Derivation of a formula describing the saturation correction of plane-parallel ionization chambers in pulsed fields with arbitrary repetition rate

Gas-filled ionization chambers are widely used radiation detectors in radiotherapy. A quantitative description and correction of the recombination effects exists for two cases, for continuous radiation exposure and for pulsed radiation fields with short single pulses. This work gives a derivation of a formula for pulsed beams with arbitrary pulse rate for which the prerequisites of the two existing descriptions are not fulfilled. Furthermore, an extension of the validity of the two known cases is investigated. The temporal evolution of idealized charge density distributions within a plane parallel chamber volume is described for pulsed beams of vanishing pulse duration and arbitrary pulse repetition rate. First, the radiation induced release, movement and collection of the charge carriers without recombination are considered. Then, charge recombination is calculated basing on these simplified charge distributions and the time dependent spatial overlap of positive and negative charge carrier distributions. Finally, a formula for the calculation of the saturation correction factor is derived by calculation and simplification of the first two terms of a Taylor expansion for small recombination. The new formula of saturation correction contains the two existing cases, descriptions for exposure by single pulses and continuous irradiation, as limiting cases. Furthermore, it is possible to determine the pulse rate range for which each of the three descriptions is applicable by comparing the dependencies of the new formula with the two existing cases. As long as the time between two pulses is lower than one third of the collection time of the chamber, the formalism for a continuous exposure can be used. The known description for single pulse irradiation is only valid if the repetition rate is less than 1.2 times the inverse collection time. For all other repetition rates in between the new formula should be used. The experimental determination by Jaffe diagrams can be applied for all repetition rates which are not single pulse irradiation but will result in a small deviation from the correct saturation correction value. (paper)