The difference between the microscopic cross-section and macroscopic cross-section is very important and is restated for clarity. The microscopic cross section represents the effective target area of a single target nucleus for an incident particle. The units are given in barns or cm2.
While the macroscopic cross-section represents the effective target area of all of the nuclei contained in the volume of the material. The units are given in cm-1.
A macroscopic cross-section is derived from microscopic cross-section and the atomic number density:
Here σ, which has units of m2, is the microscopic cross-section. Since the units of N (nuclei density) are nuclei/m3, the macroscopic cross-section Σ have units of m-1, thus in fact is an incorrect name, because it is not a correct unit of cross-sections. In terms of Σt (the total cross-section), the equation for the intensity of a neutron beam can be written as
-dI = N.σ.Σt.dx
Dividing this expression by I(x) gives
-dΙ(x)/I(x) = Σt.dx
Since dI(x) is the number of neutrons that collide in dx, the quantity –dΙ(x)/I(x) represents the probability that a neutron that has survived without colliding until x, will collide in the next layer dx. It follows that the probability P(x) that a neutron will travel a distance x without any interaction in the material, which is characterized by Σt, is:
P(x) = e-Σt.x
From this equation, we can derive the probability that a neutron will make its first collision in dx. It will be the quantity P(x)dx. If the probability of the first collision in dx is independent of its past history, the required result will be equal to the probability that a neutron survives up to layer x without any interaction (~Σtdx) times the probability that the neutron will interact in the additional layer dx (i.e. ~e-Σt.x).
P(x)dx = Σtdx . e-Σt.x = Σt e-Σt.x dx