On some necessary conditions for p-11B ignition in the hot spots of a plasma focus

  • Andrea Di Vita
Regular Article


Recently, it has been predicted that hydrogen-boron (p-11B) nuclear fusion may attain ignition in the hot spots observed in a plasma focus (PF) pinch, due to their huge values of particle density, magnetic field and (reportedly) ion temperature. Accordingly, large magnetic fields should raise electronic Landau levels, thus reducing collisional exchange of energy from ion to electrons and Bremsstrahlung losses. Moreover, large particle densities, together with ion viscous heating, should allow fulfilment of Lawson criterion and provide effective screening of cyclotron radiation. We invoke both well-known, empirical scaling laws of PF physics, Connor-Taylor scaling laws, Poynting balance of electromagnetic energy and the balance of generalised helicity. We show that the evolution of PF hot spots is a succession of relaxed states, described by the double Beltrami solutions of Hall-MHD equations of motion. We obtain some necessary conditions for ignition, which are violated in most realistic conditions. Large electromagnetic fields in the hot spot accelerate electrons at supersonic velocities and trigger turbulence, which raises electric resistivity and Joule heating, thus spoiling further compression. Ignition is only possible if a significant fraction of the Bremsstrahlung-radiated power is reflected back into the plasma. Injection of angular momentum decreases the required reflection coefficient.


Plasma Physics 


  1. 1.
    A. Bernard, H. Bruzzone, P. Choi, H. Chuaqui, V. Gribkov, J. Herrera, K. Hirano, A. Krej íč, S. Lee, C. Luo, F. Mezzetti, M. Sadowski, H. Schmidt, K. Ware, C.S. Wong, W. Zoita, J. Moscow Phys. Soc. 8, 93 (1998)Google Scholar
  2. 2.
    L. Soto, Plasma Phys. Control. Fusion 47, A361 (2005)ADSCrossRefGoogle Scholar
  3. 3.
    J.P. Rager, in Unconventional Approaches to Fusion, edited by B. Brunelli, G. Leotta (Plenum Press, New York, 1982)Google Scholar
  4. 4.
    A. Di Vita, J. Plasma Phys. 50, 1 (1993)ADSCrossRefGoogle Scholar
  5. 5.
    S. Lee, Appl. Phys. Lett. 95, 151503 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    H. Herold, A. Jerzykiewicz, M. Sadowski, H. Schmidt, Nucl. Fusion 29, 1255 (1989)CrossRefGoogle Scholar
  7. 7.
    M. Sadowski, H. Herold, H. Schmidt, M. Shakatre, Phys. Lett. A 105, 117 (1984)ADSCrossRefGoogle Scholar
  8. 8.
    G.J. Morales, J.E. Maggs, A.T. Burke, J.R. Penano, Plasma Phys. Control. Fusion 41, A519 (1999)ADSCrossRefGoogle Scholar
  9. 9.
    S. Cable, T. Tajima, Phys. Rev. A 46, 3413 (1992)ADSCrossRefGoogle Scholar
  10. 10.
    J.W. Mather, Phys. Fluids 8, 366 (1965)ADSCrossRefGoogle Scholar
  11. 11.
    L. Jakubowski, M.J. Sadowski, Braz. J. Phys. 32, 1871 (2002)CrossRefGoogle Scholar
  12. 12.
    W.H. Bostick, V. Nardi, W.J. Prior, J. Plasma Phys. 8, 7 (1972)ADSCrossRefGoogle Scholar
  13. 13.
    K.N. Koshelev, V.I. Krauz, N.G. Reshetnyak, R.G. Saluvadze, Y.V. Sidel’nikov, E.Y. Khautiev, J. Phys. D 21, 1827 (1988)ADSCrossRefGoogle Scholar
  14. 14.
    J.M. Byaley, G. Decker, W. Kies, M. Maelzig, F. Mueller, P. Roewekamp, J. Westheide, Y.V. Sidel’nikov, J. Appl. Phys. 69, 613 (1991)ADSCrossRefGoogle Scholar
  15. 15.
    W. Bostick, Int. J. Fusion Energy 3, 68 (1985)Google Scholar
  16. 16.
    W.H. Bostick, V. Nardi, W.J. Prior, F. Rodriguez-Trelles, in Proceeding 2nd Intl. Conf. Pulsed High Beta Plasmas, Garching, Germany, 1972 Google Scholar
  17. 17.
    F. Casanova, A. Tarifeño-Saldivia, F. Veloso, C. Pavez, A. Clausse, L. Soto, J. Fusion Energy 31, 3 (2012)CrossRefGoogle Scholar
  18. 18.
    E. Angeli, A. Tartari, M. Frignani, V. Molinari, D. Mostacci, F. Rocchi, M. Sumini, Nucl. Technol. Radiat. Protect. 20, 33 (2005), URLCrossRefGoogle Scholar
  19. 19.
    V. Nardi, in Proceeding 2nd Intl. Conf. Pulsed High Beta Plasmas, Garching, Germany, 1972 Google Scholar
  20. 20.
    J.J.E. Herrera, F. Castillo, On The Magnetohydrodynamic Evolution Of The m = 0Instability In The Dense Z-Pinch (1998),
  21. 21.
    R.K. Rout, A. Shyam, Pramana J. Phys. 37, 93 (1991)ADSCrossRefGoogle Scholar
  22. 22.
    H.R. Yousefi, T. Haruki, J.I. Sakai, A. Lumanta, K. Masugata, Phys. Lett. A 373, 27 (2009)CrossRefGoogle Scholar
  23. 23.
    W.H. Bostick, V. Nardi, W.J. Prior, R. Trelles, in Proceeding 4th Conf. Contr. Fusion Plasma Phys. Grenoble, France, 1972 (Commissariat à l’Énergie atomique, 1972), Vol. 1, p. 70Google Scholar
  24. 24.
    J.S. Brzosko et al., Nuclear Reactivity In Submillimetric Domains Of Focussed Discharges, in 16th EPS, Venezia, Italy, 1989 Google Scholar
  25. 25.
    E.Yu. Khautiev, V.I. Krauz, N.G. Reshetnyak, R.G. Salukvadze, A.A. Batenyuk, A.Ch. Chkhaidze, K.N. Koshelev, Yu.V. Sidelnikov, V.A. Gribkov, O.N. Krokhin, in Plasma Physics and Controlled Nuclear Fusion Research 1988 (International Atomic Energy Agency, Vienna, 1989), Vol. 2, pp. 579–586,
  26. 26.
    V. Nardi, L. Bilbao, A. Bortolotti, J.S. Brzosko, C. Powell, D. Zeng, in Plasma Physics and Controlled Nuclear Fusion Research 1988 (International Atomic Energy Agency, Vienna, 1989), Vol. 2, pp. 743–750, VOL2.pdf
  27. 27.
    Plasma Focus Apparatus With Field Distortion Elements, European Patent Office Patent No. EP0312587B1Google Scholar
  28. 28.
    E.J. Lerner, Laser Particle Beams 4, 193 (1986)CrossRefGoogle Scholar
  29. 29.
    E.J. Lerner, in Conf. Current Trends in International Fusion Research, Washington, USA, 2003 Google Scholar
  30. 30.
    E.J. Lerner, S.K. Murali, A. Haboub, J. Fusion Energy 30, 367 (2011)CrossRefADSGoogle Scholar
  31. 31.
    E.J. Lerner, S.K. Murali, D. Shannon, A.M. Blake, F. Van Roessel, Phys. Plasmas 19, 032704 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    S. Son, Reaction rates and other processes in dense plasma UMI Number: 3180070 (2005)Google Scholar
  33. 33.
    J.M. Martinez-Val et al., Phys. Lett. A 216, 142 (1996)ADSCrossRefGoogle Scholar
  34. 34.
    S. Abolhasani, M. Habibi, R. Amrollahi, J. Fusion Energy 32, 189 (2012)CrossRefADSGoogle Scholar
  35. 35.
    S. Stave, M.W. Ahmed, R.H. France III, S.S. Henshaw, B. Muller, B.A. Perdue, R.M. Prior, M.C. Spraker, H.R. Weller, Phys. Lett. B 696, 26 (2011)ADSCrossRefGoogle Scholar
  36. 36.
    National Nuclear Data Center Web Site,
  37. 37.
    W.L. Harries, J.H. Lee, Plasma Phys. 20, 95 (1978)ADSCrossRefGoogle Scholar
  38. 38.
    F. Castillo-Mejía, M. Milanese, M. Moroso, J. Pouzo, M. Santiago, in Proceeding Intl. Conf. Plasma Physics, European Physical Society, Praha, Czech Republic, 1998 Google Scholar
  39. 39.
    N.K. Neog, S.R. Mohanty, E. Hotta, J. Appl. Phys. 99, 013302 (2006)ADSCrossRefGoogle Scholar
  40. 40.
    Y. Ono, M. Yamada, T. Akao, T. Tajima, R. Matsumoto, Phys. Rev. Lett. 76, 3328 (1996)ADSCrossRefGoogle Scholar
  41. 41.
    M.G. Haines, P.D. LePell, C.A. Coverdale, B. Jones, C. Deeney, J.P. Apruzese, Phys. Rev. Lett. 96, 075003 (2006)ADSCrossRefGoogle Scholar
  42. 42.
    S. Son, N.J. Fisch, Phys. Lett. A 356, 65 (2006)ADSCrossRefGoogle Scholar
  43. 43.
    J.R. McNally, Nucl. Fusion 15, 344 (1975)ADSCrossRefGoogle Scholar
  44. 44.
    G.S. Miller, E.E. Salpeter, I. Wassermann, Astrophys. J. 314, 215 (1987)ADSCrossRefGoogle Scholar
  45. 45.
    U. Das, B. Mukhopadyay, Phys. Rev. Lett. 110, 071102 (2013)ADSCrossRefGoogle Scholar
  46. 46.
    S. Son, N.J. Fisch, Phys. Lett. A 329, 76 (2004)ADSCrossRefGoogle Scholar
  47. 47.
    M. Halper, Iranian Team To Collaborate With US Company On Nuclear Fusion Project, The Guardian, 25th May 2012,
  48. 48.
    A. Di Vita, Eur. Phys. J. D 54, 451 (2009)ADSCrossRefGoogle Scholar
  49. 49.
    D. Butler, Nature 480, 19 (2011)ADSCrossRefGoogle Scholar
  50. 50.
    D. Clery, Science 337, 1444 (2012)ADSCrossRefGoogle Scholar
  51. 51.
    L. Turner, IEEE Trans. Plasma Sci. 14, 849 (1986)ADSCrossRefGoogle Scholar
  52. 52.
    S. Lee, Radiation Enhancement And Applications - Scaling From The UNU/ICTP PF, Invited Paper, in International Meeting on Frontiers of physics, Satellite Meeting: 12 Years of UNU/ICTP PFF - A Review, 1998, Kuala Lumpur (International Centre for Theoretical Physics, Italy), pre-print 33–46,
  53. 53.
    Th.J. Dolan, Fusion Research (Pergamon, New York, 1980)Google Scholar
  54. 54.
    S. Lee, in Proceedings 1st Cairo Conf. Plasma Physics & Applications, Cairo, Egypt, 2003, edited by H.J. Kunze, T. El-Khalafawy, H. Hegazy, volume 34 of International Cooperation Bilateral Seminars (Bilateral Seminars of the International Bureau, Forschungszentrum Jülich GmbH), pp. 27–33,
  55. 55.
    J.A. Shercliff, A Textbook of Magnetohydrodynamics (Pergamon, New York, 1965)Google Scholar
  56. 56.
    A. Sestero, B.V. Robouch, S. Podda, Plasma Phys. 22, 1039 (1990)ADSCrossRefGoogle Scholar
  57. 57.
    R. Deutsch, W. Grauf, H. Herold, H. Schmidt, Plasma Phys. 25, 833 (1983)ADSCrossRefGoogle Scholar
  58. 58.
    H.A.B. Bodin, Nucl. Energy 29, 57 (1990)Google Scholar
  59. 59.
    H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, 1906)Google Scholar
  60. 60.
    A.H. Boozer, Phys. Fluids B 4, 2845 (1992)ADSCrossRefGoogle Scholar
  61. 61.
    E.T. Jaynes, Annu. Rev. Phys. Chem. 31, 579 (1980)ADSCrossRefGoogle Scholar
  62. 62.
    F. Herrmann, Eur. J. Phys. 7, 130 (1986)CrossRefGoogle Scholar
  63. 63.
    L.D. Landau, E. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960)Google Scholar
  64. 64.
    J.B. Taylor, Phys. Rev. Lett. 33, 1139 (1974)ADSCrossRefGoogle Scholar
  65. 65.
    M.R. Brown, J. Plasma Phys. 57, 203 (1997)ADSCrossRefGoogle Scholar
  66. 66.
    P.M. Bellan, Spheromaks (Imperial College Press, London, 2000)Google Scholar
  67. 67.
    J.B. Taylor, Commun. Plasma Phys. 14, 127 (1991)Google Scholar
  68. 68.
    Z. Yoshida, S.M. Mahajan, Phys. Rev. Lett. 88, 095001 (2002)ADSCrossRefGoogle Scholar
  69. 69.
    S.K.H. Auluck, Phys. Plasmas 18, 032508 (2011)ADSCrossRefGoogle Scholar
  70. 70.
    M.S. Janaki, B. Dasgupta, Phys. Plasmas 19, 032113 (2012)ADSCrossRefGoogle Scholar
  71. 71.
    J.W. Connor, J.B. Taylor, Nucl. Fusion 17, 1047 (1977)ADSCrossRefGoogle Scholar
  72. 72.
    H. Krompholz, G. Herziger, in Chaos and Order in Nature, edited by H. Haken (Springer-Verlag, Berlin, 1981)Google Scholar
  73. 73.
    G.A. Navratil, in 43rd American Physical Society Division of Plasma Physics Meeting, Long Beach, CA, USA, 2001,
  74. 74.
  75. 75.
    V.V. Vikhrev et al., Sov. J. Plasma Phys. 8, 688 (1982)ADSGoogle Scholar
  76. 76.
    P. Antsiferov, D. Franz, R. Pfau, A. Rupasov, H. Schmidt, D. Schulz, in Proceeding 2nd Intl. Conf. High-Beta Plasmas, Garching, Germany, 1972 Google Scholar
  77. 77.
    S. Goudarzi, A. Raeisdana, J. Fusion Energy 29, 103 (2010)CrossRefADSGoogle Scholar
  78. 78.
    A. Di Vita, Eur. Phys. J. D 56, 205 (2010)ADSCrossRefGoogle Scholar
  79. 79.
    N. Ryusuke, Y. Zensho, H. Takaya, J. Plasma Fusion Res. Ser. 6, 130 (2004)Google Scholar
  80. 80.
    S.M. Mahajan, S. Yoshida, Phys. Rev. Lett. 81, 4863 (1998)ADSCrossRefGoogle Scholar
  81. 81.
    V.I. Krauz, Phys. Control. Fusion 48, B221 (2006)CrossRefGoogle Scholar
  82. 82.
    M. Iqbal, P.K. Shukla, Phys. Plasmas 19, 033517 (2012)ADSCrossRefGoogle Scholar
  83. 83.
    D. Kagan, S.M. Mahajan, Mon. Not. R. Astron. Soc. 406, 1140 (2010)ADSGoogle Scholar
  84. 84.
    S. Ohsaki, N.L. Shatashvili, Z. Yoshida, S.M. Mahajan, ApJ 559, L61 (2001)ADSCrossRefGoogle Scholar
  85. 85.
    L. Jakubowski, M. Sadowski, in Proceeding 26th EPS Conf. Contr. Fus. Plasma Phys., Maastricht, The Netherlands, 1999 Google Scholar
  86. 86.
    R. Thomas, Y. Yang, G.H. Miley, F.B. Mead, AIP Conf. Proc. 746, 536 (2005)ADSCrossRefGoogle Scholar
  87. 87.
    K.D. Joensen, P. Gorenstein, P. Høghøj, J. Susini, F. Christensen, J. Wood, G. Gutman, E. Ziegler, A. Freund, Nucl. Instrum. Methods Phys. Res. B 132, 221 (1997)ADSCrossRefGoogle Scholar

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© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.DICCAUniversità di GenovaGenovaItaly

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