The thin target approximation in particle physics In the first paragraph of chapter 2 in this paper, the authors say that an effective volume is reduced to an effective area, when the thin target approximation is valid.
What is the thin target approximation, and why this approximation is valid when the interaction length of the particle is much longer than the thickness of the medium?  When the target is thick, do we expect different types of  interactions?
It would be appreciated, if you could explain qualitatively with formulas.
 A: When a beam of $N$ particles hits a target of density $n/A$ (e.g. per square cm) with a reaction cross section of $\sigma$, the number of scattered particles is:
$$N_S = N\frac n A \sigma $$
It is standard to use an areal density for the target because that is what the beam sees: stuff in its way. The depth isn't relevant. That is the thin target approximation.
Note that if the material has a volumetric density of $\rho$, then the mean free path of the beam is:
$$ \lambda = \frac 1 {\rho\sigma} $$
If the target is of length of order $\lambda$, then then a significant fraction of the beam is going to be scattered by the time it reaches the back for the target, and that part of the target won't see $N$ incident particles, it will see $N/e$ particles, as the number of particles at depth $z$ will be:
$$ N(z) = Ne^{-z/\lambda} $$
Your linked article is dealing with neutrinos, where $\lambda$ is on the order of light years, meaning the approximation:
$$ N(z) = Ne^{-z/\lambda} \approx N $$
is valid.
