## Multigroup Diffusion Equations

In previous sections we have used a very important **assumption** that all neutrons are lumped into a **single energy group**. These monoenergetic neutrons are emitted and diffuse at thermal energy (0.025 eV). **In a thermal reactor** the neutrons actually have a **distribution in energy**. In fact, the spectrum of neutron energies produced by fission vary significantly with certain **reactor design**. The figure illustrates the difference in **neutron flux spectra between a thermal reactor and a fast breeder reactor**. Note that, the neutron spectra in fast reactors also vary significantly with a given reactor coolant.

See also: Neutron Flux Spectra

In general, free neutrons can be divided into many energy groups. The reactor physics does not need fine division of neutron energies. The neutrons can be roughly (for purposes of reactor physics) divided into three energy ranges:

**Thermal neutrons**(0.025 eV – 1 eV)**Resonance neutrons**(1 eV – 1 keV)**Fast neutrons**(1 keV – 10 MeV)

Even there are reactor computing codes that use only two neutron energy groups:

**Slow neutrons group**(0.025 eV – 1 keV).**Fast neutrons group**(1 keV – 10 MeV).

**Cold Neutrons**(0 eV; 0.025 eV). Neutrons in thermal equilibrium with very cold surroundings such as liquid deuterium. This spectrum is used for neutron scattering experiments.**Thermal Neutrons**. Neutrons in thermal equilibrium with a surrounding medium. Most probable energy at 20°C (68°F) for Maxwellian distribution is**0.025 eV**(~2 km/s). This part of neutron’s energy spectrum constitutes most important part of spectrum**in thermal reactors**.**Epithermal Neutrons**(0.025 eV; 0.4 eV). Neutrons of kinetic energy greater than thermal. Some of reactor designs operates with epithermal neutron’s spectrum. This design allows to reach higher fuel breeding ratio than in thermal reactors.-
**Cadmium Neutrons**(0.4 eV; 0.5 eV). Neutrons of kinetic energy below the**cadmium cut-off**energy. One cadmium isotope,^{113}Cd, absorbs neutrons strongly only if they are below ~0.5 eV (cadmium cut-off energy). **Epicadmium Neutrons**(0.5 eV; 1 eV). Neutrons of kinetic energy above the cadmium cut-off energy. These neutrons are not absorbed by cadmium.**Slow Neutrons**(1 eV; 10 eV).**Resonance Neutrons**(10 eV; 300 eV).**The resonance neutrons**are called resonance for their special bahavior. At resonance energies the cross-sections can reach peaks more than 100x higher as the base value of cross-section. At this energies the neutron capture significantly exceeds a probability of fission. Therefore it is very important (for thermal reactors) to quickly overcome this range of energy and operate the reactor with thermal neutrons resulting in increase of probability of fission.**Intermediate Neutrons**(300 eV; 1 MeV).**Fast Neutrons**(1 MeV; 20 MeV). Neutrons of kinetic energy greater than 1 MeV (~15 000 km/s) are usually named fission neutrons. These neutrons are produced by nuclear processes such as nuclear fission or (ɑ,n) reactions. The fission neutrons a mean energy (for^{235}U fission) of 2 MeV. Inside a nuclear reactor the fast neutrons are slowed down to the thermal energies via a process called neutron moderation.**Relativistic Neutrons**(20 MeV; ->)

**neutron diffusion**in thermal reactors is by the

**multigroup diffusion method**. In this method, the entire range of neutron energies is divided into

**N intervals**. All of neutrons within each interval are l

**umped into a group**and in this group

**all parameters**such as the diffusion coefficients or cross-sections are averaged.

As an illustrative example we will show a **two group diffusion equation** and we will briefly demonstrate its solution. In this example we consider a thermal energy group, and combine all neutrons of higher energy into a fast energy group.

In steady state, the diffusion equations for the fast and thermal energy groups are:

The equations are coupled through the **thermal fission term** the **fast removal term**. In this system of equations we assume that **neutrons appear in the fast group** as the result of fission induced by thermal neutrons (therefore Φ_{2}(x)). In the fission term, **k _{∞}** is to infinite multiplication factor and

**p**is the resonance escape probability. The fast absorption term expresses actually neutrons that are lost from the fast group

**by slowing down**.

**Σ**is equal to the

_{a1}Φ_{1}**thermal slowing down density**.

Consider the second equation (thermal energy group). Neutrons enter the thermal group as a result of **slowing down** out of the fast group, therefore the term ** pΣ_{a1}Φ_{1}** in this equation comes from fast group. It represents the source of

**neutrons that escaped to resonance absorption**.

To solve this system of equations we assume for a uniform reactor, that both groups of the fluxes in the core have a **geometrical buckling B _{g}** satisfying:

Since the geometrical buckling is the same for both the thermal and fast fluxes, the diffusion equations can be rewritten as:

## Criticality Equation for Two-group Theory and Bare Reactor

The previous equation is usually referred to as the **criticality equation**. In this equation the terms

is known as the fast non-leakage factor and

is known as the thermal non-leakage factor.

For weakly absorbing media and according to Fermi Theory, the following relation can be aplied:

- the
**source distribution**, whether it is external source of neutrons or it is a multiplying environment - the
**geometry**(in a finite medium), - the
**neutron diffusion length**, L^{2}= D/Σ_{a}, in fact L^{2}is the diffusion area. It is proportional to the distance thermal neutrons travel before they are absorbed. - the
**slowing-down length**, L_{s}, of a neutron. It is proportional to the distance fast neutrons travel from the point where they are born to the point where they become thermalized. Since it can be derived from**Fermi age theory**, a parameter**τ**, called the**“age”**(often called the**“Fermi age”**) is often used.

Let us focus on the **diffusion length** and the **slowing-down length**.

The physical meaning of the diffusion length is that:

*L*^{2}* is equal to one-sixth of the square of the average distance (in all dimension) between the neutron’s birth point (as a thermal neutron) and its absorption**.*

**The Fermi age** is related to the distance traveled during moderation just as the diffusion length is for thermal neutrons. The Fermi age is the same quantity as the slowing-down length squared, **L _{s}^{2}**, but the slowing down length is the square root of the Fermi age,

**τ**. The physical meaning of the slowing-down length is:

_{th }= L_{s}^{2}*L*_{s}^{2}* is equal to one-sixth of the square of the average distance (in all dimension) between the neutron’s birth point (as a fast neutron) and the point, where it has become thermalized**.*

Let us define the quantity, M^{2}, where:

*M*^{2}* = L*^{2}* + L*_{s}^{2}* or M*^{2}* = L*^{2}* + τ*_{th}

This quantity is called the migration area or square of the migration length. The physical meaning of the migration area is simply:

*M*^{2}* is equal to one-sixth of the square of the average distance (in all dimension) between the neutron’s birth point (as a fast neutron) and its absorption (as a thermal neutron)**.*

The distance traveled by fast neutrons during moderation and the distance traveled by thermal neutrons during diffusion in a reactor are important to reactor design because of their effect on the **critical size** and because of their effect on the **neutron leakage**.

**Effect on the Neutron Leakage**

It can be derived the **total non-leakage probability** of large reactors is primarily a function of **migration area**.

**Fast Non-leakage Probability**

It can be derived from the **Fermi age theory**, the probability that a neutron will remain in the core and become a thermal neutron without being lost by fast leakage, is also represented by following equation:

where τ is the Fermi age of a neutron, B is the **geometrical buckling** (in case of critical state B_{g} = B_{m}), which depends only on the shape and size of the core. The value of B for small cores is higher than the value for large cores. So that, it is obvious, the fast neutrons leakage is higher for small cores and also depends of the macroscopic slowing down power of neutron moderator (leakage is higher for poor moderators).

**Thermal Non-leakage Probability**

It can be derived from the **neutron diffusion theory**, the probability that a thermal neutron will remain in the core is also represented by following equation:

in which **L _{d}** is the

**diffusion length**, B is the

**geometrical buckling**(in case of critical state B

_{g}= B

_{m}), which depends only on the shape and size of the core. The value of B for small cores is higher than the value for large cores.

**Total Non-leakage Probability**

**The fast non-leakage probability** (P_{f}) and **the thermal non-leakage probability** (P_{t}) may be combined into one term that gives the fraction of **all neutrons** that do not leak out of the reactor core. This term is called** the total non-leakage probability **and is given the symbol P_{NL}, and may be expressed by following equation:

**For large reactors**, we can rewrite this equation without a substantial loss of accuracy simply by replacing the **diffusion length L _{d}** and the fermi age

**τ**by the

**migration length M**in the one group equation. The term

**B**

**is very small for large reactors and therefore it can be neglected. We may then write.**

^{4}where M is the **migration area (m**^{2}**).** The migration length is defined as the square root of the migration area. As can be seen the total non-leakage probability of large reactors is primarily a function of migration area.

## Flux Distribution for Two-group Theory and Bare Reactor

**both groups**satisfies:

Since the **geometrical buckling** is the same for both the thermal and fast fluxes, the **thermal flux** and the **fast flux** are then **proportional** everywhere for the** bare reactor**. It can be derived that:

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