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This is a followup question to this pretty good answer regarding deriving the Boltzmann equation.

What if the center of the target particle is actually not the same with the scattering center (or may not have a scattering center)? For example, see the classical problem of hard-sphere scattering.

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lphys
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  • I'm confused. For hard sphere scattering one can choose to place the measurement apparatus such that the center of the hard sphere corresponds to the center of the apparatus. Could you clarify your question? – joshphysics Aug 04 '13 at 17:41
  • Yes, the center of the apparatus is not the problem here. What I means is that, for example, we draw the trajectory of a scattered particle with impact parameter b which results in scattering angle teta. Draw another particle with another impact parameter. Because they have different impact parameter and point of collision with the sphere, their directions are not pointed into the center of the sphere, causing a scattering which doesn't have single center. Sorry, it's very difficult to explain it by using words but I hope you could understand my point. Thank you very much :) – lphys Aug 05 '13 at 10:31

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The validity of the reasoning leading to the scattering formula $$ D(\theta) = \frac{b(\theta)}{\sin\theta}\left|b'(\theta)\right| $$ derived in the referenced post does not require that the scattering of all incident particles happens at a single point. One does, however, require that the impact parameters $b$ of the incoming particles can be written as a function of their outgoing angles $\theta$.

In fact, there are types of targets for which there is no single point at which the particles change direction, the point that is described as the "point of collision" in the comment above. For example, in Coulomb scattering, unless an incoming particle has zero impact parameter, it will simply continuously change direction as it moves closer to the force center and as it moves away from the force center. In such cases, the impact parameter and scattering angle are defined asymptotically.

joshphysics
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  • Yeah.. if that is the case, How would you put the detector? It makes no sense because actually a particle with a scattering angle of θ may not reach the detector in the polar angle of θ because of the off-center scattering. – lphys Aug 06 '13 at 11:53
  • The detector, which is basically assumed to be a very large sphere, should be positioned such that the incoming beam of particles is radial. Other than that, it doesn't really matter how you position the detector. If the target is a bit off-center, then the outgoing particles will simply scatter differently which is fine. It's not clear to me what your concern is in your second sentence. – joshphysics Aug 07 '13 at 18:02