First, the symmetry properties of the solution do not change in time because (4.1) is linear. This is not true for nonlinear equations. Secondly, the equations in (3.3) need to be satisfied. That is, the operations of the equation (4.4) and the boundary conditions must commute with the symmetry group elements. The symmetries of equation (4.4) are determined by the operators, the material parameters (cross-sections) and the geometry of V. The first term involves derivatives:

V(Dk (r)VYk (r, t)) = VDk (r)VYk (r) + Dk (r)V2 Y (r).

Here the first term contains a dot product which is invariant under rotations and reflections. The second term involves the laplace operator, which is also invariant under rotations and reflections. Thus, the major limiting symmetry factors are the material distributions, or the associated cross-sections as functions of space, and the shape of V. We assume the material distribution to be completely symmetric, thus for any cross-section E(r) we assume the transformation property

Og 4r) = Z(D(g)r) = £(r’) = E(r). (4.7)

Here Og is an operator applicable to the possible solutions. D(g) is a matrix representation of the symmetry group of the diffusion equation applicable to r. The following operators are encountered in diffusion theory. The general form of a reaction rate at point r Є V can be expressed as

R(r) = Еададад. (4.8)

k1

Here subscript 1 refers to the symmetric component. Since

OgR(r) = Og £Еи (г)Ти (r) = £ Og (Еи(г)Ти(г)) = £Еи(г^Ти(г) k1 k1 k1

because the material distribution is assumed symmetric hence Og E(r) = Е(г) for every symmetry g, the transformation properties of a reaction rate are completely determined by the transformation properties of the flux Yj-1 (г) . The normal component of the net current at r Є 9V is

Jnk (г) = — Dk (r)(nV)Y (г), (4.9)

where n is the normal vector at r. We apply Og to Jnk (г) to obtain:

OgJnk(г) = — Og (Dk(r)(nV)Y(г)) = — Dk(r)(nV)OgYk(г). (4.10)

Thus, the transformation properties of the normal component of the net current agree with the transformation properties of the flux. In diffusion theory, the partial currents are defined as

11

Ik (г) = 4 (Y (г) — 2Jnk (г)) ; Jk (г) = 4 (Y (г) + 2 Jnt (г)) . (4.11)

From (4.9) it follows that the transformation properties of the partial currents correspond to the transformation properties of the flux.

The boundary condition (4.3) commutes with rotations and reflections provided the material properties do. The same is true for the diffusion equation (4.1). Our first conclusion is that the material distribution may set a limit to the symmetry properties. As to the symmetries, the volume V under consideration may also be a limiting factor. Let Og be an operator that commutes with the operations of the diffusion equation (4.1) and (4.3). Furthermore, the representation D(g) maps V into itself. The set of operators form a group; the group operation is the repeated application. That group is called the symmetry group of the diffusion equation.

Example 4.1 (Symmetries in a homogeneous square). This symmetry group has eight elements, four rotations: E, C4, C|, C| and four reflections ax, ay, called of type av and a^, a^2 called of type а^. Characters of a given class have identical values. This group is known as the symmetry group of the square and denoted as C4v. The first column of a character table gives a mnemonic name to each representation, and a typical expression transforming according to the given representation. The first line is reserved for the most symmetric representation called unit representation. From the character table of the group C4v, we learn that there are groups with the same character tables, there are five irreducible representations labeled A, A2, B1, B2, E where As and Bs are one. dimensional and E is two-dimensional, it has two linearly independent components transforming as the x and y coordinates. □

Example 4.2 (Symmetries in a homogeneous equilateral triangle). The group has six elements, three rotations: E, C3, C2, and three reflections through axis passing one edge: aa, аь, ac called type av. The symmetry group is isomorphic to the C3v group and its character table is the same as that of the group D3. The C3v group is the symmetry group of the equilateral triangle, it has two one-dimensional and one two-dimensional representations. □

The key observation concerning the applications of symmetry considerations in boundary value problems is as follows. For a homogeneous problem (4.4) where there is no external source, the boundary condition is homogeneous, and every macroscopic cross-section Е(г), г Є V is such that

for all Og mapping V into itself. When the boundary conditions kg(r) in the expressions (4.3) transform according to an irreducible subspace fa (r) then the neutron flux Ф(г), the partial currents I(r), J(r), the reaction rate

G

R(r) = £Eg (r)Yg (r) k=1

all transform under the automorphism group of V as do the boundary conditions kg (r).

The symmetry group of the volume V makes it possible to reduce the domain on which we have to determine the solution of the diffusion theory problem. Once we know the transformation rule of the flux, for example, it suffices to calculate the flux in a part of V and exploit the transformation rules. That observation is formulated in the following concise way. Let r Є V a point in V and let g ■ r be the image of r under g Є G. Then the set of points g ■ r, g Є G is called the orbit of r under the group G. If there is a set V0 Є V such that the orbits of r0 Є V) give every point[16] of V we call V0 the fundamental domain of V. It is thus sufficient to solve the problem on the fundamental domain V), and "continue" the solution to the whole volume V.

When the boundary condition is not homogeneous or there is an external source, we exploit the linearity of the diffusion equation. The general solution is the sum of two terms: one with external source but homogeneous boundary condition and one with no external source but with non-homogeneous boundary condition. In either case, it is the external term that determines the transformation properties of the respective solution component.