High-precision -value measurement of the superallowed emitter Mg and an ab-initio evaluation of the isobaric triplet
A direct -value measurement of the superallowed emitter Mg was performed using TRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN). The direct ground-state to ground-state atomic mass difference between Mg and Na was determined to be keV, representing the most precise single measurement of this quantity to date. In a continued push towards calculating superallowed isospin-symmetry-breaking (ISB) corrections from first principles, ab-initio shell-model calculations of the IMME are also presented for the first time using the valence-space in-medium similarity renormalization group formalism. With particular starting two- and three-nucleon forces, this approach demonstrates good agreement with the experimental data.
High-precision measurements of nuclear decay properties have proven to be a critical tool in the quest to understand possible physics beyond the Standard Model (BSM) Naviliat-Cuncic and González-Alonso (2013). Superallowed nuclear -decay data are among the most important to these tests, as they currently provide the most precise determination of the vector coupling strength in the weak interaction, Hardy and Towner (2015); Patrignani and Group (2016). This is only possible in this unique electroweak decay mode, since the transition operator that connects the initial and final states is independent of any axial-vector contribution to the weak interaction. In fact, the up-down element of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix, , is the most precisely known () Patrignani and Group (2016), and relies nearly entirely on the high-precision superallowed -decay values determined through measurements of the half-life, decay -value, and branching fraction of the superallowed decay mode Hardy and Towner (2015).
In order to obtain the level of precision required for Standard Model tests, corrections to the experimental -values must also be made to obtain nucleus-independent values,
where is a transition-dependent radiative correction, is a transition-independent radiative correction, and is a nucleus-dependent isospin-symmetry-breaking (ISB) correction. Although relatively small (), these corrections are crucial due to the very precise () experimental values Hardy and Towner (2015). The uncertainty on , and consequently , is presently dominated by the precision of these theoretical corrections, specifically and . With a value of 2.361(38)% Marciano and Sirlin (2006), the largest fractional uncertainty of any individual correction term is due to the transition-independent radiative correction, . Despite the large uncertainty, the QED formalism that is used in the calculation of this quantity is well understood, suggesting that the central value is accurate. This situation is not as clear for the ISB corrections however, which have a similarly large uncertainty contribution in the extraction of , but require complex nuclear-structure calculations on a case-by-case basis Satuła et al. (2009); Towner and Hardy (2010a).
The current extraction of and from the superallowed data uses the shell-model-calculated ISB corrections of Towner and Hardy (TH). This is largely due to the impressive efforts towards experimental testing Bhattacharya et al. (2008); Melconian et al. (2011); Park et al. (2015) and guidance Leach et al. (2013a, b); Molina et al. (2015) that their formalism has been exposed to. However, as the experimental values have become increasingly more precise - particularly in the last decade - the model-space truncations Towner and Hardy (2010b) and small deficiencies that exist in the TH formalism Miller and Schwenk (2009) need to be investigated further. Perhaps the most important future work may result from efforts towards quantifying any overall model-dependent uncertainty or possible shifts in the central values, which still remain elusive due to the extreme complexity of this phenomenological approach to the nuclear shell model.
With increasing computational power, more exact theoretical treatments which were out of reach during the early superallowed reviews, have been under investigation for the past 10 years Satuła et al. (2009); Liang et al. (2009); Auerbach (2009); Rodin (2013); Satuła and Nazarewicz (2016). So far, these new methods have provided useful insight into where some of the older phenomenological approaches may be incomplete Towner and Hardy (2010b), but have not yet reached the level of refinement needed for testing the Standard Model. These new approaches are nonetheless intriguing, as they may offer some insight into quantifying any elusive model-dependent uncertainties, particularly using ab-initio many-body approaches based on nuclear forces from chiral effective field theory (-EFT) Epelbaum et al. (2009); Machleidt and Entem (2011); Carlsson et al. (2016). These efforts are critical due to the dramatic implications of a deviation from unity in the top-row sum of the CKM matrix resulting from a shift in the calculations Satuła et al. (2009).
These modern methods are now beginning to reach levels of accuracy comparable to that of phenomenological models, including within the and shells Leach et al. (2016). As these theoretical techniques continue to evolve, they must be exposed to increasingly stringent experimental tests before they can be reliably applied to the superallowed data to extract . In particular, a reproduction of the excitation energies of the , isobaric-analogue-states (IAS), and the coefficients of the isobaric-multiplet-mass-equation (IMME) for the respective superallowed systems are critical to providing confidence in the accuracy of the calculated ISB corrections. The coefficients of the IMME are very sensitive to the subtle relative differences in binding energies of the isobaric triplet, and have been used to guide and adjust the superallowed calculations in the past Towner and Hardy (2008). This is due to the assumption that the ISB effects that shift the IAS energies is, to first order, due entirely to the Coulomb interaction, and any small deviations are due to linear and quadratic terms, represented by the and coefficients. This article presents the progress of this theoretical work in the isobaric triplet, as well reporting the most-precise -value of the superallowed positron emitter Mg.
|Measurement||Excitation||Systematic Uncertainties ()|
The experiments were conducted at TRIUMF’s Isotope Separator and ACcelerator (ISAC) facility Dilling and Krücken (2014), in Vancouver, Canada. The rare-isotope beams (RIBs) were produced via spallation reactions from a 35 A, 480-MeV proton beam incident on a SiC target. Non-ionized reaction products were subsequently released into the Ion-Guide Laser Ion Source (IG-LIS), which selectively ionized magnesium Raeder et al. (2014). The use of IG-LIS provided a suppression of surface-ionized contaminants by nearly 6 orders of magnitude, without which this measurement would not have been possible due to high levels of contamination from the surface-ionized Na. Following ionization and mass selection, the continuous 20 keV beam, consisting of roughly ions/s of Mg was delivered to TRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN) Dilling et al. (2006). The remainder of the ISAC beam consisted primarily of Na, with a rate of ions/s.
The TITAN facility consists of four primary components: (i) A Radio-Frequency Quadrupole (RFQ) linear Paul trap Smith et al. (2006); Brunner et al. (2012), (ii) a Multi-Reflection Time-of-Flight (MR-ToF) isobar separator Jesch et al. (2015), iii) an Electron-Beam Ion Trap (EBIT) for generating Highly Charged Ions (HCIs) Sikler et al. (2005); Lapierre et al. (2010) and performing decay spectroscopy Leach et al. (2015); Lennarz et al. (2014), and (iv) a 3.7 T, high-precision mass Measurement PEnning Trap (MPET) Brodeur et al. (2012). Following the delivery of the continuous ISAC beam to TITAN, ions were injected into the RFQ where they were cooled using a He buffer gas. The resulting ion bunches were then transported with a kinetic energy of 2 keV to the Penning trap, where individual singly charged ions were captured for study.
In MPET, the mass of a single ion is determined by measuring its characteristic cyclotron frequency using the Time-of-Flight Ion-Cyclotron-Resonance (ToF-ICR) technique Gräff et al. (1980); KÃ¶nig et al. (1995). To further improve the measurement uncertainties, TITAN’s stable ion source was also used to deliver Na ions in addition to the RIB from ISAC. Reference measurements were taken both before and after each Mg run in cycles of Na-Mg-Na, which were then repeated. For the determination of the resonance frequency ratios, only cycles with 1 detected ion/cycle were used in order to reduce effects on the measurement which may result from ion-ion interactions (), which was the largest systematic uncertainty in this work. The error estimate for multiple ion interactions in the Penning trap during RF excitation was determined through a count-class analysis Kellerbauer et al. (2003). The measured frequency ratios, as well as the error budgets for each systematic in the TITAN system, are given in Table 1, based largely on the studies of Refs. Brodeur et al. (2012).
Systematic effects related to time-dependent magnetic field fluctuations () were the second largest systematic, and thus the time between measurements was kept between 30-45 minutes. Two smaller systematics related to the magnetic field in TITAN Brodeur et al. (2009) were also included: the magnetic field decay of the MPET solenoid (, and the field alignment (). For referencing to the well-known Na mass, a small mass-dependent frequency shift () was therefore accounted for between Mg and Na. Finally a small relativistic systematic was applied using the prescription of Ref. Brodeur et al. (2009).
Both quadrupole and Ramsey resonance schemes were used (Fig. 1) with excitation times in the Penning trap of 500 ms for Mg, Na, and Na. Using these measurements the extracted mass-excess for Mg and Na, along with the direct MgNa -value measurement, are presented and compared to the most recent atomic mass evaluation (AME16) Wang et al. (2017) and Hardy/Towner superallowed review (HT15) Hardy and Towner (2015) in Table 2. The values presented in this work agree with the respective experimental reviews, but provide an increase in precision to the evaluated data in each case. In fact, using the prescription for the superallowed -value review outlined in Ref. Hardy and Towner (2015), the inclusion of the work reported here results in a slightly lower Mg -value, with a 30% increase in precision.
To further push the precision limits of the extraction of from the superallowed data, benchmarking of state-of-the art ab-initio theoretical methods to the IMME in these heavier systems was also performed, following first attempts in systems using many-body perturbation theory Gallant et al. (2014). IAS energies of the multiplet were calculated within the ab-initio valence-space in-medium similarity renormalization group (VS-IMSRG) Tsukiyama et al. (2012); Bogner et al. (2014); Stroberg et al. (2016, 2017). Calculations begin from two different sets of two-nucleon (NN) and three-nucleon (3N) forces derived from -EFT Epelbaum et al. (2009); Machleidt and Entem (2011). The first method, NN+3N(400), uses the standard NN interaction at order NLO of Ref. Entem and Machleidt (2003); Machleidt and Entem (2011) combined with the NLO 3N force of Ref. Navratil (2007) with momentum cutoff MeV. These interactions are simultaneously evolved with the free-space SRG Bogner et al. (2007) to a low-momentum scale of . This Hamiltonian reproduces experimental data in the upper and lower shells, making it a potentially good choice for the nuclei studied here. The second NN+3N interaction, 1.8/2.0(EM) Hebeler et al. (2011); Simonis et al. (2016, 2017), uses the same initial NN interaction as above but is SRG-evolved to , with undetermined 3N force couplings fit to reproduce both the triton binding and alpha particle charge radius at . This Hamiltonian reproduces ground-state energies across the nuclear chart from the shell to the tin region Hagen et al. (2016); Garcia Ruiz et al. (2016); Simonis et al. (2017); Lascar et al. (2017). The resulting calculations of the IAS states are compared to the experimental data from this work and Ref. Wang et al. (2017) in Fig. 2, including the ground state in Na.
|Ref. Lam et al. (2013)||-4524.36(21)||-3812.39(16)||312.03(26)|
The energies caluclated using the NN+3N(400) approach are somewhat overbound in these systems (ranging from 3-5 MeV), and show relatively poor agreement with subtle differences in the nuclear structure. Of particular note for the results presented here, the excitation energy of the IAS in Na is overestimated by several hundred keV in these calculations. For the 1.8/2.0(EM) set, however, the agreement is significantly improved. While it does give a consistent underbinding of roughly 2 MeV, the subtle relative differences due to nuclear shell effects are now well reproduced. This includes the excitation energy of the , IAS in Na which is at an 8 keV level of agreement with experiment. Of course, the full theoretical uncertainties are likely larger than this (and a subject of current study Carlsson et al. (2016)), however along with the strong agreement across the nuclear chart using the 1.8/2.0(EM) approach, this indicates that these methods are nonetheless approaching the level of accuracy achievable with currently adopted phenomenological methods.
The results of the calculations are also compared to the experimental atomic masses from this work and Ref. Wang et al. (2017) within the framework of the IMME, shown in Table 3. In both calculations, the coefficient is lower than the experimentally observed value, however both deviate by less than 15%. For the coefficient, however, the NN+3N(400) calculations yield a value that is greater than experiment by more than a factor of 4, while the 1.8/2.0(EM) calculations are less than a factor of two higher. As the coefficient is particularly sensitive to the Coulomb contribution of the pairing force, and largely responsible for the breaking of isospin symmetry Towner and Hardy (2008), this result suggests that future work related to ab-initio calculations of can be reliably based on the 1.8/2.0(EM) theoretical approach.
In summary, the most precise value of the superallowed emitter Mg was measured using Penning-trap mass spectrometry with TITAN at TRIUMF. This value, along with previous measurements evaluated in Ref. Hardy and Towner (2015), yield an updated keV value that is 30% more precise. When combined with a very recent high-precision measurement of the half-life performed at TRIUMF ( s) Dunlop (2017), an updated value of 3077.0(71) s is extracted, which is in agreement with the value quoted in the most recent review of Ref. Hardy and Towner (2015).
The measured mass-excess value for Mg was also measured to a higher precision than the previous evaluation of Ref. Wang et al. (2017), and remains in good agreement. Using this value, along with the evaluated IAS energies for Na and Ne, new coefficients of the IMME were also derived. State-of-the-art ab-initio shell-model calculations of the IAS energies were used to compute the and coefficients of the IMME with a comparison to the high-precision experimental data in a continued push towards calculating from first principles across all superallowed cases. The VS-IMSRG approach based on the 1.8/2.0 (EM) NN+3N interaction reproduced experimental values well and was able to reproduce the excitation energy of the IAS state in Na. With the improved binding-energy reproduction in the triplet and the spectroscopic agreement seen in Na, as well as across the medium-mass region of the nuclear chart Simonis et al. (2017), these calculations suggest that extracting sensitive ISB corrections to superallowed decays from ab-initio methods can now be considered and explored in a more controlled manner.
The authors thank J. Simonis and A. Schwenk for providing the 1.8/2.0 (EM) 3N matrix elements, and A. Calci for providing the NN+3N(400) 3N matrix elements used in this work. K.G.L. would like to thank G.F. Grinyer and J.C. Hardy for useful discussions on the multiplet and superallowed review. This work is supported in part by the National Sciences and Engineering Research Council of Canada (NSERC), the U.S. Department of Energy Office of Science under grant DE-SC0017649, the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344, and the U.S. National Science Foundation under grant PHY-1419765. TRIUMF receives federal funding via a contribution agreement with the National Research Council of Canada (NRC). Computations were performed with an allocation of computing resources at the Jülich Supercomputing Center (JURECA). M.P.R. is funded by Justus-Liebig-Universität Giessen and GSI under the JLU-GSI strategic Helmholtz partnership agreement. A.T.G. and E.L. acknowledge student support from the NSERC CGS-D program, and Brazil’s Conselho Nacional de Desenvolvimento Científico e Technológico (CNPq), respectively.
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