An analytical expression is developed for calculating the average energy deposited by an ion in a volume with dimensions less than the range of the secondary electrons produced. The expression is obtained by including two additional terms in an energy-restricted linear-energy-transfer equation. The usual energy-restricted expression accounts for energy deposited in the volume by energy transfers less than a certain cutoff value. The modified expression, due to the two additional terms, also accounts for energy deposited in the volume which results from energy transfers greater than the cutoff value. The additional terms therefore convert the energy-restricted equation to a distanceor spatially restricted equation. The method is used to obtain radial dose profile information on ion tracks and to calculate the energy deposited by an ion randomly incident upon spherical and hemispherical target sites. Results are in close agreement with more complex methods reported previously for ions with energies from 0.25 to 1000 MeV/amu in volumes of water vapor with dimensions from 1 nm to 10 μm. There are no fitted parameters in this general approach, and all the necessary input data are readily available.

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