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Need some help with cross sections

  1. Mar 29, 2006 #1
    Hello all,

    I have a fellow grad student who is convinced that the differential cross section:
    [tex] \frac{d\sigma}{d\Omega}[/tex]
    is truly a ratio of differentials. That is you have an infinitesimal cross section divided by an infinitesimal solid angle.

    I contend that when we write
    [tex] \frac{d\sigma}{d\Omega}[/tex]
    we really mean
    [tex] \sigma(\Omega) [/tex] and that we only call it the differential cross section because the integral of the differential cross section over the physical range of the variable gives the total cross section.
    I stated this along with the fact that the total cross section is simply a number. You sample an event (lets say pp->pp is the event in question) at different energies and record the number of particles that come out of the reaction at the energy. This gives you a number.
    The derivative of the number with respect to any variable is zero. So the differential cross section- is not the derivative of the cross section.
    He contends this is not true because when you do the "experiment" you have a finite width detector and this smears out the solid angle so it is no longer an infinitesimal.

    Can anyone find a very clear discussion somewhere about this fact? Or maybe present one? Or am I simply wrong and it truly can be thought of as a ratio of differentials?
  2. jcsd
  3. Mar 29, 2006 #2


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    Science Advisor

    Mathematics answer. There is no such concept as a ratio of differentials. What you have is a derivative, defined in the usual way, and it is the cross section as a function of direction.

    Physics contribution. The numbers that are used for the cross sections are obtained by experiments as described.
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