The Scope and Methods oF Perturbation Theory

As an illustration of such problems, consider the situation wherein a smaJl lump of absorbing material is introduced at some point within a critical assembly. Since the precise nuclear balance of the system has been upset, it can no longer sustain itself in steady-state operation, and it is to be expected that the over-all flux level in the reactor would fall in time, eventually to zero. In order to restore the steady-state condition, it would be necessary to add a certain amount of positive reactivity to the system (presumably by means of the control system). If it were possible to calculate the required additional reactivity, the time-dependent behavior (at least after long time) of the uncompensated case would be predicted as an exponential decay of the flux level with a time constant given by the ratio of the neutron lifetime in the system to the required extra multiplication [see, for example, Eq. (9.23)]. (It would be supposed that the small amount of absorber added would have a negligible effect on the neutron lifetime.) Thus the central problem is the calculation of the reactivity decrease resulting from the introduction of the absorber. It may be seen at once that the result is dependent upon the position in the system at which the absorber is placed. Clearly, if the lump were placed near the outer boundary of the assembly it would absorb much fewer neutrons than if it were placed near the center. The effect 

Suppose that the reactor equations for the critical assembly can be written in the general form 

Here 0o(r,w) denotes the neutron flux as a function of space and letharg , and O symbolizes an operator chosen appropriately for the system at hand. For example, in the one-velocity model the operator O would have the form 

However, much more complicated operators are also encompassed in the general treatment e.g., in a later section the operator O will be taken as a matrix. It will be convenient for the analysis which follows to define operators Oi and O2 such that 

Here the intent is to bring the quantity v explicitly into the calculation. In the one-velocity case, for example, we have from (13.2) 

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