Derivatives of the Classical Deflection Function

Abstract

Integral expressions have been derived for the second and third derivatives with respect to the impact parameter of the classical deflection function. These are then given in a useful form for numerical quadrature and their evaluation is discussed. The expressions are investigated for their behavior as various quantities become zero, and in particular, formulas valid at zero impact parameter are obtained. Tables of results are presented for (12,6) and (12,4,6) potentials, the former given by V(r) = 4ε[(σ / r)12 − (σ / r)6], and the latter by V(r) = (ε / 2) [(1 + γ) (rm / r)12 − 4γ(rm / r)6 − 3(1 − γ) (rm /r )4], in which ε, σ, rm, and γ are adjustable parameters.