[Paper Review] Thermodynamic coherence of the Variational Average-Atom in Quantum Plasmas (VAAQP) approach

R. Piron, T. Błeński+1 more|arXiv (Cornell University)|Feb 17, 2009

High-pressure geophysics and materials14 references7 citations

TL;DR

This paper presents the VAAQP code, a fully variational average-atom model for quantum plasmas that ensures thermodynamic coherence by satisfying the virial theorem and eliminating the need to impose Wigner-Seitz sphere neutrality. It derives a simple analytical expression for electronic pressure without numerical differentiation, demonstrating consistent mean ionization and pressure across the aluminum 10 eV isotherm compared to non-variational and Inferno-type models.

ABSTRACT

A new code called VAAQP (Variational Average-Atom in Quantum Plasmas) is reported. The model as well as main results of previous studies are briefly recalled. The code is based on a new fully variational model of dense plasmas at equilibrium with quantum treatment of all electrons. The code can calculate the Average Atom structure and the mean ionization from the variational equations respecting the virial theorem and without imposing the neutrality of the Wigner-Seitz sphere. The formula obtained for the electronic pressure is simple and does not require any numerical differentiation. A description of the principal features of the code is given. The thermodynamic consistency of the results obtained with VAAQP is shown by a comparison with another approach on the example of the aluminium 10 eV isotherm EOS curve. A first comparison to an INFERNO-type model is also presented.

Motivation & Objective

• Develop a thermodynamically consistent quantum average-atom model for dense plasmas that respects the virial theorem.
• Address the limitations of non-variational models like Inferno and NWS, which rely on artificial neutrality constraints and numerical differentiation for pressure.
• Enable accurate calculation of mean ionization and electronic pressure without assuming Wigner-Seitz sphere neutrality.
• Provide a variational framework that captures Friedel oscillations beyond the Wigner-Seitz radius, improving accuracy in warm dense matter regimes.
• Establish a benchmark for comparing thermodynamically coherent results with existing models like Inferno and non-variational NWS approaches.

Proposed method

• Formulate a fully variational model for dense plasmas at thermodynamic equilibrium, treating all electrons quantum-mechanically.
• Use a variational principle to derive self-consistent equations for electron density and potential, ensuring compliance with the virial theorem.
• Eliminate the assumption of Wigner-Seitz sphere neutrality, allowing the average ionization to emerge naturally from the variational solution.
• Derive a closed-form analytical expression for electronic pressure (Eq. 17), avoiding numerical differentiation of the free energy.
• Implement the model in the VAAQP code to compute average atom structure, mean ionization, and equation of state (EOS) for aluminum at 10 eV.
• Compare results with non-variational NWS and Inferno-type models using the same thermodynamic conditions to assess consistency and accuracy.

Experimental results

Research questions

• RQ1 Can a fully variational average-atom model for quantum plasmas achieve thermodynamic consistency without imposing Wigner-Seitz sphere neutrality?
• RQ2 How does the analytical pressure expression derived from the variational principle compare to numerically differentiated pressure in non-variational models?
• RQ3 To what extent do Friedel oscillations beyond the Wigner-Seitz radius affect the electron density and mean ionization in warm dense matter?
• RQ4 How do the electron density and mean ionization differ between the variational VAAQP model and non-variational or Inferno-type models at the same thermodynamic state?
• RQ5 Can thermodynamically coherent results be achieved in the warm dense matter regime using a variational approach that captures long-wavelength charge oscillations?

Key findings

• The VAAQP model produces thermodynamically consistent results, as confirmed by perfect agreement between the analytical pressure (Eq. 17) and numerically differentiated free energy along the aluminum 10 eV isotherm.
• In contrast, the non-variational NWS model shows significant disagreement between analytical and numerical pressure, indicating thermodynamic inconsistency.
• Despite similar electronic pressures, the mean ionization values from VAAQP and NWS differ by up to 10%, highlighting that pressure alone is insufficient for model validation.
• The electron density profiles from VAAQP and NWS differ substantially in the region of Friedel oscillations, especially in the warm dense matter regime.
• The Inferno-type model, despite its neutrality assumption, produces electron density closer to the variational VAAQP solution than to the non-variational NWS solution, suggesting unintended stabilization effects.
• The variational approach successfully captures long-wavelength charge oscillations beyond the Wigner-Seitz radius, which are neglected in models like Inferno and NWS, leading to more accurate atomic structure predictions.