What is the "Critical impact parameter" for photons of a Black hole with a Radius $r$? Here I'm defing the Critical impact parameter $C$ as the value such that.
A photon with an impact parameter > C will be deflected by the black hole.
A photon with am impact parameter < C will be pulled into the black hole.
The impact parameter $b$ of a scattering orbit is given by (in units with $G=c=1$)
$$ b= \frac{L}{E}$$
A critical photon trajectory will have the same ratio $L/E$ as the photon orbit. This we can calculate by taking the expressions for $E$ and $L$ for circular orbits in Schwarschild spacetime:
$$ E= M\frac{r-2M}{\sqrt{r(r-3M)}}$$
and
$$ L= M^{3/2}\frac{r}{\sqrt{(r-3M)}}$$
Taking the ratio and the limit $r\to 3M$ (i.e. the photon radius) you find
$$ b= 3\sqrt{3} M$$
or (restoring $G$ and $c$),
$$ b= 3\sqrt{3} \frac{GM}{c^2} $$.
Using Schwarzschild metrics, the circular orbit implies $ \frac{dr}{d\varphi}=0 $ which means $ \frac{r^3}{b^2}-r+R_s=0 $, with $ R_s=\frac{2GM}{c^2} $ and $ b $ impact parameter.
The discriminant of this equation is $ \Delta=4b^2-27R_s^2 $, which is zero for $ b=\frac{3\sqrt{3}}{2}R_s=3\sqrt{3}\frac{GM}{c^2} $.
Hoping to have answered your question,
Best regards.
