The diffusion equation is one of the most widely used reactor physical models. It describes the neutron balance in a volume V, the neutron energy may be continuous or discretized (multi group model). The multi group version is:

where the processes leading to energy change are collected in Tkk’:

where subscripts k, k’ label the energy groups, Vk is the speed of neutrons in energy group

k, Yk(r) is the space dependent neutron flux in group k, and keff = 1. In general, the cross-sections Dk, Etk, Ek’^k, Efk’ are the space dependent diffusion constant, the total cross-section, the scattering cross-section, and the fission cross-section. Xk is called the fission spectrum. Equation (4.1) is a set of partial differencial equations, to which the initial condition Yk(r,0),r Є V and a suitable boundary condition, e. g. Yk(r, t),r Є 9V are given for every energy group k and every time t. The boundary conditions used in diffusion problems are of the type

(Vn)Yk(r) + bk(r)Yk(r) = hk(r) k = 1,…, G. (4.3)

for r Є dV. Here bk(r) depends on the boundary condition and may contain material properties, for example albedo.

The diffusion equation is a relationship between the cross-sections in V and the neutron flux Yk(r, t). The equation is linear in Yk(r, t). The main variants of equation (4.1) that are of interest in reactor physics are: l.

where the eigenvalue keff introduced as a parameter in Tkk’ thus allowing for a non-trivial solution Yk(r). That usage is typical in core design calculations.

2. Time dependent solution allowing time dependence in some cross-sections. A typical application is transient analysis.

3. Equation (4.1) is homogeneous but it is possible to add an external source and to seek the response of V to the source.

The structure of the diffusion equation is simple. Mathematical operations, like summation

and differentiation, and multiplication by material parameters (cross-sections) are applied

to the neutron flux. In such equations the symmetries are mostly determined by the space dependence of the material properties. In the next subsection we investigate the possible symmetries of equation (4.4) and the exploitation of those symmetries.

When the solutions (r), k = 1,…, G are known, not only the reaction rates, and net-

and partial currents can be determined, but also matrices can be created to transform these quantities into each other. From diffusion theory it is known that the solution is determined by specifying the entering current along the boundary dV. Thus the boundary flux is also determined. But the given boundary flux also determines the solution everywhere in V. The solution is given formally by a Green’s function as follows:

t G

Yk(r) = E Gk0,k(r0 ^ r)fk0 (r0)dr0. (4.5)

JdVk0=1

Here Gk0,k (r0 ^ r) is the Green’s function, it gives the neutron flux created at point r in energy group k by one neutron entering V at r0 in energy group k0; and fk0 (r0) is the given flux in energy group k0 at boundary point r0. Similarly the net current is obtained as

r G

Jnk(r) = — DkV E Gk0,k(r0 ^ r)fk0 (r0)dr0 (4.6)

JdVk0 = 1

where the V operator acts on variable r.