## Perturbation Theory

The change in reactivity caused by a small change (perturbation) occurring in reactors can be expressed by perturbation theory. Suppose that a perturbation (denoted by £) appears in the multi-group diffusion operators as

M’ = M + 8M (1.98)

F’ = F + 8F (1.99)

where M and F are the operators before the perturbation and M0 and F0 are those after the perturbation.

The multi-group diffusion equation before the perturbation and its adjoint equation are written as

(1.100)

My = 2-Fy (1.101)

and the multi-group diffusion equation after the perturbation is written as

(1.102)

The dagger symbol {is superscripted for the adjoint operator and adjoint neutron flux, and the prime symbol’ is used for the operator, flux, and effective multiplication factor after the perturbation. It should also be remembered that the effective multiplication factor in the adjoint equation is identical to that in the original diffusion equation.

The change in reactivity is calculated from the change from k to k’. This reactivity change becomes the expression of the perturbation theory through a mathematical formulation with 8M and £F. Taking the inner product with ф’ on both sides of Eq. (1.102) gives

Substituting M’ and F’ by Eqs. (1.98) and (1.99) respectively gives

Шф’Жф 8Мф’)=у(ф ‘)

Using the definition of the adjoint operator [Eq. (1.90)] and the adjoint equation [Eq. (1.101)], the first term in the left-hand side can be transformed as

<y, М0′)=(МУ, f)=2-(Fy, ф’)=^г(ф F

!Z iz

and then

is given. Hence, the change in reactivity caused by the perturbation can be found as

By approximating 1/k0 multiplied by SF in the numerator of the right-hand side as unity, the reactivity change becomes

(1.103)

This expression is called the exact perturbation theory and the non-matrix form is given by Eq. (1.104).

(1.104)

To evaluate the reactivity change using the exact perturbation theory, it is necessary to know the neutron flux after the perturbation as well as the adjoint flux before the perturbation and changes of the macroscopic cross sections and diffusion coefficient. In other words, the change in neutron flux caused by the perturbation should be considered.

If the perturbation is small enough and its effect on neutron flux is similarly small, the neutron flux after the perturbation can be approximated to that before the perturbation. Equation (1.103) can be written as Eq. (1.105) which is called the first-order perturbation theory.

(1.105)

Further, this equation becomes simpler in one-group theory because the neutron flux is self-adjoint. The equation in the one-group first-order perturbation theory can be detailed as Eq. (1.106).

J IS(vXf (r ))ф2(r)—SD (г )(Уф Cr ))2—£Za (г)ф2(г )1 d3r [2]

Fig. 1.15 Control rod partially inserted along the axis of a cylindrical reactor