Analysis of the Distribution of the Spacings Between Nuclear Energy Levels

An empirical spacing distribution is always based on a finite, and usually small, number of observed levels. Thus, even if the spacing of levels were described exactly by the random-matrix model, the observed distribution would necessarily fluctuate about the theoretical mean-the Gaudin-Mehta distribution. A statistic $\Lambda \mathrm{}\left(n\right)\mathrm{}$ is defined to enable one to judge whether the magnitude of the observed fluctuations about the Gaudin-Mehta distribution is compatible with the random-matrix model. It is found that the correlations between the spacings implied by the model tend to reduce the expected fluctuations significantly. The statistical properties of $\Lambda \mathrm{}\left(n\right)\mathrm{}$ are studied by means of a Monte Carlo calculation with matrices of order 100 sampled from the Gaussian orthogonal ensemble. An illustrative analysis of the published neutron resonances observed in ${\mathrm{U}}^{239}$ by Garg et al., reveals no obvious discrepancy between theory and experiment up to neutron kinetic energies of about 2 keV.