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logic smogic

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**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?

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