TEMPEST · interactive · atomic physics

Mixture mean ionization

Multispecies Thomas–Fermi average atom. In a mixture the species are coupled: each claims the subvolume that puts it in pressure equilibrium with its neighbors, so Z depends on what else is in the cell.

⟨Z⟩—
Zeff—
ne (cm⁻³)—
solve—

Mixture

SpeciesZAn (cm⁻³)

Conditions

 

Compare

Each species evaluated alone at its own partial density, ignoring the coupling. This is the shortcut the mixture solver replaces — the gap between the two is the effect Appendix A exists to capture.

Ion spheres at their solved radii. Dashed circles are the equal-volume starting guess. Higher-Z species end up with larger effective volumes — this is Fig. 13 of the paper, solved live.

Derived

Which average to use

⟨Z⟩ ⟨Z⟩ = Σa xa Z*a = ne / nion Number-weighted. This is the one that conserves electrons, so use it for quasineutrality, equation of state, and anywhere you need ne.
Zeff Zeff = ⟨Z²⟩ / ⟨Z⟩ Collisional weighting. Scattering goes as Z², so this is what transport coefficients, coupling strength and bremsstrahlung respond to.

They coincide only for a single species. For a mixture Zeff is always the larger of the two, and the gap grows with the spread in Z — quoting one where the other is meant is a quiet way to be wrong.

Model, verification and the (A6) correction

What is solved

A cell is tiled by atomic subvolumes, one per ion, and four conditions hold at once:

Z*_a = Zbar(Z_a, v_a, Te) (A1) Z*_a = v_a n_e (A2) n_e = sum_a n_a Z*_a (A3) sum_a n_a v_a = 1 (A4)

(A2) is pressure equilibrium in the Thomas–Fermi picture: two subvolumes balance when they present the same electron density at their boundary. A higher-Z species therefore claims a larger effective volume at the same number density, which is why a mixture cannot be treated one species at a time.

The (A6) correction

The paper's Newton iteration updates the electron density as n_e ← n_e + (sum n_a v_a − 1). Read literally this adds a dimensionless residual to a number density. With n_e of order 1023 the correction is of order 1, so n_e never moves and the iteration stalls at a state violating (A4) by about 2%.

This app uses the dimensionally consistent form with the same fixed point, n_e ← n_e × sum n_a v_a, which satisfies (A1)–(A4) to 10−14 and agrees to 10−14 with an independent nested-bisection solve. Presumed a typo; the physics in (A1)–(A4) is unaffected.

Verified against

  • Single species reduces to the More fit exactly (relative difference 0 to 10−16).
  • All four constraints hold to ~10−14 from 0.1 eV to 10 keV.
  • Two independent solvers agree to 4×10−14 or better.
  • H, D and T return bit-identical Z*, as the paper notes they must.
  • Fig 14, left: HDTCO gives C → 5.99 and O → 7.98 at 10 keV.
  • Fig 14, right: the Z = 1..10 ladder gives 1.00 to 9.93 at 10 keV, and finite pressure-ionized values of 0.81 to 4.65 at 0.1 eV.

Which average

“Mean ionization” is ambiguous for a mixture, so three are reported:

<Z> = sum_a x_a Z*_a = n_e / n_ion <Z^2> = sum_a x_a Z*_a^2 Z_eff = <Z^2> / <Z>

They differ, sometimes a lot. <Z> conserves electrons and is the one to use for quasineutrality; Z_eff is the collisional weighting and is what transport and bremsstrahlung respond to. Quoting one where the other is meant is a common way to be quietly wrong.

Limits

  • Thomas–Fermi has no shell structure, so it misses shell effects entirely and smooths over ionization thresholds. Above roughly 10 eV the paper argues the ionization is insensitive to model choice; below that, treat with care.
  • TF underestimates Z for low-Z species near full ionization — hydrogen reaches about 0.99, not 1, even at 10 keV.
  • No exchange and correlation, no relativity, no gradient corrections. Table I of the paper lists these as the next obvious improvements.

Feeding SMT

The per-species Z* here is the input the transport calculator needs, and the two use the same ion-sphere convention: SMT's Eq (34), a_i = (3 Z_i / 4π n_e)1/3, is identical to (A2) above.

Worth noting: computing each species' ionization independently at its own partial density is not the same thing, and the isolated-species comparison in the sidebar shows how far apart they get.