Differential Cross Section for scattering by Hard Sphere

In summary: The Born approximation is not applicable to hard spheres. Partial wave expansions are not applicable either.
  • #1
logic smogic
56
0
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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  • #3
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.
 
  • #4
malawi_glenn said:
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.
 
  • #5
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

:-)
 
  • #6
malawi_glenn said:
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.

olgranpappy said:
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.
 
  • #7
logic smogic said:
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.
 

1. What is the Differential Cross Section for scattering by Hard Sphere?

The Differential Cross Section for scattering by Hard Sphere is a mathematical quantity used to describe how particles are scattered by a hard sphere. It is a function of the scattering angle and the properties of the hard sphere, such as its size and composition.

2. How is the Differential Cross Section calculated?

The Differential Cross Section is calculated by taking the ratio of the scattered particle flux to the incident particle flux. This ratio is then multiplied by a correction factor that takes into account the size and shape of the hard sphere.

3. What is the significance of the Differential Cross Section in particle physics?

The Differential Cross Section is an important tool in understanding the interactions between particles in particle physics. It allows scientists to study the properties of particles and the forces that govern their interactions.

4. How does the Differential Cross Section change with different scattering angles?

The Differential Cross Section typically increases as the scattering angle increases, reaching a maximum at a certain angle before decreasing again. This behavior is known as the diffraction peak and is a characteristic feature of scattering by hard spheres.

5. Can the Differential Cross Section be used to study other types of scattering besides hard sphere scattering?

Yes, the Differential Cross Section can be used to study various types of scattering, including elastic and inelastic scattering. It is a versatile tool that can be applied to different scattering systems, making it a valuable tool in many areas of physics.

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