Repulsive spherical barrier scattering length
= Repulsive spherical barrier scattering length
{title2=$a_s=a-\tanh(\gamma a)/\gamma$}
For a barrier of radius $a$ and potential $\hbar^2\gamma^2/(2m)$, matching the regular inner hyperbolic-sine wave to the outer sine wave gives $a_s=a-\tanh(\gamma a)/\gamma$. The low-energy $s$-wave <scattering amplitude> is $-a_s$.