### Article

## Scattering power of radiotherapy protons

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Published: | September 24, 2009 |
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### Outline

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Moliere theory predicts accurately the total multiple scattering in a slab of arbitrary thickness, without using the concept of scattering power T=d<theta^{2}>/dx. However, some such concept (local rate of change) is needed for any nontrivial transport calculation, whether deterministic or Monte-Carlo. Therefore we are motivated to find an approximate integrable differential description of Moliere theory.

Well known formulas for T that depend only on *local* variables at x (kinetic energy and material properties), when integrated, do not give the correct total <theta^{2}> because they ignore the single scattering correction (SSC), the effect on T of the increasing dominance of the Gaussian core over the single scattering tail of the multiple scattering angular distribution.

Recently two *nonlocal *formulas have been proposed which do considerably better. The first, by Schneider et al, parametrizes the SSC for mixed slabs in terms of a generalized normalized depth variable. The second, by Kanematsu, uses an auxiliary "radiative pathlength" integral. We propose a third nonlocal formula which corresponds still more closely to Moliere theory. It parametrizes the SSC in terms of the logarithm of the decrease in the kinematic factor pv from the stack entry to the point of interest.

We will derive the formula, demonstrate its accuracy in the single slab case (the only case for which extensive experimental data exist), and discuss the kinds of transport calculations for which the improvement does or does not matter. One in particular, the upstream modulator problem in passive beam line design, provides a good way of discriminating between the six existing formulas for T experimentally.

The derivation and discussion are carried out for protons, where multiple scattring is more of a problem, but generalization to heavier ions is trivial.