Spectral Fluctuations in A=32 Nuclei Using the Framework of the Nuclear Shell Model

Chaotic properties of nuclear energy spectra in A=32 nuclei are investigated via the framework of the nuclear shell model. The energies (the main object of this investigation) are calculated through accomplishing shell model calculations employing the OXBASH computer code with the realistic effective interaction of W in the isospin formalism. The A=32 nuclei are supposed to have an inert 16 O core with 16 nucleons move in the 1d5/2, 2s1/2 and 1d3/2 orbitals. For full hamiltonian calculations, the spectral fluctuations (i.e., the nearest neighbor level spacing distributions ( ) P s and the ∆ statistics) are well characterized by the Gaussian orthogonal ensemble of random matrices. Besides, they show no dependency on the spin J and isospin . T For unperturbed hamiltonian calculations, we find a regular behavior for the distribution of ( ) P s and an intermediate behavior between the GOE and the Poisson limits for the ∆ statistics.


Introduction
Chaos in quantum system was studied extremely throughout the last three decades [1]. Bohigas et al. [2] proposed a relation between chaos in a classical system and the spectral fluctuations of the analogous quantum system, where an analytical proof of the Bohigas et al. conjecture is found in [3]. It is now typically known that quantum analogs of most classically chaotic systems demonstrate spectral fluctuations that agree with the random matrix theory (RMT) [4,5] while quantum analogs of classically regular systems reveal spectral fluctuations that agree with a Poisson distribution. For time-reversal-invariant systems, the suitable form of RMT is the Gaussian orthogonal ensemble (GOE). RMT was firstly utilized to characterize the statistical fluctuations of neutron resonances in compound nuclei [6]. RMT has become a standard tool for analyzing the universal statistical fluctuations in chaotic systems [7][8][9][10].
The chaotic behavior of the single particle dynamics in the nucleus can be analyzed via the mean field approximation. Nevertheless, the nuclear residual interaction mixes different mean field configurations and affects the statistical fluctuations of the many particle spectrum and wave functions. These fluctuations may be investigated with different nuclear structure models. The statistics of the lowlying collective part of the nuclear spectrum were studied in the framework of the interacting boson model [11,12], in which the nuclear fermionic space is mapped onto a much smaller space of bosonic degrees of freedom. Because of the relatively small number of degrees of freedom in this model, it was also possible to relate the statistics to the underlying mean field collective dynamic. At higher excitations, additional degrees of freedom (such as broken pair) become important [13], and the effects of interactions on the statistics must be studied in larger model spaces. The nuclear shell model offers an attractive framework for such studies. In this model, realistic effective interactions are available and the basis states are labeled by exact quantum numbers of angular momentum ( J ), isospin ( T ) and parity ( π ) [14].
The distribution of eigenvector components [15][16][17][18][19] was examined by the framework of the shell model. Brown and Bertsch [17] found that the basis vector amplitudes are consistent with Gaussian distribution (which is the GOE prediction) in regions of high level density but deviated from Gaussian behavior in other regions unless the calculation employs degenerate single particle energies. Later studies [19] also suggested that calculations with degenerate single particle energies are chaotic at lower energies than more realistic calculations.
The electromagnetic transition intensities in a nucleus are observables that are sensitive to the wave functions, and the study of their statistical distributions should complement [11,12] the more common spectral analysis and serve as another signature of chaos in quantum systems. Hamoudi et al carried out [20] the fp-shell model calculations to study the statistical fluctuations of energy spectrum and electromagnetic transition intensities in A=60 nuclei using the F5P [21] interaction. The calculated results were in agreement with RMT and with the previous finding of a Gaussian distribution for the eigenvector components [15][16][17][18][19]. Hamoudi studied [22] the effect of the one-body hamiltonian on the fluctuation properties of energy spectrum and electromagnetic transition intensities in 136 Xe using a realistic effective interaction for the N82-model space defined by 5

β < <
In the present study, the spectral fluctuations in 32 A nuclei are analyzed by two statistical measures: the nearest neighbor level spacing distribution ( ) P s and the Dyson-Mehta statistics ( ∆ statistics). For calculations with the full diagonalization of the hamiltonian, the spectral fluctuations are found to be consistent with the Gaussian orthogonal ensemble of random matrices. In addition, they are independent of the spin J and isospin .
T For calculations with the unperturbed hamiltonian, we find a regular behavior for the ( ) P s distribution and an intermediate behavior between the GOE and the Poisson limits for the ∆ statistics.

Theory
The many-body system can be described by an effective shell-model hamiltonian [14] 0 , where 0 H and H ′ are the independent particle (one body) part and the residual two-body interaction of .
characterizes non-interacting fermions in the mean field of the appropriate spherical core. The single-particle orbitals total angular momentum ( j ), projection z j m = and isospin projection .
τ The antisymmetrized two-body interaction H ′ of the valence particles is written as The many-body wave functions with good spin J and isospin T quantum numbers are constructed via the m − scheme determinants which have, for given J and , T the maximum spin and isospin projection [14], 3 , ; , where m span the m − scheme subspace of states with M J = and 3 .

T T =
The matrix of the many-body hamiltonian ; ; is eventually diagonalized to obtain the eigenvalues E α and the eigenvectors ; ; Here, the eigenvalues E α are considered as the main object of the present investigation.
The fluctuation properties of nuclear energy spectrum are obtained via two statistical measures: the nearest-neighbors level spacing distribution ( ) P s and the Dyson-Mehta or 3 ∆ statistics [4,24]. The staircase function of the nuclear shell model spectrum ( ) N E is firstly build. Here, ( ) N E is defined as the number of levels with excitation energies less than or equal to . E In this study, a smooth fit to the staircase function is performed with polynomial fit. The unfolded spectrum is then defined by the mapping [12] ( ) The real spacings reveal strong fluctuations whereas the unfolded levels i E ɶ have a constant average spacing. The level spacing distribution (which exemplifies the fluctuations of the short-range correlations between energy levels) is defined as the probability of two neighboring levels to be a distance s apart. The spacings i s are determined from the unfolded levels by 1 .
It measures the deviation of the staircase function (of the unfolded spectrum) from a straight line. A rigid spectrum corresponds to smaller values of 3 ∆ whereas a soft spectrum has a larger 3 . ∆ To get a smoother function 3 ( ), L ∆ we The successive intervals are taken to overlap by / 2. L In the Poisson limit, 3

Results and Discussion
Shell model calculations are accomplished, using the OXBASH code [25], for A=32 nuclei with 0, 1 T = and 2. These nuclei are supposed to have an inert core of 16 O with 16 active nucleons (8 protons and 8 neutrons) move in the sdshell (1d 5/2 , 2s 1/2 and 1d 3/2 orbitals) model space. The W interaction [26] is selected as a realistic effective interaction in the isopspin formalism together with realistic spe's. Manybody basis states k were constructed with good total angular momentum J (its projection ), M parity π and isospin T (its projection 3 ).
T Table 1 displays the dimensions of all considered J T π states for 16 particles move in the sd-shell model space. The fluctuation properties of energy spectra in A=32 nuclei are analyzed by two statistical measures: the nearest neighbor level spacing distribution ( ) P s and the Dyson-Mehta statistics (∆ statistics).  = − levels agree well with the GOE distribution. The level repulsion at small spacings produced through the mixing by the off-diagonal hamiltonian (which is considered as a distinctive feature of chaotic level statistics) is clearly seen in the calculated histograms. In spite of the level repulsion at small spacings in histograms 8 0 J T π + = and 9 0 + levels is slightly decreased, the general performance of these histograms is still very close to the GOE limit. It is obvious from Figure 1 that the ( ) P s (histograms) distribution is independent of the spin J (universal for different spins).

Conclusions
The spectral fluctuations in A=32 nuclei are studied via the nuclear shell model. The sd-shell model calculations are accomplished by the OXBASH computer code with the isospin formalism interaction of W. The spectral fluctuations obtained with full hamiltonian calculations are found to be consistent with the GOE of random matrices (which characterizes the chaotic systems). Moreover, the distributions of ) (s P and ∆ statistics are found to be independent of the spin J and isospin . T For unperturbed hamiltonian calculations, we find a regular behavior for the distribution of ) (s P and an intermediate behavior between the GOE and the Poisson limits (closer to the Poisson limit) for the ∆ statistics. This regularity is attributed to the absence of the mixing and repulsion between levels as a result of the nonexistence of the off-diagonal residual interaction.