Plasma Bound States in Grand Canonical and Mixed Representations

  • Werner EbelingEmail author
  • Vladimir E. Fortov
  • Vladimir Filinov
Part of the Springer Series in Plasma Science and Technology book series (SSPST)


Plasmas with deep bound states are not well described by density expansions. Here fugacity expansions are a very useful alternative to density expansions. The quantum-statistical theory is formulated in the grand canonical ensemble. This provides representations of the pressure and the densities in terms of the fugacities, which serve as implicit variables. This has already been demonstrated for real gases in Chap.  2, where we showed the equivalence with chemical descriptions. Fugacity expansions are, in principle, completely equivalent to density expansions. They are just a useful alternative, but with a quite different range of convergence, and different applications. For systems with deep bound states, fugacity expansions are more quickly convergent and very useful for EOS calculations. Furthermore, we develop mixed representations which combine the advantages of density and fugacity expansions.


  1. A. Alastuey, V. Ballenegger, Pressure of a partially ionized hydrogen gas: numerical results from exact low temperature expansions. Contrib. Plasma Phys. 50, 4653 (2010)CrossRefGoogle Scholar
  2. A. Alastuey, V. Ballenegger, Contrib. Plasma Phys. 52, 95 (2012)Google Scholar
  3. A. Alastuey, A. Perez, Virial expansion of the equation of state of a quantum plasma. Europhys. Lett. 20, 19 (1992)ADSCrossRefGoogle Scholar
  4. A. Alastuey, V. Ballenegger, F. Cornu et al., Exact results for thermodynamics of the hydrogen plasma: low-temperature expansions beyond Saha theory. J. Stat. Phys. 130, 1119–1176 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. A. Alastuey, V. Ballenegger, W. Ebeling, Comment on direct linear term in the equation of state of plasmas. Phys. Rev. E 62, 047101–1 (2015)Google Scholar
  6. S. Arndt, W.D. Kraeft, J. Seidel, Phys. Stat. Sol. B 194, 601 (1996)Google Scholar
  7. V. Ballenegger, Ann. Phys. (Berlin) 254, 103 (2012)Google Scholar
  8. G.P. Bartsch, Ebeling, Quantum statistical fugacity expansions for partially ionized plasmas in equilibrium, Contr. Plasma Phys. 11, 393–403 (1971); see also Contrib. Plasma Phys. 15, 25–32 (1975)Google Scholar
  9. H.A. Bethe, E.E. Salpeter, A relativistic equation for bound state problems. Phys. Rev. 82, 309 (1951)MathSciNetzbMATHGoogle Scholar
  10. D. Beule, W. Ebeling, A. Förster, H. Juranek, S. Nagel, R. Redmer, G. Röpke, Equation of state for hydrogen below 10 000 K: from the fluid to the plasma. Phys. Rev. B 59, 14177 (1999)ADSCrossRefGoogle Scholar
  11. D. Beule, W. Ebeling, R. Redmer, G. Röpke, Electrical conductivity in dense hydrogen fluid and metal plasmas. Control. Plasma Phys. 39, 25–28 (1999)Google Scholar
  12. D. Beule,W. Ebeling, A. Förster, H. Juranek, R. Redmer, G. Röpke, Isentropes and Hugoniots for dense hydrogen and deuterium. Phys. Rev. E 63, 060202 R (2001)Google Scholar
  13. L.S. Brown, L.G. Yaffe, Effective field theory of highly ionized plasmas. Phys. Rep. 340, 1 (2001)ADSCrossRefzbMATHGoogle Scholar
  14. W. Däppen, Accurate and versatile equations of state for the sun and sunlike stars. Astrophys. Space Sci. 328, 139146 (2010)CrossRefzbMATHGoogle Scholar
  15. W. Däppen, D. Mihalas, D.G. Hummer, Astrophys. J. 332, 261 (1988)ADSCrossRefGoogle Scholar
  16. H.E. DeWitt, M. Schlanges, A.Y. Sakakura, W.D. Kraeft, Low density expansion of the equation of state for a quantum electron gas. Phys. Lett. A 197, 326 (1995)ADSCrossRefGoogle Scholar
  17. M.W.C. Dharma-Wardana, F. Perrot, Level shifts, continuum lowering, and the mobility edge in dense plasmas. Phys. Rev. A 45, 5883 (1992)ADSCrossRefGoogle Scholar
  18. W. Ebeling, Statistical derivation of the mass action law of interacting gases and plasmas. Physica 73, 573–584 (1974)ADSCrossRefGoogle Scholar
  19. W. Ebeling, Free energy and ionization of dense plasma of light elements, Contr. Plasma Phys. 29, 238–242 (1989); 30, 553–561 (1990)Google Scholar
  20. W. Ebeling, H.D. Hoffmann, G. Kelbg, Quantenstatistik des Hochtemperaturplasmas. Contrib. Plasma Phys. 7, 233–248 (1967)Google Scholar
  21. W. Ebeling, W.-D. Kraeft, K. Kilimann, D. Kremp, Coexisting phases in an electron–hole plasma, Phys. Stat. Sol. 78, 241–253 (1976)Google Scholar
  22. W. Ebeling, W.-D. Kraeft, D. Kremp, Nonideal Plasmas (Proc. ICPIG XIII, Invited papers, Berlin, 1977)Google Scholar
  23. W. Ebeling, W.D. Kraeft, D. Kremp, Theory of Bound States and Ionization Equilibrium in Plasmas and Solids (Akademie-Verlag, Berlin, 1976) (Extended Russ. translation Mir, Moscow, 1979)Google Scholar
  24. W. Ebeling, W.D. Kraeft, D. Kremp, G. Röpke, Energy levels in hydrogen plasmas and the Planck–Larkin partition function–a comment. Astrophys. J. 290, 24–27 (1985)ADSCrossRefGoogle Scholar
  25. W. Ebeling, A. Förster, R. Redmer, T. Rother, M. Schlanges, ICPIG 18, Invited Lectures, Univ. of Wales (1987)Google Scholar
  26. W. Ebeling, Free energy and ionization in dense plasmas of the light elements. Control. Plasma Phys. 30, 53–561 (1990)Google Scholar
  27. W. Ebeling, S. Hilbert, On Saha’s equation for partially ionized plasmas and Onsager’s bookkeeping rule, Eur. Phys. J. D 20, 93–101 (2002)Google Scholar
  28. W. Ebeling, S. Hilbert, H. Krienke, On Bjerrum’s mass action law for electrolytes and Onsager’s bookkeeping rule. J. Mol. Liq. 96/97, 409–423 (2002)Google Scholar
  29. W. Ebeling, H. Hache, H. Juranek, R. Redmer, G. Röpke, Pressure ionization and transitions in dense hydrogen. Contrib. Plasma Phys. 45, 160–167 (2005)ADSCrossRefGoogle Scholar
  30. W. Ebeling, D. Blaschke, R. Redmer, H. Reinholz, G. Röpke, The influence of Pauli blocking effects on the properties of dense hydrogen. J. Phys. A Math. Theor. 42, 214033 (2009), arXiv:0810.3336v2
  31. W. Ebeling, D. Blaschke, R. Redmer, H. Reinholz, G. Röpke, Pauli blocking effects and Mott transition in dense hydrogen. In: Redmer et al. (2010)Google Scholar
  32. W. Ebeling, W.D. Kraeft, G. Röpke, Bound states in Coulomb systems—old problems and new solutions. Contrib. Plasma Phys. 52, 7–16 (2012); Ann. Physik (Berlin) 524, 311–326 (2012)Google Scholar
  33. W. Ebeling, Work of Baimbetov on nonideal plasmas and developments, Contr. Plasma Phys. 56, 163–175 (2016)Google Scholar
  34. G. Ecker, W. Weizel, Zustandssumme und effektive Ionisierungsspannung eines Atoms im Inneren des Plasmas. Ann. Phys. (Leipzig) 452, 126 (1956)Google Scholar
  35. M. Fisher, D.M. Zuckerman, Exact thermodynamic formulation of chemical association. J. Chem. Phys. 109, 7961–7981 (1998)ADSCrossRefGoogle Scholar
  36. H.L. Friedman, Ionic Solution Theory (Interscience, New York, London, 1962)Google Scholar
  37. H.L. Friedman, W. Ebeling, Theory of a fluid of interacting and reacting particles. Rostocker Physikal. Manuskripte 4, 35–51 (1979)Google Scholar
  38. A. Förster, T. Kahlbaum, W. Ebeling, Equation of state and pase diagram of fluid helium in the region of partial ionization. Laser Part. Beams 10, 253–262 (1992)Google Scholar
  39. H.C. Graboske, D.J. Harwood, F.J. Rogers, Phys. Rev. 186, 210 (1969)ADSCrossRefGoogle Scholar
  40. H.R. Griem, Spectral Line Broadening in Plasmas (Academic Press, New York, 1974)Google Scholar
  41. T.L. Hill, Statistical Mechanics (Mc Graw Hill, New York, 1956). Russ. transl, Moskva, 1960zbMATHGoogle Scholar
  42. S. Ichimaru, Statistical Plasma Physics (Addison-Wesley, Redwood City, 1992)Google Scholar
  43. G. Kelbg, W. Ebeling, Quantum statistics of thermal plasmas in equilibrium (in Russ.), 1–2, printed by Inst. Theor. Phys. Ukrain. Acad. Sci, Kiev (1970)Google Scholar
  44. T. Kahlbaum, The quantum-diffraction term in the free energy for Coulomb plasma, effective potential approach. J. Phys. France 10, 5-455 (2000)Google Scholar
  45. A.L. Khomkin, I.A. Mulenko: High Temp. 41, 275 (2003),
  46. A.L. Khomkin, A.S. Shumikhin, Features of the chemical models of nonideal atomic plasma at high temperatures. Plasma Phys. Rep. 34, 251–256 (2008)ADSCrossRefGoogle Scholar
  47. A. Khomkin, A. Shumikhin, I. Mulenko, Basic chemical models of partially ionized plasma, Czech. J. Phys. 54, Suppl. C 143, 1b (2004)Google Scholar
  48. K. Kilimann, W. Ebeling, Energy gap and line shifts for H-like ions in dense plasmas. Z. Naturforsch. A 45, 613–617 (1990)Google Scholar
  49. Y.L. Klimontovich, Statistical theory of nonequilibrium processes in plasmas, Izdat MGU 1964 (Engl. Pergamon, Oxford, 1967)Google Scholar
  50. Y.L. Klimontovich, Statistical Physics, In Russ. (Nauka, 1982); Engl. (Harwood, New York, 1986)Google Scholar
  51. W.D. Kraeft, D. Kremp, W. Ebeling, Complex representations of the quantum statistical second virial coefficient. Phys. Lett. A 29, 466 (1969)ADSCrossRefGoogle Scholar
  52. W.D. Kraeft, D. Kremp, W. Ebeling, G. Röpke, Quantum Statistics of Charged Particle Systems (Pergamon Press, New York, 1986)CrossRefGoogle Scholar
  53. W.-D. Kraeft, J. Vorberger, D.O. Gericke, M. Schlanges, Thermodynamic functions for plasmas beyond Montroll-Ward. Contrib. Plasma Phys. 47, 253 (2007)ADSCrossRefGoogle Scholar
  54. D. Kremp, M. Schlanges, W.D. Kraeft, Quantum Statistics of Nonideal Plasmas (Springer, Berlin, 2005)Google Scholar
  55. L.P. Kudrin, Statistical Physics of Plasmas (in Russ.) (Nauka, Moskva, 1974)Google Scholar
  56. J.E. Mayer, Theory of ionic solutions. J. Chem. Phys. 18, 1426–1436 (1950)ADSCrossRefGoogle Scholar
  57. E. Montroll, J. Ward, Quantum statistics of interacting particles. Phys. Fluids 1, 55 (1958)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. R. Redmer, B. Holst, F. Hensel (eds.), Metal to nonmetal transitions (Springer, Berlin, 2010)zbMATHGoogle Scholar
  59. F.J. Rogers, Statistical mechanics of Coulomb gases of arbitrary charge. Phys. Rev. A 10, 2441–2456 (1974)Google Scholar
  60. F.J. Rogers, Statistical mechanics of Coulomb gases of arbitrary charge. Phys. Rev. A 38, 5007 (1988)Google Scholar
  61. G. Röpke, K. Kilimann et al., Phys. Lett. A 69, 329 (1978)Google Scholar
  62. D. Saumon, G. Chabrier, Phys. Rev. Lett. 62, 2397 (1989)ADSCrossRefGoogle Scholar
  63. J. Seidel, S. Arndt, W.D. Kraeft, Energy spectrum of hydrogen atoms in dense plasmas. Phys. Rev. E 52, 5387 (1995)ADSCrossRefGoogle Scholar
  64. A.N. Starostin, V.C. Roerich, R.M. More, How correct is the EOS of weakly nonideal hydrogen plasmas? Contrib. Plasma Phys. 43, 369–372 (2003)ADSCrossRefGoogle Scholar
  65. O. Theimer , P. Kepple, Statistical Mechanics of a Partially Ionized Hydrogen Plasma, Phys. Rev. A 1, 957 (1970)Google Scholar
  66. A.A. Vedenov, A.I. Larkin, Equation of state of plasmas (in Russ.). Zh. eksp. teor. Fiz. 36, 1133 (1959)zbMATHGoogle Scholar
  67. J. Vorberger, D.O. Gericke, Phys. Plasmas 16, 082702 (2009)ADSCrossRefGoogle Scholar
  68. J. Vorberger, D.O. Gericke, Th Bornath, M. Schlanges, Phys. Rev. E 81, 046404 (2010)ADSCrossRefGoogle Scholar
  69. J. Vorberger, D.O. Gericke, W.D. Kraeft, Equation of state for high density hydrogen, Poster SCCS 2011, (2011), arXiv:1108.4826
  70. V.S. Vorobjov, I.A. Mulenko, A.L. Khomkin, The importance of excited states in the thermodynamics of partially ionized plasma, High Temp. 38, 509–514 (2000), Teplofiz. Vyssokikh Temp. 38, 533–538 (2000)Google Scholar
  71. R. Zimmermann, K. Kilimann, W.D. Kraeft, D. Kremp, G. Röpke, Phys. Stat. Sol. (b) 90, 175 (1978)Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  • Werner Ebeling
    • 1
    Email author
  • Vladimir E. Fortov
    • 2
  • Vladimir Filinov
    • 3
  1. 1.Institut für PhysikHumboldt Universität BerlinBerlinGermany
  2. 2.Russian Academy of SciencesMoscowRussia
  3. 3.Joint Institute for High TemperaturesRussian Academy of SciencesMoscowRussia

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