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| Department of Theoretical Astrophysics |
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Note:
non-magnetic conductivities have been updated 18.07.2006, as described in Ref.[18], and 18.02.2008, according to Ref.[19] (the last correction - a fixed typo in CONDI - is uploaded on 22.12.2008). A complete 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]
Introduction
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.
Finally, 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.
To summarize, the numerical data and Fortran programs at our Web site are based on the theoretical results published in Refs.[14-19]. 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.
The results presented here have been used in various astrophysical applications by our and other scientific groups - for example, in the papers listed in the Addendum below.
There are three main modifications. The first and second ones concern the contribution from electron-electron (ee) scattering into the nonmagnetic thermal conductivities. The third modification concerns the contribution from ion (ie and ii) scattering into the nonmagnetic and magnetic thermal conductivities.
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 "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 is now realized also in the "short" version of the code
("condeg", but not in "conduct").
A technical error (erroneously deleted line)
has been discovered and corrected in the code "condegin"
thanks to Nicolas
Chamel on 12.11.2007.
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