Exact Shell-model Calculations Research Articles

Isovector pairing in self-conjugate nuclei in a formalism of quartets

The isovector proton-neutron pairing in self-conjugate nuclei is treated in a formalism of quartets. Quartets are four-body correlated structures built from two neutrons and two protons coupled to total isospin T = 0. The ground state of the isovector pairing Hamiltonian is described as a product of quartets. We review both the case in which the quartets are constrained to be all identical and the case in which they are allowed to be distinct from one another. The quality of the two approaches is tested by making comparisons with exact shell model calculations for N = Z nuclei with valence nucleons outside the 16O, 40Ca, and 100 Sn cores. We consider both spherical and axially deformed mean fields. Both approaches are found to be very accurate. In the applications to a deformed mean field, in particular, the formalism with distinct quartets gives rise to results which are basically exact.

High-performance algorithm for calculating non-spurious spin- and parity-dependent nuclear level densities

A new high-performance algorithm for calculating the spin- and parity-dependent shell model nuclear level densities using methods of statistical spectroscopy in the proton–neutron formalism was recently proposed. When used in valence spaces that cover more than one major harmonic oscillator shell, this algorithm mixes the genuine intrinsic states with spurious center-of-mass excitations. In this Letter we present an advanced algorithm, based on the recently proposed statistical moments method, that eliminates the spurious states. Results for unnatural parity states of several sd-shell nuclei are presented and compared with those of exact shell model calculations and experimental data.

Padé-resummed high-order perturbation theory for nuclear structure calculations

We apply high-order many-body perturbation theory for the calculation of ground-state energies of closed-shell nuclei using realistic nuclear interactions. Using a simple recursive formulation, we compute the perturbative energy contributions up to 30th order and compare to exact no-core shell model calculations for the same model space and Hamiltonian. Generally, finite partial sums of this perturbation series do not show convergence with increasing order, but tend to diverge exponentially. Nevertheless, through a simple resummation via Padé approximants it is possible to extract rapidly converging and highly accurate results for the ground-state energy once perturbative contributions beyond 5th order are included.

Validity of pair truncation of the nuclear shell model inCa46

We study the validity of pair truncation of the nuclear shell model by using the semimagic nucleus $^{46}\mathrm{Ca}$. We present low-lying states and their $E2$ transition rates based on both nucleon pair approximation (NPA) and exact shell-model (SM) calculations. We also calculate overlaps between wave functions of low-lying states calculated by using the NPA and those calculated by using the SM. Our calculated results show a remarkable agreement between the NPA results and the SM results, although the NPA is a drastic truncation of the SM.

Dyson Boson Mapping and Shell-Model Calculations for Even-Even Nuclei

In order to truncate a high-dimensional Hilbert space for a many-nucleon shell model to a small collective subspace, we carry out a Dyson-type boson mapping and select a small number of collective bosons. In this way, we can divide the Dyson boson Hamiltonian into collective and non-collective parts; the latter includes the renormalization of the coupling effects between the collective and non-collective degrees of freedom. The boson Hamiltonian thus obtained is written in terms of the selected collective bosons only. Diagonalizing this boson Hamiltonian and comparing the numerical results with those obtained from exact shell-model calculations for some sd -a ndpf -shell nuclei, we find that our method gives a good approximation of the exact shell model. It is shown that the coupling effect between the collective and non-collective degrees of freedom plays an essential role. This method is also applied to some heavier pf g-shell nuclei for which it is unfeasible to carry out exact shell-model calculation because of the very large number of dimensions of the Hilbert space. Our numerical results are in very good agreement with experiment.

A boson approach to the structure of A=22 nuclei

AbstractWe discuss a procedure to transfer the description of a fermion system from a subspace of the full shell model space built in terms of collective pairs onto a space of corresponding bosons. We apply the procedure to systems of six nucleons in the 1s0d major shell. We perform exact shell model calculations and compare them with calculations in the collective pair and boson approximations. The effects of the truncation of the boson Hamiltonian and of the consequent violation of the Pauli principle are examined.

What is the collective subspace in the many-nucleon Hilbert space?

The shell-model calculation in the full shell-model Hilbert space of a many-nucleon system is, in general, very difficult when the nucleon number is large. For such a large-dimensional system, we have to truncate the Hilbert space to a small size of collective subspace. To do this, we carry out the Dyson-type boson mapping and truncate the boson degrees of freedom to some small number of collective bosons; for example, s-, d- and g-bosons only. Thereby we can divide the Dyson boson Hamiltonian into the collective part and the non-collective part; the latter includes the renormalization of the effects from the non-collective degrees of freedom. Comparing numerical calculations under this boson approximation with the exact shell-model calculations, we show that our truncation gives a very good approximation to the exact shell model. Thus we can study the meaning of the collective subspace and establish the foundation of the basic idea of the interacting boson model (IBM).

Coulomb energy differences in t = 1 mirror rotational bands in (50)Fe and (50)Cr.

Gamma rays from the N = Z-2 nucleus (50)Fe have been observed, establishing the rotational ground state band up to the state J(pi) = 11+ at 6.994 MeV excitation energy. The experimental Coulomb energy differences, obtained by comparison with the isobaric analog states in its mirror (50)Cr, confirm the qualitative interpretation of the backbending patterns in terms of successive alignments of proton and neutron pairs. A quantitative agreement with experiment has been achieved by exact shell model calculations, incorporating the differences in radii along the yrast bands, and properly renormalizing the Coulomb matrix elements in the pf model space.

Improvement of the extended [formula omitted] interaction by modifying the monopole field

The extended pairing plus QQ interaction with the J-independent isoscalar proton–neutron force as an average monopole field, which has succeeded in describing collective yrast states of N≈Z even- A nuclei, is improved. The improvement is accomplished by adding small monopole terms (relevant to spectroscopy) to the average monopole field (indispensable to the binding energy). This modification extends the applicability of the interaction to nuclei with N≈28 such as 48Ca, and moreover improves energy levels of nonyrast states. The modified interaction successfully describes not only even- A but also odd- A nuclei in the f 7/2 shell region. Results of exact shell model calculations in the model space ( f 7/2 , p 3/2 , p 1/2 ) (its usefulness has been demonstrated previously) are shown for A=47, 48, 49, 50 and 51 nuclei.

Aredundancy-free approach to the use ofcorrelated bases in the truncation oflarge shell-model eigenproblems

In this paper we describe a method for solving shell-model problems in aseverely truncated space spanned by correlated states. The method isbased on a partition of the full configuration space in orthogonalsubspaces, in which exact shell-model calculations can be done. Themethod allows a practically exact description of the low-energyspectroscopic properties of nuclei, as exemplified both in a testcalculation involving a schematic pairing Hamiltonian and in calculationsfor systems of neutrons and protons in which a realistic effectiveinteraction derived from the Bonn-A potential is used.

Application of the extended [formula omitted] force model to [formula omitted][formula omitted] shell nuclei

To study collective motion, the extended pairing plus QQ force model proposed recently is applied to A=46, 48 and 50 nuclei in the fp shell region. Exact shell model calculations in the truncated model space (f 7/2,p 3/2,p 1/2) prove the usefulness of the interaction. The simple model with the pairing plus quadrupole pairing plus QQ force and J-independent isoscalar proton–neutron force unexpectedly well reproduces observed binding energies, energy levels of collective (yrast) states and reduced E2 transition probabilities in 46 Ti, 46 V, 48 V, 48 Cr, 50 Cr and 50 Mn. The correspondence between theory and experiment is almost comparable to that attained by the full fp shell model calculations with realistic effective interactions. Some predictions are made for energy levels and variations of B(E2) in the yrast bands, in these nuclei. Characteristics of the interaction are investigated by comparing with the realistic effective interactions and by the backbending plot for the yrast band of 48 Cr.

Transition strength sums and quantum chaos

For the embedded Gaussian orthogonal ensemble of random matrices, the strength sums generated by a transition operator acting on an eigenstate vary with the excitation energy as the ratio of two Gaussians. This general result is compared to exact shell-model calculations of Gamow-Teller strength sums in nuclei. Good agreement is obtained in the chaotic domain of the spectrum, and strong deviations are observed as nuclear motion approaches a regular regime. Thus transition strength sums seem to be a new statistic sensitive to the chaoticity of the system.

Isoscalar and isovector pair correlations in an isospin-invariant pairing plus quadrupole model

Isovector and isoscalar correlations are investigated using an isospin-invariant pairing plus quadrupole model with inclusion of an isoscalar pairing force. An exact shell model calculation is carried out on the ${(g}_{9/2}{)}_{p}{(g}_{9/2}{)}_{n}$ shell, as a schematic example. We study isoscalar $(\ensuremath{\tau}=0)$ and isovector $(\ensuremath{\tau}=1)$ pair correlations in the ground state and excited states, and examine the interplay between isoscalar and isovector correlations. Under the single ${g}_{9/2}$ shell approximation, it is found that, for the $N=Z$ even-even system, the number of $\ensuremath{\tau}=1, J=8$ pairs increases drastically in a state with total spin $I=8$ along the yrast line. For $N=Z$ odd-odd systems, there is a crossing of an even-spin ground band by an odd-spin band at $I=1$ along the yrast line. The $\ensuremath{\tau}=0, J=9$ proton-neutron correlations increase suddenly at $I=7.$ This can be interpreted as the alignment of a proton-neutron pair. Finally, we examine the $J=0$ pair correlations in ground states of even-even and odd-odd systems.

Cranked shell model and isospin symmetry near N = Z

A cranked shell model approach for the description of rotational bands in N ≈ Z nuclei is formulated. The isovector neutron-proton pairing is taken into account explicitly. The concept of spontaneous breaking and subsequent restoration of the isospin symmetry turns out to be crucial. The general rules to construct the near yrast spectra for rotating nuclei are presented. For the model case of particles in a j-shell, it is shown that excitation spectra and the alignment processes are well described as compared to the exact shell model calculation. Realistic cranked shell model calculations are able to describe the experimental spectra of 72,73Kr and 74Rb isotopes.

E0 transitions and O+ levels in 136Xe.

The ${\mathrm{\ensuremath{\beta}}}^{\mathrm{\ensuremath{-}}}$ decay of $^{136}\mathrm{I}$ isomers to levels in $^{136}\mathrm{Xe}$ was studied using sources isolated at the on-line mass separator TRISTAN. Conversion-electron and gamma-ray singles data have been collected. Evidence for excited ${\mathrm{O}}^{+}$ states at 2582.4 and 4320 keV has been obtained. A shell-model study elucidating the salient features related to the first excited ${\mathrm{O}}^{+}$ state in the N=82 nuclei has been carried out. Comparing two-particle, four-particle exact shell-model and two-quasiparticle shell-model calculations using both a Gaussian and surface-delta interaction in the latter case, an explanation could be given for the Z dependence in the ${\mathit{E}}_{\mathit{x}}$(${\mathrm{O}}_{2}^{+}$) excitation energy. We also calculate the E0 reduced matrix elements for $^{134}\mathrm{Te}$.

Spin quenching and orbital enhancement in the Ti isotopes

Exact shell model calculations with a minimally modified form of the Kuo-Brown interaction yield detailed quantitative agreement for spin and M1 strength functions measured in (p, p′) and (e, e′) scattering in 46,48Ti. The correlations in the exact results procedure strong spin quenching, orbital enhancement and surprising modifications of the orbital collective modes with respect to the McCullen, Bayman and Zamick model and its low order extensions. Intruder states show up conspicuously and are held responsible for the residual discrepancies between data and calculations.

Pair approximation and the OAI mapping in the deformed limit

The pair subspaces — the SD- and SDG-subspaces — are constructed. Eigenstates for a quadrupole force and transition rates for a quadrupole operator are calculated in the single j-shell-model. The SDG-pair approximation is found to be excellent in describing the low-spin states of the ground bands compared to exact shell-model calculations. The fermion interactions are mapped onto the corresponding boson ones using the mapping procedure by Otsuka, Arima and Iachello (OAI). The OAI approximation in zeroth-order fails in reproducing the ground-state energies in the deformed limit.

Sdg Interacting boson hamiltonian in the seniority scheme

The sdg interacting boson hamiltonian is derived in the seniority scheme. We use the method of Otsuka, Arima and Iachello in order to derive the boson hamiltonian from the fermion hamiltonian. To examine how good is the boson approximation in the zeroth-order, we carry out the exact shell model calculations in a single j-shell. It is found that almost all low-lying levels are reproduced quite well by diagonalizing the sdg interacting boson hamiltonian in the vibrational case. In the deformed case the introduction of g-bosons improves the reproduction of the spectra and of the binding energies which are obtained by diagonalizing the exact shell model hamiltonian. In particular the sdg interacting boson model reproduces well-developed rotational bands.

Fermion calculations in the boson space using the Dyson boson mapping.

An algorithm for obtaining a recursive algebraic expression for the various expansion coefficients appearing in the physical boson basis states is presented, using the Dyson boson mapping. It is observed that only the first few terms of the physical boson basis states, expressed in the present form, contribute to the matrix element of the Hamiltonian. Explicit numerical calculations have been performed for Ni and Sn isotopes in the obtained physical boson basis states. The pairing Hamiltonian has been used as the model interaction. The results are more than 90% in agreement with exact shell model calculations.

Yrast bands in light nuclei and the effective interactions inJ=0, 2states

The dominant role played by the effective interactions in angular momentum states with $J=0, 2$ vis-\'a-vis the structure of the yrast states in the $2s\ensuremath{-}1d$ and $2p\ensuremath{-}1f\ensuremath{-}1g$ shell nuclei is pointed out.NUCLEAR STRUCTURE Effective interactions in $J=0, 2$ states. Structure of the yrast bands. Exact shell model calculations for $^{20\ensuremath{-}22}\mathrm{O}$, $^{20}\mathrm{Ne}$, and $^{60}\mathrm{Ni}$.