Generator Coordinate Method Research Articles

Consistent description for cluster dynamics and single-particle correlation

Cluster dynamics and single-particle correlation are simultaneously treated for the description of the ground state of $^{12}\mathrm{C}$. The recent development of the antisymmetrized quasicluster model (AQCM) makes it possible to generate $jj$-coupling shell-model wave functions from $\ensuremath{\alpha}$ cluster models. The cluster dynamics and the competition with the $jj$-coupling shell-model structure can be estimated rather easily. In the present study, we further include the effect of single-particle excitation; the mixing of the two-particle--two-hole excited states is considered. The single-particle excitation is not always taken into account in the standard cluster model analyses, and the two-particle--two-hole states are found to strongly contribute to the lowering of the ground state owing to the pairing-like correlations. By extending AQCM, all of the basis states are prepared on the same footing, and they are superposed based on the framework of the generator coordinate method (GCM).

Microscopic description of quadrupole-octupole coupling in neutron-rich actinides and superheavy nuclei with the Gogny-D1M energy density functional

The interplay between quadrupole and octupole degrees of freedom is discussed in a series of neutron-rich actinides and superheavy nuclei with $92 \le$ Z $\le 110$ and $186 \le$ N $\le 202$. In addition to the static Hartree-Fock-Bogoliubov approach, dynamical beyond-mean-field correlations are taken into account via both parity restoration and symmetry-conserving Generator Coordinate Method calculations based on the Gogny-D1M energy density functional. Physical properties such as correlation energies, negative-parity excitation energies as well as reduced transition probabilities $B(E1)$ and $B(E3)$ are discussed in detail. It is shown that, for the studied nuclei, the quadrupole-octupole coupling is weak and to a large extent the properties of negative parity states can be reasonably well described in terms of the octupole degree of freedom alone.

Generator coordinate method with a conjugate momentum: Application to particle number projection

We discuss an extension of the generator coordinate method (GCM) by taking simultaneously a collective coordinate and its conjugate momentum as generator coordinates. To this end, we follow the idea of the dynamical GCM (DGCM) proposed by Goeke and Reinhard. We first show that the DGCM method can be regarded as an extension of the double projection method for the center of mass motion. As an application of DGCM, we then investigate the particle number projection, for which we not only carry out an integral over the gauge angle as in the usual particle number projection but also take a linear superposition of BCS states which have different mean particle numbers. We show that the ground state energy is significantly lowered by such effect, especially for magic nuclei for which the pairing gap is zero in the BCS approximation. This suggests that the present method makes a good alternative to the variation after projection (VAP) method, as the method is much simpler than the VAP.

Low-energy dipole excitation mode in O18 with antisymmetrized molecular dynamics

Low-energy dipole (LED) excitations in $^{18}\mathrm{O}$ were investigated using a combination of the variation after $K$ projection in the framework of antisymmetrized molecular dynamics with $\ensuremath{\beta}$ constraint with the generator coordinate method. We obtained two LED states, namely, the ${1}_{1}^{\ensuremath{-}}$ state with a dominant shell-model structure and the ${1}_{2}^{\ensuremath{-}}$ state with a large $^{14}\mathrm{C}+\ensuremath{\alpha}$ cluster component. Both these states had significant toroidal dipole (TD) and compressive dipole (CD) strengths, indicating that the TD and CD modes are not separated but mixed in the LED excitations of $^{18}\mathrm{O}$. This is unlike the CD and TD modes for well-deformed nuclei such as $^{10}\mathrm{Be}$, where the CD and TD modes are generated as ${K}^{\ensuremath{\pi}}={0}^{\ensuremath{-}}$ and ${K}^{\ensuremath{\pi}}={1}^{\ensuremath{-}}$ excitations, respectively, in a largely deformed state.

Quasi-SU(3) coupling of ( 1h11/2,2f7/2 ) across the N=82 shell gap: Enhanced E2 collectivity and shape evolution in Nd isotopes

The essence of the quasi-SU(3) coupling scheme suggested by Zuker et al. is that the high-$j$ intruder orbit alone is insufficient to induce required large collectivity, and it is necessary for the high-$j$ orbit to correlate through the quadrupole interaction with another higher-lying orbit with $\mathrm{\ensuremath{\Delta}}j=2$. To extend this idea to the medium-heavy mass region, we investigate the systematics of energy levels and $B(E2)$ values for Sn, Te, Xe, Ba, Ce, Nd, and Sm isotopes by applying the recently proposed realistic PMMU shell model. The calculations are performed by using the projected Hartree-Fock-Bogolyubov plus generator coordinate method in the model space of ($1{g}_{9/2},1{g}_{7/2},2{d}_{5/2},2{d}_{3/2},3{s}_{1/2},1{h}_{11/2}$). The calculations describe well the experimental data over a wide range of nuclei. However, it is found that, in some nuclei close to the neutron midshell, $^{124,126}\mathrm{Ce}$, $^{130,132}\mathrm{Nd}$, and $^{134}\mathrm{Sm}$, the calculated $B(E2;{2}_{1}^{+}\ensuremath{\rightarrow}{0}_{1}^{+})$ values underestimate the data considerably. This problem can be resolved by inclusion of the $2{f}_{7/2}$ orbit into the model space, which, with $1{h}_{11/2}$, forms a quasi-SU(3) coupling scheme. It is shown that the $QQ$ interaction between the SU(3)-partner $1{h}_{11/2}\text{\ensuremath{-}}2{f}_{7/2}$ is a driving force for enhanced $E2$ collectivity in the above nuclei and is responsible for the shape evolution in Nd isotopes. In addition, the participation of the $2{f}_{7/2}$ orbit in the model space favors prolate shape in the ground state of $N\ensuremath{\le}76$ isotopes, thus subverting the oblate result from the same calculation but without $2{f}_{7/2}$.

Root Mean Square Radius of 6Li Nucleus using Cluster Model Wavefunction

The calculation of root mean square nuclear charge radius is one of the most important nuclear parameters regarding the size and structure of the nucleus. In this paper calculations for root mean square (r.m.s.) radius of the ground state of 6Li nucleus using high energy electron scattering is presented. The initial work involves writing first the cluster model wavefunction employing the resonating group method, generator coordinate method and complex generator coordinate technique. The wavefunction is written with definite parity, spin, total angular momentum and relative motion between the alpha cluster and deuteron cluster and the center-of-mass of the two clusters. The application of complex generator coordinate technique transforms the cluster model wavefunction into antisymmetrized products of single particle wavefunction written in terms of single particle co-ordinates, the centerof-mass coordinates, parameter coordinates and generator coordinates. The width parameters of alpha and deuteron clusters are adjusted to obtain predictions close to experimental values.

Use of Microscopic Theoretical Methods for The Construction of The Nuclear Wavefunction of 7Li

Microscopic theoretical studies of scattering and reaction problems for light nuclei have been extensively carried out using resonating group method. In this paper we have used the nuclear cluster model, the resonating group method, the generator coordinate method and complex generator coordinate technique for the construction of microscopic antisymmetrized nuclear wavefunction of 7Li nucleus. This wavefunction can be further used to calculate the structural properties of the nucleus. The 7Li nucleus in ground state is considered as a nuclear system consisting of three clusters namely an alpha cluster, a deuteron cluster and a neutron cluster. We have chosen spatial, spin and isospin function of cluster internal functions. The arguments of internal wavefunction include the parameter coordinates. These parameters can be adjusted to some extent to obtain predictions close to experimental results. The wavefunction is written using shell model with definite parity and angular momentum. The complex generator coordinate technique allows this wavefunction to write it as an antisymmetrized product of seven single particle functions after inclusion of the wavefunction for the center- of -mass motion

Generation, contraction, and polarisation of Gaussian basis sets for atomic and molecular calculations using the generator coordinate method with polynomial discretisation: atoms from Na through Cl

The pGCHF basis sets for second-row atoms were generated using the CG method based on polynomial integral expansion to discretise the GWHF equations. These new basis sets can achieve competitive accuracy while describing atomisation energies.

Microscopic description of quadrupole-octupole coupling in actinides with the Gogny-D1M energy density functional

The interplay between quadrupole and octupole degrees of freedom is discussed in a series of U, Pu, Cm and Cf isotopes both at the mean-field level and beyond. In addition to the static Hartree–Fock–Bogoliubov approach, dynamical beyond-mean-field correlations are taken into account via both parity restoration and symmetry-conserving generator coordinate method calculations based on the parametrization D1M of the Gogny energy density functional. Physical properties such as correlation energies, negative-parity excitation energies as well as reduced transition probabilities B(E1) and B(E3) are discussed in detail and compared with the available experimental data. It is shown that, for the studied nuclei, the quadrupole-octupole coupling is weak and to a large extent the properties of negative parity states can be reasonably well described in terms of the octupole degree of freedom alone.

Predominance of Triaxial Shapes in Transitional Super-Heavy Nuclei: Ground-State Deformation and Shape Coexistence along the Flerovium (Z=114) Chain of Isotopes.

We present the first triaxial beyond-mean-field study of even-even super-heavy nuclei. Calculations for the even Flerovium isotopes towards the supposed N=184 neutron shell closure were performed using the effective finite-range density-dependent Gogny force. They include the restoration of the particle-number and angular-momentum symmetries and the mixing of different shapes using the generator coordinate method. The importance of the γ degree of freedom is highlighted by comparing the triaxial to axial-symmetric calculations performed within the same framework. For the three even Fl isotopes between the prolate ^{288}Fl and the oblate ^{296}Fl triaxial ground-state shapes are predicted, whereas axial-symmetric calculations suggest a sharp prolate-oblate shape transition between ^{290}Fl and ^{292}Fl. A novel type of shape coexistence, namely, that between two different triaxial shapes, is predicted to occur in ^{290}Fl. Finally, the existence of a neutron shell closure at N=184 is confirmed, while no evidence is found for Z=114 being a proton magic number.

Microscopic calculations for Be isotopes within real-time evolution method

The Be isotopes with $$\alpha $$ + $$\alpha $$ + valence neutrons clustering structure are studied within the Real-time Evolution Method (REM). By solving the equation-of-motion (EOM) of the Gaussian wave packets involving the complex spatial coordinates and spin variables, various molecular configurations are obtained and then are superposed in the Generator Coordinate Method (GCM). The obtained energy spectra from Be isotopes are better than those from other cluster models and also basically consistent with experiments. It is found the real-time evolution method provides us with an effective way to choose basis wave functions for dealing with the multi-neutron molecular structures of Be isotopes.

Time-dependent generator coordinate method for many-particle tunneling

It has been known that the time-dependent Hartree-Fock (TDHF) method, or the time-dependent density functional theory (TDDFT), fails to describe many-body quantum tunneling. We overcome this problem by superposing a few time-dependent Slater determinants with the time-dependent generator coordinate method (TDGCM). We apply this method to scattering of two α particles in one dimension, and demonstrate that the TDGCM method yields a finite tunneling probability even at energies below the Coulomb barrier, at which the tunneling probability is exactly zero in the TDHF. This is the first case in which a many-particle tunneling is simulated with a microscopic real-time approach.

The Time-Dependent Generator Coordinate Method in Nuclear Physics

The emergence of collective behaviors and the existence of large amplitude motions are both central features in the fields of nuclear structure and reactions. From a theoretical point of view, describing such phenomena requires increasing the complexity of the many-body wavefunction of the system to account for long-range correlations. One of the challenges, when going in this direction, is to keep the approach tractable within our current computational resources while gaining a maximum of predictive power for the phenomenon under study. In the Generator Coordinate Method (GCM), the many-body wave function is a linear superposition of (generally non-orthogonal) many-body states (the generator states) labeled by a few collective coordinates. Such a method has been widely used in structure studies to restore the symmetries broken by single-reference approaches. In the domain of reactions, its time-dependent version (TDGCM) has been developed and applied to predict the dynamics of heavy-ion collisions or fission where the collective fluctuations play an essential role. In this review, we present the recent developments and applications of the TDGCM in nuclear reactions. We recall the formal derivations of the TDGCM and its most common approximate treatment, the Gaussian Overlap Approximation. We also emphasize the Schr\"odinger Collective-Intrinsic Model (SCIM) variant focused on the inclusion of quasiparticle excitations into the description. Finally, we highlight several exploratory studies related to a TDGCM built on time-dependent generator states.

AbInitio Treatment of Collective Correlations and the Neutrinoless Double Beta Decay of ^{48}Ca.

Working with Hamiltonians from chiral effective field theory, we develop a novel framework for describing arbitrary deformed medium-mass nuclei by combining the in-medium similarity renormalization group with the generator coordinate method. The approach leverages the ability of the first method to capture dynamic correlations and the second to include collective correlations without violating symmetries. We use our scheme to compute the matrix element that governs the neutrinoless double beta decay of ^{48}Ca to ^{48}Ti, and find it to have the value 0.61, near or below the predictions of most phenomenological methods. The result opens the door to abinitio calculations of the matrix elements for the decay of heavier nuclei such as ^{76}Ge, ^{130}Te, and ^{136}Xe.

Microscopic description of induced fission dynamics with nuclear energy density functionals

Static and dynamic aspects of the fission process are analyzed in a self-consistent framework based on energy density functionals. Multidimensionally constrained mean-field calculations in the collective space determine the potential energy surface of the fissioning nucleus, the scission line, the single-nucleon wave functions, energies, and occupation probabilities. Induced fission dynamics is described using the time-dependent generator coordinate method in the Gaussian overlap approximation. The position of the scission line is analyzed as a function of the strength of the pairing interaction, as well as the effect of static pairing correlations on charge yields and total kinetic energy of fission fragments.

A stochastic microscopic approach to the $^{10}{\rm Be}$ and $^{11}{\rm Be}$ nuclei

Abstract We use a microscopic multicluster model to investigate the structure of $^{10}{\rm Be}$ and of $^{11}{\rm Be}$. These nuclei are described by $\alpha +\alpha+n+n$ and $\alpha +\alpha+n+n+n$ configurations, respectively, within the Generator Coordinate Method (GCM). The 4- and 5-body models raise the problem of a large number of generator coordinates (6 for $^{10}{\rm Be}$ and 9 for $^{11}{\rm Be}$), which requires specific treatment. We address this issue by using the Stochastic Variational Method (SVM), which is based on an optimal choice of the basis functions, generated randomly. The model provides good energy spectra for low-lying states of both nuclei. We also compute rms radii and densities, as well as electromagnetic transition probabilities. We analyze the structure of $^{10}{\rm Be}$ and of $^{11}{\rm Be}$ by considering energy curves, where one of the generator coordinates is fixed during the minimization procedure.

Cluster structures and monopole transitions of C14

Cluster structures of $^{14}$C were investigated with a method of antisymmetrized molecular dynamics (AMD) combined with a $3\alpha+nn$ cluster model while focusing on the monopole excitations and linear-chain $3\alpha$ band. Variation after parity and angular momentum projections was performed in the AMD framework, and the generator coordinate method was applied to take into account various $3\alpha+nn$ cluster configurations. Energy spectra and monopole and $E2$ transition strengths of $0^+$, $2^+$, and $4^+$ states were calculated to assign band structures. The $0^+_3$ state with remarkable monopole transition was obtained as a vibrational mode of the triangle $3\alpha$ configuration. In addition, the linear-chain $3\alpha$ band from the band-head $0^+_4$ state was obtained. $^{10}$Be$+\alpha$ decay widths of $0^+$, $2^+$, and $4^+$ states were evaluated. $\alpha$ inelastic scattering off $^{14}$C was also investigated by the microscopic coupled-channel calculation with the $g$-matrix folding model to propose possible observation of the $0^+_3$ state via $\alpha$ scattering experiments.

Decay widths at the scission point in nuclear fission

An outstanding problem in the theory of nuclear fission is understanding the Hamiltonian dynamics at the scission point. Here we apply the Generator Coordinate Method to calculate decay widths for pre-scission configurations into the two-fragment continuum. Transitions that are allowed under diabatic dynamics can have widths up to several MeV. For non-diabatic decays through the pairing interaction, typical widths to a specific final state channel are 2-3 orders of magnitude smaller. The nucleus U-236 is taken as a representative example in the calculations.

Variational approximations to exact solutions in shell-model valence spaces: Calcium isotopes in the pf shell

We study the performance of self-consistent mean-field and beyond-mean-field approximations in shell-model valence spaces. In particular, Hartree-Fock-Bogolyubov, particle-number variation after projection and projected generator coordinate methods are applied to obtain ground-state and excitation energies for even-even and odd-even Calcium isotopes in the pf-shell. The standard (and non-trivial) KB3G nuclear effective interaction has been used. The comparison with the exact solutions -- provided by the full diagonalization of the Hamiltonian -- shows an outstanding agreement when particle-number and angular-momentum restorations are performed and both quadrupole and neutron-neutron pairing degrees of freedom are explicitly explored as collective coordinates.

Union of rotational and vibrational modes in generator-coordinate-type calculations, with application to neutrinoless double- β decay

Good many-body methods for medium and heavy nuclei are important. Here we combine ideas from standard generator-coordinate methods (GCM) and the so-called Monte Carlo shell model, and set forth a novel approach: starting from a mean-field solution (Hartree-Fock-Bogoliubov), we create a set of non-orthogonal basis states, by using low-lying vibrational quasiparticleTamm-Dancoff modes, and then project onto states of good angular momentum and particle number. The results we benchmark against full shell model calculations. Even with just a few such modes we find improvement over standard GCM calculations in excitation spectra. We also find significant improvement in $0\nu\beta\beta$ nuclear matrix elements.