Unusual statistics of interference effects in neutron scattering from compound nuclei

We consider interference effects between p-wave resonance scattering amplitude and background s-wave amplitude in low-energy neutron scattering from a heavy nucleus which goes through the compound nucleus stage. The first effect is in the difference between the forward and backward scattering cross sections (the 𝐩𝑖⋅𝐩𝑓 correlation). Because of the chaotic nature of the compound states, this effect is a random variable with zero mean. However, a statistical consideration shows that the probability distribution of this effect does not obey the standard central limit theorem. That is, the probability density for the effect averaged over n resonances does not become a Gaussian distribution with the variance decreasing as 𝑛^−1/2 (“violation” of the theorem). We derive the probability distribution of the effect and the limit distribution of the average. It is found that the width of this distribution does not decrease with the increase of n, i.e., fluctuations are not suppressed by averaging. Furthermore, we consider the 𝜎⋅(𝐩𝑖 × 𝐩𝑓) correlation and find that this effect, although much smaller, shows fluctuations which actually increase upon averaging over many measurements. This behavior holds for 𝜀 >𝛤_𝑝 where 𝜀 is the distance to the resonance, and 𝛤_𝑝 is the resonance width. Limits of the effects due to finite resonance widths are also considered. In the Appendix we present a simple derivation of the limit theorem for the average of random variables with infinite variances.