Zeros of the Born amplitude of a spherical step potential ($\tan (q{r}_0)=q{r}_0$)

For a weak constant spherical potential of radius $r_0$, the scattering amplitude is proportional to $[\sin x-x\cos x]/x^3$, where $x=q{r}_0$. Its forward limit is nonzero. Positive zeros satisfy $\tan x=x$, with first root $x_1\simeq 4.49341$, so an elastic incident wavenumber needs $k\ge x_1/(2r_0)$ to access a zero. The asymptotic roots are $x_n=\pi (n+1/2)-1/[\pi (n+1/2)]+O(n^{-3})$; increasing $k$ alone does not remove the low-order root correction.