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I'm working on a problem where I'm trying to find the differential cross section of a surface described by $f(z)=Az^n$, $n>1$, with $f(z)$ rotated around the z axis to make a 3D shape. Particles travel parallel to the z axis.

I figure I need to get a relation between the impact parameter $b$ and the scattering angle $\theta$. A particle scatters off the target when $b=f(z_0)$. I also know that the particle hits the surface when the surface has a slope of $Anz_0^{n-1}$, but I don't know how to use that to my advantage.

This feels like a trig problem I can't wrap my head around.

Any help or hints?

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You know the gradient of the tangent at $z=z_{\rm o}$ and so then you can find the gradient of the normal to the tangent at $z=z_{\rm o}$.

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The gradient of the normal can be related to $\tan \theta$.

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