Monte Carlo Shell Model Calculations Research Articles

Particle-number projection in the finite-temperature mean-field approximation

Calculation of statistical properties of nuclei in a finite-temperature mean-field theory requires projection onto good particle number, since the theory is formulated in the grand canonical ensemble. This projection is usually carried out in a saddle-point approximation. Here we derive formulas for an exact particle-number projection of the finite-temperature mean-field solution. We consider both deformed nuclei, in which the pairing condensate is weak and the Hartree-Fock (HF) approximation is the appropriate mean-field theory, and nuclei with strong pairing condensates, in which the appropriate theory is the Hartree-Fock-Bogoliubov (HFB) approximation, a method that explicitly violates particle-number conservation. For the HFB approximation, we present a general projection formula for a condensate that is time-reversal invariant and a simpler formula for the Bardeen-Cooper-Schrieffer (BCS) limit, which is realized in nuclei with spherical condensates. We apply the method to three heavy nuclei: a typical deformed nucleus $^{162}$Dy, a typical spherical nucleus $^{148}$Sm, and a transitional nucleus $^{150}$Sm in which the pairing condensate is deformed. We compare the results of this projection with results from the saddle-point approximation and exact shell model Monte Carlo calculations. We find that the approximate canonical HF entropy in the particle-number projection decreases monotonically to zero in the limit when the temperature goes to zero. However, in a nucleus with a strong pairing condensate, the approximate canonical HFB entropy in the particle-number projection decreases monotonically to a negative value, reflecting the violation of particle-number conservation. Computationally, the exact particle-number projection is more efficient than calculating the derivatives required in the saddle-point approximation.

Shell Evolution towards ^{78}Ni: Low-Lying States in ^{77}Cu.

The level structure of the neutron-rich ^{77}Cu nucleus is investigated through β-delayed γ-ray spectroscopy at the Radioactive Isotope Beam Factory of the RIKEN Nishina Center. Ions of ^{77}Ni are produced by in-flight fission, separated and identified in the BigRIPS fragment separator, and implanted in the WAS3ABi silicon detector array, surrounded by Ge cluster detectors of the EURICA array. A large number of excited states in ^{77}Cu are identified for the first time by correlating γ rays with the β decay of ^{77}Ni, and a level scheme is constructed by utilizing their coincidence relationships. The good agreement between large-scale MonteCarlo shell model calculations and experimental results allows for the evaluation of the single-particle structure near ^{78}Ni and suggests a single-particle nature for both the 5/2_{1}^{-} and 3/2_{1}^{-} states in ^{77}Cu, leading to doubly magic ^{78}Ni.

Abrupt shape transition at neutron number N=60 : B(E2) values in Sr94,96,98 from fast γ−γ timing

Lifetimes of low-lying yrast states in neutron-rich 94,96,98Sr have been measured by Germanium-gated γ−γ fast timing with LaBr3(Ce) detectors using the EXILL&FATIMA spectrometer at the Institut Laue-Langevin. Sr fission products were generated using cold-neutron-induced fission of 235U and stopped almost instantaneously within the thick target. The experimental B(E2) values are compared with results of Monte Carlo shell-model calculations made without truncation on the occupation numbers of the orbits spanned by eight proton and eight neutron orbits and show good agreement. Similarly to the Zr isotopes, the abrupt shape transition in the Sr isotopes near neutron number N=60 is identified as being caused by many-proton excitations to its g9/2 orbit.

Multifaceted Quadruplet of Low-Lying Spin-Zero States in ^{66}Ni: Emergence of Shape Isomerism in Light Nuclei.

A search for shape isomers in the ^{66}Ni nucleus was performed, following old suggestions of various mean-field models and recent ones, based on state-of-the-art MonteCarlo shell model (MCSM), all considering ^{66}Ni as the lightest nuclear system with shape isomerism. By employing the two-neutron transfer reaction induced by an ^{18}O beam on a ^{64}Ni target, at the sub-Coulomb barrier energy of 39MeV, all three lowest-excited 0^{+} states in ^{66}Ni were populated and their γ decay was observed by γ-coincidence technique. The 0^{+} states lifetimes were assessed with the plunger method, yielding for the 0_{2}^{+}, 0_{3}^{+}, and 0_{4}^{+} decay to the 2_{1}^{+} state the B(E2) values of 4.3, 0.1, and 0.2 Weisskopf units (W.u.), respectively. MCSM calculations correctly predict the existence of all three excited 0^{+} states, pointing to the oblate, spherical, and prolate nature of the consecutive excitations. In addition, they account for the hindrance of the E2 decay from the prolate 0_{4}^{+} to the spherical 2_{1}^{+} state, although overestimating its value. This result makes ^{66}Ni a unique nuclear system, apart from ^{236,238}U, in which a retarded γ transition from a 0^{+} deformed state to a spherical configuration is observed, resembling a shape-isomerlike behavior.

Type II shell evolution in A = 70 isobars from the N ≥ 40 island of inversion

Abstract The level structures of 70Co and 70Ni, populated from the β decay of 70Fe, have been investigated using β-delayed γ-ray spectroscopy following in-flight fission of a 238U beam. The experimental results are compared to Monte-Carlo Shell-Model calculations including the p f + g 9 / 2 + d 5 / 2 orbitals. The strong population of a ( 1 + ) state at 274 keV in 70Co is at variance with the expected excitation energy of ∼1 MeV from near spherical single-particle estimates. This observation indicates a dominance of prolate-deformed intruder configurations in the low-lying levels, which coexist with the normal near spherical states. It is shown that the β decay of the neutron-rich A = 70 isobars from the new island of inversion to the Z = 28 closed-shell regime progresses in accordance with a newly reported type of shell evolution, the so-called Type II, which involves many particle-hole excitations across energy gaps.

Quantum Phase Transition in the Shape of Zr isotopes.

The rapid shape change in Zr isotopes near neutron number N=60 is identified to be caused by type II shell evolution associated with massive proton excitations to its 0g_{9/2} orbit, and is shown to be a quantum phase transition. MonteCarlo shell-model calculations are carried out for Zr isotopes of N=50-70 with many configurations spanned by eight proton orbits and eight neutron orbits. Energy levels and B(E2) values are obtained within a single framework in good agreement with experiment, depicting various shapes in going from N=50 to 70. The novel coexistence of prolate and triaxial shapes is suggested.

Direct experimental evidence for a multiparticle-hole ground state configuration of deformedMg33

The first direct experimental evidence of a multiparticle-hole ground state configuration of the neutron-rich Mg-33 isotope has been obtained via intermediate energy (400 A MeV) Coulomb dissociation measurement. The major part similar to(70 +/- 13)% of the cross section is observed to populate the excited states of Mg-32 after the Coulomb breakup of Mg-33. The shapes of the differential Coulomb dissociation cross sections in coincidence with different core excited states favor that the valence neutron occupies both the s(1/2) and p(3/2) orbitals. These experimental findings suggest a significant reduction and merging of sd-pf shell gaps at N similar to 20 and 28. The ground state configuration of Mg-33 is predominantly a combination of Mg-32(3.0,3.5MeV; 2(-), 1(-)) circle times nu(s1/2), Mg-32(2.5MeV; 2(+)) circle times nu(p3/2), and Mg-32(0; 0(+)) circle times nu(p3/2). The experimentally obtained quantitative spectroscopic information for the valence neutron occupation of the s and p orbitals, coupled with different core states, is in agreement with Monte Carlo shell model (MCSM) calculation using 3 MeV as the shell gap at N = 20.

Benchmarking mean-field approximations to level densities

We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus $^{162}$Dy, representing a heavy deformed nucleus, and $^{148}$Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo (SMMC) calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are seen to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point approximation to extract the number-projected densities is not a significant source of error compared to other errors inherent to the mean-field theory. We also present an alternative formulation of the saddle-point approximation that makes direct use of an approximate particle-number projection and avoids computing the usual three-dimensional Jacobian of the saddle-point integration. We find that the pairing condensate is less amenable to approximate particle-number projection methods due to the explicit violation of particle-number conservation in the pairing condensate. Nevertheless, the Hartree-Fock-Bogoliubov theory is accurate to less than one unit of entropy for $^{148}$Sm at the neutron threshold energy, which is above the pairing phase transition.

New low-energy0+state and shape coexistence inNi70

In recent models, the neutron-rich Ni isotopes around $N=40$ are predicted to exhibit multiple low-energy excited ${0}^{+}$ states attributed to neutron and proton excitations across both the $N=40$ and $Z=28$ shell gaps. In $^{68}\mathrm{Ni}$, the three observed ${0}^{+}$ states have been interpreted in terms of triple shape coexistence between spherical, oblate, and prolate deformed shapes. In the present work a new $({0}_{2}^{+})$ state at an energy of 1567 keV has been discovered in $^{70}\mathrm{Ni}$ by using $\ensuremath{\beta}$-delayed, $\ensuremath{\gamma}$-ray spectroscopy following the decay of $^{70}\mathrm{Co}$. The precipitous drop in the energy of the prolate-deformed ${0}^{+}$ level between $^{68}\mathrm{Ni}$ and $^{70}\mathrm{Ni}$ with the addition of two neutrons compares favorably with results of Monte Carlo shell-model calculations carried out in the large $fp{g}_{9/2}{d}_{5/2}$ model space, which predict a ${0}_{2}^{+}$ state at 1525 keV in $^{70}\mathrm{Ni}$. The result extends the shape-coexistence picture in the region to $^{70}\mathrm{Ni}$ and confirms the importance of the role of the tensor component of the monopole interaction in describing the structure of neutron-rich nuclei.

Ab initio Monte Carlo shell model calculations for 7Li and 9Li low-lying spectra**Supported by Fundamental Research Funds for the Central Universities (JUSRP1035), National Natural Science Foundation of China (11305077)

The low-lying spectra of 7Li and 9Li are investigated within an ab initio Monte Carlo Shell Model (MCSM) employing a realistic potential obtained via the Unitary Correlation Operator Method (UCOM). The MCSM calculations in a 4-major-shells model space for the binding energy and mass quadrupole moment of 7,9Li show good convergence when the MCSM dimension reaches 20. The excitation energy of the Jπ = 1/2− state for 7Li and the magnetic moments for 7,9Li ground states in the MCSM with a treatment of spurious center-of-mass motion are close to the experimental data. Correct level ordering of Jπ = 3/2− and 1/2− states for 7,9Li can be reproduced due to the inclusion of three-body correlations in the MCSM+UCOM. However, the excitation energy of Jπ = 1/2− state for 9Li is not reproduced in the MCSM mainly due to the lack of larger model space.

No-core Monte Carlo shell model calculations with unitary correlation operator method and similarity renormalization group**Supported by Fundamental Research Funds for the Central Universities (JUSRP1035), National Natural Science Foundation of China (11305077)

The unitary correlation operator method (UCOM) and the similarity renormalization group theory (SRG) are compared and discussed in the framework of the no-core Monte Carlo shell model (MCSM) calculations for 3H and 4He. The treatment of spurious center-of-mass motion by Lawson's prescription is performed in the MCSM calculations. These results with both transformed interactions show good suppression of spurious center-of-mass motion with proper Lawson's prescription parameter βc.m. values. The UCOM potentials obtain faster convergence of total energy for the ground state than that of SRG potentials in the MCSM calculations, which differs from the cases in the no-core shell model calculations (NCSM). These differences are discussed and analyzed in terms of the truncation scheme in the MCSM and NCSM, as well as the properties of the potentials of SRG and UCOM.

Identification of deformed intruder states in semi-magicNi70

The structure of semi-magic Ni-70(28)42 was investigated following complementary multinucleon-transfer and secondary fragmentation reactions. Changes to the higher-spin, presumed negative-parity states based on observed gamma-ray coincidence relationships result in better agreement with shell-model calculations using effective interactions in the neutron f(5/2)pg(9/2) model space. The second 2(+) and (4(+)) states, however, can only be successfully described when proton excitations across the Z = 28 shell gap are included. Monte Carlo shell-model calculations suggest that the latter two states are part of a prolate-deformed intruder sequence, establishing an instance of shape coexistence at low excitation energies similar to that observed recently in neighboring Ni-68.

Nuclear magnetic moments for J π = 2 1 + state of 10Be with ab initio Monte Carlo shell model calculation

The magnetic moment of 2 1 + state for 10Be are calculated and investigated in terms of single particle orbits for protons and neutrons under the framework of ab initio Monte Carlo shell model method in an e max = 3 model space. The reduced matrix elements of orbital and spin angular momentum are evaluated. It is found that the orientations of orbital angular momentum in different single particle orbits are consistent. Conversely, the orientations of spin in different single particle orbits tend to be chaotic. The nuclear magnetic moment of 2 1 + state for 10Be is obtained as 1.006 µ N and is discussed in regards to the contribution of orbital and spin angular momentum both for protons and neutrons. The corresponding g-factor is also given.

Density profiles of light nuclei in Monte Carlo shell-model calculation

The wave functions of the ground states for 8,10Be which are obtained from the Monte Carlo shell model (MCSM) are investigated. A method to define an intrinsic state in the MCSM is discussed. The appearance of two-α-cluster structure in 8Be and the property of the valence neutrons in 10Be are also discussed.

Study of nuclei aroundZ= 28 by large-scale shell model calculations

We study Cr and Ni isotopes by using Monte Carlo shell model (MCSM) cal- culations in a p fg9d5 model space. In the MCSM, a wave function is represented as a linear combination of angular-momentum- and parity-projected deformed Slater determi- nants (MCSM bases) and we can obtain eigenstates in a large model space such as p fg9d5 space. We study intrinsic shapes of nuclei by using deformations of MCSM bases before projection. We show results of MCSM calculations in Cr and Ni isotopes and a level scheme of 68 Ni and discuss the magicity of N = 40 in Cr and Ni isotopes and shape coexistence in 68 Ni.

Density profiles in the intrinsic frame of light nuclei obtained from Monte Carlo shell model calculation

The wave functions of the ground states for 8,10Be which are obtained from the Monte Carlo shell model (MCSM) are investigated. A method to define an intrinsic state in the MCSM is discussed. The appearance of two-α-cluster structure in 8Be and the property of the valence neutrons in 10Be are discussed.

Study of nuclei around Z = 28 by large-scale shell model calculations

We study Cr and Ni isotopes by Monte Carlo shell model (MCSM) calculations in a pfg9d5 shell. In the MCSM, the wave function is represented as a linear combination of angular-momentum- and parity-projected deformed Slater determinants (MCSM bases), and we can calculate eigenstates in a large model space such as the pfg9d5 shell. We study the intrinsic shape of nuclei by using deformations of the unprojected MCSM bases. We show calculated results of Cr and Ni isotopes and a level scheme of 68Ni, and discuss the strength of the magicity of N = 40 in Cr and Ni isotopes and shape coexistence in 68Ni.

No-Core MCSM calculation for 10Be and 12Be low-lying spectra

The low-lying excited states of 10Be and 12Be are investigated within a no-core Monte Carlo Shell Model (MCSM) framework employing a realistic potential obtained via the Unitary Correlation Operator Method. The excitation energies of the 2+1 and 2+2 states and the B(E2; 2+1,2 → 0+g.s.) for 10Be in the MCSM with a standard treatment of spurious center-of-mass motion show good agreement with experimental data. Some properties of low-lying states of 10Be are studied in terms of quadrupole moments and E2 transitions. The E2 transition probability of 10C, the mirror nucleus of 10Be, is also presented with a good agreement to experiment. The triaxial deformation of 10Be and 10C is discussed in terms of the B(E2) values.

Intrinsic Structure of Light Nuclei in Monte Carlo Shell Model Calculation

Wavefunctions obtained from the Monte Carlo shell model (MCSM) calculation are investigated. It has been difficult to discuss the intrinsic structure of nuclei in conventional shell-model calculations. We propose a way to describe an intrinsic state in the MCSM and demonstrate the appearance of two-α-cluster structure in the 8Be ground state. The changes of cluster shape with respect to the number of major shell and basis is discussed. The shape of the 10Be ground state is also investigated. The behavior of valence neutron is consistent with the picture of molecular orbit state. The method is also applied to the 6He ground state, which is expected to have the asymmetric distribution of valence neutrons.

Narrowing of the neutron [formula omitted] shell gap in 29Na

The wave-function composition for the low-lying states in 29Na was explored by measuring their electromagnetic properties using the Coulomb-excitation technique. A beam of 29Na ions, postaccelerated to 70 MeV, bombarded a 110Pd target with a rate of up to 600 particles per second at the recently commissioned ISAC-II facility at TRIUMF. Six segmented HPGe clover detectors of the TIGRESS γ-ray spectrometer were used to detect deexcitation γ rays in coincidence with scattered or recoiling charged particles in the segmented silicon detector, BAMBINO. The reduced transition matrix element |〈521+||E2||32gs+〉| in 29Na was derived to be 0.237(21) e b from the measured γ-ray yields for both projectile and target. This first-time measured value is consistent with the most recent Monte Carlo shell-model calculation, indicating a significant admixture of both sd and pf components in the wave function, and also providing evidence for the narrowing of the neutron sd–pf shell gap from ∼6 MeV for stable nuclei to ∼3 MeV for 29Na.