Hydrogen, helium and lithium plasmas at high pressures

Abstract

The equations of state (EoS) and other thermodynamic properties of plasmas of the light elements H, He, and Li, are calculated using inverted fugacity expansions. Fugacity expansions are known as an alternative to density expansions but show often an inferior convergence. If, however, the inversion can be solved, the fugacity representations may be very efficient. In particular, the contributions of deeply bound states are included in the fugacity expansion in a very effective way. The mathematical problems on nonlinearity connected with the inversion of fugacities to densities are reduced to solvable algebraic problems. The inversion of fugacities to densities is solved separately for two density regions: (i) In the low density, non-degenerate region we consider ring contributions describing screening effects and ladder contributions describing bound state formation. (ii) In the high density, degenerate region the electrons are described by the Fermi–Dirac distribution. Hartree–Fock contributions and Pauli blocking have to be taken into account. The ions are considered as classical, strongly correlated subsystem eventually forming a Wigner lattice. We solve the inversion problem for each of the regions. Near the crossing point, the separate solutions are connected to each other, either by smooth concatenation at the crossing point or by Padé approximations.

Equation of state of hydrogen, helium, and solar plasmas

“…In the case that the asymptotic behaviour for one variable is finite and the other one goes to infinity, we have simple asymptotic solutions. Taking into account information about the asymptotes of the Z ‐function, [ 4 ] we find the following Padé approximation for the fugacity function: Note that this is an estimate, obtained by using the analytical expressions of the limiting cases and comparison with numerical results. For still more precise values, we may use numerical iterations of the polynomials, which may start from the given approximation.…”

“…[ 14,15,17 ] This is an approximation, which is, from the results, near to the present calculations (see ref. [4] and Figure 3). In order to give an impression how the isentropic EOS of the sun looks like, we calculated the adiabatic EOS of a sun‐like system consisting only of hydrogen with the adiabatic exponent .…”

“…[ 1–3 ] It is based on the method of inverted fugacity expansions developed in previous works (see ref. [4] and references therein). Fugacity expansions are known as an alternative to density expansions for calculating the equation of state (EOS).…”

“…The transition from the chemical potential or fugacity as a natural variable of the grand canonical ensemble to the density, often used as thermodynamic variable, is denoted as inversion problem. With respect to the EOS of the light elements, we have shown in [4] that the inversion problem in the grand ensemble can be solved, and fugacity representations may be very efficient.The EOS of the plasmas of light elements is a key problem for astrophysics and in particular for the physics of the sun, which consists mainly of hydrogen and helium. [1] The sun is a big plasma ball with plasmas in a wide range of EOS parameters.…”

“…We have more than 10 25 particles per cm 3 in the centre where T > 10 7 K, and less dense plasmas in the outer regions with T ∼ 10 4 K. For a crude approach, we may model the sun as a spherical plasma with (90-93)% mole fraction hydrogen and (6-8)% mole fraction helium and <2% heavier nuclei.The light elements, which are dominant in the sun, have lowest lying bound states, hydrogen atoms, hydrogen molecules, helium atoms, and helium ions, consisting of two to four charged particles. Our method [4] based on the relation between the so-called chemical picture and physical picture is heavily using this fact. In each case, the outer electrons are only loosely bound with ionization energies of I (1) H = 13.59 eV for atomic hydrogen, I (1) He = 24.56 eV for helium.…”