Degrees of Ionization
© Charles Chandler
Determining the degree of ionization due to pressure will be a non-trivial undertaking, as it gets deep into supercritical theory, and the laboratory confirmations are sparse at best, as it is extremely different to generate such pressures, and measuring the conditions inside such matter is not possible.
Simplistically, we might think that we could just look up the radii of the electron shells for the various elements. (There isn't a fixed radius for each shell, because the number of protons in the nucleus varies, and this produces an attractive force that varies. See http://chemistry.osu.edu/~woodward/ch121/ch7_radius.htm.) An element should be compressible with a force predicted by the ideal gas laws until the outer electron shells overlap. Further compression requires the expulsion of the electrons, producing a repulsion calculable by the Coulomb force between the ions, given their distance, and the net charges. So the compressive force needs to fare from the ideal gas laws to that plus the Coulomb force as the distance between atoms crosses this threshold. For atoms heavier than helium, with more than one electron shell, there will be additional threshold, as each shell is evacuated. For instance, if hydrogen atoms are pressed together until the k-shells overlap, the atoms will be 1.0584e-10 m apart. Up to that density, we have only the ideal gas laws. Past that density, it's that plus the Coulomb forces.
But that implies that the k-shell radius determines the liquid density, and the numbers don't match up. The mass of a hydrogen atom is 1.67e-24 g. At the 1.0584e-10 m spacing, we'd get a density of 1.4085328418683 g/cm3. Aspden and others have concluded that the Sun is all hydrogen compressed to the k-shell radius, which seems convincing as it's a heckuva coincidence that those numbers match the density of the Sun so precisely. And yet the known density of liquid hydrogen is 0.07085 g/cm3, or 1/20 of the density defined by the k-shell radius. So some other force kicks in, long before the electrons are getting expelled.
Some have suggested that it's magnetic pressure between conflicting electron spins. This would seem to make sense, in that it explains why the density within each elemental period varies. It density was just a function of the radius of the outermost shell, all of the elements in each period would have the same liquid density. But the density is greater in the middle of the period. This is because the atomic mass is increasing, but once the outer shell is halfway populated, you start getting spin conflicts, and the density starts going down, even as the atomic mass increases.
Whether or not spin conflicts could expel electrons is unknown. It's possible that this simply puts the electrons into higher energy states, and the back-pressure is the difference between the lower and higher states. So perhaps we could use the ideal gas laws down to the liquid density. From there, we could fare from no additional force, to the full Coulomb force between ions once past the shell radii, totally ignoring the nature of the repulsion that emerges between the liquid density and the shell conflict density, and simply interpolating the unknown force.
But this would oversimplify more than just that. If the temperatures are great enough that there are no bound electrons, the shells just aren't there, and the physics gets a lot more complicated. There will still be ionization due to gravity, but it will be more a matter of gravity acting 1836+ times more forcefully on the atomic nuclei than on the electrons. The ions still repel each other, as do the free electrons. But the ions are subject to 1836+ more gravitational force, and therefore, there will be more of them nearer the center of gravity.
The last complexity that needs to be considered is that while hydrogen and helium might become fully ionized, and thereafter behave as so many disassociated atoms and electrons, the heavier elements are not likely to become fully ionized, no matter the temperature and pressure. So it will be a mixed environment, in which some electrons have been liberated by temperature, and some by pressure. The problem becomes fully non-linear when the temperature cannot be known until the liquid lines have been set, and the electrostatic potentials estimated, which requires that the densities and pressures be known.

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