Electromagnetic character of the competitive γγ/γ -decay from 137m Ba
P.-A. Söderström, L. Capponi, E. Açıksöz, T. Otsuka, N. Tsoneva, Y. Tsunoda, D. L. Balabanski, N. Pietralla, G. L. Guardo, D. Lattuada, H. Lenske, C. Matei, D. Nichita, A. Pappalardo, T. Petruse
EElectromagnetic character of the competitive γγ/γ -decay from Ba P.-A. Söderström ∗ , L. Capponi , E. Açıksöz , T. Otsuka , , , N. Tsoneva , Y. Tsunoda , D. L. Balabanski , N. Pietralla ,G. L. Guardo , , D. Lattuada , , , H. Lenske , C. Matei , D. Nichita , , A. Pappalardo & T. Petruse , Extreme Light Infrastructure-Nuclear Physics (ELI-NP)/Horia Hulubei National Institute for Physics and NuclearEngineering (IFIN-HH), Str. Reactorului 30, 077125 Bucharest-M˘agurele, Romania Center for Nuclear Study, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Institut für Kernphysik, TU Darmstadt, 64289 Darmstadt, Germany Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, 95125 Catania, Italy Universitá degli Studi di Enna KORE, Viale delle Olimpiadi, 94100 Enna, Italy Institut für Theoretische Physik, Universität Gießen, 35392 Gießen, Germany Politehnica University of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania
AbstractSecond-order processes in physics is a research topic focusing attention from several fields worldwide includ-ing, for example, non-linear quantum electrodynamics with high-power lasers, neutrinoless double- β decay,and stimulated atomic two-photon transitions. For the electromagnetic nuclear interaction, the observation ofthe competitive double- γ decay from Ba has opened up the nuclear structure field for detailed investigationof second-order processes through the manifestation of off-diagonal nuclear polarizability. Here we confirmthis observation with an . σ significance, and an improved value on the double-photon versus single-photonbranching ratio as . × − (30) . Our results, however, contradict the conclusions from the original experi-ment, where the decay was interpreted to be dominated by a quadrupole-quadrupole component. Here, we finda substantial enhancement in the energy distribution consistent with a dominating octupole-dipole characterand a rather small quadrupole-quadrupole element in the decay, hindered due to an evolution of the internalnuclear structure. The implied strongly hindered double-photon branching in Ba opens up the possibilityof the double-photon branching as a feasible tool for nuclear-structure studies on off-diagonal polarizability innuclei where this hindrance is not present.1 Introduction
Polarizability is a fundamental concept in physics and chemistry defined from the principles of electromagnetic in-teraction. It describes how applied electric or magnetic fields induce an electric or magnetic dipole, or higher-ordermultipole, moment in the matter under investigation . In nuclear physics, the simple concept of polarizability influ-ences observables over a broad range of topics. For example, the static dipole polarisation of the shape of the groundand excited states in atomic nuclei is influenced by the coupling to high-energy collective modes like the giant dipoleresonance (GDR) via virtual excitations. In this case the nuclear static dipole polarizability, α d , is obtained from thephotonuclear population of excited states, α d;E1 = 2 e (cid:88) n |(cid:104) I (cid:107) E1 (cid:107) I n (cid:105)| E n − E , (1)where the transition matrix elements of the wave functions correspond to the electric dipole transition, E1 , betweenthe ground state, I , and an excited state, I n , with e the elementary unit charge and E n the energy of the state.1 a r X i v : . [ nu c l - e x ] J un y expanding the concept of polarizability beyond the scalar case, one can divide the polarizability tensorinto separate components. Typically, these are either spatial components like the birefringence properties of crystalsor electric and magnetic multipole components. Within the nuclear structure framework, this type of off-diagonalpolarizabilities can appear in very weak second order processes. In the electromagnetic case, the off-diagonal nu-clear polarizability can be defined analogous to equation (1) in terms of either electric and magnetic components, orcomponents of different multipolarities as α M2E2 = (cid:88) n (cid:104) I f (cid:107) E2 (cid:107) I n (cid:105)(cid:104) I n (cid:107) M2 (cid:107) I i (cid:105) E n − ω (2)or, α E3M1 = (cid:88) n (cid:104) I f (cid:107) M1 (cid:107) I n (cid:105)(cid:104) I n (cid:107) E3 (cid:107) I i (cid:105) E n − ω . (3)Due to the parity conserving properties of the strong force, these decays can only be observed between two differentstates, I i and I f . In the definition above, the denominator depends on the interference frequency, ω , of the emitted γ rays and is assumed to be half of the initial state energy. This type of second-order electromagnetic processes of atomswas discussed in the doctorate dissertation of Maria Göppert-Meyer where she estimated a probability for an atomictwo-photon absorption process relative to the single-photon process to be approximately − , later to be confirmedwith the observation of this effect in CaF :Eu crystals .For many years, double- γ decay was only observed in exceptional cases where both the ground state and theinitial state have a spin-parity J π = 0 + character for the doubly magic nuclei O , Ca , and Zr . Here single γ -emission is blocked, and only conversion-electron decay and double- γ decay are allowed. In these experiments,the obtained information consists of correlations between energies and angles of these γ -rays, used to determine thedecay probabilities of electric and magnetic dipoles. For a generalization of this phenomenon and the possibilityto use it as a spectroscopic tool for more fundamental understanding of the underlying physics, large state-of-the-art high-purity germanium (HPGe) detector systems have been used to search for the competitive γγ/γ decaywhere also the single γ decay is allowed. Even though unsuccessful in that respect, these experiments successfullymeasured an E5 transition with the branching of . × − . It is only with instrumentation developments ofdetector materials that can provide both the energy and time resolution required that the observation of the γγ/γ decay mode was announced . The setup used for that experiment consisted of five LaBr :Ce detectors arranged ina planar configuration with relative angles of 72 ◦ between the detectors, providing angular distribution data points at72 ◦ and 144 ◦ . Thus, the collaboration could announce a γγ/γ decay signal with 5.5 σ (standard deviations) statisticalsignificance, near but above the typical discovery limit of 5 σ . From the two angular data points as well as the energyspectrum of the individual γ rays at 72 ◦ angle, the off-diagonal polarizabilities α M2E2 = 33 . . e fm /MeV andthe α E3M1 = 10 . . e fm /MeV polarizabilities were extracted. While the observation of the peak associatedwith γγ/γ decay was statistically clear, the nature of this decay was more uncertain, having the two dominatingmultipolarity combinations separated only by a small statistical difference, favouring the α M2E2 component . Thedecay diagram of this process is shown in Figure 1.Given the nature of this experiment to observe a longstanding prediction of a the fundamental concepts inquantum mechanics and quantum electro-dynamics, and the possibility to extract nuclear structure observables fromthis, it is highly desirable to independently confirm this observation. Some possibilities that have been under discussionto perform this independent confirmation is to either return to the HPGe approach with complex detector systems andevent processing like the Advanced GAmma Tracking Array (AGATA) setup or highly charged radioactive ions . Here we report on an experiment using the ELI Gamma Above Neutron Threshold (ELIGANT) detector system at the Extreme Light Infrastructure – Nuclear Physics (ELI-NP) facilities in a configuration similar to what wasused in reference . The experimental setup was optimised for obtaining a clean signal over a wide angular range based on the expected intensities of the decay mode. Here, we can confirm the existence of the competitive double-photon decay process in atomic nuclei with an . σ significance. We, however, find a significant octupole-dipole,E3M1, matrix element product contribution to the double- γ decay mode of Ba , contradicting the conclusions of2 + + + M2 E2E3M1 γγ γγγ
M4 0 keV1252 keV1294 keV7/2 + Q β = 1176 keV βν e 137 Ba Cs30.08 years 2.552 minutes94.70% 662 keV
Figure 1:
Decay diagram from the
Cs ground state to the
Ba ground state.
Illustration of the single- γ andthe two types of double- γ decay, as fed by the β decay of Cs, including half lives of
Cs and
Ba. Theenergy of the
Cs ground state ( Q β ) is given relative to the Ba ground state. Here, M4 corresponds to the single-photon decay. The blue and pink decays show the lowest octupole-dipole and quadrupole-quadrupole compoenents,respectively.the original experiment . From our calculations using the energy-density-functional (EDF)+quasiparticle-phononmodel (QPM) and the Monte Carlo shell model (MCSM), we find that both models reproduce the octupole-dipolecomponent consistently, but the nature and the strength of the quadrupole-quadrupole component, differ significantly.It is interesting to note that this additional hindrance suggests a reduction of the γγ/γ branching with almost an orderof magnitude in the most extreme case of Table 1, which is also the case that best reproduce the α E3M1 polarizability.This opens for the possibility of a significant increase of the γγ/γ branching in nuclei in this region that do notexhibit this hindrance. In this case, experiments would be feasible also with more exotic sources , or even in-beamexperiments within reasonable beam times, to follow the evolution of the quadrupole-quadrupole strength.Table 1: Experimental and calculated α coefficients and γγ/γ decay branching ratios. B (M4) Γ exp γγ / Γ exp γ Γ th γγ / Γ th γ Γ th γγ / Γ exp γ α M2E2 α E3M1 ( e fm ) ( − ) ( − ) ( − ) (e fm /MeV) (e fm /MeV)This work 2.62(30) ± . ± . EDF+QPM (0.6 g bares ) 1.15 g bares ) 3.30 1.34 15.2 104 MCSM (0.6 g bares ) 1.18 0.579 0.840 -2.14 -21.2MCSM ( g bares ) 3.28 0.196 Literature The Γ γγ / Γ γ decay branching ratio is shown both with unquenched ( g effs = g bares ) and quenched gyromagnetic spin factors ( g effs = 0 . g bares ).The latter limit was chosen based on the reproduction of individual reduced transition probabilities. Depending on the calculation the values of g effs to best reproduce nuclear data are typically within this range. Thus, these limits should be representative of the uncertainties in the theoreticalcalculations, giving a range of ∼ % for both the α M2E2 and α E3M1 values for both models between the two extremes. The listed values closestto the measured branching are shown in bold font. The best fit for the decay branching ratio for the EDF+QPM calculations, not listed here, isobtained when choosing g effs = 0 . g bares as Γ γγ / Γ th γ = 2 . . ResultsExperimental setup.
The experiment was performed using eleven (cid:48)(cid:48) × (cid:48)(cid:48) CeBr detectors from ELIGANT, shownin Figure 2a. While ELIGANT consists of both LaBr :Ce and CeBr detectors, the CeBr detectors were chosento remove any possible source of background contribution from the natural radioactivity in lanthanum. The detectorconfiguration was a circle with an inner radius to the front-face of the scintillators of 40 cm. This distance was enoughto separate true coincidences from multiple Compton scattering of single γ rays using the photon time-of-flight (TOF),see Figure 2b. The relative angles between the eleven detectors were 32.7 ◦ , with an opening angle, given by the leadshielding, of ± . ◦ . This gave five independent γγ -correlation angles centered at: 32.7 ◦ , 65.5 ◦ , 98.2 ◦ , 130.9 ◦ , and163.6 ◦ . The detectors were separated with a minimum of approximately 15 cm of effective lead shielding betweentwo neighbouring detectors to remove any contribution from single Compton scattering between detector pairs atlow angles. The setup was characterized both with an in-house toolkit based on the G EANT , anda Eu source with an activity of 460 kilo Becquerel (kBq) and a Co source with an activity of 60 kBq. For acomprehensive overview, see reference . The γγ/γ -decay data on Ba were collected using a
Cs source withan activity of 336 kBq for 49.5 days active data taking. The source had a thin circular active area with a diameter of3 mm, encapsulated in the center of a cylindrical polymethylmetacrylate capsule with a diameter of 25 mm and 3 mmthickness, see Figure 2.
Energy spectra.
From the data set obtained with the
Cs source a ( γ , γ ) coincidence matrix was constructedwhere the γ rays were considered coincident if the time difference between them were less than one standard deviationfrom the prompt time distribution, ∆ t , ≤ ps. This condition was obtained from the coincident 444 kilo electron-volt (keV) and 245 keV γ rays from the +2 → +1 → +1 decay chain in Sm following the electron capturedecay of
Eu. Corrections for detector efficiencies were done on an event-by-event basis . A time difference of ≤ ∆ t , ≤ ns was used to estimate the uncorrelated background events with two detected γ rays and subtractedafter applying an appropriate scaling factor. To remove the background contribution from electron-positron pairsproduced by cosmic rays a multiplicity-two condition was assigned together with an additional energy condition that | E − E | < − ( E + E ) keV. The full data set, as well as the different angular groups, were used to constructthe summed energy spectra. The peaks were fitted assuming a quadratic background both with a Gaussian distributionas well as G EANT E + E with these conditions imposed is shown in Figure 3. Branching.
As experimental observable to evaluate the relative decay probability we use the definition of the inte-grated differential branching ratio , δ ( E , E , θ , ) = (4 π ) Γ γ (cid:90) E E d ω dΓ γγ d ω dΩdΩ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ , . (4)In this definition Γ γ is the total single-gamma decay width, proportional to the size of the single-gamma peak. Givenan angle, θ , , the differential decay is integrated over the frequency of the γ ray, ω . The frequency is proportionalto the energy, and the integration limits are taken as the edges of the energy bin of interest. In the experimentalspectrum a natural low-energy limit comes from the low-energy threshold of the detectors around 120 keV. However,to reduce the contamination from the 511 keV γ -rays originating from electron-positron annihilation, the integrationlimits E = 180 keV and E = 331 keV were chosen. The upper limit was chosen as the half of the total energy aswe are not able to distinguish any relative ordering of the γ rays. This procedure was performed for all combinationsof θ , and δ was evaluated as a function of angle. The results from this evaluation is shown in Figure 4.This data can be directly fitted to the generalized polarizability functions of equation (6) discussed in themethods section, using only α M2E2 and α E3M1 as free parameters. Other components like α E2M2 or α M3E1 couldin principle also contribute. However, the general polarizability functions are linearly dependent in the exchangeof terms, weighted by the coefficients given by the Wigner j symbols, and this experiment is not sensitive to this4 mm ⌀ mm ⌀ mm m (a) t (ns) ∆ − − − − − C oun t s / . s backscattering °
180 655 ps ± = σ Time gate: 1~180 - ~480 keV (b)
Figure 2:
Experimental setup. (a) Coincident γ rays could originate either from true double- γ decay eventsillustrated with red cones, or from multiple Compton scattering between detectors illustrated with blue cones.(b) Multiple Compton scattering events were rejected by the time difference ( ∆ t ) between the γ -ray interactions,shown in the blue histogram. The time condition for prompt γ -rays are shown as red dashed lines and verifiedwith a Eu source. ordering. These additional components are, furthermore, expected to be small. Thus, we restrict the discussion to the α M2E2 and α E3M1 polarizabilities from here on.
Energy sharing distributions.
The angular distributions themselves are not enough to completely distinguish be-tween the contribution from the different polarizabilities. When calculating the goodness-of-fit ( χ ), two local minimacorresponding to either a large α M2E2 component or a large α E3M1 component appear. Instead, it is necessary to studythe energy-sharing distributions between the two individual γ -rays. From equations (6) and (7) in the methods sectionit is clear that the energy dependence of the decay for the two different cases follows dΓ γγ d ω ∝ ω ω (cid:48) for M2E2 and as dΓ γγ d ω ∝ ω ω (cid:48) for E3M1 with ω (cid:48) = 662 − ω . It is clear from these relations that the energy sharing distributions areexpected to have a maximum at E γ = E (cid:48) γ = 331 keV for the M2E2 type transitions, while an asymetric maximum isexpected at E γ = 200 keV and E (cid:48) γ = 442 keV for the E3M1 type transitions.5
00 400 500 600 700 800 900(keV) ,2 γ +E ,1 γ E50010001500200025003000 C oun t s / . k e V Multiplicity = 2Multiplicity > 2Best fitWithout signal2 × (a)
300 400 500 600 700 800 900(keV) ,2 γ +E ,1 γ E5001000150020002500 C oun t s / . k e V Cosmic-rayinducedbackground2 × (b)
300 400 500 600 700 800 900(keV) ,2 γ +E ,1 γ E5001000150020002500 C oun t s / . k e V × (c) Figure 3:
Summed double- γ energy spectrum and data reduction. Black data points show the summed energy oftwo coincident photons detected in the CeBr detectors for events with a multiplicity of two. Gray data points showthe sum energy spectrum when the multiplicity is larger than two, which mainly correspond to the background inducedby cosmic ray showers. We also show the fit to the data of a quadratic background as a dashed red line and the fit ofthe background plus a Gaussian peak as a solid red line. (a) Raw data before any conditions. (b)
Reduced data with acondition that the energy difference between two γ rays is | E γ, − E γ, | < keV. (c) Final data with the additionalcondition that | E γ, − E γ, | < − ( E γ, + E γ, ) keV to remove cosmic-ray induced background. The error barsrepresent the one standard-deviation statistical uncertainty.6
20 40 60 80 100 120 140 160 180 (degree) q - · ) , q ( k e V , k e V , d This workWalz et al.FitM2E2E3M1
Figure 4:
Angular distribution.
The angular correlation of the two photons emitted in the double- γ decay from thiswork and reference , compared to the expected angular distributions of pure M2E2 and E3M1 decay. The error barsrepresent the one standard-deviation statistical uncertainty.7
100 200 300 400 500 600(keV) ,low γ E020406080 − × ) - / d E ( k e V δ d , θ ∑ This workFitM2E2E3M1E5+M1 (a) − -1 MeV fm (e M2E2 α − ) - M e V f m ( e E M α (b) Figure 5:
Multipole nature of the γγ/γ decay. (a)
Energy sharing distribution for the two photons in the double- γ decay compared to the expected energy distributions of pure M2E2 and E3M1 decay, as well as the two-step E5+M1decay where the measured intensity has been subtracted from the 300 keV data point. The data points correspond tothe sum of the differential branching ratio defined in equation (4) over all the available angles. (b) Two-dimensionalgoodness-of-fit, χ , plot for the two α parameters with the experimental data. The contours are separated by onestandard deviation. The data from the present work is shown as a black point, data from reference is shown asa red point, and a fit with the two data sets combines is shown as a purple point. The error bars represent the onestandard-deviation statistical uncertainty, except the error bars in E γ, low that represent the width of the energy bin.For this purpose, δ from equation (4) was evaluated in separate slices of 30 keV energy difference between thelow- and high-energy limit of E γ . Figure 5a shows the results of these evaluations. A χ value was then calculatedbased on (4) for different values of α M2E2 and α E3M1 simultaneously using the energy-integrated angular data pointsand the angle-summed energy data points as χ = (cid:88) θ i , E =181 E =331 δ ( E , E , θ i , ) σ δ ( E , E , θ i , ) + (cid:88) E = E low E = E high δ ( E , E , Σ θ , ) σ δ ( E , E , Σ θ , ) , (5)where σ δ is the statistical uncertainty in each data point, including both the signal and the subtracted background. Thesystematic uncertainty mainly originates from uncertainties in the intrinsic and geometric efficiencies of the setup andis expected to be on the order of a few %, much smaller than the statistical uncertainties, and have been neglected inthis expression. When including the data from Walz et al., only the energy distribution of the 72 ◦ data was includedin the second part of equation (5), and the lower energy summation limit for the 144 ◦ data point was set to 206 keVin the first part of equation (5). The resulting χ surface is shown in figure 5b. As seen here, the χ analysis fromthis data favours a large α E3M1 component, in contradiction with both the experimental interpretation and theoreticalconclusions reported in reference . To understand these results we performed theoretical calculations of the polarization functions from equation (2),using the QPM approach. The application of the QPM in the case of odd-mass spherical nuclei is discussed in8etail in reference . In particular, the nuclear structure of Ba was studied within the framework of this model inreferences and in reference . In the work presented here, the calculations were built on the EDF theory coupledwith the QPM to obtain magnetic and electric spectral distributions. The model parameters of the EDF+QPMapproach are firmly determined from nuclear structure data or derived fully microscopically . The theoreticalresults are shown in table 1 and agrees with the data in terms of absolute branching strength, Γ γγ / Γ γ . In addition, theEDF+QPM used here predicts a significantly larger α E3M1 than the value reported in reference from the QPM, closeto our experimental observations. However, the relative magnitude of the α M2E2 and α E3M1 coefficients obtained fromthe EDF+QPM theory, as well as from reference , are different than the experimental results obtained in this work.In particular, the present measurement indicates that the α M2E2 coefficient is significantly smaller than previouslyreported, and at this level of complexity EDF+QPM it is not able to account for the apparent discrepancy with theexperimental data.To understand the origin of this discrepancy, the properties of the dominant, low-lying, states were investigatedfrom another perspective using the state-of-the-art nuclear MCSM . These calculations were used to extract in-formation from the three lowest-energy J π = 7 / + states, the five lowest-energy J π = 5 / + states, as well as theground J π = 3 / + and isomeric J π = 11 / − states. The neutron component in the MCSM wave function of theisomeric J π = 11 / − state is dominated by a single neutron hole in νh / . The J π = 7 / + state is different,however, with most of the neutron hole occupation is in νd / , coupled to a + state of six valence protons. The νg / orbital itself is almost full. This is in contrast with the EDF+QPM results where the + ⊗ νd / contributionis 38.7% and the νg / single-particle component is 51.3%. Thus, the odd-neutron contribution to the M2 transitionrate in the MCSM would require a highly hindered transition between νh / and νd / , or by utilising a minor νg / vacancy. This, gives rise to a strongly hindered M2 transition within the MCSM, with a reduced transition probabil-ity, B (M2) = 13 . × − µ fm , three orders of magnitudes less than predicted by the EDF+QPM model where B (M2) = 14 . µ fm . This can explain the observed suppression of α E2M2 . It is interesting to note that with increas-ing excitation energy, the MCSM predict a smooth change in orbital occupation from νd / to νd / , constructivelyadding to the M2 transition strength for all the calculated / + transitions in contrast to the EDF+QPM where allhigher-lying states act destructively. Table 2 lists the contributing low-lying matrix elements discussed here.Table 2: Calculated matrix elements.
Matrix element EDF+QPM MCSM Matrix element EDF+QPM MCSM e · fm L e · fm L e · fm L e · fm L (cid:104) / +1 (cid:107) M1 (cid:107) / +1 (cid:105) -0.11 -0.139 (cid:104) / +1 (cid:107) E3 (cid:107) / − (cid:105) -168 57.2 (cid:104) / +1 (cid:107) M1 (cid:107) / +2 (cid:105) -0.03 (cid:104) / +2 (cid:107) E3 (cid:107) / − (cid:105) -57.3 -81.9 (cid:104) / +1 (cid:107) M1 (cid:107) / +3 (cid:105) -0.04 (cid:104) / +3 (cid:107) E3 (cid:107) / − (cid:105) -90.2 -128 (cid:104) / +1 (cid:107) E2 (cid:107) / +1 (cid:105) (cid:104) / +1 (cid:107) M2 (cid:107) / − (cid:105) (cid:104) / +1 (cid:107) E2 (cid:107) / +2 (cid:105) -46.4 (cid:104) / +2 (cid:107) M2 (cid:107) / − (cid:105) -0.112 Transition matrix elements of the lowest-energy transitions, in each model, calculated using the EDF+QPM andMCSM models for the states that contribute to the double- γ decay in Ba . The EDF+QPM values for the mag-netic transitions correspond to g effs = 0 . g bares while the MCSM values correspond to g effs = g bares . The states thatdominates the decay in each model have been highlighted with bold font. Regarding the α E3M1 component of the decay, the main components obtained from the EDF+QPM calculationsare from the coupling of the single-particle mode with the surface vibrations of the even-even core. As a consequence,due to the exchange of the collective − octupole phonon, we obtain a rather strong E3 transition, consistent with ourexperimental observations. For these states the EDF+QPM and the MCSM give a consistent picture with a constructiveaddition to the strength for each successive state among the first three excited states with the main difference that inthe EDF+QPM, the main contribution comes from the / +1 state while the MCSM predicts that the / +2 , states aredominating. 9 ethods Experimental setup.
The set-up consisted of eleven 3” ×
3” CeBr detectors coupled with Hamamatsu R6233 photomultiplier tubes and built-involtage dividers. The high voltage for the photomultiplier tubes were provided by a CAEN SY4527 power supply. The signals were read out usingone CAEN V1730 digitizer operating with a 14-bit resolution at a 500 MS/s sampling rate and a dynamic range of 0.5 V pp , running PSD firmware.The digitizers were controlled using the Multi Instance Data Acquisition System (MIDAS) software and triggered individually. Each event consistedof the energy, the time-stamp, and the the digitized voltage pulse from the detector. The sub-nanosecond time information was obtained from thevalue of the time-stamp corrected by a digital interpolation of the sampling points in the recorded pulse, at half of the maximum value of the pulseand interpolated using a quadratic polynomial. Polarization functions.
To obtain the nuclear polarizabilites, α S (cid:48) L (cid:48) SL , from the differential decay probability we follow the theoretical treatmentin references . Here the differential decay probability can be expressed in terms of generalized polarization functions, P (cid:48) J ( S, L, S (cid:48) , L (cid:48) ) , andLegendre polynomials, P l (cos θ ) , as d Γ γγ d ω ΩΩ (cid:48) = ωω (cid:48) π (cid:88) P (cid:48) J ( S (cid:48) L (cid:48) S L ) P (cid:48) J ( S (cid:48) L (cid:48) S L ) (cid:88) a ξl P l (cos θ ) , (6)where the generalized polarization functions are defined as P (cid:48) J ( S (cid:48) L (cid:48) SL ) =( − S + S (cid:48) π ( − I i + I f ω L ω (cid:48) L (cid:48) · (cid:114) L + 1 L (cid:114) L (cid:48) + 1 L (cid:48) √ L + 1 √ L (cid:48) + 1(2 L + 1)!!(2 L (cid:48) + 1)!! · (cid:18)(cid:26) L L (cid:48) JI f I i I (cid:27) α S (cid:48) L (cid:48) SL + ( − S + S (cid:48) (cid:26) L (cid:48) L JI f I i I (cid:27) α SLS (cid:48) L (cid:48) (cid:19) . (7)The sums in equation (6) run over all the permutations of electric, S = 0 , and magnetic, S = 1 , combinations with multipolarity, L , allowed inthe decay, and over all Legendre polynomials with non-zero coefficients. The general polarizability functions in equation (7) consist of a linearcombination of the off-diagonal polarizabilities of the nucleus weighted by coefficients determined by the corresponding angular momentum algebraof the decay. The quasiparticle-phonon model.
The QPM Hamiltonian includes mean field, pairing interaction and separable multipole and spin-multipoleinteractions . The mean field for protons and neutrons is defined as a Woods-Saxon potential with parameter sets derived self-consistentlyfrom a fully microscopic Hartree-Fock-Bogoljubov (HFB) calculations described in . The method assures a good description of nuclearground-state properties by enforcing that measured separation energies and nuclear radii are reproduced as close as possible . The pairing andresidual interaction parameters are fitted to reproduce the odd-even mass differences of neighbouring nuclei as well as the experimental values ofthe excitation energies and reduced transition probabilities of low-lying collective and non-collective states in the even-even core nucleus . Ofparticular importance in these studies is the determination of the isovector spin-dipole coupling constant which is extracted from comparison to datafrom and fully self-consistent quasiparticle random phase approximation (QRPA) calculations using the microscopic EDF of . Single-particle(s.p.) energies of the lowest-lying excited states in Ba are fine-tuned to experimental values to achieve the highest accuracy in the descriptionof the experimental data. We point out that the s.p. energies problem is not a matter of the interaction parameters but originate in the quasiparticlespectrum, which indicates the necessity to go beyond the static mean-field formalism .In the QPM the wave functions of the excited states of an even-odd nucleus are constructed from a combination of quasiparticles originatingfrom the single-particle orbitals and excitation phonons that are constructed from the excited states in the neighboring even-even core nucleus : Ψ ν ( JM ) = C νJ α + JM + (cid:88) λµi D λij ( Jν )[ α + jm Q + λµi ] JM Ψ (8)The notation α + jm is the quasiparticle creation operator with shell quantum numbers j ≡ [( n, l, j )] and projection m ; Q + λµi denotes the phononcreation operator with the angular momentum λ , projection m and QRPA root number i ; Ψ is the ground state of the neighboring even-evennucleus and ν stands for the number within a sequence of states of given angular momentum J π and projection M . The coefficients C νJ and D λij ( Jν ) are the quasiparticle and ’quasiparticle ⊗ phonon’ amplitudes for the ν state. The coefficients of the wave function (8) and the energyof the excited states are found by diagonalisation of the model Hamiltonian within the approximation of the commutator linearization . Thecomponents [ α + jm Q + λµi ] JM of the wave function (8) may violate the Pauli principle. The exact commutation relations between quasiparticleand phonon operators are used to solve this problem. The properties of the phonons are determined by solving QRPA equations from Refs. .The model basis includes one-phonon states with spin and parity J π = 1 ± , ± , ± , ± , ± and excitation energies up to E x = 20 MeV . Thecalculations of the α -coefficients of the double- γ decay probability of Ba include all low-energy excited states with spin and parity J π =1 / ± , / ± , / ± , / ± , / ± and excitation energies up to E x = 10 MeV .In the case of the E1 transitions, we have used effective charges e effp = ( N/A ) e (for protons) and e effn = − ( Z/A ) e (for neutrons) toseparate the center of mass motion and ’bare’ values for E2 and E3 transitions e p = e (for protons) and e n = 0 (for neutrons), where e is the lectron charge. Following previous QPM calculations , the magnetic transitions are calculated with a quenched effective spin-magnetic factor g effs . The influence of the g effs parameter on the experimental observables related to electromagnetic transitions of lowest-lying states and double- γ decay probability coefficients was investigated by carrying out EDF+QPM calculations for several choices of this parameter between 0.6 and 1of the value of the ’bare’ spin-magnetic moment, g bares . The theoretical observations indicate that the values g effs = 0 . − . g bares which arein agreement with our previous findings reproduce quite well the experimental data on M1 and M2 transition strengths and the angulardistribution of the two photons of the double- γ decay. Monte Carlo shell model.
In the MCSM, the approximated wave functions, (cid:12)(cid:12) Ψ N b (cid:11) , are obtained as a superposition of spin ( I ) and parity ( π )projected Slater determinant basis states, | φ n (cid:105) , (cid:12)(cid:12) Ψ N b (cid:11) = N b (cid:88) n =1 I (cid:88) K = − I f N b n,K P IπMK | φ n (cid:105) , (9)where N b is the number of basis states, P IπMK is the spin-parity projection operator, and the f N b n,K coefficients are obtained from diagonalizingthe Hamiltonian. The set of basis states are selected by Monte Carlo methods and iteratively refined to minimize the ground state energy. Themodel space for these calculations included the g / , g / , d / , d / , and s / even-parity orbitals, as well as the h / , f / , and p / odd-parity orbitals. The two-body matrix elements were obtained from the JUN45 and SNBG3 data sets , and the V MU interaction .To obtain the transition matrix elements effective proton and neutron charges e p = 1 . and e n = 0 . , and gyromagnetic factors g (cid:96), p = 1 , g (cid:96), n = 0 , g s , p = 5 . , and g s , n = − . was used. The calculations followed the procedure for the tin isotope chain closely . Said referenceand references within contains a detailed description of the procedure. Author contributions
P.-A.S., L.C., E.A., D.L.B., C.M., and A.P. designed the experimental setup. P.-A.S., L.C., E.A., G.L.G., D.L., D.N., andT.P. collected the data. L.C., and D.N. wrote the software for data conversion. P.-A.S. wrote the software for data analysis and analysed the data.T.O., N.T., Y.T., and H.L. performed the theoretical calculations. P.-A.S. and D.L. performed the G
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Competing Interests
The authors declare that they have no competing interests.
Data availability
Raw data were obtained at the Extreme Light Infrastructure – Nuclear Physics facility, Romania. All the data used to supportthe findings of this study are available from the authors upon reasonable request. The final data points can be obtained from http://dx.doi.org/10.17632/skhmjshxdj . Code availability
Sorting codes were developed at the Extreme Light Infrastructure – Nuclear Physics facility, Romania. All the codes for theexperimental data used in this study are available from the authors upon reasonable request.
Acknowledgements
We acknowledge A. Imreh from ELI-NP for the CAD drawings of the detector system used for the G
EANT
Correspondence
Correspondence and requests for materials should be addressed to P.-A.S. (email: [email protected]).
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