# Reflected Reactor – Two-group Diffusion Method

## Flux in a Reflected Thermal Reactor – Two-group Method

It was pointed out, the one-group method may provide reasonable results for the reactor, but this method does not accurately predict the flux. To be more accurate the energy of neutrons must be grouped into at least two groups. The two-group energy model leads to very interesting and important results, that must be considered in the design of nuclear reactors.

In this method, the entire range of neutron energies is divided into 2 intervals (fast group and thermal group). All of neutrons within each interval are lumped into a group and in this group all parameters such as the diffusion coefficients or cross-sections are averaged.

To find solution in a reactor core and in a reflector, it is necessary to solve the following diffusion equations for the fast and thermal energy groups are:

The solution of this system of homogeneous algebraic equations leads to a determinant of the coefficients. Such solution is shown in:

Reference: Ragheb, M. TWO GROUP DIFFUSION THEORY FOR BARE AND REFLECTED REACTORS, University of Illinois, 2006.

One of the striking results of such solution is that the thermal flux reaches local maximum near the core-reflector interface. This behaviour cannot be derived using one-group diffusion method, because it is caused just by thermalisation of fast neutrons. The fast neutrons, which are produced in the core can enter the reflector at high energy, are not absorbed as quickly in the reflector as neutrons thermalizing in the core, because absorption cross-sections in the reflector are much smaller than in the core (due to the absence of fuel). The thermal neutrons accumulates then near the core-reflector interface, resulting in the local maximum, that is usually known as the reflector peak. This also reduces the non-uniformity of the power distribution in the peripheral fuel assemblies and also reduces neutron leakage, i.e. increases keff of the system (or reduces the critical size of the reactor). This effect can be seen in the following figure.

The reflector peak can be seen only in thermal flux within the reflector. It is found that the fast flux does not show recovery peaks in the reflector, but rather drops off sharply inside the reflector (due to thermalization and absorption).

References:
Nuclear and Reactor Physics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
6. Kenneth S. Krane. Introductory Nuclear Physics, 3rd Edition, Wiley, 1987, ISBN: 978-0471805533
7. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
8. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
9. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

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

## See above:

Reflected Reactor