## Neutron Leakage

In an infinite multiplication system, the leakage of the system is logically neglected. But all multiplying systems, all realistic reactor cores are finite multiplying system and the leakage may not be neglected. In general neutrons may leak out of the system during:

- During the
**slowing down process**, some of the neutrons leak out of the boundaries of the reactor core before they become thermalized. This process and its impact on the effective multiplying factor is characterized by the fast non-leakage factor, P_{f}, which is defined as the ratio of the number of fast neutrons that do not leak from the reactor core during the**slowing down process**to the number of fast neutrons produced by fissions at all energies. - During the
**neutron diffusion**, some of the neutrons leak out of the boundaries of the reactor core before they are absorbed. This process and its impact on the effective multiplying factor is characterized by the thermal non-leakage factor, Pt, which is defined as the ratio of the number of thermal neutrons that do not leak from the reactor core during the**neutron diffusion**process to the number of neutrons that reach thermal energies.

Note that, there is the consistency between the numerator in the definition of

**ε**and the denominator in the definition of P

_{f}.

The fast non-leakage probability is for large reactor cores about **0.92 – 0.98** and this value is minimally affected (in comparison with the other factors) by operational changes except changes in the moderator temperature. 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).

**The thermal non-leakage probability**is for large reactor cores about**0.95 – 0.98**and this value is minimally affected (in comparison with the other factors) by operational changes except changes in the**moderator temperature**. The only parameter, that influences**the thermal non-leakage probability**is the moderator temperature. 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**), 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 thermal neutrons

_{g}= B_{m}**leakage is higher for small cores**and also depends of the macroscopic slowing down power of neutron moderator (leakage is higher for poor moderators).

**The diffusion length**is given by following equation:

## 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.

_{inf}## Neutron Reflectors – Leakage Reduction

It is well known that each reactor core is surrounded by a **neutron reflector** or **reactor core baffle**. 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.

Except research reactors, practically all power reactor cores are designed to minimize the **neutron leakage**. In order to minimize the leakage, **neutron reflectors** surround reactor cores. The neutron reflector scatters back (or reflects) into the core many neutrons that would otherwise escape. By **reducing neutron leakage**, the reflector increases k_{eff} and reduces the amount of fuel necessary to maintain the reactor critical for a long period. In LWRs the neutron reflector is installed for following purposes:

- The neutron flux distribution is “
**flattened**“, i.e., the ratio of the average flux to the maximum flux is increased. Therefore reflectors**reduce the non-uniformity**of the power distribution. - Because of the higher flux at the edge of the core, there is much
**better utilization**in the peripheral fuel assemblies. This fuel, in the outer regions of the core, now contributes much more to the total power production. - The neutron reflector scatters back (or reflects) into the core many neutrons that would otherwise escape. The neutrons reflected back into the core are available for chain reaction. This means that the
**minimum critical size**of the reactor is reduced. Alternatively, if the core size is maintained, the reflector makes additional reactivity available for higher fuel burnup. The decrease in the critical size of core required is known as the**reflector savings**. - Neutron reflectors reduce neutron leakage i.e. to reduce the neutron fluence on a reactor pressure vessel.
- Neutron reflectors reduce a coolant flow bypass of a core.
- Neutron reflectors serve as a thermal and radiation shield of a reactor core.

**Nuclear and Reactor Physics:**

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**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.
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