Coulomb Displacement Energies Research Articles

Measurements and computational analysis of the natural decay of Lu176

Background: Mainly because of its long half-life and despite its scientific relevance, spectroscopic measurements of $^{176}\mathrm{Lu}$ forbidden $\ensuremath{\beta}$ decays are very limited and lack formulation of shape factors. A direct precise measurement of its Q value is also presently unreported. In addition, the description of forbidden decays provides interesting challenges for nuclear theory. The comparison of precise experimental results with theoretical calculations for these decays can help to test underlying models and can aid the interpretation of data from other experiments.Purpose: Perform the first precision measurements of $^{176}\mathrm{Lu}\phantom{\rule{4pt}{0ex}}\ensuremath{\beta}$-decay spectra and attempt the observation of its electron capture decays, as well as perform the first precision direct measurement of the $^{176}\mathrm{Lu}\phantom{\rule{4pt}{0ex}}\ensuremath{\beta}$-decay Q value. Compare the shape of the precisely determined experimental $\ensuremath{\beta}$ spectra to theoretical calculations, and compare the end point energy to that obtained from an independent Q value measurement.Method: The $^{176}\mathrm{Lu}\phantom{\rule{4pt}{0ex}}\ensuremath{\beta}$-decay spectra measurements and the search for electron capture decays were performed with an experimental setup that employed lutetium-containing scintillator crystals and a NaI(Tl) spectrometer for coincidence counting. The $\ensuremath{\beta}$ decay Q value was determined via high-precision Penning trap mass spectrometry (PTMS) with the LEBIT facility at the National Superconducting Cyclotron Laboratory. The $\ensuremath{\beta}$-spectrum calculations were performed within the Fermi theory formalism with nuclear structure effects calculated using a shell model approach.Results: Both $\ensuremath{\beta}$ transitions of $^{176}\mathrm{Lu}$ were experimentally observed and corresponding shape factors formulated in their entire energy ranges. The search for electron capture decay branches led to an experimental upper limit of $6.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ relative to its $\ensuremath{\beta}$ decays. The $^{176}\mathrm{Lu}\phantom{\rule{4pt}{0ex}}\ensuremath{\beta}$-decay and electron capture Q values were measured using PTMS to be 1193.0(6) and 108.9(8) keV, respectively. This enabled precise $\ensuremath{\beta}$ end point energies of 596.2(6) and 195.3(6) keV to be determined for the primary and secondary $\ensuremath{\beta}$ decays, respectively. The conserved vector current hypothesis was applied to calculate the relativistic vector matrix elements. The $\ensuremath{\beta}$-spectrum shape was shown to significantly depend on the Coulomb displacement energy and on the value of the axial vector coupling constant ${g}_{A}$, which was extracted according to different assumptions.Conclusion: The implemented self-scintillation method has provided unmatched observations of $^{176}\mathrm{Lu}$, independently validated by the first direct measurements of its $\ensuremath{\beta}$-decay Q value by Penning trap mass spectrometry. Theoretical study of the main $\ensuremath{\beta}$ transition led to the extraction of very different effective ${g}_{A}$ and ${log}_{10}\phantom{\rule{0.16em}{0ex}}f$ values, showing that a high-precision description of this transition would require a realistic nuclear structure with nucleus deformation.

Binding energies of proton-rich nuclei determined from their mirror pairs* *Supported by the National Natural Science Foundation of China (12075121 and 11605089), and by the Natural Science Foundation of Jiangsu Province (BK20190067 and BK20150762)

In addition to the Coulomb displacement energy, the residual differences between the binding energies of mirror nuclei (a pair of nuclei with the same mass number plus interchanged proton and neutron numbers) contribute to the shell effect via the valence scheme in this study. To this end, one linear combining type of valence nucleon number, namely, , is chosen to tackle this shell correction, in which and are the valence proton and neutron numbers with respect to the nearest shell closure, respectively. The mass differences of mirror nuclei, as the sum of the empirical Coulomb displacement energy and shell effect correction, are then used to obtain the binding energies of proton-rich nuclei through the available data of their mirror partners to explore the proton dripline of the nuclear chart.

Systematic study of Coulomb displacement energy based on the Coulomb energy empirical formula and mass model

In this study, starting from the nuclear binding energy, we studied the empirical formula of the Coulomb energy systematically. Beginning with the simplest empirical formula, we added the Coulomb exchange term, pairing correlation, and shell correction, to calculate the Coulomb energy more precisely. In addition, we investigated the Coulomb displacement energy of widely used mass models. The discrepancies between the calculation results and experimental data might be caused by the missing of the isospin breaking terms of strong interactions in theory. By investigating these discrepancies, we found that the results obtained by microscopic theories presented a distinctive systematical behavior. Thus, these theories can be improved by including isospin breaking terms. However, in the results obtained by phenomenological models, the systematical behavior is unclear. This occurs because it is difficult for the information of the microscopic nuclear structure to be absorbed into the parameters of phenomenological models.

Experimentally well-constrained masses of 27P and 27S: Implications for studies of explosive binary systems

The mass of 27P is expected to impact the X-ray burst (XRB) model predictions of burst light curves and the composition of the burst ashes, but large uncertainties and inconsistencies still exist in the reported 27P masses. We have used the β-decay spectroscopy of 27S to determine the most precise mass excess of 27P to date to be −659(9) keV, which is 63 keV (2.3σ) higher and a factor of 3 more precise than the value recommended in the 2016 Atomic Mass Evaluation. Based on the new 27P mass, the Si26(p,γ)27P reaction rate and its uncertainty were recalculated using Monte Carlo techniques. We also estimated the previously unknown mass excess of 27S to be 17678(77) keV, based on the measured β-delayed two-proton energy and the Coulomb displacement energy relations. The impact of these well-constrained masses and reaction rates on the modeling of the explosive astrophysical scenarios has been investigated by post-processing XRB and hydrodynamic nova models. Compared to the model calculations based on the masses and rates from databases, the abundance of A=26 in the burst ashes is increased by a factor of 2.4, while no substantial change was found in the XRB energy generation rate or the light curve. Our calculation also suggests that 27S is not a significant waiting point in the rapid proton capture process, and the change of the Si26(p,γ)27P reaction rate is not sufficiently large to affect the conclusion previously drawn on the nova contribution to the synthesis of galactic 26Al.

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.

Calculating Energy Levels in 25Mg/25Al Mirror Nuclei


 
 
 
 Coulomb Displacement Energies in mirror nuclei 25Mg, 25Al have been calculated using shell model code OXBASH [1] and compared with experimental results. The code calculations were done in the USD model space with the W Hamiltonian [2]. The OXBASH code which is based on famous nuclear model, the shell model, deals with evaluating energy levels in nuclei. A comparison had been made between our results and the available experimental data [3] to test theoretical shell model description of nuclear structure in mirror nuclei. The energy states of mirror nuclei are almost identical, except for the small effects due to Coulomb interaction where the symmetry in being broken. Energy spectrum calculated with this code was in good agreement with the published experimental data [3].
 
 
 
 

Constraints on Coulomb energy, neutron skin thickness in Pb208 , and symmetry energy

The charge-symmetry-breaking (CSB) effect in a nuclear medium that gives rise to the first-order symmetry energy in finite nuclei is discussed in detail in the present paper. For heavy and superheavy nuclei with large neutron excesses, it should be nonnegligible in high-precision mass predictions, and importantly it affects the stability of these nuclei. Combined with this CSB effect, the Coulomb energy is constrained by using the experimental Coulomb displacement energy of mirror nuclei, and then the mass-dependent symmetry energy coefficients of heavy nuclei are reextracted with the experimental ${\ensuremath{\beta}}^{\ensuremath{-}}$-decay energies of heavy odd-$A$ nuclei and with the experimental mass differences. Based on these results, we probe the neutron skin thickness $\mathrm{\ensuremath{\Delta}}{R}_{np}$ of $^{208}\mathrm{Pb}$ and the density-dependent symmetry energy coefficient of nuclear matter. $\mathrm{\ensuremath{\Delta}}{R}_{np}$ in $^{208}\mathrm{Pb}$ is found to be $0.158\ifmmode\pm\else\textpm\fi{}0.014\phantom{\rule{0.28em}{0ex}}\mathrm{fm}$, and the slopes $L$ of the symmetry energy coefficient at densities of $\ensuremath{\rho}=0.16$ and $\ensuremath{\rho}=0.11\phantom{\rule{0.28em}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$ are estimated to be $42\ifmmode\pm\else\textpm\fi{}8$ and $42\ifmmode\pm\else\textpm\fi{}3\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$, respectively. These results would be meaningful to discriminate between the models and the predictions that are relevant for the investigations on properties of nuclei and of neutron stars.

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.

Energy Levels Calculations in <sup>23</sup>Mg/<sup>23</sup>Na and <sup>47</sup>Cr/<sup>47</sup>V Mirror Nuclei

Coulomb Displacement Energies in mirror nuclei ²³Mg, ²³Na and ⁴⁷Cr, ⁴⁷V have been calculated using shell model code OXBASH [1] and compared with experimental results. The calculations were carried out in the USD model space with the W Hamiltonian [2]. This code which is based on one of the most applicable nuclear models, the shell model, deals with evaluating energy levels in nuclei. A comparison had been made between computational results in this paper and the available experimental data to test theoretical shell model description of nuclear structure in mirror nuclei. The energy states of mirror nuclei are almost identical, except for the small effects due to Coulomb interaction where the symmetry in being broken. The calculated energy spectrum is in good agreement with the available experimental data.

Isospin-symmetry breaking in superallowed Fermi β-decay due to isospin-nonconserving forces

We investigate isospin-symmetry breaking effects in the sd-shell region with large-scale shell-model calculations, aiming to understand the recent anomalies observed in superallowed Fermi β-decay. We begin with calculations of Coulomb displacement energies (CDE's) and triplet displacement energies (TDE's) by adding the T=1,J=0 isospin nonconserving (INC) interaction into the usual isospin-invariant Hamiltonian. It is found that CDE's and TDE's can be systematically described with high accuracy. A total number of 122 one- and two-proton separation energies are predicted accordingly, and locations of the proton drip-line and candidates for proton-emitters are thereby suggested. However, attempt to explain the anomalies in the superallowed Fermi β-decay fails because these well-fitted T=1,J=0 INC interactions are found no effects on the nuclear matrix elements. It is demonstrated that the observed large isospin-breaking correction in the 32Cl β-decay, the large isospin-mixing in the 31Cl β-decay, and the small isospin-mixing in the 23Al β-decay can be consistently understood by introducing additional T=1,J=2 INC interactions related to the s1/2 orbit.

Calculating Energy Levels in <sup>49</sup>Mn/<sup>49</sup>Cr Mirror Nuclei with OXBASH Code

Coulomb Displacement Energies in mirror nuclei 49 Mn and 49 Cr have been calculated using shell model code OXBASH and compared with experimental results. The calculations were carried out in the F7PN model space with the F748BPN Hamiltonian. This code which is based on one of the most applicable nuclear models, the shell model, deals with evaluating energy levels in nuclei. A comparison had been made between calculated results and the available experimental data to test theoretical shell model description of nuclear structure in mirror nuclei. The energy states of mirror nuclei are almost identical, except for the small effects due to Coulomb interaction where the symmetry in being broken. The calculated energy spectrum is in good agreement with the available experimental data.

Unified studies of structure and reactions in light unstable nuclei

The generalized two-center cluster model (GTCM), which can treat covalent, ionic and atomic configurations in general systems with two inert cores plus valence nucleons, is formulated in the basis of the microscopic cluster model. In this model, the covalent configurations constructed by the molecular orbital (MO) method and the atomic (or ionic) configuration obtained by the valence bonding (VB) method can be described in a consistent manner. GTCM is applied to the light neutron-rich system, 10,12 Be = α + α + XN (X = 2,4), and the unified studies of the structural changes and the reaction problem are performed. In the structure study, the calculated energy levels are characterized in terms of the chemical bonding like structures, such as the covalent MO or ionic VB structures. The chemical bonding structures changes from level to level within a small energy interval. In the unbound region, the structure problem with the total system of α + α + XN and the reaction problem, induced by the collision of an asymptotic VB state of α +6,8 He, are combined by GTCM. The properties of unbound resonant states are discussed in a close connection to the reaction mechanism, and some enhancement factors originated from the properties of the intrinsic states are predicted in the reaction observables. The unified calculation of the structures and the reactions is applied to the Coulomb shift problem in the mirror system, such the 10 Be and 10 C nuclei. The Coulomb displacement energy of the mirror systems are discussed.

Phenomenological description of the Coulomb energies of medium-heavy and superheavy nuclei

A phenomenological description of Coulomb energies is presented for more than 400 nuclei in the range of mass numbers A = 5–244. The Coulomb displacement energies between neighboring isobaric nuclei ΔE C (A, Z) are approximated using a two-parameter formula. The influence of deformation is taken into account for heavy nuclei. It is shown that the mean difference is less than 100 keV, which is better than for microscopic calculations. Magnitudes of ΔE C (A, Z) are calculated for heavy and superheavy nuclei up to mass number A = 300.

Thomas-Ehrman effect in a three-body model: TheNe16case

The dynamic mechanism of the Thomas-Ehrman shift is studied in three-cluster systems by example of $^{16}$Ne and $^{16}$C isobaric mirror partners. We predict configuration mixings for $0^+$ and $2^+$ states in $^{16}$Ne and $^{16}$C. Large isospin symmetry breaking on the level of wave function component weights is demonstrated for these states and discussed as three-body mechanism of Thomas-Ehrman shift. It is shown that the description of the Coulomb displacement energies requires a consistency among three parameters: the $^{16}$Ne decay energy $E_T$, the $^{15}$F ground state energy $E_r$, and the configuration mixing parameters for the $^{16}$Ne/$^{16}$C $0^+$ and $2^+$ states. Basing on this analysis we infer the $^{15}$F $1/2^+$ ground state energy to be $E_r=1.39-1.42$ MeV.

Coulomb displacement energies as a probe for nucleon pairing in thef7/2shell

Coulomb displacement energies of $T=1/2$ mirror nuclei have been studied via a series of high-precision $Q_\mathrm{EC}$-value measurements with the double Penning trap mass spectrometer JYFLTRAP. Most recently, the $Q_\mathrm{EC}$ values of the $f_{7/2}$-shell mirror nuclei $^{45}$V ($Q_\mathrm{EC}=7123.82(22)$ keV) and $^{49}$Mn ($Q_\mathrm{EC}=7712.42(24)$ keV) have been measured with an unprecedented precision. The data reveal a 16-keV ($1.6\sigma$) offset in the adopted Atomic Mass Evaluation 2012 value of $^{49}$Mn suggesting the need for further measurements to verify the breakdown of the quadratic form of the isobaric multiplet mass equation. Precisely measured $Q_\mathrm{EC}$ values confirm that the pairing effect in the Coulomb energies is quenched when entering the $f_{7/2}$ shell and reaches a minimum in the midshell.

A survey of Coulomb displacement energies and questions on the anomalous behavior in the upper -shell

The recent advances in nuclear mass measurement have sparked discussions on the isospin-symmetry breaking reflected in the Coulomb displacement energy (CDE). The current data suggested that the regular phase of the odd–even staggering in CDE for the T = 1/2 mirror nuclei persists up to A = 67 and changes at A = 69. Shell-model calculations using the modern GXPF1A and JUN45 effective interactions with a proper treatment of the Coulomb and isospin-nonconserving forces cannot describe the observation. Inspired by recent work (Kaneko 2013 Phys. Rev. Lett. 110 172505), we investigate the systematic behavior of CDE along the N = Z line up to the heaviest available masses. Starting from A ≈ 65, a systematic deviation is observed between the experimental data and the model estimations assuming the nucleus as a homogeneously charged sphere. Possibilities that may resolve the conflict between the experimental mass and theoretical expectations for the 69Br-region are discussed, and new mass experiments are called for.

Variation in Displacement Energies Due to Isospin-Nonconserving Forces

For mirror nuclei with masses A=42-95, the effects of isospin-nonconserving nuclear forces are studied with the nuclear shell model using the Coulomb displacement energy and triplet displacement energy as probes. It is shown that the characteristic behavior of the displacement energies can be well reproduced if the isovector and isotensor nuclear interactions with J=0 and T=1 are introduced into the f(7/2) shell. These forces, with their strengths being found consistent with the nucleon-nucleon scattering data, tend to modify nuclear binding energies near the N=Z line. At present, no evidence is found that these forces are needed for the upper fp shell. Theoretical one- and two-proton separation energies are predicted accordingly, and locations of the proton drip line are thereby suggested.

Level structure of10C

The level structure of ${}^{10}$C below 7 MeV is discussed. We suggest the spin-parity assignments for the unbound states in ${}^{10}$C. The assignments are based on the observed widths, Coulomb displacement energies, the reported decay modes, the potential model prediction, and the shell-model calculations.

Mass prediction of proton-rich nuclides with the Coulomb displacement energies in the relativistic point-coupling model

The masses, one- and two-proton separation energies of proton-rich nuclei with Z = 20–55, are computed using the measured masses of mirror neutron-rich nuclei and the Coulomb displacement energies calculated from the relativistic point-coupling model. The implications for the proton drip lines, candidates for two-proton emitters, as well as the impact on the astrophysical rp-process are discussed.

Mass measurements in the vicinity of the doubly magic waiting pointNi56

Masses of $^{56,57}\mathrm{Fe}$, $^{53}\mathrm{Co}$${}^{m}$, $^{53,56}\mathrm{Co}$, $^{55,56,57}\mathrm{Ni}$, $^{57,58}\mathrm{Cu}$, and $^{59,60}\mathrm{Zn}$ have been determined with the JYFLTRAP Penning trap mass spectrometer at the Ion-Guide Isotope Separator On-Line facility with a precision of $\ensuremath{\delta}m/m\ensuremath{\leqslant}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$. The ${Q}_{\mathrm{EC}}$ values for $^{53}\mathrm{Co}$, $^{55}\mathrm{Ni}$, $^{56}\mathrm{Ni}$, $^{57}\mathrm{Cu}$, $^{58}\mathrm{Cu}$, and $^{59}\mathrm{Zn}$ have been measured directly with a typical precision of better than $0.7 \mathrm{keV}$ and Coulomb displacement energies have been determined. The $Q$ values for proton captures on $^{55}\mathrm{Co}$, $^{56}\mathrm{Ni}$, $^{58}\mathrm{Cu}$, and $^{59}\mathrm{Cu}$ have been measured directly. The precision of the proton-capture $Q$ value for $^{56}\mathrm{Ni}$$(p,\ensuremath{\gamma})$$^{57}\mathrm{Cu}$, ${Q}_{(p,\ensuremath{\gamma})}=689.69(51) \mathrm{keV}$, crucial for astrophysical $\mathit{rp}$-process calculations, has been improved by a factor of 37. The excitation energy of the proton-emitting spin-gap isomer $^{53}\mathrm{Co}$${}^{m}$ has been measured precisely, ${E}_{x}=3 174.3(10) \mathrm{keV}$, and a Coulomb energy difference of $133.9(10) \mathrm{keV}$ for the $19/{2}^{\ensuremath{-}}$ state has been obtained. Except for $^{53}\mathrm{Co}$, the mass values have been adjusted within a network of 17 frequency ratio measurements between 13 nuclides, which allowed also a determination of the reference masses $^{55}\mathrm{Co}$, $^{58}\mathrm{Ni}$, and $^{59}\mathrm{Cu}$.