HOME

TheInfoList



OR:

Plasma oscillations, also known as Langmuir waves (after
Irving Langmuir Irving Langmuir (; January 31, 1881 – August 16, 1957) was an American chemist, physicist, and engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry. Langmuir's most famous publication is the 1919 ar ...
), are rapid oscillations of the
electron density In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial va ...
in conducting media such as plasmas or
metal A metal (from ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, e ...
s in the
ultraviolet Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30  PHz) to 400 nm (750  THz), shorter than that of visible light, but longer than X-rays. UV radiation ...
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 In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum. For exa ...
resulting from the quantization of these oscillations is the
plasmon In physics, a plasmon is a quantum of plasma oscillation. Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quantiz ...
. Langmuir waves were discovered by American
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
s
Irving Langmuir Irving Langmuir (; January 31, 1881 – August 16, 1957) was an American chemist, physicist, and engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry. Langmuir's most famous publication is the 1919 ar ...
and Lewi Tonks in the 1920s. They are parallel in form to
Jeans instability In stellar physics, the Jeans instability causes the collapse of interstellar gas clouds and subsequent star formation, named after James Jeans. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of ...
waves, which are caused by gravitational instabilities in a static medium.


Mechanism

Consider an electrically neutral plasma in equilibrium, consisting of a gas of positively charged ions and negatively charged
electrons 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 n ...
. If one displaces by a tiny amount an electron or a group of electrons with respect to the ions, the
Coulomb force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
pulls the electrons back, acting as a restoring force.


'Cold' electrons

If the thermal motion of the electrons is ignored, it is possible to show that the charge density oscillates at the ''plasma frequency''
: \omega_ = \sqrt, \left mathrm\right/math> (
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
), :\omega_ = \sqrt, \left mathrm\right/math> ( cgs units), where ''n_\mathrm '' is the
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric num ...
of electrons, ''e'' is the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
, ''m^* '' is the effective mass of the electron, and \varepsilon_0 is the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
. Note that the above
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
is derived under the
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
that the ion mass is infinite. This is generally a good approximation, as the electrons are so much lighter than ions. Proof using Maxwell equations. Assuming charge density oscillations \rho(\omega)=\rho_0 e^ the continuity equation: : \nabla \cdot \mathbf = - \frac = i \omega \rho(\omega) the Gauss law : \nabla \cdot \mathbf(\omega) = 4 \pi \rho(\omega) and the conductivity : \mathbf(\omega) = \sigma(\omega) \mathbf(\omega) it remains: : i \omega \rho(\omega) = 4 \pi \sigma(\omega) \rho(\omega) which is always true only if : 1+ \frac = 0 But this is also the dielectric constant (see
Drude Model The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltag ...
) \epsilon(\omega) = 1+ \frac and the condition of transparency (i.e. \epsilon \ge 0 from a certain plasma frequency \omega_ and above), the same condition here \epsilon = 0 apply to make possible also the propagation of density waves in the charge density. This expression must be modified in the case of electron-
positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collide ...
plasmas, often encountered in
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...
. Since the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
is independent of the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
, these
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendul ...
s have an
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
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, ...
and zero
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'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
. Note that, when m^*=m_\mathrm, the plasma frequency, \omega_, depends only on
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant ...
s and electron density n_\mathrm. The numeric expression for angular plasma frequency is :f_\text = \frac~\left text\right/math> Metals are only transparent to light with a frequency higher than the metal's plasma frequency. For typical metals such as aluminium or silver, ''n_\mathrm'' is approximately 1023 cm−3, which brings the plasma frequency into the ultraviolet region. This is why most metals reflect visible light and appear shiny.


'Warm' electrons

When the effects of the
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 n ...
thermal speed v_ = \sqrt are taken into account, the electron pressure acts as a restoring force as well as the electric field and the oscillations propagate with frequency and
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
related by the longitudinal Langmuir* wave: : \omega^2 =\omega_^2 +\frack^2=\omega_^2 + 3 k^2 v_^2 , called the BohmGross
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 t ...
. If the spatial scale is large compared to the
Debye length In plasmas and electrolytes, the Debye length \lambda_ (also called Debye radius), is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. With each Debye length the charges are in ...
, the
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendul ...
s are only weakly modified by the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
term, but at small scales the pressure term dominates and the waves become dispersionless with a speed of \sqrt \cdot v_. For such waves, however, the electron thermal speed is comparable to 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, ...
, i.e., : v \sim v_ \ \stackrel\ \frac, so the plasma waves can
accelerate In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
electrons that are moving with speed nearly equal to the phase velocity of the wave. This process often leads to a form of collisionless damping, called Landau damping. Consequently, the large-''k'' portion in 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 t ...
is difficult to observe and seldom of consequence. In a bounded plasma, fringing electric fields can result in propagation of plasma oscillations, even when the electrons are cold. In a
metal A metal (from ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, e ...
or
semiconductor A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way ...
, the effect of the ions' periodic potential must be taken into account. This is usually done by using the electrons' effective mass in place of ''m''.


Plasma oscillations and the effect of the negative mass

Plasma oscillations may give rise to the effect of the “
negative mass In theoretical physics, negative mass is a type of exotic matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg. Such matter would violate one or more energy conditions and show some strange properties such as t ...
”. The mechanical model giving rise to the negative effective mass effect is depicted in Figure 1. A core with mass m_2 is connected internally through the spring with constant k_2 to a shell with mass m_1. The system is subjected to the external sinusoidal force F(t)=\widehat\sin\omega t. If we solve the equations of motion for the masses m_1 and m_2 and replace the entire system with a single effective mass m_ we obtain: m_=m_1+, where \omega_0=\sqrt. When the frequency \omega approaches \omega_0 from above the effective mass m_ will be negative. The negative effective mass (density) becomes also possible based on the electro-mechanical coupling exploiting plasma oscillations of a free electron gas (see Figure 2). Text was copied from this source, which is available under
Creative Commons Attribution 4.0 International License
The negative mass appears as a result of vibration of a metallic particle with a frequency of \omega which is close the frequency of the plasma oscillations of the electron gas m_2 relatively to the ionic lattice m_1. The plasma oscillations are represented with the elastic spring k_2=\omega_^2m_2, where \omega_ is the plasma frequency. Thus, the metallic particle vibrated with the external frequency ''ω'' is described by the effective mass m_=m_1+, which is negative when the frequency \omega approaches \omega_ from above. Metamaterials exploiting the effect of the negative mass in the vicinity of the plasma frequency were reported.


See also

* Electron wake * List of plasma physics articles *
Plasmon In physics, a plasmon is a quantum of plasma oscillation. Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quantiz ...
*
Relativistic quantum chemistry Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of ...
*
Surface plasmon resonance Surface plasmon resonance (SPR) is the resonant oscillation of conduction electrons at the interface between negative and positive permittivity material in a particle stimulated by incident light. SPR is the basis of many standard tools for measu ...
* Upper hybrid oscillation, in particular for a discussion of the modification to the mode at propagation angles oblique to the magnetic field * Waves in plasmas


References


Sources

*


Further reading

*{{Citation , last=Longair , first=Malcolm S. , title=Galaxy Formation , year=1998 , publisher=Springer , location=Berlin , isbn=978-3-540-63785-1 Waves in plasmas Plasma physics Plasmonics