Is the Differential Cross Section Truly a Ratio of Differentials?

[Norman]

Hello all,

I have a fellow grad student who is convinced that the differential cross section:
$$\frac{d\sigma}{d\Omega}$$
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
$$\frac{d\sigma}{d\Omega}$$
we really mean
$$\sigma(\Omega)$$ 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?
Thanks,
Ryan

 

[mathman]

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.