What is a differential cross section?

[Orlando]

In particle physics, what is a differential cross section? I think I understand what a cross section is, but I can't find any information on what a differential cross section is. I don't see how a cross section could be differentiated.

 

[gsal]

I don't know what you mean by particle physics, and so, I run the risk if giving the wrong answer to your question...but just in case...

...I would think that a differential cross section is the one that you have as you integrate to determine a total cross sectional area. For example, if you have an oddly shaped surface where the profile along the bottom is straight but the one along the top is not, you cannot simply multiply width times height to calculate the cross section; instead, you need to describe the line along the top as a function of the width, say x, multiply by a delta length, say dx, and now you have a differential area that can be integrated...is this what you are talking about?

 

[jtbell]

Maybe this earlier post might help:

https://www.physicsforums.com/showpost.php?p=3239196&postcount=5

I think of $d\sigma / d\Omega$ not as a derivative, but as a sort of notational quirk that reflects the fact that we often have to integrate it over scattering angle to get the actual scattering cross-section into your particular detector:

$$\sigma = \int {\frac{d\sigma}{d\Omega}d\Omega} = \int {\frac{d\sigma}{d\Omega} \sin \theta d\theta d\phi}$$

 

[vanhees71]

Sure, but that's just the total cross section. Sometimes you also like to know an scattering-angle distribution, and then you need the differential cross section. Then you know (on average), how many particles are scattered (or newly produced) in the reaction and are found in a certain solid angle. Knowing the differential cross section means more information about the scattering process than to just have the total cross section at hand. Of course, if you have the complete knowledge of the differential cross section, you get the total cross section by integration, but you cannot get the differential cross section, if you only know the total cross section.

 

[Orlando]

Thank you all, I think I understand it a little better now but am still a bit unsure (i'm not very well read in particle physics or maths). What I really need to know is the significance of the numerical value produced by the Klein-Nishina formula (http://en.wikipedia.org/wiki/Klein–Nishina_formula). From what I understand, this number is dimensionless and is used to quantify the probability of a photon-electron interaction that results in a specified photon scattering angle for a specified incident photon energy level. If this is correct, would I be able to determine the mean (the most probable) scattering angle for a photon if I knew the incident energy level? How exactly would I do that?

 

[Bill_K]

Orlando, It's not dimensionless, note the factor of r_c², which means it has dimensions of an area.

There's a standard picture used to explain what a scattering cross-section means. I'm surprised I can't quickly locate it on the web, so here it is in words: "dσ/dΩ is the effective area surrounding the scattering center which the incident particle must hit in order to be scattered through an angle between Θ and Θ + dΘ." (dΩ is the solid angle subtended by the ring-shaped region that lies between Θ and Θ + dΘ.)

In other words (classically at least) if you see the scattered particle come out at a particular angle, you know it must have passed by the scatterer at a certain distance b say, and the differential cross-section expresses the relationship between b and Θ (actually db and dΘ)

 

[Orlando]

Thank you for your response. So if the particle must hit somewhere in this area around the scattering centre to be scattered at an angle Θ, then is the probability of the incident particle being scattered at the angle Θ directly proportional to this area? If so, would I be able to describe the probabilities of certain scattering angles given an incident photon energy using this formula?

 

[Orlando]

A more direct question would be how can I use the Klein-Nishina formula to determine the probability of a photon scattering at a particular angle after a Compton interaction? I would appreciate any further help