Repulsive spherical barrier scattering length ($a_s=a-\tanh (\gamma a)/\gamma$)

For a barrier of radius $a$ and potential ${\hslash }^2{\gamma }^2/(2m)$, 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 $-a_s$.