## Migration Length – Migration Area

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

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

The physical meaning of the diffusion length is that:

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

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

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

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

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

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

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

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

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

**Effect on the Neutron Leakage**

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

**Fast Non-leakage Probability**

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

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

**Thermal Non-leakage Probability**

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

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

**diffusion length**, B is the

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

_{g}= B

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

**Total Non-leakage Probability**

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

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

**τ**by the

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

**B**

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

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

**P**) are affected by a change in

_{f}and P_{t}**moderator temperature**in a heterogeneous water-moderated reactor and the directions of the feedbacks is the same, the resulting

**total non-leakage probability**is also sensitive on the change in the moderator temperature. In result, an

**increase in the moderator temperature**causes that the probability of

**leakage increases**. This effect is one of two main effects causing the

**moderator temperature coefficient (MTC)**of most PWRs to be negative.

**The thermal neutron leakage** is dependent on the core temperature (or moderator temperature). The moderator temperature influences macroscopic cross-sections for elastic scattering reaction, especially the atomic number density – **N _{H2O}**(

**Σ**) due to the

_{s}=σ_{s}.N_{H2O}**thermal expansion**of water. Also the microscopic cross-section (

**σ**) for neutron absorption changes with core temperature. Both processes have the same direction. As the temperature of the core increases,

_{a}**the diffusion coefficient**(

**D = 1/3.Σ**) increases and the absorption cross-section decreases and this together causes the increase in the thermal neutron leakage. This physical process is a part of the

_{tr}**moderator temperature coefficient (MTC).**

**The fast neutron leakage** is also dependent on the core temperature (or moderator temperature). **The moderator temperature** influences macroscopic cross-sections for elastic scattering reaction (**Σ _{s}=σ_{s}.N_{H2O}**) due to the

**thermal expansion**of water. As the temperature of the core increases, the

**fast neutron leakage increases**. This physical process is a part of the

**moderator temperature coefficient**(MTC) and it is responsible for an increase in neutron flux measured by neutron detectors, which are situated around the reactor vessel.

**In power reactors**, **the total non-leakage probability** also significantly changes with **fuel burnup**. This dependency is not associated with any of the parameters like the diffusion coefficient or the geometrical buckling. In power reactors, the total non-leakage probability strongly depends on the certain **fuel loading pattern** and also on the reload strategy. The neutron leakage is one of key parameters in the** neutron and fuel economy**.

In order to enhance the neutron and fuel economy, core designers designs the **low leakage loading patterns**, in which fresh fuel assemblies are not situated in the peripheral positions of the reactor core. The peripheral positions are loaded with the fuel with highest fuel burnup. These “high” burnup assemblies have inherently lower relative power (due to the lower **k _{inf}** and due to the fact they feel the presence of non-multiplying environment) in comparison with the average assemblies. In short, this parameter is significantly dependent on the certain loading pattern. During fuel burnup, the neutron leakage usually increases, especially in low leakage loading patterns. This process is caused by reducing the differences in

**k**between fresh fuel assemblies and peripheral high-burnup assemblies.

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