Bennett Pinch
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A pinch (or: Bennett pinch (after
Willard Harrison Bennett Willard Harrison Bennett (June 13, 1903 – September 28, 1987) was an American scientist and inventor, born in Findlay, Ohio. Bennett conducted research into plasma physics, astrophysics, geophysics, surface physics, and physical chemistry. T ...
), electromagnetic pinch, magnetic pinch, pinch effect, or plasma pinch.) is the compression of an electrically conducting
filament The word filament, which is descended from Latin ''filum'' meaning " thread", is used in English for a variety of thread-like structures, including: Astronomy * Galaxy filament, the largest known cosmic structures in the universe * Solar filament ...
by
magnetic Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, m ...
forces, or a device that does such. The conductor is usually a plasma, but could also be a solid or liquid
metal A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, electricity and thermal conductivity, heat relatively well. These properties are all associated wit ...
. Pinches were the first type of device used for experiments in controlled nuclear
fusion power Fusion power is a proposed form of power generation that would generate electricity by using heat from nuclear fusion reactions. In a fusion process, two lighter atomic nuclei combine to form a heavier nucleus, while releasing energy. Devices d ...
. Pinches occur naturally in electrical discharges such as lightning bolts, planetary
aurora An aurora ( aurorae or auroras), also commonly known as the northern lights (aurora borealis) or southern lights (aurora australis), is a natural light display in Earth's sky, predominantly observed in high-latitude regions (around the Arc ...
s,
current sheet A current sheet is an electric current that is confined to a surface, rather than being spread through a volume of space. Current sheets feature in magnetohydrodynamics (MHD), a model of electrically conductive fluids: if there is an electric cu ...
s, and
solar flare A solar flare is a relatively intense, localized emission of electromagnetic radiation in the Sun's atmosphere. Flares occur in active regions and are often, but not always, accompanied by coronal mass ejections, solar particle events, and ot ...
s.


Basic mechanism


Types

Pinches exist in nature and in laboratories. Pinches differ in their geometry and operating forces. These include: * ''Uncontrolled'' – Any time an electric current moves in large amounts (e.g., lightning, arcs, sparks, discharges) a magnetic force can pull together plasma. This can be insufficient for fusion. * '' Sheet pinch'' – An astrophysical effect, this arises from vast sheets of charged particles. * ''
Z-pinch In fusion power research, the Z-pinch (zeta pinch) is a type of plasma confinement system that uses an electric current in the plasma to generate a magnetic field that compresses it (see pinch). These systems were originally referred to simpl ...
'' – The current runs down the axis, or walls, of a cylinder while the
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 ...
is
azimuth An azimuth (; from ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system. Mathematically, the relative position vector from an observer ( origin) to a point ...
al * ''
Theta pinch Theta-pinch, or θ-pinch and sometimes known as a thetatron, is a type of fusion power reactor design. The name refers to the configuration of currents used to confine the plasma (physics), plasma fuel in the reactor, arranged to run around a cy ...
'' – The magnetic field runs down the axis of a cylinder, while the
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 ...
is in the
azimuth An azimuth (; from ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system. Mathematically, the relative position vector from an observer ( origin) to a point ...
al direction (also called a thetatron) * ''Screw pinch'' – A combination of a Z-pinch and theta pinch (also called a stabilized Z-pinch, or θ-Z pinch) * ''
Reversed field pinch A reversed-field pinch (RFP) is a device used to produce and contain near-thermonuclear plasmas. It is a toroidal pinch that uses a unique magnetic field configuration as a scheme to magnetically confine a plasma, primarily to study magnetic ...
'' or ''toroidal pinch'' – This is a Z-pinch arranged in the shape of a
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
. The plasma has an internal magnetic field. As distance increases from the center of this ring, the magnetic field reverses direction. * ''Inverse pinch'' – An early fusion concept, this device consisted of a rod surrounded by plasma. Current traveled through the plasma and returned along the center rod. This geometry was slightly different than a z-pinch in that the conductor was in the center, not the sides. * ''Cylindrical pinch'' * ''Orthogonal pinch effect'' * ''Ware pinch'' – A pinch that occurs inside a Tokamak plasma, when particles inside the banana orbit condense together. * ''Magnetized liner inertial fusion'' (MagLIF) – A Z-pinch of preheated, premagnetized fuel inside a metal liner, which could lead to ignition and practical fusion energy with a larger pulsed-power driver.


Common behavior

Pinches may become
unstable In dynamical systems instability means that some of the outputs or internal state (controls), states increase with time, without bounds. Not all systems that are not Stability theory, stable are unstable; systems can also be marginal stability ...
. They radiate energy across the whole
electromagnetic spectrum The electromagnetic spectrum is the full range of electromagnetic radiation, organized by frequency or wavelength. The spectrum is divided into separate bands, with different names for the electromagnetic waves within each band. From low to high ...
including
radio wave Radio waves (formerly called Hertzian waves) are a type of electromagnetic radiation with the lowest frequencies and the longest wavelengths in the electromagnetic spectrum, typically with frequencies below 300 gigahertz (GHz) and wavelengths g ...
s,
microwave Microwave is a form of electromagnetic radiation with wavelengths shorter than other radio waves but longer than infrared waves. Its wavelength ranges from about one meter to one millimeter, corresponding to frequency, frequencies between 300&n ...
s,
infrared Infrared (IR; sometimes called infrared light) is electromagnetic radiation (EMR) with wavelengths longer than that of visible light but shorter than microwaves. The infrared spectral band begins with the waves that are just longer than those ...
,
x-ray An X-ray (also known in many languages as Röntgen radiation) is a form of high-energy electromagnetic radiation with a wavelength shorter than those of ultraviolet rays and longer than those of gamma rays. Roughly, X-rays have a wavelength ran ...
s,
gamma ray A gamma ray, also known as gamma radiation (symbol ), is a penetrating form of electromagnetic radiation arising from high energy interactions like the radioactive decay of atomic nuclei or astronomical events like solar flares. It consists o ...
s,
synchrotron radiation Synchrotron radiation (also known as magnetobremsstrahlung) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in some types ...
, and
visible light Light, visible light, or visible radiation is electromagnetic radiation that can be perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400–700 nanometres (nm ...
. They also produce
neutron The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
s, as a product of fusion.


Applications and devices

Pinches are used to generate
X-rays An X-ray (also known in many languages as Röntgen radiation) is a form of high-energy electromagnetic radiation with a wavelength shorter than those of ultraviolet rays and longer than those of gamma rays. Roughly, X-rays have a wavelength ran ...
and the intense magnetic fields generated are used in
electromagnetic forming Electromagnetic forming (EM forming or magneforming) is a type of high-velocity, cold forming process for electrically conductive metals, most commonly copper and aluminium. The workpiece is reshaped by high-intensity pulsed magnetic fields tha ...
of metals. They also have applications in
particle beam A particle beam is a stream of charged particle, charged or neutral particles other than photons. In Particle accelerator, particle accelerators, these particles can move with a velocity close to the speed of light. There is a difference between ...
s including
particle beam weapon A particle-beam weapon uses a high-energy beam of atomic or subatomic particles to damage the target by disrupting its atomic and/or molecular structure. A particle-beam weapon is a type of space-based directed-energy weapon, which directs fo ...
s, astrophysics studies and it has been proposed to use them in space propulsion. A number of large pinch machines have been built to study
fusion power Fusion power is a proposed form of power generation that would generate electricity by using heat from nuclear fusion reactions. In a fusion process, two lighter atomic nuclei combine to form a heavier nucleus, while releasing energy. Devices d ...
; here are several: *
MAGPIE Magpies are birds of various species of the family Corvidae. Like other members of their family, they are widely considered to be intelligent creatures. The Eurasian magpie, for instance, is thought to rank among the world's most intelligent c ...
A Z-pinch at Imperial College. This dumps a large amount of current across a wire. Under these conditions, the wire becomes plasma and compresses to produce fusion. *
Z Pulsed Power Facility The Z Pulsed Power Facility, informally known as the Z machine or simply Z, is the largest high frequency electromagnetic wave generator in the world, operated by Sandia National Laboratories in Albuquerque, New Mexico. It has primarily been ...
at
Sandia National Laboratories Sandia National Laboratories (SNL), also known as Sandia, is one of three research and development laboratories of the United States Department of Energy's National Nuclear Security Administration (NNSA). Headquartered in Kirtland Air Force B ...
. *
ZETA Zeta (, ; uppercase Ζ, lowercase ζ; , , classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived from the Phoenician alphabet, Phoenician letter zay ...
device in Culham, England *
Madison Symmetric Torus The Madison Symmetric Torus (MST) is a reversed field pinch (RFP) physics experiment with applications to both fusion energy research and astrophysical plasmas. MST is located at the Center for Magnetic Self Organization (CMSO) at the Univers ...
at the University of Wisconsin, Madison *
Reversed-Field eXperiment The Reversed-Field eXperiment (RFX) is the largest reversed field pinch device presently in operation, situated in Padua, Italy. It was constructed from 1985 to 1991, and has been in operation since 1992. The experiments carried out in the last tw ...
in Italy. *
Dense plasma focus A dense plasma focus (DPF) is a type of Plasma (physics), plasma generating system originally developed as a fusion power device, starting in the early 1960s. The system demonstrated Power law, scaling laws that suggested it would not be useful in ...
in New Jersey *
University of Nevada, Reno The University of Nevada, Reno (Nevada, the University of Nevada, or UNR) is a Public university, public Land-grant university, land-grant research university in Reno, Nevada, United States. It is the state's flagship public university and prim ...
(USA) *
Cornell University Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...
(USA) *
University of Michigan The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...
(USA) *
University of California, San Diego The University of California, San Diego (UC San Diego in communications material, formerly and colloquially UCSD) is a public university, public Land-grant university, land-grant research university in San Diego, California, United States. Es ...
(USA) *
University of Washington The University of Washington (UW and informally U-Dub or U Dub) is a public research university in Seattle, Washington, United States. Founded in 1861, the University of Washington is one of the oldest universities on the West Coast of the Uni ...
(USA) *
Ruhr University The Ruhr University Bochum (, ) is a public research university located in the southern hills of the central Ruhr area, Bochum, Germany. It was founded in 1962 as the first new public university in Germany after World War II. Instruction began i ...
(Germany) *
École Polytechnique (, ; also known as Polytechnique or l'X ) is a ''grande école'' located in Palaiseau, France. It specializes in science and engineering and is a founding member of the Polytechnic Institute of Paris. The school was founded in 1794 by mat ...
(France) *
Weizmann Institute of Science The Weizmann Institute of Science ( ''Machon Weizmann LeMada'') is a Public university, public research university in Rehovot, Israel, established in 1934, fourteen years before the State of Israel was founded. Unlike other List of Israeli uni ...
(Israel) *
Universidad Autónoma Metropolitana The Metropolitan Autonomous University ( Spanish: ''Universidad Autónoma Metropolitana'') also known as UAM, is a Mexican public research university. Founded in 1974 with the support of then-President Luis Echeverria Alvarez, the institution a ...
(Mexico). *
Zap Energy Zap Energy is an American privately held company that aims to commercialize fusion power through use of a sheared-flow-stabilized Z-pinch. The firm is based in Seattle Washington, with research facilities nearby in Everett, Washington, Everett and ...
Inc. (USA)


Crushing cans with the pinch effect

Many high-voltage electronics enthusiasts make their own crude electromagnetic forming devices. They use
pulsed power Pulsed power is the science and technology of accumulating energy over a relatively long period of time and releasing it instantly, thus increasing the instantaneous power. They can be used in some applications such as food processing, water treatme ...
techniques to produce a theta pinch able to crush an aluminium soft drink can using the
Lorentz force In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...
s created when large currents are induced in the can by the strong magnetic field of the primary coil. An electromagnetic aluminium can crusher consists of four main components: a
high-voltage High voltage electricity refers to electrical potential large enough to cause injury or damage. In certain industries, ''high voltage'' refers to voltage above a certain threshold. Equipment and conductors that carry high voltage warrant spe ...
DC
power supply A power supply is an electrical device that supplies electric power to an electrical load. The main purpose of a power supply is to convert electric current from a source to the correct voltage, electric current, current, and frequency to power ...
, which provides a source of
electrical energy Electrical energy is the energy transferred as electric charges move between points with different electric potential, that is, as they move across a voltage, potential difference. As electric potential is lost or gained, work is done changing the ...
, a large ''energy discharge''
capacitor In electrical engineering, a capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the condenser, a term st ...
to accumulate the electrical energy, a high voltage switch or
spark gap A spark gap consists of an arrangement of two Conductor (material), conducting electrodes separated by a gap usually filled with a gas such as air, designed to allow an electric spark to pass between the conductors. When the potential differenc ...
, and a robust coil (capable of surviving high magnetic pressure) through which the stored electrical energy can be quickly discharged in order to generate a correspondingly strong pinching magnetic field (see diagram below). In practice, such a device is somewhat more sophisticated than the schematic diagram suggests, including electrical components that control the current in order to maximize the resulting pinch, and to ensure that the device works safely. For more details, see the notes.


History

The first creation of a Z-pinch in the laboratory may have occurred in 1790 in Holland when
Martinus van Marum Martin(us) van Marum (; 20 March 1750 – 26 December 1837) was a Dutch physician, inventor, scientist and teacher, who studied medicine and philosophy in Groningen. Van Marum introduced modern chemistry in the Netherlands after the theories o ...
created an explosion by discharging 100
Leyden jar A Leyden jar (or Leiden jar, or archaically, Kleistian jar) is an electrical component that stores a high-voltage electric charge (from an external source) between electrical conductors on the inside and outside of a glass jar. It typically co ...
s into a wire. The phenomenon was not understood until 1905, when Pollock and BarracloughPollock J A and Barraclough S (1905) ''Proc. R. Soc. New South Wales'' 39 131 investigated a compressed and distorted length of copper tube from a
lightning rod A lightning rod or lightning conductor (British English) is a metal rod mounted on a structure and intended to protect the structure from a lightning strike. If lightning hits the structure, it is most likely to strike the rod and be conducted ...
after it had been struck by lightning. Their analysis showed that the forces due to the interaction of the large current flow with its own magnetic field could have caused the compression and distortion. A similar, and apparently independent, theoretical analysis of the pinch effect in liquid metals was published by Northrup in 1907. The next major development was the publication in 1934 of an analysis of the radial pressure balance in a static Z-pinch by Bennett (see the following section for details). Thereafter, the experimental and theoretical progress on pinches was driven by
fusion power Fusion power is a proposed form of power generation that would generate electricity by using heat from nuclear fusion reactions. In a fusion process, two lighter atomic nuclei combine to form a heavier nucleus, while releasing energy. Devices d ...
research. In their article on the "Wire-array Z-pinch: a powerful x-ray source for ICF", M G Haines ''et al.'', wrote on the "Early history of Z-pinches". :In 1946 Thompson and Blackman submitted a patent for a
fusion reactor Fusion power is a proposed form of power generation that would generate electricity by using heat from nuclear fusion reactions. In a fusion process, two lighter atomic nuclei combine to form a heavier nucleus, while releasing energy. Devices ...
based on a toroidal Z-pinch with an additional vertical magnetic field. But in 1954 Kruskal and Schwarzschild published their theory of MHD instabilities in a Z-pinch. In 1956, Kurchatov gave his famous Harwell lecture showing nonthermal neutrons and the presence of ''m'' = 0 and ''m'' = 1 instabilities in a deuterium pinch. In 1957 Pease and Braginskii independently predicted radiative collapse in a Z-pinch under pressure balance when in hydrogen the current exceeds 1.4 MA. (The viscous rather than resistive dissipation of magnetic energy discussed above and in would however prevent radiative collapse). In 1958, the world's first controlled thermonuclear fusion experiment was accomplished using a theta-pinch machine named Scylla I at the
Los Alamos National Laboratory Los Alamos National Laboratory (often shortened as Los Alamos and LANL) is one of the sixteen research and development Laboratory, laboratories of the United States Department of Energy National Laboratories, United States Department of Energy ...
. A cylinder full of deuterium was converted into a plasma and compressed to 15 million degrees Celsius under a theta-pinch effect. Lastly, at Imperial College in 1960, led by R Latham, the
Plateau–Rayleigh instability In fluid dynamics, the Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same total volume but less surface area per droplet. It is ...
was shown, and its growth rate measured in a dynamic Z-pinch.


Equilibrium analysis


One dimension

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 ...
three pinch geometries are commonly studied: the θ-pinch, the
Z-pinch In fusion power research, the Z-pinch (zeta pinch) is a type of plasma confinement system that uses an electric current in the plasma to generate a magnetic field that compresses it (see pinch). These systems were originally referred to simpl ...
, and the screw pinch. These are cylindrically shaped. The cylinder is symmetric in the axial (''z'') direction and the azimuthal (θ) directions. The one-dimensional pinches are named for the direction the current travels.


The θ-pinch

The θ-pinch has a magnetic field directed in the z direction and a large diamagnetic current directed in the θ direction. Using
Ampère's circuital law In classical electromagnetism, Ampère's circuital law, often simply called Ampère's law, and sometimes Oersted's law, relates the circulation of a magnetic field around a closed loop to the electric current passing through the loop. James ...
(discarding the displacement term) :\begin \vec &= B_z(r)\hat \\ \mu_0 \vec &= \nabla \times \vec \\ &= \frac \fracB_z(r) \hat - \fracB_z(r) \hat \end Since ''B'' is only a function of ''r'' we can simplify this to : \mu_0 \vec = -\fracB_z(r) \hat So ''J'' points in the θ direction. Thus, the equilibrium condition (\nabla p = \mathbf\times\mathbf) for the θ-pinch reads: : \frac \left( p + \frac \right) = 0 θ-pinches tend to be resistant to plasma instabilities; This is due in part to
Alfvén's theorem In ideal magnetohydrodynamics, Alfvén's theorem, or the frozen-in flux theorem, states that electrically conducting fluids and embedded magnetic fields are constrained to move together in the limit of large magnetic Reynolds numbers. It is name ...
(also known as the frozen-in flux theorem).


The Z-pinch

The Z-pinch has a magnetic field in the θ direction and a current ''J'' flowing in the ''z'' direction. Again, by electrostatic Ampère's law, :\begin \vec &= B_(r)\hat \\ \mu_0 \vec &= \nabla \times \vec \\ &= \frac\frac\left(r B_\theta(r)\right) \hat - \fracB_\theta(r) \hat \\ &= \frac\frac\left(r B_\theta(r)\right) \hat \end Thus, the equilibrium condition, \nabla p = \mathbf\times\mathbf, for the Z-pinch reads: : \frac \left( p + \frac \right) + \frac = 0 Although Z-pinches satisfy the MHD equilibrium condition, it is important to note that this is an unstable equilibrium, resulting in various instabilies such as the m = 0 instability ('sausage'), m = 1 instability ('kink'), and various other higher order instabilities.


The screw pinch

The screw pinch is an effort to combine the stability aspects of the θ-pinch and the confinement aspects of the Z-pinch. Referring once again to Ampère's law, :\nabla \times \vec = \mu_0 \vec But this time, the ''B'' field has a θ component ''and'' a ''z'' component :\begin \vec &= B_\theta \hat + B_z \hat \\ \mu_0 \vec &= \frac\frac\left(r B_\theta\right) \hat - \fracB_z \hat \end So this time ''J'' has a component in the ''z'' direction and a component in the θ direction. Finally, the equilibrium condition (\nabla p = \mathbf\times\mathbf) for the screw pinch reads: : \frac \left(p + \frac \right) + \frac = 0


The screw pinch via colliding optical vortices

The '' screw pinch '' might be produced in laser plasma by colliding optical vortices of ultrashort duration.A.Yu.Okulov. "Laser singular Theta-pinch", Phys.Lett.A, v.374, 4523-4527, (2010)
For this purpose optical vortices should be phase-conjugated. Optical phase conjugation and electromagnetic momenta The magnetic field distribution is given here again via Ampère's law: :\nabla \times \vec = \mu_0 \vec


Two dimensions

A common problem with one-dimensional pinches is the end losses. Most of the motion of particles is along the magnetic field. With the θ-pinch and the screw-pinch, this leads particles out of the end of the machine very quickly, leading to a loss of mass and energy. Along with this problem, the Z-pinch has major stability problems. Though particles can be reflected to some extent with
magnetic mirror A magnetic mirror, also known as a magnetic trap or sometimes as a pyrotron, is a type of magnetic confinement fusion device used in fusion power to trap high temperature Plasma (physics), plasma using magnetic fields. The mirror was one of the e ...
s, even these allow many particles to pass. A common method of beating these end losses, is to bend the cylinder around into a torus. Unfortunately this breaks θ symmetry, as paths on the inner portion (inboard side) of the torus are shorter than similar paths on the outer portion (outboard side). Thus, a new theory is needed. This gives rise to the famous
Grad–Shafranov equation The Grad–Shafranov equation ( H. Grad and H. Rubin (1958); Vitalii Dmitrievich Shafranov (1966)) is the equilibrium equation in ideal magnetohydrodynamics (MHD) for a two dimensional plasma, for example the axisymmetric toroidal plasma in a toka ...
. Numerical solutions to the Grad–Shafranov equation have also yielded some equilibria, most notably that of the
reversed field pinch A reversed-field pinch (RFP) is a device used to produce and contain near-thermonuclear plasmas. It is a toroidal pinch that uses a unique magnetic field configuration as a scheme to magnetically confine a plasma, primarily to study magnetic ...
.


Three dimensions

, there is no coherent analytical theory for three-dimensional equilibria. The general approach to finding such equilibria is to solve the vacuum ideal MHD equations. Numerical solutions have yielded designs for
stellarator A stellarator confines Plasma (physics), plasma using external magnets. Scientists aim to use stellarators to generate fusion power. It is one of many types of magnetic confinement fusion devices. The name "stellarator" refers to stars because ...
s. Some machines take advantage of simplification techniques such as helical symmetry (for example University of Wisconsin's Helically Symmetric eXperiment). However, for an arbitrary three-dimensional configuration, an equilibrium relation, similar to that of the 1-D configurations exists: : \nabla_\perp \left( p + \frac \right) - \frac\vec = 0 Where κ is the curvature vector defined as: : \vec = \left(\vec \cdot \nabla\right)\vec with ''b'' the
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
tangent to ''B''.


Formal treatment


The Bennett relation

Consider a cylindrical column of fully ionized quasineutral plasma, with an axial electric field, producing an axial current density, j, and associated azimuthal magnetic field, B. As the current flows through its own magnetic field, a pinch is generated with an inward radial force density of j x B. In a steady state with forces balancing: :\nabla p = \nabla(p_e+p_i) = \mathbf\times\mathbf where ∇''p'' is the magnetic pressure gradient, and ''p''e and ''p''i are the electron and ion pressures, respectively. Then using
Maxwell's equation Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circ ...
\nabla\times\mathbf = \mu_0\mathbf and the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
p=NkT, we derive: :2 N k(T_e + T_i) = \frac I^2 (the Bennett relation) where ''N'' is the number of electrons per unit length along the axis, ''Te'' and ''Ti'' are the electron and ion temperatures, ''I'' is the total beam current, and ''k'' is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
.


The generalized Bennett relation

The ''generalized Bennett relation'' considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω. Along the axis of the plasma cylinder flows a current density jz, resulting in an azimuthal magnetic field Βφ. Originally derived by Witalis, the generalized Bennett relation results in: :\begin \frac \frac = &W_ + \Delta W_ + \Delta W_ + \Delta W_k - \frac I^2 (a) \\ pt & - \fracG\overline^2 N^2 (a) + \frac\pi a^2 \epsilon_0 \left(E_r^2 (a) - E_\phi^2 (a) \right)\\ \end *where a current-carrying, magnetic-field-aligned cylindrical plasma has a radius ''a'', *''J''0 is the total moment of inertia with respect to the z axis, *''W''⊥kin is the
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
per unit length due to beam motion transverse to the beam axis *''W''Bz is the self-consistent Bz energy per unit length *''W''Ez is the self-consistent Ez energy per unit length *''W''k is thermokinetic energy per unit length *''I''(''a'') is the axial current inside the radius ''a'' (''r'' in diagram) *''N''(''a'') is the total number of particles per unit length *''E''r is the radial electric field *''E''φ is the rotational electric field The positive terms in the equation are expansional forces while the negative terms represent beam compressional forces.


The Carlqvist relation

The Carlqvist relation, published by Per Carlqvist in 1988, is a specialization of the generalized Bennett relation (above), for the case that the kinetic pressure is much smaller at the border of the pinch than in the inner parts. It takes the form :\frac I^2 (a) +\fracG\overline^2 N^2 (a) = \Delta W_ + \Delta W_k and is applicable to many space plasmas. The Carlqvist relation can be illustrated (see right), showing the total current (''I'') versus the number of particles per unit length (''N'') in a Bennett pinch. The chart illustrates four physically distinct regions. The plasma temperature is quite cold (''T''i = ''T''e = ''T''n = 20 K), containing mainly hydrogen with a mean particle mass 3×10−27 kg. The thermokinetic energy ''W''''k'' >> ''Ï€a''2 ''p''''k''(a). The curves, ΔWBz show different amounts of excess magnetic energy per unit length due to the axial magnetic field Bz. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside. Chart regions: (a) In the top-left region, the pinching force dominates. (b) Towards the bottom, outward kinetic pressures balance inwards magnetic pressure, and the total pressure is constant. (c) To the right of the vertical line Δ''W''''B''z = 0, the magnetic pressures balances the gravitational pressure, and the pinching force is negligible. (d) To the left of the sloping curve Δ''W''''B''z = 0, the gravitational force is negligible. Note that the chart shows a special case of the Carlqvist relation, and if it is replaced by the more general Bennett relation, then the designated regions of the chart are not valid. Carlqvist further notes that by using the relations above, and a derivative, it is possible to describe the Bennett pinch, the Jeans criterion (for gravitational instability, in one and two dimensions), force-free magnetic fields, gravitationally balanced magnetic pressures, and continuous transitions between these states.


References in culture

A fictionalized pinch-generating device was used in ''
Ocean's Eleven ''Ocean's Eleven'' is a 2001 heist comedy film directed by Steven Soderbergh and written by Ted Griffin. A remake of the 1960 film of the same name, it serves as the first installment in the ''Ocean's'' franchise. The film features an ense ...
'', where it was used to disrupt Las Vegas's power grid just long enough for the characters to begin their heist.


See also

*
Electromagnetic forming Electromagnetic forming (EM forming or magneforming) is a type of high-velocity, cold forming process for electrically conductive metals, most commonly copper and aluminium. The workpiece is reshaped by high-intensity pulsed magnetic fields tha ...
*
Explosively pumped flux compression generator An explosively pumped flux compression generator (EPFCG) is a device used to generate a high-power electromagnetic pulse by compressing magnetic flux using high explosives. EPFCGs are physically destroyed during operation, making them single-us ...
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Fusion power Fusion power is a proposed form of power generation that would generate electricity by using heat from nuclear fusion reactions. In a fusion process, two lighter atomic nuclei combine to form a heavier nucleus, while releasing energy. Devices d ...
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Madison Symmetric Torus The Madison Symmetric Torus (MST) is a reversed field pinch (RFP) physics experiment with applications to both fusion energy research and astrophysical plasmas. MST is located at the Center for Magnetic Self Organization (CMSO) at the Univers ...
(reversed field pinch)


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Examples of electromagnetically shrunken coins and crushed cansThe MAGPIE project at Imperial College London
is used to study wire array Z-pinch implosions. {{Authority control Fusion power Plasma phenomena Dutch inventions