aa r X i v : . [ nu c l - e x ] S e p Si β + -decay spectroscopy ∗ L. J. Sun ∗ ,
1, 2, † M. Friedman ∗ ,
1, 3, ‡ T. Budner,
D. P´erez-Loureiro, E. Pollacco, C. Wrede,
1, 4, § B. A. Brown,
M. Cortesi, C. Fry,
B. E. Glassman,
J. Heideman, M. Janasik,
1, 4 and P. Tiwari
1, 4 National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA Canadian Nuclear Laboratories, Chalk River, Ontario K0J 1J0, Canada IRFU, CEA, Universit´e Paris-Saclay, F-91191, Gif-sur-Yvette, France Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA (Dated: September 15, 2020)
Background : β -decay spectroscopy provides valuable information on exotic nuclei and a stringenttest for nuclear theories beyond the stability line. Purpose : To search for new β -delayed protons and γ rays of Si to investigate the prop-erties of Al excited states.
Method : Si β decays were measured by using the Gaseous Detector with Germanium Taggingsystem at the National Superconducting Cyclotron Laboratory. The protons and γ rays emittedin the decay were detected simultaneously. A Monte Carlo method was used to model theDoppler broadening of Mg γ -ray lines caused by nuclear recoil from proton emission. Shell-modelcalculations using two newly-developed sd -shell Hamiltonians, USDC and USDI, were performed. Results : The most precise Si half-life to date has been determined. A new proton branchat 724(4) keV and new proton- γ -ray coincidences have been identiﬁed. Three Mg γ -ray linesand eight Al γ -ray lines are observed for the ﬁrst time in Si decay. The ﬁrst measurement ofthe Si β -delayed γ ray intensities through the Al unbound states is reported. All the boundstates of Al are observed to be populated in the β decay of Si. Several inconsistencies betweenthe previous measurements have been resolved, and new information on the Al level scheme isprovided. An enhanced decay scheme has been constructed and compared to the mirror decay of Na and the shell-model calculations.
Conclusions : The measured excitation energies, γ -ray and proton branchings, log ft val-ues, and Gamow-Teller transition strengths for the states of Al populated in the β decay of Siare in good agreement with the shell model calculations, oﬀering gratifyingly consistent insightsinto the ﬁne nuclear structure of Al.
The investigation of exotic nuclei lying far from the sta-bility line has been one of the attractive topics of nuclearphysics during the past few decades . β -decay stud-ies have proved to be a powerful tool to obtain a varietyof spectroscopic information on nuclei far from stabilitythat are diﬃcult to obtain otherwise [2, 3], which pro-vides an excellent and stringent test of nuclear structuretheories and fundamental symmetries  and also deep-ens our understanding of the astrophysical rapid protoncapture process  and p -process .Nuclei near the proton drip line with large Q values for β + decay and low proton separation energies often decay ∗ These authors contributed equally to this work and should beconsidered co-ﬁrst authors. † [email protected] ‡ [email protected] § [email protected] by β -delayed proton emission ( βp ). Since the discoveryof the ﬁrst βp emitter Si in 1963, a total of 196 βp emitters (including isomers) have been identiﬁed rangingfrom C ( Z = 6) to Lu ( Z = 71) . The β decay of Sihas been one of the most studied cases [8–17]. All the β -decay measurements of Si were focused on the pro-ton spectrum, whereas the γ -ray spectrum has not beenmeasured with high statistics. Construction of the de-cay scheme based solely on proton spectra could lead toinaccurate assignments. Thomas et al .  reported themost comprehensive measurement but with very limited γ -ray information. They may have missed some of thelow-intensity and high-energy γ rays due to low statis-tics and low eﬃciency. The existing information on Sidecay properties is still incomplete and therefore moti-vates new experiments to search for new β -delayed par-ticles and γ rays. Detecting protons and γ rays in Si β decay and the coincidence between them allows one toreliably construct the decay scheme. Si β -decay spec-troscopy provides a sensitive and selective means to probethe properties of Al excited states as well as a goodveriﬁcation of the information on the structure of Alpreviously collected by other experimental approaches.It should be noted that most of the information onthe β -delayed proton decay of Si was obtained with sil-icon implantation detectors. A major problem for thismethod is the strong β -summing eﬀect caused by energydeposited by β particles . Robertson et al . employeda gas-silicon detector telescope to detect Si β -delayedprotons for the ﬁrst time. Despite the small solid an-gle coverage and the existence of dead layers for inci-dent particles, they were able to identify several newlow-energy proton peaks . Hence, the development ofcomplementary experimental tools for the clean detectionof low-energy β -delayed proton branches is particularlyvaluable.In this work, the emitted particles and γ rays in the β decay of Si were measured simultaneously with high ef-ﬁciency and high energy resolution. Combining all avail-able experimental information yields an improved decayscheme of Si, which is compared to theoretical calcula-tions and to the β − decay of the mirror nucleus, Na.A comparison between the mirror Gamow-Teller decaysalso provides an opportunity to investigate isospin asym-metry. A nonzero mirror asymmetry parameter impliesabnormal nuclear structure, such as halo structure in theinitial and/or ﬁnal state. In view of the asymmetries re-ported in the nearby sd -shell nuclei Mg − O [18–21], Si − O [22, 23], Si − Al [24–26] P − Na [27, 28], S − Na [29–32], it is desirable to extend this test to Si and its mirror partner nucleus Na [33–35].
II. EXPERIMENTAL TECHNIQUES
The experiment was conducted at the National Super-conducting Cyclotron Laboratory (NSCL) in May 2018.The experimental procedure has been detailed in Ref. and is brieﬂy repeated here for completeness. A Ar primary beam was accelerated by the K500 and K1200Coupled Cyclotron Facility to 150 MeV/nucleon at abeam current of ∼ p nA. The secondary Si beam wasproduced via the projectile fragmentation of the Arbeam impinging on a 1363 mg / cm thick Be targetand puriﬁed using the A1900 fragment separator .The Gaseous Detector with Germanium Tagging (GAD-GET) , comprised of the Proton Detector and theSegmented Germanium Array (SeGA) , has beenbuilt and successfully commissioned to measure the de-cays for the nuclei near the proton-drip line. In the cur-rent experiment, a total of 3 × Si ions were im-planted into the gaseous Proton Detector with an aver-age beam rate of approximately 1800 particles per sec-ond. The Proton Detector was ﬁlled with P10 gas mix-ture at a pressure of 780 Torr, which is ideally suitedfor low-energy proton detection because the backgroundcontributed by β particles was mitigated. The charged-particle measurement was carried out under a pulsed-beam mode, i.e., the beam ions were delivered for 500 ms, then the decays were detected during the 500-ms beam-oﬀ period. The Proton Detector was mounted at the cen-ter of SeGA, which consists of 16 high-purity germaniumdetectors arranged into two rings surrounding the ProtonDetector. These two rings of 8 detectors will be referredto as “upstream” and “downstream”. The detection forthe γ rays emitted from decays was done over both thebeam on/oﬀ periods. The preampliﬁer signals from theProton Detector and SeGA were read into Pixie-16 cards(16-Channel 250 MHz PXI Digital Processor) and pro-cessed by the NSCL digital data acquisition system . III. ANALYSISA. γ -ray energy and eﬃciency calibration To create a cumulative γ -ray energy spectrum, thespectrum of each SeGA detector was linearly gain-matched run by run using room background lines at1460 . ± .
005 keV and 2614 . ± .
010 keV from the β decays of K  and
Tl , respectively. Anexponentially modiﬁed Gaussian (EMG) function of theform f ( x ; N, µ, σ, τ ) = N τ exp (cid:20) (cid:16) στ (cid:17) + x − µτ (cid:21) × erfc (cid:20) √ (cid:18) στ + x − µσ (cid:19)(cid:21) , (1)was used to ﬁt each β -delayed γ -ray line in the spectrum.The EMG is characterized by an exponential decay con-stant τ , width σ , mean µ , energy x , and area below thecurve N . Also, a linear function is added to this formulato model the local background. Four Si β -delayed γ -raylines with known energies and the corresponding absoluteintensities shown in brackets: 451.7(5) keV [18.4(42)%],493.3(7) keV [15.7(34)%], 944.9(5) keV [10.4(23)%], and1612.4(5) keV [14.7(32)%] [17, 42, 43], were observed withhigh statistics and used as energy calibration standards.The maximum values from the ﬁts of these γ -ray lineswere plotted against the standard energies to provide aninternal energy calibration of each SeGA detector. In thiswork, all the γ -ray energies are reported in the labora-tory frame, and all the excitation energies are reported inthe center-of-mass frame with recoil corrections applied.One of the 16 SeGA detectors malfunctioned during theexperiment and three of the others displayed relativelypoor resolutions, so these four detectors are excludedfrom the subsequent analysis. A cumulative spectrum in-corporating the other 12 SeGA detectors was generatedfor analysis. The characteristic resolution for the cumu-lative SeGA spectrum is 0.19% FWHM at 1612 keV.To reduce the systematic uncertainty associated withextrapolation, further energy calibration was applied byincluding four Si( βpγ ) Mg lines known with very goodprecision at 1368.626(5), 2754.007(11), 2869.50(6), and4237.96(6) keV as standards . These γ rays areemitted from the recoiling Mg after the β -delayed pro-ton emission of Si. Therefore, the γ -ray line shape isDoppler-broadened, and the regular EMG function is notsuited to ﬁt the peak. To accurately extract informa-tion from each peak, we applied Doppler broadening lineshape analysis. The detailed procedure will be describedin Sec. III D. B. Proton energy and eﬃciency calibration
As detailed in Ref. , the anode plane of the Pro-ton Detector is segmented into 13 readout pads, labeledA − M. The β -delayed proton spectrum is usually pro-duced by event-level summing of the ﬁve central pads(A − E) and the eight surrounding pads (F − M) are usu-ally used to veto the high-energy protons that escapethe active volume and deposit only part of their en-ergy in the active volume. In the current experiment,four veto pads (F, G, L, M) were not instrumented, sothe resulting background caused by the escaping high-energy protons hindered the identiﬁcation of low-energyprotons. Instead, we could obtain the proton spectrummeasured by three central pads (A+C+D) and used theother six neighboring pads (B, E, H, I, J, K) as veto trig-gers. Our Proton Detector is not sensitive to protonsabove 2.4 MeV. The strong β -delayed proton peaks at402, 1268, and 1924 keV were used for the energy cal-ibration of the Proton Detector. We took a weightedaverage of the literature proton energies [12, 15–17] ascalibration standards. The proton-detection eﬃciencysimulated for full utilization of all 13 readout pads cannot be used in this case. Even though the proton in-formation in our work is incomplete, we can normalizethe literature relative intensities for each proton branchto the Si( βpγ ) Mg intensities measured in this work(Sec. IV B).
In Ref. , we investigated the longitudinal beam dis-tribution via the proton drift time distribution, and thebeam in the radial direction was estimated as a Gaussianbeam with the transverse distribution determined basedon the distribution of proton counts in diﬀerent pads ofthe Proton Detector. The investigation showed that the Si beam ions were mainly contained in the active vol-ume of the Proton Detector. We modeled the Brownianmotion of the Si atoms using a Monte Carlo simulation.The diﬀusion of the Si atoms is estimated to be lessthan 1 cm within four lifetimes, and there was very littledrift of Si ions to the cathode of the Proton Detector.The β -delayed γ rays and protons from subsequent de-cays were detected by the SeGA detectors and the ProtonDetector, respectively. The geometry of our experimentalsetup and the beam spatial distribution were used as in-puts for a geant γ -ray photopeak eﬃciency curve for the SeGAdetectors. We veriﬁed the simulated eﬃciency curve byusing a Eu calibration source between 122 keV and1408 keV and Al data  up to 7801 keV. Althoughthe
Eu source was absolutely calibrated, our procedurefor determining the absolute intensities of the γ rays onlyrequires relative eﬃciencies. The simulated eﬃciency ismatched with the measured eﬃciency when scaled by aconstant factor on the order of unity. The uncertaintiesassociated with the scaling factors are 0.7% for γ -ray en-ergies < γ -ray energies > γγ summingeﬀect . Ultimately, we adopt a conservative 3% un-certainty envelope for γ -ray energies < γ -ray energies > γ -rayintensity.We adopt an I gs = 21 . β feeding for the Alground state based on our shell-model calculated I gs =20 .
9% and 22.2%. The diﬀerence between the two the-oretical I gs represents the uncertainty coming from theHamiltonian (Sec. IV F). We can perform the normaliza-tion by requiring the intensity of all decay paths sum to100%: I βp + I βγ + I gs = 100% ,I βpγ I βp = 59 . , (2)where I βp is the total intensity of all Si( βp ) Mgtransitions, and I βγ is the total intensity of all Si( βγ ) Al transitions. The intensity of the Si( βpγ ) Mg ( I βpγ ) accounts for 59.0(5)% of the total Si( βp ) Mg intensity based on the previous β -delayedproton measurements [15–17]. The remainder of ournormalization procedure is entirely based on the γ -rayintensities. Using the simulated relative eﬃciency andthe number of counts in each peak extracted from theEMG ﬁts yields the intensity for each γ ray. Multiple γ rays in one cascade are treated as one transition. Thus,we determine the total intensities of the β -delayed pro-ton and β -delayed γ decays to be I βγ = 40 . I βp = 38 . β feedings to all the unbound Al states I unb = 39 . Al states I bnd =60 . γ -ray in-tensities originating from the unbound states (Sec. IV C).Our values may be compared with the previous litera-ture values of I unb = 40 . I bnd = 58 . D. Doppler broadening analysis
When a proton is emitted from a nucleus, the daughternucleus will recoil with equal and opposite momentum asthe ejected proton due to the conservation of momentum.If a γ ray is emitted while the nucleus is still recoiling, itwill be Doppler-shifted in the laboratory frame. For anensemble of such events, the resulting γ -ray line shapein the measured energy spectrum will be Doppler broad-ened. In this experiment, we observed four γ -ray linesemitted from the Mg recoiling in the gas after the β -delayed proton emission of Si. Detailed Monte Carlosimulations have been developed to model the Dopplerbroadening. The results are then compared to the actual γ -ray data [47–52]. The simulation takes into accountthe energy and relative intensity of each proton branchpopulating the Mg excited state, the energy of the γ ray deexciting the Mg excited state, the lifetime of the Mg excited state, the stopping power of the implanta-tion material (780-Torr P10 gas), and the response func-tion of each SeGA detector.Robertson et al .  and Thomas et al .  reportedthe most comprehensive Si( βp ) Mg assignments andthey are generally in agreement. Hence, we adoptedtheir proton energies and proton-feeding intensities inthe simulation. The stopping power of the recoiling Mg in P10 gas is estimated as a function of energyusing the code srim , which is expected to be accurateto within 10% . The lifetimes for the three low-lying Mg excited states at 1368, 4123, and 4238 keVhave been precisely measured to be 1.92(9) ps, 31.7(3) fs,and 59.2(6) fs, respectively . An isotropic distribu-tion of γ rays with respect to the proton distributionis assumed in each simulation. Another input of thesimulation is the intrinsic response function for each ofthe SeGA detectors. By ﬁtting unbroadened β -delayed γ -ray peaks with the EMG function Eq. (1) at ener-gies of 451.7(5), 493.3(7), 944.9(5), and 1612.4(5) keV( Si( βγ ) Al)  and 450.70(15), 1599(2), 2908(3), and7801(2) keV ( Al( βγ ) Mg)  measured using thesame detection setup in a subsequent experiment in thesame campaign , the parameters τ and σ were char-acterized as a function of energy for each SeGA detector.Every detector has a diﬀerent contribution to the totalnumber of counts in the peak depending on its detectioneﬃciency, and the simulation accounts for this by normal-izing the number of counts simulated for each detector.A linear function is adopted to model the localbackground and added to each simulated peak whencompared to the actual data. Then, the simulation-data comparison can be done using the classical χ -minimization method. Because of the relatively lowstatistics collected in the present experiment comparedto the Mg( βpγ ) Ne experiment , the constructionof a simulated peak shape follows the same method ofRef.  with one major change. Although least-squaresbased χ statistics (e.g., Neyman’s χ or Pearson’s χ )are widely used for this type of analysis, they do not always give reliable results for low-statistics data. An al-ternative method better suited for low-statistics analysisis to derive a χ statistic from a Poisson maximum like-lihood function [56, 57]. The “likelihood χ ” is deﬁnedin the equation: χ = − λ = 2 N X i =1 y i − n i + n i ln n i y i , (3)where λ is the likelihood ratio, N is the number ofbins, n i is the number of counts in the i th bin of themeasured spectrum, and y i is the number of counts inthe i th bin predicted by the simulation. The minimiza-tion of χ is equivalent to the maximization of λ . Thebinned maximum likelihood method is known to miss theinformation with feature size smaller than the bin size ofthe spectrum. It is therefore necessary to use a ﬁne bin-ning for line shape analysis. We used a 0.1-keV bin sizefor the γ -ray spectrum. The application of this likelihoodratio χ method to the Si( βpγ ) Mg line shape analysisis discussed in Sec. IV B.
IV. RESULTS AND DISCUSSIONA. Half-life of Si The Si half-life has been previously measured tobe 225(6) ms , 218(4) ms , 232(15) ms , and222.6(59) ms . A weighted average of all previouslypublished values gives t / = 221 . Si by using the countrate of the 402-keV proton as a function of time duringthe decay period of the implant-decay cycle. Here, wefurther investigated the systematics to provide a half-lifemeasurement. The Si half-life is extracted by ﬁttingthe count rate of all the protons within 350-2400 keVrecorded by the ﬁve central pads as a function of timeelapsed since the beginning of each implant-decay cycle.The decay in the count rate is enhanced by diﬀusion of Si out of the active volume. This eﬀect is modeledby a Monte Carlo simulation of the Brownian motion ofthe Si atoms. The eﬀect of Si losses due to diﬀusionout of the central pads is parameterized by a 4 th degreepolynomial P ( t − t ) where t is the clock time within thecycle and t is the beginning of the decay period of thecycle. The data are ﬁt using the function: f ( t ; N, t / , t , B ) = N e − ln(2)( t − t t / P ( t − t ) + B, (4)where N is the initial count rate of protons at the be-ginning of the decay period of the cycle. We measuredthe background during the interval between each run andestimated the background level B to be 0 . t is estimated to be 0.05 ms, and the systematic eﬀect TABLE I. Uncertainty budget for the measured half-life of Si.Source of uncertainty Uncertainty (ms)Statistics ± . Si atoms +0 . − . Starting time of decay period ± . ± . +1 . − . associated with the ﬁt range is estimated to be ± . t and the ﬁt range within reasonable val-ues. The diﬀusion is estimated to decrease the decaylifetime by 1.8 ms. However, this assumes that the Siis in the atomic form. In reality, it is plausible that Siatoms bond with hydrogen and carbon atoms that ex-ist in the P10 mixture. As a result, the diﬀusion is ex-pected to decrease in a non-trivial manner. We thenestimate other systematic eﬀects due to the diﬀusion byvarying the initial Si beam distribution in the volumeand the gas pressure of the simulation. The total un-certainty associated with diﬀusion is determined to be +0 . − . ms. Other eﬀects, such as the trigger threshold,the time window for the trigger, and the contributionof the beam contaminants, are found to be negligible.The eﬀects contributing to the uncertainty are summa-rized in Table I. The ﬁnal result is determined to be t / = 218 . ± . +0 . − . (syst) ms, where the uncer-tainties are statistical and systematic, respectively. Thisvalue can be written as t / = 218 . +1 . − . ms with thestatistical and systematic uncertainties added in quadra-ture. As can be seen from Fig. 1, our result is consis-tent with, and more precise than, all the literature val-ues [9, 11, 15, 16]. We have reevaluated the half-life tobe t / = 219 . +0 . − . ms by taking a weighted average ofall published values. B. Si( βpγ ) Mg A 1369-keV γ ray originating from the ﬁrst excitedstate of Mg was observed in the previous Si β -decaymeasurements [14, 17]. In this work, we observed threeadditional Mg γ -ray lines following Si β -delayed pro-ton emissions. Figure 2 shows four γ -ray lines at 1369,2754, 2870, and 4238 keV, corresponding to the deexci-tations from the three lowest-lying Mg excited statespopulated by Si( βp ). The placement of γ rays is alsoveriﬁed using γγ coincidences. To remove the distur-bance from a room background γ -ray line near the 1369-keV peak, the Proton-Detector-gated γ -ray spectrum wasgenerated for the 1369-keV peak. This coincidence gateis set by any signal above 15 keV recorded by the Proton +1.01.4 Present218(4)Reeder1966
Measurements H a l f- li f e ( m s ) FIG. 1. Half-life of Si measured in the present work com-pared with the values previously measured by McPherson etal . , Reeder et al . , Hatori et al . , and Robertson et al . . The weighed average of all published values isindicated by the solid red band.
Detector, and all the protons emitted from decays essen-tially have equal probabilities to trigger a 10- µ s back-ward time window and select the coincident γ -ray sig-nals. Therefore, the Proton-Detector gate only reducesthe number of counts in the 1369-keV γ -ray line and doesnot alter its relative proton feedings and the resultingpeak shape. Distributions of χ values from the simu-lated and experimental spectra were constructed for eachpeak. An example of the χ distribution of the 1369-keV γ -ray line is shown in Fig. 3, where the γ -ray centroidis considered a free parameter for χ minimization. Thebest-ﬁt peak centroid and integral as well as their statis-tical uncertainties ( χ +1) were taken from a quadraticpolynomial ﬁt of the χ distribution. We obtained the re-duced χ value ( χ ν ) by dividing the χ value by the num-ber of degrees of freedom. Each statistical uncertainty isthen inﬂated by the square root of the χ ν value for thecorresponding ﬁt. We are able to achieve a minimum inthe χ ν distribution close to 1 for all four Mg γ -ray lines,using the proton energies and the relative proton feedingintensities measured by Thomas et al . . The resultantbest ﬁts from the χ minimization are shown in Fig. 2.Replacing the proton energies and the relative protonfeeding intensities with the values measured by Robert-son et al .  in our simulation yields very similar χ values. Our Doppler-broadening analysis does not havesuﬃcient sensitivity to distinguish discrepancies in theintensities of weak proton branches in this relatively low-statistics case. Nevertheless, their proton inputs both ﬁtthe γ -ray data equally well, indicating that both of theprevious measurements placed the majority of the protonintensity in the decay scheme correctly.The γ -ray intensities per Si β decay ( I βpγ ) are de-rived from the integral of each ﬁt corrected for the SeGAeﬃciency and normalized to the aforementioned total γ -ray intensity. The lifetime and the ﬁt parameters τ and σ are varied by their one standard deviation uncertaintyand the stopping power is varied by ±
50% to investigate Al Simulation Background p4545 p4252 p3610 p3463 p3231 p2980 p2162 p1268 p943 p555 p1804 p1489 p13770200400600800 C oun t s p e r . k e V (a) EMG MC R e s i du a l s Energy (keV) Al Simulation Background C oun t s p e r . k e V (b) EMG MC R e s i du a l s Energy (keV) (c) EMG MC Al p1377 Simulation Background0102030405060 C oun t s p e r . k e V R e s i du a l s Energy (keV)010203040 p1377 Al Simulation Background C oun t s p e r . k e V (d) EMG MC R e s i du a l s Energy (keV)
FIG. 2. γ -ray spectrum measured by the SeGA detectors magniﬁed at (a) 1369 keV, (b) 2754 keV, (c) 2870 keV, and (d)4238 keV. We show the raw spectrum for panel (b), (c), and (d) and the Proton-Detector-gated spectrum for panel (a) tosuppress the contribution from a room background line near the 1369-keV peak. Four upper panels: The Monte Carlo (MC)simulations of the (a) 1369-keV, (b) 2754-keV, (c) 2870-keV, and (d) 4238-keV γ -ray peaks are produced by using lifetimesadopted from the data evaluation  and the proton energies and relative proton feeding intensities measured by Thomas etal . . The black dots represent the data, the green lines denote the background model, the red lines denote the simulatedline shapes including diﬀerent contributions of proton feedings. Each proton feeding is represented by a colored line, and inthe legend it is labeled with a letter p followed by its center-of-mass energy given in units of keV. The orange lines denote thesmall unbroadened contribution of the contaminant decay Al( βγ ) Mg. Four lower panels: The residual plots show the datasubtracted from the simulation. Compared with a regular EMG ﬁt of each peak, our Doppler-broadening analysis substantiallyimproved the χ ν and p -values, which are shown in the top-left corner of each panel. values Parabola fit Energy (keV)
FIG. 3. χ distribution of the 1368.672(5)-keV γ -ray line asa function of input γ -ray energy (black squares). A quadraticpolynomial ﬁt (solid red line) was used to determine the bestﬁt energy and uncertainty. the systematic uncertainty. The uncertainties associatedwith the aforementioned simulated eﬃciency, the stop-ping power of the P10 gas, the lifetime of Mg states, theproton feedings, the parameters τ and σ , and the devi-ation caused by adopting proton energies and intensitiesfrom diﬀerent literature [16, 17] were added in quadra-ture to obtain the total systematic uncertainty. Addingthe systematic uncertainties with the statistical uncer-tainty in quadrature yields the total uncertainty for each γ -ray intensity.The Al( βγ ) Mg decay from the beam contami-nant Al might yield a small portion of counts in the Si( βpγ ) Mg peaks as they both produce Mg ex-cited states. The Al( βγ ) Mg lines are unbroadenedand should also be included in the Doppler broadeningsimulation. A 7070-keV γ -ray peak is identiﬁed in thespectrum, and it can only be produced by Al( βγ ) Mg.Based on the number of counts and intensity of the 7070-keV γ ray , we estimate that the beam contaminant Al comprised 0.13(4)% of the implanted beam ions.Thus, the number of counts in the 1369-, 2754-, 2870-,and 4238-keV γ -ray peaks contributed by Al( βγ ) Mgare quantiﬁed based on the SeGA eﬃciency at these ener-gies and their known intensities relative to the 7070-keV γ -ray peak [44, 58].For the 4238-keV Mg state, a further correction is re-quired. Since the 4238-keV state has two γ decay pathsto the ground state, the 2870-1369 cascade that does notdirectly decay to the ground state will yield a small por-tion of counts in the 4238-keV peak due to summing ina single SeGA detector. The number of counts in the4238-keV peak due to the summing eﬀect is calculatedfrom the number of counts in the 2870-keV peak and theSeGA eﬃciency for a 1369-keV γ ray. The loss of pho-topeak counts for the 1369- and 2870-keV γ rays due tothe summing eﬀect is corrected likewise. After correct-ing the contaminant counts and the summing counts for TABLE II. Si( βpγ ) Mg measured in the present work. The Mg ground state and three lowest-lying excited states areobserved to be populated by Si( βp ). The well-known Mgexcitation energies (column one) and γ -ray energies (columnthree) are adopted from the data evaluation . Columntwo reports the measured Si( βp )-feeding intensities to each Mg state. Column four reports the intensity of each γ -ray transition per Si decay. Column ﬁve reports the γ -raybranching ratios for each Mg excited state. E x (keV)  I βp (%) E γ (keV)  I βpγ (%) B.R. (%)0 15.7(7) − − − the Si( βpγ ) Mg peak integral, we determine the ﬁnal Si( βpγ ) Mg intensities and γ -ray branching ratios (seeTable II).The two γ rays emitted from the 4238.24(3)-keV Mgstate at 4237.96(6) and 2869.50(6) keV are measured tohave branching ratios of 75(6)% and 25(3)%, respectively.The branching ratios are in agreement with the evalu-ated values of 78.2(10)% and 21.8(10)% , which tooka weighed average of the results in Refs. [58, 60] with in-ﬂated uncertainty. We obtain the β -delayed proton feed-ings to the 1369-, 4123-, 4238-keV Mg states per Sidecay of I βp = 21 . γ -ray decays originating from eachstate and subtracting feeding from higher lying states.The proton feeding to the Mg ground state accountsfor 41.0(5)% of the total Si( βp ) Mg intensity [15–17]. Combining this branching ratio and our measured Si( βpγ ) Mg intensities yields an I βp = 15 . Mg ground state. Thomas et al .  reported I βp = 14.3(10)%, 18.7(15)%, 1.06(20)%, and 0.41(12)%and Robertson et al .  reported I βp = 14.96(5)%,20.27(5)%, 1.105(9)%, and 0.378(8)% for the Mgground state and excited states at 1369, 4123, and4238 keV, respectively. The consistency of intensitiesfurther conﬁrms the literature interpretation of Si β -delayed proton branches. C. Si( βγ ) Al Figure 4 shows the full γ -ray spectrum measured bythe SeGA detectors. The Proton-Detector-coincident γ -ray spectrum is also shown for comparison. This coinci-dence gate reduced the statistics for the Si( βpγ ) Mgpeaks approximately by a factor of 4, which is relatedto the implant-decay cycle and the trigger eﬃciencies ofthe Proton Detector for protons. As can be seen fromFig. 4, the relative statistics for the Si( βγ ) Al peaksare even lower. This can be understood by consideringthe low trigger eﬃciency of the Proton Detector for β particles. The coincidence condition suppresses the roombackground lines substantially and helps verify the ori-gins of the γ -ray lines. Eight new β -delayed γ rays areclearly observed in the β -decay of Si, and the resultsare summarized in Table III. The uncertainty associatedwith the energy calibration of the SeGA detector and thestatistical uncertainty from peak ﬁtting were added inquadrature to obtain the total uncertainty of each γ -ray.For all the Si( βγ ) Al peaks reported in this work, thedominant source of the γ -ray energy uncertainty is thestatistical uncertainty. The absolute intensity of each γ ray in the β decay of Si is determined using the num-ber of counts in the γ -ray peak, the γ -ray detection ef-ﬁciency of the SeGA detectors, and the aforementionedtotal γ -ray intensity. A further correction for the sum-ming eﬀect is applied whenever necessary. The statisti-cal uncertainty associated with the peak area is obtainedfrom the peak-ﬁtting procedure. The statistical uncer-tainty and the systematic uncertainties associated withthe SeGA eﬃciency simulation are propagated throughthe calculation of each γ -ray intensity.The 452- and 1612-keV γ rays correspond to the 100%transitions from the 452- and 1612-keV states to the Alground state, respectively. The intensity of the 1612-keV γ ray is corrected for the contribution of a nearby 1611.7-keV β -delayed γ ray of Al [42, 61]. There are two γ rayswhich are emitted from the 945-keV state at 493 and945 keV, and they are expected to have branching ratiosof 61(4)% and 39(4)%, respectively [42, 43]. We haveimproved these branching ratios to be 58.4(22)% and41.6(16)% in this work. The 1789-keV state is observedto be populated by the β -decay of Si for the ﬁrst time.There are three γ rays which are emitted from this statewith energies of 844.6(7), 1337.4(16), and 1789.4(9)-keV,and their branching ratios are measured to be 44(3)%,30.6(20)%, and 25.2(19)%, respectively. The measuredenergies are consistent with the evaluated literature val-ues of 844.6(7), 1337.8(7), and 1789.4(5) keV . Thethree branching ratios are consistent with the literaturevalues of 39.6(21)%, 36.1(18)%, and 23.3(10)%, whichare the weighted averages of ﬁve previous measurementswith inﬂated uncertainty [43, 62–65]. The excitation en-ergy is determined to be 1789.2(6) keV by combining thethree γ -ray energies. This value is of comparable preci-sion to the excitation energy of 1789.5(5) keV reported inthe data evaluation , and we have reevaluated the ex-citation energy to be 1789.4(4) keV by taking a weightedaverage of the two values.In all the previous Si decay measurements [8–17], theproton-unbound states in Al were observed to decayonly by proton emission. For the ﬁrst time, we haveobserved the β -delayed γ rays through two unbound Alstates at 2673 and 7902 keV.There are four known γ rays which are emitted fromthe 2673-keV state at 883.8(8), 1728.3(8), 2221.5(8),2673.1(6) keV , and they are expected to have branch-ing ratios of 42.8(8)%, 0.5(2)%, 31.4(7)%, and 25.3(5)%, respectively . We have observed three γ -ray branchesfrom this state. As can be seen from Table III, wehave measured their energies and branching ratios tobe 883.8(6) keV [37(5)%], 2221.4(18) keV [36(4)%], and2673.6(6) keV [26(3)%], respectively, which agree withthe literature values [42, 66] at the 2 σ level. However, oursensitivity does not allow us to see the weakest 1728.3(8)-keV γ ray from this state. The excitation energy is ob-tained to be 2673.4(5) keV from the three γ -ray ener-gies. Combining our result with the excitation energyof 2673.3(6) keV from the data evaluation  yields aweighted average of 2673.4(4) keV.The 5 / + isobaric analog state (IAS) with isospin T = 3 / Al is predicted to decay by 36 or 37 γ -ray branches by our shell-model calculations using theUSDC or USDI Hamiltonian, respectively. The threemost intense γ -ray branches at 6288(2), 6955(2), and7900(2) keV account for 92.4(15)% of its total theoreti-cal γ -ray branch. Their branching ratios measured by a Mg( p, γ ) Al reaction experiment [42, 67] are normal-ized to be 34(3)%, 12.0(19)%, and 46(3)%, respectively.A theoretical percentage of 80.7(13)% is used to normal-ize the branching ratios for the two γ rays at 6289(3) and7902(3) keV observed in our work. Their branching ratiosare determined to be 36(10)% and 45(10)%, respectively,in agreement with the previous measurement . Thehighest-energy γ ray at 7902(3) keV is assigned as the de-excitation from the IAS to the ground state of Al. Itssingle escape and double escape peaks are also observed,and the excitation energy of the IAS is determined to be7903(2) keV by combining the full photopeak energy andescape-peak energies. This value is of comparable preci-sion to the excitation energy of 7901(2) keV reported inthe data evaluation , and we have reevaluated the ex-citation energy of the IAS to be 7902.0(14) keV by takinga weighted average of the two values. The weakest 6955-keV γ -ray line is less than 3 σ above the background levelin our spectrum, and we estimate the 90% conﬁdence up-per limit for its branching ratio to be < D. Si( βp ) Mg The β -delayed proton spectrum of Si is shown inFig. 5. The event-level summing of three central pads(A+C+D) and an individual spectrum for pad A areshown for comparison. The single pad spectrum is gener-ated with anti-coincidence cuts on all other pads, result-ing in a lower background and a fast-declining eﬃciencyas a function of proton energy. Robertson et al .  ob-served 13 proton peaks below 2310 keV. Hatori et al . did not observe six of them, and Thomas et al .  didnot observe two of them at 1037 and 1684 keV. In thepresent work, all 13 known proton peaks below 2310 keVhave been observed. We reevaluate the proton energiesby taking a weighted average of literature results withinﬂated uncertainty. A total of 34 proton energies and Ungated spectrum B i ( ) P b ( ) A c ( ) B i ( ) A c ( ) B i ( ) B i ( ) B i ( ) A c ( ) A c ( , ) ( + ) P b ( ) P b ( ) B i ( ) T l ( ) C oun t s p e r k e V A l ( ) A l ( ) B i ( ) P b ( ) B i ( ) A l ( ) T l ( ) A c ( ) A l ( ) A l ( ) B i ( ) ( + ) B i ( ) B i ( ) B i ( ) A l ( ) M g ( ) B i ( ) K ( ) A l ( ) A c ( ) A c ( ) B i ( ) A l ( ) B i ( ) B i ( ) A c ( ) A l ( ) B i ( ) (1369+511) B i ( ) B i ( ) B i ( ) B i ( ) Proton-Detector-gated spectrum B i ( ) T l ( ) B i ( ) T l ( ) M g ( ) M g ( ) M g ( ) M g ( ) A l ( ) A l ( ) Mg(2754*)(1612+511) B i ( ) B i ( ) Mg(4238) overflow Al(6289*) Al(7902) Al(7902*) Al(6289)
Energy (keV) Al(7902**) Al(7070)
FIG. 4. γ -ray spectrum measured by the SeGA detectors showing the assignments for the photopeaks used to construct the Si decay scheme as well as those from room background. To reduce the room background contribution, a Proton-Detector-coincident γ -ray spectrum is produced by requiring coincidences with particle signals from the Proton Detector. Each photopeakis labeled by the emitting nucleus and its energy rounded to the closest integer in units of keV. Peaks labeled with one or twoasterisks (*) correspond to single and double escape peaks, respectively. Peaks labeled with a single dagger ( † ) are sum peakswith the summation noted. The bump at ∼ TABLE III. Results from the Si( βγ ) Al decays and Si( βp ) Mg obtained in the present work. Columns one through threereport the spin and parity ( J πi ), excitation energies ( E i ), and β -feeding intensities ( I β ) of each Al level populated by the β decay of Si, respectively. Columns four through seven report the excitation energies of the ﬁnal states populated by γ -raytransitions ( E f ) from each Al state, the laboratory frame energies of each γ -ray branch ( E γ ), relative γ -ray branching ratios(B.R.), and γ -ray intensities per Si β decay ( I γ ), respectively. Columns eight through ten report the excitation energies ( E f )of the Mg states fed by proton emissions from each Al state, the energies of the emitted protons in the center-of-mass frame( E p (c . m . )), and proton intensities ( I p ) per Si β decay, respectively. J πi  E i (keV) I β (%) γ -ray transition Proton emission E f (keV) E γ (keV) B.R. (%) I γ (%) E f (keV) E p (c . m . ) (keV) I p (%)5 / + c / + a − a
100 15.0(6)3 / + a a a / + a a
100 15.3(9)5 / + / + f f
945 1728.3(8) a a − d
452 2221.4(18) 36(4) 0.25(3)0 2673.6(6) 26(3) 0.184(17)5 / + a − − − − e e / + a − − − − j j e e / + a − − − − e e e e (7 / + a − − − − g g l l (3 / , / , / + a − − − − e e (3 / , / , / + a − − − − e e − a − − − − m m (3 / , / , / + a − − − − m m f f / + a − − − − e e (3 / , / , / + a − − − − e e / + a − − − − l l / + a − − − − e e e e / + a − − − − e e e e (7 / + a − − − − k k (3 / , / , / + a − − − − e e − a − − − − m m / + g g
945 6955(2) a < d < d g g e e e e − a − − − − h h (3 / , / , / + a − − − − k k e e e e (3 / , / , / + a − − − − i i − b − − − − b b − b − − − − b ba Adopted from the data evaluation . b Adopted from Ref. . c Average of the theoretical I β calculated using the USDC and USDI interactions. d γ -ray branch indicated by the data evaluation  but not observed in this work due to limited sensitivity. e Average of Refs. [12, 15–17]. f Average of Refs. [15–17]. g Average of Refs. [12, 16, 17]. h Average of Refs. [12, 15, 16]. i Average of Refs. [13, 15, 17]. j Average of Refs. [16, 17]. k Average of Refs. [12, 16]. l Average of Refs. [15, 16]. m Adopted from Ref. . etal .  were unrealistically small , and therefore, wetake an unweighted average of literature relative inten-sities and assign an uncertainty that covers all litera-ture central values. For the three proton emissions at2453, 2486, and 5549 keV only observed by Robertson etal . , the uncertainties evaluated in this way becomezero. Hence, we extract the residuals between the aver-aged literature relative intensities and those measured byRobertson et al .  based on all other proton emissions.We derive a standard deviation of all the residuals, andthis standard deviation is then factored into the uncer-tainties of the intensities for the 2453-, 2486-, and 5549-keV protons. As shown in Fig. 5, the β -particle back-ground in our proton spectrum is suppressed to as low as100 keV, enabling the clear identiﬁcation of a new pro-ton peak at 724(4) keV. We derive a detection eﬃciencycurve for protons based on the number of counts in eachpeak and its corresponding intensity. We then interpo-late the eﬃciency at 724 keV and determine the intensityfor the 724-keV proton emission to be 0.036(15)%. pad A C oun t s p e r k e V Energy (keV)
402 1268 1380 1492 1584 1684 1794 1924 2164 2310 pads A+C+D
FIG. 5. Proton spectra measured by three central padsA+C+D (blue) and central pad A (red). Each proton peakfrom the β -delayed proton decay of Si is labeled with itscenter-of-mass energy rounded to the closest integer in unitsof keV.
E. Proton- γ coincidences and decay scheme In order to reliably construct the decay scheme, it isdesirable to conduct a pγ coincidence analysis. Only twoprevious measurement showed a handful of pγ coinci-dences. Garc´ıa et al . reported the coincidences betweenthe 1369-keV γ ray and 3464-, 3606-, and 4257-keV pro-tons . Thomas et al . conﬁrmed the 4257-1369 pγ coincidence . In the present work, much more pγ co-incidences have been directly observed. Figure 6 showsthree regions of the two-dimensional coincidence spec-trum between the protons and γ rays from Si decay.
Proton energy (keV)500 1000 1500 2000 2500 -r a y e n e r gy ( k e V ) γ Proton energy (keV)500 1000 1500 2000 2500 -r a y e n e r gy ( k e V ) γ Proton energy (keV)500 1000 1500 2000 2500 -r a y e n e r gy ( k e V ) γ FIG. 6. Coincidence spectrum between the Proton Detectorand SeGA detection for Si decay. The γ -ray spectrum ismagniﬁed at 4238 keV (top panel), 2754 and 2870 keV (mid-dle panel), and 1369 keV (bottom panel), corresponding tothe four γ rays originating from the three lowest-lying Mgstates.
The protons and γ rays detected in coincidence are sum-marized in Table 6 in the form of a coincidence matrix.The newly identiﬁed 724-keV proton is observed in co-incidence with the 2870- and 4238-keV γ rays. Hence,2 FIG. 7. Simpliﬁed decay scheme of Si. The mass excesses, separation energies, Q values, spins, and parities are adopted fromthe data evaluations [42, 44, 77]. The half-life is the weighted average of Refs. [9, 11, 15, 16] and the present work. The γ -rayenergies and the excitation energies deduced from these γ -ray energies are rounded to the nearest keV. Each γ -ray transitionis denoted by a vertical arrow followed by its γ -ray energy, and the corresponding γ -ray transition intensity is denoted by thethicknesses of the arrow. Each β -decay transition is depicted by an arrow on the right side of the ﬁgure followed by its feedingintensity. The 2673- and 7902-keV Al states are observed to decay by both proton and γ -ray emissions. The newly observed724-keV proton is emitted from the 7240-keV Al state. Each proton transition is denoted by an arrow between its initial andﬁnal states labeled alongside by its center-of-mass energy. For the sake of brevity, we omit other unbound Al states. All theenergies and masses are given in units of keV. See Table III for details. we assign it as a proton transition to the 4238-keV ex-cited state of Mg and obtain an excitation energyof E x = 7234(4) keV for its proton-emitting statein Al. The excitation energy is consistent with a5 / + proton-emitting state, which was previously mea-sured to be 7240(7) keV , 7240(3) keV , and7239(5) keV , 7243(12) keV , 7248(5) keV ,7245(8) keV , 7255(7) keV , respectively. A de-cay width of 19(4) keV was reported in a polarized protonscattering experiment , which explains the broad peakshape at 724 keV observed in our proton spectrum. Pre-vious Si decay experiments [12, 15–17] observed twoproton peaks at 3606 and 4980 keV, corresponding tothe proton transitions from this state to the ﬁrst excitedstate and ground state of Mg, respectively. Our Pro-ton Detector is not sensitive to those high-energy pro- tons. Hence, we have determined the I p = 1 . Si decay experiments [12, 15–17] and the aforementioned proton feedings to each Mgstate (Table II). No γ -ray branches populating or deexcit-ing the 7240-keV state have been observed; therefore, the β feeding of the 7240-keV state is determined by addingup the intensities of the three proton branches from thisstate.The γ -ray transitions are placed in the decay schemeshown in Fig. 7 based on the known level scheme in thedatabase [42, 44], as well as including consideration ofspin and parity selection rules and the γ -ray energy re-lationships. The level scheme is also veriﬁed using γγ coincidences. Except for the γγ coincidences associated3 TABLE IV. Coincidence matrix of the protons and γ raysmeasured in the β decay of Si. The ﬁrst row corresponds tothe γ -ray energy on which the gate is set. The following rowsindicate the protons observed in the gated spectrum. Protonsobserved in coincidence are indicated with a checkmark ( X )if the signal is statistically signiﬁcant. All the energies arerounded to the closest integer and are given in units of keV.1369 2754 2870 4238402554 X X X X X X X X X X
X X X with the two relatively weak γ rays originating from theIAS, all the expected γγ coincidences between other γ -ray transitions are observed in our work. All the boundstates of Al are observed to be populated in the β de-cay of Si. The β -feeding intensity to a Al bound stateis determined by subtracting the intensity of the γ raysfeeding this level from the intensity of the γ rays deex-citing this level. The feeding of the ﬁrst excited state of Al with J π = 1 / + is consistent with its populationby βγ decay rather than directly by a second-forbidden β transition. It is possible that there exist weak, unob-served γ feedings from high-lying states, and the appar-ent β feedings for low-lying states are thus higher thanthe true β feedings due to the Pandemonium eﬀect .We have assessed the extent of this eﬀect based on theshell-model calculations, and the unobserved γ feedingsfrom high-lying states for each state are expected to benegligible ( < − ) due to the dominance of proton emis-sion. The excitation energies and β -feeding intensities( I β ) of Al levels populated by Si β decay measuredin the present work are tabulated in Table III. F. Shell-model calculations
We have performed the theoretical calculations us-ing the shell-model code
NuShellX  in the sd -shellmodel space involving the π d / , π s / , π d / , ν d / , ν s / , and ν d / valence orbits. Two modiﬁed univer-sal sd (USD) Hamiltonian , USDC and USDI, whichdirectly incorporate Coulomb and other isospin-breakinginteractions  were used. A quenching factor q = 0 . sd shell.Given the quenching factors in sd shell ranging from 0.5near Ca to 0.7 near O, the theoretical uncertaintyassociated with the A = 25, q = 0 . ± .
1. The theoretical log f t and B (GT) values are re-ported in Table V. In general, the characteristics of thedecay scheme measured in the present work including theexcitation energies, β -feeding intensities, log f t , B (GT), γ -ray and proton partial widths for the states of Al canbe reproduced well within the framework of the nuclearshell model.Low-lying states and the T = 3 / Al havebeen unambiguously identiﬁed and their excitation ener-gies have been well measured. Given that decay widthsand intensities are very sensitive to energies, we haveapplied a correction to the theoretical β feedings, γ -raypartial widths (Γ γ ), and proton partial widths (Γ p ) basedon the experimental energies. The I β = T /t is deter-mined using the half-life of Si, T , and the individualpartial half-life for each transition, t . The latter is scaledfrom the theoretical t by the phase space factor f usingthe experimental β -decay energy of Si and the excita-tion energy of each Al state under the assumption ofconstant f t value. Each theoretical Γ p is calculated byusing the theoretical spectroscopic factor and the barrier-penetration factor  corrected for the experimental res-onance energy. Each theoretical Γ γ is obtained using theeﬀective M E E L +1 γ energy dependence, where L denotes the multipolarity of the radiation. G. Mirror asymmetry
With the β -decay energy of Si Q EC ( Si) =12743(10) keV , the Si half-life of 219 . +0 . − . ms,the excitation energies of Al states, and the β -feedingintensities to Al states measured in the present work,the corresponding log f t values for each Al state can becalculated through the logft analysis program providedby the NNDC website . The corresponding Gamow-Teller transition strengths, B (GT), are calculated fromthe f t values using the following relation: f t = Kg V B (F) + g A B (GT) , (5)where K/g V = 6144 . g A /g V ) =( − . , with g V and g A being the free vectorand axial-vector coupling constants of the weak interac-tion. Our shell-model calculations predict that the Fermitransition strengths B (F) are negligible for low-lying Alstates.The degree of isospin-symmetry breaking can be quan-tiﬁed by the mirror-asymmetry parameter δ = f t + /f t − −
1, where the f t + and f t − values are associated with the β + decay of Si and the β − decay of Na, respectively. δ = 0 denotes perfect isospin symmetry. The log f t and4 TABLE V. Comparison of the mirror transitions in Si and Na β decays. Column one lists the excitation energies of each Alstate. Columns two through three report the log ft and B (GT) values for the each Si β transition. Column four shows the J π assignments . Columns ﬁve through seven list the results of the mirror Na β -decay transitions. The mirror-asymmetryparameters δ are reported in the last column. The USDC and USDI shell-model calculated results for both Si and Nadecays are shown for comparison. Si → Al Present experiment Na → Mg [34, 35] Al E x (keV) log ft B (GT) J π  Mg E x (keV) log ft B (GT) δ / + / + / + / + / + Si → Al USDC Na → Mg USDC Al E x (keV) log ft B (GT) Mg E x (keV) log ft B (GT) δ / + / + / + / + − / + − Si → Al USDI Na → Mg USDI Al E x (keV) log ft B (GT) Mg E x (keV) log ft B (GT) δ / + / + / + / + − / + − B (GT) values for each β -decay transition and the corre-sponding mirror-asymmetry parameter are summarizedin Table V. Limited by the Q β − = 3835 . Mg states were observed to be populated by Na β decay [33–35], and each one of them can be matched witha speciﬁc Al state measured in our work. Thomas etal .  compared four transitions between the mirror nu-clei Si and Na, and their mirror-asymmetry param-eters for three bound states are consistent with but lessprecise than our values. We did not observe mirror asym-metry between the transitions to the second 3 / + state.We observed some small but signiﬁcant asymmetries forthe other four low-lying states. The theoretical B (GT)values for Si decay are in agreement with our exper-imental values considering the theoretical uncertainties.Our shell-model calculations somewhat underestimatedthe B (GT) value for the 7 / + state but slightly overes-timated that for the second 3 / + state compared with Na β -decay measurements [33–35], suggesting that amore careful theoretical treatment is needed, e.g., use theshell model in conjunction with more realistic radial wavefunctions and sums over parentages in the A − H. Al 2673-keV state
The β feeding of the 2673-keV state of Al is mea-sured to be I β = 6 . β -delayed γ rays deexciting the 2673-keV state I γ = 0 . I p = 6 . β feeding of the 2673-keV stateis in agreement with the previous measured values of I β = 6 . I β = 8 . etal . reported a smaller I β = 4 . I γ /I p is equal to the ratio Γ γ / Γ p .We determine an experimental value of Γ γ / Γ p = 0 . γ / Γ p = 0 . Mg( p, γ ) Al reaction measurement .Another Mg( p, γ ) Al reaction measurement deter-mined the resonance strength of the 2673-keV state tobe ωγ = 41 . tot is the sum of the Γ p and Γ γ since they represent theonly two open decay channels for the 402-keV resonancein Al. The Γ tot and ωγ are related by the followingexpression: ωγ = 2 J r + 1(2 J p + 1)(2 J T + 1) Γ p × Γ γ Γ tot , (6)where J r = 3 / J p = 1 / J T = 0 is the spin of5 TABLE VI. Decay properties of the 2673-keV 3 / + state in Al.
Reference Γ γ (meV) Γ p (meV) Γ γ / Γ p ωγ (meV) τ (fs)Refs. [66, 80] 23.8(15) 166(16) 0.143(16) 41.6(26) 6 . +4 . − . USDC 20.6 173 0.119 36.8 3.4USDI 21.2 173 0.123 37.7 3.4Present 23(11) a a b a Deduced from the ωγ measured by Refs. [66, 80] and the I γ and I p measured in the present work. b Adopted from Refs. [66, 80]. the ground state of Mg. The lifetime of the 2673-keV Al state was previously measured to be τ = 6 . +4 . − . fsusing the Doppler shift attenuation method . Thisvalue was converted to a half-life of 4(3) fs and adoptedby the evaluation . The lifetime is inversely propor-tional to the decay width by τ = ~ / Γ tot , where ~ is thePlanck constant. Combining the branching ratio Γ γ / Γ p measured in this work with the literature ωγ value yieldsa lifetime for the 2673-keV state of 2.9(12) fs, which isconsistent with, as well as more precise than, the pre-viously measured lifetime . The decay properties ofthe 2673-keV state in Al are summarized and comparedto the USDC and USDI shell-model calculations in Ta-ble VI, and good agreement is obtained for all the quan-tities. I. Al T = 3 / IAS at 7902 keV
The proton partial width of the lowest T = 3 / Al was determined to be Γ p = 155(50) eV [81, 82]and 105(18) eV , respectively, in two proton scatter-ing measurements with polarized-proton beams. Thesetwo results agree, and a weighted average Γ p is deducedto be 111(17) eV. The γ -ray partial width of the IASwas previously determined to be Γ γ = 2 . Mg( p, γ ) Al reaction yield measurement  by adopt-ing a proton branching ratio I p /I p tot = 0 .
17 from the Si β -delayed proton measurement . I p is the in-tensity of the proton emission from the IAS proceedingto the ground state of Mg. I p tot is the total inten-sity of the proton branches of the IAS. However, an-other Mg( p, γ ) Al reaction study  reported a muchsmaller Γ γ = 0 . ωγ = 0 . I p /I p tot = 0 . Si β -delayed proton measurement . The ratio of the γ -raypartial width to the proton partial width is deduced to beeither Γ γ / Γ p = 0 . γ of Ref. or Γ γ / Γ p = 0 . γ of Ref. .The decay properties of the IAS obtained in the presentwork are shown in Table VII. The sum of the intensi-ties for the 7902- and 6289-keV β -delayed γ rays throughthe IAS is measured to be I γ = 0 . TABLE VII. Decay properties of the 7902-keV 5 / + , T = 3 / Al.Reference Γ γ (eV) Γ p (eV) Γ γ / Γ p ωγ (eV)Ref.  2.0(10) 111(17) a a a a Weighted average of Γ p reported in Refs. [81–83]. model predicts a 19.3(13)% branch for unobserved weak γ rays, so we obtain a corrected I γ = 0 . Si decay experiments [12, 15–17] and normalizedto our Si( βpγ ) Mg intensities, we have determined an I p = 13 . I γ and I p yields thetotal β feeding intensity I β = 13 . f t value of 3.23(6). The USDC andUSDI shell-model calculations predicted the log f t = 3 . γ / Γ p = 0 . I γ and I p values. Combining our Γ γ / Γ p ratio with the liter-ature Γ p value [81–83] yields a Γ γ = 2 . ωγ = 1 . I p /I p tot = 0 . et al .  and aremore precise, but they deviate from the values reportedby Rogers et al .  roughly by a factor of 4.The USDC and USDI shell-model calculations esti-mate the Γ γ to be 2.98 and 2.45 eV, respectively, inagreement with the Γ γ = 2 . γ / Γ p ratio and the Γ p from Refs. [81–83]. The shellmodel also indicates that Γ p of the IAS depends onthe mixing with a predicted nearby 5 / + , T = 1 / p of 50 keV. Unfortunately, this statehas not yet been identiﬁed experimentally. The sumof Fermi and Gamow-Teller contributions is derived tobe B (F) + ( g A /g V ) B (GT) = 3 . f t = 3 . B (GT) ≈ . B (F) should ful-ﬁll the sum rule P B (F) = 3, suggesting that the Fermistrength is mainly concentrated on the IAS. The frag-mentation of the Fermi strength via isospin mixing israther small compared with the strong mixing observedfor some special cases [85–87]. The USDC and USDIshell-model calculations predict that this state is 23 and9 keV above the IAS, respectively, but there is an un-certainty of about 150 keV for the predicted energy ofeach state. It has been shown that this energy uncer-tainty leads to uncertainties of about an order of mag-nitude for the proton and neutron decay width of IAS6 l og [ p ( e V )] (a) USDC-0.2 -0.1 0.0 0.1 0.20123456 (b) USDI Energy difference (MeV) l og [ p ( e V )] FIG. 8. Γ p for the T = 3 / Al and its neighboring T = 1 / E x / − E x / ) calculated by the (a) USDCand (b) USDI shell models. The decay width for the statedominated by the T = 3 / T = 1 / p valuefor the T = 3 / in the sd shell . In order to assess the results for Al, we move the relative location of the T = 3 / T = 1 / b ˆ T to the Hamiltonian, whereˆ T is the isospin operator. For states with good isospin, b ˆ T | T i = bT ( T +1) | T i . The T = 3 / . b , and the T = 1 / . b . Theresults obtained for the IAS and its neighboring T = 1 / b are shown inFig. 8. If the IAS is moved down by a few keV, the Γ p forthe IAS comes into agreement with the well-measured Γ p value [81–83]. This is equivalent to moving the T = 1 / T = 1 / T = 3 / ∼ T = 1 / V. CONCLUSION
By using the GADGET system at NSCL, simultane-ous measurements of Si β -delayed proton and γ decayswere carried out. We have reported the most precisehalf-life of Si to date. Eight new β -delayed γ -ray tran-sitions were detected, leading to the population of three Al states that have not been previously observed via Si β -delayed γ decay. A total of 14 β -delayed pro-ton branches have been identiﬁed, including a new pro-ton peak at 724 keV. We have conﬁrmed the placementof protons in the decay scheme of Si reported by pre-vious literature [16, 17] using both Doppler-broadeningline shape analysis and proton- γ -ray coincidence analy-sis. We have reevaluated the energies and intensities for34 Si β -delayed proton emissions. A more precise life-time for the Al 2673-keV state has been extracted, andthe discrepancy involving the γ -ray partial width of the7902-keV T = 3 / Al in the literature has beenresolved, which demonstrates the potential of utilizingcomplementary experimental approaches. The mirror-asymmetry parameters have been deduced for ﬁve tran-sitions in the mirror β decays of Si and Na, whichwill contribute to the systematic understanding of thenature of mirror-symmetry breaking. Shell-model cal-culations using the USDC and USDI Hamiltonians bothreproduce the experimental data well and predict a 5 / + , T = 1 / T = 3 / T = 1 / Al state that exhibits weak isospin mixingwith the IAS.
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