= Zeros of the Born amplitude of a spherical step potential
{c}
{title2=$\tan(qr_0)=qr_0$}
For a weak constant spherical potential of radius $r_0$, the <scattering amplitude> is proportional to $[\sin x-x\cos x]/x^3$, where $x=qr_0$. Its forward limit is nonzero. Positive zeros satisfy $\tan x=x$, with first root $x_1\simeq4.49341$, 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.
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