Moment-based effective transport equations for energy straggling.

Ion energy straggling is accomodated in condensed history (CH) Monte Carlo simulation by sampling energy-losses at the end of a fixed spatial step from precomputed, pathlength dependent energy-loss distributions. These distributions are essentially solutions to a straight ahead transport equation given by {partial_derivative}{psi}(s,E)/{partial_derivative}s = {integral}{sub Q{sub min}}{sup Q{sub max}} dQ {sigma}{sub e}(E,Q){psi}(s, E + Q) - {sigma}{sub e}(E){psi}(s,E), 8 {ge} 0, with monoenergetic incidence {psi}(0, E) = {delta}(E{sub 0} - E). In Eq.(1), s is the pathlength variable, {sigma}{sub e}(E,Q) is the differential cross section for energy loss Q, typically given by the relativistic Rutherford cross section for hard collisions, {sigma}{sub e}(E) is the total ion-electron scattering cross section, and Q{sub min} and Q{sub max} are, respectively, the minimum and maximum energy transfer per collision. Direct solution of Eq.( 1) by stochastic or deterministic numerical techniques is not feasible because of the very small energy transfers and very small mean free paths that characterize charged particle interactions. Condensed history codes typically employ an approximate solution due to Vavilov, obtained assuming a constant mean free path and thus restricted to short step sizes. This solution is formal and its numerical evaluation can be computationally laborious, especially for small step sizes. In practice, Monte Carlo codes have incorporated the Vavilov theory through elaborate numerical approximations, such as truncated Edgeworth expansions, curve-fitting approximations using Moyal functions for small penetration depths or higher energies, and special treatments for the large energy-loss tail of the distribution. In this paper we propose an alternative approach which is also valid under the conditions of the Vavilov theory but has the potential of being computationally more efficient