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.

 

[sir-pinski]

Hello Ryan,

Thank you for reaching out for help with cross sections. This is a common misconception among students, so you are not alone in your confusion. Let me try to clarify things for you.

First, let's define what a cross section is. A cross section is a measure of the probability of a particular interaction occurring between two particles. It is typically represented by the Greek letter sigma (σ) and has units of area. This is because it represents the effective area of the target that the incident particle must hit in order for the interaction to occur.

Now, let's look at the differential cross section. As you correctly pointed out, it is defined as the ratio of the infinitesimal cross section to the infinitesimal solid angle. However, this is just a mathematical representation and does not necessarily mean that the differential cross section is a true physical quantity. In fact, the differential cross section is not a physical quantity at all. It is a mathematical construct that is used to describe the behavior of the cross section as a function of the scattering angle.

The reason we use the term "differential" is because the total cross section is obtained by integrating the differential cross section over the range of scattering angles. This is a common practice in physics and is not unique to cross sections. For example, in calculus, we use the term "differential" to represent the infinitesimal change in a function. But when we integrate over that infinitesimal change, we get the total change in the function.

To address your friend's argument about the finite width detector, it is true that a detector will have some finite resolution. However, this does not change the fact that the differential cross section is a mathematical construct. The finite resolution of the detector simply means that our measurements will have some uncertainty, but it does not change the fundamental definition of the differential cross section.

In summary, the differential cross section is not a physical quantity, but rather a mathematical construct that is used to describe the behavior of the cross section as a function of the scattering angle. I hope this helps clarify things for you and your friend. If you need any further assistance, please don't hesitate to ask. Best of luck with your studies!