ISQ derivation ("derivation in SI units") of a formula for the electrostatic field ionization rate-constant for a hydrogenic atom in its ground electronic state

TL;DR

This paper provides a detailed derivation in SI units of a formula for the electrostatic field ionization rate-constant of hydrogenic atoms, emphasizing the dependence on ionization energy and clarifying theoretical approaches.

Contribution

It offers the first comprehensive ISQ derivation of the ESFI rate-constant formula, connecting it with established models and highlighting differences in theoretical treatments.

Findings

Ionization energy appears as I^(5/2) in the pre-exponential.

Defines a universal 'field ionization constant'.

Relates ISQ formula to Gurney and Condon's attempt frequency.

Abstract

Technical applications using electrostatic field ionization (ESFI) need a formula for the rate-constant K_e for the free-space ESFI of a hydrogenic atom in its ground electronic state. This formula must indicate the dependence on ionization energy I. Existing formulae were derived using atomic units. However, many scientists work with ESFI as an important technical process, but are not familiar with the Gaussian or atomic units systems. In the 1970s, the present International System of Quantities (ISQ), which includes the equations behind SI units, was designated as the main system for university teaching and the communication of scientific equations. 40 years on, ISQ derivations of ESFI rate-constant formulae are still not easily found, but are now needed. This tutorial paper gives a detailed ISQ derivation of a formula for K_e. The derivation is closely modelled on the Landau and Lifshitz (LL) approach used in their 1958 work (in atomic units) on hydrogen atom ESFI. The ISQ derivation confirms that, for hydrogenic atoms, ionization energy appears in the pre-exponential as I^(5/2), and defines a universal "field ionization constant". It also shows how the ISQ formula relates to the Gurney and Condon "attempt frequency" form for tunnelling rate-constants, and an ISQ formula is given for the motive energy in the related JWKB integral. The ISQ derivation uses a motive-energy transformation analogous to a transformation used by LL. The need for this transformation in ESFI theory raises questions as to the correctness of theoretical treatments of field electron emission from non-planar emitters, which do not make this transformation.