NONLINEAR COLLECTIVE DYNAMICS IN ATOMIC NUCLEI

           The nonlinear dynamics of giant monopole resonances is investigated in the time-dependent relativistic mean-field model. The time-series analysis of dynamical variables that characterize nucleon distributions indicate regular motion for the isoscalar mode, and more complex dynamics for the isovector oscillations. Information entropy functionals disclose the underlying nonlinear collective dynamics in quantum systems that have spatial as well as temporal structure. In the relativistic mean field theory the starting point is the following form of the Lagrangian density:

$${\cal L} = \bar\psi\left(i\gamma\cdot\partial-m\right)\psi ~+~\frac{1}{2}(\partial\sigma)^2-U(\sigma )$$

$$-~\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu} +\frac{1}{2}m^2_\omeg... ...frac{1}{2}m^2_\rho\vec\rho^{\,2} ~-~\frac{1}{4}{\rm F}_{\mu\nu}{\rm F}^{\mu\nu}$$

$$-~g_\sigma\bar\psi\sigma\psi~-~ g_\omega\bar\psi\gamma\cdot\omega... ...a\cdot\vec\rho\vec\tau\psi~-~ e\bar\psi\gamma\cdot A \frac{(1-\tau_3)}{2}\psi\;$$ (1)

Collective motion of protons and neutrons in atomic nuclei is studied in the framework of the time-dependent relativistic mean-field theory. The nucleus is described as a system of Dirac nucleons that interact through the exchange of virtual mesons and photons. Nuclear dynamics is described by the simultaneous evolution of single particle Dirac spinors in the time-dependent mean fields. For an initial monopole deformation of the nucleus ²⁰⁸Pb, time-dependent selfconsistent calculations are performed in order to describe the collective dynamics of the giant monopole resonances. The frequencies of eigenmodes are found to be in agreement with experimental data on monopole resonances in ²⁰⁸Pb.

The model has been used to describe the isoscalar and isovector mode of monopole vibrations in nuclei, and the difference in their dynamics has been investigated. The self-consistent system is nonlinear, and therefore chaotic regime of motion is to be expected for certain sets of parameters. Collective dynamics in the nuclei has been analysed with the fast Fourier transforms and the autocorrelation function. The response of system is described in the reconstructed phase space.

The time delays are determined from the autocorrelation function, and the average mutual information, while the embedding dimension is determined by the false nearest neighbours method. The reconstructed phase-spaces have been represented by recurrence plots, that shows for isoscalar mode a pattern characteristic for regular oscillations, while for the isovector mode it indicates non-stationarity.

Correlation integrals and dimensions have been determined from the reconstructed phase-spaces of the monopole moments. As a function of the embedding dimension, the correlation dimension saturates at the integer value 2 for the isoscalar mode, while for isovector mode, it has fractional value that slowly increase above 2.0. The results show that the collective coordinate of the isoscalar mode is regular, while for the isovector mode, oscillations have properties of the low dimensional deterministic chaos.

For better understanding of nuclear collective dynamics, the time series prediction with the artificial neural networks with backpropagation has been applied. Dynamic predictions of isoscalar oscillations is succesfull for longer period of time, while for the isovector mode, modelling becomes difficult. Only short-time predictions are possible for the oscillations in the isovector regime of deterministic chaos. The system is higly sensitive to initial conditions, within the isovector initial monopole deformation, while for the isoscalar deformation, the response of the sysem is independent on the initial conditions.

Collective dynamics in atomic nuclei is studied in the framework of the information theory. The averaged information entropy from one body time dependent nucleon densities was calculated. The Fourier analysis has shown that the entropy of the isoscalar mode contains the same information as the dynamical variable, while for the isovector mode the peaks are found both in the regions of isoscalar and isovector eigenfrequencies. Information entropy defined from a two-body nucleon density enables the study of the influence of spatial motion on temporal chaos. It shows a lower degree of spatial correlation in earlier stage of evolution in the isovector mode of oscillations. Mutual information between the collective dynamical variable of the proton and neutron density is more than a factor three larger for the isoscalar mode compared with isovector mode. Mutual information between time-dependent proton and neutron densities showed interesting radial dependence. It showed the difference in the dynamics of the monopole motion in the volume and on the surface of the nucleus.

Atomic nuclei in the time dependent relativistic mean field model have shown interesting regular and chaotic dynamics in quantum systems. But only the most simple modes of collective motion, monopole oscillations, have been investigated. More complicated excitations, like those involving spin and isospin degrees of freedom would certainly disclose more interesting properties of the underlying nonlinear dynamics.

References:

1. D. Vretenar, P. Ring, G. A. Lalazissis, and N. Paar, "Relativistic mean-field description of the dynamics of giant resonances", Nucl. Phys.A649, 29-36 (1999).

2. D. Vretenar, N. Paar, P. Ring, and G. A. Lalazissis, "Nonlinear dynamics of giant resonances in atomic nuclei", Phys. Rev. E 60, 308-319 (1999).

3. N. Paar, "Nonlinear dynamics of collective vibrations in nuclei in the time-dependent relativistic mean-field model (in Croatian) ", Diploma work, University of Zagreb, 1998.

4. G. A. Lalazissis, D. Vretenar, N. Paar, and P. Ring "Relativistic description of regular and chaotic dynamics in the giant monopole resonances ", Chaos, Solitons & Fractals. 17, 585-590 (2003).

Nils Paar, 2000.