Zero-range Force Research Articles

Microscopic study on low-energy quadrupole states in Ni isotope chain atomic nuclei

This work mainly investigates the properties of the low-energy quadrupole strength in Ni isotopes, especially the evolution of the pygmy quadrupole states with the increase of neutron number. And the effect of shell evolution on the pygmy resonance is also discussed in detail. Based on the Skyrme Hartree-Fock+Bardeen-Cooper-Schrieffer (HF+BCS) theory and the self-consistent quasiparticle random phase approximation (RPA) method, the evolution of the nucleus of the nickel isotope chain with the increase of neutron number is studied. and In the calculations, three effective Skyrme interactions, namely SGII, SLy5 and SKM*, and a density-dependent zero-range type force are adopted. The properties of the first 2<sup>+</sup> state in Ni isotopes are studied.)The calculated excited energy values of the first 2<sup>+</sup> states can accurately accord with the experimental values, and the SGII and SLy5 can well describe the reduced electric transition probabilities for <inline-formula><tex-math id="M3">\begin{document}$^{58-68}{\rm{Ni}}$\end{document}</tex-math></inline-formula>. It is found that the energy value of the first 2<sup>+</sup> state for <inline-formula><tex-math id="M4">\begin{document}$^{68}{\rm{Ni}}$\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$^{78}{\rm{Ni}}$\end{document}</tex-math></inline-formula> are obviously high than those of other states, reflecting the obvious shell effect. In addition to the first 2<sup>+</sup> states, pygmy quadrupole states between 3 and 5 MeV with relatively large electric transition probabilities are evidently found for <inline-formula><tex-math id="M6">\begin{document}$^{70-76}{\rm{Ni}}$\end{document}</tex-math></inline-formula> in the isoscalar quadruple strength distribution [see Fig. (b)]. The pygmy quadrupole states have the energy values decreasing with the number of neutrons increasing, but their strengths increase gradually. Therefore, they are more sensitive to the change in the shell structure. This is due to the fact that the gradual filling of the neutron level 1<inline-formula><tex-math id="M7">\begin{document}$g_{9/2}$\end{document}</tex-math></inline-formula> has a significant effect on the pygmy quadrupole states of <inline-formula><tex-math id="M8">\begin{document}$^{70-76}{\rm{Ni}}$\end{document}</tex-math></inline-formula>, and it leads to switching from proton-dominated excitations to neutron-dominated ones. The pygmy quadrupole states for <inline-formula><tex-math id="M9">\begin{document}$^{70-76}{\rm{Ni}}$\end{document}</tex-math></inline-formula> are sensitive to the proton and neutron shell gaps, so they can provide the information about the shell evolution in neutron-rich nuclei.

STUDY OF OCTUPOLE DEFORMATION OF NUCLEI IN CHAIN OF ISOTONS IN THE REGION OF ACTINIDES IN THE MEAN FIELD APPROXIMATION

Calculations of the properties of even-even isotones N = 134 of nuclei with Z = 86–96 were carried out in the Hartree-Fock-Bogolyubov approximation, taking into account the axial symmetry of nuclei with forces Skyrme (SkM*, SLy4). Pairing of nucleons in nuclei is described by zero-range pairing forces of mixed type with different sets of pairing force constants. In the calculations we used constrained conditions on the parameters of the quadrupole and octupole deformations of the nuclei. It is shown that for the considered chain of isotones the octupole deformation of nuclei strongly depends on the choice of parameters of the nucleon pairing force and weakly depends on the type of parameterization of the Skyrme forces used in this study.

Sensitivity impacts owing to the variations in the type of zero-range pairing forces on the fission properties using the density functional theory

Sensitivity impacts owing to the variations in the type of zero-range pairing forces on the fission properties using the density functional theory

Zero-Range Hamiltonian for a Bose Gas with an Impurity

We study the Hamiltonian for a system of N identical bosons interacting with an impurity, i.e., a different particle, via zero-range forces in dimension three. It is well known that, following the standard approach, one obtains the Ter-Martirosyan Skornyakov Hamiltonian which is unbounded from below. In order to avoid such instability problem, we introduce a three-body force acting at short distances. The effect of this force is to reduce to zero the strength of the zero-range interaction between two particles, i.e., the impurity and a boson, when another boson approaches the common position of the first two particles. We show that the Hamiltonian defined with such regularized interaction is self-adjoint and bounded from below if the strength of the three-body force is sufficiently large. The method of the proof is based on a careful analysis of the corresponding quadratic form.

Three-Body Hamiltonian with Regularized Zero-Range Interactions in Dimension Three

We study the Hamiltonian for a system of three identical bosons in dimension three interacting via zero-range forces. In order to avoid the fall to the center phenomenon emerging in the standard Ter-Martirosyan–Skornyakov (TMS) Hamiltonian, known as Thomas effect, we develop in detail a suggestion given in a seminal paper of Minlos and Faddeev in 1962 and we construct a regularized version of the TMS Hamiltonian which is self-adjoint and bounded from below. The regularization is given by an effective three-body force, acting only at short distance, that reduces to zero the strength of the interactions when the positions of the three particles coincide. The analysis is based on the construction of a suitable quadratic form which is shown to be closed and bounded from below. Then, domain and action of the corresponding Hamiltonian are completely characterized and a regularity result for the elements of the domain is given. Furthermore, we show that the Hamiltonian is the norm resolvent limit of Hamiltonians with rescaled non-local interactions, also called separable potentials, with a suitably renormalized coupling constant.

Causality and dimensionality in geometric scattering

The scattering matrix which describes low-energy, non-relativistic scattering of spin-1/2 fermions interacting via finite-range potentials can be obtained from a geometric action principle in which space and time do not appear explicitly [1]. In the case of zero-range forces, causality leads to constraints on scattering trajectories in the geometric picture. The effect of spatial dimensionality is also investigated by considering scattering in two and three dimensions. In the geometric formulation it is found that dimensionality is encoded in the phase of the harmonic potential that appears in the geometric action.

Systematic application of the M3Y NN forces for describing the capture process in heavy-ion collisions involving deformed target nuclei

We present results of a systematic study of the capture process through the barrier penetration model. The nucleus-nucleus interaction potential is calculated using the double-folding model (DFM) with the M3Y Paris $\mathit{NN}$ forces. The nucleon densities entering the model are generated from the experimental three-parameter Fermi charge densities. The DFM has been extended to the case of deformed target nuclei. It is shown that the density-dependent M3Y $\mathit{NN}$ forces with the finite range exchange part can be mimicked successfully by the zero-range density-independent forces. The latter option significantly reduces the required computer time. For the nucleon densities and target nuclei deformations, we employ the values from the commonly used data bases. Thus, we do not vary any parameters to reach a better agreement with the data. The resulting cross sections are compared with data for 20 reactions with the product of the charge numbers ${Z}_{P}{Z}_{T}$ ranging from 216 up to 2576. We discuss the opinion met in the literature that the M3Y $\mathit{NN}$ forces provide a poorer description of the capture cross sections in heavy-ion collisions in comparison to the $\mathit{NN}$ forces coming from the relativistic mean-field approach. Our calculations show that the M3Y $\mathit{NN}$ forces give an agreement with the data which is not perfect yet is not worse than the one resulting from the relativistic mean-field approach $\mathit{NN}$ forces.

Probing nuclear forces beyond the nuclear drip line: the cases of $$^{16}$$F and $$^{15}$$F

The unbound proton-rich nuclei $^{16}$F and $^{15}$F are investigated experimentally and theoretically. Several experiments using the resonant elastic scattering method were performed at GANIL with radioactive beams to determine the properties of the low lying states of these nuclei. Strong asymmetry between $^{16}$F-$^{16}$N and $^{15}$F-$^{15}$C mirror nuclei is observed. The strength of the $nucleon-nucleon$ effective interaction involving the loosely bound proton in the $s_{1/2}$ orbit is significantly modified with respect to their mirror nuclei $^{16}$N and $^{15}$C. The reduction of the effective interaction is estimated by calculating the interaction energies with a schematic zero-range force. It is found that, after correcting for the effects due to changes in the radial distribution of the single-particle wave functions, the mirror symmetry of the $n-p$ interaction is preserved between $^{16}$F and $^{16}$N, while a difference of 63\% is measured between the $p-p$ versus $n-n$ interactions in the second excited state of $^{15}$F and $^{15}$C nuclei. Several explanations are proposed.

Neutron-proton pairing in nuclear matter

The self-energy effect on the neutron-proton (np) pairing gap is investigated up to the third order within the framework of the extend Bruecker-Hartree-Fock (BHF) approach combined with the BCS theory. The self-energy up to the second-order contribution turns out to reduce strongly the effective energy gap, while the \emph{renormalization} term enhances it significantly. In addition, the effect of the three-body force on the np pairing gap is shown to be negligible. To connect the present results with the np pairing in finite nuclei, an effective density-dependent zero-range pairing force is established with the parameters calibrated to the microscopically calculated energy gap.

Interplay of quasiparticle-vibration coupling and pairing correlations on β-decay half-lives

The nuclear β-decay half-lives of Ni and Sn isotopes, around the closed shell nuclei 78Ni and 132Sn, are investigated by computing the distribution of the Gamow–Teller strength using the Quasiparticle Random Phase Approximation (QRPA) with quasiparticle-vibration coupling (QPVC), based on ground-state properties obtained by Hartree–Fock–Bogoliubov (HFB) calculations. We employ the effective interaction SkM⁎ and a zero-range effective pairing force. The half-lives are strongly reduced by including the QPVC. We study in detail the effects of isovector (IV) and isoscalar (IS) pairing. Increasing the IV strength tends to increase the lifetime for nuclei in the proximity of, but lighter than, the closed-shell ones in QRPA calculations, while the effect is significantly reduced by taking into account the QPVC. On the contrary, the IS pairing mainly plays a role for nuclei after the shell closure. Increasing its strength decreases the half-lives, and the effect at QRPA and QRPA+QPVC level is comparable. The effect of IS pairing is particularly pronounced in the case of the Sn isotopes, where it turns out to be instrumental to obtain good agreement with experimental data.

Axially deformed solution of the Skyrme–Hartree–Fock–Bogolyubov equations using the transformed harmonic oscillator basis (III) hfbtho (v3.00): A new version of the program

We describe the new version 3.00 of the code hfbtho that solves the nuclear Hartree–Fock (HF) or Hartree–Fock–Bogolyubov (HFB) problem by using the cylindrical transformed deformed harmonic oscillator basis. In the new version, we have implemented the following features: (i) the full Gogny force in both particle–hole and particle–particle channels, (ii) the calculation of the nuclear collective inertia at the perturbative cranking approximation, (iii) the calculation of fission fragment charge, mass and deformations based on the determination of the neck, (iv) the regularization of zero-range pairing forces, (v) the calculation of localization functions, (vi) a MPI interface for large-scale mass table calculations. PROGRAM SUMMARYProgram title:hfbtho v3.00Program Files doi:http://dx.doi.org/10.17632/c5g2f92by3.1Licensing provisions: GPL v3Programming language: FORTRAN-95Journal reference of previous version: M.V. Stoitsov, N. Schunck, M. Kortelainen, N. Michel, H. Nam, E. Olsen, J. Sarich, and S. Wild, Comput. Phys. Commun. 184 (2013).Does the new version supersede the previous one: YesSummary of revisions:1. the Gogny force in both particle–hole and particle–particle channels was implemented;2. the nuclear collective inertia at the perturbative cranking approximation was implemented;3. fission fragment charge, mass and deformations were implemented based on the determination of the position of the neck between nascent fragments;4. the regularization method of zero-range pairing forces was implemented;5. the localization functions of the HFB solution were implemented;6. a MPI interface for large-scale mass table calculations was implemented.Nature of problem:hfbtho is a physics computer code that is used to model the structure of the nucleus. It is an implementation of the energy density functional (EDF) approach to atomic nuclei, where the energy of the nucleus is obtained by integration over space of some phenomenological energy density, which is itself a functional of the neutron and proton intrinsic densities. In the present version of hfbtho, the energy density derives either from the zero-range Skyrme or the finite-range Gogny effective two-body interaction between nucleons. Nuclear super-fluidity is treated at the Hartree–Fock–Bogolyubov (HFB) approximation. Constraints on the nuclear shape allows probing the potential energy surface of the nucleus as needed e.g., for the description of shape isomers or fission. The implementation of a local scale transformation of the single-particle basis in which the HFB solutions are expanded provide a tool to properly compute the structure of weakly-bound nuclei.Solution method: The program uses the axial Transformed Harmonic Oscillator (THO) single-particle basis to expand quasiparticle wave functions. It iteratively diagonalizes the Hartree–Fock–Bogolyubov Hamiltonian based on generalized Skyrme-like energy densities and zero-range pairing interactions or the finite-range Gogny force until a self-consistent solution is found. A previous version of the program was presented in M.V. Stoitsov, N. Schunck, M. Kortelainen, N. Michel, H. Nam, E. Olsen, J. Sarich, and S. Wild, Comput. Phys. Commun. 184 (2013) 1592–1604 with much of the formalism presented in the original paper M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring, Comput. Phys. Commun. 167 (2005) 43–63.Additional comments: The user must have access to (i) the LAPACK subroutines dsyeevr, dsyevd, dsytrf and dsytri, and their dependencies, which compute eigenvalues and eigenfunctions of real symmetric matrices, (ii) the LAPACK subroutines dgetri and dgetrf, which invert arbitrary real matrices, and (iii) the BLAS routines dcopy, dscal, dgemm and dgemv for double-precision linear algebra (or provide another set of subroutines that can perform such tasks). The BLAS and LAPACK subroutines can be obtained from the Netlib Repository at the University of Tennessee, Knoxville: http://netlib2.cs.utk.edu/.

Regularized perturbative series for the ionization potential of atomic ions

We study N-electron atoms with nuclear charge Z. It is well known that, in the cationic (Z > N) large-Z region, the atom behaves as a weakly interacting system. The anionic (Z < N) regime, on the other hand, is characterized by an instability threshold at [Formula: see text], below which the atom spontaneously emits an electron. We construct a regularized perturbative series (RPS) for the ionization potential that is based on the behaviors for large Z and Z near Zc. The large-Z expansion coefficients are analytically computed from perturbation theory, whereas the slope of the energy curve at Z = N − 1 is computed from a kind of zero-range forces theory that uses as input the electron affinity and the covalent radius of the neutral atom with N − 1 electrons. Relativistic effects in the one-particle Hamiltonian are considered at the level of first-order perturbation theory. Our RPS formula is to be used to check the consistency of the ionization potential values for atomic ions contained in the NIST database.

Cluster decay half-lives of trans-lead nuclei based on a finite-range nucleon–nucleon interaction

Nuclear cluster radioactivity is investigated using microscopic potentials in the framework of the Wentzel–Kramers–Brillouin approximation of quantum tunneling by considering the Bohr–Sommerfeld quantization condition. The microscopic cluster–daughter potential is numerically constructed in the well-established double-folding model. A realistic M3Y-Paris NN interaction with the finite-range exchange part as well as the ordinary zero-range exchange NN force is considered in the present work. The influence of nuclear deformations on the cluster decay half-lives is investigated. Based on the available experimental data, the cluster preformation factors are extracted from the calculated and the measured half lives of cluster radioactivity. Some useful predictions of cluster emission half-lives are made for emissions of known clusters from possible candidates, which may guide future experiments.

Scaling Limit Analysis of Borromean Halos

The analysis of the core recoil momentum distribution of neutron-rich isotopes of light exotic nuclei is performed within a model of the halo nuclei described by a core and two neutrons dominated by the $s-$wave channel. We adopt the renormalized three-body model with a zero-range force, that accounts for the universal Efimov physics. This model is applicable to nuclei with large two-neutron halos compared to the core size, and a neutron-core scattering length larger than the interaction range. The halo wave function in momentum space is obtained by using as inputs the two-neutron separation energy and the energies of the singlet neutron-neutron and neutron-core virtual states. Within our model, we obtain the momentum probability densities for the Borromean exotic nuclei Lithium-11 ($^{11}$Li), Berylium-14 ($^{14}$Be) and Carbon-22 ($^{22}$C). A fair reproduction of the experimental data was obtained in the case of the core recoil momentum distribution of $^{11}$Li and $^{14}$Be, without free parameters. By extending the model to $^{22}$C, the combined analysis of the core momentum distribution and matter radius suggest (i) a $^{21}$C virtual state well below 1 MeV; (ii) an overestimation of the extracted matter $^{22}$C radius; and (iii) a two-neutron separation energy between 100 and 400 keV.

Momentum distributions in light halo nuclei and structure constraints

The core recoil momentum distribution of neutron-rich isotopes of light exotic nuclei is studied within a three-body model, where the nuclei are described by a core and two neutrons, with interactions dominated by the s-wave channel. In our framework, the two-body subsystems should have large scattering lengths in comparison with the interaction range allowing to use a three-body model with a zero-range force. The ground-state halo wave functions in momentum space are obtained by using as inputs the two-neutron separation energy and the energies of the singlet neutron-neutron and neutron-core virtual states. Within our model, we obtain the momentum probability densities for the Borromean exotic nuclei 11 Li and 22 C. In the case of the core recoil momentum distribution of 11Li, a fair reproduction of the experimental data was obtained, without free parameters, considering only the two-body low-energies. By analysing the obtained core momentum distribution in face of recent experimental data, we verify that such data are constraining the 22 C two-neutron separation energy to a value between 100 and 400 keV.

A Class of Hamiltonians for a Three-Particle Fermionic System at Unitarity

We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass $ m $, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for $ m $ larger than a critical value $ m^* \simeq (13.607)^{-1} $ a self-adjoint and lower bounded Hamiltonian $ H_0 $ can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for $ m \in( m^*,m^{**}) $, where $ m^{**} \simeq (8.62)^{-1} $, there is a further family of self-adjoint and lower bounded Hamiltonians $ H_{0,\beta} $, $ \beta \in \mathbb{R} $, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.

Equation of State and Symmetry Energy at High Densities for Zero and Finite Temperatures

The equation of state of isospin asymmetric nuclear matter within the framework of the Brueckner theory is used to calculate the energy per particle for nuclear and neutron matter. Pressure and symmetry energy are also presented. Here we extended our work to include Skyrme-like zero-range density-dependent two-body forces, which could mimic three-body forces. A three-body forces are shown to be necessary for reproducing the empirical saturation properties of symmetric nuclear matter. We also studied the effect of extending the calculation to finite temperatures.

Shape coexistence and α-decay chains of 293Lv

Two recently observed 293Lv (Z = 116) α-decay chains [Eur. Phys. J. A 48, 62 (2012)] are investigated in the framework of covariant density functional theory with PC-PK1, where the pairing correlations are treated by the Bardeen-Cooper-Schrieffer method with a density-independent zerorange force. From the calculated potential energy curves, it is found that two minima always occur, with one having an almost spherical shape and the other exhibiting a large deformed prolate shape. Originating from the ground state and the shape-isomeric state of 293Lv, the two observed α-decay chains are constructed and the calculated Qα values are found to be in good agreement with the data.

Hyperspherical treatment of strongly-interacting few-fermion systems in one dimension

We examine a one-dimensional two-component fermionic system in a trap, assuming that all particles have the same mass and interact through a strong repulsive zero-range force. First we show how a simple system of three strongly interacting particles in a harmonic trap can be treated using the hyperspherical formalism. Next we discuss the behavior of the energy for the N-body system.

Regularization of zero-range effective interactions in finite nuclei

The problem of the divergences which arise in beyond mean-field calculations, when a zero-range effective interaction is employed, has not been much considered so far. Some of us have proposed, quite recently, a scheme to regularize a zero-range Skyrme-type force when it is employed to calculate the total energy, at second-order perturbation theory level, in uniform matter. Although this scheme looked promising, the extension for finite nuclei is not straightforward. We introduce such procedure in the current paper, by proposing a regularization procedure that is similar, in spirit, to the one employed to extract the so-called V_{\rm low-k} from the bare force. Although this has been suggested already by B.G. Carlsson and collaborators, the novelty of our work consists in setting on equal footing uniform matter and finite nuclei; in particular, we show how the interactions that have been regularized in uniform matter behave when they are used in a finite nucleus with the corresponding cutoff. We also address the problem of the validity of the perturbative approach in finite nuclei for the total energy.