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arXiv:nucl-th/0305092 · Physical Review C

Fluctuation properties of strength functions associated with giant resonances

Hirokazu Aiba, Masayuki Matsuo, Shigeru Nishizaki, T. Suzuki

Published Nov 24, 2003• 13 citations
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Abstract

We performed fluctuation analysis by means of the local scaling dimension for the strength function of the isoscalar (IS) and the isovector (IV) giant quadrupole resonances in $^{40}\mathrm{Ca}$, where the strength functions are obtained by the shell model calculation within up to the $2p2h$ configurations. It is found that at small energy scale, fluctuation of the strength function almost obeys the Gaussian orthogonal ensemble (GOE) random matrix theory limit. On the other hand, we found a deviation from the GOE limit at the intermediate energy scale of about $1.7\phantom{\rule{0.3em}{0ex}}\text{MeV}$ for the IS and at $0.9\phantom{\rule{0.3em}{0ex}}\text{MeV}$ for the IV. The results imply that different types of fluctuations coexist at different energy scales. Detailed analysis strongly suggests that GOE fluctuation at small energy scale is due to the complicated nature of $2p2h$ states and that fluctuation at the intermediate energy scale is associated with the spreading width of the Tamm-Dancoff $1p1h$ states.

DOI
10.1103/physrevc.68.054316

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We performed fluctuation analysis by means of the local scaling dimension for the strength function of the isoscalar (IS) and the isovector (IV) giant quadrupole resonances in 40Ca^{40}\mathrm{Ca}40Ca, where the strength functions are obtained by the shell model calculation within up to the 2p2h2p2h2p2h configurations. It is found that at small energy scale, fluctuation of the strength function almost obeys the Gaussian orthogonal ensemble (GOE) random matrix theory limit. On the other hand, we found a deviation from the GOE limit at the intermediate energy scale of about 1.7MeV1.7\phantom{\rule{0.3em}{0ex}}\text{MeV}1.7MeV for the IS and at 0.9MeV0.9\phantom{\rule{0.3em}{0ex}}\text{MeV}0.9MeV for the IV. The results imply that different types of fluctuations coexist at different energy scales. Detailed analysis strongly suggests that GOE fluctuation at small energy scale is due to the complicated nature of 2p2h2p2h2p2h states and that fluctuation at the intermediate energy scale is associated with the spreading width of the Tamm-Dancoff 1p1h1p1h1p1h states.
We performed fluctuation analysis by means of the local scaling dimension for the strength function of the isoscalar (IS) giant quadrupole resonance (GQR) in 208Pb^{208}\mathrm{Pb}208Pb where the strength function is obtained by the shell model calculation including 1p1h and 2p2h configurations. It is found that at almost all energy scales, fluctuation of the strength function obeys the Gaussian orthogonal ensemble (GOE) random matrix theory limit. This is contrasted with the results for the GQR in 40Ca^{40}\mathrm{Ca}40Ca, where at the intermediate energy scale of about 1.7 MeV, a deviation from the GOE limit was detected. It is found that the physical origin for this different behavior of the local scaling dimension is ascribed to the difference in the properties of the damping process.
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.
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.
The exact solution of the many-body problem in the framework of the nuclear shell model with a realistic residual Hamiltonian makes it possible to study the fragmentation of simple configurations as a function of excitation energy and interaction strength. The analysis is performed for 839 states with quantum numbers ${\mathit{J}}^{\mathrm{\ensuremath{\pi}}}$T=${0}^{+}$0 in a system of 12 valence particles within the sd shell. Our statistics allow us to establish the generic shape of the strength function in the region of strong mixing. For the realistic interaction, the strength function is close to Gaussian in the central part and has exponential wings. The spreading width is larger than predicted by the standard golden rule. At the artificially suppressed interaction strength, we recover the Breit-Wigner shape and the golden rule for the spreading width. The transition between these regimes agrees with theoretical considerations based on the idea of chaotic dynamics. \textcopyright{} 1996 The American Physical Society.
Employing hamiltonians defined by two-body embedded Gaussian orthogonal ensemble of random matrices(EGOE(2)) plus a mean-field producing one-body part, strength functions (for states defined by the one-body part) are constructed for various values of the strength of the chaos generating two-body part. Numerical calculations for six and seven fermion systems clearly demonstrate Breit-Wigner to Gaussian transition, in the chaotic domain, in strength functions as found earlier in nuclear shell model and Lipkin-Meshkov-Glick model calculations.
Some recent advancements of the nuclear many-body theory and selected results on nuclear giant and pygmy resonances are presented. The theory is compactly reviewed, with a special focus on the emergent scale of the quasiparticle-vibration coupling (qPVC), which carries the order parameter associated with the qPVC vertex, and an efficient treatment of the nuclear many-body problem organized around the qPVC hierarchy. Self-consistent numerical solutions of the relativistic Bethe-Salpeter-Dyson equation for the nuclear response function in medium-heavy nuclei are discussed. The presented update on the pygmy dipole resonance focuses on establishing the formation of its two-component structure as a result of the fragmentation of the low-energy dipole mode due to the qPVC and its mixing with the similarly fragmented giant dipole resonance. The centroid of the isoscalar giant monopole resonance is also linked to qPVC effects, particularly to its sensitivity to the coupling of the collective breathing mode to the lowest quadrupole vibrations, which is enhanced by quadrupole collectivity. The resolution of the long-standing “fluffiness” puzzle regarding the compressibility of open-shell tin isotopes is summarized. The recently developed thermal variant of the superfluid response theory is briefly introduced and continuously linked to the description of the isoscalar monopole response at finite temperature with the prospect of refining the temperature-dependent nuclear equation of state.