**Application of Finite Symmetry Groups to Reactor Calculations**

Yuri Orechwa1,\* and Mihály Makai2 *1NRC, Washington DC 2BME Institute of Nuclear Techniques, Budapest 1USA 2Hungary* 

#### **1. Introduction**

284 Nuclear Reactors

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Group theory is a vast mathematical discipline that has found applications in most of physical science and particularly in physics and chemistry. We introduce a few of the basic concepts and tools that have been found to be useful in some nuclear engineering problems. In particular those problems that exhibit some symmetry in the form of material distribution and boundaries. We present the material on a very elementary level; an undergraduate student well versed in harmonic analysis of boundary value problems should be able to easily grasp and appreciate the central concepts.

The application of group theory to the solution of physical problems has had a curious history. In the first half of the 20th century it has been called by some the "Gruppen Pest" , while others embraced it and went on to win Noble prizes. This dichotomy in attitudes to a formal method for the solution of physical problems is possible in light of the fact that the results obtained with the application of group theory can also be obtained by standard methods. In the second half of the 20th century, however, it has been shown that the formal application of symmetry and invariance through group theory leads in complicated problems not only to deeper physical insight but also is a powerful tool in simplifying some solution methods.

In this chapter we present the essential group theoretic elements in the context of crystallographic point groups. Furthermore we present only a very small subset of group theory that generally forms the first third of the texts on group theory and its physical applications. In this way we hope, in short order, to answer some of the basic questions the reader might have with regard to the mechanical aspects of the application of group theory, in particular to the solution of boundary value problems in nuclear engineering, and the benefits that can accrue through its formal application. This we hope will stimulate the reader to look more deeply into the subject is some of the myriad of available texts.

The main illustration of the application of group theory to Nuclear Engineering is presented in Section 4 of this chapter through the development of an algorithm for the solution of the neutron diffusion equation. This problem has been central to Nuclear Engineering from the very beginning, and is thereby a useful platform for demonstrating the mechanics of bringing group theoretic information to bear. The benefits of group theory in Nuclear Engineering are

<sup>\*</sup>The views expressed are those of the authors and do not reflect those of any government agency or any part thereof.

An extensive discussion of the mathematics and application of group theory to engineering problems in general and nuclear engineering in particular is presented in (Makai, 2011).

Application of Finite Symmetry Groups to Reactor Calculations 287

Although the basic mathematical definition of a group and much of the abstract algebraic machinery applies to finite, infinite, and continuous groups, our interest for applications in nuclear engineering is limited to finite point groups. Furthermore, it should be kept in mind that most of the necessary properties of the crystallographic point groups for applications, such as the group multiplication tables, the class structures, irreducible representations, and characters are tabulated in reference books or can be obtained with modern software such as

An abstract group *G* is a set of elements for which a law of composition or "product" is defined. For illustrative purposes let us consider a simple set of three elements {*E*, *A*, *B*}. A law of composition for these three elements can be expressed in the form of a multiplication table, see Table 1. In position *i*, *j* of Table 1. we find the product of element *i* and element *j* with the numbering 1 → *E*, 2 → *A*, 3 → *B*. From the table we can read out that *B* = *AA* because element 2, 2 is *B* and the third line contains the products *AE*, *AA*, *AB*. The table is symmetric therefore *AB* = *BA*. Such a group is formed for example by the even permutations of three objects: *E* = (*a*, *b*, *c*), *A* = (*c*, *a*, *b*), *B* = (*b*, *c*, *a*). The multiplication table reflects four necessary **E A B E** E A B **A** A B E **B** B E A

conditions that a set of elements must satisfy to form a group *G*. These four conditions are: 1. The product of any two elements of *G* is also an element of *G*. Such as for example *AB* = *E*.

3. *G* contains a unique element *E* called the identity element, such that for example *AE* =

4. For every element in *G* there exists another element in *G*, such that their product is the identity element. In our example *AB* = *E* therefore *B* is called the inverse of *A* and is

The application of group theory to physical problems arises from the fact that many characteristics of physical problems, in particular symmetries and invariance, conform to the definition of groups, and thereby allows us to bring to bear on the solution of physical

For example, if we consider a characteristic of an equilateral triangle we observe the following with regard to the counter clockwise rotations by 120 degrees. Let us give the operations

The group operation is the sequential application of these operations, the leftmost operator should be applied first. The reader can easily check the multiplication table 2. applies to the

<sup>3</sup>-rotation by 240*o*.

the following symbols: *E*-no rotations, *C*3-rotation by 120*<sup>o</sup>*, *C*3*C*<sup>3</sup> = *C*<sup>2</sup>

**2. Basic group theoretic tools**

**2.1 Group definition**

denoted *B* = *A*−1.

MAPLE or MATHEMATICA for example.

Table 1. Multiplication table for elements {*E*, *A*, *B*}

problems the machinery of abstract group theory.

2. Multiplication is associative. For example (*AB*)*E* = *A*(*BE*).

*EA* = *A* and the same holds for every element of *G*.

not restricted to solving the diffusion equation. We wish to also point the interested reader to other areas of Nuclear Engineering were group theory has proven useful.

An early application of group theory to Nuclear Engineering has been in the design of control systems for nuclear reactors (Nieva, 1997). Symmetry considerations allow the decoupling of the linear reactor model into decoupled models of lower order. Thereby, control systems can be developed for each submodel independently.

Similarly, group theoretic principles have been shown to allow the decomposition of solution algorithms of boundary value problems in Nuclear Engineering to be specified over decoupled symmetric domain. This decomposition makes the the problem amenable to implementation for parallel computation (Orechwa & Makai, 1997).

Group theory is applicable in the investigation of the homogenization problem. D. S. Selengut addressed the following problem (Selengut, 1960) in 1960. He formulated the following principle: If the response matrix of a homogeneous material distribution in a volume *V* can be substituted by the response matrix of a homogeneous material distribution in *V*, then there exists a homogeneous material with which one may replace *V* in the core. The validity of this principle is widely used in reactor physics, was investigated applying group theoretic principles (Makai, 1992),(Makai, 2010). It was shown that Selengut's principle is not exact; it is only a good approximation under specific circumstances. These are that the homogenization recipes preserve only specific reaction rates, but do not provide general equivalence.

Group theory has also been fruitfully applied to in-core signal processing (Makai & Orechwa, 2000). Core surveillance and monitoring are implemented in power reactors to detect any deviation from the nominal design state of the core. This state is defined by a field that is the solution of an equation that describes the physical system. Based on measurements of the field at limited positions the following issues can be addressed:


The solution to these problems requires a complex approach that incorporates numerical calculations incorporating group theoretic considerations and statistical analysis.

The benefits of group theory are not restricted to numerical problems. In 1985 Toshikazu Sunada (Sunada, 1985) made the following observation: If the operator of the equation over a volume *V* commutes with a symmetry group *G*, and the Green's function for the volume *V* is known and volume *V* can be tiled with copies tile *t* (subvolumes of *V*), then the Green's function of *t* can be obtained by a summation over the elements of the symmetry group *G*. Thus by means of group theory, one can separate the solution of a boundary value problem into a geometry dependent part, and a problem dependent part. The former one carries information on the structure of the volume in which the boundary value problem is studied, the latter on the physical processes taking place in the volume. That separation allows for extending the usage of the Green's function technique, as it is possible to derive Green's functions for a number of finite geometrical objects (square, rectangle, and regular triangle) as well as to relate Green's functions of finite objects, such as a disk, or disk sector, a regular hexagon and a trapezoid, etc. Such relations are needed in problems in heat conduction, diffusion, etc. as well.

An extensive discussion of the mathematics and application of group theory to engineering problems in general and nuclear engineering in particular is presented in (Makai, 2011).
