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Iraqi Journal of Physics

Statistical Fluctuations of Energy Spectrum, Electromagnetic Transitions and Electromagnetic Moments in 136Xe Nucleus Using the Framework of Nuclear Shell Model

Adel Khalaf Hamoudi

Published Oct 1, 2010• 0 citations
OpenAlex

Abstract

The fluctuation properties of energy spectrum, electromagnetic transition intensities and electromagnetic moments in nucleus are investigated with realistic shell model calculations. We find that the spectral fluctuations of are consistent with the Gaussian orthogonal ensemble of random matrices. Besides, we observe a transition from an order to chaos when the excitation energy is increased and a clear quantum signature of the breaking of chaoticity when the single-particle energies are increased. The distributions of the transition intensities and of the electromagnetic moments are well described by a Porter-Thomas distribution. The statistics of electromagnetic transition intensities clearly deviate from a Porter-Thomas distribution (i.e., a transition towards regularity is observed) when the single-particle energies are increased whereas the statistics of electromagnetic moments are not affected by the change of the single-particle energies.

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The fluctuation properties of energy spectrum, electromagnetic transition intensities and electromagnetic moments in nucleus are investigated with realistic shell model calculations. We find that the spectral fluctuations of are consistent with the Gaussian orthogonal ensemble of random matrices. Besides, we observe a transition from an order to chaos when the excitation energy is increased and a clear quantum signature of the breaking of chaoticity when the single-particle energies are increased. The distributions of the transition intensities and of the electromagnetic moments are well described by a Porter-Thomas distribution. The statistics of electromagnetic transition intensities clearly deviate from a Porter-Thomas distribution (i.e., a transition towards regularity is observed) when the single-particle energies are increased whereas the statistics of electromagnetic moments are not affected by the change of the single-particle energies.
The energies and wave functions of stationary many-body states are analyzed to look for the signatures of quantum chaos. Shell model calculations with the Wildenthal interaction are performed in the J−TJ-TJ−T scheme for 12 particles in the sdsdsd-shell. The local level statistics are in perfect agreement with the GOE predictions. The analysis of the amplitudes of the eigenvectors in the shell model basis with the aid of the informational entropy and moments of the distribution function shows evidence for local chaos with a localization length reaching 90\% of the total dimension in the middle of the spectrum. The degree of chaoticity is sensitive to the the strength of the residual interaction as compared to the single particle energy spacing.
Shell-model results for the electric quadrupole (E2),(E2),(E2), magnetic dipole (M1),(M1),(M1), and Gamow-Teller transition strength sums and occupation numbers are calculated using different valence spaces and compared to the embedded Gaussian orthogonal ensemble (EGOE) predictions. It is shown that transition strength sums can be considered as a new statistic able to distinguish between regular and chaotic motion. Moreover it is established that the EGOE (but not the GOE) provides the good description of the shell-model strength sums in the chaotic domain. To get this result we have studied the behavior of strength sums in order to chaos transitions generated by means of a family of Hamiltonians $H(\ensuremath{\lambda})=h(1)+\ensuremath{\lambda}V(2),$ built from realistic one- and two-body interactions.
Abstract The spectral fluctuations and transition intensity fluctuations in the excited-state quantum phase transitions (ESQPTs) have been investigated within the framework of the interacting boson model (IBM) by adopting three statistical measures, including the nearest neighbor level spacing distribution P ( S ) measuring the chaoticity (regularity) in energy spectra, the Δ 3 ( L ) statistics of Dyson and Mehta measuring the spectral rigidity and the intensity distribution P ( y ) measuring the chaoticity (regularity) in B ( E 0) transitions. The results indicate that that the ESQPT as a function of the excitation energy may occur as a transition from regular (or semiregular) to highly chaotic if only the associated whole spectrum is chaotic, which fits most of the deformed situations in the IBM including those in the U(5)–SU(3) and SU(3)–O(6) transitional regions. Otherwise, the ESQPT will appear as a transition from regular (or semiregular) to regular such as the cases in the U(5)–O(6) transitional region or those on the ‘Alhassid–Whelan arc’, which represents a nearly regular parameter region connecting the U(5) and SU(3) limits in the IBM.
The present-day shell-model computer programmes of nuclear physics can handle very large model spaces, and often in practice the upper bounds on dimensionalities are set by the capacity of the computer. However, still larger configuration spaces can be treated successfully by the methods of statistical spectroscopy to obtain average properties of many-particle systems. These methods are based on the use of distribution functions with parameters that can be evaluated without the explicit calculation and diagonalization of a high-dimensional hamiltonian matrix. The basic philosophy behind this method is that the average properties depend on the particular model at hand, whereas the fluctuations about the average are universal. The latter remark means that the fluctuations can be described by the gaussian orthogonal ensemble. Of course, the results are less detailed than those one would obtain, if possible, from the explicit diagonalization of the energy matrix and the use of the resulting many-dimensional
Recent high-resolution nuclear resonance fluorescence experiments performed on the even-even Chromium isotopes 50,52,54Cr^{50,52,54}\mathrm{Cr}50,52,54Cr have lead to the identification (energy, spin, parity, and transition strength) of altogether 108 nuclear levels of spin J=1J=1J=1 (70 levels with ${J}^{\ensuremath{\pi}}={1}^{\ensuremath{-}}$ and 38 with ${J}^{\ensuremath{\pi}}={1}^{+}$) at excitation energies Ex{E}_{x}Ex​ ranging roughly from 4.5 to 9.7 MeV. In this region just above the orbital magnetic-dipole scissors mode, sizable spin-flip magnetic-dipole strength as well as electric-dipole strength belonging to the pygmy dipole resonance (PDR) is expected. Using statistical measures for short- and long-range correlations, we perform an analysis of the fluctuation properties in the subspectra of the energy levels and also of the distributions of their respective dipole transition strengths. We compare the results with those of a random matrix ensemble interpolating between Poisson statistics generally describing the fluctuation properties in the energy spectra of many-body systems with collective, i.e., regular motion of the particles and the Gaussian orthogonal ensemble (GOE) for complex (i.e., chaotic) behavior. This comparison reveals that the spectral properties of the 1+{1}^{+}1+ states are close to the GOE results while those of the ${1}^{\ensuremath{-}}$ states are closer to Poisson. This is confirmed by an analysis of the spectral fluctuations based on the method of Bayesian inference and corroborated by large-scale shell-model and quasiparticle-phonon model calculations, respectively. The nearly Poissonian behavior of the ${1}^{\ensuremath{-}}$ levels suggests a sizable collectivity of the PDR indeed.