The any-angle formalism

Goudsmit and Saunderson [2,3] presented a formal solution of the multiple-scattering problem in the form of an expansion in Legendre polynomials $P_l(\cos \theta)$, which is valid for any-angle scattering:

$$F_{{\rm GS}}(\cos \theta) = \sum_{l=0}^\infty \left(l + {1 \over 2} \right) P_l(\cos \theta) \exp(- \lambda Q_l)~,$$ (3)

where $Q_l$ denotes the moments of the single-scattering distribution,

$$Q_l = \int_{-1}^1 {\rm d}(\cos\chi)\,\tilde{\sigma}(\cos \chi) \Big[ 1 - P_l (\cos \chi) \Big]~.$$ (4)

Bethe [7] and Winterbon [17] have discussed some of the approximations required to obtain the small-angle expression, Eq. (1), from the Goudsmit-Saunderson (GS) series. Bethe has proposed a correction factor $\sqrt{\sin \theta/\theta}$ to improve the small-angle multiple-scattering approximation at large angles, while Winterbon discusses higher-order corrections.