Non-ideality and Deep Bound States in Plasmas

  • Werner Ebeling
  • Thorsten Pöschel
Part of the Lecture Notes in Physics book series (LNP, volume 953)


With increasing density of plasmas, non-ideality effects become more and more apparent. This is of particular importance for plasmas with deep bound states, that is, when the thermal energy is smaller than or of the same order as the ground state energies of the atoms in the plasma. For hydrogen this is the case already for T < 105 K. Then, except for very low density, the convergence of the expansions becomes worse and we have to consider higher order terms. Sometimes even the transition to other expansion parameters, as the fugacity is recommended.


  1. Abrikosov, A.A.., L.P. Gor’kov, and I.E. Dzyaloshinskii. 1962. Quantum Field Theory Techniques in Statistical Physics (in Russian). Moscow: Fizmatgiz.Google Scholar
  2. Abrikosov, A.A., L.P. Gor’kov, and I.E. Dzyaloshinskii. 1965. Methods of Quantum Field Theory in Statistical Physics. Oxford: Pergamon Press.Google Scholar
  3. Alastuey, A., and V. Ballenegger. 2010. Pressure of a Partially Ionized Hydrogen Gas: Numerical results From Exact Low Temperature Expansions. Contributions to Plasma Physics 50: 46–53.ADSCrossRefGoogle Scholar
  4. Alastuey, A., and A. Perez. 1992. Virial Expansion of the Equation of State of a Quantum Plasma. Europhysics Letters 20: 19–24.ADSCrossRefGoogle Scholar
  5. Alastuey, A., and A. Perez. 1996. Virial Expansions for Quantum Plasmas: Fermi-Bose Statistics. Physical Review E 53: 5714–5728.ADSCrossRefGoogle Scholar
  6. Alder, B., and T. Wainwright. 1957. Phase Transition for a Hard Sphere System. The Journal of Chemical Physics 27: 1208–1209.ADSCrossRefGoogle Scholar
  7. Alastuey, A., V. Ballenegger, F. Cornu, and Ph.A. Martin. 2008. Exact Results for Thermodynamics of the Hydrogen Plasma: Low-Temperature Expansions Beyond Saha Theory. Journal of Statistical Physics 130: 1119–1176.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. Alastuey, A., V. Ballenegger, and W. Ebeling. 2015. Comment on ‘Direct Linear Term in the Equation of State of Plasmas’ by Kraeft et al. (2015). Physical Review E 92 (see also Kraeft et al. 2015), 047101.Google Scholar
  9. Bartsch, G.P., and W. Ebeling. 1971. Quantum Statistical Fugacity Expansions for Partially Ionized Plasmas in Equilibrium. Contributions to Plasma Physics 11.5: 393–403.Google Scholar
  10. Binder, K., ed. 1979. Monte Carlo Methods in Statistical Physics. Berlin: Springer.Google Scholar
  11. Brown, L.S., and L.G. Yaffe. 2001. Effective Field Theory of Highly Ionized Plasmas. Physics Reports 340: 1–164.ADSzbMATHCrossRefGoogle Scholar
  12. Ceperley, D.M. 1995. Path Integrals in the Theory of Condensed Helium. Reviews of Modern Physics 67: 279–356.ADSCrossRefGoogle Scholar
  13. Deutsch, C. 1977. Nodal Expansion in a Real Matter Plasma. Physics Letters A 60: 317–318.ADSCrossRefGoogle Scholar
  14. DeWitt, H.E. 1961. Thermodynamic Functions of a Partially Degenerate, Fully Ionized Gas. International Journal of Nuclear Energy Part C: Plasma Physics 2: 27–45.CrossRefGoogle Scholar
  15. DeWitt, H.E. 1976. Asymptotic Form of the Classical One-Component Plasma Fluid Equation of State. Physical Review A 14: 1290–1293.ADSCrossRefGoogle Scholar
  16. DeWitt, H.E., M. Schlanges, A.Y. Sakakura, and W.D. Kraeft. 1995. Low Density Expansion of the Equation of State for a Quantum Electron Gas. Physics Letters A 197: 326–329.ADSCrossRefGoogle Scholar
  17. Ebeling, W. 1968. Ableitung der freien Energie von Quantenplasmen kleiner Dichte aus den exakten Streuphasen. Annals of Physics 477: 33–39.CrossRefGoogle Scholar
  18. Ebeling, W. 1969. Zur Quantenstatistik der Bindungszustände in Plasmen. I Cluster-Entwicklungen. Annals of Physics 22: 383–391.zbMATHCrossRefGoogle Scholar
  19. Ebeling, W. 1974. Statistical Derivation of the Mass Action Law or Interacting Gases and Plasmas. Physica 73: 573–584.ADSCrossRefGoogle Scholar
  20. Ebeling, W. 2016. The Work of Baimbetov on Nonideal Plasmas and Some Recent Developments. Contributions to Plasma Physics 56: 163–175.ADSCrossRefGoogle Scholar
  21. Ebeling, W., and G.E. Norman. 2003. Coulombic Phase Transitions in Dense Plasmas. Journal of Statistical Physics 110: 861–877.zbMATHCrossRefGoogle Scholar
  22. Ebeling, W., W.D. Kraeft, and D. Kremp. 1976. Theory of Bound States and Ionisation Equilibrium in Plasmas and Solids. Berlin: Akademie-Verlag.Google Scholar
  23. Ebeling, W., W.D. Kraeft, and D. Kremp. 1977. Nonideal Plasmas. In Proceedings of the XIIIth International Conference on Phenomena in Ionized Gases, ed. P. Bachmann, 73–90. Berlin: Physikalische Gesellschaft der DDR.Google Scholar
  24. Ebeling, W., A. Förster, V.F. Fortov, V.K. Gryaznov, and A. Ya. Polishchuk. 1991. Thermophysical Properties of Hot Dense Plasmas. Stuttgart: Teubner-Verlag.Google Scholar
  25. Ebeling, W., D. Blaschke, R. Redmer, H. Reinholz, and G. Röpke. 2009. The Influence of Pauli Blocking Effects on the Properties of Dense Hydrogen. Journal of Physics A: Mathematical and Theoretical 42: 214033.ADSzbMATHCrossRefGoogle Scholar
  26. Ebeling, W., W.D. Kraeft, and G. Röpke. 2012. On the Quantum Statistics of Bound States within the Rutherford Model of Matter. Annals of Physics 524: 311–326.zbMATHCrossRefGoogle Scholar
  27. Ebeling, W., V.E. Fortov, and V.S. Filinov. 2017. Quantum Statistics of Dense Gases and Nonideal Plasmas. Springer Series in Plasma Science and Technology. Berlin: Springer.zbMATHGoogle Scholar
  28. Ecker, G., and W. Weizel. 1956. Zustandssumme und effektive Ionisierungsspannung eines Atoms im Inneren des Plasmas. Annals of Physics 452: 126–140.zbMATHCrossRefGoogle Scholar
  29. Falkenhagen, H. 1971. Theorie der Elektrolyte. Leipzig: Hirzel.zbMATHGoogle Scholar
  30. Falkenhagen, H., and W. Ebeling. 1971. Equilibrium Properties of Ionized Dilute Electrolytes. Ionic Interactions, ed. S. Petrucci, Vol. 1, 1–59. New York, London: Academic Press.Google Scholar
  31. Filinov, V.S., M. Bonitz, W. Ebeling, and V.E. Fortov. 2001. Thermodynamics of Hot Dense H-Plasmas: Path Integral Monte Carlo Simulations and Analytical Approximations. Plasma Physics and Controlled Fusion 43 (6): 743–759.ADSCrossRefGoogle Scholar
  32. Filinov, A.V., M. Bonitz, and W. Ebeling. 2003. Improved Kelbg Potential for Correlated Coulomb Systems. Journal of Physics A: Mathematical and General 36 (22): 5957–5962.ADSzbMATHCrossRefGoogle Scholar
  33. Fortov, V.E. 2009. Extreme States of Matter (in Russian). Moskva: Fiz-MatGis.Google Scholar
  34. Fortov, V.E. 2011. Extreme States of Matter: On Earth and in the Cosmos. Berlin: Springer.zbMATHCrossRefGoogle Scholar
  35. Fortov, V.E. 2013. Equation of State From the Ideal Gas to the Quark-Gluon Plasma (in Russian). Moskva: FizMatGis.Google Scholar
  36. Fortov, V.E., R.I. Ilkaev, V.A. Arinin, V.V. Burtzev, V.A. Golubev, I.L. Iosilevskiy, et al. 2007. Phase Transition in a Strongly Nonideal Deuterium Plasma Generated by Quasi-Isentropical Compression at Megabar Pressures. Physical Review Letters 99: 185001.ADSCrossRefGoogle Scholar
  37. Friedman, H.L. 1962. Ionic Solution Theory. New York: Interscience.Google Scholar
  38. Friedman, H.L., W. Ebeling. 1979. Theory of Interacting and Reacting Particles. Rostocker Physikalische Manuskripte 4: 33–48.Google Scholar
  39. Gombás, P. 1965. Pseudopotentiale. Fortschritte der Physik 13: 137–156.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. Hansen, J.P., and I.R. McDonald. 1981. Microscopic Simulation of a Strongly Coupled Hydrogen Plasma. Physical Review A 23: 2041–2059.ADSCrossRefGoogle Scholar
  41. Hansen, J.P., I.R. McDonald, and E.L. Pollock. 1975. Statistical Mechanics of Dense Ionized Matter. III. Dynamical Properties of the Classical One-Component Plasma. Physical Review A 11: 1025–1039.ADSCrossRefGoogle Scholar
  42. Heine, V. 1970. The Pseudopotential Concept. Solid State Physics 24: 1–36.CrossRefGoogle Scholar
  43. Hellmann, H. 1935. A New Approximation Method in the Problem of Many Electrons. The Journal of Chemical Physics 3: 61–61.ADSGoogle Scholar
  44. Hill, T.L. 1956. Statistical Mechanics. New York: McGraw Hill.zbMATHGoogle Scholar
  45. Hoffmann, H.J., and W. Ebeling. 1968a. On the Equation of State of Fully Ionized Quantum Plasmas. Physica 39: 593–598.ADSGoogle Scholar
  46. Hoffmann, H.J., and W. Ebeling. 1968b. Quantenstatistik des Hochtemperatur-Plasmas im thermodynamischen Gleichgewicht. II. Die freie Energie im Temperaturbereich 106 bis 108 K. Contributions to Plasma Physics 8 (1): 43–56.Google Scholar
  47. Inglis, D.R., and E. Teller. 1939. Ionic Depression of Series Limits in One-Electron Spectra. Astrophysics 90: 439–448.zbMATHCrossRefGoogle Scholar
  48. Kahlbaum, T. 2000. The Quantum-Diffraction Term in the Free Energy for Coulomb Plasma and the Effective-Potential Approach. Journal de Physique IV France 10: 455–459.CrossRefGoogle Scholar
  49. Kalman, G.J., J.M. Rommel, and K. Blagoev, eds. 1998. Strongly Coupled Coulomb Systems. New York: Springer.Google Scholar
  50. Kelbg, G. 1963. Quantenstatistik der gase mit Coulomb-Wechselwirkung. Annalen der Physik 467: 354–360.ADSMathSciNetCrossRefGoogle Scholar
  51. Kelbg, G. 1964. Klassische statistische Mechanik der Teilchen-Mischungen mit sortenabhängigen weitreichenden zwischenmolekularen Wechselwirkungen. Annals of Physics 14: 394–403.MathSciNetzbMATHCrossRefGoogle Scholar
  52. Kelbg, G. 1972. Einige Methoden der statistischen Thermodynamik hochionisierter Plasmen, Ergebnisse der Plasmaphysik und Gaselektronik, Vol. Bd. III. Berlin: Akademie-Verlag.Google Scholar
  53. Knudson, M.D., M.P. Desjarlais, A. Becker, R.W. Lemke, K.R. Cochrane, M.E. Savage, et al. 2015. Direct Observation of an Abrupt Insulator-to-Metal Transition in Dense Liquid Deuterium. Science 348(6242): 1455–1460.ADSCrossRefGoogle Scholar
  54. Kraeft, W.D., D. Kremp, W. Ebeling, and G. Röpke. 1986. Quantum Statistics of Charged Particle Systems. Berlin: Akademie-Verlag.CrossRefGoogle Scholar
  55. Kraeft, W.D., D. Kremp, and G. Röpke. 2015a. Direct Linear Term in the Equation of State of Plasmas. Physical Review E 91: 013108.ADSCrossRefGoogle Scholar
  56. Kraeft, W.D., D. Kremp, and G. Röpke. 2015b. Reply to Alastuey et al. (2015). Physical Review E 92: 047102.Google Scholar
  57. Kremp, D., M. Schlanges, and W.D. Kraeft. 2005. Quantum Statistics of Nonideal Plasmas. Berlin: Springer.zbMATHGoogle Scholar
  58. Lorenzen, W., B. Holst, and R. Redmer. 2010. First-Order Liquid-Liquid Phase Transition in Dense Hydrogen. Physical Review B 82: 195107.ADSCrossRefGoogle Scholar
  59. Meeron, E. 1962. On Cluster Theory. In Electrolytes. Proceedings of an International Symposium held in Trieste, June 1959, 7, ed. B. Pesce. Oxford: Pergamon.Google Scholar
  60. Militzer, B., and E.L. Pollock. 2000. Variational Density Matrix Method for Warm Condensed Matter and Application to Dense Hydrogen. Physical Review E 61: 3470–3482.ADSCrossRefGoogle Scholar
  61. Montroll, E., and J. Ward. 1958. Quantum Statistics of Interacting Particles. Physics of Fluids 1: 55–72.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  62. Morita, T. 1959. Equation of State of High Temperature Plasma. Progress in Theoretical Physics 22: 757–774.ADSzbMATHCrossRefGoogle Scholar
  63. Mott, N. 1961. The Transition to the Metallic State. Philosophical Magazine 6: 287–309.ADSCrossRefGoogle Scholar
  64. Norman, G.E., I.M. Saitov, and V.V. Stegailov. 2015. Plasma-Plasma and Liquid-Liquid First-Order Phase Transitions. Contributions to Plasma Physics 55: 215–221.ADSCrossRefGoogle Scholar
  65. Norman, G.E., and A.N. Starostin. 1968. Description of Nondegenerate Dense Plasma. Soviet Physics High Temperature 6: 394–408.Google Scholar
  66. Norman, G.E., and A.N. Starostin. 1970. Thermodynamics of Strongly Nonideal Plasma. Soviet Physics High Temperature 8: 381–395.Google Scholar
  67. Pines, D. 1961. The Many Body Problem—A Lecture Note. New York: Benjamin.zbMATHGoogle Scholar
  68. Pines, D., P. Nozieres. 1966. The Theory of Quantum Liquids. New York: Benjamin.zbMATHGoogle Scholar
  69. Rahman, A. 1964. Correlations in the Motion of Atoms in Liquid Argon. Physics Review 136: A405–A411.ADSCrossRefGoogle Scholar
  70. Redmer, R. 1997. Physical Properties of Dense, Low-Temperature Plasmas. Physics Reports 282: 35–157.ADSCrossRefGoogle Scholar
  71. Redmer, R., and G. Röpke. 2010. Progress in the Theory of Dense Strongly Coupled Plasmas. Contributions to Plasma Physics 50: 970–985.ADSCrossRefGoogle Scholar
  72. Riemann, J., M. Schlanges, H.E. DeWitt, and W.D. Kraeft. 1995. Equation of State of the Weakly Degenerate One-Component Plasma. Physica A 219: 423–435.ADSCrossRefGoogle Scholar
  73. Rogers, F.J. 1974. Statistical Mechanics of Coulomb Gases of Arbitrary Charge. Physical Review A 10: 2441–2456.ADSCrossRefGoogle Scholar
  74. Rogers, F.J., B.G. Wilson, and C.A. Iglesias. 1988. Parametric potential method for generating atomic data. Physical Review A 38: 5007–5020.ADSCrossRefGoogle Scholar
  75. Saumon, D., and C. Chabrier. 1989. Fluid Hydrogen at High Density: The Plasma Phase Transition. Physical Review Letters 62: 2397–2400.ADSCrossRefGoogle Scholar
  76. Trigger, S.A., W. Ebeling, V.S. Filinov, V.E. Fortov, and M. Bonitz. 2003. Internal Energy of High Density Hydrogen: Analytic Approximations Compared with Path Integral Monte Carlo Calculations. Journal of Experimental and Theoretical Physics 96: 465–479.ADSCrossRefGoogle Scholar
  77. Vedenov, A.A., and A.I. Larkin. 1959. Equation of State of Plasmas. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 36: 1133.MathSciNetGoogle Scholar
  78. Zamalin, V.M., G.E. Norman, and V.S. Filinov. 1977. The Monte Carlo Method in Statistical Thermodynamics (in Russian). Moscow: Nauka.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Werner Ebeling
    • 1
  • Thorsten Pöschel
    • 2
  1. 1.FB PhysikHumboldt Universität zu BerlinBerlinGermany
  2. 2.Friedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

Personalised recommendations