Nolen-Schiffer Anomaly Research Articles

Precision calculation of isospin-symmetry-breaking corrections to T=1/2 mirror decays using configuration-interaction framework built upon multireference charge-dependent density functional theory

We present systematic study of isospin impurities ($\alpha_{\rm ISB}$) to the wave functions of $T=1/2$, $11\leq A \leq 47$ mirror nuclei and the isospin-symmetry-breaking (ISB) corrections ($\delta_{\rm ISB}^{\rm V}$) to their ground state vector $\beta$-decays using, for the first time, multi-reference charge-dependent density functional theory (MR-DFT) that includes strong-force-rooted class-III interaction adjusted to correct for the Nolen-Schiffer anomaly in nuclear masses. We demonstrate that, unexpectedly, the strong-force-rooted isovector force gives rise to a large systematic increase of $\alpha_{\rm ISB}$ and $\delta_{\rm ISB}^{\rm V}$ as compared to the results obtained within MR-DFT that uses Coulomb interaction as the only source of ISB. This, in turn, increases a central value of the $V_{\rm ud}$ element of the CKM matrix extracted from the $T=1/2$ mirrors bringing it closer to the value obtained form the purely vector superallowed $0^+ \to 0^+$ transitions. In order to compute the value of $V_{\rm ud}$, we performed precision calculation of the Fermi matrix elements in $A=19, 21, 35$, and 37 mirror nuclei using DFT-rooted configuration-interaction model that includes all relevant axially-deformed particle-hole configurations built upon Nilsson orbitals originating from the spherical $sd$ shell. Our calculations yield $|V_{\rm ud}|=0.9736(16)$.

Mass relations of mirror nuclei in terms of Coulomb energies based on relativistic continuum Hartree-Bogoliubov calculations

In this paper we study mass relations of mirror nuclei in terms of Coulomb energies calculated by the microscopic relativistic continuum Hartree-Bogoliubov theory. An additional term corresponding to the Nolen-Schiffer anomaly is assumed to take the same form as the well-known empirical formula. The resultant root-mean-square deviations of these mass relations are typically less than 100 keV. Based on these relations, 61 unknown proton-rich nuclear masses are predicted for $18\ensuremath{\leqslant}A\ensuremath{\leqslant}87$ with positive two-proton separation energies and are tabulated in the Supplemental Material of this paper.

An effective Coulomb interaction in nuclear energy density functionals

It is well-known that the charge-violating interaction is usually underestimated in nuclear many-body approaches. In the framework of the Skyrme–Hartree–Fock (SHF) method, the effective two-body charge-symmetry breaking (CSB) and charge-independent breaking (CIB) interactions in nuclear medium based on Brueckner theory are included, then we constrain the effective Coulomb interaction in turn with the help of experimental Coulomb displacement energy (CDE), i.e., the binding-energy difference between mirror nuclei. Accordingly, we introduce a new (effective) Coulomb coupling constant e02=e2(1+aexcZ−2/3) to replace the original one e2 between protons (note that the original coupling constant just applies to point-like charge), where Z is proton number. This effective coupling constant e02 is phenomenologically embodying the effects of many complicated corrections beyond mean-field method or even nuclear structure physics such as core polarization, nucleon finite size and vacuum polarization. With e02=e2(1+0.45Z−2/3) for SLy4 interaction, the experimental CDE for T=1/2,1,3/2,2 isobaric multiplets, together with the excitation energy of isobaric analog states for heavy nuclei such as 208Pb, can be rather well reproduced, indicating the validity of such a treatment. Moreover, the c coefficients in the isobaric multiplet mass equation for isobaric quartets are computed, and the results based on Coulomb force turns out to be around 10%–15% lower than the experimental ones persistently, just as the Nolen–Schiffer anomaly. Yet, the introduction of both the CIB effect and the e02 systematically improves the agreement with experimental data substantially.

Generalized isobaric multiplet mass equation and its application to the Nolen-Schiffer anomaly

The Wigner Isobaric Multiplet Mass Equation (IMME) is the most fundamental prediction in nuclear physics with the concept of isospin. However, it was deduced based on the Wigner-Eckart theorem with the assumption that all charge-violating interactions can be written as tensors of rank two. In the present work, the charge-symmetry breaking (CSB) and charge-independent breaking (CIB) components of the nucleon-nucleon force, which contribute to the effective interaction in nuclear medium, are established in the framework of Brueckner theory with AV18 and AV14 bare interactions. Because such charge-violating components can no longer be expressed as an irreducible tensor due to density dependence, its matrix element cannot be analytically reduced by the Wigner-Eckart theorem. With an alternative approach, we derive a generalized IMME (GIMME) that modifies the coefficients of the original IMME. As the first application of GIMME, we study the long-standing question for the origin of the Nolen-Schiffer anomaly found in the Coulomb displacement energy of mirror nuclei. We find that the naturally-emerged CSB term in GIMME is largely responsible for explaining the Nolen-Schiffer anomaly.

NATURALNESS, ISOSPIN BREAKING IN NUCLEAR MATTER AND THE OKAMOTO–NOLEN–SCHIFFER ANOMALY

A recently developed relativistic theory for nuclear matter, which shows consistency with the concept of naturalness and conservation of chiral symmetry, is applied to the description of the Okamoto–Nolen–Schiffer anomaly. The results of our study show significant improvements in the description of the anomaly when compared to the corresponding results of other authors. In particular, our predictions for the Okamoto–Nolen–Schiffer anomaly include the known anomalous growth, namely that the anomaly not necessarily grow with the nuclear mass, since the anomaly is lower, for instance, for the nuclei 39 C and 39 K when compared to the 17 F and 17 O nuclei. As far as we know, this trend was not described by any other model found in the literature.

Neutron-Proton Mass Difference in Nuclear Matter and in Finite Nuclei and the Nolen-Schiffer Anomaly

The neutron-proton mass difference in (isospin asymmetric) nuclear matter and finite nuclei is studied in the framework of a medium-modified Skyrme model. The proposed effective Lagrangian incorporates both the medium influence of the surrounding nuclear environment on the single nucleon properties and an explicit isospin-breaking effect in the mesonic sector. Energy-dependent charged and neutral pion optical potentials in the s- and p-wave channels are included as well. The present approach predicts that the neutron-proton mass difference is mainly dictated by its strong part and that it markedly decreases in neutron matter. Furthermore, the possible interplay between the effective nucleon mass in finite nuclei and the Nolen-Schiffer anomaly is discussed. In particular, we find that a correct description of the properties of mirror nuclei leads to a stringent restriction of possible modifications of the nucleon’s effective mass in nuclei.

Neutron-proton mass difference in finite nuclei and the Nolen-Schiffer anomaly

The neutron-proton mass difference in finite nuclei is studied in the framework of a medium-modified Skyrme model. The possible interplay between the effective nucleon mass in finite nuclei and the Nolen-Schiffer anomaly is discussed. In particular, we find that a correct description of the properties of mirror nuclei leads to a stringent restriction of possible modifications of the nucleon's effective mass in nuclei.

Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. VIII. Role of Coulomb exchange

Following suggestions that the energy associated with Coulomb correlations and a possible charge-symmetry breaking of nuclear forces might largely cancel the Coulomb-exchange term, we refit the HFB-14 mass model without the Coulomb-exchange term to essentially all the mass data. The resulting mass model, HFB-15, gives a better fit to the 2149 mass data, ${\ensuremath{\sigma}}_{\mathrm{rms}}$ falling from 0.729 to 0.678 MeV. The improvement in the energy differences between mirror nuclei is particularly striking: the Nolen-Schiffer anomaly, which is strong for HFB-14, is essentially eliminated. As for the extrapolation to highly neutron-rich nuclei, the HFB-15 model differs significantly from HFB-14, with up to 15 MeV less binding being predicted. However, the differences in the predicted values of differential quantities such as the neutron-separation energies, \ensuremath{\beta}-decay energies and fission barriers are very much smaller.

LOCAL ENERGY-DENSITY FUNCTIONAL APPROACH TO MANY-BODY NUCLEAR SYSTEMS WITH S-WAVE PAIRING

The ground-state properties of superfluid nuclear systems with 1S0 pairing are studied within a local energy-density functional (LEDF) approach. A new form of the LEDF is proposed with a volume part which fits the Friedman-Pandharipande and Wiringa-Fiks-Fabrocini equation of state at low and moderate densities and allows an extrapolation to higher densities which preserves causality. For inhomogeneous systems, a surface term is added, with two free parameters, which has a fractional form like a Padé approximant containing the square of the density gradient in both the numerator and denominator. In addition to the direct and exchange Coulomb interaction energy, an effective density-dependent Coulomb-nuclear correlation term is included with one more free parameter. A three-parameter fit to the masses and radii of about 100 spherical nuclei has shown that the latter term gives a contribution of the same order of magnitude as the Nolen-Schiffer anomaly in the Coulomb displacement energy. The root-mean-square deviations from experimental masses and radii with the proposed LEDF come out about a factor of two smaller than those obtained with the conventional functionals based on the Skyrme or finite-range Gogny force, or on relativistic mean-field theory. The generalized variational principle is formulated leading to the self-consistent Gor'kov equations which are sovled exactly, with physical boundary conditions both for the bound and scattering states. The method is used to calculate the differential observables such as odd-even mass differences and staggering in charge radii. With a zero-range density-dependent cutoff pairing interaction incorporating a density-gradient term, the evolution of these observables is reproduced reasonably well, including the kinks at magic neutron numbers and the sizes of the associated staggering. An extrapolation from the pairing properties of finite nuclei to pairing in infinite nuclear matter is discussed. A "reference" value of the pairing gap ΔF≈ 3.3 MeV is found for subsaturated nuclear matter at about 0.65 of the equilibrium density. With the formulated LEDF approach, we study also the dilute limit in both the weak and strong coupling regimes. Within the sum rules approach it is shown that the density-dependent pairing may also induce sizeable staggering and kinks in the evolution of the mean energies of multipole excitations.

PROTON SINGLE PARTICLE ENERGY SHIFTS DUE TO COULOMB CORRELATIONS

A theoretically consistent approach to the calculation of the Coulomb energy of nuclei is presented. New contributions to the single-particle energies are taken into account. We show that the interplay between the Coulomb interaction and the strong interaction leads to an upward shift of the proton single-particle levels, affecting the position of the calculated proton drip line. The same contributions are responsible for significant corrections to the mass difference of the mirror nuclei (Nolen-Schiffer anomaly) and to the effective proton mass.

Charge symmetry breaking of the nucleon-nucleon interaction: ρ-ω mixing versus nucleon mass splitting

We investigate three models for the charge symmetry breaking (CSB) of the nucleon-nucleon ($NN$) interaction (based upon $\rho$-$\omega$ mixing, nucleon mass splitting, and phenomenology) that all reproduce the empirical value for the CSB of the $^1S_0$ scattering length ($\Delta a_{CSB}$) accurately. We reveal that these models make very different predictions for CSB in $^3P_J$ waves and examine the impact of this on some observable quantities of $A\geq 3$ nuclear systems. It turns out that the $^3$H-$^3$He binding energy difference is essentially ruled by $\Delta a_{CSB}$ and not very sensitive to CSB from $P$ waves. However, the Coulomb displacement energies (which are the subject of the Nolen-Schiffer anomaly) receive about 50% of their CSB contribution from $NN$ partial waves beyond $^1S_0$. Consequently, the predictions by the various CSB models differ here substantially (10-20%). Unfortunately, the evaluation of the leading Coulomb contributions carry a large uncertainty such that no discrimination between the competing CSB models can presently be made. To decide the issue we suggest to look into nuclear few-body reactions that are sensitive to CSB of the nuclear force.

Displacement energies with the Skyrme Hartree–Fock method

The binding energy differences of mirror nuclei are calculated with the Skyrme Hartree–Fock method and compared to experiment. The Nolen–Schiffer anomaly can be accounted for either by dropping the Coulomb exchange term or by introducing a charge-symmetry breaking interaction. The displacement energy systematics can be quantitatively reproduced with either of these modifications. The results are important for an accurate description of the proton and neutron drip lines. The possible origins of these modifications in terms of nuclear correlations and nucleon-nucleon charge-symmetry breaking interactions are discussed.

Proton single-particle energy shifts due to Coulomb correlations

A theoretically consistent approach to the calculation of the Coulomb correlation corrections to the single-particle energies is presented. New contributions to the single-particle energies previously overlooked in the literature are now identified and taken into account. We show that the interplay between the Coulomb interaction and the strong interaction, which is enhanced in the nuclear surface, leads to an upward shift of the proton single-particle levels. This shift affects the position of the calculated proton drip line, a shift towards decreasing Z. We describe briefly a similar mechanism which is at work for neutron levels. The same mechanism is responsible for significant corrections to the mass difference of the mirror nuclei (Nolen–Schiffer anomaly) and to the effective proton mass.

MASS PREDICTIONS IN THE INFINITE NUCLEAR MATTER MODEL

We present here the binding energies and mass excesses of 7208 nuclei in the ranges 4 ≤ Z ≤ 120 and 8 ≤ A ≤ 270. Relative to our 1986–1987 mass table, the present results are obtained with the recently improved infinite nuclear matter model which has the desirable features of natural decoupling of the infinite part from the finite part of the ground state energy, together with the cancellation of the exchange Coulomb, finite-size proton form factor, and Nolen–Schiffer anomaly terms. In addition, we have developed a new scheme of an interactive network covering the entire nuclear chart to obtain the local energy (comprising shell, deformation, etc.) of a nucleus in a consistent manner, using the technique of ensemble averaging of a large number of values predicted with the help of recursion relations derived in the model. This has widened the scope for predictions of masses of nuclei far into the drip-line regions of the nuclear chart. On the basis of systematics of the two-neutron separation energies, several new islands of stability in the exotic regions are predicted. This model has only five parameters representing the properties of infinite nuclear matter, the surface tension, and the Coulomb and pairing terms, which are determined once and for all in least-squares fits to known nuclear masses. The root-mean-square deviation of the fit to 1884 known masses is 401 keV, while the mean deviation is a remarkably low 9 keV, indicating that remanent systematic effects are vanishingly small.

Modern nucleon–nucleon interactions and charge-symmetry breaking in nuclei

Coulomb displacement energies, i.e., the differences between the energies of corresponding nuclear states in mirror nuclei, are evaluated using recent models for the nucleon–nucleon (NN) interaction. These modern NN potentials account for breaking of isospin symmetry and reproduce pp and pn phase shifts accurately. The predictions by these new potentials for the binding of 16O are calculated. A particular focus of our study are effects due to nuclear correlations and charge-symmetry breaking (CSB). We find that the CSB terms in the modern NN interactions substantially reduce the discrepancy between theory and experiment for the Coulomb displacement energies; however, our calculations do not completely explain the Nolen–Schiffer anomaly. Potential sources for the remaining discrepancies are discussed.

Towards a universal nuclear density functional

A new form of the nuclear energy-density functional for describing the ground-state properties of finite nuclei up to the drip lines and beyond is proposed. The volume part in this functional is fit to the Friedman-Pandharipande and Wiringa-Fiks-Fabrocini equation of state for the UV14 plus TNI model up to densities ρ≈1 fm−3 by a fractional expression in ρ which can be extended to higher densities while preserving causality. For inhomogeneous systems, a surface energy-density term is added, with two free parameters, which also has a fractional form like the Pade approximant containing (∇ρ)2 in both the numerator and denominator. In addition to the Coulomb direct and exchange interaction energy, an effective ρ-dependent Coulomb-nuclear correlation term is included with one more free parameter. A three-parameter fit to the masses and radii of real nuclei shows that the latter term gives a contribution to the Coulomb displacement energy of the same order of magnitude as the Nolen-Schiffer anomaly. The first self-consistent run with the proposed functional, performed for about 100 spherical nuclei, gives rms deviations from experiment of ≈1.2 MeV in the masses and ≈0.01 fm in the radii, which are about a factor of two better than those obtained with the Skyrme functionals or with the Gogny force. The extrapolation to the drip lines lies in between the ETFSI and the macroscopic-microscopic model predictions.

New Skyrme interaction for normal and exotic nuclei

A new set of Skyrme parameters is obtained from a fit to the binding energies, rms charge radii, and single-particle energies of both normal and exotic spherical nuclei. Nuclear matter and neutron matter properties are used to put constraints on the parameters which are not well determined from the nuclear data. Special problems with the Nolen-Schiffer anomaly and the spin-orbit interaction are discussed.

Medium effect on charge symmetry breaking

We examine the nuclear medium effect on charge symmetry breaking (CSB) caused by isospin mixing of two neutral vector mesons interacting with nucleons in the nuclear medium. Isospin mixing is assumed to occur through the transition between isoscalar and isovector mesons. We use a quantum hadrodynamic nuclear model in the mean-field approximation for the meson fields involved. We find that (i) charge symmetry is gradually restored in nuclear matter in $\ensuremath{\beta}$ equilibrium as the nucleon density increases; (ii) when the system departs from $\ensuremath{\beta}$ equilibrium, CSB is much enhanced because the isospin mixing depends strongly on the nucleon isovector density; (iii) this leads to the symmetry energy coefficient of $32$MeV, of which more than 50 percent arises from the mesonic mean fields; (iv) the Nolen-Schiffer anomaly regarding the masses of neighboring mirror nuclei can be resolved by considering these aspects of CSB in nuclear medium.

Nuclear sizes and the Coulomb Displacement Energy

Data on Coulomb Displacement Energies in the mass rangeA=40−240 are analyzed in the deformed Liquid Drop model and in the independent particle model. Reduced half-widths of Woods-Saxon mean-field potential of the resulting neutron-excess distributions are deduced. It is argued that the Nolen-Schiffer anomaly may be lifted by allowing for a slight binding-energy dependence of the mean-field potential geometry.

Rho-omega mixing and the Nolen-Schiffer anomaly in relativistic nuclear models

We calculate within the framework of relativistic nuclear models the contribution of the ϱ 0 - ω mixing interaction to the binding energy differences of the mirror nuclei in the neighborhood of A = 16 and A = 40. We use two relativistic models for the nuclear structure, one with scalar and vector Woods-Saxon potentials, and the Walecka model. The ϱ 0 - ω interaction is treated in first order perturbation theory. When using the Walecka model the ϱ- and ω-nucleon coupling constants are the same for calculating bound state wave functions and the perturbation due to the mixing. We find that the relativistic results on the average are of the same order as the ones obtained with nonrelativistic calculations.