Theoretical definition and pratical mesurement of differential cross section In Sakurai's book, the definition of differential cross section is:
$$d\sigma/d\Omega= transition \;rate / probability\; flux $$
However this def doesn't contain any information about the thickness of the material or the density of target particle. How do one compare the experiment with the theory?
I checked Wikipedia but didn't find anything useful.
 A: This for me was the best place to start:
Rutherford Scattering
Differential Crosssections are a way of expressing the results of a scattering experiment. You have some initial state of the particles to be scattered, you have the target that they will be scattered off of, and you have some final state of your test particles in which they end up because of the scattering. That final state can be designated in many different ways. It can be a final energy state, a final momentum state. It could be some position. For example in the Stern-Gerlach experiment, spint up particles end up in one container, spin-down in another after passing through a magnetic field. 
Consider the following experiment: 
Suppose you're blind, but when you hear a sound, you know how far way it is from you, and you know how far away it is from whatever you can touch in front of you.  Suppose someone has glued billiard balls of varying, non-standard sizes down on a pool table and you want to know where they are. You're at one of the narrower ends. Along this side in front of you are a series of buttons evenly spaced each with a marker in Braille withe letters A,B,C, and so on. Each is also  some distance away from the zero point. Push one of these buttons and it shoots out a small ball bearing parallel to the adjacent walls. 
As these ball bearings are shot out, you hear a click if they hit one of the object balls. The ball bearings are deflected from their initial path. They are scattered elsewhere on the billiard table along a path deviating from their initial direction. When they hit the wall of the Billiard table, they make another sound telling you where and how long after the previous click they hit. 
You know the intial starting points of the ball bearings. You know where they are intially deflected, and you know where they end up. With all that information, combined with the conservation of energy and momentum, you can construct a map of where the object balls are, as well as their size and shape. However, just the initial point of emission of the ball bearings and their final location is going to give you a lot of that information. A cross section, in particular a differential cross section, represents information about initial and final states of a scattering experiment in such a way that you have some understanding of various features of the targets the ball bearings were scattered from, their size and shape, as well as location.  
In the Rutherford Experiment, they fired highly energetic helium nuclei, alpha particles, at a very thin gold foil. Sufficiently thin, the particles would scatter only once from the gold atoms in the foil. The alpha particles where shot along parallel lines and it was expected that the gold atoms would have roughly the same shape and size throughout the foil. They would also be evenly distributed. This implies some uniformity of final states of the alpha particle regardless of where or when an interaction occurs. If there is a pattern behind the transition from initial to final states, its due specifically to the interaction between the alphaparticle and the gold nucleus and not to incidentals as to where the nuclei are located in the foil or specific geometric features of the target nuclei. 
In other words, you know the initial magnitude and direction of your alpha particles' velocities. You also have detectors that can tell you the final magnitude and direction of their velocities. The final state is a function of the initial state and that function represents certain properties of what caused the transition. That's a "cross section". It has units of area while conveying information about the relevant interaction. For example, if the cross section stays the same at high or low energies of the alpha beam, then the interaction is mechanical, like two billiard balls bouncing off of each other. All that matters is point of impact, center's of mass and normals at the boundary point. If there's an action at a distance force like gravity or electromagnetism, then initial energy will gradually over come the non-mechanical forces involved. you'll find the cross section shrinks at higher energies, isolating the nuclei to a specific location. The cross section varies with energy. It could also vary with the relative position of the initial trajectory of the incoming beam. 
If you have the differential cross section instead of the cross section itself, you remove incidentals as to how the interaction works at different entities and have some picture of what features are common to every interaction and not just those specific cases. But a differential cross section is just another corss section. It's a representation of final states as they relate to initial states of the particle to be scattered. 
There are further geometric implications to scattering cross sections. Check out Mean Free Paths and Cross-sections as they apply to scattering theory: Scattering Theory. 
A: Correct, the scattering cross-section is a measure of the intrinsic probability for a given process. It doesn't know anything about the experimental conditions under which the process is actually observed (density of the target material, flux of incoming particles, etc.). 
These are instead encoded in what is called the luminosity of the experiment. The rate of a process in a given experiment is given by cross-section times luminosity:
$$R=\sigma \mathcal L.$$
