A semi-analytical procedure to determine the ion recombination correction factor in high dose-per-pulse beams

Background: The conventional theories and methods of determining the ion recombination correction factor, such as Boag theory and the related two voltage method and Jaffe plot extrapolation, do not seem to yield accurate results in FLASH /high dose per pulse (DPP) beams (>$>$10 mGy DPP). This is due to the presence of a large free electron fraction that distorts the electric field inside the chamber sensitive volume. To understand the influence of these effects on the ion recombination correction factor and to develop new expressions for it, it is necessary to re-visit the underlying physics. Purpose: To present a mathematical procedure to develop an analytical expression for the ion recombination correction factor. The expression is the basis for an extrapolation method so the correction factor can be determined in a clinical setting. Methods: A semi-analytical solution method, the homotopy perturbation method (HPM), is used to solve the partial differential equations (PDEs) describing the charge carrier physics, including space charge and free electrons. The electron velocity and attachment rate are modeled as functions of the electric field strength. An expression for the charge collection efficiency and ion recombination correction factor are developed. A fit procedure based on this expression is used to compare it to measured data from previously published articles. Another fit procedure using a general equation is also proposed and compared to the data. Results: The series obtained for the charge collection efficiency and the ion recombination correction factor are determined to be asymptotic series and the optimal truncation established. The ion recombination correction factor exhibits a 1/V2$1/V<^>2$ dependency due to the free electron presence. The fit using this expression agrees well with measured data as long as (1) the DPP is below 1 Gy for chambers with a 1 mm plate separation and (2) when the DPP is below 3 Gy for chambers with a 0.5 mm plate separation. In these DPP ranges, the deviation between measured and fit value did not exceed 6%. In both chamber cases the voltage range where the fit applies decreases as DPP increases. The general equation yielded comparable results. Conclusions: The HPM was shown to be applicable to a complex system of PDEs and generate meaningful and novel solutions, as they include both space charge and free electrons. The HPM also lends itself to other chamber geometries. The fit procedure was also shown to yield accurate results for the ion recombination correction up to the 1 Gy DPP level.