## 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.

**Neutron transport theory**is concerned with the transport of neutrons through various media. As was discussed neutrons are neutral particles, therefore they travel in

**straight lines**, deviating from their path only when they actually

**collide**with a nucleus to be scattered into a new direction or absorbed.

Transport theory is relatively **simple in principle** and an exact transport equation governing this phenomenon can **easily be derived**. This equation is called the **Boltzmann transport equation** and entire study of transport theory focuses on the study of this equation. In general, the neutron balance can be expressed graphically as:

Boltzman Transport Equation

**The Boltzman transport equation** is a balance statement that **conserves neutrons**. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It must be added it is **much easier to derive** the Boltzmann transport equation **than it is to solve it**. **Deterministic methods** solve the Boltzmann transport equation in a numerically approximated manner everywhere throughout a modeled system. This task demands enormous computational resources because the problem has many dimensions. But the Boltzmann transport equation can be treated in a rather straightforward way. This simplified version of the Boltzmann transport equation is just the **neutron diffusion equation**. Naturally, there are many assumptions that have to be fulfilled when using the diffusion equation, but the use of the diffusion equation usually provides sufficient accurate approximation to the exact transport equation.

Nowadays **reactor core analyses** and design can be performed using nodal two-group diffusion methods. These methods are based on **pre-computed** assembly homogenized **cross-sections** and assembly **discontinuity factors** (pin factors) obtained by single assembly calculation with reflective boundary conditions (infinite lattice). 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).

## 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**.

**neutron flux**is often used, because it expresses the

**total path length**covered by

**all neutrons**. The total distance these neutrons can travel each second is determined by their velocity and therefore the

**neutron flux density**value is calculated as the

**neutron density (n)**multiplied by

**neutron velocity (v)**.

**Ф = n.v**

where:

**Ф – neutron flux (neutrons.cm ^{-2}.s^{-1})**

**n – neutron density (neutrons.cm**

^{-3})**v – neutron velocity (cm.s**

^{-1})**The neutron flux**, which is the number of neutrons crossing through some arbitrary cross-sectional unit area in **all directions** per unit time, is a **scalar quantity**. Therefore it is also known as the **scalar flux**. The expression **Ф(E).dE** is the total distance traveled during one second by all neutrons with energies between E and dE located in 1 cm^{3}.

The connection to the **reaction rate**, respectively the reactor power, is obvious. Knowledge of the **neutron flux** (the **total path length** of all the neutrons in a cubic centimeter in a second) and the** macroscopic cross sections** (the probability of having an interaction **per centimeter path length**) allows us to compute the rate of interactions (e.g. rate of fission reactions). **The reaction rate** (the number of interactions taking place in that cubic centimeter in one second) is then given by multiplying them together:

where:

**Ф – neutron flux (neutrons.cm ^{-2}.s^{-1})**

**σ – microscopic cross section (cm**

^{2})**N – atomic number density (atoms.cm**

^{-3})This flux of neutron flux is called the neutron current density. We have to distinguish between the** neutron flux** and the **neutron current density**. Although both physical quantities have the** same units**, namely, neutrons.cm^{-2}.s^{-1}, their physical interpretations are different. In contrast to the neutron flux, the neutron current density is the number of neutrons crossing through some arbitrary cross-sectional unit area** in a single direction** per unit time (a surface is perpendicular to the direction of the beam). The **neutron current density** is a **vector quantity**.

The vector **J is defined as the following integral:**

A typical **thermal reactor** contains about **100 tons** of uranium with an average enrichment of **2%** (do not confuse it with the enrichment of the **fresh fuel**). If the reactor power is 3000MW_{th}, determine the **reaction rate** and the **average core thermal flux**.

**Solution:**

Multiplying the reaction rate per unit volume (RR = Ф . Σ) by the total volume of the core (V) gives us the** total number of reactions** occurring in the reactor core per unit time. But we also know the amount of energy released per one fission reaction to be about **200 MeV/fission**. Now, it is possible to determine the **rate of energy release** (power) due to the fission reaction. It is given by following equation:

**P = Ф . Σ _{f} . E_{r} . V = Ф . N_{U235} . σ_{f}^{235} . E_{r} . V**

where:

**P – reactor power (MeV.s ^{-1})**

**Ф – neutron flux (neutrons.cm**

^{-2}.s^{-1})**σ – microscopic cross section (cm**

^{2})**N – atomic number density (atoms.cm**

^{-3})**Er – the average recoverable energy per fission (MeV / fission)**

**V – total volume of the core (m**

^{3})The amount of fissile ^{235}U per the volume of the reactor core.

m_{235} [g/core] = 100 [metric tons] x 0.02 [g of ^{235}U / g of U] . 10^{6} [g/metric ton]
= **2 x 10 ^{6} grams of ^{235}U** per the volume of the reactor core

The atomic number density of ^{235}U in the volume of the reactor core:

N_{235} . V = m_{235} . N_{A} / M_{235}

= 2 x 10^{6} [g 235 / core] x 6.022 x 10^{23} [atoms/mol] / 235 [g/mol]
= **5.13 x 10 ^{27} atoms / core**

The microscopic fission cross-section of

^{235}U (for thermal neutrons):

**σ _{f}^{235} = 585 barns**

The average recoverable energy per ^{235}U fission:

**E _{r} = 200.7 MeV/fission**

## Fick’s Law

The derivation of the diffusion equation depends on **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

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.

**diffusion coefficient**is expressed in terms of the macroscopic cross-section and mean free path as:

where **Σ _{s}** is the

**macroscopic scattering cross-section**and

**λ**is the

_{s}**scattering mean free path**.

However, we can express the **diffusion coefficient** from the more advanced transport theory in terms of transport and **absorption cross-sections**:

where:

**λ**is the transport mean free path_{tr}**Σ**is the macroscopic absorption cross-section_{a}**Σ**is the macroscopic transport cross-section_{tr}**μ**is average value of the cosine of the angle in the lab system_{0}

In a weakly absorbing medium where **Σ**_{a}** << Σ**** _{s}** the diffusion coefficient can be approximately calculated as:

**The transport mean free path (λ _{tr})** is an

**average distance**a neutron will move in its original direction

**after infinite number of scattering collisions**.

is average value of the cosine of the angle in the lab system at which neutrons are scattered in the medium. It can be calculated for most of the neutron energies as (A is the mass number of target nucleus):

**Operational changes that affect the diffusion length**

The **diffusion coefficient** is very important parameter in **thermal reactors** and its magnitude can be changed during reactor operation. Since the diffusion coefficient is dependent on the **macroscopic scattering cross-section**,** Σ _{s}**, we will study impacts of operational changes on this parameter.

**Change in the moderator temperature**

The diffusion coefficient, D, is sensitive especially on the change in the **moderator temperature**.

*In short, as the moderator temperature increases, the diffusion coefficient also slightly increases.*

This increase in the diffusion coefficient is especially due (**Σ _{s}=σ_{s}.N_{H2O}**) to a decrease in the macroscopic scattering cross-section, Σ

_{s}=σ

_{s}.N

_{H2O}, caused by the

**thermal expansion of water**(a decrease in the

**atomic number density**).

**Calculation of Diffusion Coefficient**

The scattering cross-section of carbon at 1 eV is 4.8 b (4.8×10^{-24} cm^{2}). **Calculate the diffusion coefficient and the transport mean free path**.

**Solution:**

We will calculate the diffusion according to the advanced formula:

First, we have to determine the **atomic number density** of carbon and then the **scattering macroscopic cross-section.**

**Density:**

M_{C} = 12

N_{C} = ρ . N_{a} / M_{C}

= (2.2 g/cm^{3})x(6.022×10^{23} nuclei/mol)/ (12 g/mol)

= **1.1×10**^{23}** nuclei / cm**^{3}

σ_{s}^{12C} = 4.8 b

**Σ**_{s}** ^{12C}** = 4.8×10

^{-24}x 1.1×10

^{23}=

**0.528 cm**

^{-1}**the diffusion coefficient is then:**

**D = **1 / (3 x 0.528 x 0.9445)** = 0.668 cm**

**the transport mean free path**

**λ**_{tr}** = 3 x D = 2.005 cm**

## 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**.

The derivation of diffusion equation is based on Fick’s law which is derived under **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.

See also : Microscopic Cross-section

The extent to which neutrons interact with nuclei is described in terms of quantities known as **cross-sections**. **Cross-sections** are used to express the **likelihood of particular interaction** between an incident neutron and a target nucleus. It must be noted this likelihood do not depend on real target dimensions. In conjunction with the neutron flux, it enables the calculation of the reaction rate, for example to derive the **thermal power of a nuclear power plant**. The standard unit for measuring the **microscopic cross-section** (σ-sigma) is the **barn**, which is equal to **10 ^{-28} m^{2}**. This unit is very small, therefore barns (abbreviated as “b”) are commonly used.

**The cross-section σ** can be interpreted as the **effective ‘target area’** that a nucleus interacts with an incident neutron. The larger the effective area, the greater the probability for reaction. This cross-section is usually known as **the microscopic cross-section**.

The concept of the microscopic cross-section is therefore introduced to represent the probability of a neutron-nucleus reaction. It can be shown that whether a neutron will interact with a certain volume of material depends not only on **the microscopic cross-section** of the individual nuclei but also on **the density of nuclei** within that volume. It depends on the **N.σ factor**. This factor is therefore widely defined and it is known **as the macroscopic cross section**.

The difference between the microscopic and macroscopic cross sections is extremely important. The **microscopic cross section** represents the effective target area of a **single nucleus**, while the **macroscopic cross section** represents the effective target area of **all of**

**the nuclei** contained in certain volume.

**nuclear cross-sections**can be measured for all possible interaction processes together, in this case they are called

**total cross-sections (σ**. The total cross section is the sum of all the partial cross sections such as:

_{t})- elastic scattering cross-section (σ
_{s}) - inelastic scattering cross-section (σ
_{i}) - absorption cross-section (σ
_{a}) - radiative capture cross-section (σ
_{γ}) - fission cross-section (σ
_{f})

**σ _{t} = σ_{s} + σ_{i} + σ_{γ} + σ_{f} + ……**

The total cross-section measures the probability that an interaction of any type will occur when neutron interacts with a target.

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**

**control rod**usually contains solid

**boron carbide**with natural boron. Natural boron consists primarily of two stable isotopes,

**(80.1%) and**

^{11}B**(19.9%). Boron carbide has a density of**

^{10}B**2.52 g/cm**.

^{3}Determine the **total macroscopic cross-section** and the **mean free path**.

Density:

M_{B} = 10.8

M_{C} = 12

M_{Mixture} = 4 x 10.8 + 1×12 g/mol

N_{B4C} = ρ . N_{a} / M_{Mixture}

= (2.52 g/cm^{3})x(6.02×10^{23} nuclei/mol)/ (4 x 10.8 + 1×12 g/mol)

= **2.75×10 ^{22} molecules of B4C/cm^{3}**

N_{B} = 4 x 2.75×10^{22} atoms of boron/cm^{3}

N_{C} = 1 x 2.75×10^{22} atoms of carbon/cm^{3}

N_{B10} = 0.199 x 4 x 2.75×10^{22} = 2.18×10^{22} atoms of 10B/cm^{3}

N_{B11} = 0.801 x 4 x 2.75×10^{22} = 8.80×10^{22} atoms of 11B/cm^{3}

N_{C} = 2.75×10^{22} atoms of 12C/cm^{3}

**the microscopic cross-sections**

σ_{t}^{10B} = 3843 b of which σ_{(n,alpha)}^{10B} = 3840 b

σ_{t}^{11B} = 5.07 b

σ_{t}^{12C} = 5.01 b

**the macroscopic cross-section**

**Σ _{t}^{B4C} **= 3843×10

^{-24}x 2.18×10

^{22}+ 5.07×10

^{-24}x 8.80×10

^{22}+ 5.01×10

^{-24}x 2.75×10

^{22}

= 83.7 + 0.45 + 0.14 =

**84.3 cm**

^{-1}**the mean free path**

**λ _{t} **= 1/Σ

_{t}

^{B4C}= 0.012 cm =

**0.12 mm**(compare with B4C pellets diameter in control rods which may be around 7mm)

**λ**

_{a}≈ 0.12 mm## 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:

The diffusion equation is mostly solved in media with high densities such as neutron moderators (H_{2}O, D_{2}O or graphite). The problem is usually bounded by air. The **mean free path** of neutron in air is **much larger** than in the moderator, so that it is possible to treat it** as a vacuum** in neutron flux distribution calculations. The vacuum boundary condition supposes that** no neutrons are entering a surface**.

If we consider that no neutrons are reflected from the vacuum back to the volume, the following condition can be derived from the **Fick’s law**:

Where **d ≈ ⅔ λ _{tr}** is known as the

**extrapolated length**. For homogeneous, weakly absorbing media, an exact solution of the mono-energetic transport equation in this case yields

**d ≈ 0.7104 λ**. The geometric interpretation of the previous equation is that the relative neutron flux near the boundary has a slope of

_{tr}**-1/d**, i.e., the flux would extrapolate linearly to 0 at a distance d beyond the boundary. This

**zero flux boundary**condition is more straightforward and is can be written mathematically as:

If d is not negligible, physical dimensions of the reactor are increased by d and extrapolated boundary is formulated with dimension **R _{e} = R + d** and this condition can be written as

**Φ(R + d) = Φ(R**

_{e}) = 0.It may seem the flux goes to 0 at an **extrapolated length** beyond the boundary. **This interpretation is not correct.** The flux **cannot go to zero** in a vacuum, because there are no absorbers to absorb the neutrons. The flux only appears to be heading to the zero value at the extrapolation point.

Note that, the equation **d ≈ 0.7104 λ _{tr}** is applicable to plane boundaries only. The formulas for curved boundaries can differ slightly, however, the difference is small unless the radius of curvature of the boundary is of the same order of magnitude as the extrapolated length.

**Typical values of the extrapolated length:**

The most common moderators have following diffusion coefficients (for thermal neutrons):

D(H_{2}O) = 0.142 cm

D(D_{2}O) = 0.84 cm

D(Be) = 0.416 cm

D(C) = 0.916 cm

The thermal neutron **extrapolated lengths** are given by:

d ≈ 0.7104 λ_{tr} = 0.7104 x 3 x D

**therefore:**

**H _{2}O: d ≈ 0.30 cm**

**D _{2}O: d ≈ 1.79 cm**

**Be: d ≈ 0.88 cm**

**C: d ≈ 1.95 cm**

As can be seen, this approximation is valid when the dimension L of the diffusing medium is much larger than the extrapolated length, **L >> d**.

**reasonable values**i.e. must be real, non-negative, and single valued. Also the solution must be finite in those regions where the equation is valid, except perhaps at artificial singular points of a source distribution. This boundary condition can be written mathematically as:

**This conditions are often used to eliminate **unnecessary functions **from solutions.**

**interface**between

**two different media**. At interfaces between two different diffusion media (such as between the reactor core and the neutron reflector), on physical grounds the neutron flux and the normal component of the neutron current must be

**continuous**. In other words,

**φ and J**are not allowed to show a jump.

It must be added, as **J** must be continuous, the flux gradient will show a jump if the diffusion coefficients in both media differ from each other.

**neutron source**. But the presence of the neutron source can be used as a

**boundary condition**, because it is necessary that all neutrons flowing through

**bounding area**of the source must come from the neutron source. This boundary condition depends on the source geometry and can be written mathematically as:

**neutron reflector**. The reflector

**reduces the non-uniformity**of the power distribution in the peripheral fuel assemblies,

**reduces neutron leakage**and reduces a coolant flow bypass of the core. The neutron reflector is a

**non-multiplying medium**, whereas the reactor core is a multiplying medium.

On this special interface we shall apply an **albedo boundary condition** to represent the neutron reflector. Albedo, the latin word for “whiteness”, was defined by Lambert as the fraction of the incident light reflected diffusely by a surface.

In reactor engineering, **albedo**, or the **reflection coefficient**, is defined as the **ratio** of **exiting to entering neutrons** and we can express it in terms of **neutron currents** as:

For sufficiently thick reflectors, it can be derived, that albedo becomes

where **D _{refl}** is the

**diffusion coefficient**in the reflector and the

**L**is the

_{refl}**diffusion length**in the reflector.

If we are not interested in the neutron flux distribution in the reflector (let say in the slab B) but only in the effect of the reflector on the neutron flux distribution in the medium (let say in the slab A), the albedo of the reflector can be used as a boundary condition for the diffusion equation solution. This boundary condition is similar to the **vacuum boundary condition**, i.e. **Φ(R _{albedo}) = 0**, where

**R**and

_{albedo}= R + d_{e}## 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.

**diffusion length**is very important parameter in thermal reactors and its magnitude

**can be changed**during reactor operation. Since the diffusion length is dependent on the

**diffusion coefficient**(~Σ

_{s}) and

**Σ**, we will study impacts of operational changes on these parameters. For illustration, here are some examples of these operational changes and their impacts on L

_{a}^{2}, that may take place in PWRs.

**Change in the moderator and fuel temperature**

The diffusion length, L^{2}, is sensitive especially on the change in the** moderator temperature**. Since diffusion in heterogeneous reactors occurs especially in the **moderator**, the change in the moderator temperature dominates over the change in the fuel temperature.

*In short, as the moderator temperature increases, the diffusion length also increases.*

The moderator temperature influences all macroscopic cross-sections (e.g. Σ_{s}=σ_{s}.N_{H2O}) especially due to the **thermal expansion of water**, which results in a decrease in the atomic number density. In fact, the microscopic cross-sections also slightly changes with temperature, but not so much in thermal spectrum.

For **the diffusion length** there are two effects. Both processes have the same direction and together causes the increase in the diffusion lengths as the temperature increases. Since **the diffusion length** significantly influences the thermal non-leakage probability, it is of importance in reactor dynamics (it influences moderator temperature feedback).

- Macroscopic cross-sections for elastic scattering reaction
**Σ**_{s}**=σ**_{s}**.N**which significantly changes due to the_{H2O,}**thermal expansion**of water. As the temperature of the core increases,**the diffusion coefficient**(**D = 1/3.Σ**) increases._{tr} - The macroscopic cross-section for neutron absorption also decreases as the moderator temperature increases. This is especially due to thermal expansion of water, but also due to the changes in the microscopic cross-section (
**σ**) for neutron absorption. As the temperature of the core increases, the absorption cross-section decreases._{a}

**Change in the boron concentration**

The concentration of boric acid diluted in the primary coolant influences the **diffusion length**. For example, an increase in the concentration of boric acid (chemical shim) causes an addition of new absorbing material into the core and this causes an increase in the macroscopic absorption cross-section and this, in turn, causes a decrease in the diffusion length.

## 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

As was previously discussed the **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.

**neutron source**(with strength

**S**) as an

_{0}**infinite plane source**(in y-z plane geometry). In this geometry the flux varies so slowly in y and z allowing us to eliminate the y and z derivatives from ∇

^{2}. The flux is then a

**function of x only**, and therefore the

**Laplacian**and

**diffusion equation**(outside the source) can be written as (everywhere except x = 0):

For x > 0, this diffusion equation has **two possible solutions** **exp(x/L)** and **exp(-x/L)**, which give a general solution:

**Φ(x) = Aexp(x/L) + Cexp(-x/L)**

Note that, B is not usually used as a constant, because B is reserved for a **buckling parameter**. To determine the coefficients A and C we must apply boundary conditions.

From **finite flux condition** (0≤ Φ(x) < ∞), that required only reasonable values for the flux, it can be derived, that **A must be equal to zero**. The term exp(x/L) goes to ∞ as x ➝∞ and therefore cannot be part of a physically acceptable solution for x > 0. The physically acceptable solution for x > 0 must then be:

**Φ(x) = Ce ^{-x/L}**

where C is a constant that can be determined from source condition at x ➝0.

If S_{0} is the source strength per unit area of the plane, then the number of neutrons crossing outwards per unit area in the positive x-direction **must tend to S _{0} /2 as x ➝0**.

**neutron source**(with strength S

_{0}) as an

**isotropic point source**situated in spherical geometry. This point source is placed at the origin of coordinates. In order to solve the

**diffusion equation**, we have to replace the

**Laplacian**by its

**spherical form**:

We can replace the **3D Laplacian** by its **one-dimensional spherical form**, because there is no dependence on angle (whether polar or azimuthal). The source is assumed to be an **isotropic source** (there is the spherical symmetry). The flux is then a function of **radius – r** only, and therefore the diffusion equation (outside the source) can be written as (everywhere except r = 0):

If we make the **substitution** **Φ(r) = 1/r ψ(r)**, the equation simplifies to:

For r > 0, this differential equation has** two possible solutions** **exp(r/L)** and **exp(-r/L)**, which give a general solution:

Note that, B is not usually used as a constant, because B is reserved for a **buckling parameter**. To determine the coefficients A and C we must apply **boundary conditions**.

To find constants A and C we use the identical procedure as in the case of infinite planar source. From** finite flux condition** (0≤ Φ(r) < ∞), that required only reasonable values for the flux, it can be derived, that **A must be equal to zero**. The term exp(r/L)/r goes to ∞ as r ➝∞ and therefore cannot be part of a physically acceptable solution for r > 0. The physically acceptable solution for r > 0 must then be:

Φ(r) = Ce^{-r/L}/r

where C is a constant that can be determined from **source condition** at x ➝0.

If S_{0} is the source strength, then the number of neutrons crossing a sphere outwards in the positive r-direction **must tend to S _{0} as r ➝0**.

So that the solution may be written:

**neutron source**(with strength S

_{0}) as an isotropic line source situated in an infinite

**cylindrical geometry**. This line source is placed at r = 0. In order to solve the

**diffusion equation**, we have to replace the Laplacian by its

**cylindrical form**:

Since there is no dependence on angle Θ and z-coordinate, we can replace the 3D Laplacian by its **one-dimensional form**, and we can solve the problem only in **radial direction**. The source is assumed to be an isotropic source. Since the flux is a function of radius – r only, the diffusion equation (outside the source) can be written as (everywhere except r = 0):

This differential equation is called the **Bessel’s equation** and it is well known to mathematicians. In this case, the solutions to the Bessel’s equation are called the **modified Bessel functions** (or occasionally the **hyperbolic Bessel functions**) **of the first and second kind, **I_{α}(x) and K_{α}(x) respectively.

For r > 0, this differential equation has two possible solutions **I _{0}(r/L)** and

**K**, the modified Bessel functions of order zero, which give a general solution:

_{0}(r/L)To find constants A and C we use the identical procedure as in the case of infinite planar source. From** finite flux condition** (0≤ Φ(r) < ∞), that required only reasonable values for the flux, it can be derived, that A must be equal to zero. The term I_{0}(r/L) goes to ∞ as r ➝∞ and therefore cannot be part of a physically acceptable solution for r > 0. The physically acceptable solution for r > 0 must then be:

**Φ(r) = C.K _{0}(r/L)**

where C is a constant that can be determined from** source condition** at r ➝0.

If S_{0} is the source strength, then the number of neutrons crossing a cylinder outwards in the positive r-direction must tend to S_{0} as r ➝0.

So that the solution may be written:

**various zones**of different composition. The consequence of this is that the

**diffusion coefficient**,

**absorption macroscopic cross-section**, and therefore, the neutron flux distribution, will vary per zone. For the determination of the flux distribution in various zones, the

**diffusion equations**in zone 1 and zone 2 need to be solved:

where **a** is the real width of zone 1 and** b** the outer dimension of the diffusion environment including the **extrapolated distance**. With problems involving two different diffusion media, the interface boundary conditions play crucial role and must be satisfied:

*At interfaces between two different diffusion media (such as between the reactor core and the neutron reflector), on physical grounds the neutron flux and the normal component of the neutron current must be continuous. In other words, φ and J are not allowed to show a jump.*

1., 2. Interface Conditions

It must be added, as **J** must be continuous, the flux gradient will show a jump if the **diffusion coefficients** in both media differ from each other. Since the solution of these two diffusion equations requires four boundary conditions, we have to use two boundary conditions more.

3. Finite Flux Condition

The solution must be finite in those regions where the equation is valid, except perhaps at artificial singular points of a source distribution. This boundary condition can be written mathematically as:

The presence of the **neutron source** can be used as a boundary condition, because it is necessary that all neutrons flowing through bounding area of the source must come from the neutron source. This boundary condition depends on the source geometry and for planar sourve can be written mathematically as:

For x > 0, these diffusion equations have the following appropriate solutions:

**Φ _{1}(x) = A_{1}exp(x/L_{1}) + C_{1}exp(-x/L_{1})**

and

**Φ _{2}(x) = A_{2}exp(x/L_{2}) + C_{2}exp(-x/L_{2})**

where the four constants must be determined with use of the four boundary conditions. The typical neutron flux distribution in a simple two-region diffusion problem is shown at the picture below.

## Solutions of the Diffusion Equation – Multiplying Systems

In previous section it has been considered that the environment is **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**.

**stable, self-sustained fission chain reaction**in a multiplying system (in a nuclear reactor) is that

**exactly every fission initiate another fission**. The minimum condition is for each nucleus undergoing fission to produce, on the average, at least one neutron that causes fission of another nucleus. Also the number of fissions occurring per unit time (the reaction rate) within the system must be constant.

This condition can be expressed conveniently in terms of **the multiplication factor**. The infinite multiplication factor is the ratio of the **neutrons produced by fission** in one neutron generation to the number of **neutrons lost through absorption** in the preceding neutron generation. This can be expressed mathematically as shown below.

It is obvious** the infinite multiplication factor** in a multiplying system is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation.

**k**. If the multiplication factor for a multiplying system is_{∞}< 1**less than 1.0**, then the**number of neutrons is decreasing**in time (with the mean generation time) and the chain reaction will never be self-sustaining. This condition is known as**the subcritical state**.

**k**. If the multiplication factor for a multiplying system is_{∞}= 1**equal to 1.0**, then there is**no change in neutron population**in time and the chain reaction will be**self-sustaining**. This condition is known as**the critical state**.

**k**. If the multiplication factor for a multiplying system is_{∞}> 1**greater than 1.0**, then the multiplying system produces**more neutrons**than are needed to be self-sustaining. The number of neutrons is exponentially increasing in time (with the mean generation time). This condition is known as**the supercritical state**.

In this section, we will solve the following **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.

It is known the fission neutrons are of importance in any chain-reacting system. Neutrons trigger the nuclear fission of some nuclei (^{235}U, ^{238}U or even ^{232}Th). What is crucial the fission of such nuclei produces **2, 3 or more** free neutrons.

But not all neutrons are released **at the same time following fission**. Even the nature of creation of these neutrons is different. From this point of view we usually divide the fission neutrons into two following groups:

**Prompt Neutrons.**Prompt neutrons are emitted**directly from fission**and they are emitted within**very short time of about 10**.^{-14}second**Delayed Neutrons.**Delayed neutrons are emitted by**neutron rich fission fragments**that are called**the delayed neutron precursors**. These precursors usually undergo beta decay but a small fraction of them are excited enough to undergo**neutron emission**. The fact the neutron is produced via this type of decay and this happens**orders of magnitude later**compared to the emission of the prompt neutrons, plays an extremely important role in the control of the reactor.

**Radiative Capture.**Most absorption reactions result in the loss of a neutron coupled with the production of one or more gamma rays. This is referred to as a**capture reaction**, and it is denoted by**σ**._{γ}

**Neutron-induced Fission Reaction.**Some nuclei (fissionable nuclei) may undergo a fission event, leading to two or more fission fragments (nuclei of intermediate atomic weight) and a few neutrons. In a fissionable material, the neutron may simply be captured, or it may cause nuclear fission. For fissionable materials we thus divide the absorption cross section as**σ**._{a}= σ_{γ}+ σ_{f}

**uniform infinite reactor**, i.e. uniform infinite multiplying system without an external neutron source. This system is in a Cartesian coordinate system and under these assumptions (no neutron leakage, no changes in diffusion parameters) the

**neutron flux**must be

**inherently constant**throughout space.

Since the neutron current is equal to zero (**J** = -D∇Ф, where Ф is constant), the diffusion equation in the infinite uniform multiplying system must be:

The only solution of this equation is a **trivial solution**, i.e., Ф = 0, **unless** **Σ**_{a}** = **ν**Σ**_{f}**. **This equation (**Σ**_{a}** = **ν**Σ**** _{f}**) is known as the

**criticality condition**for an infinite reactor and expresses the perfect balance (critical state) between neutron absorption and neutron production. This balance must be continuously maintained in order to have

**steady state neutron flux**.

**Infinite Multiplication Factor**

In this section the **infinite multiplication factor, k _{∞}**, will be defined from another point of view than in section – Nuclear Chain Reaction.

As can be seen we can rewrite the diffusion equation in following way and we can define a new factor – **k _{∞} = νΣ_{f }/ Σ_{a}**:

A non-trivial solution of this equation is guaranteed when **k _{∞} = νΣ_{f }/ Σ_{a }= 1**. On the other hand we have no information about the

**neutron flux**in such critical reactor. In fact the neutron flux can have

**any value**and the critical uniform infinite reactor can operate at any power level. It should be noted this theory can be used for a reactor

**at low power levels**, hence “

**zero power criticality**”.

In power reactor core, the power level does not influence the criticality of a reactor unless** thermal reactivity feedbacks** act (operation of a power reactor without reactivity feedbacks is between 10E-8% – 1% of rated power).

See also: Reactor Criticality

**uniform reactor**(multiplying system) in the shape of a slab of physical

**width**

**a**in the x-direction and infinite in the y- and z-directions. This reactor is situated in the center at x=0. In this geometry the flux does not vary in y and z allowing us to eliminate the y and z derivatives from ∇

^{2}. The flux is then a

**function of x only**, and therefore the Laplacian and diffusion equation can be written as:

The quantity **B _{g}^{2}** is called the

**geometrical buckling**of the reactor and depends only on the geometry. This term is derived from the notion that the

**neutron flux distribution**is somehow

**‘‘buckled’’**in a homogeneous finite reactor. It can be derived the geometrical buckling is the negative relative curvature of the neutron flux (

**B**). In a small reactor the

_{g}^{2}= ∇^{2}Ф(x) / Ф(x)**neutron flux**have more concave downward or ‘‘buckled’’ curvature (

**higher B**) than in a large one. This is a very important parameter and it will be discussed in following sections.

_{g}^{2}For x > 0, this diffusion equation has two possible solutions **sin(B _{g}x)** and

**cos(B**, which give a general solution:

_{g}x)**Φ(x) = A.sin(B _{g}x) + C.cos(B_{g }x)**

From finite flux condition (**0≤ Φ(x) < ∞**), that required only reasonable values for the flux, it can be derived, that A must be equal to zero. The term **sin(B _{g}x)** goes to negative values as x goes to negative values and therefore it cannot be part of a physically acceptable solution. The physically acceptable solution must then be:

**Φ(x) = C.cos(B _{g }x)**

where **B _{g}** can be determined from

**vacuum boundary condition**.

The vacuum boundary condition requires the relative neutron flux near the boundary to have a **slope** of **-1/d**, i.e., the flux would extrapolate linearly to** 0 at a distance d** beyond the boundary. This **zero flux boundary condition** is more straightforward and is can be written mathematically as:

If d is not negligible, physical dimensions of the reactor are increased by d and extrapolated boundary is formulated with dimension **a _{e}/2 = a/2 + d** and this condition can be written as

**Φ(a/2 + d) = Φ(a**.

_{e}/2) = 0Therefore, the solution must be **Φ(a _{e}/2) = C.cos(B_{g }.a_{e}/2) = 0** and the values of geometrical buckling, B

_{g}, are limited to

**B**, where n is any

_{g}=^{nπ}/_{a_e}**odd integer**. The only one physically acceptable odd integer is

**n=1**, because higher values of n would give cosine functions which would become negative for some values of x. The solution of the diffusion equation is:

It must be added the **constant C cannot be obtained** from this diffusion equation, because this constant gives the absolute value of neutron flux. In fact the** neutron flux** can have** any value** and the **critical reactor** can operate at any power level. It should be noted the **cosine flux shape** is only in theoretical case in a uniform homogeneous reactor at low power levels (at “**zero power criticality**”).

In power reactor core (at full power operation), the neutron flux can reach, for example, about **3.11 x 10**^{13 }**neutrons.cm**^{-2}**.s**^{-1}**, **but this values depends significantly on many parameters (type of fuel, fuel burnup, fuel enrichment, position in fuel pattern, etc.).

The power level does not influence the criticality (k_{eff}) of a power reactor unless thermal reactivity feedbacks act (operation of a power reactor without reactivity feedbacks is between 10E-8% – 1% of rated power).

Let assume a **uniform reactor** (multiplying system) in the shape of a **sphere** of **physical radius R. **The spherical reactor is situated in **spherical geometry** at the origin of coordinates. In order to solve the diffusion equation, we have to replace the Laplacian by its spherical form:

We can replace the 3D Laplacian by its one-dimensional spherical form, because there is **no dependence on angle** (whether polar or azimuthal). The source term is assumed to be isotropic (there is the spherical symmetry). The flux is then a **function of radius – r only**, and therefore the diffusion equation can be written as:

The solution of the diffusion equation is based on a **substitution Φ(r) = 1/r ψ(r)**, that leads to equation for ψ(r):

For r > 0, this differential equation has two possible solutions **sin(B _{g}r)** and

**cos(B**, which give a general solution:

_{g}r)From finite flux condition (**0≤ Φ(r) < ∞**), that required only reasonable values for the flux, it can be derived, that **C must be equal to zero**. The term **cos(B _{g}r)/r** goes to ∞ as r ➝0 and therefore cannot be part of a physically acceptable solution. The physically acceptable solution must then be:

**Φ(r) = A sin(B _{g}r)/r**

The vacuum boundary condition requires the relative **neutron flux** near the boundary to have a **slope** of **-1/d**, i.e., the flux would extrapolate linearly to **0 at a distance d** beyond the boundary. This **zero flux boundary condition** is more straightforward and is can be written mathematically as:

If d is not negligible, physical dimensions of the reactor are increased by d and extrapolated boundary is formulated with dimension **R _{e} = R + d** and this condition can be written as

**Φ(R + d) = Φ(R**.

_{e}) = 0Therefore, the solution must be **Φ(R _{e}) = A sin(B_{g}R_{e})/R_{e} = 0** and the values of geometrical buckling, B

_{g}, are limited to

**B**, where n is any

_{g}=^{nπ}/R_{e}**odd integer**. The only one physically acceptable odd integer is

**n=1**, because higher values of n would give sine functions which would become negative for some values of x before returning to 0 at R

_{e}. The solution of the diffusion equation is:

It must be added the constant **A cannot be obtained** from this diffusion equation, because this constant gives the absolute value of neutron flux. In fact the **neutron flux can have any value** and the critical reactor can operate at any power level. It should be noted this flux shape is only in theoretical case in a uniform homogeneous spherical reactor at low power levels (at “**zero power criticality**”).

In power reactor core, the neutron flux can reach, for example, about **3.11 x 10**^{13 }**neutrons.cm**^{-2}**.s**^{-1}**, **but this values depends significantly on many parameters (type of fuel, fuel burnup, fuel enrichment, position in fuel pattern, etc.).

The power level does not influence the criticality (k_{eff}) of a power reactor unless thermal reactivity feedbacks act (operation of a power reactor without reactivity feedbacks is between 10E-8% – 1% of rated power).

**uniform reactor**(multiplying system) in the shape of a cylinder of physical radius

**R and height H.**This finite cylindrical reactor is situated in cylindrical geometry at the origin of coordinates. In order to solve the

**diffusion equation**, we have to replace the Laplacian by its cylindrical form:

**Since there is no dependence on angle Θ, we can replace the 3D Laplacian by its two-dimensional form, and we can solve the problem in radial and axial directions. Since the flux is a function of radius – r and height – z only (Φ(r,z)), the diffusion equation can be written as:**

The solution of this diffusion equation is based on use of the **separation-of-variables technique**, therefore:

where R(r) and Z(z) are functions to be determined. Substituting this into the diffusion equation and dividing by **R(r)Z(z)**, we obtain:

Because the first term depends only on r, and the second only on z, both terms must be **constants** for the equation to have a solution. Suppose we take the constants to be **v ^{2}** and

**к**, sum of these constants must be equal to

^{2}**B**. Now we can

_{g}^{2}= v^{2}+ к^{2}**separate variables**, and the

**neutron flux**must satisfy the following differential equations:

**Solution for radial direction**

The differential equation for radial direction is called the **Bessel’s equation** and it is well known to mathematicians. In this case, the solutions to the Bessel’s equation are called the **Bessel functions** **of the first and second kind, **J_{α}(x) and Y_{α}(x) respectively.

For r > 0, this differential equation has two possible solutions **J _{0}(vr)** and

**Y**, the Bessel functions of order zero, which give a general solution:

_{0}(vr)From finite flux condition (**0≤ Φ(r) < ∞**), that required only reasonable values for the flux, it can be derived, that C must be equal to zero. The term **Y _{0}(vr)** goes to -∞ as r ➝0 and therefore cannot be part of a physically acceptable solution. The physically acceptable solution for must then be:

**R(r) = AJ _{0}(vr)**

The **vacuum boundary condition** requires the relative neutron flux near the boundary to have a **slope of -1/d**, i.e., the flux would extrapolate linearly to **0 at a distance d** beyond the boundary. This **zero flux boundary condition** is more straightforward and is can be written mathematically as:

If d is not negligible, physical dimensions of the reactor are increased by d and extrapolated boundary is formulated with dimension **R _{e} = R + d** and this condition can be written as

**Φ(R + d) = Φ(R**.

_{e}) = 0Therefore, the solution must be **R(R _{e}) = A J_{0}(vR_{e}) = 0**. The function of

**J**has several zeroes, the first is at

_{0}(r)**r**, and the second at r

_{1}= 2.405_{2}= 5.6. However, because the neutron flux cannot have regions of negative values, the only physically acceptable value for

**v**is

**2.405/R**. The solution of the diffusion equation is:

_{e}**Solution for axial direction**

The solution for axial direction have been solved in previous sections (**Infinite Slab Reactor**) and therefore it has the same solution. The solution of in axial direction is:

**Solution for radial and axial directions**

The **full solution** for the **neutron flux distribution** in the finite cylindrical reactor is therefore:

where **B _{g}^{2} **is the total geometrical buckling.

It must be added the constants A and C cannot be obtained from the diffusion equation, because these constant give the **absolute value of neutron flux**. In fact the neutron flux can have **any value** and the critical reactor can operate at any power level. It should be noted this flux shape is only in theoretical case in a uniform homogeneous cylindrical reactor at low power levels (at “**zero power criticality**”).

In power reactor core, the neutron flux can reach, for example, about **3.11 x 10**^{13 }**neutrons.cm**^{-2}**.s**^{-1}**, **but this values depends significantly on many parameters (type of fuel, fuel burnup, fuel enrichment, position in fuel pattern, etc.).

The power level does not influence the criticality (k_{eff}) of a power reactor unless thermal reactivity feedbacks act (operation of a power reactor without reactivity feedbacks is between 10E-8% – 1% of rated power).

See also: Multigroup Method

See also: Reflected Reactor

## 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**.

**Nuclear and Reactor Physics:**

- J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
- J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
- W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
- Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
- W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
- G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
- Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
- U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

**Advanced Reactor Physics:**

- K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
- K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
- D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
- E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

### See above:

Contents

- Neutron Diffusion Theory
- Derivation of One-group Diffusion Equation
- Fick’s Law
- Validity of Fick’s Law
- Neutron Balance – Continuity Equation
- The Diffusion Equation
- Boundary Conditions
- Diffusion Length of Neutron
- Physical Meaning of the Diffusion Length
- Applicability of Diffusion Theory
- Solutions of the Diffusion Equation – Non-multiplying Systems
- Solutions of the Diffusion Equation – Multiplying Systems
- Power Distribution in Conventional Reactor Cores
- Reactor Physics