Scattering length and effective range for charged-particle scattering in a plane and in higher dimensions

The concepts of scattering length a and effective range re previously introduced for low-energy scattering from a potential V(r) in a plane and in higher dimensions are extended to include a 1/r potential (strength parameter γ). Both a and re have the physical significance of being equal to the radius of an equivalent hard sphere giving rise to the same O(k0) and O(k2) terms in the expression for the phase shift. The method used is based on the properties of the ``local scattering length'' a(r,γ) for the potential V(r) cut off at radius r and an ``equivalent hard-sphere radius'' a(r,k,γ) for wave number k≠0. It is shown that these quantities have a smooth behavior for γ-->0 and for dimension n-->2