8.7 Experimental Observation of RMT Predictions
285
8.7 Experimental Observation of RMT Predictions
In Chap. 7, we have already shown some experimental data for the spectral statistics
of chaotic cavities constructed from microwave waveguides. These are, in fact,
open systems that are driven into resonance by external energy sources. However,
the antennas used to couple the interior of the microwave cavity to the external
radiation source are designed to have minimum effect on the energy distribution
inside the cavity. Thus, the cavity resonances lie very close to the energy eigenvalues
of the closed cavity. These microwave experiments are more recent than the nuclear
scattering experiments described Sect. 8.2. As we saw in Fig. 8.2, as far back as
1964, scattering of slow neutrons off of heavy nuclei, such as 238 U, showed similar
behavior. It is hoped that the energies at which the scattering resonance takes place
are in direct correspondence with the eigenvalue spectrum of the internal states of
nuclei or molecules being studied. To keep the distinction clear, in this section we
will refer to resonance peaks as “levels.”
In the subsections below, we discuss in more detail the comparison between
RMT predictions, the results of nuclear scattering experiments, and the results of
molecular scattering experiments.
8.7.1 Experimental Nuclear Spectral Statistics
In order to study the spectral statistics of a level sequence, it is important that all
neighboring levels be included in the segment of the level sequence being studied.
One of the main difficulties in obtaining good sequences experimentally is that
of resolving closely spaced levels. Fortunately, resonances found in the scattering
of slow neutrons on heavy nuclei provide good sequences. “The resonances are
narrowed because of the strong surface reflection of long wavelength neutrons and
as a consequence one can often observe up to a few hundred resonances, essentially
all (if the target state is even-even) with the same exact quantum numbers (angular
momentum, J , parity, π ; isospin is also good but is usually irrelevant)” (Brody et al.
1981).
In Fig. 8.14, we show three experimentally obtained sequences of 50 levels taken
from the spectra of three different nuclei, and we show a level sequence of 50 levels
with Poisson random spacing. The levels have been rescaled to lie in the same
spectral span and have the same average spacing, D (Brody et al. 1981). The arrows
in Fig. 8.14 indicate level spacings smaller than one-quarter of the average spacing.
Such spacings cannot be seen on the scale shown. It is interesting that the pure
sequences have far fewer close spacings than do the Poisson or mixed sequences.
In Fig. 8.15, we show the nearest neighbor spacing distributions for these four
cases but using all available data for each nucleus. Figures 8.14a and 8.15a show
data from the s-wave scattering of slow neutrons on the erbium isotope 166 Er Liou
et al. (1972). Figures 8.14b and 8.15b show experimental data for proton scattering
on the titanium isotope 48 Ti (Prochnow et al. 1972). Both the erbium and titanium
285
8.7 Experimental Observation of RMT Predictions
In Chap. 7, we have already shown some experimental data for the spectral statistics
of chaotic cavities constructed from microwave waveguides. These are, in fact,
open systems that are driven into resonance by external energy sources. However,
the antennas used to couple the interior of the microwave cavity to the external
radiation source are designed to have minimum effect on the energy distribution
inside the cavity. Thus, the cavity resonances lie very close to the energy eigenvalues
of the closed cavity. These microwave experiments are more recent than the nuclear
scattering experiments described Sect. 8.2. As we saw in Fig. 8.2, as far back as
1964, scattering of slow neutrons off of heavy nuclei, such as 238 U, showed similar
behavior. It is hoped that the energies at which the scattering resonance takes place
are in direct correspondence with the eigenvalue spectrum of the internal states of
nuclei or molecules being studied. To keep the distinction clear, in this section we
will refer to resonance peaks as “levels.”
In the subsections below, we discuss in more detail the comparison between
RMT predictions, the results of nuclear scattering experiments, and the results of
molecular scattering experiments.
8.7.1 Experimental Nuclear Spectral Statistics
In order to study the spectral statistics of a level sequence, it is important that all
neighboring levels be included in the segment of the level sequence being studied.
One of the main difficulties in obtaining good sequences experimentally is that
of resolving closely spaced levels. Fortunately, resonances found in the scattering
of slow neutrons on heavy nuclei provide good sequences. “The resonances are
narrowed because of the strong surface reflection of long wavelength neutrons and
as a consequence one can often observe up to a few hundred resonances, essentially
all (if the target state is even-even) with the same exact quantum numbers (angular
momentum, J , parity, π ; isospin is also good but is usually irrelevant)” (Brody et al.
1981).
In Fig. 8.14, we show three experimentally obtained sequences of 50 levels taken
from the spectra of three different nuclei, and we show a level sequence of 50 levels
with Poisson random spacing. The levels have been rescaled to lie in the same
spectral span and have the same average spacing, D (Brody et al. 1981). The arrows
in Fig. 8.14 indicate level spacings smaller than one-quarter of the average spacing.
Such spacings cannot be seen on the scale shown. It is interesting that the pure
sequences have far fewer close spacings than do the Poisson or mixed sequences.
In Fig. 8.15, we show the nearest neighbor spacing distributions for these four
cases but using all available data for each nucleus. Figures 8.14a and 8.15a show
data from the s-wave scattering of slow neutrons on the erbium isotope 166 Er Liou
et al. (1972). Figures 8.14b and 8.15b show experimental data for proton scattering
on the titanium isotope 48 Ti (Prochnow et al. 1972). Both the erbium and titanium