Mixture mean ionization -- multispecies Thomas-Fermi average atom

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Mixture

SpeciesZAn (cm⁻³)

Preset mixture

Conditions

Electron temperature eV

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Show isolated-species Z

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⟩ = Σ_a x_a Z*_a = n_e / n_ion Number-weighted. This is the one that conserves electrons, so use it for quasineutrality, equation of state, and anywhere you need n_e.

Z_eff Z_eff = ⟨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 Z_eff 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.

Dependency-free solver. The per-species Z* it returns is exactly what the SMT transport calculator takes as input.

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.