Neutron-Resonance Spectroscopy. VIII. The Separated Isotopes of Erbium: Evidence for Dyson's Theory Concerning Level Spacings

Results are given for high-resolution neutron-resonance-spectroscopy studies of the separated erbium isotopes (164, 166, 167, 168, 170) using the Nevis Synchrocyclotron. These results, particularly ${\mathrm{Er}}^{166}$ to 4200 eV ($n=109\mathrm{}$ levels), give the first excellent agreement with the Dyson-Mehta (DM) $\Delta$ statistic which applies for the statistical orthogonal ensemble (O.E.) of Wigner and Dyson. The observed ${\mathrm{Er}}^{166}$ levels seem to be a nearly pure and complete $s$ population, fitting Wigner's nearest-neighbor spacing law and the Porter-Thomas (PT) single-channel ${\Gamma }_n^0$ distribution. The weakest ${\mathrm{Er}}^{168}$ and ${\mathrm{Er}}^{170}$ levels include some $p$ levels (excess of weak ${\Gamma }_n^0$ values) which are separated by a Bayes-theorem method yielding resulting $s$ populations which also give good fits to all statistical O.E. tests. Unexpectedly, the ${\mathrm{Er}}^{166}$ levels give a nonzero correlation coefficient between adjacent ${\Gamma }_n^0$ values of $\rho ({\Gamma }_{\mathrm{nj}}^0,{\Gamma }_{nj+1\mathrm{}}^0)=-0.21\mathrm{}\pm 0.08$, and with less fluctuation in $\Sigma {\Gamma }_n^0$ over adjacent intervals than expected for an uncorrelated PT series of ${\Gamma }_n^0$ values. Assuming a true $p$ level density 3 times that for $s$ levels, $S_1$ values are calculated on the basis of the observed excess of weak levels and our known level detection threshold. We obtain ${10}^4{S}_0=1.70\mathrm{}\pm 0.23, 1.89\pm 0.20, 1.50\pm 0.21, \mathrm{and} 1.54\pm 0.22$ for 166, 167, 168, 170; and ${10}^4{S}_1=\le 0.75, 0.70\pm 0.20, \mathrm{and} 0.80\pm 0.25$ for 166, 168, 170. We find $\Delta =0.455\mathrm{}$ ($n=109\mathrm{}$) for ${\mathrm{Er}}^{166}$, 0.287 ($n=50\mathrm{}$) for ${\mathrm{Er}}^{168}$, and 0.359 ($n=31\mathrm{}$) for ${\mathrm{Er}}^{170}$ vs DM's predicted $\Delta =0.47\mathrm{}\pm 0.11, 0.39\pm 0.11, \mathrm{and} 0.34\pm 0.11$. For a large $n$, $\Delta$ is much larger for an uncorrelated Wigner (U.W.) spacing sequence or for an incomplete or impure $s$ level O.E. set. The correlation coefficient for adjacent spacings, $\rho (S_j,S_{j+1\mathrm{}})\equiv \rho =-0.22, -0.29, -0.09\mathrm{}$ for 166, 168, 170 vs $\rho \approx -0.27\mathrm{}$ (±0.09, 0.13, 0.17 for 166, 168, 170) (O.E.). The probability is 0.0004 for $[\Delta +\rho ]\le \mathrm{the} {\mathrm{Er}}^{166}(4.2\mathrm{}-\mathrm{keV})$ value for a U.W. set. The $〈D〉$ values for $s$ levels, if our $s$ level count is correct, are 38.4, 4.06, 95.3, and 155 eV [$\pm \frac{0.9}{n}$ fractional uncertainty (O.E.)] for $n=109, 30, 50, \mathrm{and} 31$ for 166, 167, 168, 170.