Single-nucleon Wave Functions Research Articles

The nuclear symmetry energy from relativistic Brueckner-Hartree-Fock model * *Supported by in part by the National Natural Science Foundation of China (11775119, 12175109), the Natural Science Foundation of Tianjin

The microscopic mechanisms of the symmetry energy in nuclear matter are investigated in the framework of the relativistic Brueckner-Hartree-Fock (RBHF) model with a high-precision realistic nuclear potential, pvCDBonn A. The kinetic energy and potential contributions to symmetry energy are decomposed. They are explicitly expressed by the nucleon self-energies, which are obtained through projecting the G-matrices from the RBHF model into the terms of Lorentz covariants. The nuclear medium effects on the nucleon self-energy and nucleon-nucleon interaction in symmetry energy are discussed by comparing the results from the RBHF model and those from Hartree-Fock and relativistic Hartree-Fock models. It is found that the nucleon self-energy including the nuclear medium effect on the single-nucleon wave function provides a largely positive contribution to the symmetry energy, while the nuclear medium effect on the nucleon-nucleon interaction, i.e., the effective G-matrices provides a negative contribution. The tensor force plays an essential role in the symmetry energy around the density. The scalar and vector covariant amplitudes of nucleon-nucleon interaction dominate the potential component of the symmetry energy. Furthermore, the isoscalar and isovector terms in the optical potential are extracted from the RBHF model. The isoscalar part is consistent with the results from the analysis of global optical potential, while the isovector one has obvious differences at higher incident energy due to the relativistic effect.

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.

Single-particle spatial dispersion and clusters in nuclei

The spatial dispersion of the single-nucleon wave functions is analyzed using the self-consistent mean-field framework based on nuclear energy density functionals, and with the harmonic oscillator approximation for the nuclear potential. It is shown that the dispersion depends on the radial quantum number n, but displays only a very weak dependence on the orbital angular momentum. An analytic expression is derived for the localization parameter that explicitly takes into account the radial quantum number of occupied single-nucleon states. The conditions for single-nucleon localization and formation of cluster structures are fulfilled in relatively light nuclei with $A \leq 30$ and $n=1$ states occupied. Heavier nuclei exhibit the quantum liquid phase of nucleonic matter because occupied levels that originate from $n > 1$ spherical states are largely delocalized. Nevertheless, individual $\alpha$-like clusters can be formed from valence nucleons filling single-particle levels originating from $n=1$ spherical mean-field states.

Microscopic study of induced fission dynamics of Th226 with covariant energy density functionals

Static and dynamic aspects of the fission process of $^{226}\mathrm{Th}$ are analyzed in a self-consistent framework based on relativistic energy density functionals. Constrained relativistic mean-field calculations in the collective space of axially symmetric quadrupole and octupole deformations, based on the energy density functional PC-PK1 and a $\ensuremath{\delta}$-force pairing, are performed to determine the potential energy surface of the fissioning nucleus, the scission line, the single-nucleon wave functions, energies, and occupation probabilities, as functions of deformation parameters. Induced fission dynamics is described using the time-dependent generator coordinate method in the Gaussian overlap approximation. A collective Schr\"odinger equation, determined entirely by the microscopic single-nucleon degrees of freedom, propagates adiabatically in time the initial wave packet built by boosting the ground-state solution of the collective Hamiltonian for $^{226}\mathrm{Th}$. The position of the scission line and the microscopic input for the collective Hamiltonian are analyzed as functions of the strength of the pairing interaction. The effect of static pairing correlations on the preneutron emission charge yields and total kinetic energy of fission fragments is examined in comparison with available data, and the distribution of fission fragments is analyzed for different values of the initial excitation energy.

Isoscalar monopole and dipole excitations of cluster states and giant resonances inC12

The isoscalar monopole(ISM) and dipole(ISD) excitations in $^{12}$C are theoretically investigated with the shifted antisymmetrized molecular dynamics(AMD) plus $3\alpha$-cluster generator coordinate method(GCM). The small amplitude vibration modes are described by coherent one-particle one-hole excitations expressed by small shift of single-nucleon Gaussian wave functions within the AMD framework, whereas the large amplitude cluster modes are incorporated by superposing $3\alpha$-cluster wave functions in the GCM. The coupling of the excitations in the intrinsic frame with the rotation and parity transformation is taken into account microscopically by the angular-momentum and parity projections. The present calculation describes the ISM and ISD excitations in a wide energy region covering cluster modes in the low-energy region and the giant resonances in the high-energy region, though the quantitative description of the high-energy part is not satisfactory. The low-energy ISM and ISD strengths of the cluster modes are enhanced by the radial motion of $\alpha$ clusters, and they split into a couple of states because of the angular motion of $\alpha$ clusters. The low-energy ISM strengths exhaust 26% of the EWSR, which is consistent with the experimental data for the $^{12}$C($0^+_2$;7.65 MeV) and $^{12}$C($0^+_3$;10.3 MeV) measured by $(e,e')$, $(\alpha,\alpha')$, and ($^6$Li,$^6$Li$'$) scatterings. In the calculated low-energy ISD strengths, two $1^-$ states (the $1^-_1$ and $1^-_2$) with the significant strengths are obtained in $E=10-15$ MeV. It is indicated that the ISD excitations can be a good probe to experimentally search for new cluster states such as the $^{12}$C($1^-_2$) obtained in the present calculation.

Quenching of magnetic form factors ofs−dshell nucleiO17,Mg25,Al27,Si29, andP31within the relativistic mean field model

The magnetic form factors of the odd-A s-d shell nuclei O-17, Mg-25, Al-27, Si-29, and P-31 are studied within the relativistic frame with single-nucleon wave functions generated using the relativistic mean field model. Single valence-nucleon contributions to the nuclear magnetic form factors are calculated, and two sets of quenching ratios of the nuclear magnetic form factors relative to the single valence-nucleon contributions are extracted for each nucleus. It is found that the single valence-nucleon contributions can generally give a good approximate description for the shapes of the nuclear magnetic form factors, including the positions of the minima and maxima, and after the quenching ratios are applied the experimental data can be very well reproduced. The quenching ratios are compared with those obtained within some other nuclear models, and the physical implications for each nucleus are discussed. Calculations also show that no reasonable quenching ratios could be found for Al-27 when mu(expt.)/mu(s.n.) is chosen as the M1 multipole quenching ratio. This implies that the common way of choosing mu(expt.)/mu(s.n.) as the M1 multipole quenching ratio for the odd-A nuclei is not always valid.

DIRHB—A relativistic self-consistent mean-field framework for atomic nuclei

The DIRHB package consists of three Fortran computer codes for the calculation of the ground-state properties of even–even atomic nuclei using the framework of relativistic self-consistent mean-field models. Each code corresponds to a particular choice of spatial symmetry: the DIRHBS, DIRHBZ and DIRHBT codes are used to calculate nuclei with spherical symmetry, axially symmetric quadrupole deformation, and triaxial quadrupole shapes, respectively. Reflection symmetry is assumed in all three cases. The latest relativistic nuclear energy density functionals are implemented in the codes, thus enabling efficient and accurate calculations over the entire nuclide chart. Program summaryProgram title: DIRHB package (codes DIRHBS, DIRHBZ and DIRHBT)Catalogue identifier: AESN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: noneNo. of lines in distributed program, including test data, etc.: 44516No. of bytes in distributed program, including test data, etc.: 317161Distribution format: tar.gzProgramming language: Fortran 77.Computer: All computers.Operating system: All operating systems. The Makefiles are specific for a Unix OS and have to be modified for Windows.RAM: Depends on the imposed symmetry and number of oscillator shells. For the triaxial test case it takes 300 MB.Classification: 17.22.Nature of problem:Ground-state properties of even–even open-shell nuclei can be calculated using the framework of self-consistent mean-field models based on relativistic energy density functionals. The structure of arbitrary heavy nuclei with spherical symmetry, axially symmetric quadrupole deformation, and triaxial quadrupole shapes, is modeled using the latest zero- and finite-range relativistic effective interactions. The particle–particle channel of the effective inter-nucleon interaction is described by a separable finite-range pairing force.Solution method:The current implementation of the model computes the mean-field solution of the nuclear many-body problem for even–even open-shell spherical and quadrupole deformed nuclei. The codes are used to solve the stationary relativistic Hartree–Bogoliubov equations in a self-consistent iteration scheme. At each iteration the matrix elements of the equations are updated using the modified Broyden method or the linear mixing method. The single-nucleon wave functions are expanded in a basis of spherical, axially symmetric or triaxial harmonic oscillator, depending on the assumed symmetry of the nuclear shape. For calculations that constrain the shape to specific values of the deformation parameters, the augmented Lagrangian method is used.Restrictions:Time-reversal and reflection symmetries are assumed. Open-shell even–even spherical and quadrupole deformed nuclei are considered.Running time:Depends on the imposed symmetry and number of oscillator shells. For the test cases it runs from few seconds (spherical) up to few hours (triaxial).

Localization and clustering in the nuclear Fermi liquid

Using the framework of nuclear energy density functionals we examine the conditions for single-nucleon localization and formation of cluster structures in finite nuclei. We propose to characterize localization by the ratio of the dispersion of single-nucleon wave functions to the average inter-nucleon distance. This parameter generally increases with mass and describes the gradual transition from a hybrid phase in light nuclei, characterized by the spatial localization of individual nucleon states that leads to the formation of cluster structures, toward the Fermi liquid phase in heavier nuclei. Values of the localization parameter that correspond to a crystal phase cannot occur in finite nuclei. Typical length and energy scales in nuclei allow the formation of liquid drops, clusters, and halo structures.

High-energy α-nucleus scattering with microscopic wave functions

Alpha-nucleus scattering at high energies is analyzed using an eikonal approximation which considers the α internal wave function. It is constructed by antisymmetrizing 4 single nucleon wave functions with a 0S shell as a spatial part of each one. Differential elastic cross sections are calculated at different oscillator parameters and a strong dependence of them is observed.

Nuclear Halo-Like Phenomena and Short-Range Nuclear Correlation**The project supported in part by National Natural Science Foundation of China under Grant Nos. 19975053, 19835010, and 10075081

Based on Glauber Multiple Scattering Theory, high-energy proton elastic scattering on halo-like nucleus 13C is studied in a single nucleon wave function with low angular momentum configurations. A great agreement with experimental data is obtained and the theoretical prediction clearly shows that 13C has a neutron halo-like structure. Then, the origin and nature of nuclear halo phenomena are explained in terms of nuclear short-range correlations. Our conclusion shows that the origin of nuclear halo-like phenomena originates from short range nuclear correlation.

Nuclear wave functions for spin and pseudospin partners

Using relations between wave functions obtained in the framework of the relativistic mean field theory, we investigate the effects of pseudospin and spin symmetry breaking on the single nucleon wave functions in spherical nuclei. In our analysis, we apply both relativistic and nonrelativistic self-consistent models as well as the harmonic oscillator model.

Pseudospin symmetry and relativistic single-nucleon wave functions

We show that the occurrence of approximate pseudospin symmetry in nuclei is connected with certain similarities in the relativistic single-nucleon wave functions of the corresponding pseudospin doublets. We perform a case study in which several examples and the systematics of this connection are explored.

Electron scattering from 10B.

Electron scattering measurements have been made in order to determine the longitudinal and transverse form factors of low-lying level in $^{10}\mathrm{B}$. With the exception of the broad 5.18 MeV level, results were obtained for all known levels up to 6.7 MeV. The measurements span the momentum transfer range q=0.48--2.58 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$. The primary objective of this work was to improve the data on the pure isovector M3 form factor of the 1.740 MeV excitation, the transform of which yields the 1${\mathit{p}}_{3/2}$ single-nucleon wave function. A Woods-Saxon potential was found to provide a much better representation of the data than the harmonic oscillator model, and the rms size of this orbit was determined to be 2.79\ifmmode\pm\else\textpm\fi{}0.11 fm in the relative core-particle coordinate frame. Nevertheless, confidence in the quantitative details of this interpretation is hindered by conflicting evidence regarding the contribution of core polarization. Our analysis of the Coulomb elastic form factor gave an rms radius for the ground-state charge distribution equal to 2.58\ifmmode\pm\else\textpm\fi{}0.05\ifmmode\pm\else\textpm\fi{}0.05 fm, slightly larger than values previously published. Longitudinal and transverse form factors deduced for inelastic transitions were compared with theoretical results of conventional 1p-shell models, models with 1\ensuremath{\Elzxh}\ensuremath{\omega} and 2\ensuremath{\Elzxh}\ensuremath{\omega} configurations involving the 1s, 2s1d, and 2p1f shells, and finally, a model that included core polarization. Although restricted 1p-shell models were found to provide good predictions for the $^{10}\mathrm{B}$ natural-parity level spectrum and transverse form factors, they were less successful for C2 form factors: not only is there a considerable dependence on the 1p-shell interaction, but these models give just 45% of the total observed C2 transition strength. Only a 10% improvement was realized by expanding the shell model space to include 2\ensuremath{\Elzxh}\ensuremath{\omega} configurations. The inclusion of even higher-excited configurations by means of core polarization calculations was essential to remove the remaining shortfall.

Two-step processes and short-range correlations in the reaction 12C( gamma,p).

One-nucleon knockout calculations have been performed for the reaction $^{12}\mathrm{C}$(\ensuremath{\gamma},p${)}^{11}$B at incident photon energies of 60, 80, and 100 MeV. Both two-step processes and high-momentum components in the single-nucleon wave function due to short-range correlations have been included. It is shown that direct knockout leading to the $^{11}\mathrm{B}$ ground state followed by inelastic excitation of the 7/${2}^{\mathrm{\ensuremath{-}}}$ state at 6.743 MeV in $^{11}\mathrm{B}$ exceeds the direct excitation of this state by 1 to 2 orders of magnitude. For the ground-state transition the influence of high-momentum components becomes increasingly important going from ${\mathit{E}}_{\ensuremath{\gamma}}$=60 to 100 MeV. Constraining the parameters entering the calculation by recently measured quasielastic $^{12}\mathrm{C}$(e,e'p${)}^{11}$B data a discrepancy is found between the calculations and the data, particularly at low photon energies, which is attributed to exchange currents and random-phase approximation correlations.

Determination of single-nucleon wave functions by transverse electron scattering

Transverse form factors have been measured for elastic electron scattering from $^{10}\mathrm{B}$ and $^{11}\mathrm{B}$, as well as for the electroexcitation of the 1.74-MeV (${J}^{\ensuremath{\pi}}$=${O}^{+}$, T=1) and 5.17-MeV (${2}^{+}$, 1) levels of $^{10}\mathrm{B}$. These results were utilized to determine the radial shape of the 1${p}_{3/2}$ single-particle wave function within the nuclear interior.

Meson exchange potentials in an NN quark model with quark-antiquark excitations

In a recent quark-model study of the NN interaction, we explicitly incorporated the qq̄ excitations inherent in the quark-gluon interaction lagrangian into the model space, and examined the contributions to the NN potential of particular (3 q)( q q ) components of the single-nucleon wave function, with specific baryon and meson quantum numbers, through the Wigner transforms of the resultant coupling terms of the quark exchange kernels. The present investigation focuses on the long-range parts of the potentials. It is shown that these arise almost entirely through two of the 25 possible types of exchange terms, and that these two correspond to the simple exchange of a q q pair between the two nucleons and contribute only through coupling terms for which the (3q) components have the quantum numbers of a nucleon and the ( q q ) component the color-singlet character of a real pseudoscalar or vector meson. The potentials arising from these simple color-singlet q q exchanges have been evaluated in explicit analytic form. In their dependence on nucleon ( σ 1, · σ 2) and ( τ 1 · τ 2) factors and the relative importance and signs of spin-spin, spin-independent central, LS, and tensor terms they have all the characteristics of conventional OBEP's. They are in remarkably good agreement with conventional OBEP's in the r ⩾ 1.2 fm range and have the same qualitative radial features over an even wider range, the one exception being the steep short-range repulsive rise of the spin-independent central term of the ω-exchange potential. The corresponding q q exchange potential instead changes sign to become attractive at r < 0.8 fm.

Microscopic relativistic nucleon-nucleus inelastic scattering

We formulate a microscopic relativistic treatment of nucleon-nucleus inelastic scattering. Both the continuum and bound state single nucleon wave functions are obtained by solving the Dirac equation with scalar and timelike vector potentials. The interaction which drives the transition is expressed in terms of invariant combinations of Dirac matrices. The structure of the resulting amplitudes is presented, and certain of their features, such as selection rules, are discussed.

Rotational bands from shell-model Hamiltonians

Shell-model Hamiltonians with eigenstates forming rotational bands are considered. Such states have eigenvalues proportional to J( J+ 1) and can be projected from a Slater determinant of deformed orbitals. The latter are linear combinations of single-nucleon wave functions of j-orbits in a major shell. Conditions on matrix elements and single-nucleon energies are obtained in terms of the deformation parameters. An actual effective interaction is constructed yielding exact ground-state rotational bands for 20Ne and 24Mg which gives reasonable agreement with energies of other sd shell nuclei. Unlike the case of SU(3) symmetry, spin-orbit interaction and different single-nucleon energies can be accommodated and the procedure is not confined to oscillator major shells. Other welcome departures of our effective interaction from the SU(3) picture are the absence of rotational spectra in oxygen isotopes and that the 24Mg ground-state band is projected from a Nilsson-type deformed state with axial symmetry.

A new DWBA method for two-nucleon transfer with heavy ions

A formalism is developed for calculating two-nucleon transfer cross sections. It is intended mainly for heavy-ion induced reactions. The single-nucleon wave functions are expanded in a harmonic oscillator series. A Moshinsky transformation then yields a series of terms containing the relative and c.m. coordinates. It is found that only a limited number of these terms are needed.

Mirror asymmetry in Gamow-Teller transitions of large ft value

Abstract The several single-nucleon wave functions appropriate to the several parent states of a β-decaying nucleus are different in that they must reflect the several separation energies. These differences engender differences between the overlap integrals that enter into the β-transition amplitudes which will therefore themselves differ as between the negaton- and positon-emitting members of an isobaric mirror since the separation energies differ on the two sides of the mirror. The question is asked as to how large this effect may be in cases where, because of strong cancellation between parents, the ft value is large and so where this asymmetry due to binding energy differences will be largest. The question is answered via a Monte Carlo approach: it is shown that large log ft value differences may indeed be expected but that the effect is not large enough to account for the differences found in the A = 14 and 32 systems. This conclusion agrees with that made earlier in relation to the small ft value differences found between mirror transitions of normal ft value.