Pinch (plasma physics)

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Lightning bolts illustrating electromagnetically pinched plasma filaments
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Lightning bolts illustrating electromagnetically pinched plasma filaments

A pinch is a phenomenon that may occur in a current-carrying plasma whose magnetic field constricts or "pinches" the plasma, and is associated with filamenation and particle beams. Examples are readily seen in electrical discharges such as a lightning bolt[1], in the aurora[2], current sheets[3], and solar flares[4]. It is also known as the "Bennett pinch"[5] (after Willard Harrison Bennett), "Z-pinch", "electromagnetic pinch"[6], "magnetic pinch"[7], "pinch effect"[8] and "plasma pinch[9]"

Pinches not only occur naturally, but are produced in the laboratory by scientists in the nuclear industry, and immortalised in metal by adventuous amateurs with potentially lethal electrical equipment and a soft drinks can (see below). The phenomenon has been recognised in the experiments of 18th century Dutch scientist Martinus van Marum, and subsequently explained in the early 20th century by Australians James Arthur Pollock and Samuel Barraclough (see below). Further work by American scientist Willard Bennett and Swedish plasma physicist Per Carlqvist provided a formal mathematical treatment of the pinch.

Contents

[edit] Pinch production and types

Pinches are created in the laboratory in equipment related to nuclear fusion, such as the Z-pinch machine and high-energy physics, such as the dense plasma focus. Pinches may also become unstable resulting in instabilities[10], and generate radiation across the electromagnetic spectrum, including radio waves, x-rays[11] and gamma rays[12], and also neutrons[13] and synchrotron radiation[14]. Types of pinches, that may differ in geometry and operating forces[15], include the Cylindrical pinch, Inverse pinch, Orthogonal pinch effect, Reversed field pinch, Sheet pinch, Screw pinch[16] (also called stabalized z-pinch, or θ-z pinch)[17], Theta pinch (or thetatron[18]), Toroidal pinch, Ware pinch [19] and Z-pinch.

Pinches are used to generate X-rays, the intense magnetic fields generated are used in electromagnetic forming of metals (they have been demonstrated in crushing aluminium soft drinks cans[20]), and applications to particle beams[21] including particle beam weapons[22], and astrophysics[23].

[edit] Pinch and crush cans

Pinched aluminium can, produced from a pulsed magnetic field created by rapidly discharging 2 kilojoules from a high voltage capacitor bank into a 3-turn coil of heavy gauge wire. Source: Bert Hickman, Stoneridge Engineering.
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Pinched aluminium can, produced from a pulsed magnetic field created by rapidly discharging 2 kilojoules from a high voltage capacitor bank into a 3-turn coil of heavy gauge wire. Source: Bert Hickman, Stoneridge Engineering.

Many high-voltage electronics enthusiasts make their own devices using pulsed power techniques to produce an electromagnetic pinch capable of crushing an aluminium soft drink can. (Warning! High-voltage electric shocks may be lethal).

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 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
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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 [24].

Sam Barros's PowerLabs' can crusher cost about $500, uses the second largest semiconductor in production, generates a magnetic field a quarter of million times the strength of the Earth's magnetic field, from 22 megawatts of electricity, enough to kill 250 people or power 220,000 100W light bulbs. [25]

[edit] History

The IEEE Emblem of the Institute of Electrical and Electronics Engineers shows the basic features of an azimuthal magnetic pinch (depicted by the arc-shaped arrow), and a vertical current depicted by the vertical arrow (in the Z-direction)
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The IEEE Emblem of the Institute of Electrical and Electronics Engineers shows the basic features of an azimuthal magnetic pinch (depicted by the arc-shaped arrow), and a vertical current depicted by the vertical arrow (in the Z-direction)[26]
A stream of water pinching into droplets has been suggested as an analogy of the electromagnetic pinch . The surface tension which causes the water to constrict and bead, is analogous to a magnetic field which has been suggested as the cause of pinching in bead lightning . Like plasma, the water accelerates and moves faster through the pinches. The morphology (shape) is similar to the so-called sausage instability in plasma (Click image for detail)
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A stream of water pinching into droplets has been suggested as an analogy of the electromagnetic pinch [27]. The surface tension which causes the water to constrict and bead, is analogous to a magnetic field which has been suggested as the cause of pinching in bead lightning [28]. Like plasma, the water accelerates and moves faster through the pinches. The morphology (shape) is similar to the so-called sausage instability in plasma (Click image for detail)

In 1984, fusion program pioneer R. S. "Bas" Pease, wrote in an article "The Electromagnetic Pinch: From Pollock to the Joint European Torus"[29]:

"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. Watson-Munro. 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)[30] 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."

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"[31]:

"The z-pinch has a remarkably old history. In 1790 Martinus van Marum in Holland discharged 1 kJ of energy from 100 Leyden jars into a 1m long wire, causing an explosion [39][32]. In 1905, Pollock and Barraclough in Australia [40][33] proposed the pinch effect to explain the implosion of a copper tube used as a lightning conductor. In 1907 Northrupp proposed a continuous flow liquid metal z-pinch [41][34].
    A most important analytic result was derived by Bennett [42][35] in 1934 describing the radial pressure balance in a static z-pinch,
8πNikB(Ti + ZTe) = μ0I2. (2)
On the basis of this energy dependence per unit length depending on the square of the current, the idealized yield of x-rays is considered to scale as I2.
    In 1946 Thompson and Blackman [43] submitted a patent for a fusion reactor based on a toroidal z-pinch [43][36] with an additional vertical magnetic field. But in 1954 Kruskal and Schwarzchild [44][37] 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 [45][38]. In 1957 Pease [46][39] and Braginskii [47][40] 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 [32][41] would however prevent radiative collapse). Lastly, at Imperial College in 1960, led by R Latham, the Rayleigh–Taylor (RT) instability was shown, and its growth rate measured in a dynamic z-pinch [48][42].
    Thus many of the phenomena that dominate in the wire-array pinch had their origins 40 to 200 years ago, namely exploding wires, shell implosions and RT instabilities."

[edit] Formal treatment

[edit] The Bennett Relation

Consider a cylindrical column of plasma (fully ionized), 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 j x B. In a steady state with forces balancing:

∇p = ∇(pe + pi) = j x Β

where ∇p is the magnetic pressure gradient, pe and pi is the electron and ion pressures. Then using Maxwell's equation ∇ x B = μ0 j and the ideal gas law p = N k T, we derive:

2 N k(T_e + T_i) = \frac{{\mu_0}} {4 \pi} 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 Boltzmann's constant.

[edit] The Generalized Bennett Relation

The Generalized Bennett Relation considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω
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The Generalized Bennett Relation considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω

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 a toroidal magnetίc field Βφ. Originally derived by Witalis [43] the Generalized Bennett Relation results in [44]:

\frac{1}{4} \frac{\partial^2 J_0}{\partial t^2} = W_{\perp k i n} + \Delta W_{E_z} + \Delta W_{B_z} + \Delta W_k - \frac{{\mu_0}} {8 \pi} I^2 (a)

.\qquad \qquad - \frac{1}{2}G\overline{m}^2 N^2 (a) + \frac{1}{2}\pi a^2 \epsilon_0 \left(E_r^2 (a) - E_\phi^2 (a) \right )

  • where a cuπent-carrying, magnetic-field-aligned cylindrical ρlasma has a radius a,
  • J0 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
  • WBz is the self-consistent Bz energy per unit length
  • WEz is the self-consistent Ez energy per unit length
  • Wk 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
  • Er is the radial electric field
  • Eφ is the rotational electric field

The positive terms in the equation are expansional forces while the negative terms represent beam compressional forces.

[edit] The Carlqvist Relation

\frac{{\mu_0}} {8 \pi} I^2 (a) +\frac{1}{2}G\overline{m}^2 N^2 (a) = \Delta W_{B_z} + \Delta W_k
The Bennett pinch showing the total current (I) versus the number of particle per unit length (N). The chart illustrates four physically distinct regions. The plasma temperature is 20 K, the mean particle density 3x10-27kg, and ΔWBz is the excess magnetic energy per unit length due to the axis magnetic field Bz. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.
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The Bennett pinch showing the total current (I) versus the number of particle per unit length (N). The chart illustrates four physically distinct regions. The plasma temperature is 20 K, the mean particle density 3x10-27kg, and ΔWBz is the excess magnetic energy per unit length due to the axis magnetic field Bz. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.

This is the Carlqvist Relation (name after Swedish physicist Per Carlqvist), a derivation of the Generalized Bennett Relation, which applies to a non-rotating cyclindrical plasma in a steady-state condition, whose kinetic pressure is much smaller at the border of the pinch than in the inner parts, and consequently has applications to space plasmsa (see below).

The Carlqvist Relation can be illustrated (see right), showing the total current (I) versus the number of particle per unit length (N) in a Bennett pinch. The chart illustrates four physically distinct regions. The plasma temperature is quite cold (Ti = Te = Tn = 20 K), containing mainly hydrogen with a mean particle density 3x10-27kg. The thermokinetic energy Wk >> π a2 pk(a). The curves, ΔWBz show different amounts of excess magnetic energy per unit length due to the axis 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 veritical line ΔWBz=0, the magnetic pressures balances the gravitational pressure, and the pinching force is negligible. (d) To the left of the sloping curve ΔWBz=0, the gravitational force is neglibible. Note that the chart shows a special case of the Carlqvist relation, and if it is replace 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 [45], in one and two dimensions), force-free magnetic fields, gravitationally balanced magnetic pressures, and continuous transitions between these states. [46].

[edit] Trivia

Pinch-generating equipment 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.

[edit] Notes

  1. ^ 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. 240-242.
  2. ^ Galperin, Iu. I.; Zelenyi, L. M.; Kuznetsova, M. M. "Pinching of field-aligned currents as a possible mechanism for the formation of raylike auroral forms" (1986) Kosmicheskie Issledovaniia (ISSN 0023-4206), vol. 24, Nov.-Dec. 1986, p. 865-874. In Russian.
  3. ^ Syrovatskii, S. I. "Pinch sheets and reconnection in astrophysics" (1981) In Annual review of astronomy and astrophysics. Volume 19. (A82-11551 02-90) Palo Alto, CA, Annual Reviews, Inc., 1981, p. 163-229
  4. ^ 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
  5. ^ See for example, Buneman, O., "The Bennett Pinch" (1961) Plasma Physics, Edited by James E. Drummond. LOC 60-12766. Publ. McGraw-Hill, Inc., New York, 1961, p.202
  6. ^ Lee, S., "Energy balance and the radius of electromagnetically pinched plasma columns" (1983) Plasma Physics, Volume 25, Issue 5, pp. 571-576 (1983).
  7. ^ 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
  8. ^ Severnyi, A. B., "On the Appearance of Cosmics Rays in the Pinch Effect in Solar Flares" (1959) Soviet Astronomy, Vol. 3, p.887
  9. ^ 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
  10. ^ Hardee, P. E., "and pinching instability of supersonic expanding jets in extragalactic radio sources" (1982) Astrophysical Journal, Part 1, vol. 257, June 15, 1982, p. 509-526
  11. ^ Pereira, N. R., et al, "[X rays from z-pinches on relativistic electron-beam generators]" (1988) Journal of Applied Physics (ISSN 0021-8979), vol. 64, Aug. 1, 1988, p. R1-R27
  12. ^ Wu, Mei; Chen, Li; Li, Ti-Pei, "Polarization in Gamma-Ray Bursts Produced by Pinch Discharge" (2005) Chinese Journal of Astronomy & Astrophysics, Vol. 5, p. 57-64
  13. ^ Anderson, Oscar A., et al, "Neutron Production in Linear Deuterium Pinches" (1958) Physical Review, vol. 110, Issue 6, pp. 1375-1387
  14. ^ 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. 259-346.
  15. ^ Carlqvist, Per, "Cosmic electric currents and the generalized Bennett relation" (1988) Astrophysics and Space Science (ISSN 0004-640X), vol. 144, no. 1-2, May 1988, p. 73-84
  16. ^ Srivastava, K. M.; Vyas, D. N., "Non-linear analysis of the stability of the screw pinch", (1982) Astrophysics and Space Science, vol. 86, no. 1, Aug. 1982, p. 71-89
  17. ^ See "[http://silas.psfc.mit.edu/introplasma/chap4.html#tth_sEc4.7 MHD Equilibria" in Introduction to Plasma Physics by I.H.Hutchinson (2001)
  18. ^ See Dictionary of Material Science and High Energy Physics p.315 ISBN 0-8493-2889-6
  19. ^ Helander, P. et al "The effect of non-inductive current drive on tokamak transport" (2005) Plasma Physics and Controlled Fusion, Volume 47, Issue 12B, pp. B151-B163
  20. ^ For example, see "Electromagnetic Crusher"
  21. ^ 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
  22. ^ Andre Gsponer, "Physics of high-intensity high-energy particle beam propagation in open air and outer-space plasmas" (2004) http://arxiv.org/abs/physics/0409157
  23. ^ Peratt, Anthony L., "The role of particle beams and electrical currents in the plasma universe" (1988) Laser and Particle Beams (ISSN 0263-0346), vol. 6, Aug. 1988, p. 471-491.
  24. ^ 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
  25. ^ Sam Borros's PowerLabs' Solid State Can Crusher
  26. ^ See also the IEEE History Center, "Evolution of the IEEE Logo" March 1963; see also the comments in "Laboratory Astrophysics"
  27. ^ Trubnikov, Boris A., "A new hypothesis of cosmic ray generation in plasma pinches" (1992) IEEE Transactions on Plasma Science (ISSN 0093-3813), vol. 20, no. 6, p. 898-904.
  28. ^ "The PLASMAK™ Configuration and Ball Lightning" (PDF) presented at the International Symposium on Ball Lightning; July 1988
  29. ^ R. S. Pease, "Pollock Memorial Lecture for 1984 delivered at the University of Sydney, 28 November, 1984"
  30. ^ Pollock J A and Barraclough S, 1905 Proc. R. Soc. New South Wales 39 131
  31. ^ M G Haines, T W L Sanford and V P Smirnov, "Wire-array z-pinch: a powerful x-ray source for ICF" (2005) Plasma Phys. Control. Fusion 47 B1-B11 (online in full, click PDF).
  32. ^ [39] van Marum M 1790 Proc. 4th Int. Conf. on Dense Z-Pinches (Vancouver 1997) (Am. Inst. Phys. Woodbury, New York, 1997) Frontispiece and p ii
  33. ^ [40] Pollock J A and Barraclough S 1905 Proc. R. Soc. New South Wales 39 131
  34. ^ [41] 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 non-electrolytic, liquid conductor contained in a trough, that the liquid contracted in cross-section 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'.
  35. ^ [42] Bennett W H "Magnetically Self-Focussing Streams" 1934 Phys. Rev. 45 890
  36. ^ [43] 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
  37. ^ [44] Kruskal M D and Schwarzchild "Some Instabilities of a Completely Ionized Plasma" 1954 Proc. R. Soc. Lond. A 223 348
  38. ^ [45] Kurchatov I V 1957 J. Nucl. Energy 4 193
  39. ^ [46] Pease R S "Equilibrium Characteristics of a Pinched Gas Discharge Cooled by Bremsstrahlung Radiation" 1957 Proc. Phys. Soc. Lond. 70 11
  40. ^ [47] Braginskii S I 1957 Zh. Eksp. Teor. Fiz 33 645; Braginskii S I 1958 Sov. Phys.—JETP 6 494
  41. ^ [32] Haines M G et al 2005 Phys. Rev. Lett. submitted; see also EPS Conf. on Plasma Physics 2004 (London, UK) paper 73
  42. ^ [48] 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
  43. ^ Witalis, E. A. "Plasma-physical aspects of charged-particle beams" (1981) Physical Review A - General Physics, 3rd Series, vol. 24, Nov. 1981, p. 2758-2764
  44. ^ Anthony L . Peratt, "Physics of the Plasma Universe", 1992 Springer-Verlag, ISBN 0-387-97575-6
  45. ^ J. H. Jeans, "The stability of a spherical nebula" Phil. Trans. R. Soc. Lond. A 199 (1902)
  46. ^ Carlqvist, Per, "Cosmic electric currents and the generalized Bennett relation" (1988) Astrophysics and Space Science (ISSN 0004-640X), vol. 144, no. 1-2, May 1988, p. 73-84

[edit] See also

[edit] External Links

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