Collection efficiencies of ionization chambers in pulsed radiation beams: an exact solution of an ion recombination...

Objective.Boaget 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. Boaget al(1996) therefore derived collection efficiency formulaef',f″ andf'″ based on three approximate descriptions of the exponentially-varying negative ion cloud, each uniform within a region. Collection efficiencies calculated by Boaget 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 withf,f',f″ andf″' values and results from an ion transport code.Main results.f_expvalues agreed to 5 decimal places with ion transport code results. The maximum relative difference betweenf_expandf″', which was often closest tof_exp, was 0.78% for the chamber properties and doses-per-pulse studied by Boaget al(1996), rising to 6.1% at 1 Gy dose-per-pulse and 2 mm electrode separation.Significance.Use off_expshould reduce ambiguities in collection efficiencies calculated using the approximate formulae, although like themf_expdoes not account for electric field distortion, which becomes substantial at doses-per-pulse ≥100 mGy.