In analogy with the application of group theory in particle physics, where group theory leads to insights into the relationships between elementary particles, we present an application of group theory to the solution to a specific reactor physics problem. The question is whether it is possible to replace a part of a heterogeneous core by a homogeneous material so that the solution outside the homogeneous region remains the same? This old problem is known as homogenization (Selengut, 1960).

In particular, for non-uniform lattices, asymptotic theory has shown that a lattice composed of identical cells has a solution that is composed of a periodic microflux and a slowly varying macroflux. What happens if the cell geometry is the same but the material composition varies?

In reactor calculations, we solve an equation derived from neutron balance. In that equation, we encounter reaction rates, currents or partial currents. It is reasonable to derive all the quantities we need from one given basic quantity, say from the neutron flux at given points of the boundary. The archetype of such relation is the exiting current determined from the entering current by a response matrix. We show that by using irreducible components of the partial currents, the response matrix becomes diagonal.

The Selengut principle is formulated: if the response matrix of a given heterogeneous material in V can be substituted by the response matrix of a homogeneous material in V, there exist an equivalent homogeneous material with which one may replace V. This principle simplifies calculations considerably, and, therefore, has been widely used in reactor physics. We investigate the Selengut principle more closely(Makai, 2010),(Makai, 1992).

The analysis is based on the analytical solution of the diffusion equation derived in the previous Section. The problem is considered in a few energy groups, the boundary flux F is a vector, as well as the volume averaged flux Ф. Using that solution, we are able to derive matrices mapping into each other the volume integrated fluxes, the surface integrated partial and net currents. The derivation of the corresponding matrices is as follows. Our basis is the boundary flux, that we derive for each irrep i from (4.13). The expression (4.13) has three components. The first one is vector tk which is independent of the position r and is multiplied by an exponential function with r in the exponent. The third component is the weight Wk which is independent of r but varies with subscript k. The product is summed for subscript k, that labels the eigenvalues of the cross-section matrix in (4.14). That expression can be put into the following concise form:

Fi = T < fi > Ci, (5.1)

where Ci comprises the third component. Here < fi(r) > is a diagonal matrix. Note that position dependent quantities like reaction rates, follow that structure. The normal component of the net current is Jnet obtained from the flux by taking the derivative and is given in irrep i as

Leti = -DT < gi > c-i. (“)

We eliminate a to get

JnetA = — DT < gt/fi > T-1 Fi = RiFi. (5.3)

Here n is the outward normal to face Fi,

gi = — Vnfi (r). (5.4)

The volume integrated flux Ф is obtained after integration from (4.13) as

ф = T < FA1 > CAUFA1 = J(5.5)

and the integration runs over volume V of the node. Note that only irrep Al (i. e. complete symmetry) contributes to the average flux because of the orthogonality of the irreducible flux components. After eliminating cai from (5.1), we get the response matrix for determining the volume integrated flux Ф from the face integrated flux Fai:

Ф = T < Fai//ai > T-1 < F > = W < F > . (5.6)

This assures that V is completely described by matrix W and the diagonal matrices < F(r) > , < f (r) >, < g(r) > for each irrep. For example, we are able to reconstruct the cross-section matrix 2 from them. Note that WT = T < F/f >, the eigenvectors of matrix W are the eigenvectors of A. Now we need only a numerical procedure to find the eigenvalues Ak from

< F/f >.

The question is, under what conditions are the above calculations feasible. We count the number of response matrices. The matrix elements we need to characterize V may be all different and the number of matrices depends on the shape of V, since the number of irreducible components of the involved matrices depends on the geometry. In a square shaped homogeneous V, we have four Ri matrices and one W. Altogether we have to determine 5 * G * G elements. In an inhomogeneous hexagonal volume, there are 6 * G * G matrix elements, whereas the homogeneous material is described by G * (G + 1) parameters as in a homogeneous material there are altogether G * G cross-sections and G diffusion constants8. Therefore the Selengut principle is not exact it may only be a good approximation under specific circumstances. Homogenization recipes preserve only specific reaction rates, but they do not provide general equivalence.

5. Conclusions

The basic elements of the theory of finite symmetry groups has been introduced. In particular, the use of the machinery associated with the decomposition into irreducible representations, in analogy with harmonic analysis of functions in function space, in the analysis of Nuclear Engineering problems. The physical settings of many Nuclear Engineering problems exhibit symmetry, as for example in the solution of the multi-group neutron diffusion equation. This symmetry can be systematically exploited via group theory, and elicit information that leads to more efficient numerical algorithms and also to useful insights. This is a result due to the added information inherent in symmetry, and the ability of group theory to define the "rules" of the symmetry and allows one to exploit them.