Effect of Small Perturbations on Pile Period

The effect of small perturbations (localized impurities, etc.) on the pile is important not only because of the theory of danger coefficient measurements but also because the effect of certain influences cannot be taken into account easily by any other than the perturbation method. The present report endeavors to go beyond Fermi s simple theory which applies only to a uniform bare pile and recognizes only one kind of neutrons (thermal). 

It was customary so far to express the effect of small perturbations as a change in the multiplication constant. However, this is a possible procedure only in the case of a uniform, bare pile. A composite pile, or even a simple pile with a reflector, has no single multiplication constant and it was thought best to express the effect of perturbations in terms of the change in reciprocal pile period. However, in order to avoid unnecessary complications, the effect of the delayed neutrons was omitted. Their inclusion would not cause any fundamental difficulty but would complicate the formulae and make them more cumbersome. They were omitted for this reason and the results for the change in period apply as they stand, only for very short perturbations (short as compared with the period of the delayed neutrons) or if the infinitely small perturbations cause a very much larger change in some sort of multiplication constant than the about 1% for which the delayed neutrons are responsible. However, as was stated before, it would not be difficult at all to include the delayed neutrons into 

In many of the calculations one is interested not so much in the change of the pile period caused by one perturbation as in the amount of another t3rpe of perturbation which would coimteract the perturbation of the first kind. This can be calculated most simply by determining the joint effect of both perturbations and choosing the second one so large that the joint effect should vanish. The delayed neutrons do not play ordinarily any role in such calculations. 

All pile equations considered heretofore have the form 

In this, the neutron density n is a function of the position and, in most theories, also of the energy. It could be made to be a function of the direction of the velocity also without the present section losing its applicability. 

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E.P. Wigner, Effect of Small Perturbations on Pile Period , CP-3048, June 13, 1945. 

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