{-i\omega t}$, Gaussian units give $\mathbf{ϵ}=\mathbf{I}+4\pi i\mathbf{\sigma }/\omega$, where $\mathbf{\sigma }$ is the summed conductivity tensor. With the equilibrium magnetic field along $z$, write ${ϵ}_{xx}={ϵ}_{yy}=S$, ${ϵ}_{zz}=P$, and ${ϵ}_{xy}=-{ϵ}_{yx}=ig$. For signed cyclotron frequencies ${\Omega }_s=q_{s}B/(m_{s}c)$,Here ${\omega }_{ps}$ is the species plasma frequency. The two circular eigenvalues are $S\mp g$; defining the sign convention before naming their polarizations prevents ambiguity.

Collisionless damping of a collective plasma perturbation by phase mixing and resonant exchange with particles near its phase velocity. In linear electrostatic response, the imaginary part of the dielectric function at real frequency is proportional to the derivative of the equilibrium phase-space distribution function at resonance. A velocity plateau removes this contribution on an interval; compact velocity support also permits undamped waves whose phase velocities lie outside that support.

For freely streaming particles, a spatial Fourier perturbation of a velocity distribution evolves with factors $e^{-ikvt}$. Contributions from different velocities acquire different phases and can cancel in a density or field moment even while the fine-grained distribution retains structure. In a self-consistent plasma, resonant field–particle exchange gives Landau damping. This is distinct from collisional thermalization and from orbital phase mixing in a planetary ensemble.

A broad electron Maxwell-Boltzmann distribution of width ${\sigma }_p$ plus a narrow beam of width ${\sigma }_b$ centred at speed $V$ can have a positive resonant velocity slope. The beam slope is largest at $V-{\sigma }_b$. Treating the background slope as constant over the beam width gives the displayed density threshold. Replacing the Maxwellian tail at $V-{\sigma }_b$ by its value at $V$ also requires $V{\sigma }_b/{\sigma }_p^2\ll 1$; width separation alone does not ensure that exponentially sensitive approximation. A positive slope is the local resonant-drive criterion, while an exact unstable mode also requires the kinetic dispersion relation.

A pole of the causally continued collective kinetic response, usually a zero of its scalar response denominator or response-matrix determinant. With time dependence $e^{-i{\omega }_{j}t}$, negative imaginary part gives damping, positive imaginary part growth, and zero imaginary part an undamped oscillation. Damped poles need not be regular discrete eigenmodes of the original collisionless operator: their contribution is obtained by continuing the initial-value response. The least damped excited pole can set the waiting time for a stationary forced response.

The velocity-integration contour implementing causal continuation of a kinetic response from the upper complex phase velocity half-plane. As a pole approaches the real axis, the contour passes below it. With denominator $v-u$, the real-axis limit is $\mathrm{PV}\int f_0^{\prime }(v)/(v-u)dv+i\pi f_0^{\prime }(u)$. Continuing the pole below the axis keeps this causal prescription rather than simply replacing the contour by an undeformed real-axis integral.

An inductively coupled plasma receives power through electromagnetic induction. An alternating coil electric current produces a changing magnetic field, which induces an electric field in the gas. Electrons accelerated by that field transfer energy through collisions, sustaining ionization and driving atomic transitions.

A typical analytical source surrounds a gas-carrying dielectric torch with a radiofrequency coil. Excited atoms emit spectral lines useful for atomic spectroscopy, while extracted ions can feed mass spectrometry. The coil supplies energy without requiring it to touch the plasma. Coupling, gas composition and particle losses determine the discharge response. ^[ref]

On spatial scales much larger than the Debye length, a plasma may have nearly cancelling positive and negative electric charge densities while still carrying an electric current and having a nonzero electric field. Quasineutrality is an approximation on appropriate slow macroscopic scales, not an assertion that each fluid element has exactly zero charge or that charge separation can be ignored at every boundary.

A plasmon is a quantum of a collective oscillation of electric charge density in a plasma or conducting material. Long-range Coulomb interaction can give its long-wavelength longitudinal mode a nonzero frequency.

A plasma oscillation is a collective oscillation of charge density and its restoring electric field. In a three-dimensional bulk charged superfluid, long-range Coulomb forces give the longitudinal collective mode a nonzero long-wave plasma frequency. Different geometry and electromagnetic boundary conditions can change that limit.

A low-frequency electrostatic plasma wave with ion inertia and a pressure restoring force, usually dominated by electron pressure. Weak ion Landau damping requires the phase velocity to exceed typical ion speeds. Electron Landau damping depends on the electron parallel marginal distribution and need not vanish even when ion damping is absent.

For equal densities of singly charged ions and electrons, a completely degenerate electron equation of state gives $d{P}_e/d{n}_e=m_e{v}_{Fe}^2/3$. Coupling this restoring force to ion inertia yields the sound speed in the title. If both species are degenerate and $m_e\ll m_i$, then $v_{Fi}\ll c_s\ll v_{Fe}$. Ion resonances are absent, while the electron parallel marginal distribution has a small slope at the phase velocity, producing relative Landau damping of order $\sqrt{m_e/m_i}$.

An electron-supported electrostatic branch with approximately linear small-wavenumber dispersion relation $\omega \simeq k{u}_a$. A velocity plateau can support such a completely undamped mode even when ion response is neglected. A single Maxwellian electron population generally damps the corresponding acoustic response strongly. Distinct electron populations provide another setting for electron acoustic behavior, with different assumptions and damping properties.

A longitudinal electron plasma oscillation whose long-wavelength frequency approaches the electron plasma frequency when ions can be treated as stationary. Its dispersion relation and Landau damping depend on the electron phase-space distribution function. Compact velocity support can permit exactly undamped branches outside the support.

The long-wavelength root of the degenerate-plasma dielectric response. Expanding its logarithm at large $s=\omega /(k{v}_F)$ gives $ϵ=1-{\omega }_p^2/{\omega }^2-3{\omega }_p^2{k}^2{v}_F^2/(5{\omega }^4)+\cdots$. The occupied Fermi sphere has $⟨v_{\parallel }^2⟩=v_F^2/5$, accounting for the dispersive coefficient. The static electron response instead defines the Thomas-Fermi screening length $v_F/(\sqrt{3}{\omega }_p)$.

A transverse magnetic polarization electromagnetic wave bound to an interface through coupling to the material's collective charge response. Its fields propagate along the interface and decay away from it on both sides. In an ideal nonmagnetic lossless planar vacuum-material interface, a real negative relative permittivity below $-1$ supports the bound branch described by the lossless planar surface-plasmon dispersion. Material losses generally make the propagation constant complex.

For a vacuum-material interface at $z=0$, write the magnetic phasor as $H_y{e}^{ikx-{\beta }_{j}z}$, with ${\beta }_1>0$ in vacuum and ${\beta }_2<0$ in the material. The Ampere-Maxwell law gives $E_x=i{\beta }_j{H}_y/(\omega {\epsilon }_0{ϵ}_j)$ and $E_z=-k{H}_y/(\omega {\epsilon }_0{ϵ}_j)$. Tangential field continuity gives ${\beta }_2=ϵ{\beta }_1$. Together with $k^2-{\beta }_j^2={ϵ}_j{\omega }^2/c^2$, this gives the displayed relation and requires $ϵ<-1$ for real propagating $k$ and decay on both sides. For $ϵ=1-{\omega }_p^2/{\omega }^2$, the branch has $0<\omega <{\omega }_p/\sqrt{2}$ and approaches that upper frequency only as $k\to \infty$. At low frequency, $k=(\omega /c)[1+{\omega }^2/(2{\omega }_p^2)+O({\omega }^4/{\omega }_p^4)]$.

Collective electric charge rearrangement modifies particle interactions in a plasma. In stellar nuclear fusion, this response can alter the Coulomb barrier used to infer a thermonuclear reaction rate.

For mobile thermal electrons with a fixed neutralizing background, the Debye length is the characteristic distance over which plasma screening suppresses an electrostatic perturbation. Linearizing the Boltzmann density response and using Poisson's equation gives the displayed scale. Other mobile species contribute additively to ${\lambda }_D^{-2}$ through $n_s{q}_s^2/({ϵ}_0{k}_B{T}_s)$.

A sphere of radius equal to the Debye length. Its particle count defines one common normalization of the plasma parameter.

For a three-dimensional ideal degenerate electron gas, let $r=\hslash {\omega }_{\mathrm{pe}}/E_F$. If one heuristically defines a Debye length by replacing $k_{\mathrm{B}}T$ with the Fermi energy, then $k_F{\lambda }_D=\sqrt{2}/r$ and the associated electron count is the displayed expression. A count below one signals failure of the classical many-particle screening interpretation. It does not prohibit plasmons. The zero-temperature compressibility instead gives the Thomas-Fermi screening length squared $2{ϵ}_0{E}_F/(3n_e{e}^2)$.

The plasma frequency is the characteristic electrostatic oscillation frequency of a charged species displaced against a neutralizing background. For number density $n_s$, charge $q_s$ and mass $m_s$, the displayed formula uses SI units; in Gaussian units it is ${\omega }_{ps}=\sqrt{4\pi n_s{q}_s^2/m_s}$. The ion skin depth is $c/{\omega }_{pi}$ for the ion species. Different species have different plasma frequencies.

The electron plasma frequency is the characteristic angular frequency of small collective electron displacements against an effectively stationary neutralizing ion background. For electron number density $n_e$, its cold, unmagnetized value in SI units isHere $e$ is the magnitude of the electron charge and $m_e$ the electron mass. Angular frequency and cycles per second differ by $2\pi$.^[ref]

A displaced slab gives a simple derivation. Shift a uniform electron population by a small distance $x$ relative to fixed positive ions. The exposed charge produces a restoring electric field $E=n_{e}ex/{ϵ}_0$. Newton's equation for an electron then giveswhich is a harmonic oscillator with the frequency above. This argument describes a collective charge-separation mode, not an individual electron's orbital frequency.

For $n_e=10^6{\mathrm{m}}^{-3}=1{\mathrm{cm}}^{-3}$, $f_{pe}\simeq 8.98\mathrm{kHz}$. Increasing density by a factor of four doubles the frequency. An identified plasma oscillation can therefore diagnose density, while interpreting an arbitrary spectral peak this way would be unjustified.

Finite temperature changes the wave's dispersion relation; magnetic fields, collisions and ion motion introduce additional modes or damping. The same cold model also gives the transverse wave cutoff in a cold unmagnetized plasma, but an observed electromagnetic cutoff and a local electrostatic oscillation are distinct measurements.

With immobile ions, negligible collisions and electron density $n_e$, the plasma frequency is ${\omega }_p^2=n_e{e}^2/(m_e{ϵ}_0)$. Combining inertial electron current with vacuum displacement current gives ${\omega }^2={\omega }_p^2+c^2{k}^2$. Transverse fields are evanescent below ${\omega }_p$ and propagate above it.

A carrier response must appear once in an electromagnetic constitutive model. With harmonic convention $e^{-i\omega t}$, absolute permittivity $\epsilon$ describing the remaining polarization and an explicit conduction electric current density ${\mathbf{J}}_c=\sigma \mathbf{E}$, the Ampère-Maxwell equation containsThus an alternative description incorporates that carrier contribution into the complex permittivity ${\epsilon }_{\mathrm{eff}}=\epsilon +i\sigma /\omega$ and omits the explicit ${\mathbf{J}}_c$. Using this effective quantity and adding the same conduction current again doubles that response. For relative permittivity, the added term is $i\sigma /({\epsilon }_0\omega )$; changing the harmonic sign convention changes the displayed sign. Stating which charges are included in each term prevents a mathematical rearrangement from becoming an extra physical mechanism.

In an unmagnetized plasma, an electron displacement driven by $\mathbf{E}{e}^{-i\omega t}$ obeys $-m_e{\omega }^2{\mathbf{x}}_e=-e\mathbf{E}$. Its dipole moment is $-e{\mathbf{x}}_e$, giving electric polarization $\mathbf{P}=-n{e}^2\mathbf{E}/(m_e{\omega }^2)$. With $\mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}$, this yields the displayed relative permittivity and ${\omega }_p^2=n{e}^2/(m_e{\epsilon }_0)$. Ions contribute a further negative term smaller by $Z{m}_e/m_i$ for one ion species of charge $Ze$. Neglecting it requires heavy ions at electron-wave frequencies; collisions, background magnetization and pressure dispersion are excluded. The transverse electromagnetic wave relation is $c^2{k}^2={\omega }^2-{\omega }_p^2$.

For a downward transverse wave in an isotropic plasma, $E_z/E_x=\tan {\theta }_t$ fixes the transmitted direction. Tangential phase matching gives $\sin {\theta }_i=n_p\sin {\theta }_t$. Continuity of tangential electric and magnetic fields then separates incident and reflected amplitudes. A measured transmitted electric amplitude alone cannot be identified with the incident one; mean power for peak phasors includes a factor of one half, and interface-normal flux includes a cosine.

For vacuum incidence on a sharp planar, collisionless plasma interface, the transmitted normal wavenumber obeys $k_z^2=({\omega }^2/c^2)({\cos }^2\theta -{\omega }_p^2/{\omega }^2)$. Negative $k_z^2$ produces an evanescent field and total reflection in this lossless model.

For a singly charged ion species in Gaussian units, ${\omega }_{pi}=(4\pi n{e}^2/m_i{)}^{1/2}$ is its plasma frequency and $d_i=c\sqrt{m_i/(4\pi n{e}^2)}$. The ion skin depth marks a characteristic scale below which ion and electron motion can separate in a magnetized plasma. Together with the Alfvén speed, it satisfies $v_A{d}_i=c{B}_0/(4\pi en)$, the coefficient of guide-field propagation in electron magnetohydrodynamics.