# Classical scattering off a paraboloid

## Homework Statement

Particles are scattered (classically) from a paraboloid shape. The surface is given by the relation:

$$z = a \left(\frac{y^2+x^2}{R^2}-1 \right)$$
for x^2 +y^2 leq R^2 where a and R are constants. The particle is incident from z = -infinity with impact parameter s.
Show that there a minimum angle below which there is no scattering occurs given by

$$tan(\theta_{min}/2) = R/2a$$

where theta is the scattering angle.

## The Attempt at a Solution

I found $\tan(\theta/2) = R^2/2sa$. The problem implies that s cannot be greater than 1, which I fail to understand.

I am not sure why the restriction "x^2 +y^2 leq R^2" was included. Maybe that has something to do with this...

EDIT: I got it! x^2 +y^2 leq R^2 effectively cuts off the paraboloid at the x-y plane. The cross-section of the paraboloid on the x-y plane is the circle x^2+y^2 = R^2. Therefore the impact paramater cannnot be greater than R, which implies that $\tan(\theta/2) \geq R/2a$ as desired.

Last edited:

The last part of this problem asks: Does the $\sin^{-4}(\theta/4)$ dependence have anything to do with Rutherford scattering?