Mass of astrophysically relevant $^{31}$Cl and the breakdown of the isobaric multiplet mass equation

The mass of $^{31}$Cl has been measured with the JYFLTRAP double Penning trap mass spectrometer at the Ion-Guide Isotope Separator On-Line (IGISOL) facility. The determined mass-excess value, -7034.7(34) keV, is 15 times more precise than in the Atomic Mass Evaluation 2012. The quadratic form of the isobaric multiplet mass equation for the T=3/2 quartet at A=31 fails ($\chi^2_n$=11.6) and a non-zero cubic term, d=-3.5(11) keV, is obtained when the new mass value is adopted. $^{31}$Cl has been found to be less proton-bound with a proton separation energy of $S_p$=265(4) keV. Energies for the excited states in $^{31}$Cl and the photodisintegration rate on $^{31}$Cl have been determined with significantly improved precision using the new $S_p$ value. The improved photodisintegration rate helps to constrain astrophysical conditions where $^{30}$S can act as a waiting point in the rapid proton capture process in type I x-ray bursts.

The proton captures on 30 S are dominated by resonant captures to the two lowest excited states in 31 Cl.These have been studied via beta-delayed proton decay of 31 Ar [26][27][28][29] with observed laboratory energies of 446 (15) and 1415 (5) keV [26] and 1416 (2) keV [28].Recently, 31 Cl has been studied via Coulomb breakup of 31 Cl at high energy in inverse kinematics using the R 3 B-LAND setup at GSI [30].The two lowest-lying levels, 1/2 + at 782 (32) keV and 5/2 + at 1793 (26) keV [30], were found to be in a reasonable agreement with the estimations, 745 (17) and 1746 (9) keV [31], based on the IMME and beta-delayed proton data.However, also lower excitation energies, 726 (37) keV and 1731(82) keV, have been reported from R 3 B-LAND [32].A similar Coulomb dissociation study of 31 Cl performed at RIKEN resulted in resonance energies of around 0.45 and 1.3 MeV [33].In order to compare the results from R 3 B-LAND with the beta-delayed proton data, and to verify the excitation energies of the lowest resonance states in 31 Cl, the proton separation energy of 31 Cl, i.e. its mass, has to be known more precisely.
To estimate the waiting-point conditions for 30 S, also the rate for photodisintegration reactions on 31 Cl (λ γ,p ) has to be taken into account.The ratio of λ γ,p to the proton-capture reaction rate N A σv depends exponentially on the proton-capture Q value on 30 S (i.e. the proton separation energy S p of 31 Cl) [34]: where m i are the masses in atomic mass units, g i the statistical factors g i = 2J i + 1 and G i normalized partition functions for 30 S, p and 31 Cl.The normalized partition functions [35] are close to one in the relevant energy region.The uncertainty in the present Q value has been shown to significantly affect XRB nucleosynthesis calculations in a high-temperature (T peak = 2.50 GK) scenario with normal burst duration (≈100 s) as well as in a short burst (≈ 10 s) scenario with T peak = 1.36 GK [36]. 31Cl + ions were produced via 32 S(p, 2n) 31 Cl reactions using a 40-MeV proton beam impinging on a 1.8mg/cm 2 -thick ZnS target at the IGISOL facility [37].The reaction products were stopped in helium gas and extracted with a sextupole ion guide [38] and accelerated to 30 keV before mass-separation with a 55 • dipole magnet.A radiofrequency quadrupole cooler and buncher [39] was implemented to convert the continuous A = 31 beam into short ion bunches which are released into the JYFLTRAP double Penning trap mass spectrometer [40].Simultaneous magnetron and cyclotron excitations were applied for the ions in the purification trap for 40 ms to select the 31 Cl + ions using the mass-selective buffer gas cooling method [41].In the precision trap, a 10-ms magnetron excitation was followed by a short, 50-ms cyclotron excitation to minimize the decay losses of 31 Cl.The ion's cyclotron resonance frequency ν c = qB/(2πm), where q and m are the charge and mass of the ion, respectively, was determined using the time-of-flight ion cyclotron resonance (TOF-ICR) technique [42] (see Fig. 1).The magnetic field strength B was calibrated using 31 P + ions as a reference (m( 31 P) =30.9737619984(7) u [6]).Thus, the atomic mass of 31  The weighted mean of the measured frequency ratios was r = 1.00060330 (12) resulting in a mass-excess value ∆ = −7034.7 (34) keV (see Fig. 2), which is 31 keV higher than the value in the Atomic Mass Evaluation 2012 (AME12) [6].The uncertainty is dominated by the statistical error of the frequency fit.The systematic uncertainties, as described in Ref. [43], have a negligible contribution to the final result.
The IMME was studied at A = 31 using the new mass value for 31 Cl.The ground-state masses for the other members of the multiplet have been taken from AME12 [6] (see Table I).The mass values of 31 S and 31 P are based on Penning-trap measurements at JYFLTRAP [18] and the Florida State University trap [44].The mass of 31 Si is linked via (n, γ) measurements (see, e.g.Refs.[45][46][47][48]) to 29 Si, which has been precisely measured with a  Penning-trap at Massachusetts Institute of Technology [49].The excitation energy for the T = 3/2 IAS in 31 S is based on data from beta-delayed γ-rays of 31 Cl [1, 5] as well as from 31 P( 3 He, t) [50] and 33 S(p, t) reactions [51].
The energy for the IAS in 31 P has been determined with high precision using the Gammasphere detector array [52].A similar excitation energy, E x = 6380.0(20)keV, has also been obtained via 30 Si(p, γ) measurements [53][54][55].Thus, the data for the IMME are based on various independent measurements which do not show any significant discrepancies.
Table II summarizes the IMME fit results.With the new 31 Cl mass value, the quadratic IMME fails (χ 2 n = 11.6) and a significant non-zero cubic coefficient d = −3.5 (11) keV is obtained.The more precise mass for 31 Cl reveals the breakdown of the IMME: with the  [17], and the rest have been adopted from Ref. [13].The lower panel shows the significance of the deviation from zero.
AME12 mass value for 31 Cl the quadratic IMME fits perfectly well (χ 2 n = 0.08).So far, only A = 9 [16], A = 35 [20], A = 53 [21], and recently A = 21 [17], of the known T = 3/2 quartets have shown significant non-zero cubic coefficients (see Fig. 3).New precision measurements pave the way towards more fundamental understanding of the reasons behind the breakdown.Isospin mixing has successfully explained the breakdown of the IMME at A = 9 [16] and A = 32 [22] but failed at A = 21 [17] albeit detailed shell-model calculations were carried out.
The breakdown of the IMME at A = 31 is a crucial finding since the IMME prediction from Ref. [1] has been used to establish level energies in 31 Cl [60] from the betadelayed proton data of 31 Ar [26][27][28][29].The new mass value of 31 Cl shows it is less bound than previously expected.The proton separation energy S p = 265(4) keV is 31 keV lower and ≈13 times more precise than the AME12 value (S p = 296(50) keV [6]).The new mass measurement shifts all levels based on beta-delayed proton data [26][27][28][29] 18 keV lower in energy and reduces the inherent sys- tematic uncertainties from 50 keV to 4 keV.The revised energy for the J π = 5/2 + , T = 5/2 IAS in 31 Cl, the member of the T = 5/2 sextet at A = 31, is 12292.2(23)keV based on Refs.[26,28] and the S p and S 2p values from this work.The two lowest excited states in 31 Cl are relevant for the radiative resonant proton captures in the rp process.By combining the new S p value with the betadelayed proton data of Refs.[26,28], excitation energies of 726 (16) and 1728(4) keV are obtained for the 1/2 + and 5/2 + states, respectively.These are about 15 keV lower than the presently recommended values (740 (50) and 1746 (5) keV [60]) and in a perfect agreement with the R 3 B results 726 (37) and 1731(82) keV [32].The first excited state also agrees well with the USDB shell-model value of 724 keV [30].The weighted mean for the resonance energies was calculated from Refs.[26,28,32] using the S p value from this work for Ref. [32].The values from Ref. [30] deviate by ≈ 2σ and were not included.The resulting resonance energies, E r = 461 (15) keV and 1463(2) keV, are very close to the beta-delayed proton data [26,28] and do not change the calculated protoncapture rates from Ref. [31].
The new S p value was used to compute the ratio of photodistegration rate on 31 Cl to the proton capture rate on 30 S according to Eq. 1 and using λ p,γ = N A σv r ρ Xi mH for typical XRB conditions with density ρ = 10 6 g/cm 3 and hydrogen mass fraction of X H = 0.73.The uncertainty related to the Q value has been significantly reduced and photodisintegration rate takes over at lower temperatures compared to the ratio calculated with the AME12 Q value (see Fig. 4).Above 0.44(1) GK, at least 20 % of the reaction and decay flow has to wait for β + decay of 30  per temperature limit for 30 S waiting point, 1.0(3) GK, comes from the rate of the unmeasured 30 S(α, p) 33 Cl reaction [31].
The JYFLTRAP Penning-trap mass measurement of 31 Cl has shown that the quadratic IMME fails at A = 31 and the cubic term is non-zero.Theoretical calculations are anticipated to explain the deviation from the quadratic form and to explore possible underlying reasons for similarities in the cubic coefficients for A = 31 and A = 35.Isospin mixing between T = 1/2 and T = 3/2 states is plausible as there are candidates for 3/2 + states lying nearby the IAS.The improved precision in the proton separation energy of 31 Cl has reduced the uncertainties related to excitation energies in 31 Cl and the photodisintegration rate of 31 Cl.Photodisintegration starts to dominate at lower temperatures than previously thought.The improved rate will be useful for future XRB model calculations helping to interpret the observed double-peaked structure in the luminosity curves.
Cl was determined using m( 31 Cl) = r(m ref − m e ) + m e , where r = ν c,ref νc is the cyclotron frequency ratio of 31 P + and 31 Cl + , m ref and m e are the 31 P and electron masses, respectively.

FIG. 1 .
FIG. 1. (Color online) TOF-ICR spectrum of 31 Cl + with a quadrupolar RF excitation of 50 ms.The spectrum represents a typical resonance of 31 Cl obtained in 140 minutes.The blue squares indicate the number of ions in each time-of-flight bin: the darker the color, the greater the number of ions.

FIG. 2 .
FIG. 2. (Color online) Mass-excess values determined in this work.The red line shows the weighted mean of the results and the dashed blue lines 1σ error bands.

FIG. 3 .
FIG.3.Cubic coefficients for the known (lowest) T = 3/2 isobaric quartets.The value for A = 31 (red square) is from this work, for A = 21 from Ref.[17], and the rest have been adopted from Ref.[13].The lower panel shows the significance of the deviation from zero.

FIG. 4 .
FIG. 4. (Color online)The ratio of (γ, p) to (p, γ) rates for typical XRB conditions.The uncertainties related to the JYFLTRAP Q value are shown by the blue lines and to the AME12 value by the grey-shaded area.

TABLE I .
[6]s-excess values ∆ and excitation energies Ex for the J π = 3/2 + , T = 3/2 isobaric analog states at A = 31.The mass-excess value of31Cl is from this work, the others are from the AME12[6].

TABLE II .
Coefficients for the quadratic and cubic IMME fits (in keV) for the T = 3/2 quartet at A = 31.
a The parameter uncertainty has been scaled with χ 2 n .