- #1
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?
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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