Dear Xueyan,

The following text was taken from a recent publication entitled " Kubo-Greenwood approach to conductivity in dense plasmas with average atom "  by C. E. Starrett published in 2016 counts the advantages and disadvantages of both methods in calculating a thermal conductivity of materials.

An important aspect of modeling warm and hot dense matter is the calculation of electron thermal and electrical conductivities. The former is of particular relevance

in the field of inertial confinement fusion [1, 2] where it is the main phenomena that determines the ablation of the cold deuterium/tritium fuel. Currently we have no reliable model that can predict accurate thermal and electrical conductivities across all temperature and density regimes of interest. In particular, as we move out of the

degenerate electron regime the gold standard method of Kohn-Sham density functional theory molecular dynamics (KS-DFT-MD) coupled with the Kubo-Greenwood formalism [3–6] quickly becomes computationally prohibitive. In the degenerate, or nearly degenerate regimes, this method is thought to be accurate and agrees with experiments for materials under normal conditions [7].

Average atom models provide an computationally efficient alternative at the cost of physical accuracy. The central idea is that one tries to calculate the properties of

one atom in the plasma that is supposed to represent the average of all atoms in the plasma. Average atom models have been used successfully for many years for equation of state calculations [8–13]. They have also been used for electrical conductivity calculations, primarily by coupling to the Ziman-Evans (ZE) formula [14–20]. Recently, a systemic comparison of calculations of electrical conductivity

using this method against Kubo-Greenwood KSDFT-MD calculations [14] showed generally very good agreement between the methods provided that a judicious

choice was made when coupling the average atom model to the ZE formula. However, the ZE formula, unlike the KG method, is not easily generalized to thermal

conductivity or optical conductivity. The latter is useful as it can by used to calculate other optical properties, including the opacity and reflectivity [21]. A formulation of the Kubo-Greenwood method for average atoms models has been developed by Johnson and co-workers [22–24]. However, a subsequent systematic analysis of the method compared to KS-DFT-MD showed some serious inaccuracies [25]. Unlike the ZE formulation, Johnson’s KG formulation does make not explicit

account of the ion-ion structure factor S(k). In this work, we give an alternative derivation of the KG formulation for average atom models that explicitly accounts

for S(k). The new formulation recovers Johnson’s result when S(k) = 1. We also give the equations for thermal and optical conductivity.

References:

[1] Flavien Lambert, Vanina Recoules, Alain Decoster, Jean Clerouin, and Michael Desjarlais. On the transport coefficients of hydrogen in the inertial confinement fusion regime a). Physics of Plasmas (1994-present), 18(5):056306, 2011.

[2] S. X. Hu, L. A. Collins, T. R. Boehly, J. D. Kress, V. N. Goncharov, and S. Skupsky. First-principles thermal conductivity of warm-dense deuterium plasmas for inertial

confinement fusion applications. Phys. Rev. E, 89:043105, Apr 2014.

[3] Ryogo Kubo. Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems. Journal of the Physical Society of Japan, 12(6):570–586, 1957.

[4] D.A. Greenwood. The boltzmann equation in the theory of electrical conduction in metals. Proceedings of the Physical Society, 71(4):585, 1958.

[5] David E. Hanson, Lee A. Collins, Joel D. Kress, and Michael P. Desjarlais. Calculations of the thermal conductivity of national ignition facility target materials at

temperatures near 10 ev and densities near 10 g/cc using finite-temperature quantum molecular dynamics. Physics of Plasmas, 18(8), 2011.

[6] M. P. Desjarlais, J. D. Kress, and L. A. Collins. Electrical conductivity for warm, dense aluminum plasmas and liquids. Phys. Rev. E, 66:025401, Aug 2002.

[7] Travis Sjostrom and J´erˆome Daligault. Ionic and electronic transport properties in dense plasmas by orbitalfree density functional theory. Phys. Rev. E, 92:063304,

2015.

[8] R. P. Feynman, N. Metropolis, and E. Teller. Equations of state of elements based on the generalized fermithomas theory. Phys. Rev., 75:1561–1573, May 1949.

[9] David A. Liberman. Self-consistent field model for condensed matter. Phys. Rev. B, 20:4981–4989, Dec 1979.

[10] R. Piron and T. Blenski. Variational-average-atomin-quantum-plasmas (vaaqp) code and virial theorem: Equation-of-state and shock-hugoniot calculations for

warm dense al, fe, cu, and pb. Phys. Rev. E, 83:026403, Feb 2011.

[11] B. Wilson, V. Sonnad, P. Sterne, and W. Isaacs. Purgatorioa new implementation of the inferno algorithm. Journal of Quantitative Spectroscopy and Radiative Transfer, 99(13):658 – 679, 2006. Radiative Properties of Hot Dense Matter.

[12] G´erald Faussurier, Christophe Blancard, Philippe Coss´e, and Patrick Renaudin. Equation of state, transport coefficients, and stopping power of dense plasmas from the average-atom model self-consistent approach for astrophysical

and laboratory plasmas. Physics of Plasmas, 17(5), 2010.

[13] Balazs F. Rozsnyai. Relativistic hartree-fock-slater calculations for arbitrary temperature and matter density. Phys. Rev. A, 5:1137–1149, Mar 1972.

[14] D.J. Burrill, D.V. Feinblum, M.R.J. Charest, and C.E. Starrett. Comparison of electron transport calculations in warm dense matter using the ziman formula. High

Energy Density Physics, 19:1 – 10, 2016.

[15] G´erald Faussurier and Christophe Blancard. Resistivity saturation in warm dense matter. Phys. Rev. E, 91:013105, Jan 2015.

[16] P.A. Sterne, S.B. Hansen, B.G. Wilson, and W.A. Isaacs. Equation of state, occupation probabilities and conductivities in the average atom purgatorio code. High Energy Density Physics, 3(12):278 – 282, 2007. Radiative Properties

of Hot Dense Matter.

[17] Fran¸cois Perrot and M. W. C. Dharma-wardana. Electrical resistivity of hot dense plasmas. Phys. Rev. A, 36:238–246, Jul 1987.

[18] J.C. Pain and G. Dejonghe. Electrical resistivity in warm dense plasmas beyond the average-atom model. Contributions to Plasma Physics, 50(1):39–45, 2010.

[19] M. W. C. Dharma-wardana. Static and dynamic conductivity of warm dense matter within a density-functional approach: Application to aluminum and gold. Phys. Rev. E, 73:036401, Mar 2006.

[20] Balazs F. Rozsnyai. Electron scattering in hot/warm plasmas. High Energy Density Physics, 4(1):64–72, 2008.

[21] S. Mazevet, L.A. Collins, N.H. Magee, J.D. Kress, and J.J. Keady. Quantum molecular dynamics calculations of radiative opacities. Astronomy & Astrophysics,

405(1):L5–L9, 2003.

[22] W.R. Johnson, C. Guet, and G.F. Bertsch. Optical properties of plasmas based on an average-atom model. Journal of Quantitative Spectroscopy and Radiative Transfer, 99(13):327 – 340, 2006. Radiative Properties of Hot Dense Matter.

[23] W.R. Johnson. Low-frequency conductivity in the average-atom approximation. High Energy Density Physics, 5(12):61 – 67, 2009.

[24] M. Yu. Kuchiev and W. R. Johnson. Low-frequency plasma conductivity in the average-atom approximation. Phys. Rev. E, 78:026401, Aug 2008.

[25] C. E. Starrett, J Cl´erouin, V Recoules, J. D. Kress, L. A. Collins, and D. E. Hanson. Average atom transport properties for pure and mixed species in the hot and warm dense matter regimes. Physics of Plasmas (1994- present), 19(10):102709, 2012.

https://arxiv.org/pdf/1603.06925.pdf

Hoping this will be helpful,

Rafik