The radiative width of the Hoyle state from γ -ray spectroscopy
T. Kibédi, B. Alshahrani, A.E. Stuchbery, A.C. Larsen, A. Görgen, S. Siem, M. Guttormsen, F. Giacoppo, A.I. Morales, E. Sahin, G.M. Tveten, F.L. Bello Garrote, L. Crespo Campo, T.K. Eriksen, M. Klintefjord, S. Maharramova, H.-T. Nyhus, T.G. Tornyi, T. Renstrøm, W. Paulsen
TThe radiative width of the Hoyle state from γ -ray spectroscopy T. Kib´edi, ∗ B. Alshahrani,
1, 2, † A.E. Stuchbery, A.C. Larsen, A. G¨orgen, S. Siem, M. Guttormsen, F. Giacoppo, ‡ A.I. Morales, § E. Sahin, G.M. Tveten, F.L. Bello Garrote, L. Crespo Campo, T.K. Eriksen, M. Klintefjord, S. Maharramova, H.-T. Nyhus, T.G. Tornyi,
3, 5, ¶ T. Renstrøm, and W. Paulsen Department of Nuclear Physics, Research School of Physics,The Australian National University, Canberra, ACT, Australia Department of Physics, Faculty of Science, King Khalid University, Abha, Kingdom of Saudi Arabia Department of Physics, University of Oslo, Oslo, Norway Dipartimento di Fisica dell’Universit´a degli Studi di Milano and INFN-Milano, Milano, Italy Institute of Nuclear Research, MTA ATOMKI, Debrecen, Hungary (Dated: September 24, 2020)The cascading 3.21 MeV and 4.44 MeV electric quadrupole transitions have been observed fromthe Hoyle state at 7.65 MeV excitation energy in C, excited by the C(p,p (cid:48) ) reaction at 10.7 MeVproton energy. From the proton- γ - γ triple coincidence data, a value of Γ rad / Γ = 6 . × − wasobtained for the radiative branching ratio. Using our results, together with Γ E π / Γ from Eriksen etal. [1] and the currently adopted Γ π ( E
0) values, the radiative width of the Hoyle state is determinedas Γ rad = 5 . × − eV. This value is about 34% higher than the currently adopted value andwill impact on models of stellar evolution and nucleosynthesis. The triple-alpha reaction, which produces stable Cin the universe, is a fundamental processes of heliumburning stars. The entry state of the triple-alpha pro-cess, the second excited state in C, is a 0 + state at 7.65MeV. It has attracted significant attention [2–4] since itwas first proposed in 1953 by Fred Hoyle [5]. The exis-tence of the state was confirmed in the same year fromthe analysis of the α -spectrum from the N(d, α ) C re-action [6]. The Hoyle state is α unbound and the domi-nant decay process ( > . Be, which then disintegrates into two alpha particles.Stable carbon will only be produced either if the Hoylestate decays directly to the ground state via an electricmonopole (E0) transition or by a cascade of two electricquadrupole (E2) transitions.Due to its unusual structure, the Hoyle state has at-tracted continuous attention; see the recent review ofFreer and Fynbo [2] and other recent works [3, 7, 8]. Thediscussion includes nuclear clustering, a spacial arrange-ment of the three α particle clusters of which the state isbelieved to be composed, and discussion on a new formof nuclear matter, in analogy with the Bose-Einstein con-densates. The characterization of the 2 + and 4 + stateson top of the 7.65 MeV 0 + state, forming the Hoyle band [9], together with much improved ab initio calculations[10] are important steps forward. ∗ [email protected] † Present address Department of Physics, Faculty of Science, KingKhalid University, Abha, Kingdom of Saudi Arabia ‡ Present address GSI Helmholtzzentrum f¨ur Schwerionen-forschung, Darmstadt, Germany, and Helmholtz-Institut Mainz,Mainz, Germany § Present address IFIC, CSIC-Universitat de Val´encia, Val´encia,Spain ¶ Present address Institute of Nuclear Research, MTA ATOMKI,Debrecen, Hungary
The production rate of stable carbon in the universeis cardinal for many aspects of nucleosynthesis. The re-action rate is closely related to the decay properties ofthe Hoyle state. The triple-alpha reaction rate can beexpressed as: r α = Γ rad exp( − Q α /kT ) [11]. Here Γ rad is the total electromagnetic (radiative) decay width, Q α is the energy release in the three α breakup of the Hoylestate, and T is the stellar temperature. Γ rad has con-tributions from the 3.21-MeV E2 and the 7.65-MeV E0transitions. The contributions of electron conversion arenegligible, so including photon ( γ ) and pair conversion( π ), Γ rad = Γ E γ + Γ E π + Γ E π . Based on current knowl-edge, 98.4% of the electromagnetic decay width is fromthe E2 photon emission and 1.5% is from the E0 pairdecay [12]. The Γ E π contribution is less than 0.1%.The value of Γ rad cannot be directly measured. It isusually evaluated as a product of three independentlymeasured quantities:Γ rad = (cid:20) Γ rad Γ (cid:21) × (cid:20) ΓΓ π ( E (cid:21) × [Γ π ( E , (1)where Γ is the total decay width of the Hoyle state, whichincludes the α , as well as the E2 and E0 electromagneticdecays.The only absolute quantity in Eq. (1) is Γ π ( E σ . Following the recommenda-tion of Freer and Fynbo [2], we have adopted a value of62.3(20) µ eV from the latter study.The least precisely known quantity is Γ π ( E / Γ. Com-bining all previous measurements [21–25], a value ofΓ π ( E / Γ = 6 . × − was adopted [2]. This valuehas been further improved by a new pair conversionmeasurement at the ANU [1] and a Γ π ( E / Γ ratio of7 . × − was recommended.The third term, Γ rad / Γ, has been measured 8 times be-tween 1961 and 1976 [26–33]. By excluding the value of a r X i v : . [ nu c l - e x ] S e p
0 (4.98) + FIG. 1. Singles proton events recorded in the SiRi E (hori-zontal axis) vs. ∆ E (vertical axis) telescopes using the SiO plus carbon target. Events corresponding to the excitationof the 4.98 MeV 0 + state are indicated by the ellipse. Theinsert shows ∆ E + E total energy spectrum around the pro-ton group of the 4.98 MeV 0 + state, together with the energygate (dashed lines). × − by Seeger and Kavanagh [27], the weightedmean value is 4.13(11) × − . In Ref. [2], a slightly highervalue of 4.19(11) × − was recommended. Γ rad / Γ isclaimed to be the most precise term in Eq. (1).In the present paper we report a new measurementof Γ E γ / Γ, which was deduced from the rate of proton- γ - γ triple coincidences, N . , corresponding to the de-excitation of the Hoyle state through the emission of the3.21 and 4.44 MeV γγ -cascade, to the rate of singles pro-ton events, N . , exciting the Hoyle state:Γ E γ Γ = N . N . × (cid:15) . × (cid:15) . × W . , (2)where (cid:15) . and (cid:15) . are the photon detection efficiencies,and W . is the angular correlation correction for a 0-2-0 cascade. Our approach is similar to that of Obst andBraithwaite [33], but with much improved experimentalapparatus and analysis techniques.The experiments were carried out at the CyclotronLaboratory of the University of Oslo. The Hoyle statewas populated in inelastic scattering of 10.7 MeV protonson a 180 µ g/cm natural carbon target. This energy wasslightly higher than the notional optimum energy of 10.5MeV, where the C(p,p (cid:48) ) reaction has a relatively broadresonance [34]. The higher proton energy was employedto shift the inelastically scattered protons to ∼
1 2 3 4 5 6 7 8 9 10 C g.s.7.65 4.44 O g . s . . ( O ) . ( O ) . ( O ) | | Energy [MeV] (a) Singles protons x1/5
1 2 3 4 5 6 7 8 9 104.44 | | | | C oun t s Energy [MeV] (b) Singles γ -rays FIG. 2. Singles spectra of (a) protons and (b) γ -rays usingthe C(p,p (cid:48) ) reaction. The proton (7 . p ) and γ -ray energy(3 . γ and 4 . γ ) gates used for the analysis are indicated byred lines. were carried out using a target consisting of a layer of140 µ g/cm SiO on a 32 µ g/cm natural carbon back-ing. The Si(p,p (cid:48) ) reaction was used to determine thephoton detection efficiencies. The 0 + state at 4.98 MeVin Si decays with a 100% branching ratio to the groundstate through the emission of a 3.20 MeV - 1.78 MeV cas-cade. In addition, the 4.50 MeV - 1.78 MeV cascade fromthe 6.28 MeV 3 + state was also analyzed. The branchingratio of this cascade is BR . γ = 88 . γ - γ coincidences were measured with the SiRiparticle telescope [36] and the CACTUS γ -ray detectorarray [37]. The 64 ∆ E − E telescopes of SiRi were placedin the backward direction covering angles between 126 ◦ and 140 ◦ relative to the beam direction. The solid angleof the particle detection was around 6% of 4 π . The front(∆ E ) and back ( E ) particle detectors have thicknesses of130 µ m and 1550 µ m, respectively. γ -rays were recordedwith the CACTUS array consisting of 26 collimated 5” ×
5” NaI(Tl) detectors, placed at 22 cm from the tar-get. Each detector had a 10 cm lead collimator to ensureillumination of the center of the detector. The total pho-ton efficiency of the array is ≈ π at 1.33 MeVenergy.Signals in the ∆ E detectors were used as triggers andto start the time-to-digital-converter (TDC). The stopsignal was generated when any NaI(Tl) detector fired. Inthis way prompt proton- γ - γ coincidences could be sortedfrom the event-by-event data. Fig. 1 shows the energydeposition in the E vs. ∆ E detectors recorded with theSiO plus carbon target. The fraction of the particleenergy deposited in the front detector depends on Z , A and the particle energy. This relation, visible in Fig. 1as a “banana” shaped region, can be used to identify thedetected particles, and also to filter events of incompleteenergy deposition (horizontal and vertical bands), as well | | pr | | bgL1 | | bgL2 | | bgL3 | | bgL4 | | bgR1 | | bgR2 | | bgR3 | | bgR4 C oun t s P ea k a r ea [ c oun t s ] Δ T(p γ ) [ns] gate: 7.65 p & 3.21 γ & 4.44 γ FIG. 3. Time differences between protons exciting the Hoylestate and 3.21- and 4.44-MeV γ -rays. The prompt ( pr ) andfour background gates on each side ( bgLx , bgRx ) are marked inred. The average counts in the background peaks is 318(18). as other beam related background events. The ∆ E − E spectrum can be used to select the population of specificstates. Protons exciting the Hoyle state fully stop inthe ∆ E detector. In this case the ∆ E − E telescope wasoperated in anti-coincidence to reject high energy particleevents depositing only partial energy in the ∆ E detector.Fig. 2 shows the spectra of singles proton and γ -rayevents from the C(p,p (cid:48) ) reaction collected over a pe-riod of 12 days. The peak at 1.5 MeV proton energy,labelled as “7.65”, represents the excitation of the Hoylestate. It contains N . = 2 . × events, howeveronly 1 out of ∼ C. In comparison, the number of protons ex-citing the 4.44 MeV 2 + state is about 4.7 times higher,and this state always decays to the ground state with anE2 γ -ray transition. The singles γ -ray spectrum, shownin panel (b) of Fig. 2, is dominated by the 4.44 MeVphoton events. Beside the full energy peak, there is abroad distribution of events of single (at ∼ ∼ γ -rays produced by two unrelatedreactions and observed in prompt coincidence is 7 × − per second, which is about three times lower than thetrue coincidence rate and can be considered as high.Fig. 3 shows the time differences between protons ex-citing the Hoyle state and a pair of 3.21 and 4.44 MeV γ -rays. The main peak at ∆ T ( pγ )= 0 ns ( “pr” ) cor-responds to γ -rays in prompt coincidence with protons.The secondary peaks ( “bgLx” and “bgRx” ) occurring ev-ery 72 ns are from accidental coincidences where oneof the two gamma-rays was produced in another beamburst. The 4 background gates either side of the promptand equal width to the prompt peak were averaged over.Panel (a) of Fig. 4 shows protons (“7.65”) in promptcoincidence with a 3.21 and a 4.44 MeV γ -rays with-out subtraction of accidental coincidences. In the samespectrum N p (2 +1 ), the number of protons exciting the C g . s . . ( O ) . ( O ) Energy [MeV] (a) gate: pr-pr& 3.21 γ & 4.44 γ x1/50 C oun t s Proton energy [MeV](b) random subtracted
FIG. 4. Protons in prompt coincidence with 3.21 and 4.44MeV γ -rays cascade. Panel (a) : both γ -rays observed in the inprompt ( “pr-pr” ) TDC window; (b) : random events from the “pr-bgLx” and “pr-bgRx” TDC gates (Fig. 3) are subtracted. +1 state (“4.44”), is due to accidental coincidences andis nearly 50 times higher. Using TDC gates of “pr-pr” , “pr-bgLx” , “pr-bgRx” and “bgLx-bgRx” the num-bers of N p (2 +1 ) events in the corresponding proton spec-tra are 8251(91), 7697(88), 7914(89) and 54(9), respec-tively. Protons exciting the 2 +1 state will only pro-duce single photon events, therefore the N p (2 +1 ) ratescan be used to remove the random events. Using theabove N p (2 +1 ) rates the scaling factor was obtained as8251(91) / [[7697(88) + 7914(89) − /
2] = 1 . N p (0 + ) rates in the same TDC gates were 249(16),158(13), 197(14) and 66(8), respectively. This gives N . = 212(22) counts. The final proton spectrum intriple coincidence with the 3.21 and 4.44 MeV γ -rays isshown in panel (b) of Fig. 4.Fig. 5 shows the γ - γ coincidence events gated by pro-tons exciting the Hoyle state, where the horizontal axis isthe γ -ray energy and the vertical axis is the summed en-ergy of the two gamma-rays in coincidence. The numberof random events has been evaluated using the accidentalcoincidences of the 4.44 MeV gamma-ray with itself, in-dicated as “4.44/4.44”. The number of such events in thevarious TDC gates were 131(12), 157(13), 134(12), 63(8),which gives a subtraction factor of 1.15(11), a value con-sistent with the one obtained from the proton spectra.To deduce the final γγ coincidence spectra, the scalingfactor of 1.061(12) was adopted. Fig. 5 also shows thefinal matrix of γγ coincidence events. A small residueof the 4.44-4.44 random coincidences is visible, but thenumber of related events under the peaks of interest isnegligible.The final γ -ray spectrum of the 3.21-4.44 MeV cascadeis shown in Fig. 6. The areas of the 3.21 and 4.44 MeVphoton peaks, 208(21) and 213(21) counts, were obtainedby fitting Gaussian functions to these data.Using the scaling factor of 1.061(12), the true triplecoincidence events in the prompt pγ peak in Fig. 3 wasevaluated as N . = 237(23). The adopted value of the
2 3 4 5 6 7 8 9 0 5 10 15 E γ [ M e V ] S u mm ed E γ [ M e V ] C oun t s E γ [MeV] S u mm e d E γ [ M e V ] FIG. 5. γ -ray energy vs. summed γ -ray energy matrixconstructed from γ - γ coincidence events gated by protons ex-citing the Hoyle state. Random events have been removed.The gate representing the 3.21 plus 4.44 MeV summed energy(7 . sum ) is indicated with red horizontal lines. The insertshows the region around the 3.21 and 4.44 MeV transitionsin 3D. Data have been compressed by factor 4. The locationof the random coincidences of the 4.44 MeV γ -ray with itselfis also marked. N . = 217(21) was obtained as the weighted mean ofthe three values deduced from the different projections.The absolute photon detection efficiency, (cid:15) , was evalu-ated using the Penelope code [38]. The same simulationswere used to evaluate the correction factors, W and W , for the γ -ray angular correlation, including geo-metrical attenuation coefficients [39], listed in Table I.To confirm the accuracy of the simulations, the protongated spectrum of the 1.78 and 4.50 MeV γ -rays fromthe 6.28 MeV 3 + state in Si was used. The ratio of thepeak areas of the 1.78 MeV and 4.50 MeV transitions is1.58(3), which after applying the 1.0170(15) correctionfor the angular correlation, is very close to the value of1.63(4) from the simulations.By evaluating Eq. 2 with values from Table I and con-sidering all 325 NaI detector combinations, we obtainedΓ E γ / Γ = 6 . × − .To reduce dependence on the Monte Carlo evaluationof the absolute efficiencies and perform an analysis simi-lar to that of Obst and Braithwaite [33], the Γ E γ / Γ ratiowas deduced using:Γ E γ Γ = N . N . × N . N . × (cid:15) . γ (cid:15) . γ × (cid:15) . γ (cid:15) . γ × W . W . . (3)The symbols are as given for Eq. (1). An alternative C C oun t s E γ [MeV] gate: 7.65 p & pr-pr & 7.65 sum FIG. 6. Random subtracted γ -rays from the Hoyle state.The fit to the spectrum including the 3.21 and 4.44 MeVtransitions is shown in red. equation can be obtained using the 6.28 MeV 3 + statein Si. Using the singles proton and pγγ triple co-incidence rates of the 4.98 MeV and 6.28 MeV states,the ratio of the proton to photon efficiencies could bedetermined. Combining the results from Eq. 3 andusing numerical values from Table I, we again obtainΓ E γ / Γ=6.1(6) × − .Using the theoretical total conversion coefficient, α tot ( E , .
21 MeV) = 8 . × − [40] and the rec-ommended value of Γ π ( E / Γ [1], we obtain Γ rad / Γ =6 . × − . This value is more than 3 σ away from thecurrently recommended Γ rad / Γ value [2]. Most of theprevious measurements [29–32] were based on countingthe number of C atoms surviving after the Hoyle statewas formed in various nuclear reactions. To achieve highstatistics, the particle detection was carried out withoutmagnetic selection and often with reported count ratesabove 10 kHz. Under these conditions the elimination ofaccidental coincidences is very challenging.The investigation by Obst and Braithwaite [33] de-duced the Γ E γ / Γ ratio using a similar procedure to thepresent study. Their final result, which was obtained us-ing Eq. (14) of their paper, contains five ratios ( A to E ).Despite some differences between their experiment andours, various combinations of these ratios should agreewithin a few percent. The largest difference occurs for B × D = ( N . × N . ) / ( N . × N . ), Ref. [33]reports 0.409(15) whereas our value is 0.80(4). Thus mostof the difference between Obst and Braithwaite [33] andour work stems from the N . /N . ratio in the Sicalibration data. Our results were independently checkedin Canberra and Oslo using different analysis software.Moreover, the data of Obst and Braithwaite for B × D are not self consistent. Using the photon efficiencies, thecorrection factors for the γγ angular correlations and the γ -ray branching ratio from the 3 + , BR . γ state we have: B × D × (cid:15) . (cid:15) . × W . W . = BR . γ = 0 . . (4)The data of [33] are in disagreement with Eq. (4) by afactor of two; the present data (Table I) agree within 2%. TABLE I. Quantities used to evaluate Γ E γ / Γ ratio.0 + (7.65 ) 0 + (4.98) 3 + (6.28) N
020 or 320 N singles . × . × . × γ -ray (cid:15) . =0.221(3) (cid:15) . =0.222(3) (cid:15) . =0.186(3)efficiency [%] (cid:15) . =0.187(3) (cid:15) . =0.304(3) W
020 or 320
Finally, using the recommended Γ π ( E / Γ [1], theadopted Γ E π and our Γ rad / Γ values, the radiative widthof the Hoyle state is Γ rad = 5 . × − eV. This re-sult suggests a significantly higher radiative width thancurrently adopted.The triple-alpha reaction together with C( α, γ ) arethe two most important helium burning nuclear reactionswith a significant impact on nucleosynthesis and the evo-lution of massive stars [41–43]. In the core-He burningcycle these reactions compete to determine the relativecarbon and oxygen abundances before the core-C burningstarts. The uncertainties due to production rates growat every step. This makes the uncertainty of the triple-alpha and the C( α, γ ) reaction rates crucial for the pro-duction of heavy elements. Recent calculations [42, 43]have explored variations within the uncertainties of theproduction rates: ±
10% for the triple-alpha and ± C( α, γ ) reactions. West, Heger and Austin [42] pointed out that a 25% increase in the triple alpha ratewould be consistent with a 33% larger C( α, γ ) rate.Here we report a 34% change in the triple-alpha reactionrate, which is outside of the parameter space of the cal-culations. This scenario needs to be explored, as it couldchange many of the model predictions.In summary, a new measurement of the Γ rad / Γ ratioof the Hoyle state has been performed using a muchimproved experimental setup than used in the laststudy, more than 40 years ago, giving a value thatis significantly higher. The accurate determinationof the triple-alpha rate remains a challenge for lowenergy nuclear physics. The present experiment onlyfocused on one of the three terms defined in Eq. (1).Confirmation of the new result, using higher resolutionphoton spectrometers is well warranted. Additionalexperiments of the Γ π ( E / Γ ratio, as well as of the E0width, Γ π ( E
0) are equally important.
ACKNOWLEDGMENTS
The project was supported by the Australian ResearchCouncil Discovery Grants DP140102986, DP170101673and by the Research Council of Norway, Grant 263030.TK, BA and AES acknowledge the hospitality of the Uni-versity Oslo during the experiments. ACL gratefully ac-knowledges funding from the European Research Councilthrough ERC-STG-2014, Grant Agreement no. 637686. [1] T.K. Eriksen, T. Kib´edi, M.W. Reed, A.E. Stuchbery,K.J. Cook, A. Akber, et al. , Phys. Rev.
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