What is cross section? I am learning about unit Barn, its unit of cross section. I know it means how likely a particle is to interact with material, its like opacity or transparency in optics.
What I dont understand is what cross section means. Lets say material X has cross section of 10 barns for neutron of certain energy. How far will that neutron go inside the material before hitting something? Meter? Micron? Femtometer? Mile? 69 parsecs? Two and half planck lenghts?
I hate the usage of this Barn unit in nuclear physics, who cares about how thicc atoms are to neutrons, we arent trying to play atomic pool, we are not trying to snipe single atom, mean free path is much more usefull and simple, so explain to me why is this Barn unit used instead?
 A: "Cross section" simply means that the barn is a measure of area; $1$ barn is equal to $10^{-28}$ square metres. It measures the size of the "target" that a neutron (or any other beam particle) must hit in order to interact with a specific nucleus (or other target particle). The "target" area is measured perpendicular to the direction of the beam particles.
It is possible to convert the cross-sectional area of a scattering process into the mean free path of beam particles between interactions, as long as you know the density of target nuclei. If the cross-sectional area is $\sigma$ square metres and there are $n$ target nuclei per cubic metre then the mean free path $\lambda$ in metres is
$$\lambda = \frac 1 {n \sigma}$$
However, the calculated mean free path is often large compared to the target thickness; in other words, most particles in the beam will not interact with target nuclei at all. So mean free path is of limited use in particle physics.
A: "Who cares"? You care. You are shooting at a swarm of bees of known number density n and depth d (the foil thickness) trying to infer the size (surface area) of each combined with your pellet, σ, from the number of hits! I assume you want intuition on WP or your modern physics text.
Assume you are shooting at a swarm of bees of a given unknown size/radius  a with pellets of size b, and you wish to determine the unknown bee-pellet cross-section
$$
\sigma=\pi (a+b)^2,
$$
a fundamental property of bees & pellets. The swarm is very sparse, say with $n= 1/m^3$, and has thickness/depth $d= 10m$; and, moreover, you can detect the kill ratio (probability of interaction), say from the number of smeared pellets recovered.
For $a+b= 10^{-2} m$, your pellet sweeps a cylinder of volume σd through the swarm, so it encounters dσn~ π/1000 bees. Thus,  from this kill ratio, you may determine σ, a fundamental quantity related to the size of bees and pellets, never having examined a bee closely. If you double the thickness of the swarm, you double your kill ratio.

How far will that  [pellet] go inside the [swarm] before hitting [a bee]?

Note for an almost certain kill, on average, you'll need a huge depth, d~λ=10 km/π. Whether you choose to parameterize your kill ratio by d/λ is optional, but why not work in the units of the small object you are investigating, in this case the bees (cum-pellet)? You are not studying the swarm (whose bee density your know): you are studying the bee size. In HEP, people investigate femtobarns, $fb= 10^{-43}m^2$, so they investigate distances of the order of millionths of a fermi.
