Collection efficiencies of ionization chambers in pulsed radiation beams: an exact solution of an ion recombination model including free electron effects

Abstract

Abstract Objective. Boag et al (1996) formulated a key model of collection efficiency for ionization chambers in pulsed radiation beams, in which some free electrons form negatively charged ions with a density that initially varies exponentially across the chamber. This non-uniform density complicates ion recombination calculations, in comparison with Boag’s 1950 work in which a collection efficiency formula, f, was straightforwardly obtained assuming a uniform negative ion cloud. Boag et al (1996) therefore derived collection efficiency formulae f′, f″ and f′″ based on three approximate descriptions of the exponentially-varying negative ion cloud, each uniform within a region. Collection efficiencies calculated by Boag et al (1996) using these formulae differed by a maximum of 5.1% relative (at 144 mGy dose-per-pulse with 212 V applied over a 1 mm electrode separation) and all three formulae are often used together. Here an exact solution of the exponentially-varying model is obtained. Approach. The exact solution was derived from a differential equation relating the number of negative ions collected from within some distance of the anode to numbers of ions initially located within that region. Using the resulting formula, f exp, collection efficiencies were calculated for a range of ionization chamber properties and doses-per-pulse, and compared with f, f′, f″ and f″′ values and results from an ion transport code. Main results. f exp values agreed to 5 decimal places with ion transport code results. The maximum relative difference between f exp and f″′, which was often closest to f exp, was 0.78% for the chamber properties and doses-per-pulse studied by Boag et al (1996), rising to 6.1% at 1 Gy dose-per-pulse and 2 mm electrode separation. Significance. Use of f exp should reduce ambiguities in collection efficiencies calculated using the approximate formulae, although like them f exp does not account for electric field distortion, which becomes substantial at doses-per-pulse ≥100 mGy.