Wigner Term Research Articles

A close look at the competition of isovector and isoscalar pairing in A=18 and 20 even-even N≈Z nuclei * * Supported by the National Natural Science Foundation of China (11675071 and 11375080), the U. S. National Science Foundation (OIA-1738287 and ACI -1713690), U. S. Department of Energy (DE-SC0005248), the Southeastern Universities Research Association, the China-U. S. Theory Institute for Physics

The competition of isovector and isoscalar pairing in A=18 and 20 even-even N≈Z nuclei is analyzed in the framework of the mean-field plus the dynamic quadurpole-quadurpole, pairing and particle-hole interactions, whose Hamiltonian is diagonalized in the basis in the L = 0 configuration subspace. Besides the pairing interaction, it is observed that the quadurpole-quadurpole and particle-hole interactions also play a significant role in determining the relative positions of low-lying excited 0+ and 1+ levels and their energy gaps, which can result in the ground state first-order quantum phase transition from J = 0 to J = 1. The strengths of the isovector and isoscalar pairing interactions in these even-even nuclei are estimated with respect to the energy gap and the total contribution to the binding energy. Most importantly, it is shown that although the mechanism of the particle-hole contribution to the binding energy is different, it is indirectly related to the Wigner term in the binding energy.

Symmetry energies for A=24 and 48 and the USD and KB3 shell model Hamiltonians

Calculations in the sd and pf shells reported some time ago by Satu\l a\etal\ [Phys.~Lett.~B~407, 103 (1997)] are redone for an extended analysis of the results. As in the original work, we do calculations for one mass number in each shell and consider in each case the sequence of lowest energies for isospins 0, 2, and 4, briefly the symmetry spectrum. Following further the original work we study how this spectrum changes when parts of the two-nucleon interaction are turned off. The variation of its width is explored in detail. A differential combination $\epsilon_\text{W}$ of the three energies was taken in the original work as a measure of the so-called Wigner term in semi-empirical mass formulas, and it was found to decrease drastically when the two-nucleon interaction in the channel of zero isospin is turned off. Our analysis shows that the width of the symmetry spectrum experiences an equally drastic decrease, which can be explained qualitatively in terms of schematic approximations. We therefore suggest that the decrease of $\epsilon_\text{W}$ be seen mainly as a side effect of a narrowing of the symmetry spectrum rather than an independent manifestation of the two-nucleon interaction in the channel of zero isospin.

Coefficients of different macro–microscopic mass formulae from the AME2012 atomic mass evaluation

Abstract The coefficients of different possible macro–microscopic mass formulae previously proposed have been adjusted on 2264 experimental atomic masses extracted from the AME2012 atomic mass evaluation [1] assuming N , Z ⩾ 8 and the one standard deviation uncertainty on the mass lower than 150 keV. All the formulae include the volume and surface energies, the Coulomb energy, the diffuseness correction to the sharp radius Coulomb energy, the shell and pairing energies and take into account or not the curvature energy, different forms of the Wigner term, a free charge radius, the experimental equivalent rms charge radius or a fixed short central radius. Masses of 976 more exotic nuclei are extrapolated from the most accurate formula.

Decay studies of59Cu*formed in the35Cl +24Mg reaction using the dynamical cluster-decay model

The reformulated dynamical cluster-decay model (DCM) is applied to study the decay of odd-A and non-$\ensuremath{\alpha}$ structured ${}^{59}{\text{Cu}}^{*}$ formed in the ${}^{35}\text{Cl}+{}^{24}\text{Mg}$ reaction at ${E}_{\text{lab}}=275$ MeV. Here, the temperature ($T$)--dependent binding energies due to Krappe are used. The roles of Wigner and pairing energies in the fragmentation potential are explicitly shown in this work. For the temperature $T=4.1898$ MeV corresponding to ${E}_{\text{lab}}=275$ MeV, the contribution of pairing vanishes and the resulting structure of the fragmentation potential due to Wigner term is shown. In addition to this, we have studied the role of factor $\ensuremath{\alpha}$ appearing in the inertia part of the equation of motion dictating the mass-transfer process. It is shown that this factor has significant effect in the structure of preformation probability values and hence in turn we see significant changes in the cross sections. We compare the cross sections of the measured charge distributions of the fission fragments for two limiting values of the parameter $\ensuremath{\alpha}$ with the experimental data. In order to fit the total cross-section values, a linear relation is obtained between the free parameter of the model $▵R$ and the factor $\ensuremath{\alpha}$ appearing in the hydrodynamical mass.

Wigner term generated by isovector p-n pairing

The linear term proportional tojN Zj in the nuclear symmetry energy (Wigner energy) is obtained in a model that uses isovector pairing on single particle levels from a deformed potential combined with a ~ 2 interaction. The pairing correlations are calculated by numerical diagonalization of the pairing Hamiltonian acting on the six or seven levels nearest the N = Z Fermi surface. The experimental binding energies of nuclei with N Z are well reproduced. The Wigner energy emerges as a consequence of restoring isospin symmetry.

Reexamination and extension of the liquid drop model: Correlation between liquid drop parameters and curvature term

The relationship between the volume and surface energy coefficients in the liquid drop A^{-1/3} expansion of nuclear masses is discussed. The volume and surface coefficients in the liquid drop expansion share the same physical origin and their physical connection is used to extend the expansion with a curvature term. A possible generalization of the Wigner term is also suggested. This connection between coefficients is used to fit the experimental nuclear masses. The excellent fit obtained with a smaller number of parameters validates the assumed physical connection.

Mass predictions of exotic nuclei within a macroscopic-microscopic model

Different Liquid Drop Model mass formulae have been studied. They include a Coulomb diffuseness correction Z2/A term and pairing and shell energies of the Thomas-Fermi model. The influence of the selected charge radius, the curvature energy and different forms of the Wigner term has been investigated. Their coefficients have been determined by a least square fitting procedure to 2027 experimental atomic masses. The different fits lead to a surface energy coefficient of 17-18 MeV. A large equivalent rms radius (r0 = 1.22 − 1.24 fm) or a shorter central radius may be used. A rms deviation of 0.54 MeV can be reached between the experimental and theoretical masses. The remaining differences come from the determination of the shell and pairing energies. Mass predictions are given for exotic nuclei.

Macro-microscopic mass formulae and nuclear mass predictions

Different mass formulae derived from the liquid drop model and the pairing and shell energies of the Thomas–Fermi model have been studied and compared. They include or not the diffuseness correction to the Coulomb energy, the charge exchange correction term, the curvature energy, different forms of the Wigner term and powers of the relative neutron excess I = ( N − Z ) / A . Their coefficients have been determined by a least square fitting procedure to 2027 experimental atomic masses (G. Audi et al. (2003) [1]). The Coulomb diffuseness correction Z 2 / A term or the charge exchange correction Z 4 / 3 / A 1 / 3 term plays the main role to improve the accuracy of the mass formula. The Wigner term and the curvature energy can also be used separately but their coefficients are very unstable. The different fits lead to a surface energy coefficient of around 17–18 MeV. A large equivalent rms radius ( r 0 = 1.22 – 1.24 fm ) or a shorter central radius may be used. An rms deviation of 0.54 MeV can be reached between the experimental and theoretical masses. The remaining differences come probably mainly from the determination of the shell and pairing energies. Mass predictions of selected expressions have been compared to 161 new experimental masses and the correct agreement allows to provide extrapolations to masses of 656 selected exotic nuclei.

On the coefficients of the liquid drop model mass formulae and nuclear radii

The coefficients of different mass formulae derived from the liquid drop model and including or not the curvature energy, the diffuseness correction to the Coulomb energy, the charge exchange correction term, different forms of the Wigner term and different powers of the relative neutron excess I = ( N − Z ) / A have been determined by a least square fitting procedure to 2027 experimental atomic masses. The Coulomb diffuseness correction Z 2 / A term or the charge exchange correction Z 4 / 3 / A 1 / 3 term plays the main role to improve the accuracy of the mass formula. The Wigner term and the curvature energy can also be used separately for the same purpose. The introduction of an | I | dependence in the surface and volume energies improves slightly the efficiency of the expansion and is more effective than an I 4 dependence. Different expressions reproducing the experimental nuclear charge radius are provided. The different fits lead to a surface energy coefficient of around 17–18 MeV and a relative equivalent rms charge radius r 0 of 1.22–1.23 fm.

Correlation energy contribution to nuclear masses

During the last few years, much effort has been made to develop a microscopic description of the nuclear masses based on mean field theory. The accuracy achieved, when phenomenological parameters are added to take specific effects into account (Wigner term, cut-off in pairing space, etc.), leads to a rms of 0.6–0.7 MeV [S. Goriely, F. Tondeur, J.M. Pearson, Atom Data Nucl. Data Tables 77 (2001) 311] (see also [M. Bender, P.H. Heenen, P.G. Reinhard, Rev. Modern Phys. 75 (2003) 121]). We present evidence that further progress can be made by taking into account medium polarization effects associated with surface and pairing vibrations [S. Baroni, M. Armati, F. Barranco, R.A. Broglia, G. Colò, G. Gori, E. Vigezzi, J. Phys. G: Nucl. Part. Phys. 30 (2004) 1353; S. Baroni, F. Barranco, P.F. Bortignon, R.A. Broglia, G. Colò, E. Vigezzi, Phys. Rev. C 74 (2006) 024305] (see also [M. Bender, G.F. Bertsch, P.-H. Heenen, Phys. Rev. C 73 (2006) 034322]).

Hartree-Fock mass formulas and extrapolation to new mass data

The two previously published Hartree-Fock (HF) mass formulas, HFBCS-1 and HFB-1 (HF-Bogoliubov), are shown to be in poor agreement with new Audi-Wapstra mass data. The problem lies first with the prescription adopted for the cutoff of the single-particle spectrum used with the $\ensuremath{\delta}$-function pairing force, and second with the Wigner term. We find an optimal mass fit if the spectrum is cut off both above ${E}_{F}+15 \mathrm{MeV}$ and below ${E}_{F}\ensuremath{-}15 \mathrm{MeV},$ ${E}_{F}$ being the Fermi energy of the nucleus in question. In addition to the Wigner term of the form ${V}_{W} \mathrm{exp}(\ensuremath{-}\ensuremath{\lambda}|N\ensuremath{-}Z|/A)$ already included in the two earlier HF mass formulas, we find that a second Wigner term linear in $|N\ensuremath{-}Z|$ leads to a significant improvement in lighter nuclei. These two features are incorporated into our new Hartree-Fock-Bogoliubov model, which leads to much improved extrapolations. The 18 parameters of the model are fitted to the 2135 measured masses for $N,Z>~8$ with an rms error of 0.674 MeV. With this parameter set a complete mass table, labeled HFB-2, has been constructed, going from one drip line to the other, up to $Z=120.$ The new pairing-cutoff prescription favored by the new mass data leads to weaker neutron-shell gaps in neutron-rich nuclei.

Interpretation of the Wigner energy as due to RPA correlations

In a schematic model with equidistant fourfold degenerate single-nucleon levels, a conventional isovector pairing force and a symmetry force, the RPA correlation energy rises almost linearly with the isospin T, thus producing a Wigner term in accordance with the empirical proportionality of the symmetry energy to T( T+1).

A Hartree–Fock–Bogoliubov mass formula

In order to have more reliable predictions of nuclear masses at the neutron drip line, we here go beyond the recent mass formula HFBCS-1 and present a new mass formula, HFB-1, based on the Hartree–Fock–Bogoliubov method. As with the HFBCS-1 mass formula, we use a 10-parameter Skyrme force along with a 4-parameter δ-function pairing force and a 2-parameter phenomenological Wigner term. However, with the original HFBCS-1 Skyrme force (MSk7), the rms error becomes unacceptably large and a new force fit is required. With the isoscalar and isovector effective masses constrained to be equal, the remaining 15 degrees of freedom are fitted to the masses of all the 1754 measured nuclei with A⩾16, | N− Z|>2 given in the 1995 Audi–Wapstra compilation. The rms error with respect to the masses of all the 1888 measured nuclei with Z, N⩾8 is 0.764 MeV. A complete mass table, HFB-1 (available on the Web), has been constructed, giving all nuclei lying between the two drip lines over the range Z, N⩾8 and Z⩽120. A comparison between HFB-1 and HFBCS-1 mass tables shows that the HFBCS model is a very good approximation of the HFB theory, in particular for masses, the extrapolated masses never differing by more than 2 MeV below Z⩽110. We also find that the behaviour of shell gaps far away from the region of beta stability does not depend on whether the HFBCS or HFB methods are used, in particular, no quenching of astrophysical interest arises from replacing the BCS method by the Bogoliubov method.

A HARTREE–FOCK NUCLEAR MASS TABLE

We present the first complete nuclear mass table, HFBCS-1, to be based on the Hartree–Fock–BCS method. The force used, MSk7, is a 10-parameter Skyrme force, along with a 4-parameter δ-function pairing force and a 2-parameter phenomenological Wigner term. Our tabulation presents 9200 nuclei, including all those lying between the drip lines over the range Z, N≥8 and Z≤120. The root-mean-square error of our fit to the 1888 nuclei in this range for which measured masses are given in the 1995 Audi–Wapstra compilation is 0.738 MeV. In addition to the calculated masses, we show the calculated neutron- and proton-separation energies, and beta-decay energies. We also give for each nucleus in the table the calculated values for the deformation parameters and deformation energy (with axial and left–right symmetry assumed), and for the charge radius.

The Wigner term in heavy nuclei

We note the origin of the Wigner term in T = 0 and J = 0 nucleon pair correlations. Empirical evidence indicates that its energy is continuous through a shell region, and it persists through the region 82-126. Z = 126 might well be the next spherical proton magic number after Z = 82.

The physical origin of the Wigner term and its persistence in heavy nuclei

Abstract The physical origin of the Wigner term in terms of correlated nucleon pairs is pointed out. Empirical evidence is shown for the distribution of the Wigner term energy throughout a shell and for its persistence to heavy nuclei. The data indicate that Z=126 might well be the next proton spherical major magic number after Z=82.

On the origin of the Wigner energy

Surfaces of experimental masses of even-even and odd-odd nuclei exhibit a sharp slope discontinuity at N=Z. This cusp (Wigner energy), reflecting an additional binding in nuclei with neutrons and protons occupying the same shell model orbitals, is usually attributed to neutron-proton pairing correlations. A method is developed to extract the Wigner term from experimental data. Both empirical arguments and shell-model calculations suggest that the Wigner term can be traced back to the isospin T=0 part of nuclear interaction. Our calculations reveal the rather complex mechanism responsible for the nuclear binding around the N=Z line. In particular, we find that the Wigner term cannot be solely explained in terms of correlations between the neutron-proton J=1, T=0 (deuteron-like) pairs.

A nuclear mass formula based onSU(4) symmetry

It is argued that mass anomalies at the N≈Z line are associated with SU(4) isospin-spin symmetry. Drawing on these arguments, a Weizsacker-type nuclear mass formula is investigated which has the eigenvalue of the quadratic Casimir operator of SU(4) as a Wigner term. This SU(4)-based mass formula yields a better agreement than the one with the usual Wigner term |N—Z|/A. In addition, the SU(4) eigenvalue expression adequately replaces the usual pairing term of the Weizsacker formula giving a lower overall rms deviation than the latter.

Nuclear properties according to the Thomas-Fermi model

In order to formulate a statistical model of nuclear properties we combine the Thomas-Fermi assumption of two fermions per h 3 of phase space with an effective interaction between nucleons that contains seven adjustable parameters. After allowing for shell effects, an even-odd correction and a congruence energy (“Wigner Term”), six of the seven parameters were fitted to 1654 ground state masses of nuclei with N , Z ⩾ 8, together with a constraint that ensures agreement with measured values of the nuclear surface diffuseness. The RMS deviation in the fit to masses was 0.655 MeV, and the calculated values exhibit no drastic discrepancies even for A = 3. Calculated sizes of nuclear charge distributions agree closely with measurements. Calculated fission barriers were compared with 40 measured values down to 75 Br. For Z ⩾ 88 the agreement is almost perfect. For Z < 88 the trend of the measurements seems to confirm the expectation that the congruence energy should double its magnitude for strongly necked-in saddle-point shapes. A seventh (density-dependence) parameter in the effective interaction can be adjusted to ensure fair agreement with the measured energy-dependence of the optical model potential in the range from −70 MeV to 180 MeV. The model is used to predict properties of nuclear and neutron matter (including their compressibilities). A table of some 9000 calculated ground state masses of nuclei up to Z = 135 has been prepared.

New mass relations and two- and four-nucleon correlations

Periodic terms in the nuclear binding energies, such as the pairing energy, are investigated. To this end we introduce a new, general set of mass relations (indicators) sensitive to specific types of periodicities in the binding-energy surface. Based on experimental data, the analysis leads to an improved expression for the (average) pairing term in the mass formulas. No trace of α-clustering is found and an upper limit for the corresponding energy gain can be deduced to be 0.4 MeV per α-particle or 1.5% of the total α-binding energy. The Wigner term, especially important for light nuclei, is accurately determined. An extra binding in odd N = Z nuclei is observed above the average of all other odd-odd nuclei.