8.7 Experimental Observation of RMT Predictions
287
level sequences are pure in that all levels in a given sequence have the same angular
momentum, J , and parity, π . Figures 8.14c and 8.15c show a level sequence and
spacing distribution, respectively, for a sequence with Poisson random spacing.
Finally, Figs. 8.14d and 8.15d show experimental data for slow neutron scattering
on the tantalum isotope 181 Ta. The data for tantalum are in fact a mixed sequence
because they consist of a mixture of states with J π = 3 + and 4 + (Hacken et al.
1978). The histograms in Fig. 8.15 reveal some interesting tendencies. It is clear
that the nuclear level spacings are not Poisson but exhibit level repulsion. The
mixed sequence, Fig. 8.15d, has a large spacing tail that is suggestive of Poissonlike behavior.
Haq et al. (1982) have obtained even more convincing results. They have
combined data from many different experimental nuclear level sequences in order
to obtain better level statistics. They computed the average 3 -statistic using 1407
resonance energy levels corresponding to 30 different sequences from 27 different
nuclei (references to these experimental data are given in Haq et al. 1982):
(i) a total of 1146 levels from slow neutron resonance data on
110,112,114 Cd,
152,154 Sm, 154,156,158,160 Gd, 160,162,164 Dy,
166,168,170 Er, 172,174,176 Yb,
182,184,186 W, 232 Th, and 238 U;
(ii) a total of 157 levels from proton resonance data on 44 Ca(J =
1
2
± ) and
48 Ti(J =
1
2
+ ); and
(iii) a total of 104 levels from (n, γ ) reaction data on 177 Hf(J = 3, 4) and
179 Hf(J = 4, 5).
Their results for the 3 -statistic are shown in Fig. 8.16. The data are in excellent
agreement with the predictions of the Gaussian orthogonal ensemble (GOE). GOE
has no parameters to fit. By using real symmetric matrices, it takes into account only
rotation and time reversal invariance. It contains no information about the detailed
form of the nuclear Hamiltonian.
8.7.2 Experimental Molecular Spectral Statistics
“Whereas the spectra of diatomic molecules are usually clearly arranged and
conceptually easy to interpret, even the smallest polyatomic molecules may exhibit
very irregular spectral sequences and high level density frustrating a priori a detailed
analysis of individual levels.” (Haller et al. 1983) In view of this, Haller, Koppel, and
Cederbaum have obtained the spectral spacing distribution for energy levels of NO 2
with vibronic symmetry, B 2 . They used 140 experimental levels from (Smalley et al.
1983) in the energy range 14,900–17,500 cm −1 . The spectral spacing histogram
is shown in Fig. 8.17a. The histogram shows evidence of level repulsion for this
molecular system.
287
level sequences are pure in that all levels in a given sequence have the same angular
momentum, J , and parity, π . Figures 8.14c and 8.15c show a level sequence and
spacing distribution, respectively, for a sequence with Poisson random spacing.
Finally, Figs. 8.14d and 8.15d show experimental data for slow neutron scattering
on the tantalum isotope 181 Ta. The data for tantalum are in fact a mixed sequence
because they consist of a mixture of states with J π = 3 + and 4 + (Hacken et al.
1978). The histograms in Fig. 8.15 reveal some interesting tendencies. It is clear
that the nuclear level spacings are not Poisson but exhibit level repulsion. The
mixed sequence, Fig. 8.15d, has a large spacing tail that is suggestive of Poissonlike behavior.
Haq et al. (1982) have obtained even more convincing results. They have
combined data from many different experimental nuclear level sequences in order
to obtain better level statistics. They computed the average 3 -statistic using 1407
resonance energy levels corresponding to 30 different sequences from 27 different
nuclei (references to these experimental data are given in Haq et al. 1982):
(i) a total of 1146 levels from slow neutron resonance data on
110,112,114 Cd,
152,154 Sm, 154,156,158,160 Gd, 160,162,164 Dy,
166,168,170 Er, 172,174,176 Yb,
182,184,186 W, 232 Th, and 238 U;
(ii) a total of 157 levels from proton resonance data on 44 Ca(J =
1
2
± ) and
48 Ti(J =
1
2
+ ); and
(iii) a total of 104 levels from (n, γ ) reaction data on 177 Hf(J = 3, 4) and
179 Hf(J = 4, 5).
Their results for the 3 -statistic are shown in Fig. 8.16. The data are in excellent
agreement with the predictions of the Gaussian orthogonal ensemble (GOE). GOE
has no parameters to fit. By using real symmetric matrices, it takes into account only
rotation and time reversal invariance. It contains no information about the detailed
form of the nuclear Hamiltonian.
8.7.2 Experimental Molecular Spectral Statistics
“Whereas the spectra of diatomic molecules are usually clearly arranged and
conceptually easy to interpret, even the smallest polyatomic molecules may exhibit
very irregular spectral sequences and high level density frustrating a priori a detailed
analysis of individual levels.” (Haller et al. 1983) In view of this, Haller, Koppel, and
Cederbaum have obtained the spectral spacing distribution for energy levels of NO 2
with vibronic symmetry, B 2 . They used 140 experimental levels from (Smalley et al.
1983) in the energy range 14,900–17,500 cm −1 . The spectral spacing histogram
is shown in Fig. 8.17a. The histogram shows evidence of level repulsion for this
molecular system.