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Electron Conductivity of Stellar Plasmas
Department of Theoretical Astrophysics
Contact: Alexander Potekhin (palex@astro.ioffe.ru)

Basic reference (review):
Potekhin A.Y., Pons J.A., Page D. Neutron Stars - Cooling and Transport, Space Sci. Rev., 191, 239-291 (2015) [arXiv:1507.06186]

conductivities have been last updated
in 2021
A summary of the updates since 2006 is given below.

[click here to go straight to the available resources]
[click here to see the relevant papers - the Bibliography]


Electrical and thermal conductivities of dense plasmas under conditions which take place in the stars have been studied by large number of authors over the twentieth century. Milestones on the theoretical investigation are the classical works by R. E. Marshak (1940), T. D. Lee (1950), J. M. Ziman (1960), L. Spitzer (1962), W. B. Hubbard & M. Lampe (1966 - 1969), E. Flowers & N. Itoh (1970s). In 80s - 90s, significant progress was due to L. Hernquist, N. Itoh, R. Redmer, and their co-workers, as well as other theoretical groups. It would be impossible to give the appropriate credit to all of them in this short introductory note. We restrict ourselves to a brief account only of the results obtained at the Department of Theoretical Astrophysics (DTA) of the Ioffe Institute since late 70s, which are implemented in the Fortran codes presented at this site.

As early as in 1980 Yakovlev and Urpin [1,2] critically analyzed previously published results and presented simple analytic approximations to the conductivities of degenerate electrons dominated by their scattering off ions (in the ion liquid) or off phonons or charged lattice impurities (in the ion crystal). Subsequent more detailed studies of this research team [3-6], as well as other ones, have confirmed reliability of these results in the range of plasma parameters typical for the thermal-insulating envelopes of cooling neutron stars (at densities of order 103-1010 g/cc and temperatures T about 106-109 K, or more precisely, where T is much lower than the electron degeneracy temperature, but not much lower than the ion plasma temperature). At the same time, those later studies have extended the range of considered parameters to lower temperatures [3] at which quantization of ionic motion is important (which is relevant for interiors of cool white dwarfs), to lower density [4] (for outermost layers of neutron stars), and to higher densities (in the inner envelopes of neutron stars) at which the finite nuclear sizes are non-negligible [6].

Although transport properties of dense stellar plasma are mainly determined by electron-ion collisions, electron-electron collisions may also be significant for thermal conductivity under certain conditions. The collisions between degenerate electrons have been considered in [1]. They are unimportant if the ions are highly charged or electron degeneracy is strong. Both electron-ion and electron-electron collisions are taken into acount in the Fortran programs presented at this site.

Electrical and thermal conductivity tensors, as well as thermopower tensor, in strong magnetic fields - the fields in which the electron gyrofrequency is larger than the typical collision frequency - were studied first for the non-quantizing [7] and then for quantizing fields. Kaminker & Yakovlev [8] considered the electron transport across the field and Yakovlev [9] the transport along the field (strongly or weakly quantizing, for arbitrary relativism of the electron gas) as reviewed in Ref.[10]. The results [9] based on the electron distribution function formalism have been confirmed later [11,12] using a more general formalism of electron density matrix.

Recent progress in the treatment of electrical and thermal conductivities of strongly coupled Coulomb plasmas is connected with improvements of ion structure factors relevant to calculation of electron relaxation times. Physics of these improvements has been explained in Ref.[13], and the resulting impact on nonmagnetic conductivities has been explored in Ref.[14]. In the solid phase, multiphonon electron scattering (neglected previously) has been taken into account. In the Coulomb liquid, an approximate treatment has been proposed to describe reduction of the effective electron-ion collision rate due to incipient ordering of the ions in the regime of strong ion coupling. Both modifications (in the solid and liquid phases) change the kinetic coefficients near the melting point and drastically reduce their discontinuities at the solid/liquid phase interface. Simple analytic approximations for effective relaxation times determined by the electron-ion and electron-electron collisions in degenerate, fully ionized plasmas have been also published [14].

In Ref.[15], these theoretical advances have been employed to calculate longitudinal, transverse, and Hall components of tensors of electron electrical and thermal conductivities and thermopower in arbitrary magnetic fields. Fitting formulae have been devised and implemented in a Fortran code which is available on these Web pages.

The generalization of the formulae obtained in Ref.[14] to the case where the size of the ions must be taken into account, which is the case for the inner crust of a neutron star (at densities above 1011 g/cc), has been given in the Appendix to Ref.[16].

The electron-electron scattering does not contribute in the electrical conductivity, but it can be important for the thermal conductivity. This scattering has been considered by Urpin & Yakovlev [1] and used in Ref.[14]. However, Shternin & Yakovlev [17] reconsidered this mechanism, taking into account Landau damping of transverse plasmons. This effect (neglected before) is due to the difference of the components of the polarizability tensor, responsible for screening the charge-charge and current-current interactions (i.e., different screening of the timelike and spacelike components of the electron currents in the interaction matrix element in quantum electrodynamics). Shternin & Yakovlev found that the Landau damping of transverse plasmons strongly increases the effective electron-electron collision frequency at high densities (where degenerate electrons are relativistic), compared to the older results of Flowers & Itoh and Urpin & Yakovlev. In Ref.[18], a fitting formula has been presented, which reproduces Shternin & Yakovlev results for strongly degenerate plasmas and Hubbard & Lampe ones for weakly- or non-degenerate matter.

Chugunov & Haensel [19] considered an alternative heat transport by the plasma ions (phonons) and presented fitting formulae for effective Coulomb logarithms describing scattering of ions off ions and electrons in dense plasmas. (However, by default, we do not use their fit in the latest version of our code, because this correction is, as a rule, unimportant, while its use may cause certain problems.)

Chugunov [20] has shown that the "freezing-out" of the Umklapp processes is ineffective in star conditions of degenerate stellar plasmas. This result is now taken into account (see the list of updates below).

Blouin et al. [21] suggested that our formulas underestimate the conductivities around the transition from nondegenerate to strongly degenerate plasmas, which is substantial in the case where the electron-electron scattering is important, viz. for light elements (especially H and He). Cassisi et al. [22] discussed theoretical uncertainties in this region and suggested alternative ways to bridge the domains of non-degenerate and strongly degenerate plasmas, which are now implemented in the code on the present site.

To summarize, the numerical data and Fortran programs at our Web site are based on the theoretical results published in Refs.[14-18, 20, 22]. The plasma is assumed fully ionized (collisions with neutrals are neglected). This model may be still useful for evaluation of conductivities of partially ionized plasmas, if one uses a mean-ion model. Then the ion charge Z should be replaced by an effective (or average) ion charge Zeff. For nondegenerate plasmas, the results are based on a continuation from the degenerate domain (using Fermi-Dirac averaging) and can be considered as order-of-magnitude estimates. For degenerate plasmas, on the contrary, the results come from the exact theory and are expected to be much more accurate.


When you use results obtained with the aid of the resources provided here in a publication, please make a reference to the following review, where all the above-mentioned results have been briefly described and further most relevant references have been given:

Potekhin A.Y., Pons J.A., Page D. Neutron Stars - Cooling and Transport, Space Sci. Rev., 191, 239 (2015) [ArXiV:1507.06186]

  1. V. A. Urpin, D. G. Yakovlev (1980). Thermal conductivity due to collisions between electrons in a degenerate relativistic electron gas, Sov. Astron. 24, 126 [PDF-file (258 K)]
  2. D. G. Yakovlev, V. A. Urpin (1980). Thermal and electrical conductivity in white dwarfs and neutron stars, Sov. Astron. 24, 303 [PDF-file (1468 K)]
  3. M. E. Raikh, D. G. Yakovlev (1982). Thermal and electrical conductivities of crystals in neutron stars and degenerate dwarfs, Astrophys. Space Sci. 87, 193 [PDF-file (984 K)]
  4. D. G. Yakovlev (1987). Thermal and electrical conductivities of a degenerate electron gas with electron scattering on heavy ions in the liquid or gaseous phases, Sov. Astron. 31, 347
  5. D. A. Baiko, D. G. Yakovlev (1995). Thermal and electrical conductivities of Coulomb crystals in neutron stars and white dwarfs, Astronomy Letters 21, 702
  6. D. A. Baiko, D. G. Yakovlev (1996). Thermal and electric conductivities of Coulomb crystals in the inner crust of a neutron star, Astronomy Letters 22, 708
  7. V. A. Urpin, D. G. Yakovlev (1980). Thermogalvanomagnetic effects in white dwarfs and neutron stars, Sov. Astron. 24, 425
  8. A. D. Kaminker, D. G. Yakovlev (1981). Description of a relativistic electron in a quantizing magnetic field. Transverse transport coefficients of an electron gas, Theor. Math. Phys. 49, 1012 [PDF file (730 K)]
  9. D. G. Yakovlev (1984). Transport properties of the degenerate electron gas of neutron stars along the quantizing magnetic field, Astrophys. Space Sci. 98, 37
  10. D. G. Yakovlev, A. D. Kaminker (1994). Neutron star crusts with magnetic fields, in The Equation of State in Astrophysics, ed. G. Chabrier, E. Schatzman (Cambridge Univ., Cambridge), p. 214
  11. A. Y. Potekhin (1996). Electron conduction along quantizing magnetic fields in neutron star crusts. I. Theory, Astron. Astrophys. 306, 999; erratum (1997) 327, 441 [gzipped PS-file (168 K)]
  12. A. Y. Potekhin, D. G. Yakovlev (1996). Electron conduction along quantizing magnetic fields in neutron star crusts. II. Practical formulae, Astron. Astrophys. 314, 341; erratum (1997) 327, 442 [gzipped PS-file (468 K)]
  13. D. A. Baiko, A. D. Kaminker, A. Y. Potekhin, D. G. Yakovlev (1998). Ion structure factors and electron transport in dense Coulomb plasmas, Phys. Rev. Lett. 81, 5556 [gzipped PS-file (122 K)]
  14. A. Y. Potekhin, D. A. Baiko, P. Haensel, D. G. Yakovlev (1999). Transport properties of degenerate electrons in neutron star envelopes and white dwarf cores, Astron. Astrophys. 346, 345 [gzipped PS-file (256 K)]
  15. A. Y. Potekhin (1999). Electron conduction in magnetized neutron star envelopes, Astron. Astrophys. 351, 787 [gzipped PS-file (455 K)]
  16. O. Y. Gnedin, D. G. Yakovlev, A. Y. Potekhin (2001). Thermal relaxation of young neutron stars, Mon. Not. R. Astron. Soc. 324, 725 [PDF-file (815 K)] (see Appendix)
  17. P. S. Shternin, D. G. Yakovlev (2006). Electron thermal conductivity owing to collisions between degenerate electrons. Phys. Rev. D 74, 043004 [PDF-file (348 K)]
  18. S. Cassisi, A. Y. Potekhin, A. Pietrinferni, M. Catelan, M. Salaris (2007). Updated electron-conduction opacities: the impact on low-mass stellar models. Astrophys. J., 661, 1094 [astro-ph/0703011]
  19. A. I. Chugunov, P. Haensel (2007). Thermal conductivity of ions in a neutron star envelope. Mon. Not. R. Astron. Soc., 381, 1143 [arXiv:0707.4614]
  20. A. I. Chugunov (2012). Electrical conductivity of the neutron star crust at low temperatures. Astronomy Lett., 38, 25 [PDF file (1.3 MB)]
  21. S. Blouin, R. Shaffer, D. Saumon, C. E. Starrett (2020). New conductive opacities for white dwarf envelopes. Astrophys. J. 899, 46 [arXiv:2006.16390]
  22. S. Cassisi, A. Y. Potekhin, M. Salaris, A. Pietrinferni (2021). Electron conduction opacities at the transition between moderate and strong degeneracy: Uncertainties and impact on stellar models. Astron. Astrophys. 654, A149 [arXiv:2108.11653]

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Recent updates

  1. 2006: The ee contribution is now included in such a way that both the cases of strongly degenerate and nondegenerate plasmas are recovered accurately.

    The older version of the table was inaccurate for low-Z chemical elements (especially for H and He) at T around or higher than Fermi temperature, because it did not take into account electron-electron scattering.

    The "long" version of the code includes the contribution of electron-electron scattering into the thermal conductivity at magnetic field B=0. However, in this case the older version still was inapplicable at T much higher than Fermi temperature, because the contribution from electron-electron scattering previously used a fit designed for strongly degenerate electrons only.

    Now the high-T limit of our data matches the numerical tables of Hubbard & Lampe, 1968, Astrophys. J. Suppl. 18, 297 (which remain the most accurate calculations of conductive opacities for astrophysical use in nonmagnetic, nondegenerate, weakly coupled plasma).

    The ee contribution is updated in both the "long" and "short" versions of the code, as well as in the codes for the inner crust of the neutron star, according to the results by Shternin and Yakovlev [17], described above.

  2. In 2008 — 2012, the ii and ie (i.e., ion transport) contributions are included in the heat conductivities according to the results by Chugunov and Haensel [19], described above. A technical error in the latter addition was fixed on 06.10.2008. By default, however, this correction is switched off altogether since 31.01.2013, for the reason stated above.

    In early (before 2007) releases of the "long" version of the Fortran code ("conduct"), an accidental error in the ionic component of the pressure might occur (though rarely) due to a technical slip in subroutine CHEMPOT. It did not affect other data or programs presented at this site. This slip was found and eliminated on 29.04.2007.

    A technical error (erroneously deleted line) has been discovered and corrected in the code "condegin" thanks to Nicolas Chamel on 12.11.2007.

    Another technical slip in the "long" version of the code, some variables left uninitialized in ThAv99, which could lead (though quite rarely) to a bug, was discovered and fixed on 12.05.2011.

  3. 2013: The "freezing-out" of the so-called Umklapp processes of electron-ion scattering, which can occur in extreme quantum regime, leads to a switch to "normal" processes. It was treated with an error up to a factor of a few in the codes for inner neutron-star crust "condegin" and "condegsc". This is corrected on 23.05.2007 thanks to the remarks of Andrey Chugunov. This switch was realized also in the "short" version of the code ("condegen", but not in "conduct"). However, this switch has been completely commented out by default now, because it seems to be non-operative in neutron star conditions, as shown by A.I.Chugunov [20] [Ast. Lett. 38, 25 (2012)].

    A new code "condBSk" is added for the inner crust of neutron stars. It is analogous to the "condegsc" code, but uses more recent nuclear form factors. Namely, the Oyamatsu (1993) approximation is now replaced by the HFB19 (BSk19), HFB20 (BSk20) and HFB21 (BSk21) models, according to the series of papers by N.Chamel, J.M.Pearson, and collaborators (2011-2013).

  4. 2017: A possible accuracy loss in calculation of conductivities in the inner crust, i.e., with allowance for finite nuclear radius (non-pointlike atomic nuclei), has been revealed (in function COULAN2) and fixed; the function COULAN2 is upgraded to subroutine COULAN3, which provides, along with the Coulomb logarithm, also its derivative over electron energy. Another possible accuracy loss in calculation of electron-electron relaxation time (in the nonmagnetic case) is fixed by using a switch to an analytical asymptote.

  5. 2018: Calculation of thermal conductivity is improved in the case where an approximation of degenerate electrons is used. Small fixes. The code "condBSk" is updated with inclusion of new EoS models BSk22,24,25,26.

  6. 2019: A typo in Eq.(26) of Ref.[19] for the ion conduction has been found (A.I.Chugunov, private communication) and fixed in subroutine CONDI and in its descendants CONDIN, CONDIsc (by default these subroutines are not used, but may be optionally switched on). The electron-electron scattering correction for thermal conductivity is now used at weakly quantizing magnetic field (not only at nonquantizing field as in the version of 2018; but it is still ignored at strongly quantizing field because of its obvious inapplicability in this regime). The "long" version of the code (with accurate thermal averaging) is additionally improved in several ways: thermal averaging subroutine ThAv99 is upgraded to ThAv18 to conform to Fortran 2018 and to take finite size of nuclei into account; Coulomb logarithm subroutines COUL01 and COUL99I are replaced by their improved versions COUL15 and COUL18I to fix numerical stability in limiting cases.

  7. 2020-2021: Blouin et al. [21] have shown that our thermal conductivity of hydrogen or helium plasmas should be substantially enhanced in the domain of partially degenerate electrons.They derived an analytic fit to this correction. To avoid double-counting, we have not implemented this correction directly in our code, but provided the corresponding subroutine and envisioned a possibility of its use for the longitudinal thermal conductivity in the MAIN (driving) block of the "long" (that is designed to grasp the case of partial degeneracy) version of the conductivity code. We have also provided two modified ("damped") versions of this correction, called "weak" and "strong" damping, representing possible plausible alternatives of a smooth transition to the regimes of strongly degenerate electrons; details are given in Ref.[22].

    Note:This correction is available only for H and He at zero magnetic field.

    The basic numerical table for interpolation is given including the "weakly damped" version of this correction. Ancillary tables without this correction, with full (non-damped) correction, and with the "strong damping" are also provided.

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What is available:
[make your choice
and click]
Precalculated data Tables of thermal conductivities of pure nonmagnetic electron-ion plasmas, together with an interpolation tool (last updated 18.05.2021)
Fortran programs:
Coulomb logarithm Effective Coulomb logarithms for conductivities of strongly degenerate electron gas
Magnetic conductivities Electrical and thermal electron conductivities of electron-ion plasmas, including options for arbitrary degeneracy and arbitrary magnetic fields (last updated 18.05.2021)
Inner crust Electrical and thermal electron conductivities in the inner crust of the neutron star (assuming strong degeneracy and allowing for the presence of free neutrons and for finite ion sizes) (last updated 12.09.19)

When you use results obtained with the aid of the above resources in a publication, please make a reference to the following review, where all the above-mentioned results have been briefly described and further most relevant references have been given:

green ball Potekhin A.Y., Pons J.A., Page D. Neutron Stars - Cooling and Transport, Space Sci. Rev., 191, 239 (2015) [arXiv:1507.06186]

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Page design and maintenance: Alexander Potekhin
Page created in 2000; last updated on 15.06.22.
The data at this site have been last updated on 18.05.21.