## Neutron Diffusion Theory

In previous section we dealt with the multiplication system and we defined the **infinite and finite multiplication factor**. This section was about conditions for a **stable, self-sustained fission chain reaction **and how to maintain such conditions. This problem contains no information about the **spatial distribution of neutrons**, because it is a point geometry problem. We have characterized the effects of the global distribution of neutrons simply by a nonleakage probability (thermal or fast), which as stated earlier increases toward a value of one as the reactor core becomes larger.

In order to design a nuclear reactor properly, the prediction how the **neutrons** will be **distributed** throughout the system is of the **highest importance**. This is a very difficult problem, because the neutrons interacts with differently with different environments (moderator, fuel, etc.) that are in a reactor core. Neutrons undergo various interactions, when they migrate through the multiplying system. To a **first approximation **the overall effect of these interactions is that the neutrons undergo a kind of **diffusion** in the reactor core, much like the diffusion of one gas in another. This approximation is usually known as the **diffusion approximation** and it is based on the **neutron diffusion theory**. This approximation allows solving such problem using **the diffusion equation**.

In this chapter we will introduce the **neutron diffusion theory **and we will examine the **spatial migration of neutrons** to understand the relationships between **reactor size**, **shape**, and **criticality**, and to determine the spatial flux distributions within power reactors. The diffusion theory provides theoretical basis for a **neutron-physical computing** of nuclear cores. It must be added there are many neutron-physical codes that are based on this theory.

First, we will analyze the spatial distributions of neutrons and we will consider a **one group diffusion theory** (**monoenergetic neutrons**) for a **uniform non-multiplying medium**. That means that the neutron flux and cross sections have already been averaged over energy. Such a relatively simple model has the great advantage of illustrating many important features of spatial distribution of neutrons without the complexity introduced by the treatment of effects associated with the neutron energy spectrum.

See also: Neutron Flux Spectra

Moreover, mathematical methods used to analyze a **one group diffusion equation** are the same as those applied in more sophisticated and accurate methods such as **multi-group diffusion theory**. Subsequently, the one-group diffusion theory will be applied on an uniform multiplying medium (a homogeneous “nuclear reactor”) in simple geometries. Finally, the multi-group diffusion theory will be applied on an non-uniform multiplying medium (a heterogenous “nuclear reactor”) in simple geometries.

## Derivation of One-group Diffusion Equation

The derivation of the diffusion equation depends on **Fick’s law**, which states that solute diffuses **from high concentration to low**. But first, we have to define a **neutron flux **and **neutron current density**. The neutron flux is used to characterize the **neutron distribution** in the reactor and it is the main output of solutions of **diffusion equations**. The neutron flux, φ, does not characterize the flow of neutrons. There may be no flow of neutrons, yet many interactions may occur (I = Σ.φ). The neutrons move in a random directions and hence may not flow. Therefore the neutron flux φ is more closely related to **densities**. Neutrons will exhibit a net flow when there are **spatial differences** in their density. Hence we can have a **flux of neutron flux**! This flux of neutron flux is called the **neutron current density**.

## Fick’s Law

**Fick’s law**, which states that solute diffuses (

**neutron current**) from high concentration (high flux) to low concentration. As can be seen we have to investigate the relationship between the

**flux (φ)**and the

**current (J)**. This relationship between the flux (φ) and the current (J) is identical in form to a law (

**the Fick’s law**) used in the study of physical diffusion in liquids and gases.

**In chemistry, Fick’s law states that**:

*If the concentration of a solute in one region is greater than in another of a solution, the solute diffuses from the region of higher concentration to the region of lower concentration, with a magnitude that is proportional to the concentration gradient.*

In one (spatial) dimension, the law is:

where:

*J*is the diffusion flux,*D*is the**diffusion coefficient,***φ*(for ideal mixtures) is the concentration.

The use of this law in **nuclear reactor theory** leads to the **diffusion approximation**.

**The Fick’s law in reactor theory stated that**:

*The current density vector J is proportional to the negative of the gradient of the neutron flux. The proportionality constant is called the diffusion coefficient and is denoted by the symbol D.*

In one (spatial) dimension, the law is:

where:

(neutrons.cm*J*is the neutron current density^{-2}.s^{-1}) along x-direction, the net flow of neutrons that pass per unit of time through a unit area perpendicular to the x-direction.*D*is the**diffusion coefficient,**it has unit of cm and it is given by:*φ*is the neutron flux, which is the number of neutrons crossing through some arbitrary cross-sectional unit area in**all directions**per unit time.

The generalized Fick’s law (in three dimension) is:

where **J** denotes the **diffusion flux vector**. Note that the gradient operator turns the neutron flux, which is a **scalar quantity** into the neutron current, which is a **vector quantity**.

## Physical Interpretation

[xyz-ihs snippet=”diffusion”]The physical interpretation is similar to fluxes of gases. The neutrons exhibit a net flow in the direction of least density. This is a natural consequence of **greater collision densities** at positions of **greater neutron densities**.

Consider neutrons passing through the plane at x=0 from left to right as the result of collisions to the left of the plane. Since the concentration of neutrons and the flux is larger for negative values of x, there are **more collisions per cubic centimeter on the left**. Therefore more neutrons are scattered from left to right, then the other way around. Thus the neutrons naturally diffuse toward the right.

## Validity of Fick’s Law

**It must be emphasized that Fick’s law is an approximation and was derived under the following conditions:**

**Infinite medium.**This assumption is necessary to allow integration over all space but flux contributions are negligible beyond a few mean free paths (about three mean free paths) from boundaries of the diffusive medium.**Sources or sinks.**Derivation of Fick’s law assumes that the contribution to the flux is mostly from elastic scattering reactions. Source neutrons contribute to the flux if they are more than a few mean free paths from a source.**Uniform medium.**Derivation of Fick’s law assumes that a uniform medium was used. There are different scattering properties at the boundary (interface) between two media.**Isotropic scattering**. Isotropic scattering occurs at low energies, but is not true in general. Presence of anisotropic scattering can be corrected by modification of the diffusion coefficient (based on transport theory).**Low absorbing medium**. Derivation of Fick’s law assumes (an expansion in a Taylor’s series) that the neutron flux,*φ,*is slowly varying. Large variations in φ occur when Σ_{a}(neutron absorption) is large (compared to Σ_{s}).**Σ**_{a}**<< Σ**_{s}**Time – independent flux.**Derivation of Fick’s law assumes that the neutron flux is independent of time.

To some extent, these limitations are valid in every practical reactor. Nevertheless Fick’s law gives a reasonable approximation. For more detailed calculations, higher order methods are available.

## Neutron Balance – Continuity Equation

The mathematical formulation of **neutron diffusion theory** is based on the **balance of neutrons** in a differential volume element. Since neutrons do not disappear (β decay is neglected) the following neutron balance must be valid in an arbitrary volume V.

**rate of change of neutron density = production rate – absorption rate – leakage rate**

where

Substituting for the different terms in the balance equation and by dropping the integral over (because the volume V is arbitrary) we obtain:

where

**n**is the**density of neutrons**,**s**is the rate at which neutrons are emitted from sources per cm^{3 }(either from external sources (S) or from fission (**ν.Σ**)),_{f}.Ф**J**is the neutron current density vector**Ф**is the scalar neutron flux**Σ**is the macroscopic absorption cross-section_{a}

In steady state, when n is not a function of time:

## The Diffusion Equation

In previous chapters we introduced **two bases for the derivation** of the diffusion equation:

**Fick’s law:**

which states that neutrons diffuses from high concentration (high flux) to low concentration.

**Continuity equation:**

which states, that rate of change of neutron density = production rate – absorption rate – leakage rate.

We return now to the neutron balance equation and **substitute** the neutron current density vector by **J = -D∇Ф**. Assuming that ∇.∇ = ∇^{2} = Δ (therefore **div J = **-D div (∇Ф) = **-DΔФ**) we obtain the **diffusion equation**.

**many assumptions**. The diffusion equation can, therefore, not be exact or valid at places with strongly differing diffusion coefficients or in strongly absorbing media. This implies that the diffusion theory may show deviations from a more accurate solution of the transport equation in the proximity of external neutron sinks, sources and media interfaces.

## Boundary Conditions

To solve the diffusion equation, which is a second-order partial differential equation throughout the reactor volume, it is necessary to specify certain **boundary conditions**. It is very dependent on the complexity of certain problem. **One-dimensional problems** solutions of diffusion equation contain **two arbitrary constants**. Therefore, in order to solve one-dimensional one-group diffusion equation, we need two boundary conditions to determine these coefficients. The most convenient boundary conditions are summarized in following few points:

## Diffusion Length of Neutron

During solution of the diffusion equation we often meet with very important parameter that describes behavior of neutrons in a medium.

The solution of diffusion equation (let assume the simplest diffusion equation) usually starts by division of entire equation by diffusion coefficient:

The term **L ^{2}** is called the

**diffusion area**(and

**L**called the

**diffusion length**). For thermal neutrons with an energy of 0.025 eV a few values of L are given in table below.

## Physical Meaning of the Diffusion Length

It is interesting to try to interpret the **“physical” meaning** of the **diffusion length**. The physical meaning of the diffusion length can be seen by calculation of the **mean square distance** that a neutron travels in the one direction from the plane source to its absorption point.

It can be calculated, that **L ^{2}** is equal to one-half the square of the average distance (

**in one dimension**) between the neutron’s birth point and its absorption.

If we consider a **point source** of neutrons the physical meaning of the diffusion length can be seen again by calculation of the mean square distance that a neutron travels from the source to its absorption point.

It can be calculated, 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**.

This distance must not be confused with the average distance traveled by the neutrons. The average distance traveled by the neutrons is equal to the mean free path for absorption λ_{a} = 1/Σ_{a} and is much larger than the distance measured in a straight line. This is because neutrons in medium undergo many collisions and they follow a very **zig-zag path **through medium.

## Applicability of Diffusion Theory

Nowadays the **diffusion theory** is widely used in core design of the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). It provides a strictly valid mathematical description of the **neutron flux**, but it must be emphasized that the **diffusion equation** (in fact the **Fick’s law**) was derived under the following assumptions:

**Infinite medium****No sources or sinks.****Uniform medium.****Isotropic scattering.****Low absorbing medium.****Time – independent flux.**

To some extent, these limitations are valid in **every practical reactor**. Nevertheless the diffusion theory gives a reasonable approximation and makes **accurate predictions**. Nowadays reactor core analyses and design are often performed using **nodal two-group diffusion methods**. These methods are based on **pre-computed assembly homogenized cross-sections**, **diffusion coefficients** and **assembly discontinuity factors** (pin factors) obtained by single assembly calculation with reflective boundary conditions (infinite lattice). Highly absorbing control elements are represented by effective diffusion theory cross-sections which reproduce transport theory absorption rates. These **pre-computed data** (discontinuity factors, homogenized cross-sections, etc.) are calculated by **neutron trasport codes** which are based on a more accurate **neutron transport theory**. In short, neutron transport theory is used to **make diffusion theory work**.

Two methods exist for calculation of the pre-computed assembly cross-sections and pin factors.

**Deterministic methods**that solve the Boltzmann transport equation.**Stochastic methods**that are known as Monte Carlo methods that model the problem almost exactly.

These methods are very efficient and accurate when applied to the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs).

## Solutions of the Diffusion Equation – Non-multiplying Systems

**diffusion theory**is widely used in core design of the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). This section is not about such calculations, but provides an

**illustrative insights**, how can be the neutron flux distributed in any diffusion medium. In this section we will solve diffusion equations:

in various geometries that satisfy the **boundary conditions** discussed in the previous section.

We will start with simple systems and increase complexity gradually. The most important assumption is that **all neutrons** are lumped into a** single energy group**, they are emitted and diffuse at **thermal energy (0.025 eV)**.

In the first section, we will deal with neutron diffusion in **non-multiplying system**, i.e., in system where fissile isotopes are missing and therefore the **fission cross-section is zero**. The neutrons are emitted by external neutron source. We will assume that the system is uniform outside the source, i.e. **D** and **Σ _{a }**are constants.

## Solutions of the Diffusion Equation – Multiplying Systems

**non-multiplying**. In non-multiplying environment neutrons are emitted by a neutron source situated in the center of coordinate system and then they freely diffuse through media. We are now prepared to consider

**neutron diffusion**in

**multiplying system**, which contains fissionable nuclei (i.e.

**Σ**

_{f }**≠ 0**). In this multiplying system we will also study spatial distribution of neutrons, but in contrast to non-multiplying environment these neutrons can trigger

**nuclear fission reaction**. [/su_accordion]

**diffusion equation**

in various geometries that satisfy the **boundary conditions**. In this equation** ν** is number of neutrons emitted in fission and **Σ _{f}** is macroscopic cross-section of fission reaction.

**Ф**denotes a

**reaction rate**. For example a fission of

^{235}U by thermal neutron yields

**2.43 neutrons.**

It must be noted that we will solve the diffusion equation without any external source. This is very important, because such equation is a **linear homogeneous equation** in the flux. Therefore if we find one solution of the equation, then any multiple is also a solution. This means that the** absolute value** of the neutron flux **cannot possibly be deduced** from the diffusion equation. This is totally different from problems with external sources, which determine the absolute value of the neutron flux.

We will start with simple systems (planar) and increase complexity gradually. The most important assumption is that all neutrons are lumped into a **single energy group**, they are emitted and diffuse at** thermal energy** (**0.025 eV**). Solutions of diffusion equations in this case provides an illustrative insights, how can be the neutron flux distributed in a reactor core.

## Power Distribution in Conventional Reactor Cores

**In commercial reactor cores the flux distribution is significantly influenced by:**

**Heterogeneity of fuel-moderator assembly.** The geometry of the core strongly influences the **spatial and energy self-shielding**, that take place primarily in heterogeneous reactor cores. In short, the neutron flux **is not constant** due to the heterogeneous geometry of the unit cell. The flux will be different in the **fuel cell** (lower) than in the **moderator cell** due to the high absorption cross-sections of fuel nuclei. This phenomenon causes a significant increase in **the resonance escape probability** (“p” from four factor formula) in comparison with homogeneous cores.

**Reactivity Feedbacks.** **At power operation** (i.e. above 1% of rated power) the reactivity feedbacks causes the **flattening** of the flux distribution, because the feedbacks acts** stronger** on positions, where the **flux is higher**. The neutron flux distribution in commercial power reactors is dependent on many other factors as the **fuel loading pattern**, control rods position and it may also oscillate within short periods (e.g. as a result of spatial distribution of xenon nuclei). Simply, there is no cosine and J_{0} in the commercial power reactor at power operation.

**Fuel Loading Pattern. **The key feature of **PWRs fuel cycles** is that there are **many fuel assemblies** in the core and these assemblies have **different multiplying properties**, because they may have **different enrichment** and **different burnup**. Generally, a common fuel assembly contain energy for approximately **4 years of operation at full power**. Once loaded, fuel stays in the core for 4 years depending on the design of the operating cycle. During these 4 years the reactor core have to be refueled. During refueling, every 12 to 18 months, some of the fuel – usually **one third or one quarter of the core** – is removed to **spent fuel pool**, while the remainder is rearranged to a location in the core better suited to its remaining level of enrichment. The removed fuel (one third or one quarter of the core, i.e. 40 assemblies) has to be replaced by a **fresh fuel assemblies**.

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