From Wikipedia, the free encyclopedia
A pinch is the compression of an electrically
conducting filament by magnetic forces. The conductor is
usually a plasma, but could also be a solid or
liquid metal. In a zpinch, the current
is axial (in the z direction in a cylindrical coordinate
system) and the magnetic field azimuthal; in a
thetapinch, the current is azimuthal (in the
theta direction in cylindrical coordinates) and the magnetic field
is axial. The phenomenon may also be referred to as a "Bennett
pinch"^{[1]} (after
Willard Harrison Bennett),
"electromagnetic pinch",^{[2]}
"magnetic pinch",^{[3]} "pinch
effect"^{[4]} or
"plasma pinch".^{[5]}
Pinches occur naturally in electrical discharges such as lightning bolts,^{[6]} the aurora,^{[7]} current sheets,^{[8]} and solar flares.^{[9]} They
are also produced in the laboratory, primarily for research into fusion power, but
also by hobbyists in crushing aluminum cans.
Pinch
production and types
A section of the crushed lightning rod studied by Pollock and
Barraclough.
^{[10]}
The rod is in the collection of the School of Physics,
University of Sydney,
Australia.
Pinches are created in the laboratory in equipment related to nuclear fusion,
such as the Zpinch machine
and highenergy physics, such as the dense plasma focus. Pinches may also
become unstable,^{[11]} and
generate radiation across the electromagnetic spectrum,
including radio
waves, xrays^{[12]} and
gamma
rays,^{[13]} and
also neutrons^{[14]} and
synchrotron radiation.^{[15]} Types
of pinches, that may differ in geometry and operating forces,^{[16]}
include the Cylindrical pinch, Inverse pinch, Orthogonal pinch
effect, Reversed field pinch, Sheet pinch,
Screw pinch^{[17]} (also
called stabilized zpinch, or θz pinch),^{[18]} Theta
pinch (or thetatron^{[19]}), Toroidal pinch, Ware pinch^{[20]} and
Zpinch.
Pinches are used to generate Xrays, and the intense magnetic fields
generated are used in electromagnetic forming of
metals (they have been demonstrated in crushing aluminium soft
drinks cans^{[21]}).
They have applications to particle beams^{[22]}
including particle beam weapons,^{[23]} and
astrophysics.^{[24]}
History
The first creation of a zpinch in the laboratory may have
occurred in 1790 in Holland when Martinus van Marum created an
explosion by discharging 100 Leyden jars into a wire.^{[26]} The
phenomenon was not understood until 1905, when Pollock and
Barraclough^{[10]}
investigated a compressed and distorted length of copper tube from
a lightning rod
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.^{[27]} A
similar, and apparently independent, theoretical analysis of the
pinch effect in liquid metals was published by Northrupp in
1907.^{[28]}. The
next major development was the publication in 1934 of an analysis
of the radial pressure balance in a static zpinch by Bennett^{[29]} (See
the following section for details.)
Thereafter, the experimental and theoretical progress on pinches
was driven by fusion
power research. In their article on the "Wirearray zpinch: a
powerful xray source for ICF", M G Haines et
al., wrote on the "Early history of zpinches":^{[30]}
 In 1946 Thompson and Blackman [43] submitted a patent for a fusion reactor based on a toroidal zpinch
[43]^{[31]} with
an additional vertical magnetic field. But in 1954 Kruskal and
Schwarzschild [44]^{[32]}
published their theory of MHD instabilities in a zpinch. 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 [45].^{[33]} In
1957 Pease [46]^{[34]} and
Braginskii [47]^{[35]}
independently predicted radiative collapse in a zpinch 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 [32]^{[36]} would
however prevent radiative collapse). Lastly, at Imperial College in
1960, led by R Latham, the PlateauRayleigh instability was shown, and
its growth rate measured in a dynamic zpinch [48].^{[37]}"
Configurations
One Dimensional
configurations
There are three analytic one dimensional configurations
generally studied in plasma physics. These are the θpinch, the Zpinch, and the Screw Pinch.
All of the classic one dimensional pinches are cylindrically
shaped. Symmetry is assumed in the axial (z) direction and
in the azimuthal (θ) direction. It is traditional to name a
onedimensional pinch after the direction in which the current
travels.
The θpinch
A sketch of the θPinch Equilibrium. The z directed magnetic field
(shown in purple) corresponds to a θ directed plasma current (shown
in yellow).
The θpinch has a magnetic field traveling in the z direction.
Using Ampère's law (discarding the displacement
term)
Since B is only a function of r we can
simplify this to
So J points in the θ direction. θpinches tend to be
resistant to plasma instabilities. This is due in part to the
frozen in flux theorem, which is beyond the scope of this
article.
The ZPinch
A sketch of the zPinch Equilibrium. A θ directed magnetic field
(shown in purple) corresponds to a z directed plasma current (shown
in yellow).
The ZPinch has a magnetic field in the θ direction. Again, by
electrostatic Ampere's Law
So J points in the z direction. Since
particles in a plasma basically follow magnetic field lines,
Zpinches lead them around in circles. Therefore, they tend to have
excellent confinement properties.
The Screw Pinch The Screw pinch is an effort to
combine the stability aspects of the θpinch and the confinement
aspects of the Zpinch. Referring once again to Ampere's Law
But this time, the B field has a θ component
and a z component
So this time J has a component in the z
direction and a component in the θ direction.
Two
Dimensional Equilibria
A
toroidal coordinate system in common use in plasma
physics. The red arrow indicates the
poloidal
direction (θ) and the blue arrow indicates the
toroidal direction (φ)
A common problem with onedimensional equilibria based machines
is end losses. As mentioned above, most of the motion of particles
in a plasma is directed along the magnetic field. With the θpinch
and the screwpinch, this leads particles to the end of the machine
very quickly (as the particles are typically moving quite fast).
Additionally, the Zpinch has major stability problems. Though
particles can be reflected to some extent with magnetic
mirrors, even these allow many particles to pass. The most
common method of mitigating this effect 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 GradShafranov equation.
The one dimensional equilibria provide the inspiration for some
of the toroidal configurations. An example of this is the ZETA
device at Culham England (which also operated as a Reversed
Field Pinch). The most well recognized of these devices is the
toroidal version of the screw pinch, the Tokamak.
Numerical solutions to the GradShafranov equation have also
yielded some equilibria, most notably that of the Reversed
Field Pinch.
Three Dimensional
Equilibria
There does not exist a coherent analytical theory for
threedimensional equilibria. The general approach to finding three
dimensional equilibria is to solve the vacuum ideal MHD equations.
Numerical solutions have yielded designs for stellarators. Some machines take advantage
of simplification techniques such as helical symmetry (for example
University of Wisconsin's Helically Symmetric eXperiment).
Formal
treatment
A stream of water pinching into droplets has been
suggested as an analogy to the electromagnetic pinch.
^{[38]} The
gravity accelerates freefalling water which causes the water
column to constrict. Then
surface tension breaks the narrowing
water column into droplets (not shown here) (see
PlateauRayleigh instability), which is
analogous to the
magnetic field which has been suggested
as the cause of pinching in bead lightning.
^{[39]} The
morphology (shape) is similar to the socalled sausage
instability in
plasma.
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:
 ∇p = ∇(p_{e} + p_{i}) =
j x Β
where ∇p is the magnetic pressure gradient,
p_{e} and p_{i} is the electron and ion pressures.
Then using Maxwell's equation ∇ x
B = μ_{0} j and the ideal gas law p
= N k T, we derive:

(The Bennett Relation)
where N is the number of electrons per unit length
along the axis, T_{e} and T_{i}
are the electron and ion temperatures, I is the total beam
current, and k is the Boltzmann constant.
The Generalized Bennett
Relation
The Generalized Bennett Relation considers a currentcarrying
magneticfieldaligned cylindrical plasma pinch undergoing rotation
at angular frequency ω
The Generalized Bennett Relation considers a
currentcarrying magneticfieldaligned cylindrical plasma pinch
undergoing rotation at angular frequency ω. Along the axis of the
plasma cylinder flows a current density j_{z}, resulting in
a toroidal magnetίc field Β_{φ}. Originally derived by
Witalis,^{[40]} the
Generalized Bennett Relation results in:^{[41]}
 where a currentcarrying, magneticfieldaligned 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 per unit length due to
beam motion transverse to the beam axis
 W_{B}z is the selfconsistent B_{z} energy per
unit length
 W_{E}z is the selfconsistent E_{z} 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,^{[42]} 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
and is applicable to many space plasmas.
The Bennett pinch showing the total current (I) versus the number
of particles per unit length (N). The chart illustrates four
physically distinct regions. The plasma temperature is 20 K, the
mean particle mass 3×10
^{27} kg, and ΔW
_{Bz} is
the excess magnetic energy per unit length due to the axial
magnetic field B
_{z}. The plasma is assumed to be
nonrotational, and the kinetic pressure at the edges is much
smaller than inside.
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} >> π a^{2} p_{k}(a).
The curves, ΔW_{Bz} show different amounts of excess
magnetic energy per unit length due to the axial magnetic field
B_{z}. The plasma is assumed to be nonrotational, and the
kinetic pressure at the edges is much smaller than inside.
Chart regions: (a) In the topleft 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
ΔW_{Bz}=0, the magnetic pressures balances the
gravitational pressure, and the pinching force is negligible. (d)
To the left of the sloping curve ΔW_{Bz}=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 Jean's
criterion (for gravitational instability,^{[43]} in
one and two dimensions), forcefree magnetic fields,
gravitationally balanced magnetic pressures, and continuous
transitions between these states.
Crushing cans with the
pinch effect
Pinched aluminium can, produced from a
pulsed magnetic field
created by rapidly discharging 2 kilojoules from a high voltage
capacitor bank into a
3turn coil of heavy gauge wire. Source: Bert Hickman,
Stoneridge Engineering.
Many highvoltage electronics enthusiasts make their own devices
using pulsed power
techniques to produce a theta pinch capable of crushing an
aluminium soft drink can by pressure of strong magnetic field.
An electromagnetic aluminium can crusher consists of four main
components (1) A high
voltage DC
power supply which
provides a source of electrical energy
(2) A large energy discharge capacitor to accumulate the electrical energy
(3) A high voltage switch or spark gap and (4) 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).
Electromagetic pinch "can crusher": schematic diagram
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.^{[44]}
Sam Barros's can crusher cost about $500, and uses a large SCR
and a 900 Volt capacitor bank
storing about 3000 Joules of
energy. For a very short time, it generates a magnetic field B~5T
(250,000 times the strength of the Earth's magnetic field) which
has magnetic pressure P ~ 100 atm. Rate of energy conversion (from
electric into magnetic and back) in this device is about 22 megawatts.^{[45]}
Depictions
A fictionalized pinchgenerating device was used in Ocean's Eleven, where
it was used to disrupt Las Vegas's power grid just long enough for
the characters to begin their heist.^{[46]}
References
 ^
See for example, Buneman, O., "The Bennett Pinch" (1961)
Plasma Physics, Edited by James E. Drummond. LOC 6012766.
Publ. McGrawHill, Inc., New York, 1961, p.202
 ^
Lee, S., "Energy balance and the radius
of electromagnetically pinched plasma columns" (1983)
Plasma Physics, Volume 25, Issue 5, pp. 571–576
(1983).
 ^
Schmidt, Helmut, "Formation of a Magnetic Pinch
in InSb and the Possibility of Population Inversion in the
Pinch" (1966) Physical Review, vol. 149, Issue 2, pp.
564–573
 ^
Severnyi, A. B., "On the Appearance of Cosmics
Rays in the Pinch Effect in Solar Flares" (1959) Soviet
Astronomy, Vol. 3, p.887
 ^
Zueva, N. M.; Solov'ev, L. S.; Morozov, A. I. "Nonlinear instability of
plasma pinches" (1976) Journal of Experimental and
Theoretical Physics Letters, Vol. 23, p.256
 ^
Rai, J.; Singh, A. K.; Saha, S. K, "Magnetic field within the
return stroke channel of lightning" (1973) Indian Journal
of Radio and Space Physics, vol. 2, Dec. 1973, p.
240242.
 ^
Galperin, Iu. I.; Zelenyi, L. M.; Kuznetsova, M. M. "Pinching of fieldaligned
currents as a possible mechanism for the formation of raylike
auroral forms" (1986) Kosmicheskie Issledovaniia (ISSN
00234206), vol. 24, Nov.Dec. 1986, p. 865874. In Russian.
 ^
Syrovatskii, S. I. "Pinch sheets and reconnection
in astrophysics" (1981) In Annual review of astronomy and
astrophysics. Volume 19. (A8211551 0290) Palo Alto, CA,
Annual Reviews, Inc., 1981, p. 163229
 ^
Airapetyan, V. S.; Vikhrev, V. V.; Ivanov, V. V.; Rozanova, G. A.
"Pinch Mechanism of Energy
Release of Stellar Flares" (1990) Astrophsyics (Tr.
Astrofizika) v.32 No.3 Nov. p.230 1990
 ^ ^{a}
^{b}
Pollock J A and Barraclough S, 1905 Proc. R. Soc. New South
Wales 39 131
 ^
Hardee, P. E., "Helical and pinching
instability of supersonic expanding jets in extragalactic radio
sources" (1982) Astrophysical Journal, Part 1, vol.
257, June 15, 1982, p. 509526
 ^
Pereira, N. R., et al., "[X rays from zpinches on
relativistic electronbeam generators]" (1988) Journal of
Applied Physics (ISSN 00218979), vol. 64, Aug. 1, 1988, p.
R1R27
 ^
Wu, Mei; Chen, Li; Li, TiPei, "Polarization in GammaRay
Bursts Produced by Pinch Discharge" (2005) Chinese Journal
of Astronomy & Astrophysics, Vol. 5, p. 5764
 ^
Anderson, Oscar A., et al., "Neutron Production in Linear
Deuterium Pinches" (1958) Physical Review, vol. 110,
Issue 6, pp. 1375–1387
 ^
Peratt, A.L., "Synchrotron radiation from
pinched particle beams", (1998) Plasma Physics: VII Lawpp 97:
Proceedings of the 1997 Latin American Workshop on Plasma Physics,
Edited by Pablo Martin, Julio Puerta, Pablo Martmn, with reference
to Meierovich, B. E., "Electromagnetic collapse.
Problems of stability, emission of radiation and evolution of a
dense pinch" (1984) Physics Reports, Volume 104, Issue
5, p. 259346.
 ^
Carlqvist, Per, "Cosmic electric currents and
the generalized Bennett relation" (1988) Astrophysics and
Space Science (ISSN 0004640X), vol. 144, no. 12, May 1988,
p. 7384
 ^
Srivastava, K. M.; Vyas, D. N., "Nonlinear analysis of the
stability of the screw pinch", (1982) Astrophysics and
Space Science, vol. 86, no. 1, Aug. 1982, p. 7189
 ^
See "MHD Equilibria" in
Introduction to Plasma Physics by I.H.Hutchinson (2001)
 ^
See Dictionary of Material Science and High Energy Physics
p.315 ISBN
0849328896
 ^
Helander, P. et al. "The effect of noninductive
current drive on tokamak transport" (2005) Plasma Physics
and Controlled Fusion, Volume 47, Issue 12B, pp.
B151B163
 ^
For example, see "Electromagnetic
Crusher"
 ^
Ryutov, D. D.; Derzon, M. S.; Matzen, M. K, "The physics of fast Z
pinches" (2000) Reviews of Modern Physics, vol. 72,
Issue 1, pp. 167–223
 ^
Andre Gsponer, "Physics of highintensity
highenergy particle beam propagation in open air and outerspace
plasmas" (2004) http://arxiv.org/abs/physics/0409157
 ^
Peratt, Anthony L., "The role of particle beams and
electrical currents in the plasma universe" (1988) Laser
and Particle Beams (ISSN 02630346), vol. 6, Aug. 1988, p.
471491.
 ^
See also the IEEE History Center, "Evolution of the IEEE
Logo" March 1963; see also the comments in "Laboratory
Astrophysics"
 ^
van Marum M 1790 Proc. 4th Int. Conf. on Dense ZPinches
(Vancouver 1997) (Am. Inst. Phys. Woodbury, New York, 1997)
Frontispiece and p ii
 ^
R. S. Pease, "The
Electromagnetic Pinch: From Pollock to the Joint
European Torus", "Pollock Memorial Lecture for
1984 delivered at the University of Sydney, 28 November, 1984":
This review of the electromagnetic pinch starts with an exhibit
taken from Pollock's work, carefully preserved and drawn to
attention of modern research by Professor C. WatsonMunro. It is a
compressed and distorted length of copper tube originally part of
the lightning conductor on the Hartley Vale kerosene refinery in
New South Wales. It was known to have been struck by lightning.
Pollock and Barraclough (1905) from the Department of Mechanical
Engineering at Sydney University carried out an analysis to see
whether or not the compression could have arisen from the flow of
electric current. They concluded that the compressive forces, due
to the interaction of the large current flow with its own magnetic
field could have been responsible for the compression and
distortion. As far as I know, this is the first identified piece of
observational data on the electromagnetic pinch; and the first
theoretical discussion of the effect.
 ^
Northrupp E F 1907 "Some Newly Observed
Manifestations of Forces in the Interior of an Electric
Conductor" (1907) Phys. Rev. 24 474. He wrote: "Some
months ago, my friend, Carl Hering, described to me a surprising
and apparently new phenomenon which he had observed. He found, in
passing a relatively large alternating current through a
nonelectrolytic, liquid conductor contained in a trough, that the
liquid contracted in crosssection and flowed up hill lengthwise of
the trough... Mr. Hering suggested the idea that this contraction
was probably due to the elastic action of the lines of magnetic
force which encircle the conductor... As the action of the forces
on the conductor is to squeeze or pinch it, he jocosely called it
the 'pinch phenomenon'.
 ^
W.H.Bennett, "Magnetically SelfFocussing
Streams", Phys. Rev. 45 890
(1934)
 ^
M G Haines, T W L Sanford and V P Smirnov, "Wirearray zpinch: a powerful
xray source for ICF" (2005) Plasma Phys. Control.
Fusion 47 B1B11 (online in full, click PDF).
 ^
Thompson G P and Blackman M 1946 British Patent 817681. Haines M G
1996 "Historical Perspective: Fifty
years of controlled fusion research" Plasma Phys. Control.
Fusion 38 643
 ^
Kruskal M D and Schwarzchild "Some Instabilities of a
Completely Ionized Plasma" 1954 Proc. R. Soc. Lond. A
223 348
 ^
Kurchatov I V 1957 J. Nucl. Energy 4 193
 ^
Pease R S "Equilibrium Characteristics
of a Pinched Gas Discharge Cooled by Bremsstrahlung Radiation"
1957 Proc. Phys. Soc. Lond. 70 11
 ^
Braginskii S I 1957 Zh. Eksp. Teor. Fiz 33 645; Braginskii
S I 1958 Sov. Phys.—JETP 6 494
 ^
Haines M G et al. 2005 Phys. Rev. Lett.
submitted; see also EPS Conf. on Plasma Physics 2004 (London, UK)
paper 73
 ^
Curzon F L et al. "Experiments on the Growth
Rate of Surface Instabilities in a Linear Pinched Discharge"
1960 Proc. R. Soc. Lond. A 257 386
 ^
Trubnikov, Boris A., "A new hypothesis of cosmic
ray generation in plasma pinches" (1992) IEEE Transactions
on Plasma Science (ISSN 00933813), vol. 20, no. 6, p.
898904.
 ^
"The PLASMAK Configuration and Ball Lightning" (PDF) presented at the
International Symposium on Ball Lightning; July 1988
 ^
Witalis, E. A. "Plasmaphysical aspects of
chargedparticle beams" (1981) Physical Review A  General
Physics, 3rd Series, vol. 24, Nov. 1981, p. 2758–2764
 ^
Anthony L . Peratt, "Physics of the Plasma Universe", 1992
SpringerVerlag, ISBN 0387975756
 ^
Carlqvist, Per, "Cosmic electric currents and
the generalized Bennett relation" (1988) Astrophysics and
Space Science (ISSN 0004640X), vol. 144, no. 12, May 1988,
p. 7384
 ^
J. H. Jeans, "The stability of a spherical
nebula" Phil. Trans. R. Soc. Lond. A 199 (1902)
 ^
Examples of electromagnetic pinch can crushers can be found at (a)
Bob LaPointe's site on High Voltage Devices and
Experiments (b) Tristran's Electromagnetic Can
Crusher (including schematic) (c) Sam Borros's Solid State Can Crusher
 ^
Sam Borros's PowerLabs' Solid State Can Crusher
 ^
"The ConArtist Physics of
'Ocean's Eleven'.". American Physical Society. March, 2002. http://www.aps.org/publications/apsnews/200203/oceanseleven.cfm.
See also
External
links