In
plasma physics
Plasma () is a state of matter characterized by the presence of a significant portion of charged particles in any combination of ions or electrons. It is the most abundant form of ordinary matter in the universe, mostly in stars (including th ...
, an electromagnetic electron wave is a
wave
In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
in a
plasma which has a
magnetic field
A magnetic field (sometimes called B-field) is a physical 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 ...
component and in which primarily the
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s oscillate.
In an unmagnetized plasma, an electromagnetic electron wave is simply a
light
Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
wave modified by the plasma. In a magnetized plasma, there are two modes perpendicular to the field, the O and X modes, and two modes parallel to the field, the R and L waves.
Waves in an unmagnetized plasma
Langmuir Wave
The
Langmuir wave is a purely
longitudinal wave
Longitudinal waves are waves which oscillate in the direction which is parallel to the direction in which the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal ...
, that is, the wave vector is in the same direction as the E-field. It is an electrostatic wave; as such, it doesn't have an oscillating magnetic field.
A plasma consists of charged particles which react to electric fields, in contrast with dielectric matter. When electrons in a uniform, homogeneous plasma are perturbed from their equilibrium position, a charge separation occurs creating an
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
which acts as restoring force on the electrons. Since electrons have inertia the system behaves as a
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive const ...
, where the electrons oscillate at a frequency ω
pe, called electron plasma frequency. These oscillations do not propagate—the group velocity is 0.
When the thermal motion of the electrons is taken into account a shift in frequency from the electron plasma frequency ω
pe occurs. Now the electron pressure gradient acts as the restoring force, creating a propagating wave analogous to a sound wave in non-ionized gases. Combining these two restoring forces (from the electric field and electron pressure gradient)
a type of wave, named Langmuir wave, is excited. The dispersion relation is:
The first term on the right-hand side of the dispersion relation is the electron plasma oscillation related to the electric field force and the second term is related to the thermal motion of the electrons, where C
e is the electron thermal speed and k is the
wave vector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength) ...
.
Electromagnetic wave
In an unmagnetized plasma, waves above the plasma frequency propagate through the plasma according to the dispersion relation:
In an unmagnetized plasma for the high frequency or low electron density limit, i.e. for
or
where ω
pe is the
plasma frequency
Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), 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 ...
, the wave speed is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). 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 i ...
in vacuum. As the electron density increases, the
phase velocity
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, t ...
increases and the
group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space.
For example, if a stone is thro ...
decreases until the
cut-off frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than ...
where the light frequency is equal to ω
pe. This density is known as the critical density for the
angular frequency
In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
ω of that wave and is given by
:
(
SI units
The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...
)
If the critical density is exceeded, the plasma is called over-dense.
In a magnetized plasma, except for the O wave, the cut-off relationships are more complex.
O wave
The O wave is the "ordinary" wave in the sense that its
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the ...
is the same as that in an unmagnetized plasma, that is,
. It is
plane polarized with
E
1 , , B
0. It has a cut-off at the
plasma frequency
Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), 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 ...
.
X wave
The X wave is the "extraordinary" wave because it has a more complicated dispersion relation:
[D. Gary Swanson, '' Plasma Waves '', Academic Press, 1989.]
Where
.
It is partly transverse (with E
1⊥B
0)
and partly longitudinal; the E-field is of the form
Where
refer to the Stix notation.
As the density is increased, the phase velocity rises from ''c'' until the cut-off at
is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency
. Then it can propagate again until the second cut-off at
. The cut-off frequencies are given by
:
where
is the
electron cyclotron resonance frequency, and
is the electron
plasma frequency
Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), 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 ...
.
The resonant frequencies for the X-wave are:
where
and
.
R wave and L wave
The R wave and the L wave are right-hand and left-hand circularly polarized, respectively. The R wave has a cut-off at ω
R (hence the designation of this frequency) and a resonance at ω
c. The L wave has a cut-off at ω
L and no resonance. R waves at frequencies below ω
c/2 are also known as whistler modes.
Dispersion relations
The
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the ...
can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the
index of refraction
In optics, the refractive index (or refraction index) of an optical medium is the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...
''ck''/ω (squared).
See also
*
Appleton-Hartree equation
References
{{DEFAULTSORT:Electromagnetic Electron Wave
Waves in plasmas