## Fast Non-leakage Probability

**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 this process may not be neglected. 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**, 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._{f}**1030**

**↓**

**P _{f} ~ 0.95**

**↓**

**979**

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).

## 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 τ by the migration length M in the one group equation. The term **B ^{4}** is very small for large reactors and therefore it can be neglected. We may then write.

where M is the **migration area (m ^{2}).** The migration length is defined as the square root of the migration area.

## Main operational changes, that affect this factor:

**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 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.

**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.

## Social Media