Differential Cross Section for scattering by Hard Sphere

[logic smogic]

Problem
Determine the differential and total cross sections for scattering from a "hard
sphere" of radius R ("hard" means impenetrable).

Relevant Formulae

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

$$d \Omega = 2 \pi sin \Theta d \Theta$$, solid angle, with $$\Theta$$ as the scattering angle

$$s = \frac{l}{\sqrt{2mE}}$$, for impact parameters "s" and angular momentum "l"

$$\sigma (\Theta) = \frac{s}{sin \Theta} \vert \frac{ds}{d \Theta} \vert$$, differential cross section

$$\sigma_{T} = \int \sigma(\Omega)d\Omega = 2 \pi \int \sigma (\Theta) sin \Theta d\Theta$$, 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

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

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?

 

[malawi_glenn]

Here you can find all you need to calculate scattering problems;)

http://farside.ph.utexas.edu/teaching/qm/lectures/node66.html

The key-sections are "The Born approximation" and "Partial wave expansion"

 

[olgranpappy]

draw a circle at the origin (the hard sphere) and use geometry to relate the scattering parameter to the outgoing angle. sorry, it's hard to describe the proceedure w/out pictures and I don't have time to draw and upload one. But this is a pretty standard problem and is treated in a lot of texts. I think Griffiths reviews the classical calculation of hard sphere cross-section in his QM text.

 

[olgranpappy]

Here you can find all you need to calculate scattering problems;)

http://farside.ph.utexas.edu/teaching/qm/lectures/node66.html

The key-sections are "The Born approximation" and "Partial wave expansion"

The Born approximation doesn't work for a *hard* sphere.

Also, I wasn't sure whether the OP wanted the classical or the QM calcuation, but it sounded to me like he is interested in classical... in which case the partial wave expansion is also a red herring.

 

[malawi_glenn]

No it does not work in hard spheres, I should have said that Born Appro is in general where one starts with QM scattering. Then one introdues partial waves etc..

Ok well I have no idea if he wanted to do that in Classical mechanics

:-)

 

[logic smogic]

Here you can find all you need to calculate scattering problems;)

http://farside.ph.utexas.edu/teaching/qm/lectures/node66.html

The key-sections are "The Born approximation" and "Partial wave expansion"

Thanks for the link. That's a very helpful set of webpages, although I'm really just interested in the classical regime.

draw a circle at the origin (the hard sphere) and use geometry to relate the scattering parameter to the outgoing angle. sorry, it's hard to describe the proceedure w/out pictures and I don't have time to draw and upload one. But this is a pretty standard problem and is treated in a lot of texts. I think Griffiths reviews the classical calculation of hard sphere cross-section in his QM text.

Yep - on the train today I think I sketched out a rough solution. I just needed to back away from the terse way scattering was covered in my text, and think about it more basically. I'll post some of my solution if there's time later. Thanks.

 

[olgranpappy]

Yep - on the train today I think I sketched out a rough solution. I just needed to back away from the terse way scattering was covered in my text, and think about it more basically.

it's good to hear you made some progress. cheers.

 

[x3qt0r]

http://www.rpi.edu/dept/phys/Courses/phys410/lct11.pdf

Classical ( for hard sphere of radius R ) and Quantum scattering are treated.