- #1
Norman
- 895
- 4
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
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