Statistical Fluctuations of Energy Spectra in the Isobar A = 68 Nuclei

Statistical fluctuations of nuclear energy spectra for the isobar A = 68 were examined by means of the random matrix theory together with the nuclear shell model. The isobar A = 68 nuclei are suggested to consist of an inert core of 56 Ni with 12 nucleons in f5p-space (2p3/2, 1f5/2 and 2p1/2 orbitals). The nuclear excitation energies, required by this work, were obtained through performing f5p-shell model calculations using the isospin formalism f5pvh interaction with realistic single particle energies. All calculations of the present study were conducted using the OXBASH code. The calculated level densities were found to have a Gaussian shape. The distributions of level spacing P(s) and statistic for the considered classes of states, obtained with full Hamiltonian of f5pvh (absence of the off-diagonal Hamiltonian) calculations, showed a chaotic (regular) behavior and coincided well with the distribution of Gaussian orthogonal ensemble (Poisson). Moreover, these distributions were independent of spin ( J ) and isospin ( ). T


Introduction
Quantum chaos was explored vastly in the past 30 years [1]. Bohigas et al. [2] assumed a relationship amongst disorder in a classical system and the statistical fluctuations of nuclear spectrum of identical quantum system, whereas a systematic evidence of Bohigas assumption is presented in another report [3]. At the moment, it is eminent that quantum analogs of utmost classically disordered systems depict fluctuations in energies that come to an agreement with Random Matrix Theory (RMT) [4,5], but quantum analogs of classically ordered systems depict fluctuations in energies that come to an agreement with a Poisson limit. For invariant systems under time reversal, the proper formula of RMT is the Gaussian orthogonal ensemble (GOE). RMT was, at first, operated to illustrate the fluctuation features of neutron resonance in compound nucleus [6]. RMT was developed into a typical scheme for probing the common statistical fluctuations in disordered system [7 -10].
Mean field approximation may be employed to explore the disordered manners of single particle dynamic in nuclei. Nevertheless, the two-body residual interaction mixes various configurations in the mean field which in sequence leads to change the fluctuations properties of the nuclear spectrum and wave functions. Actually, one can investigate these fluctuations through utilizing different models. The nuclear shell model provides an attractive context for such investigations, where effective twobody residual interactions are obtainable and the basis states are designated by exact quantum numbers of J (total angular momentum), T (isospin) and  (parity). In earlier works [11][12][13][14][15][16], the context of the nuclear shell model was utilized to examine eigenvector component distributions. The basis vector amplitudes were found [14] to be in accordance with the Gaussian distribution (GOE prediction) in regions of large level density and diverged from Gaussian manners in further regions unless the computation employs degenerated single-particle energies. Another investigation [16] also recommended that computations by means of the degenerate single particle energies are disordered at lower excitation energy than that of realistic computations.
Electromagnetic probabilities in nuclei are observables which are related to the wave function. The examination of their fluctuations would enhance the universal spectral investigation as well as assist as an extra sign of disorder in the quantum system. In the previous investigations [17][18][19][20][21][22] we adopted the context of the RMT together with the nuclear shell model to explore the physical characteristics of chaos in nuclear spectra, electromagnetic probabilities, and moments for various nuclei located in different shell model spaces. As a whole, the results were very good depicted by the GOE limit.
There has been no comprehensive analysis for the chaotic (disordered) properties in the mass region of f5p shell nuclei. Thus, in the present analysis, we look at the statistical features of excitation energies in the isobar A = 68 (such as 68 Se, 68 As and 68 Ge) nuclei. The present shell model computations are carried out for 12 valence nucleons in f5p-model space (with 56 Ni as a core) using the f5pvh interaction [23] with the realistic single particle energies (spe's). The computed results for the considered T J  classes of states exhibit Gaussian shape for the level densities and GOE distribution for the spectral fluctuations.

Theory
The effective shell-model Hamiltonian of many particle systems can be expressed by [11] Here 0 H and H  are the unperturbed (one body) portion and the residual (two body) interaction of H, correspondingly. The one body Hamiltonian defines the non-interacting nucleons in an average field of suitable core, and  denotes the single particle orbitals. The residual interaction H  of active nucleons is given by The nuclear wave functions of many-body, with good quantum number of J and , T are built by means of the  m scheme determinants [11], The many body Hamiltonian is given by , ; ; where k and k are the many-body basis states. The energies  E as well as wave functions are computed by diagonalizing the matrix elements of Eq. (5).
The chaotic properties of nuclear spectra are typically found by the level spacing ) (s P and Dyson-Mehta ( 3  ) statistics [4,24]. We first built the staircase function ) (E N (which is the number of energy levels with energies where a smooth fit to ) (E N is made utilizing the fit of polynomial. We second expressed the unfolded spectrum through using the mapping [25] ) ( The genuine spacing show forceful fluctuations but the unfolded spectrum i Ẽ possesses a fixed average spacing. The distribution ) (s P is designated as the probability of two adjacent levels separated by a distance . s The i th spacing i s is found via .
The 3  statistics are utilized to determine the rigidity of the spectrum and expressed by [4]   It determines the divergence of the function ) (E N from a straight line. Here, ) (E N is constructed from the unfolded spectrum of Eq. (7). It is well-known that rigid (soft) spectra have small (large) values of . 3  For the purpose of obtaining a smoother distribution ), The successive intervals are taken to overlap by while that of the GOE is described by

Results and discussion
The present computations are carried out for A = 68 nuclei with T = 0 ( 68 Se), 1 ( 68 As) and 2 ( 68 Ge). The isobar of A=68 consists of the 56 Ni core and 12 active nucleons that move in the f5p-shell model space, defined by 2p 3/2 , 1f 5/2 and 2p 1/2 orbitals. The isospin formalism interaction of f5pvh [23] is chosen as an effective two-body residual interaction with realistic single particle energies (spe's). All computations of the present work are performed using the shell model code OXBASH [26].
In Table- Table-2. The histograms in the upper panel are calculated with the full Hamiltonian of f5pvh interaction together with realistic spe's, while those in the lower panel are calculated without the presence of the off-diagonal Hamiltonians of f5pvh. It is evident that the histograms in both panels are indistinguishable, with the exception of a shift in energy as a whole. The histograms in the lower panel reveal an enormous number of energy levels that accumulate at the mid-portion of ) (E  , which in sequence leads these histograms to spread with a narrow-range of excitation energy. This behavior is due to the non-considering of the off-diagonal Hamiltonian in the computations. However, the attachment of the off-diagonal mixing interaction in the computations (upper panel) leads commonly to push up the entire set of energy levels in the direction of the higher excitation energy. Therefore, the histograms (upper panel) reveal an important drop in its mid-portion. Besides, they distribute over a wider-range of energy than that of the lower panel. It is clear from both panels that the level density abruptly evolves in conjunction with the excitation energies, attains its maximum in the mid of the spectrum, and subsequently decreases once more for the greatest energy. This behavior of the great energy, and the uneven symmetry with regard to the mid of the spectrum, are non-regular features of models with restricted Hilbert space, which is in difference to real many-body systems. It is significant to denote that the calculated ) (E  (histograms), which has a Gaussian shape, is in accordance with the expectation of Brody et al. [7] that is designed for systems of many-body with two body interactions.
Actually, Figure-   Type of calculations J π T = 0 ( 68 Se) T = 1 ( 68 As) To examine the influence of varying the spin  J on the computed ), (E  we replicate the computations in Figure- Table-3. Figure-2 shows that the computed level densities      Figure-4 illustrates that the calculated nearest-neighbors level spacing distributions ) (s P [for low spin (1 + ), medium spin (4 + and 7 + ), and high spin (10 + )], are in agreement with the GOE distribution, i.e., they demonstrate no dependence on the spin .     T