**multigroup diffusion method**is one of the most effective ways of calculating

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

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

**lumped into a group,**and in this group,

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

We have used a very important **assumption** in previous sections 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 have **distribution in the energy**. The spectrum of neutron energies produced by fission varies significantly with certain **reactor designs**. 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 a 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 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 briefly demonstrate its solution. In this example, we consider a thermal energy group and combine all neutrons of a 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** due to 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 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 the 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 the 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 **proportional to** the** bare reactor**. It can be derived that: