- Key People:
- Lev Davidovich Landau
- Hannes Alfvén
- On the Web:
- Massachusetts Institute of Technology - Department of Mathematics - What is a Plasma? (PDF) (Mar. 18, 2025)
Just as a lightweight cork in water will bob up and down about its rest position, any general displacement of light electrons as a group with respect to the positive ions in a plasma leads to the oscillation of the electrons as a whole about an equilibrium state. In the case of the cork, the restoring force is provided by gravity; in plasma oscillations, it is provided by the electric force. These movements are the plasma oscillations that were studied by Langmuir and Tonks. Analogously, just as buoyancy effects guide water waves, plasma oscillations are related to waves in the electron component of the plasma called Langmuir waves. Wavelike phenomena play a critical role in the behaviour of plasmas.
The time τ required for an oscillation of this type is the most important temporal parameter in a plasma. The main spatial parameter is the Debye length, h, which is the distance traveled by the average thermal electron in time τ/2π. A plasma can be defined in terms of these parameters as a partially or fully ionized gas that satisfies the following criteria: (1) a constituent electron may complete many plasma oscillations before it collides with either an ion or one of the other heavy constituents, (2) inside each sphere with a radius equal to the Debye length, there are many particles, and (3) the plasma itself is much larger than the Debye length in every dimension.
Another important temporal parameter is the time between collisions of particles. In any gas, separate collision frequencies are defined for collisions between all different particle types. The total collision frequency for a particular species is the weighted sum of all the separate frequencies. Two basic types of collision may occur: elastic and inelastic. In an elastic collision, the total kinetic energy of all the particles participating in the collision is the same before and after the event. In an inelastic collision, a fraction of the kinetic energy is transferred to the internal energy of the colliding particles. In an atom, for example, the electrons have certain allowed (discrete) energies and are said to be bound. During a collision, a bound electron may be excited—that is, raised from a low to a high energy state. This can occur, however, only by the expenditure of kinetic energy and only if the kinetic energy exceeds the difference between the two energy states. If the energy is sufficient, a bound electron may be excited to such a high level that it becomes a free electron, and the atom is said to be ionized; the minimum, or threshold, energy required to free an electron is called the ionization energy. Inelastic collisions may also occur with positive ions unless all the electrons have been stripped away. In general, only collisions of electrons and photons (quanta of electromagnetic radiation) with atoms and ions are significant in these inelastic collisions; ionization by a photon is called photoionization.
A molecule has additional discrete energy states, which may be excited by particle or photon collisions. At sufficiently high energies of interaction, the molecule can dissociate into atoms or into atoms and atomic ions. As in the case of atoms, collision of electrons and photons with molecules may cause ionization, producing molecular ions. In general, the reaction rate for inelastic collisions is similar to that of chemical reactions. At sufficiently high temperatures, the atoms are stripped of all electrons and become bare atomic nuclei. Finally, at temperatures of about 1,000,000 K or greater, nuclear reactions can occur—another form of inelastic collisions. When such reactions lead to the formation of heavier elements, the process is called thermonuclear fusion; mass is transmuted, and kinetic energy is gained instead of lost.
All sources of energy now existing on the Earth can be traced in one way or another to the nuclear fusion reactions inside the Sun or some long-extinct star. In such energy sources, gravity controls and confines the fusion process. The high temperatures required for the nuclear fusion reactions that take place in a hydrogen, or thermonuclear, bomb are attained by first igniting an atomic bomb, which produces a fission chain reaction. One of the great challenges of humankind is to create these high temperatures in a controlled manner and to harness the energy of nuclear fusion. This is the great practical goal of plasma physics—to produce nuclear fusion on the Earth. Confinement schemes devised by scientists use magnetic fields or the inertia of an implosion to guide and control the hot plasma.
Basic plasma physics
Plasma formation
Apart from solid-state plasmas, such as those in metallic crystals, plasmas do not usually occur naturally at the surface of the Earth. For laboratory experiments and technological applications, plasmas therefore must be produced artificially. Because the atoms of such alkalies as potassium, sodium, and cesium possess low ionization energies, plasmas may be produced from these by the direct application of heat at temperatures of about 3,000 K. In most gases, however, before any significant degree of ionization is achieved, temperatures in the neighbourhood of 10,000 K are required. A convenient unit for measuring temperature in the study of plasmas is the electron volt (eV), which is the energy gained by an electron in vacuum when it is accelerated across one volt of electric potential. The temperature, W, measured in electron volts is given by W = T/12,000 when T is expressed in kelvins. The temperatures required for self-ionization thus range from 2.5 to 8 electron volts, since such values are typical of the energy needed to remove one electron from an atom or molecule.
Because all substances melt at temperatures far below that level, no container yet built can withstand an external application of the heat necessary to form a plasma; therefore, any heating must be supplied internally. One technique is to apply an electric field to the gas to accelerate and scatter any free electrons, thereby heating the plasma. This type of ohmic heating is similar to the method in which free electrons in the heating element of an electric oven heat the coil. Because of their small energy loss in elastic collisions, electrons can be raised to much higher temperatures than other particles. For plasma formation a sufficiently high electric field must be applied, its exact value depending on geometry and the gas pressure. The electric field may be set up via electrodes or by transformer action, in which the electric field is induced by a changing magnetic field. Laboratory temperatures of about 10,000,000 K, or 8 kiloelectron volts (keV), with electron densities of about 1019 per cubic metre have been achieved by the transformer method. The temperature is eventually limited by energy losses to the outside environment. Extremely high temperatures, but relatively low-density plasmas, have been produced by the separate injection of ions and electrons into a mirror system (a plasma device using a particular arrangement of magnetic fields for containment). Other methods have used the high temperatures that develop behind a wave that is moving much faster than sound to produce what is called a shock front; lasers have also been employed.
Natural plasma heating and ionization occur in analogous ways. In a lightning-induced plasma, the electric current carried by the stroke heats the atmosphere in the same manner as in the ohmic heating technique described above. In solar and stellar plasmas the heating is internal and caused by nuclear fusion reactions. In the solar corona, the heating occurs because of waves that propagate from the surface into the Sun’s atmosphere, heating the plasma much like shock-wave heating in laboratory plasmas. In the ionosphere, ionization is accomplished not through heating of the plasma but rather by the flux of energetic photons from the Sun. Far-ultraviolet rays and X rays from the Sun have enough energy to ionize atoms in the Earth’s atmosphere. Some of the energy also goes into heating the gas, with the result that the upper atmosphere, called the thermosphere, is quite hot. These processes protect the Earth from energetic photons much as the ozone layer protects terrestrial life-forms from lower-energy ultraviolet light. The typical temperature 300 kilometres above the Earth’s surface is 1,200 K, or about 0.1 eV. Although it is quite warm compared with the surface of the Earth, this temperature is too low to create self-ionization. When the Sun sets with respect to the ionosphere, the source of ionization ceases, and the lower portion of the ionosphere reverts to its nonplasma state. Some ions, in particular singly charged oxygen (O+), live long enough that some plasma remains until the next sunrise. In the case of an aurora, a plasma is created in the nighttime or daytime atmosphere when beams of electrons are accelerated to hundreds or thousands of electron volts and smash into the atmosphere.
Methods of describing plasma phenomena
The behaviour of a plasma may be described at different levels. If collisions are relatively infrequent, it is useful to consider the motions of individual particles. In most plasmas of interest, a magnetic field exerts a force on a charged particle only if the particle is moving, the force being at right angles to both the direction of the field and the direction of particle motion. In a uniform magnetic field (B), a charged particle gyrates about a line of force. The centre of the orbit is called the guiding centre. The particle may also have a component of velocity parallel to the magnetic field and so traces out a helix in a uniform magnetic field. If a uniform electric field (E) is applied at right angles to the direction of the magnetic field, the guiding centre drifts with a uniform velocity of magnitude equal to the ratio of the electric to the magnetic field (E/B), at right angles to both the electric and magnetic fields. A particle starting from rest in such fields follows the same cycloidal path a dot on the rim of a rolling wheel follows. Although the “wheel” radius and its sense of rotation vary for different particles, the guiding centre moves at the same E/B velocity, independent of the particle’s charge and mass. Should the electric field change with time, the problem would become even more complex. If, however, such an alternating electric field varies at the same frequency as the cyclotron frequency (i.e., the rate of gyration), the guiding centre will remain stationary, and the particle will be forced to travel in an ever-expanding orbit. This phenomenon is called cyclotron resonance and is the basis of the cyclotron particle accelerator.
The motion of a particle about its guiding centre constitutes a circular current. As such, the motion produces a dipole magnetic field not unlike that produced by a simple bar magnet. Thus, a moving charge not only interacts with magnetic fields but also produces them. The direction of the magnetic field produced by a moving particle, however, depends both on whether the particle is positively or negatively charged and on the direction of its motion. If the motion of the charged particles is completely random, the net associated magnetic field is zero. On the other hand, if charges of different sign have an average relative velocity (i.e., if an electric current flows), then a net magnetic field over and above any externally applied field exists. The magnetic interaction between charged particles is therefore of a collective, rather than of an individual, particle nature.
At a higher level of description than that of the single particle, kinetic equations of the Boltzmann type are used. Such equations essentially describe the behaviour of those particles about a point in a small-volume element, the particle velocities lying within a small range about a given value. The interactions with all other velocity groups, volume elements, and any externally applied electric and magnetic fields are taken into account. In many cases, equations of a fluid type may be derived from the kinetic equations; they express the conservation of mass, momentum, and energy per unit volume, with one such set of equations for each particle type.
Determination of plasma variables
The basic variables useful in the study of plasma are number densities, temperatures, electric and magnetic field strengths, and particle velocities. In the laboratory and in space, both electrostatic (charged) and magnetic types of sensory devices called probes help determine the magnitudes of such variables. With the electrostatic probe, ion densities, electron and ion temperatures, and electrostatic potential differences can be determined. Small search coils and other types of magnetic probes yield values for the magnetic field; and from Maxwell’s electromagnetic equations the current and charge densities and the induced component of the electric field may be found. Interplanetary spacecraft have carried such probes to nearly every planet in the solar system, revealing to scientists such plasma phenomena as lightning on Jupiter and the sounds of Saturn’s rings and radiation belts. In the early 1990s, signals were being relayed to the Earth from several spacecraft approaching the edge of the plasma boundary to the solar system, the heliopause.
In the laboratory the absorption, scattering, and excitation of neutral and high-energy ion beams are helpful in determining electron temperatures and densities; in general, the refraction, reflection, absorption, scattering, and interference of electromagnetic waves also provide ways to determine these same variables. This technique has also been employed to remotely measure the properties of the plasmas in the near-space regions of the Earth using the incoherent scatter radar method. The method works by bouncing radio waves from small irregularities in the electron gas that occur owing to random thermal motions of the particles. The returning signal is shifted slightly from the transmitted one—because of the Doppler-shift effect—and the velocity of the plasma can be determined in a manner similar to the way in which the police detect a speeding car. Using this method, the wind speed in space can be found, along with the temperature, density, electric field, and even the types of ions present. In geospace the appropriate radar frequencies are in the range of 50 to 1,000 megahertz (MHz), while in the laboratory, where the plasma densities and plasma frequencies are higher, microwaves and lasers must be used.
Aside from the above methods, much can be learned from the radiation generated and emitted by the plasma itself; in fact, this is the only means of studying cosmic plasma beyond the solar system. The various spectroscopic techniques covering the entire continuous radiation spectrum determine temperatures and identify such nonthermal sources as those pulses producing synchrotron radiations.
Waves in plasmas
The waves most familiar to people are the buoyancy waves that propagate on the surfaces of lakes and oceans and break onto the world’s beaches. Equally familiar, although not necessarily recognized as waves, are the disturbances in the atmosphere that create what is referred to as the weather. Wave phenomena are particularly important in the behaviour of plasmas. In fact, one of the three criteria for the existence of a plasma is that the particle-particle collision rate be less than the plasma-oscillation frequency. This in turn implies that the collective interactions that control the plasma gas depend on the electric and magnetic field effects as much as, or more so than, simple collisions. Since waves are able to propagate, the possibility exists for force fields to act at large distances from the point where they originated.
Ordinary fluids can support the propagation of sound (acoustic) waves, which involve pressure, temperature, and velocity variations. Electromagnetic waves can propagate even in a vacuum but are slowed down in most cases by the interaction of the electric fields in the waves with the charged particles bound in the atoms or molecules of the gas. Although it is important for a complete description of electromagnetic waves, such an interaction is not very strong. In a plasma, however, the particles react in concert with any electromagnetic field (e.g., as in an electromagnetic wave) as well as with any pressure or velocity field (e.g., as in a sound wave). In fact, in a plasma sound wave the electrons and ions become slightly separated owing to their difference in mass, and an electric field builds up to bring them back together. The result is called an ion acoustic wave. This is just one of the many types of waves that can exist in a plasma. The brief discussion that follows touches on the main types in order of increasing wave-oscillation frequency.
Low-frequency waves
At the lowest frequency are Alfvén waves, which require the presence of a magnetic field to exist. In fact, except for ion acoustic waves, the existence of a background magnetic field is required for any wave with a frequency less than the plasma frequency to occur in a plasma. Most natural plasmas are threaded by a magnetic field, and laboratory plasmas often use a magnetic field for confinement, so this requirement is usually met, and all types of waves can occur.
Alfvén waves are analogous to the waves that occur on the stretched string of a guitar. In this case, the string represents a magnetic field line. When a small magnetic field disturbance takes place, the field is bent slightly, and the disturbance propagates in the direction of the magnetic field. Since any changing magnetic field creates an electric field, an electromagnetic wave results. Such waves are the slowest and have the lowest frequencies of any known electromagnetic waves. For example, the solar wind streams out from the Sun with a speed greater than either electromagnetic (Alfvén) or sound waves. This means that, when the solar wind hits the Earth’s outermost magnetic field lines, a shock wave results to “inform” the incoming plasma that an obstacle exists, much like the shock wave associated with a supersonic airplane. The shock wave travels toward the Sun at the same speed but in the opposite direction as the solar wind, so it appears to stand still with respect to the Earth. Because there are almost no particle-particle collisions, this type of collisionless shock wave is of great interest to space plasma physicists who postulate that similar shocks occur around supernovas and in other astrophysical plasmas. On the Earth’s side of the shock wave, the heated and slowed solar wind interacts with the Earth’s atmosphere via Alfvén waves propagating along the magnetic field lines.
The turbulent surface of the Sun radiates large-amplitude Alfvén waves, which are thought to be responsible for heating the corona to 1,000,000 K. Such waves can also produce fluctuations in the solar wind, and, as they propagate through it to the Earth, they seem to control the occurrence of magnetic storms and auroras that are capable of disrupting communication systems and power grids on the planet.
Two fundamental types of wave motion can occur: longitudinal, like a sound or ion acoustic wave, in which particle oscillation is in a direction parallel to the direction of wave propagation; and transverse, like a surface water wave, in which particle oscillation is in a plane perpendicular to the direction of wave propagation. In all cases, a wave may be characterized by a speed of propagation (u), a wavelength (λ), and a frequency (ν) related by an expression in which the velocity is equal to the product of the wavelength and frequency, namely, u = λν. The Alfvén wave is a transverse wave and propagates with a velocity that depends on the particle density and the magnetic field strength. The velocity is equal to the magnetic flux density (B) divided by the square root of the mass density (ρ) times the permeability of free space (μ0)—that is to say, B/Square root of√μ0ρ. The ion acoustic wave is a longitudinal wave and also propagates parallel to the magnetic field at a speed roughly equal to the average thermal velocity of the ions. Perpendicular to the magnetic field a different type of longitudinal wave called a magnetosonic wave can occur.
Higher frequency waves
In these waves the plasma behaves as a whole, and the velocity is independent of wave frequency. At higher frequencies, however, the separate behaviour of ions and electrons causes the wave velocities to vary with direction and frequency. The Alfvén wave splits into two components, referred to as the fast and slow Alfvén waves, which propagate at different frequency-dependent speeds. At still higher frequencies these two waves (called the electron cyclotron and ion cyclotron waves, respectively) cause electron and cyclotron resonances (synchronization) at the appropriate resonance frequencies. Beyond these resonances, transverse wave propagation does not occur at all until frequencies comparable to and above the plasma frequency are reached.
At frequencies between the ion and electron gyrofrequencies lies a wave mode called a whistler. This name comes from the study of plasma waves generated by lightning. When early researchers listened to natural radio waves by attaching an antenna to an audio amplifier, they heard a strange whistling sound. The whistle occurs when the electrical signal from lightning in one hemisphere travels along the Earth’s magnetic field lines to the other hemisphere. The trip is so long that some waves (those at higher frequencies) arrive first, resulting in the generation of a whistlelike sound. These natural waves were used to probe the region of space around the Earth before spacecraft became available. Such a frequency-dependent wave velocity is called wave dispersion because the various frequencies disperse with distance.
The speed of an ion acoustic wave also becomes dispersive at high frequencies, and a resonance similar to electron plasma oscillations occurs at a frequency determined by electrostatic oscillations of the ions. Beyond this frequency no sonic wave propagates parallel to a magnetic field until the frequency reaches the plasma frequency, above which electroacoustic waves occur. The wavelength of these waves at the critical frequency (ωp) is infinite, the electron behaviour at this frequency taking the form of the plasma oscillations of Langmuir and Tonks. Even without particle collisions, waves shorter than the Debye length are heavily damped—i.e., their amplitude decreases rapidly with time. This phenomenon, called Landau damping, arises because some electrons have the same velocity as the wave. As they move with the wave, they are accelerated much like a surfer on a water wave and thus extract energy from the wave, damping it in the process.
Containment
Magnetic fields are used to contain high-density, high-temperature plasmas because such fields exert pressures and tensile forces on the plasma. An equilibrium configuration is reached only when at all points in the plasma these pressures and tensions exactly balance the pressure from the motion of the particles. A well-known example of this is the pinch effect observed in specially designed equipment. If an external electric current is imposed on a cylindrically shaped plasma and flows parallel to the plasma axis, the magnetic forces act inward and cause the plasma to constrict, or pinch. An equilibrium condition is reached in which the temperature is proportional to the square of the electric current. This result suggests that any temperature may be achieved by making the electric current sufficiently large, the heating resulting from currents and compression. In practice, however, since no plasma can be infinitely long, serious energy losses occur at the ends of the cylinder; also, major instabilities develop in such a simple configuration. Suppression of such instabilities has been one of the major efforts in laboratory plasma physics and in the quest to control the nuclear fusion reaction.
A useful way of describing the confinement of a plasma by a magnetic field is by measuring containment time (τc), or the average time for a charged particle to diffuse out of the plasma; this time is different for each type of configuration. Various types of instabilities can occur in plasma. These lead to a loss of plasma and a catastrophic decrease in containment time. The most important of these is called magnetohydrodynamic instability. Although an equilibrium state may exist, it may not correspond to the lowest possible energy. The plasma, therefore, seeks a state of lower potential energy, just as a ball at rest on top of a hill (representing an equilibrium state) rolls down to the bottom if perturbed; the lower energy state of the plasma corresponds to a ball at the bottom of a valley. In seeking the lower energy state, turbulence develops, leading to enhanced diffusion, increased electrical resistivity, and large heat losses. In toroidal geometry, circular plasma currents must be kept below a critical value called the Kruskal-Shafranov limit, otherwise a particularly violent instability consisting of a series of kinks may occur. Although a completely stable system appears to be virtually impossible, considerable progress has been made in devising systems that eliminate the major instabilities. Temperatures on the order of 10,000,000 K at densities of 1019 particles per cubic metre and containment times as high as 1/50 of a second have been achieved.
