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V. Zelevinsky and S. Karampagia
spectroscopic output in the form of level energies, expectation values of observables,
and transition probabilities. Currently such work that took many years of adjustment
can be considered reliable for the nuclei of sd and pf shells and a little beyond.
The possibility of the exact diagonalization is obviously limited by the size of the
huge Hamiltonian matrix, in spite of the fast computational progress. Therefore the
Hilbert space has to be truncated with the appropriate renormalization of the matrix
elements. This truncation limits the trustworthy predictions because, starting from
some excitation energy, the truncated parts of the Hilbert space enter the game with
the new particle configurations and new levels. Finally, the standard versions of
the shell model are based on the harmonic oscillator scheme and do not feel the
continuum thresholds and finite life time of the levels beyond that. (This, however,
is a common feature of current approaches to the level density problem).
With all deficiencies of the shell model, it still gives the most reliable description
of nuclear spectroscopy. In practice, even being limited by the space truncation
and increasing widths of the continuum states, the shell-model predictions of the
level density, at least up to A ≈ 60 − 70, agree with the available information
below excitation energy 12–15 MeV, and probably even beyond. It turns out to
be an important advantage of the shell model that it accounts for all (allowed by
conservation laws) interaction matrix elements. In many theoretical approaches,
only the mean field and collective interactions are accounted for. The shell model
adds here all possible incoherent collision-like processes which turn out to influence
the level density in an important way making it a smooth function of excitation
energy. There is an obvious objection based on the deficit of information from
low-lying spectroscopy about such processes that makes these matrix elements ill
defined. However, one can argue that their exact values are of less importance; there
are many of them and their action is statistically averaged.
Here we come to the role of what is called quantum chaos and thermalization in a
small mesoscopic system of interacting constituents. The equilibration here comes
without an external heat bath, just due to the interaction that becomes effectively
strong along with the growing level density, simply because of combinatorics.
Starting from some excitation energy above the pairing gap, the neighboring wave
functions within the same symmetry class become more and more mixed and similar
by their main properties as was understood long ago [10]. The excitation energy
is distributed over the growing number of degrees of freedom as in the classical
compound nucleus picture. Statistical characteristics of stationary states, such as the
level spacing distribution, informational entropy, number of principal components,
correlational measures, etc., are smoothly changing along the spectrum as functions
of energy, similarly to thermodynamic equilibrium [11–13]. Those properties are
analogous to the predictions of the extreme limit of the Gaussian Orthogonal
Ensemble (GOE) in spite of the fact that two-body interactions are not at all random
(any two-body matrix element is repeated many times in the Hamiltonian matrix for
different background of spectators). In this way, the dense set of mixed stationary
states creates chaotic properties which in turn self-consistently lead to the generic
behavior of the level density.
V. Zelevinsky and S. Karampagia
spectroscopic output in the form of level energies, expectation values of observables,
and transition probabilities. Currently such work that took many years of adjustment
can be considered reliable for the nuclei of sd and pf shells and a little beyond.
The possibility of the exact diagonalization is obviously limited by the size of the
huge Hamiltonian matrix, in spite of the fast computational progress. Therefore the
Hilbert space has to be truncated with the appropriate renormalization of the matrix
elements. This truncation limits the trustworthy predictions because, starting from
some excitation energy, the truncated parts of the Hilbert space enter the game with
the new particle configurations and new levels. Finally, the standard versions of
the shell model are based on the harmonic oscillator scheme and do not feel the
continuum thresholds and finite life time of the levels beyond that. (This, however,
is a common feature of current approaches to the level density problem).
With all deficiencies of the shell model, it still gives the most reliable description
of nuclear spectroscopy. In practice, even being limited by the space truncation
and increasing widths of the continuum states, the shell-model predictions of the
level density, at least up to A ≈ 60 − 70, agree with the available information
below excitation energy 12–15 MeV, and probably even beyond. It turns out to
be an important advantage of the shell model that it accounts for all (allowed by
conservation laws) interaction matrix elements. In many theoretical approaches,
only the mean field and collective interactions are accounted for. The shell model
adds here all possible incoherent collision-like processes which turn out to influence
the level density in an important way making it a smooth function of excitation
energy. There is an obvious objection based on the deficit of information from
low-lying spectroscopy about such processes that makes these matrix elements ill
defined. However, one can argue that their exact values are of less importance; there
are many of them and their action is statistically averaged.
Here we come to the role of what is called quantum chaos and thermalization in a
small mesoscopic system of interacting constituents. The equilibration here comes
without an external heat bath, just due to the interaction that becomes effectively
strong along with the growing level density, simply because of combinatorics.
Starting from some excitation energy above the pairing gap, the neighboring wave
functions within the same symmetry class become more and more mixed and similar
by their main properties as was understood long ago [10]. The excitation energy
is distributed over the growing number of degrees of freedom as in the classical
compound nucleus picture. Statistical characteristics of stationary states, such as the
level spacing distribution, informational entropy, number of principal components,
correlational measures, etc., are smoothly changing along the spectrum as functions
of energy, similarly to thermodynamic equilibrium [11–13]. Those properties are
analogous to the predictions of the extreme limit of the Gaussian Orthogonal
Ensemble (GOE) in spite of the fact that two-body interactions are not at all random
(any two-body matrix element is repeated many times in the Hamiltonian matrix for
different background of spectators). In this way, the dense set of mixed stationary
states creates chaotic properties which in turn self-consistently lead to the generic
behavior of the level density.