Physical meaning of Differential Cross Section in scattering

Fizicks1
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I'm learning scattering theory from Griffiths' book, and I don't really understand the meaning of differential cross section; it's one of those things that I find hard to conceptualize and give a physical meaning to. The most intuitive description he gives is "the proportionality factor between dσ and dΩ, and that isn't very satisfying.

If anyone can give me an intuitive physical interpretation to the differential cross section, I would greatly appreciate it!

Thanks!
 
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Consider the total cross-section for a process (particles -> collision -> other particles): It roughly corresponds to a probability that the process happens. But then you don't know in which directions the resulting particles fly, and with which energy. The differential cross-section gives you this information.
 

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