(adsbygoogle = window.adsbygoogle || []).push({}); Problem

Determine the differential and total cross sections for scattering from a "hard

sphere" of radius R ("hard" means impenetrable).

Relevant Formulae

[tex]\sigma (\Omega) d \Omega = \frac{number \ of \ particles \ scattered \ into \ solid \ angle \ per \ unit \ time}{incident \ intensity}[/tex],cross section of scattering

[tex]d \Omega = 2 \pi sin \Theta d \Theta[/tex],solid angle, with [tex]\Theta[/tex] as the scattering angle

[tex]s = \frac{l}{\sqrt{2mE}}[/tex],for impact parameters "s" and angular momentum "l"

[tex]\sigma (\Theta) = \frac{s}{sin \Theta} \vert \frac{ds}{d \Theta} \vert[/tex],differential cross section

[tex]\sigma_{T} = \int \sigma(\Omega)d\Omega = 2 \pi \int \sigma (\Theta) sin \Theta d\Theta[/tex],total scattering cross section

Attempt at Solution

Presumable, I need to find an equation between the impact parameter s and the scattering angle. From there I can answer both questions.

Perhaps if I found the equation of an orbit with eccentricity, and then substituted the angular momentum expression above in, I could arrive at such an expression. (This is what the author of our text did for the Coulomb potential.)

As far as I can see, the potential for a hard sphere is

[tex]V(r) = \left\{ \begin{array}{rcl} \infty & \mbox{for} & |r|<R \\ 0 & \mbox{for} & |r|>R \end{array}\right[/tex]

But I don't see how I can derive an orbit equation, as this isn't a potential that allows for orbits!

I think I need some expression relating the above potential V with angular momentum l, so that I can substitute in the expression for s, and work from there.

Any thoughts?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Differential Cross Section for scattering by Hard Sphere

**Physics Forums | Science Articles, Homework Help, Discussion**