## 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).

**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 solution of this pair of homogeneous algebraic equations leads to a determinant of the coefficients, which have following solution (using Cramer’s rule):

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:

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

For a uniform reactor, the vanishing of the neutron flux on the boundary requires that the neutron flux in **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: