Generator Coordinate Method Calculation Research Articles

Optimization of the generator coordinate method with machine-learning techniques for nuclear spectra and neutrinoless double- β decay: Ridge regression for nuclei with axial deformation

Background: The generator coordinate method (GCM) is an important tool of choice for modeling large-amplitude collective motion in atomic nuclei. Recently, it has attracted increasing interest as it can be exploited to extend ab initio methods to the collective excitations of medium-mass and heavy deformed nuclei, as well as the nuclear matrix elements (NME) of candidates for neutrinoless double-beta ($0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$) decay.Purpose: The computational complexity of the GCM increases rapidly with the number of collective coordinates. It imposes a strong restriction on the applicability of the method. We aim to exploit statistical machine-learning (ML) algorithms to speed up GCM calculations and ultimately provide a more efficient description of nuclear energy spectra and other observables such as the NME of $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ decay without loss of accuracy.Method: In this work, we propose a subspace-reduction algorithm that employs optimal statistical ML models as surrogates for exact quantum-number projection calculations for norm and Hamiltonian kernels. The model space of the original GCM is reduced to a subspace relevant for nuclear low energy spectra and the NME of ground state to ground state $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ decay based on the orthogonality condition (OC) and the energy-transition-orthogonality procedure (ENTROP), respectively. Nuclear energy spectra are determined by the GCM through the configuration mixing within this subspace. For simplicity, the polynomial ridge regression (RR) algorithm is used to learn the norm and Hamiltonian kernels of axially deformed configurations. The efficiency and accuracy of this algorithm are illustrated for $^{76}\mathrm{Ge}$ and $^{76}\mathrm{Se}$ by comparing results obtained using the optimal RR models to direct GCM calculations. The nonrelativistic Gogny force D1S and relativistic energy density functional PC-PK1, a valence-space shell-model Hamiltonian, and a modern nuclear interaction derived from chiral effective field theory are employed.Results: The low-lying energy spectra of $^{76}\mathrm{Ge}$ and $^{76}\mathrm{Se}$, as well as the $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$-decay NME between their ground states, are computed. The results show that the performance of the $\mathrm{GCM}+\text{OC/ENTROP}+\mathrm{RR}$ is more robust than that of the $\mathrm{GCM}+\mathrm{RR}$ alone, and the former can reproduce the results of the original GCM calculation accurately with a significantly reduced computational cost.Conclusions: Statistical ML algorithms, when implemented properly, can accelerate GCM calculations without loss of accuracy. In applications with axially deformed states, the computation time can be reduced by a factor of three to nine for energy spectra and NMEs, respectively. This factor is expected to increase significantly with the number of employed generator coordinates.

Operator overlaps in harmonic oscillator bases with different oscillator lengths

We apply a formalism recently developed to carry out Generator Coordinate Method calculations using a set of Hartree- Fock- Bogoliubov wave functions, where each of the members of the set can be expanded in an arbitrary basis. In this paper it is assumed that the HFB wave functions are expanded in Harmonic Oscillator (HO) bases with different oscillator lengths. General expressions to compute the required matrix elements of arbitrary operators are given. The application of the present formalism to the case of fission is illustrated with an example.

Formulation of the generator coordinate method with arbitrary bases

The existing formalism used to compute the operator overlaps necessary to carry out generator coordinate method calculations using a set of Hartree- Fock- Bogoliubov wave functions, is generalized to the case where each of the HFB states are expanded in different arbitrary bases spanning different sub-space of the Hilbert space.

Microscopic calculations of $$^6$$He and $$^6$$Li with real-time evolution method

The low-lying cluster states of $$^6$$ He ( $$\alpha $$ +n+n) and $$^6$$ Li ( $$\alpha $$ +n+p) are calculated by the real-time evolution method (REM) which generates basis wave functions for the generator coordinate method (GCM) from the equation of motion of Gaussian wave packets. The $$0^+$$ state of $$^6$$ He as well as the $$1^+$$ , $$0^+$$ and $$3^+$$ states of $$^6$$ Li are calculated as a benchmark. We also calculate the root-mean-square (r.m.s.) radii of the point matter, the point proton, and the point neutron of these states, particularly for the study of the halo characters of these two nuclei. It is shown that REM can be one constructive way for generating effective basis wave functions in GCM calculations.

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.

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.

Generator-coordinate reference states for spectra and 0νββ decay in the in-medium similarity renormalization group

We use a reference state based on symmetry-restored states from deformed mean-field or generator-coordinate-method (GCM) calculations in conjunction with the in-medium similarity-renormalization group (IMSRG) to compute spectra and matrix elements for neutrinoless double-beta ($0\nu\beta\beta$) decay. Because the decay involves ground states from two nuclei, we use evolved operators from the IMSRG in one nucleus in a subsequent GCM calculation in the other. We benchmark the resulting IMSRG+GCM method against complete shell-model diagonalization for both the energies of low-lying states in $^{48}$Ca and $^{48}$Ti and the $0\nu\beta\beta$ matrix element for the decay of $^{48}$Ca, all in a single valence shell. Our approach produces better spectra than either the IMSRG with a spherical-mean-field reference or GCM calculations with unevolved operators. For the $0\nu\beta\beta$ matrix element the improvement is slight, but we expect more significant effects in full ab-initio calculations.

Experimental study of the $$^{9}\hbox {Li}$$ 9 Li breakup reaction on Pb target

The breakup reaction of $$^{9}\hbox {Li}$$ on a Pb target has been measured at 32.7 MeV/nucleon for the first time. Two peaks of $$^{6}\hbox {He}+\hbox {t}$$ coincident fragments at 9.8 and 12.5 MeV were observed and agreed with the results of the generator coordinate method calculation. The experiment was carried out thanks to a specially arranged detection system around zero degrees at the Heavy Ion Research Facility in Lanzhou, Radioactive Ion Beam Line in Lanzhou (HIRFL–RIBLL).

Breathing-like excited state of the Hoyle state inC12

The existence of the $0_3^+$ and $0_4^+$ states around 10 MeV excitation energy in ${^{12}{\rm C}}$ is confirmed by a fully microscopic 3$\alpha$ cluster model. Firstly, a GCM (generator coordinate method) calculation is performed by superposing optimized 2$\alpha$+$\alpha$ THSR (Tohsaki-Horiuchi-Schuck-R\"{o}pke) wave functions with the radius-constraint method. The obtained two excited $0^+$ states above the Hoyle state are consistent with the recently observed states by experiment. Secondly, a variational calculation using the single 2$\alpha$+$\alpha$ THSR wave function orthogonalized to the ground and Hoyle states is made and it also supports the existence of the $0_3^+$ state obtained by the GCM calculation. The analysis of the obtained $0_3^+$ state is made by studying its 2$\alpha$-$\alpha$ reduced width amplitude, its 2$\alpha$ correlation function, and the large monopole matrix element between this state and the Hoyle state, which shows that this $0_3^+$ state is a breathing-like excited state of the Hoyle state. This character of the $0_3^+$ state is very different from the $0_4^+$ state which seems to have a bent-arm 3$\alpha$ structure.

Microscopic description of quadrupole collectivity in neutron-rich nuclei across the N = 126 shell closure

The quadrupole collectivity in Nd, Sm, Gd, Dy, Er, Yb, Hf and W nuclei with neutron numbers 122 ≤ N ≤ 156 is studied, both at the mean field level and beyond, using the Gogny energy density functional. Besides the robustness of the N = 126 neutron shell closure, it is shown that the onset of static deformations in those isotopic chains with increasing neutron number leads to an enhanced stability and further extends the corresponding two-neutron drip lines far beyond what could be expected from spherical calculations. Independence of the mean-field predictions with respect to the particular version of the Gogny energy density functional employed is demonstrated by comparing results based on the D1S and D1M parameter sets. Correlations beyond mean field are taken into account in the framework of the angular momentum projected generator coordinate method calculation. It is shown that N = 126 remains a robust neutron magic number when dynamical effects are included. The analysis of the collective wave functions, average deformations and excitation energies indicate that, with increasing neutron number, the zero-point quantum corrections lead to dominant prolate configurations in the 0 +1 , 0 +2 , 2 +1 and 2 +2 states of the studied nuclei. Moreover, those dynamical deformation effects provide an enhanced stability that further supports the mean-field predictions, corroborating a shift of the r-process path to higher neutron numbers. Beyond mean-field calculations provide a smaller shell gap at N = 126 than the mean-field one in good agreement with previous theoretical studies. However, the shell gap still remains strong enough in the two-neutron drip lines.

Deformation and clustering in excited states of 42Ca

The coexistence of various low-lying deformed states in |$^{42}$|Ca and the |$\alpha $|–|$^{38}$|Ar correlations in those deformed states have been investigated using deformed-basis antisymmetrized molecular dynamics. Wave functions of the low-lying states are obtained via parity and angular momentum projections and the generator coordinate method (GCM). Basis wave functions of the GCM calculation are obtained via energy variations with constraints on the quadrupole deformation parameter |$\beta $| and the distance between |$\alpha $| and the |$^{38}$|Ar clusters. The rotational bands built on the |$J^\pi = 0_2^+$| (1.84 MeV) state as well as the |$J^\pi = 0_3^+$| (3.30 MeV) state are both reproduced. The coexistence of two additional |$K^\pi = 0^+$| rotational bands is predicted; one band is shown to be built on the |$J^\pi = 0_3^+$| state. Members of the ground-state band and the rotational band built on the |$J^\pi = 0_3^+$| state contain |$\alpha $|–|$^{38}$|Ar cluster structure components.

Cluster states and monopole transitions inO16

Cluster structures and monopole transitions in positive parity states of $^{16}$O were investigated based on the generator coordinate method calculation of an extended $^{12}$C+$\alpha$ cluster model. The ground and excited states of a $^{12}$C cluster are taken into account by using $^{12}$C wave functions obtained with the method of antisymmetrized molecular dynamics. The $0^+_2$ state of $^{16}$O and its rotational members, the $2^+_1$ and $4^+_1$ states are described well by the cluster states dominated by the $^{12}$C($0^+_1$)+$\alpha$ structure. Above the $^{12}$C($0^+_2$)+$\alpha$ threshold energy, we obtained a $0^+$ state having the $^{12}$C($0^+_2$)+$\alpha$ cluster structure, which is considered to be a candidate for the $4\alpha$ cluster gas state. The band structures were discussed based on the calculated $E2$ transition strength. Isoscalar Monopole excitations from the ground state were also discussed.

Quadrupole-octupole coupling in the light actinides

The relevance of coupling of quadrupole and octupole collective degrees of freedom in physical observables is explored in calculations with the Gogny force for the light radon, radium, and thorium isotopes. The results of the generator coordinate method calculations for the properties of negative parity states show an improvement over the traditional ones that consider just the octupole moment.

Cluster correlations in largely deformed states

The coexistence of various deformed states in 42Ca and α-38Ar correlations have been investigated using deformed-basis antisymmetrized molecular dynamics. Wavefunctions of the low-lying states are obtained via parity and angular momentum projections and the generator coordinate method (GCM). Basis wavefunctions of the GCM calculation are obtained via energy variation with constraints on the quadrupole deformation parameter β and the distance between α and 38Ar clusters. The rotational band built on the Jπ = 0+2 (1.84 MeV) state as well as the Jπ = 0+3 (3.30 MeV) state are both reproduced. The coexistence of two additional Kπ = 0+ rotational bands is predicted; one band is shown to be built on the Jπ = 0+3 state. Members of the ground-state band and the rotational band built on the Jπ = 0+3 state contain α-38Ar cluster structure components. The α-38Ar higher-nodal Jπ = 0+ state is also obtained, and the Jπ = 0+ state has large E0 transition strength from the ground state.

Evaluation of overlaps between arbitrary fermionic quasiparticle vacua

We derive an expression that allows for the unambiguous evaluation of the overlap between two arbitrary quasiparticle vacua, including its sign. Our expression is based on the Pfaffian of a skew-symmetric matrix, extending the formula recently proposed by [L. M. Robledo, Phys. Rev. C 79, 021302(R) (2009)] to the most general case, including the one of the overlap between two different blocked n-quasiparticle states for either even or odd systems. The powerfulness of the method is illustrated for a few typical matrix elements that appear in realistic angular-momentum-restored Generator-Coordinate Method calculations when breaking time-reversal invariance and using the full model space of occupied single-particle states.

Triaxial superdeformation inAr40

Superdeformed (SD) states in $^{40}\mathrm{Ar}$ have been studied using the deformed-basis antisymmetrized molecular dynamics. Low-energy states were calculated by the parity and angular momentum projection and the generator coordinate method (GCM). Basis wave functions were obtained by the energy variation with a constraint on the quadrupole deformation parameter $\ensuremath{\beta}$, while other quantities such as triaxiality $\ensuremath{\gamma}$ were optimized by the energy variation. By the GCM calculation, an SD band was obtained just above the ground-state band. The SD band involves a ${K}^{\ensuremath{\pi}}={2}^{+}$ side band due to the triaxiality. The calculated electric quadrupole transition strengths of the SD band reproduce the experimental values appropriately. Triaxiality is significant for understanding low-lying states.

Microscopic investigation of the 12Be spectroscopy

We analyze the 12Be spectrum with two microscopic methods. Microscopic cluster calculations are performed within the Generator Coordinate Method (GCM) framework. An extended two-cluster model which includes the He 8 ( 0 + , 2 + ) + α , and He 6 ( 0 + , 2 + ) + He 6 ( 0 + , 2 + ) channels is developed. It represents an extension of a previous work by Descouvemont and Baye where only the 0 + ground states of 8He and 6He were taken into account. The GCM calculations lead to a good description of the molecular band suggested by Freer et al. and by Saito et al. The 0 2 + isomeric state and the 2 2 + state are also well described by the GCM. New bands of positive and negative parities are proposed. This approach is complemented by No-Core Shell Model (NCSM) calculations, in order to clarify the structure of the ground state and the hypothesis of an associated rotational band. The GCM and NCSM calculations do not support the existence of a band built on the ground state.

Clustering and triaxial deformations ofCa40

We have studied the positive-parity states of $^{40}\mathrm{Ca}$ using antisymmetrized molecular dynamics (AMD) and the generator coordinate method (GCM). Imposing two different kinds of constraints on the variational calculation, we have found various kinds of $^{40}\mathrm{Ca}$ structures such as a deformed-shell structure, as well as $\ensuremath{\alpha}\text{\ensuremath{-}}$$^{36}\mathrm{Ar}$ and $^{12}\mathrm{C}$-$^{28}\mathrm{Si}$ cluster structures. After the GCM calculation, we obtained a normal-deformed band and a superdeformed band together with their side bands associated with triaxial deformation. The calculated $B(E2)$ values agreed well with empirical data. It was also found that the normal-deformed and superdeformed bands have non-negligible $\ensuremath{\alpha}\text{\ensuremath{-}}$$^{36}\mathrm{Ar}$ cluster and $^{12}\mathrm{C}$-$^{28}\mathrm{Si}$ cluster components, respectively. This leads to the presence of an $\ensuremath{\alpha}\text{\ensuremath{-}}$$^{36}\mathrm{Ar}$ higher nodal band occurring above the normal-deformed band.

Generator coordinate method calculations of one-nucleon removal reactions on40Ca

An approach to the generator coordinate method (GCM) using Skyrme-type effective forces and Woods-Saxon construction potential is applied to calculate the single-particle proton and neutron overlap functions in ${}^{40}\mathrm{Ca}.$ The relationship between the bound-state overlap functions and the one-body density matrix has been used. These overlap functions are applied to calculate the cross sections of one-nucleon removal reactions such as ${(e,e}^{\ensuremath{'}}p),$ $(\ensuremath{\gamma},p),$ and $(p,d)$ on ${}^{40}\mathrm{Ca}$ on the same theoretical footing. A consistent description of data for the different reactions is achieved. The shapes of the experimental cross sections for transitions to the ${3/2}^{+}$ ground state and the first ${1/2}^{+}$ excited state of the residual nuclei are well reproduced by the overlap functions obtained within the GCM. An additional spectroscopic factor accounting for correlations not included in the overlap function must be applied to the calculated results to reproduce the size of the experimental cross sections.

Capability of the GCM calculation with spin and particle number projections for single j-shell model

Generator coordinate method (GCM) combined with particle number and triaxial angular momentum (spin) projections is applied to single j-shell model with the monopole-pairing interaction (MPI) and the quadrupole-quadrupole interaction (QQI). The GCM calculation yields excellent results, e.g. the GCM with the QQI reproduces exact energy spectra of all the excited states for 4-hole (also particle) systems in the j = 11 2 15 2 and 19 2 shells; and the GCM with the MPI plus QQI, which requires simultaneously the particle number projection, reproduces exact energy spectra for the 6-hole (also particle) systems except for the spin I = 1 state in the j = 13 2 shell. it turns out that components of triaxial deformation contribute much to reproduce correct physical states, so that the triaxial spin projection is essential in the present GCM.