Determination of the depth of 50% of maximum ionization, I50, for electron beams by the divided difference method

A new characterization of depth-ionization parameters for electron beams is empirically deduced from our data analysis based on the divided difference method (the DD method), which employs the numerical differential of an ionization curve. The important feature of the present method is that it does not necessarily require normalized percent depth-ionization (NPDI) data. The depth of 50% of maximum ionization, I50, which is an important parameter for electron beam dosimetry, can be deduced from the analysis of an unnormalized (or partial) depth-ionization (UDI) curve obtained over a short interval of depth. The values of I50 determined by the DD method are in agreement to within 0.1 mm for energies of 4, 6, and 9 MeV, compared with the ones determined by the TG-51 protocol method (or the conventional method), and the difference was 0.9 mm for 12 and 15 MeV. The dose at the reference depth, dref, calculated from I50 by the DD method, is found to be in agreement with TG-51 to within 0.1%. The field size dependence of the DD method using UDI data was studied for three field sizes: 6 x 6, 10 x 10, and 20 x 20 cm2. For all energies, the discrepancies of I50 as determined by both methods were 0.9 mm on average for the 6 x 6 cm2 fields and 0.6 mm for the other two field sizes. This dependence was remarkable for 6 x 6 cm2 fields for 12 and 15 MeV, and the discrepancies shown by the DD method were 1.2 mm for 12 MeV and 1.8 mm for 15 MeV, respectively. Since the reference field size in clinical dosimetry is usually 10 x 10 cm2, this dependence will not affect clinical dosimetry. The DD method could be an alternative option for checking beam quality in dose calibration.