## Macroscopic Cross-section

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 cm ^{2}**.

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**:

**Σ=σ.N**

Here **σ**, which has units of m^{2}, is the microscopic cross-section. Since the units of N (nuclei density) are nuclei/m^{3}, 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 Σ_{t} (the total cross-section), the equation for the intensity of a neutron beam can be written as

**-dI = N.σ.Σ _{t}.dx**

Dividing this expression by I(x) gives

**-dΙ(x)/I(x) = Σ _{t}.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 Σ_{t}, 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 (~Σ_{t}dx) times the probability that the neutron will interact in the additional layer dx (i.e. ~e^{-Σt.x}).

**P(x)dx = Σ _{t}dx . e^{-Σt.x} = Σ_{t} e^{-Σt.x} dx**

## Mean Free Path

From the equation for the probability of the** first collision in dx** we can calculate **the mean free path** that is traveled by a neutron between two collisions. This quantity is usually designated by the symbol **λ** and it is equal to the average value of x, the distance traveled by a neutron without any interaction, over the interaction probability distribution.

whereby one can distinguish** λ _{s}, λ_{a}, λ_{f}**, etc. This quantity is also known as the

**relaxation length**, because it is the distance in which the intensity of the neutrons that have not caused a reaction has decreased with a factor e.

For materials with high absorption cross-section, the mean free path is **very short** and neutron absorption occurs mostly** on the surface** of the material. This surface absorption is called **self-shielding** because the outer layers of atoms shield the inner layers.

## Macroscopic Cross-section of Mixtures and Molecules

Most materials are composed of several chemical elements and compounds. Most of chemical elements contains several** isotopes** of these elements (e.g. gadolinium with its six stable isotopes). For this reason most materials involve many cross-sections. Therefore, to include all the isotopes within a given material, it is necessary to determine the macroscopic cross section for each isotope and then sum all the individual macroscopic cross-sections.

In this section both factors (different** atomic densities** and different **cross-sections**) will be considered in the calculation of the **macroscopic cross-section of mixtures**.

First, consider the Avogadro’s number N_{0} = **6.022 x 10 ^{23}**, is the number of particles (molecules, atoms) that is contained in the amount of substance given by one mole. Thus if M is the

**molecular weight**, the ratio

**N**equals to the number of molecules in 1g of the mixture. The number of molecules per cm

_{0}/M^{3}in the material of density ρ and the macroscopic cross-section for mixtures are given by following equations:

**N _{i} = ρ_{i}.N_{0} / M_{i}**

Note that, in some cases, the cross-section of the molecule** is not equal** to the sum of cross-sections of its** individual nuclei**. For example the cross-section of neutron elastic scattering of water exhibits anomalies for thermal neutrons. It occurs, because the kinetic energy of an incident neutron is of the order or less than **the chemical binding energy** and therefore the scattering of slow neutrons by water (H_{2}O) is greater than by free nuclei (2H + O).

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