= Solution
A photon arriving from infinity crosses the barrier precisely when $c^2k^2/2$ exceeds the maximum in part (c):
$$
\boxed{k^2>\frac{h^2}{27\mu^2c^2}.}
$$
At infinity its spatial tangent has speed in affine-parameter units $ck$, while $h$ is its angular momentum per unit affine normalization. A straight asymptotic ray with <impact parameter> $b$ therefore has $|h|=bck$. Thus
$$
\boxed{b=\frac{|h|}{ck},\qquad b<3\sqrt3\mu.}
$$
At equality, the incoming ray approaches the unstable <photon sphere> asymptotically and is not captured. A larger impact parameter gives an outer turning point and scattering back to infinity.
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