Generalized Equations for the Ionization of Solid Particles

Abstract

A generalized formula is derived for the ionization of spherical solid particles of submicroscopic size by allowing for multiple ionization. A system of simultaneous equilibrium equations is set up connecting successive states of ionization. Its solution yields a generalized formula that reduces to the Saha equation when only single ionization can occur. When the particles are multiply ionized, the equation can be simplified by the method of steepest descents, $$N_e{\mathrm{/N(rkT/e}}^2)\ln {\mathrm{(K/N}}_e\mathrm{)+}\frac{1}{2}{\mathrm{,\hspace{0.17em}\hspace{0.17em}(N}}_e\text{/N≳3),}$$ where N_e is the electron concentration, N is the number density of solid particles, r their radius and ${\text{K=2(2πmkT/h}}^2{)}^{\frac{3}{2}}$ exp(—[open phi]/kT). This equation is applied to the observations of Shuler and Weber on the ionization of free carbon particles produced in rich acetylene‐oxygen flames. Their data are consistent with a work function [open phi] = 4.35 ev, identical with that of graphite, and particle radii between 55 and 85 A.