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 Er¹⁶⁶ to 4200 eV ($n=109$ levels), give the first excellent agreement with the Dyson-Mehta (DM) Δ statistic which applies for the statistical orthogonal ensemble (O.E.) of Wigner and Dyson. The observed Er¹⁶⁶ 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 Γₙ⁰ distribution. The weakest Er¹⁶⁸ and Er¹⁷⁰ levels include some p levels (excess of weak Γₙ⁰ 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 Er¹⁶⁶ levels give a nonzero correlation coefficient between adjacent Γₙ⁰ values of ρ(Γₙⱼ⁰,Γₙⱼ₊₁⁰)=-0.21±0.08, and with less fluctuation in ΣΓₙ⁰ over adjacent intervals than expected for an uncorrelated PT series of Γₙ⁰ values. Assuming a true p level density 3 times that for s levels, S₁ values are calculated on the basis of the observed excess of weak levels and our known level detection threshold. We obtain 10⁴S₀=1.70±0.23, 1.89±0.20, 1.50±0.21, and 1.54±0.22 for 166, 167, 168, 170; and 10⁴S₁=≤0.75, 0.70±0.20, and 0.80±0.25 for 166, 168, 170. We find Δ=0.455 ($n=109$) for Er¹⁶⁶, 0.287 ($n=50$) for Er¹⁶⁸, and 0.359 ($n=31$) for Er¹⁷⁰ vs DM's predicted Δ=0.47±0.11, 0.39±0.11, and 0.34±0.11. For a large n, Δ 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, ρ(Sⱼ,Sⱼ₊₁)≡ρ=-0.22, -0.29, -0.09 for 166, 168, 170 vs ρ≈-0.27 (±{}0.09, 0.13, 0.17 for 166, 168, 170) (O.E.). The probability is 0.0004 for [Δ+ρ]≤the Er¹⁶⁶(4.2-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 [±0.9/n fractional uncertainty (O.E.)] for n=109, 30, 50, and 31 for 166, 167, 168, 170.