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Density-functional calculations of transport properties in the nondegenerate limit and the role of electron-electron scattering

Michael P. Desjarlais1,*, Christian R. Scullard2, Lorin X. Benedict2, Heather D. Whitley2, and Ronald Redmer3

  • 1Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
  • 2Lawrence Livermore National Laboratory, Livermore, California 94550, USA
  • 3Institut für Physik, Universität Rostock, D-18051 Rostock, Germany

  • *mpdesja@sandia.gov
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Phys. Rev. E 95, 033203 – Published 13 March, 2017

DOI: https://doi.org/10.1103/PhysRevE.95.033203

Abstract

We compute electrical and thermal conductivities of hydrogen plasmas in the nondegenerate regime using Kohn-Sham density functional theory (DFT) and an application of the Kubo-Greenwood response formula, and demonstrate that for thermal conductivity, the mean-field treatment of the electron-electron (e-e) interaction therein is insufficient to reproduce the weak-coupling limit obtained by plasma kinetic theories. An explicit e-e scattering correction to the DFT is posited by appealing to Matthiessen's Rule and the results of our computations of conductivities with the quantum Lenard-Balescu (QLB) equation. Further motivation of our correction is provided by an argument arising from the Zubarev quantum kinetic theory approach. Significant emphasis is placed on our efforts to produce properly converged results for plasma transport using Kohn-Sham DFT, so that an accurate assessment of the importance and efficacy of our e-e scattering corrections to the thermal conductivity can be made.

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References (69)

  1. M. P. Desjarlais, J. D. Kress, and L. A. Collins, Phys. Rev. E 66, 025401 (2002).
  2. T. R. Mattsson and M. P. Desjarlais, Phys. Rev. Lett. 97, 017801 (2006).
  3. A. Kietzmann, R. Redmer, M. P. Desjarlais, and T. R. Mattsson, Phys. Rev. Lett. 101, 070401 (2008).
  4. V. Recoules, F. Lambert, A. Decoster, B. Canaud, and J. Clérouin, Phys. Rev. Lett. 102, 075002 (2009).
  5. D. E. Hansen, L. A. Collins, J. D. Kress, and M. P. Desjarlais, Phys. Plasmas 18, 082704 (2011).
  6. F. Lambert, V. Recoules, A. Decoster, J. Clérouin, and M. P. Desjarlais, Phys. Plasmas 18, 056306 (2011).
  7. B. Holst, M. French, and R. Redmer, Phys. Rev. B 83, 235120 (2011).
  8. M. Pozzo, M. P. Desjarlais, and D. Alfè, Phys. Rev. B 84, 054203 (2011).
  9. S. X. Hu, L. A. Collins, T. R. Boehly, J. D. Kress, V. N. Goncharov, and S. Skupsky, Phys. Rev. E 89, 043105 (2014).
  10. S. X. Hu, L. A. Collins, V. N. Goncharov, J. D. Kress, R. L. McCrory, and S. Skupsky, Phys. Plasmas 23, 042704 (2016).
  11. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  12. W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  13. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
  14. D. A. Greenwood, Proc. Phys. Soc. London 71, 585 (1958).
  15. L. Spitzer and R. Härm, Phys. Rev. 89, 977 (1953).
  16. L. D. Landau, Phys. Z. Sowjetunion 10, 154 (1936); Zh. Eksp. Teor. Fiz. 7, 203 (1937).
  17. A. Lenard, Ann. Phys. 10, 390 (1960); R. Balescu, Phys. Fluids 3, 52 (1960).
  18. R. H. Williams and H. E. DeWitt, Phys. Fluids 12, 2326 (1969).
  19. F. Morales, M. K. Kilimann, R. Redmer, M. Schlanges, and F. Bialas, Contrib. Plasma Phys. 29, 425 (1989).
  20. R. Redmer, G. Röpke, F. Morales, and M. K. Kilimann, Phys. Fluids B 2, 390 (1990).
  21. B. Bernu, J. Wallenborn, and V. Zehnle, J. Phys. (France) 49, 1161 (1988).
  22. H. D. Whitley, C. R. Scullard, L. X. Benedict, J. I. Castor, A. Randles, J. N. Glosli, D. F. Richards, M. P. Desjarlais, and F. R. Graziani, Contrib. Plasma Phys. 55, 192 (2015).
  23. J. M. Ziman, Philos. Mag. 6, 1013 (1961); Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems (Cambridge University Press, New York, 1979).
  24. F. Perrot and M. W. C. Dharma-wardana, Phys. Rev. E 52, 5352 (1995).
  25. P. A. Sterne, S. B. Hansen, B. G. Wilson, and W. A. Isaacs, High Ener. Dens. Phys. 3, 278 (2007).
  26. B. F. Rosznyai, High Ener. Dens. Phys. 4, 64 (2008).
  27. G. Faussurier, C. Blancard, P. Cossé, and P. Renaudin, Phys. Plasmas 17, 052707 (2010).
  28. C. E. Starrett, High Ener. Dens. Phys. 19, 58 (2016).
  29. The approximation of the shifted velocity distribution can be made in the case of nearly degenerate electrons, wherein for σ, the steady-state electron distribution function can be taken to be f(v)=f0(v−je/ne), where f0 is the equilibrium distribution and n is the number density of electrons. See C. Kittel, Introduction to Solid State Physics (John Wiley and Sons, New York, 1995).
  30. S. I. Braginskii, Zh. Eksp. Teor. Fiz. 33, 459 (1957).
  31. H. A. Lorentz, Proc. Amst. Acad. 7, 438 (1905).
  32. E. M. Epperlein and M. G. Haines, Phys. Fluids 29, 1029 (1986).
  33. S. Ichimaru and S. Tanaka, Phys. Rev. A 32, 1790 (1985).
  34. Y. T. Lee and R. M. More, Phys. Fluids 27, 1273 (1984).
  35. G. A. Rinker, Phys. Rev. A 37, 1284 (1988).
  36. M. P. Desjarlais, Contrib. Plasma Phys. 41, 267 (2001).
  37. J. Wicht and A. Tilgner, Space Sci. Rev. 152, 501 (2010).
  38. M. French, A. Becker, W. Lorenzen, N. Nettelmann, M. Bethkenhagen, J. Wicht, and R. Redmer, Astrophys. J. Suppl. 202, 5 (2012).
  39. M. Pozzo, C. Davies, D. Gubbins, and D. Alfè, Nature (London) 485, 355 (2012).
  40. B. A. Hammel et al., High Ener. Dens. Phys. 6, 171 (2010).
  41. M. K. Matzen, M. A. Sweeney et al., Phys. Plasmas 12, 055503 (2005).
  42. M. P. Desjarlais (unpublished).
  43. G. Kelbg, Ann. Phys. 467, 219 (1963).
  44. T. Dunn and A. A. Broyles, Phys. Rev. 157, 156 (1967).
  45. D. N. Zubarev, Nonequilibrium Statistical Thermodynamics (Consultants Bureau, New York, 1974).
  46. G. Röpke, Phys. Rev. A 38, 3001 (1988).
  47. R. Redmer, Phys. Rep. 282, 35 (1997).
  48. S. Kuhlbrodt and R. Redmer, Phys. Rev. E 62, 7191 (2000).
  49. H. Reinholz, G. Röpke, S. Rosmej, and R. Redmer, Phys. Rev. E 91, 043105 (2015).
  50. G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993); 49, 14251 (1994); G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996); Phys. Rev. B 54, 11169 (1996).
  51. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
  52. The very high plane-wave cutoff energy employed for the DFT-MD runs is necessary to converge the electronic contribution to the energy and pressure for the 1/r potential. Our tests indicate convergence at the 0.1% level for plane wave cutoff energies above 3000 eV.
  53. G. D. Mahan, Many-Particle Physics (Plenum Press, New York, 1981).
  54. S. Ichimaru, Statistical Plasma Physics, Vol. 1: Basic Principles (Addison-Wesley, Redwood City, CA, 1992).
  55. Indeed, it has even been shown that LB theory works quite well for considerably stronger plasma couplings when compared with classical MD, as demonstrated in the work of Ref. [21].
  56. Our Lenard-Balescu treatment is very similar to that of Ref. [18], with the notable exception that they employ an additional collision term to account for hard scattering. We have established that this additional term is of no importance for the plasma conditions discussed here. We also note that our numerical procedures are slightly different from theirs for the terms which are common to the two approaches; but again, the discrepancies resulting from these differences are very minor. See Figs. 1 and 2 of Ref. [22] for comparisons along these lines.
  57. By comparing the conductivities calculated with classical versus quantum collision integrals within the Zubarev formalism, we estimate this difference to be on the order of 20% at 500 eV and less than 10% at 900 eV.
  58. We use the notation f(v) somewhat loosely here; v refers to the velocity of the electron in the semiclassical sense. In the actual Kohn-Sham DFT calculation performed on a periodic array of supercells, v is related to the wave vector, k.
  59. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd ed. (Cambridge University Press, New York, 1970).
  60. A. Matthiessen, Philos. Trans. R. Soc. London 148, 383 (1858).
  61. It should be noted that the exchange-correlation term within the Kohn-Sham Hamiltonian plays practically no role for hydrogen at these (ρ,T) conditions and contributes only about 1% to the total electronic energy.
  62. L. V. Pourovskii, J. Mravlje, A. Georges, S. I. Simak, and I. A. Abrikosov, arXiv:1603.02287.
  63. A. Domps, P.-G. Reinhard, and E. Suraud, Phys. Rev. Lett. 81, 5524 (1998).
  64. E. K. U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1985).
  65. G. Vignale, Time-Dependent Density Functional Theory (Springer, Berlin, 2006), pp. 75–91.
  66. H. Reinholz, R. Redmer, and S. Nagel, Phys. Rev. E 52, 5368 (1995).
  67. H. Reinholz, R. Redmer, and D. Tamme, Contrib. Plasma Phys. 29, 395 (1989).
  68. J. R. Adams, N. S. Shilkin, V. E. Fortov, V. K. Gryaznov, V. B. Mintsev, R. Redmer, H. Reinholz, and G. Röpke, Phys. Plasmas 14, 062303 (2007).
  69. V. S. Karakhtanov, R. Redmer, H. Reinholz, and G. Röpke, Contrib. Plasma Phys. 53, 639 (2013).

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Phys. Rev. E 95, 033203– Published 13 March, 2017
  • Received 16 December 2016
Article part of CHORUS

©2017 American Physical Society

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