It is convenient to begin with the multi-group diffusion equation in a steady state.

(1.84)

Dg diffusion coefficient in energy group g

Ъ1д: macroscopic total cross section in group g

£sg’g: macroscopic scattering cross section from group g’ to group g

v: average number of neutrons released per fission

ILfg. macroscopic fission cross section in group g

Xg. fission spectrum in group g

фд. neutron flux in group g

k: effective multiplication factor

Equation (1.84) can be represented in the matrix form expanded in the energy group as

(1.85)

where the matrix elements of M and F, and the elements of vector ф are given by the followings.

(1.86)

Wgg^XgVg’Zfg’ (1-87)

(Ф)д = Фд (1-88)

Since M and F are matrices and the former includes differential operators, the

product order is not commutative. M and F are called the operators on ф and the effective multiplication factor k is called the eigenvalue.

The inner product between two arbitrary functions / and g which are vectors in the energy group is defined as

(Л д>Ліії(?)д,(г)<і*г (1-89)

9

where fg denotes the complex conjugate of fg. Since physical quantities used in nuclear reactor physics are normally real numbers, the complex conjugate need not actually to be considered (although necessary in mathematics).

The inner product to define the operator M{ adjoint to the operator M is used as Eq. (1.90).

(MV, g) = (J, Mg) (1.90)

The adjoint operator of F can be defined in the same way.

In general, there is a corresponding solution ф{ (the adjoint neutron flux) to the adjoint equation

(1.91)

Here, it is noted that the eigenvalue is the same as that in Eq. (1.85). This can be found as the following. First, Eq. (1.91) is written as

(Ю

The inner product of Eq. (1.85) is taken with the adjoint flux ф{ as

From the definition of the adjoint operator in Eq. (1.90), the following is given.

(MVf- 0)=НгЧрУ. )

ft

Applying Eq. (Й0 to the left-hand side gives

-^-(FV, ф)=± )

and it is found that k = k{*. Since the eigenvalue in the multi-group diffusion equation is a real number, k = k{, therefore it is evident that the eigenvalue in Eq. (1.85) agrees with that in its adjoint equation.

Next, it is necessary to discuss some characteristics of the adjoint operator. An inner product for one-group macroscopic cross section can be written as

if, !g)=/vf*I, gd3r=fvil, f)*gd3r = (‘Zf, g) (1-92)

It is found that = £ and thus the operator and its adjoint operator are identical. Such operators are said to be self-adjoint.

Then, the spatial derivatives in the leakage term of the multi-group diffusion equation are considered, that is,

where the Integration by Parts formula was used to move the operand function of the differential operator from g to /. The first term on the right-hand side can be converted into an integral over the surface using Gauss’ theorem. Since the func­tions / and g are regarded as zero on the surface, the surface integral vanishes. Repeating this procedure for the second term gives

if, ^•D%’>=-/v(^f*>D%dar

=-fvV — [( Vf*)Dg]d

=fvtf-D^f)*gd3r = (V-DVf, g) .

V • DV and thus the leakage term is also self-adjoint.

From Eqs. (1.92) and (1.93), it is apparent that the operators in the static one-group diffusion equation are self-adjoint. In the one-group theory, the diffusion equation and adjoint equation are identical, namely, the neutron flux and its adjoint flux are identical except for the normalization factor.

Not being self-adjoint is encountered in multi-group operators or the time differential operators, that is,

(1.94)

where T implies the transpose matrix. Equation (1.94) is derived in the following. Since the time differential operators are out of the scope of this book, the references [23—25] should be consulted for Eq. (1.95).

The relation between the multi-group operator M and its adjoint operator M{, which meets the definition of the adjoint operator in Eq. (1.90), can be determined to find Eq. (1.94). It can be readily expressed with the matrix elements in energy group. Disregarding the spatial dependence of the operators for simplicity, the right-hand side of Eq. (1.90) is rewritten as

(Л Mflf)=2 f* (M 9=lfg*Mgg gg’ =£ f’Yg,

9 9,9 9, 9′

and the left-hand side is rewritten as

(му, g)=1j(M Y)^ =ЪШ}„ feTg,■

9 9,9 .

Hence,

м},=м*я- мг=мг*.

The matrix elements of the adjoint operators M{ and F{ are concretely represented by the next two equations.

№)gg — = И • Dj + Ztg)8gg’ — Zsgg. (1.96)

Wgg^VahgXg’ (1.97)

As mentioned above, the multi-group diffusion equation was represented in the form of a matrix and the adjoint operators were introduced. These are very useful in deriving the perturbation theory to calculate reactivity changes.