The stochastics of the positive ion penumbra

Statistical analysis of Monte Carlo simulations of positive ion track structure is presented for 1 MeV protons passing outside an absorbing region of interest. Energy deposition and production of ionization within the region occur only by delta-ray transport from the ion's path. Depositions in spherical absorber regions of 2 to 100 nm diameter in unit density tissue were scored. It is shown that production of delta rays sufficiently energetic to reach the region is infrequent and the delta rays are therefore relatively far apart along the ion path compared to their average range. The probability for any delta-ray energy deposition in a distant site is small and proportional to the solid angle intercepted by the site. The functional dependence of the frequency density distributions in energy imparted and in ionization number, conditional on there being some interaction, is approximately exponential, and the first and second moments of the distributions are largely independent of distance from the ion path. These findings confirm similar conclusions from experimental measurements on the track structure of germanium, iron and uranium ions.