Shell Model Spectra Research Articles

Microscopic structure and mathematical background of the multiconfigurational dynamical symmetry

Background: The multiconfigurational dynamical symmetry (MUSY) is the common intersection of the shell, collective, and cluster models for the multi-major-shell problem. It is able to describe the spectra of different configurations in different energy windows in a unified way. It is based on some heuristic arguments, related to the connection of the wave functions, and energy eigenvalues. The detailed mathematical background has been worked out so far only for the simplest case of the two binary cluster configurations.Purpose: I intend to construct the exact algebraic framework for the general case of the MUSY, i.e., for any number of configurations and any number of clusters or nucleons. As an illustrative example, the spectrum of different configurations of the $^{44}\mathrm{Ti}$ nucleus is described by a simple Hamiltonian.Methods: Classification schemes defined by different algebra chains need to be combined; in particular, those of the major shell scheme and of the particle index scheme.Results: A class of Hamiltonians, which is invariant under the transformations from one configuration to the other, is determined. In case of the $^{44}\mathrm{Ti}$ low-lying shell-model spectrum, as well as $^{40}\mathrm{Ca}+^{4}\mathrm{He}$ and $^{28}\mathrm{Si}+^{16}\mathrm{O}$ cluster states are obtained in a unified way.Conclusions: The MUSY is based on two pillars: (i) a unified multiplet structure for shell, collective, and cluster model states and (ii) a Hamiltonian which is invariant with respect to the transformations from one configuration to the other.

A SYSTEMATIC STUDY OF LyαTRANSFER THROUGH OUTFLOWING SHELLS: MODEL PARAMETER ESTIMATION

Outflows promote the escape of Lyman-$\alpha$ (Ly$\alpha$) photons from dusty interstellar media. The process of radiative transfer through interstellar outflows is often modelled by a spherically symmetric, geometrically thin shell of gas that scatters photons emitted by a central Ly$\alpha$ source. Despite its simplified geometry, this `shell model' has been surprisingly successful at reproducing observed Ly$\alpha$ line shapes. In this paper we perform automated line fitting on a set of noisy simulated shell model spectra, in order to determine whether degeneracies exist between the different shell model parameters. While there are some significant degeneracies, we find that most parameters are accurately recovered, especially the HI column density ($N_{\rm HI}$) and outflow velocity ($v_{\rm exp}$). This work represents an important first step in determining how the shell model parameters relate to the actual physical properties of Ly$\alpha$ sources. To aid further exploration of the parameter space, we have made our simulated model spectra available through an interactive online tool.

Maximum likelihood method to correct for missed levels based on theΔ3(L)statistic

The $\Delta_3(L)$ statistic of Random Matrix Theory is defined as the average of a set of random numbers $\{\delta\}$, derived from a spectrum. The distribution $p(\delta)$ of these random numbers is used as the basis of a maximum likelihood method to gauge the fraction $x$ of levels missed in an experimental spectrum. The method is tested on an ensemble of depleted spectra from the gaussian orthogonal ensemble (GOE), and accurately returned the correct fraction of missed levels. Neutron resonance data and acoustic spectra of an aluminum block were analyzed. All results were compared with an analysis based on an established expression for $\Delta_3(L)$ for a depleted GOE spectrum. The effects of intruder levels is examined, and seen to be very similar to that of missed levels. Shell model spectra were seen to give the same $p(\delta)$ as the GOE.

Recent results in quantum chaos and its applications to atomic nuclei

A survey of chaotic dynamics in atomic nuclei is presented, using on the one hand standard statistics of quantum chaos studies, and on the other a new approach based on time series analysis methods. The study of shell-model spectra in the pf shell shows that nuclear chaos is strongly isopin dependent and increases with excitation energy. On the other hand, it is found that chaotic quantum systems exhibit 1/f noise and regular systems exhibit 1/f2 behaviour. It is shown that the time series approach can be used to calculate quite accurately the fraction of missing levels and the existence of mixed symmetries in experimental level spectra.

Power spectrum of nuclear spectra with missing levels and mixed symmetries

Sequences of energy levels in nuclei are often plagued with missing levels whose number and position are unknown. It is also quite usual that all the quantum numbers of certain levels cannot be experimentally determined, and thus levels of different symmetries are mixed in the same sequence. The analysis of these imperfect spectra (from the point of view of spectral statistics) is unavoidable if one wants to extract some statistical information. The power spectrum of the δq statistic has emerged in recent years as an important tool for the study of quantum chaos and spectral statistics. We derive analytical expressions for the observed power spectrum in terms of the fraction of observed levels and the number of mixed sequences. These expressions are tested with large shell model spectra simulating realistic experimental situations. A good estimation of the number of mixed symmetries and the fraction of missing levels is obtained by means of a least-squares fit in a wide set of different situations.

Nonmetal-metal transition inZnn(n=2–20)clusters

By using density functional calculation with generalized gradient approximation, we have studied the structural and electronic properties of the zinc clusters. The lowest-energy structures of Zn{sub n} (n=2-20) clusters are determined. Three kinds of growth pathways are obtained in the small zinc clusters from Zn{sub 4} to Zn{sub 8} and tetrahedron-based structures have favorable energy. The zinc clusters with 7-16 atoms are semiconductorlike. A structural transition from low coordination cagelike to high coordination compact structures is obtained around Zn{sub 17}. The Zn{sub n} clusters with n=4, 7, 9, 10, 14, 18, 20 show relatively high stability, consistent with the electron shell model and mass spectra. The ionization potentials of the Zn{sub n} clusters are calculated and compared with conducting sphere droplet model. The size evolution of zinc clusters from van der Waals to covalent and bulk metallic behavior is discussed. The Zn clusters show stronger metallicity than the Cd and Hg clusters with same size.

Complete spectroscopy of30P

The ${}^{29}\mathrm{Si}(p,\ensuremath{\gamma}{)}^{30}\mathrm{P}$ reaction was studied for 32 resonances in the energy range ${E}_{p}=1.0--2.5$ MeV. Angular distributions were measured for 230 primary and secondary transitions. Results from the analysis of these data were combined with previous measurements of the ${}^{29}\mathrm{Si}{(p,p}_{0}{)}^{29}\mathrm{Si}$ and ${}^{29}\mathrm{Si}(p,\ensuremath{\gamma}{)}^{30}\mathrm{P}$ reactions to determine the spin J, parity $\ensuremath{\pi},$ and isospin T of many levels. Remaining ambiguities were eliminated by comparison with the spectrum of the parent nucleus ${}^{30}\mathrm{Si}$ and with the calculated shell-model spectrum. J, $\ensuremath{\pi},$ and T have been assigned for 103 states up to an excitation energy of 8.0 MeV.

Influence of ΛNN forces and the nuclear core size on the Λ 7 Li level spectrum

An effective zero-range ΛNN interaction with the Λ-spin dependence peculiar to dispersive ΛNN forces is introduced in order to estimate its influence on the shell model spectra of light hypernuclei. The parameters of the three-body and pair interactions are evaluated using the values of BΛ for s-shell hypernuclei with A=4 and 5. It is shown that the description of the Λ 7 Lig.s. doublet energy ΔE≃689 keV, when fitted together with other hypernuclei, gives a reduced radius R≃2.2 fm for 6Li bound in Λ 7 Li (compared to R≃2.4 fm for the free nucleus 6Li) and a dispersive ΛNN force contribution to this energy of ∼250 keV. A more refined version of the Λ 7 Li level spectrum is proposed, including ΛN and ΛNN forces and the compression of the 6Li core.

Structure of odd- A 1s0d- and 1p0f-shell nuclei in the collective pair approximation

The structure of states of nuclei with three and five active nucleons in the 1s0d and 1p0f shells is studied in the framework of the collective pair approximation. The collective pairs determined by diagonalizing the Hamiltonian in the space of two nucleons outside closed shells are considered as building blocks to describe a nucleus with 2k+1 nucleons in terms of k pairs coupled to one nucleon. It is shown that the low-lying, shell-model spectrum can be described quite well by considering only a selected subset of all possible collective pairs.

Kinetic-energy operator in the effective shell-model interaction.

Differences in the Hartree-Fock and effective shell-model interaction arising from alternative treatments of the kinetic-energy operator in finite nuclear many-body problems are described. The Hartree-Fock single-particle energies and their relationship to experimental removal energies depend sensitively on whether or not the center-of-mass kinetic energy is retained in the nuclear Hamiltonian. Large effects in particle-hole energies are obtained which have important consequences for effective shell-model Hamiltonians. If the center-of-mass contribution of the kinetic-energy operator is removed from the Hamiltonian, substantial effects appear in a simple example of the shell-model spectra of $^{16}\mathrm{O}$ and $^{17}\mathrm{O}$ treated as four and five valence nucleons, respectively, outside a $^{12}\mathrm{C}$ core. The contributions to the energy coming from the valence, relative kinetic-energy operator push the energy spectra of both nuclei up by about 1 MeV relative to their ground states.

Pair condensates in realistic shell model states. II. M1 states in 54,56Cr and 56,58,60Fe.

The possible microscopic realization of interacting boson model (IBM) collective M1 modes is investigated for $^{54,56}\mathrm{Cr}$ and $^{56,58,60}\mathrm{Fe}$, using a shell-model space of protons in the 1${\mathit{f}}_{7/2}$ orbit and neutrons in the 2${\mathit{p}}_{3/2}$, 1${\mathit{f}}_{5/2}$, and 2${\mathit{p}}_{1/2}$ orbits. Distributions of the pair analogues of IBM-2 mixed-symmetry states, and of some g-IBM-2 states, over shell-model eigenstates are explicitly determined. For J=1, the mixed-symmetry SD strength is generally lower than that of the G-pair states, and responsible for most of the low-energy M1 strength. Structures incorporating extended sets of levels, including the M1 states, corresponding to the sd-IBM-2 U(5)\ensuremath{\supset}SO(6)\ensuremath{\supset}SO(5) dynamical symmetry representations [N-1,1] (N-1,1,0) (\ensuremath{\tau}) are manifest in the shell-model spectra for $^{56}\mathrm{Fe}$ and $^{54}\mathrm{Cr}$. The correspondence with observed levels is discussed.

Broken isospin symmetry in the shell model and chaotic behavior

>For the first time, the effect of isospin symmetry breaking on quantum chaos is demonstrated for a realistic shell-model spectrum. Breaking the isospin via the Coulomb interaction shifts the fluctuation pattern from the intermediate case between Poisson and GOE toward GOE statistics, but the latter is not fully restored. This result is in accordance with recent analysis of experimental data for 26Al.

"Universality" of Gaussian orthogonal ensemble fluctuations: the two-body random ensemble and shell model spectra

Starting from an appropriate decomposition of the level density into an average and fluctuating part, we studied the energy level fluctuations of an ensemble defined by two-body random Hamiltonians. A detailed analysis of several spectrally averaged fluctuation measures shows close agreement with the predictions of the Gaussian orthogonal ensemble (GOE). This confirms earlier indications that, except for noninteracting particles, fluctuation measures are insensitive to the rank of the interaction. Further, analysis of spectra obtained from realistic nuclear interactions agrees well with the GOE indicating that specific properties of the Hamiltonian have little or no influence on fluctuations. These results, therefore, strengthen our belief in the "universality" of GOE fluctuations.

Density and fluctuations in shell model spectra

A simple method, based on the Gram-Charlier expansion, to separate the smooth and fluctuating parts of the level density is developed. Applications to spectra arising from large shell model calculations are considered. Fluctuations in the energy levels are studied. It is found that with this proper decomposition, even long range correlations agree with the predictions of the random matrix theory.

The one-boson-exchange potential and the shell-model spectra of mass-18 nuclei

The one-boson-exchange potentials in the momentum representation of Holinde and Machleidt (see Nucl. Phys. A, vol.256, p.497, 1976) are used in a shell-model calculation of the spectra of mass-18 nuclei in order to gain an insight into the possibilities of convergence of the effective interaction expansion for these nucleon-nucleon interactions. The spectra are calculated in first order, and also in first order including the core polarisation term, which is the most important second-order correction. Comparison is made with the Reid soft-core potential results.

Realistic effective interactions for thesdshell: Global and discrete nuclear properties

Comparisons of the Kuo and Vary-Yang $\mathrm{sd}$ shell realistic effective shell-model interactions are presented through studies of the resulting global and discrete nuclear properties. The methods of spectral distributions are used to obtain energy moments for $^{21}\mathrm{Ne}$ and spin cutoff factors for $^{28}\mathrm{Si}$ for each interaction. Shell-model spectra are presented for some mass $A=18$ to $A=21$ $\mathrm{sd}$ shell nuclei. In each case we show for comparison the results using the Chung-Wildenthal empirical interaction. In addition to demonstrating the role of various approximations through resulting global and discrete properties, we show that certain conclusions drawn from global properties are consistent with those based upon discrete properties.NUCLEAR STRUCTURE Realistic effective interactions for $\mathrm{sd}$ shell, spectral distribution theory, level densities, and spin cutoff factors.

A three-nucleon force and A=18 shell-model spectra

The three-nucleon force due to two-pion exchange is approximated by an effective two-nucleon potential. Its effect on the lowest-order term of the shell-model effective interaction is calculated. The effect on nuclear levels attributed to two valence particles is found to be small. This result is discussed in the context of three-nucleon force contributions to the binding energy of nuclear matter and nuclei.

Number-projected HFB theory in the nuclear 2s-1d shell: Comparison with exact shell-model results

Within the framework of Hartree-Fock-Bogoliubov theory a method is described for number projection before variation. The method is applied to some even nuclei with neutron excess, on the assumption that for such nuclei n-n and p-p pairing dominate. For each case level spectra are produced by angular momentum projection after variation. The basis used is the 2s-1d shell, and within that basis there is good agreement with exact shell model spectra for the nuclei 24Ne, 26Mg and 32Ar. This agreement extends to states not in the ground state band.

Convergence Rate of Intermediate-State Summations in the Effective Shell-Model Interaction

We examine the convergence rate of the sum over intermediate-particle states in the second-order core-polarization diagram of the effective shell-model interaction. For a $G$ matrix resulting from the Reid soft-core interaction, convergence is achieved in an oscillator basis only after the intermediate particle is summed over five or six major-shell excitations ($10\ensuremath{\hbar}\ensuremath{\Omega}$ to $12\ensuremath{\hbar}\ensuremath{\Omega}$ excitations). The total effect of this diagram on the shell-model spectra of $^{18}\mathrm{O}$ is markedly altered from the results of Kuo and Brown and others who only include oscillator excitations of one major shell ($2\ensuremath{\hbar}\ensuremath{\Omega}$) in the second-order diagram. We further demonstrate that slow convergence is attributed to strong components of the effective tensor force in the $G$ matrix.

Off-Shell Effects in theO18andF18Shell-Model Spectra

The two-particle shell-model spectra of $^{18}\mathrm{O}$ and $^{18}\mathrm{F}$ are studied using effective interactions obtained from phase-shift-equivalent nucleon-nucleon potentials. The spectra depend significantly on the off-shell behavior of the potentials.