Breakdown of the Isobaric Multiplet Mass Equation for the A = 20 and 21 Multiplets

Breakdown of the Isobaric Multiplet Mass Equation for the A = 20 and 21 Multiplets


Using the Penning trap mass spectrometer TITAN, we performed the first direct mass measurements of Mg, isotopes that are the most proton-rich members of the and isospin multiplets. These measurements were possible through the use of a unique ion-guide laser ion source, a development that suppressed isobaric contamination by six orders of magnitude. Compared to the latest atomic mass evaluation, we find that the mass of Mg is in good agreement but that the mass of Mg deviates by 3. These measurements reduce the uncertainties in the masses of Mg by 15 and 22 times, respectively, resulting in a significant departure from the expected behavior of the isobaric multiplet mass equation in both the and multiplets. This presents a challenge to shell model calculations using either the isospin non-conserving USDA/B Hamiltonians or isospin non-conserving interactions based on chiral two- and three-nucleon forces.


Current Address: ]Rare Isotope Science Project (RISP), 309 Institute for Basic Science, Daejeon 305-811, Republic Current Address: ]TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3 Canada

The wealth of data obtained from experiments on increasingly exotic nuclei is only possible because of improved radioactive beam production and delivery (1). Experiments with these improved beams continually challenge existing theories, a challenge often led by precision mass measurements (2). High-precision atomic mass measurements are crucial in refining nucleosynthesis abundance calculations (3); (4), in deepening our understanding of fundamental aspects of the strong force (5); (6); (7); (8), pointing to nuclear structure change (9), and providing signatures of exotic phenomena such as nuclear halo formation (9). A tool from theory currently confronted by increasingly precise mass values is the isobaric multiplet mass equation (IMME) (10), which relates the mass excesses ME of isospin multiplet members as


where and are the mass number and total isospin of the multiplet, and , , are coefficients that depend on all quantum numbers except . In the isospin description of nucleons, the proton and neutron belong to a doublet with projections and . However, isospin is only an approximate symmetry, broken by electromagnetic interactions and the up and down quark mass difference in QCD. One can see large deviations from the IMME in the and and quartets (11); (12); (13) and in the and (14); (15); (16) quintets. Suggested explanations for these deviations include isospin mixing and second-order Coulomb effects, requiring cubic, , or quartic, , terms to be considered.

Many tests of the IMME on proton-rich systems are hindered by a long standing problem: excessive in-beam contamination. Substantial isobaric background from alkalis and lanthanides often prevents ground-state measurements of exotic nuclei, especially for nuclei produced at low rates. This applies even for beams produced using element-selective laser ionization in a classical hot-cavity resonant ionization laser ion source. We have developed a novel technology for on-line laser ion sources that suppresses any background contamination, enabling a substantial number of experiments to proceed. This new ion-guide laser ion source (IG-LIS) has allowed the first direct mass measurements of the most proton-rich members of the and isospin multiplets. Being the lightest isospin multiplets where all members are stable against particle emission, and the lightest isospin multiplets which can be described within the , , and orbitals ( shell), the and multiplets provide an excellent test of the IMME. This can be done experimentally by high-precision mass measurements of Mg and theoretically by shell model calculations using either the USDA/B isospin non-conserving (INC) Hamiltonian or interactions based on chiral effective field theory.

In a Penning trap, contaminants can be effectively removed if their ratio to the ions of interest remains . It is possible to clean beams with higher levels of contamination using either gas-filled Penning traps (17) or multi-reflection time-of-flight devices (18); (19), but these methods suffer from increased preparation times and transport losses. An alternative method is to suppress contamination from surface-ionized species at the source through the use of an IG-LIS. The IG-LIS is conceptually similar to the originally proposed ion-source trap (21); (20); however it is much simpler because no trap is formed and no buffer gas is used. Figure 1 shows a diagram of the IG-LIS. The isotope production target is typically operated at temperatures above 1900 K. The target production volume is coupled via a heated transfer tube to a positively biased radio-frequency quadrupole (RFQ) ion-guide. A repeller electrode (20); (22) reflects surface-ionized species, preventing them from entering the ion-guide volume. Neutral particles drift into the ion-guide volume, where element-selective resonant laser excitation and ionization creates an isobar free ion beam. A square-wave RF field radially confines the laser ionized beam. The ion beam is then extracted from the IG-LIS for subsequent mass separation and delivery to the experiment. A complete description of the IG-LIS can be found in Ref. (22).

The IG-LIS concept has been implemented and used online for the first time at TRIUMF’s isotope separator and accelerator (ISAC) facility. Beams of Mg were produced by bombarding a SiC target with 40 A of 480 MeV protons. Compared to previous SiC targets, we found that the IG-LIS decreased the magnesium yield by 50 times and the sodium background by times. This is a signal-to-background improvement of better than .

Figure 1: (Color online) A schematic drawing of the ISAC target with the ion-guide laser ion source (IG-LIS). See the text for a detailed description.
Figure 2: (Color online) Time-of-flight ion cyclotron resonance of Mg with 97 ms excitation time. The line is a fit of the theoretical line shape (27).
Nuclide (ms) Frequency Ratio ME (keV) ME(keV)
Mg 90 (6) 6 1401 97
Mg 122 (2) 5 31369 97
Table 1: Half-lives, number of measurements , detected number of ions , excitation times , measured frequency ratios and derived mass excesses of Mg. For the frequency ratios, the first two numbers in parentheses are the statistical and systematic uncertainties, while the number in brackets is the total uncertainty; otherwise the quoted uncertainty is the total uncertainty. Half-lives are from the NNDC (34).
Nuclide ME(g.s.) (keV) (keV)
= 20, ,
O +2 3796 .17 (89) 0 .0
F +1 -17 .45 (3) 6519 .0 (30)
Ne 0 -7041 .9306 (16) 16732 .9 (27)
Na -1 6850 .6 (11) 6524 .0 (97) 1
Mg -2 17477 .7 (18) 2 0 .0
Ref. (keV) (keV) (keV)
This Work 9689.79 (22) -3420.57 (50) 236.83 (61) 10.2
Ref. (13) 9693 (2) -3438 (4) 245 (2) 1.1
Fit (keV) (keV)
Cubic - 3.7
Quartic Only - 9.9
Quartic -
= 21, ,
F +3/2 -47 .6 (18) 0 .0
Ne +1/2 -5731 .78 (4) 8859 .2 (14)
Na -1/2 -2184 .6 (3) 8976 .0 (20)
Mg -3/2 10903 .85 (74) 3 0 .0
Ref. (keV) (keV) (keV)
This Work 4898.4 (13) -3651.36 (63) 235.00 (77) 28.0
Ref. (13) 4894 (1) -3662 (2) 243 (2) 3.0
Fit (keV)
Cubic -
USDB 0.3
= 21, ,
F +3/2 -47 .6 (18) 279 .93 (6)
Ne +1/2 -5731 .78 (4) 9148 .9 (16)
Na -1/2 -2184 .6 (3) 9217 .0 (20)
Mg -3/2 10903 .85 (74) 4 200 .5 (28) 5
Ref. (keV) (keV) (keV)
This Work 5170.4 (14) -3633.6 (10) 220.9 (10) 9.7
Ref. (13) 5171 (10) -3617 (2) 217 (2) 3.5
Fit (keV)
Cubic -
USDB 1.9
Table 2: Extracted IMME parameters for the and multiplets. Mass excesses are taken from (30) and excitation energies from (34) and (36), except where noted. Also shown are the and coefficients for cubic and quartic fits and the values of the fit. Shell model calculation results using the USDA/B plus INC interactions are presented.

The TITAN system (23) currently consists of three ion traps: an RFQ cooler and buncher (24), an electron beam ion trap (EBIT) (25) for charge breeding and in-trap decay spectroscopy, and a measurement Penning trap (MPET) (26) to precisely determine atomic masses. We bypassed the EBIT because the required precision could be reached without charge breeding. In the MPET the mass is determined by measuring an ion’s cyclotron frequency via the time-of-flight ion cyclotron resonance technique (27). A typical resonance for Mg is shown in Fig. 2. To eliminate any dependence on the magnetic field the ratio of cyclotron frequencies is taken between a well-known reference and the ion of interest. The reference ion used was Na. Because the magnetic field fluctuates in time, due to field decays and temperature or pressure variations, it must be monitored. This monitoring is achieved by performing a reference measurement before and after a measurement of the ion of interest and linearly interpolating to the time when the frequency of the ion of interest was measured. To further limit these fluctuations, the length of a measurement was limited to approximately one hour. The atomic mass of the nuclide of interest is then calculated from


where is the atomic mass of a nuclide, is the electron mass and we neglect electron binding energies. Although no contaminants were observed during the measurements, to be conservative we performed a “count class” analysis to account for ion-ion interactions. In the case of Mg, the statistics were too low ( ions/s) for such an analysis. Instead we follow the procedure in Ref. (28), where the difference between the frequency ratio taken with one detected ion and the ratio with all detected ions was added as a systematic. We also include the conservative mass dependent shift given in Ref. (29), which in the cases here amounts to ppb. Table 1 summarizes the measurement results. Our mass excess of 10903.85 (74) keV for Mg agrees well with the tabulated value from AME2012 (30). The mass excess of 17477.7 (18) keV for Mg deviates by 81 keV as compared to the AME2012, which is a shift of 3. In both cases the precision was increased by more than one order of magnitude.

Determining the energy level of an isospin multiplet member relies on knowing both the ground-state and excited-state energies accurately. For different experimental techniques, the measured excitation energy depends on separation energies, which can change with improved mass measurements. New mass measurements of the ground states of Na (31) and Ne (32) led to an improved proton separation energy value for Na of 2190.1(11) keV. This value is required to derive the excitation energy of the state in Na. Combining a new measurement of the excitation energy (33) with the value compiled in (31), an averaged value of 6524.0(98) keV is obtained, a result that is shifted by 10 keV relative to the tabulated value (34). In Mg, a new measurement of the state was completed (35), which, when averaged with the NNDC (36) value, yields 200.5(28) keV. Both of these new values are included in the following analysis.

Table 2 summarizes the fit results of the quadratic, and higher order forms of the IMME for the and multiplets. For each multiplet the of the fit greatly increased, as compared to the tabulated values (13). The most uncertain member of the multiplet is now Na, with nearly all of the uncertainty originating from the excitation energy of the state. The best fit is obtained when a cubic term is included, resulting in keV, and , a result that is an order of magnitude larger than the literature value of 0.2 (13). For the , multiplets, the for a quadratic fit have increased to 28 and 9.7 for the and , respectively, as compared to the literature values of 3.0 and 3.5 (13). The IMME clearly fails in both of the multiplets. Large cubic terms are required for both multiplets, with for the multiplet and for the multiplet.

Exploring the impact of isospin-symmetry breaking on nuclear structure is an exciting field of research, as it relates to the symmetries of QCD and their breaking. The shell is particularly interesting because it can be accessed by phenomenological and ab initio methods. The phenomenological isospin-symmetric USD Hamiltonians (37) generally reproduce data very well throughout the shell, but ultimately need to be supplemented with an isospin non-conserving (INC) part (38). In addition, there are valence-space calculations based on chiral two-nucleon (NN) and three-nucleon (3N) forces, without phenomenological adjustments. The resulting shell Hamiltonians are inherently isospin asymmetric and have successfully described proton- and neutron-rich systems (39); (40), but it is still an open question how well they work in systems with both proton and neutron valence degrees of freedom. Therefore the current measurements provide valuable new tests of these methods.

We first calculated the IMME in the shell with the USDA and USDB isospin-conserving Hamiltonians (37), supplemented with the INC Hamiltonian of Ref. (38). The results for the and and coefficients are presented in Tab. 2. For , the USDA value for , which comes from mixing of states with similar energy but different isospin in F, Ne, and Na, agrees with experiment only when is also included in the IMME fit. Here, the largest mixing comes from a pair of close-lying , states in Ne. With the USDB Hamiltonian, these two states are nearly degenerate resulting in an uncertainty of the energy of the good isospin states that is too large to give a meaningful result. The calculated term, on the other hand, comes from mixing in F and Na. With the USDB, the , levels in these nuclei are well separated from the isobaric analog state (IAS), leading to a small energy-mixing shift and hence a too small value.

For the systems, the USD values of also do not agree with experiment. The non-zero values come from mixing of the states with close-lying states in Ne and Na. For instance, the largest shift in the multiplet is due to a state in Ne, which for the USDA lies 372 keV below the IAS, instead of the 50 keV necessary to reproduce  keV. Experimentally, several states with unidentified spin lie around the IAS (34). This illustrates the challenge in obtaining accurate calculations capable of describing the new experimental findings.

In addition we calculated the properties of the and multiplets from valence-space Hamiltonians constructed within the framework of many-body perturbation theory (41), based on low-momentum (42) NN and 3N forces derived from chiral effective field theory (43), without empirical adjustments. These isospin-asymmetric Hamiltonians describe ground- and excited-state properties in neutron-rich oxygen isotopes (44); (39); (45) and proton-rich isotones (40). Here we use the valence-space Hamiltonians of Refs. (39) and Ref. (40) for the neutron-neutron and proton-proton parts, respectively, and include for the first time valence-space proton-neutron interactions. The ground-state energies of Mg are shown in Table 3. The calculated ground-state energy of Mg is in very good agreement with experiment, while Mg, with one neutron above the closed shell, is overbound by 1.6 MeV. For the multiplet, the coefficient is found to be keV, i.e. giving isospin-symmetry breaking larger than in experiment. As increases, other members of the multiplet become less overbound than Mg (F is only 0.8 MeV overbound). This, however, also results in larger cubic terms for the multiplets ( keV for , ). Therefore, the new experimental findings cannot be described with these Hamiltonians, but they nonetheless provide a promising first step towards understanding isospin-symmetry breaking based on electromagnetic and strong interactions.

Nuclide Exp. USDA USDB NN + 3N
Table 3: Experimental and calculated ground-state energies (in MeV) of Mg with respect to O. USDA/B results include the INC Hamiltonian discussed in the text.

In summary, we have performed the first direct mass measurement of Mg using the TITAN Penning trap, making Mg the most exotic proton-rich nuclide to be measured in a Penning trap. The new mass of Mg is 15 times more precise and deviates from the AME12 value by 3, while the new mass of Mg agrees with the AME12 but is 22 times more precise. A quadratic fit of the IMME results in a of 10.4, reducing to 3.7 or 9.9 with the inclusion of cubic or quartic terms, respectively. The increased precision of the Mg mass now gives, for the multiplet, = 28 with a quadratic IMME, making the and multiplets new members of the shell to present strong deviations from the quadratic form of the IMME. In both cases shell-model calculations are presently unable to predict the required cubic coefficients. Further Penning trap measurements of members of other multiplets will be valuable to better test some of the proposed mechanisms for cubic and quartic terms. With the on-line use of the IG-LIS, such exotic nuclei will be available contaminant free, paving the way for new experiments far from stability.

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). A. T. G. acknowledges support from the NSERC CGS-D program and A. L. from the DFG under Grant No. FR601/3-1. This work was also supported by the BMBF under Contract No. 06DA70471, the DFG through Grant SFB 634, the Helmholtz Association through the Helmholtz Alliance Program, contract HA216/EMMI “Extremes of Density and Temperature: Cosmic Matter in the Laboratory”. Computations were performed with an allocation of advanced computing resources at the Jülich Supercomputing Center. We acknowledge support from NSF grant PHY-1068217 and the NSF REU program at MSU.


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