Non-Extensive Generalization of Double Ionization by Electron Impact: Calculations of Both Cross Sections and Rate Coefficients for ${\text{H}}^{-}$, $\text{He}$, and ${\text{Li}}^{+}$

We generalize Tweed’s seminal Born~I formalism for the electron-impact double ionization of two-electron atomic systems (${\text{H}}^{-}$, $\text{He}$, ${\text{Li}}^{+}$) to non-extensive plasma environments described by Tsallis $q$-statistics. The bare Coulomb interaction is replaced by a $q$-deformed screened potential, which introduces a unique form factor envelope into the momentum-transfer integrations. We rederive the differential, partial, and total cross sections, demonstrating that this screening regularizes the classical high-energy Bethe logarithm and, in the $q<1$ regime, induces diffraction-like oscillations in the angular cross sections. The model is calibrated against digitized historical experimental data to quantify how non-extensive screening deforms the vacuum baseline. Finally, coupling these cross sections to a $q$-Maxwellian distribution yields the macroscopic rate coefficients and reveals a novel subextensive critical-temperature threshold below which ionization is energetically forbidden.