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Stanton–Murillo transport

Ionic transport coefficients for high-energy-density matter, from the effective-Boltzmann collision integrals of Phys. Rev. E 93, 043203 (2016).

Γ—
κ—
g—
regime—

Mixture

SpeciesZAfrac

Uncheck to set Z by hand. TF underestimates Z for low-Z species near full ionization.

Conditions

 

Coefficients

Screening model

Eq (35): Debye–Hückel with the ion-sphere floor that keeps λ from collapsing below ai at strong coupling. This is the recommended model.

Validity map in the Yukawa plane, Eqs (84)–(85). SMT is a binary-collision theory, so it is most trustworthy in the green band where the velocity autocorrelation decays monotonically.

Derived quantities

Model, units and verification

What is evaluated

The effective-Boltzmann approach: a screened-Coulomb (Yukawa) effective potential is used inside a Chapman–Enskog solution of the Boltzmann equation, so both strong scattering and screening are handled without an adjustable cutoff.

Everything reduces to one function of one variable — the reduced collision integral K_nm(g), where

g = Z_i Z_j e² / (λ_eff T)

evaluated from the published fits, Eqs (C22)–(C24) with Table IV.

Screening length

The central result of the paper. Plain Debye–Hückel overscreens at strong coupling, driving λ below the interionic spacing. Eq (35) floors it at the ion-sphere radius:

1/λ²_eff = 1/λ²_e + Σ_i (1/λ²_i)(1/(1+3Γ_i))

Electrons enter through the finite-temperature Thomas–Fermi length, Eq (25), so degeneracy is included. Switch models in the sidebar to see how much this matters — it is the difference between agreeing and disagreeing with MD above Γ ~ 1.

Coefficients

D = 3T^(5/2) / (16 sqrt(pi m_i) n Z^4 e^4 K11) D_ij = 3T^(5/2) / (16 sqrt(2 pi mu) n Zi^2 Zj^2 e^4 K11) eta = 5 sqrt(m_i) T^(5/2) / (16 sqrt(pi) Z^4 e^4 K22) K = 75 T^(5/2) / (64 sqrt(pi m_i) Z^4 e^4 K22)

Reduced forms use D* = D/(ω_p a²), η* = η/(m n ω_p a²), K* = K/(n ω_p a²). Note the conductivity normalizes by n, not m.

Verification

  • All four K_nm fits are continuous across g = 1 to better than 0.003%, which independently confirms the Table IV coefficients.
  • Self-diffusion reproduces the paper's MD data (Table I) to 4.7% mean absolute error for Γ ≤ 2, degrading at strong coupling exactly as Fig. 10 shows. See the Validation vs MD plot.
  • D*, η*, K* match Eqs (56), (75), (82) to 9 significant figures.
  • Thomas–Fermi gives Z = 2.99 for solid aluminium at 10 eV, and 12.3 at 1 keV.

Limits and caveats

  • Mixtures: Dij is exact, and the mixture screening length feeds every coefficient. But η and K are reported per species using the pure-species formula — the mixture-averaged η_tot and K_tot need the Chapman–Enskog matrix inversion of Appendix B, which is not implemented yet.
  • Strong coupling: above the caging boundary the binary-collision picture fails. η and K in particular do not reproduce their minima.
  • Static screening only. The velocity-dependent screening of Eq (87) changes the coefficients by ~10% near Γ ~ 1 at small κ.
  • Ionization: Thomas–Fermi is crude. Override Z by hand where you have better numbers.

Regime boundaries

From the velocity-autocorrelation classification in Sec. VI:

Γ_osc = 5.97 + 1.93k + 1.16k² + 1.44k³ Γ_cage = 34.1 + 17.1k² - 13.6k³ + 5.39k⁴

Below Γ_osc the autocorrelation decays monotonically and the Boltzmann picture is sound. Between the two it oscillates but stays positive; SMT still does well there. Above Γ_cage the ions are caged by their neighbors and a binary cross section cannot describe the dynamics.