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The differential cross section definition is often expressed as $\frac{d\sigma}{d\Omega} = \frac{n(\theta,\phi)}{N_{target} j_A}=\frac{n(\theta,\phi)}{p_N \Delta x \space j_A}$

My question concerns how we simply divide by $N_{target}$. I understand the logic of the incoming particle having 'more opportnities to scatter. However does this only work if we assume that once a particle has scattered of one target, it goes straight to a detector without any more scatters? This question is only relevant if we assume the target is thick so we use the $\Delta x$ thickness notation (so multiple successive scatters are possible).

I would think that in cases where a back scatter is incredibly unlikely (say negligible) but a 'sideways' scatter is quite likely, if we account for a particle undergoing multiple scatters then having more particles would increase the liklihood of 'two successive sideways scatters' and therefore skew the final distribution to have back scatters no longer neglible.

Therefore am I correct in my assumption that we only include $N_{target}$ as 'more opportunities to scatter one' and not 'more opportunities to also scatter multiple times'?

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    $\begingroup$ Yes, it generally is not applied to multiple scattering. $\endgroup$
    – Jon Custer
    Jan 7 at 20:12
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The microscopic scattering cross section- and the differential microscopic scattering cross section- applies to one scattering event for one target atom/nucleus. The microscopic differential cross section is that fraction of the microscopic scattering cross section for scattering into a solid angle.

The cross section for any reaction (fission, scattering, etc.) also is for one specific reaction event for one target atom/nucleus.

I like the following description of the microscopic cross section. Let $\Sigma$ be the mean free path for a specific interaction. The microscopic cross section $\sigma$ is defined as $\Sigma/N$ where $N$ is the density of atoms/nuclei in the target.

The evaluation of the incident particles as they travel through and interact with a thick target is complicated since scattering does not remove incident particles, and multiple scattering events can occur that change the angular distribution of the incident particles as they move throughout the target. To model neutrons, the transport equation is used, and if the scattering cross section is much greater than the absorption cross sections (and if other conditions apply), diffusion theory (Fick's law) can be used as an approximation.

Other unusual reactions can change the intensity of the incident particles as they interact with a target. For example, for 14 MeV neutrons incident on lead, the dominant reaction is (n, 2n) that effectively doubles the number of incident neutrons.

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