I would like to know what exactly the term differential cross section means. It comes from electron scattering experiments which are used to deduce the charge distribution of a nucleus and the cross section is given as a function of the scattering angle. But I don't understand what it means.
[Physics] Differential Cross Section
scattering-cross-section
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I don't know specifically what you're looking for, but I can give you the basic idea of the relation between the cross section and the differential cross section. Generally, the cross section $\sigma$ is defined as the integral of the differential cross section $\frac{d\sigma}{d\Omega}$ over the entire solid angle. Here, $d\Omega$ is the spherical surface element: $d\Omega\equiv \sin\theta\ d\theta\ d\phi $, with $\theta$ and $\phi$ the usual angles defined on the sphere, which parametrize the outgoing direction of the scattering object. So, we have $$\sigma=\oint d\Omega \frac{d\sigma}{d\Omega} $$ In many cases, we have cylindrical symmetry so that we can simplify: $$\sigma =b\oint \frac{d\Omega}{\sin\theta}\left|\frac{d b}{d\theta}\right|=2\pi b\int_0^\pi d\theta\left|\frac{d b}{d\theta}\right| $$ Here, $b$ is the impact parameter. The absolute value signs are there to assure that we get a positive outcome, even though one usually has lower deflection with larger impact parameters. You may also benefit from reading the relevant Wikipedia article, although it will not tell you what to do in your specific situation .
In a lot of ways what follows is only a partial answer. I think you are getting confused because you are trying to reason about infinities using the same tools you would apply to numbers.
I want to start with a set of practical considerations.
Where in the classical definition are the zero-angle deflected particles not counted in the scattering cross section?
The classical description includes particles that are scattered through a vanishingly small angle, but no real experiment need worry about those for two reasons:
The equipment include both beam-pipe and detector elements has finite size. As does the prepared beam. There is always an lower limit on the angle at which scattered and un-scattered particles can be distinguished. The result is that the integration of the cross-section should not be taken to cover $4\pi$.
The target generally does not consist of a single scattering center, but of a macroscopic quantity of matter meaning that the beam particles are in principle scattered by many centers. In most cases the measured scattering is dominated by a single hard (or at least harder) scattering event. In any case, scattering at impact parameters above about half the inter-center distance is mostly averaged out (lookup "multiple scattering" for the statistical properties of this process).
This leads to a practical notion that particles are either scattered or not scattered, and the scattering cross-section concerns itself only with the former. Now, that generates some tension with the understanding that the scattering cross-section is infinite for long-range forces. (I think this is the source of your question.)
You can dodge the tension by several means:
Noting that measured scattering cross-sections are always averaged over finite chunks of solid angle, and can't distinguish between no interaction and very little interaction. That is, the infinite value for long ranges forces is a black-board physics construct (albeit a deeply useful one).
By making a semantic distinction between "infinitesimally scattered" and "not scattered". The divergent part of the long-range cross-section goes in the former category, which the "missed the effective range" part of the short-range cross-section goes in the latter category not-withstanding that they are experimentally indistinguishable.
Best Answer
Differential cross section is defined to be:
$$ \dfrac{d\sigma}{d\Omega} $$
In plain words, this expression gives the probability that a particle passing through an area of $d\sigma$ before scattering can be found within the solid angle $ d\Omega $ after scattering.
This Wikipedia image can give a clear picture