8.2 Random Matrix Theory and Nuclear Scattering Processes
241
to the COE? It turns out that under special circumstances this connection can be
made, and we will show that in Sect. 8.6.
In Sect. 8.7, we consider the nearest neighbor spectral spacing statistics and
the 3 -statistic obtained from several nuclear level sequences and molecular level
sequences obtained from scattering experiments. We find definite evidence of level
repulsion, and the manifestations of chaos in nuclei and molecules. Finally, in
Sect. 8.8, we make some concluding remarks.
8.2 Random Matrix Theory and Nuclear Scattering
Processes
Wigner refers to the following excerpt by Dyson to summarize the motivation
behind the use of statistical methods to analyze nuclear scattering processes (Wigner
1967; Dyson 1962a): “Recent theoretical analyses have had impressive success in
interpreting the detailed structure of the low-lying excited states of complex nuclei.
Still, there must come a point beyond which such analyses of individual levels cannot
usefully go. For example, observations of levels of heavy nuclei in the neutroncapture region give precise information concerning a stretch of levels from number
N to number (N + n), where N is an integer of the order of 10 6 . It is improbable
that level assignments based on shell structure and collective or individual-particle
quantum numbers can ever be pushed as far as the millionth level. It is therefore
reasonable to inquire whether the highly excited states may be understood from
the diametrically opposite point of view, assuming as a working hypothesis that all
shell structure is washed out and that no quantum numbers other than spin and
parity remain good. The result of such an inquiry will be a statistical theory of
energy levels. The statistical theory will not predict the detailed sequence of levels
in any one nucleus, but it will describe the general appearance and the degree of
irregularity of the level structure that is expected to occur in any nucleus which
is too complicated to be understood in detail.” This view led Wigner to surmise
(Wigner 1959) a possible eigenvalue nearest neighbor spacing distribution based
on the assumption that matrix elements of the Hamiltonian matrix were unknown
and unknowable, and led to the development of the random matrix theory (RMT)
described in Chap. 6.
Support for the use of RMT to interpret scattering data was obtained in
experiments analyzed by Garg et al. (1964). They computed the nearest neighbor
spacing distribution for absorption peaks that were obtained from the scattering
of neutrons off of heavy nuclei. In Fig. 8.1, we show a segment of this nuclear
scattering data for the scattering of slow neutrons off of the 238 U nucleus. In Fig. 8.2
we show the histogram of nearest neighbor spacings obtained from these data. It is
a fairly good fit to the Wigner surmise. One should note, however, that slow neutron
scattering data for other nuclei were also examined in (Garg et al. 1964) and those
data do not always give even a qualitative fit to the Wigner surmise.
241
to the COE? It turns out that under special circumstances this connection can be
made, and we will show that in Sect. 8.6.
In Sect. 8.7, we consider the nearest neighbor spectral spacing statistics and
the 3 -statistic obtained from several nuclear level sequences and molecular level
sequences obtained from scattering experiments. We find definite evidence of level
repulsion, and the manifestations of chaos in nuclei and molecules. Finally, in
Sect. 8.8, we make some concluding remarks.
8.2 Random Matrix Theory and Nuclear Scattering
Processes
Wigner refers to the following excerpt by Dyson to summarize the motivation
behind the use of statistical methods to analyze nuclear scattering processes (Wigner
1967; Dyson 1962a): “Recent theoretical analyses have had impressive success in
interpreting the detailed structure of the low-lying excited states of complex nuclei.
Still, there must come a point beyond which such analyses of individual levels cannot
usefully go. For example, observations of levels of heavy nuclei in the neutroncapture region give precise information concerning a stretch of levels from number
N to number (N + n), where N is an integer of the order of 10 6 . It is improbable
that level assignments based on shell structure and collective or individual-particle
quantum numbers can ever be pushed as far as the millionth level. It is therefore
reasonable to inquire whether the highly excited states may be understood from
the diametrically opposite point of view, assuming as a working hypothesis that all
shell structure is washed out and that no quantum numbers other than spin and
parity remain good. The result of such an inquiry will be a statistical theory of
energy levels. The statistical theory will not predict the detailed sequence of levels
in any one nucleus, but it will describe the general appearance and the degree of
irregularity of the level structure that is expected to occur in any nucleus which
is too complicated to be understood in detail.” This view led Wigner to surmise
(Wigner 1959) a possible eigenvalue nearest neighbor spacing distribution based
on the assumption that matrix elements of the Hamiltonian matrix were unknown
and unknowable, and led to the development of the random matrix theory (RMT)
described in Chap. 6.
Support for the use of RMT to interpret scattering data was obtained in
experiments analyzed by Garg et al. (1964). They computed the nearest neighbor
spacing distribution for absorption peaks that were obtained from the scattering
of neutrons off of heavy nuclei. In Fig. 8.1, we show a segment of this nuclear
scattering data for the scattering of slow neutrons off of the 238 U nucleus. In Fig. 8.2
we show the histogram of nearest neighbor spacings obtained from these data. It is
a fairly good fit to the Wigner surmise. One should note, however, that slow neutron
scattering data for other nuclei were also examined in (Garg et al. 1964) and those
data do not always give even a qualitative fit to the Wigner surmise.