Improved precision on the experimental E0 decay branching ratio of the Hoyle state
T.K. Eriksen, T. Kibédi, M.W. Reed, A.E. Stuchbery, K.J. Cook, A. Akber, B. Alshahrani, A.A. Avaa, K. Banerjee, A.C. Berriman, L.T. Bezzina, L. Bignell, J. Buete, I.P. Carter, B.J. Coombes, J.T.H. Dowie, M. Dasgupta, L.J. Evitts, A.B. Garnsworthy, M.S.M. Gerathy, T.J. Gray, D.J. Hinde, T.H. Hoang, S.S. Hota, E. Ideguchi, P. Jones, G.J. Lane, B.P. McCormick, A.J. Mitchell, N. Palalani, T. Palazzo, M. Ripper, E.C. Simpson, J. Smallcombe, B.M.A. Swinton-Bland, T. Tanaka, T.G. Tornyi, M.O. de Vries
IImproved precision on the experimental E T.K. Eriksen, ∗ T. Kib´edi, † M.W. Reed, A.E. Stuchbery, K.J. Cook,
1, 2
A. Akber, B. Alshahrani, ‡ A.A. Avaa,
3, 4
K. Banerjee,
1, 5
A.C. Berriman, L.T. Bezzina, L. Bignell, J. Buete, I.P. Carter, B.J. Coombes, J.T.H. Dowie, M. Dasgupta, L.J. Evitts,
6, 7, § A.B. Garnsworthy, M.S.M. Gerathy, T.J. Gray, D.J. Hinde, T.H. Hoang, S.S. Hota, E. Ideguchi, P. Jones, G.J. Lane, B.P. McCormick, A.J. Mitchell, N. Palalani, ¶ T. Palazzo, M. Ripper, E.C. Simpson, J. Smallcombe, ∗∗ B.M.A. Swinton-Bland, T. Tanaka, T.G. Tornyi, †† and M.O. de Vries Department of Nuclear Physics, Research School of Physics,The Australian National University, Canberra, ACT 2601, Australia Facility for Rare Isotope Beams, Michigan State University,640 South Shaw Lane, East Lansing, MI 48824, USA iThemba LABS, National Research Foundation, P.O. Box 722, 7129 Somerset West, South Africa School of Physics, University of Witwatersrand, Johannesburg, 2000, South Africa Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata 700064 India TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka, 567-0047, Japan (Dated: July 31, 2020)
Background:
Stellar carbon synthesis occurs exclusively via the 3 α process, in which three α particles fuse toform C in the excited Hoyle state, followed by electromagnetic decay to the ground state. The Hoyle state isabove the α threshold, and the rate of stellar carbon production depends on the radiative width of this state.The radiative width cannot be measured directly, and must instead be deduced by combining three separatelymeasured quantities. One of these quantities is the E α process is an important input parameter in astrophysical calculations on stellar evolution, and a high precision isimperative to constrain the possible outcomes of astrophysical models. Purpose:
To deduce a new, more precise value for the E Method:
The E E E +2 Hoyle state and 2 +1 state in C, respectively. The excited states were populatedby the C( p, p (cid:48) ) reaction at 10.5 MeV beam energy, and the pairs were detected with the electron-positron pairspectrometer, Super-e, at the Australian National University. The deduced branching ratio required knowledge ofthe proton population of the two states, as well as the alignment of the 2 +1 state in the reaction. For this purpose,proton scattering and γ -ray angular distribution experiments were also performed. Results: An E E π / Γ = 8 . × − was deduced in the current work, and an adoptedvalue of Γ E π / Γ = 7 . × − is recommended based on a weighted average of previous literature values andthe new result. Conclusions:
The new recommended value for the E E π / Γ = 6 . × − , while the uncertainty has been reduced from 9% to 5%. The new resultreduces the radiative width, and hence 3 α reaction rate, by 11% relative to the adopted value, and the uncertaintyto 6.1%. This reduction in width and increased precision is likely to constrain possible outcomes of astrophysicalcalculations. ∗ Current address: Department of Physics, University of Oslo,N-0316 Oslo, Norway † Corresponding author: [email protected] ‡ Current address: King Khalid University, Department ofPhysics, Faculty of Science, Abha 61413, Saudi Arabia § Current address: Nuclear Futures Institute, Bangor University,Bangor, Gwynedd, LL57 2DG, United Kingdom ¶ Current address: University of Botswana, 4775 Notwane Rd.,Gaborone, Botswana ∗∗ Current address: Oliver Lodge Laboratory, University of Liver-pool, Liverpool L69 9ZE, United Kingdom †† Current address: Institute for Nuclear Research, The HungarianAcademy of Sciences, Debrecen 4026, Hungary
I. INTRODUCTION
The synthesis of heavier elements in the universe isinitiated by the pp -chain reactions in hydrogen burningstars, where four protons are ultimately converted intoone α particle with the release of energy. However, protoncapture reactions forming heavier elements are inhibitedby the rapid disintegration of Be, T / = 8 . × − s[1], into two α particles, so that no heavier elements areformed in stars at the hydrogen burning stage. It was notknown how nucleosynthesis could proceed beyond Beuntil Salpeter suggested that an equilibrium concentra-tion of Be can be sustained in a star of sufficient he-lium concentration and stellar temperature, resulting ina small probability for a third α particle to fuse with a r X i v : . [ nu c l - e x ] J u l the Be and form C [2]. Carbon production was thussuggested to occur via a sequential fusion of three α parti-cles, (cid:0) α + α → Be (cid:1) + α → C ∗ , now commonly knownas the 3 α process. The stellar conditions required forthe 3 α process are fulfilled at the end of the hydrogenburning stage, due to gravitational contraction of the he-lium produced by the pp -chain reactions. The 0 +2 stateat 7.65 MeV above the ground state in C is crucialfor the 3 α process, as it acts as a resonance for s -wave α capture at the relevant stellar temperatures. Withoutthis resonant state, the cross section for the sequential 3 α process would be too small to produce the observed car-bon abundance in the universe. The resonant state waspredicted by Fred Hoyle [3] before the first experimentalobservations [4, 5], and became known as the Hoyle state.The Hoyle state energy exceeds the α decay threshold,and it disintegrates back to Be + α or 3 α ∼ .
96% ofthe time [6]. Stable carbon is only formed in ∼ . α reaction instances, by electromagnetic decayto the ground state. Figure 1 provides a schematic illus-tration of the formation and various decay modes of theHoyle state. Direct disintegration to three α particles oc-curs very rarely, as is indicated by recent measurements,which provide upper limits of 0.043% [7, 8] and 0.019%[9] for this decay mode relative to the total α break-up.The branching ratio of direct vs. sequential decay is im-portant for structure studies of the Hoyle state, but itis not relevant in the context of stellar carbon formationbecause the contribution from direct fusion of three α particles is negligible. FIG. 1. The 3 α process and the decay modes of the Hoylestate. The carbon production rate can be described by theresonance equation [10] r α = 4 √ × N α π (cid:126) M α k T × Γ α Γ rad Γ × e − ( E α /k B T ) , (1)where N α and M α are the number density and mass ofthe interacting α particles, and Γ, Γ α , and Γ rad are thetotal, α decay, and radiative decay widths of the Hoylestate, respectively. Furthermore, E α = 0 .
38 MeV is the energy released in the break-up of the Hoyle state,and (cid:126) , k B , and T are the reduced Planck constant, theBoltzmann constant, and the temperature, respectively.Since the Hoyle state decays mainly by α emission, Γ ≈ Γ α , and Eq. (1) may be simplified into the expression r α ∝ Γ rad T × e − ( E α /k B T ) , (2)which shows that the carbon production rate dependsdirectly on the radiative width of the Hoyle state. Dueto the sequential nature of the 3 α process and the shorthalf-life of Be, Γ rad cannot be measured directly. How-ever, it can be deduced indirectly by three independentlymeasured quantities (shown in square brackets in Eq. (3))according toΓ rad = (cid:20) Γ rad Γ (cid:21) × (cid:20) ΓΓ E π (cid:21) × (cid:2) Γ E π (cid:3) , (3)where Γ E π is the partial E rad = 3 . rad / Γ [11–18], Γ / Γ E π [19–23], and Γ E π [24], respectively, hence the uncertainty onthe radiative width stems mainly from the challenges ofmeasuring Γ E π / Γ. The goal of the present work was toextract Γ E π / Γ by a new measurement with improved pre-cision.
II. METHOD
The E C, shown in Fig. 1, based on the pro-cedure reported by Alburger [23]. In a C( p, p (cid:48) ) exper-iment, the number of experimentally measured E N E π = N p (0 +2 ) × Γ E π Γ × (cid:15) E π , (4)where N p (0 +2 ) is the number of protons populating theHoyle state, Γ E π / Γ is the E (cid:15) E π is the pair detection efficiency of the 7.65 MeV E E +1 state in C, the expected number ofpairs is N E π = (cid:20) N p (2 +1 ) + (cid:18) N p (0 +2 ) × Γ rad (0 +2 → +1 )Γ (cid:19)(cid:21) (5) × α π (1 + α π ) × (cid:15) E π (cid:39) N p (2 +1 ) × α π (1 + α π ) × (cid:15) E π , where the second term in the bracket may be omitted be-cause Γ (cid:29) Γ rad (0 +2 → +1 ). The pair decay probability ofthe 4.44 MeV E α π = I π /I γ . The E E π Γ = N E π N E π × N p (2 +1 ) N p (0 +2 ) × (cid:15) E π (cid:15) E π × α π (1 + α π ) . (6)Hence, to deduce the E E γ -decay must beknown to account for alignment of the 2 +1 state, whichcan affect the observed pair decay intensity. Measure-ments of the pair transitions and the proton populationratio of the two excited states, as well as of the angu-lar distribution of the 4.44 MeV γ ray de-exciting the2 +1 state, were performed in the present work. The de-tector efficiency for pair measurements was determinedfrom Monte Carlo simulations, as described in Sec. III B. III. EXPERIMENTAL DETAILSA. Spectrometer setup
The experimental setup is located in the Heavy IonAccelerator Facility (HIAF) at The Australian NationalUniversity (ANU). Proton beams were delivered by the14 UD pelletron tandem accelerator [25]. A new spec-trometer setup was developed and optimized for pairmeasurements, based on the existing ANU 2.1 T super-conducting solenoid [26]. The main upgrades involveda new baffle system and detector array, which will bedescribed later in this section. The solenoid itself con-sists of liquid helium cooled NbTi coils, which providea highly homogeneous and axially symmetric magneticfield, with a uniformity of − . ≤ ∆ B/B ≤ +1 . l = 350 mm, respectively. Starting from the left handside of Fig. 2, it can be seen that the target is posi-tioned at 45 ◦ relative to the beam to allow electrons and FIG. 2. An illustration of the Super-e pair spectrometer,showing the target, baffle system, and detector array (fromleft to right). The setup is mounted perpendicular to thebeam, which is represented in yellow. An electron-positronpair transmission is indicated by the red and green trajecto-ries. Image courtesy of Thomas Tunningley, ANU. positrons to be emitted through the rear of the target andinto the spectrometer. Electrons and positrons emittedwithin the acceptance angles and momentum window ofthe spectrometer are transported through the baffle sys-tem and reach the detector plane after following helicaltrajectories due to the Lorentz force. The axially sym-metric baffle system is designed to shield the detector ar-ray against γ rays emitted from the target, and consistsof two axial baffles and a diaphragm made of Heavymet (W-Ni-Fe alloy) coated with a 1 mm layer of
TorrSeal (low vapor pressure epoxy), a low- Z material intendedto reduce both the amount of scattering and secondaryelectron production.The Si(Li) detector array, named Miel, consists of sixidentical, 9 mm thick sector-shaped Si(Li) segments, eachwith an active area of 236 mm [27]. When assembled,the segments form an annular array, but are separated by3-mm-thick, non-magnetic Heavymet spacers to suppresscross-scattering of electrons and positrons between seg-ments. Cross-scattering of 511 keV annihilation quantais also suppressed. The assembled detector array can beseen to the right in Fig. 2. The Miel Si(Li) array maybe operated as a single detector by summing the individ-ual spectra of the segments, or in coincidence mode byrequiring two or more segments to have fired, which isthe case for the pair measurements in the present work.The six segments of Miel provide 15 unique two-segmentcoincidence combinations. The thickness of the segmentsallows for full absorption of electrons and positrons upto a kinetic energy of 3.5 MeV, which corresponds to atransition energy of 8 MeV for internal pair formation.Thus, the array is capable of detecting the 7.65 MeV E γ emission from the tar-get. The detector is positioned at 135 ◦ relative to thebeam axis, 1.5 m away from the target, and has a crys-tal size of 81 mm ×
54 mm (length × diameter). Datameasured at different magnetic fields may then be nor-malized to relative sampling and reaction rates by usingthe peak area of a strong γ -ray transition in spectra pro-jected with gates on the respective magnetic fields. Thesame γ line is used for all normalizations in a particularexperiment. In this work, the strong 4.44 MeV 2 +1 → +1 γ -ray transition was used for normalization.The quantities recorded in the current work were theenergies and times from the six Si(Li) segments of Miel,the energy from the HPGe monitor detector, the solenoidcontrol voltage and the Hall probe reading. There weretwo trigger requirements for storing the information,namely either two Si(Li) signals in coincidence or a signalfrom the HPGe monitor detector. The data were storedevent-by-event, and sorted offline. Summed Miel ener-gies, Miel time differences, and the magnetic rigiditiesof the particles were deduced from the stored quantities.The summed electron-positron pair energy could then beprojected with gates on the physical momentum windowof the spectrometer and prompt time differences, withbackground subtraction performed by gating on the ran-dom time differences. B. Spectrometer efficiency
The overall pair detection efficiency depends on thespectrometer transmission and intrinsic detector effi-ciency. The transmission is determined by the spectrom-eter acceptance angles with respect to the symmetry axis, θ ∈ [15 . ◦ , . ◦ ], the geometry of the baffle system, andthe magnetic field strength. In addition to the directionallimits of the acceptance angles, these properties definethe physical limits in terms of momentum (the momen-tum window) for transportation of an electron or positronfrom the target through the baffle to the detector surface.Particles emitted within the acceptance angles and mo-mentum window are able to reach the detector, while par-ticles outside either will not be transmitted. The widthand centroid of the momentum window increases withmagnetic field strength, which means that the transmis-sion efficiency of the spectrometer increases with particleenergy, and that there is an optimum magnetic field fortransportation of a certain particle energy. A magneticfield vs. energy matrix from a singles conversion elec-tron measurement is displayed in Fig. 3, depicting theincreasing momentum window as a function of magneticfield and measured energy. The solid lines indicate thelimits of the momentum window. An example demon-strating the momentum window for pair measurements isprovided in the energy vs. energy matrix shown in Fig. 4.The transmission of an electron-positron pair involvesthe directional kinematics of two correlated particles, forwhich the emission is dictated by the energy-angle corre-lation between the electron and positron. More specifi-cally, the electron and positron share the available tran-sition energy, less the energy consumed in the creationof two electron masses, 2 m c , according to the double-
500 1000 1500Electron energy [keV]50010001500200025003000 B [ G au ss ] FIG. 3. Energy vs. magnetic field from a
Lu sourceconversion-electron measurement, demonstrating the increas-ing width of the momentum window as a function of magneticfield strength and measured energy. The solid lines indicatethe bounds of the momentum window, calculated as describedin Ref. [26]. The color scale indicates the number of counts.
500 1000 1500 2000 2500 3000Electron(positron) energy [keV]50010001500200025003000 P o s i t r on ( e l e c t r on ) ene r g y [ k e V ] C - 4438 keV E O - 6048 keV E FIG. 4. Energy vs. energy from a C pair conversion mea-surement, showing the 4.44 MeV and 6.05 MeV transitionsin C and O, respectively. Note that the pair distributionfor the 4.44 MeV transition is broadened due to the Dopplereffect caused by decay from moving target recoils. The solidlines indicate the bounds of the momentum window, calcu-lated as described in Ref. [26]. The color scale indicates thenumber of counts. differential pair-emission probability. The double differ-ential is defined as a function of positron energy, E + , andseparation angle of the pair, θ s , and depends on the tran-sition energy and multipolarity. Figure 5 illustrates thekinematics of a pair emission in the spectrometer frameof reference. In the present work, the double differentialpair emission probability was calculated within the Born (cid:28) (cid:882) (cid:28) (cid:1085) (cid:84) (cid:400) (cid:77) (cid:882) (cid:77) (cid:1085) (cid:84) (cid:882) (cid:84) (cid:1085) (cid:400)(cid:381)(cid:437)(cid:396)(cid:272)(cid:286) FIG. 5. Pair emission in the spectrometer frame of reference.The intersection of the beam axis and spectrometer symmetryaxis defines the origin of the coordinate system. approximation with Coulomb correction, which will beexplained in the following. Comparison of the distribu-tions calculated with the Born approximation integratedover θ s , and single differential values for finite size calcu-lations from Refs. [29] and [30], showed that the agree-ment was better than 1% for Z = 6 when E − ≈ E + .Hence, the Born approximation was considered satisfac-tory for the C pair emission simulations. The doubledifferential probability distribution for E d Ω π ( E dE + d cos θ s = (7) p + p − (cid:0) W + W − − m c + p + p − c cos θ s (cid:1) , where p ± denote momenta and W ± = E ± + m c thetotal energies for electrons ( − ) and positrons (+). Forhigher electric multipoles, EL , the double differential dis-tribution is given in terms of the pair conversion coeffi-cient of the transition [32] by d α π ( EL ) dE + d cos θ s = (cid:18) απ ( L + 1) (cid:19) (cid:18) p + p − q (cid:19) ( q/ω ) L − ( ω − q ) × (cid:34) (2 L + 1) (cid:16) W + W − + 1 − p + p − θ s (cid:17) (8)+ L (cid:18) q ω − (cid:19) ( W + W − − p + p − cos θ s ) + 13 ( L − p + p − (cid:18) q ( p − + p + cos θ s )( p + + p − cos θ s ) − cos θ s (cid:19) (cid:35) , where α is the fine structure constant, q is the magnitudeof the quantization vector, (cid:126)q = (cid:126)p + + (cid:126)p − , and ω denotesthe transition energy. It is important to note that inEq. (8), (cid:126) = m = c = 1, so all energies are in terms of m c and p = √ W − E E F = (2 πB + )(2 πB − )(exp(2 πB + ) − − exp( − πB − )) , (9)where B ± denotes the relativistic Sommerfeld parame-ter ( ZαE ± ) /p ± . The Coulomb correction, which alsodepends on the energy budget of the pair, is applied bymultiplication with the distributions provided in Eqs. (7)and (8). Double differential distributions calculated forthe 3.22 MeV E E C are shown in Figs. 6 (a) and (b),respectively. Pairs emitted in the 4.44 MeV E E +1 state of C, it is nec-essary to account for alignment of the nuclear spin statesinduced by the reaction and the effects on the correspond-ing pair emission distribution. The alignment correctionis evaluated by using the distribution coefficients, A and A , of the Legendre polynomials associated with the γ -ray angular distribution of the transition, W γ ( θ lab ) = A + A P (cos θ lab ) + A P (cos θ lab ) , (10)where P ν denotes the Legendre polynomial of order ν ,and θ lab is the γ -ray emission angle in the laboratory rel-ative to the beam axis. The procedure for applying thesecoefficients to correct Eq. (8) for alignment is explainedin Refs. [33, 34].Trajectories of electrons and positrons emitted fromthe target were simulated by solving the relativistic equa-tions of motion with the 4th order Runge-Kutta method.The equations were solved in a realistic magnetic-fieldprofile for the solenoid calculated with Poisson Superfish[35], and the particle trajectories were projected within a (a) 3215 keV E E E + [keV] s [ deg ] (c) 3215 keV E (d) 7654 keV E FIG. 6. Monte Carlo simulations of pair emission and trans-mission. Panels (a) and (b): Double differential pair emissiondistributions for the transitions de-exciting the Hoyle state.The maximum emission probabilities are for E − ≈ E + , and θ s = 30 ◦ and 60 ◦ for the E E detailed specification of the spectrometer geometry in thespectrometer frame of reference. A trajectory calculationwas terminated if the corresponding particle struck thesurface of the absorber system or the inner bore. If boththe electron and positron reached a Si(Li) segment, theevent was registered as successful and all the parameterswere stored. The pair-transmission efficiency was ulti-mately found by the ratio of pairs reaching two separatedetector segments versus the number of emitted pairs.Transmitted and detected events of the emitted 3.22 MeV E E Z C nucleus, and nosystematic uncertainties were assumed for the simulatedtransmission efficiency in the present work.The availability of sources for determining pair detec-tion efficiencies, and even singles conversion electron de-tection efficiencies, is very limited. Consequently, the in-trinsic detector efficiency was deduced from Monte Carlosimulations performed with the PENELOPE simulationtool [36]. Simulated spectra have previously been com-pared to
Ba and Co conversion electron measure-ments [37], with the conclusion that PENELOPE is re-liable for the electron and positron energies relevant for I n t r i n s i c e ff i c i en cy ElectronsPositrons
FIG. 7. Intrinsic electron (blue/upper line) and positron(red/lower line) detection efficiencies of the Miel Si(Li) array,deduced from Monte Carlo simulations (data points). Theerror bars are defined as 3 σ of the statistical uncertainties.The solid lines represent interpolations between data points,and the dashed lines indicate the uncertainties. the current work. Simulations of transmitted monoener-getic electrons and positrons between 0 . − C. Experimental conditions
The 4.44 MeV 2 +1 and 7.65 MeV 0 +2 levels in C werepopulated by using the C( p, p (cid:48) ) reaction at 10.5 MeVproton energy, which is a resonant bombarding energyfor population of the Hoyle state [38]. Target foils of1 mg/cm and 2 × natural carbon containing98.9% C and 1.1% C were used. The beam inten-sity varied between 0 . − . µ A, but was mostly stablearound 500 −
600 nA. For the chosen target and beamenergy, the cross sections for populating the 4.44 MeVand 7.65 MeV levels are reported to be σ . = 291 mb[39] and σ . = 86 . and 2 × target foils posi-tioned at 45 ◦ relative to the beam are 56 keV and 110 keV[40], respectively. A simple reaction rate calculation with10.5 MeV monoenergetic protons and a beam intensity of500 nA impinging on a 1 mg/cm target, yields rates of r . = 6 . × s − and r . = 2 . × s − for pop-ulation of the two excited states. By taking into accountthe relevant conversion coefficients, branching ratios, andspectrometer transmission, the rates of pair constituentsstriking different detector segments in coincidence werededuced. The deduced rates are 0 . E E . E O,which allowed the 6.05 MeV E +2 state to be sampled and conveniently used for energycalibration in conjunction with the strong 4.44 MeV tran-sition in C. The optimum magnetic field for measuringthe 6.05 MeV E IV. RESULTS
Four transitions were sampled during the pair mea-surements of the present work. They were the 4.44 MeV2 +1 → +1 , 3.22 MeV 0 +2 → +1 , and 7.65 MeV 0 +2 → +1 transitions in C, and the 6.05 MeV 0 +2 → +1 transitionin O. An initial objective was to detect the 3.22 MeV E ( O ) ( O ) Cinternal pair spectrumTransition energy [keV] C oun t s FIG. 8. The summed pair spectrum of the three C experi-mental runs. The transitions are normalized to the peak areaof the 4.44 MeV γ -ray transition measured by the monitordetector. Note that the 6048 keV and 7654 keV lines havebeen scaled up for visualization purposes. rendered the observation of this weak transition impos-sible. Instead, the focus turned to the 4.44 MeV andthe 7.65 MeV transitions, which were clearly visible.These two pair transitions in C, as well as the Oline used for energy calibration, are shown in Fig. 8.Note that the spectrum has been shifted up in energyby 2 m c = 1022 keV to reflect the transition energy.The spectrum in Fig. 8 corresponds to 9 days of beamon target, from three experimental runs. To account forsampling time and beam intensity, the individual spectrawere normalized to the peak area of the 4.44 MeV γ -raytransition measured by the monitor detector before sum-mation. Furthermore, the spectra have been random sub-tracted by applying gates on prompt and random timedifferences.Since the 4.44 MeV E +1 state, the pair emission distribution for the trans-mission efficiency calculation had to be corrected for nu-clear alignment effects. In order to obtain the distri-bution coefficients needed for the correction, the γ -rayintensities of the 4.44 MeV transition were measured at θ lab = 20 ◦ − ◦ in 10 ◦ steps, using a HPGe detectorwith a crystal size of 81 mm ×
54 mm (length × diam-eter) positioned 41.5 cm away from the target. The at-tenuation factors for this setup were found to be close tounity. For these measurements, a 1 mg/cm thick naturalcarbon target was used, and the 2 +1 state was populatedby the C( p, p (cid:48) ) reaction at 10.5 MeV. The resulting an-gular distribution is shown with fitted distribution coef-ficients in Fig. 9, and corresponds very well with the onemeasured by Alburger in 1977 [23]. By comparing MonteCarlo simulations for pair transitions from unaligned andaligned cases of the 4.44 MeV state, a 7.45% reduction intransmission efficiency was revealed for the aligned casewith the measured distribution coefficients. lab [deg] W ( l ab ) C - 4438 keV E = +0.170(10)A = -0.246(15) FIG. 9. The angular distribution of γ rays from the 2 +1 → +1 transition in C. The results are in good agreement with theprevious measurement performed by Alburger [23].
The proton population ratio, N p (2 +1 ) /N p (0 +2 ), is alsoneeded to extract Γ E π / Γ from the pair measurements ac-cording to the method described in Sec. II. For this rea-son, scattering measurements of C( p, p (cid:48) ) were carriedout using the ANU BALiN double sided silicon strip de-tector array [41–43]. The proton scattering distributionsof the 2 +1 and 0 +2 states were measured simultaneouslyfor scattering angles between 20 ◦ − ◦ . Measurementswere performed using both a 50 µ g/cm and the same1 mg/cm thick C target foil used in the pair con-version measurements. The 50 µ g/cm thick target wasbombarded over several runs with proton beams of en-ergies ranging between 10 . − . thick target was bombarded with 10.5 MeV protons toobtain the proton angular distributions under the sameconditions as in the C pair measurements of the presentwork. The angular distributions will be discussed in de-tail in a separate paper [44]. Angular distribution func-tions were fitted to the data, and the ratio of the integralsover the full solid angle were used to deduce the protonpopulation ratio of the 2 +1 and 0 +2 states. The 50 µ g/cm target measurements provided N p (2 +1 ) /N p (0 +2 ) as a func-tion of proton energy, which are shown in Fig. 10 for E lab [MeV] N p ( + ) / N p ( + ) C - 2 +1 / 0 +2 proton population ratio FIG. 10. Proton population ratio of the 2 +1 and 0 +2 states in C as a function of proton beam energy. The dashed linerepresents a linear interpolation between the data points. energies relevant to the present work. By averaging thepopulation ratio over the proton energy loss in the tar-gets used in the pair measurements, proton populationratios of N p (2 +1 ) /N p (0 +2 ) = 3 . . and 2 × thick tar-gets, respectively. The corresponding ratio obtained fromthe proton scattering measurement with the 1 mg/cm thick target yields N p (2 +1 ) /N p (0 +2 ) = 3 . N p (2 +1 ) /N p (0 +2 ) = 3 . target. These results are consistent withthe previous value of N p (2 +1 ) /N p (0 +2 ) = 3 . thicktarget.The E E π / Γ values are listed in the fourth column ofTable I. An average E E π / Γ = 8 . × − was found using AveTools [46],which utilizes three different methodologies to evaluatethe average. These are the Limitation of Relative Statis-tical Weight, Normalized Residual Method, and the Ra-jeval Technique, which are explained in detail in Ref. [47].The three methods returned the same average value anduncertainty. A summary of the previous, current, and aweighted average of the E AveTools . TABLE I. The experimental quantities used to deduce the E AveTools [46].Run N p (2 +1 ) / N p (0 +2 ) N E π / N E π × E π / Γ × E (cid:15) E π (cid:15) E π α π . × − . × − . × − ( E0 ) / [10 -6 ] Ajzenberg et al . (1960) Obst et al . (1972) Robertson et al . (1977) Alburger (1977)Present workWeighted average: 7.6(4)
FIG. 11. Previous, current, and weighted average values ofΓ E π / Γ. Further information about the previous measure-ments can be found in Refs. [20–23] (listed in chronologicalorder).
V. DISCUSSION
As can be seen in Fig. 11, there are four previous pub-lished values for the E et al. [20] andObst et al. [21] come from measurements of the neu-tron population ratio, N n (2 +1 ) /N n (0 +2 ), in the reaction Be( α, n ) C at E α = 5 .
81 MeV, while their results forthe E C oun t s / k e V ( A NU ) ( O ) ( O ) C(p,p ) @ 10.5 MeV pair conversion Run 1 Run 2 C oun t s pe r i n t eg r a t o r un i t s ( A l bu r ge r) ANUAlburger
FIG. 12. A comparison of the 7.65 MeV E O peaks.The energy region containing the O peaks was excluded inAlburger’s second run to save time. based on the pair intensity ratio, N E π /N E π , measuredby Alburger in 1960 [19] under the same experimentalconditions. Robertson et al. [22] applied an independentand direct approach to deduce Γ E π / Γ, by measuring theratio of protons in coincidence with a 7.65 MeV pair tran-sition over the singles proton rate N E p,π (0 +2 ) /N tot p (0 +2 ) in a C( p, p (cid:48) ) experiment at E p = 10 .
56 MeV. The pair tran-sitions were detected with a plastic scintillator detectorcovering nearly the full solid angle around the target,thus providing close to 100% pair detection efficiency.However, due to the nature of the experimental setup, anumber of corrections and uncertainties had to be consid-ered in their analysis. In 1977, Alburger performed a pairintensity ratio measurement using the C( p, p (cid:48) ) reactionat 10.5 MeV [23]. The advantages of this approach arethe resonant reaction for populating the Hoyle state, andthe relative ease of measuring population ratios of pro-tons as compared to neutrons. Alburger then deducedthe E E E π / Γ = 8 . × − , agrees with that deduced by Al-burger, Γ E π / Γ = 7 . × − , within the uncertainties.We recommend a weighted average of the previous andcurrent measurements of the E E π / Γ = 7 . × − , for calculation of the ra-diative width of the Hoyle state. As a result, the presentwork reduces the uncertainty of the E r = N A v [ ( c m m o l - ) / s ] -10 NACRE rad =3.70(40) meV rad =3.28(20) meV
FIG. 13. The 3 α reaction rate calculated within the tempera-ture range of helium burning red giant stars using the NACRElibrary value [48] (solid red line with dashed lines indicatingthe range of uncertainty), previous recommended value (bluecircles), and new recommended value (black triangles) of theradiative width of the Hoyle state. one adopted in the recent review by Freer and Fynbo [6],Γ E π / Γ = 6 . × − . The new value of Γ E π / Γ providesa radiative width of Γ rad = 3 . E rad / Γ = 4 . × − [11–18]and Γ E π = 62 . µ eV [24]. Compared to the previ-ously adopted value of the radiative width of the Hoylestate, Γ rad = 3 . α reaction rates, r α , usingthe previous and current radiative widths are providedin Fig. 13. The figure also includes rates calculated withthe standard NACRE library value [48]. The reactionrates agree well within the uncertainties, and it is clearthat the new value on Γ rad would not significantly changeour astrophysical models and predictions. However, thereduced uncertainty will constrain possible scenarios andoutcomes of the calculations, and facilitates advances inthe research on stellar evolution and element synthesisin the universe. A major implication of r α is its effecton the carbon-to-oxygen abundance ratio at the end ofthe helium burning phase of stars, in which the 3 α pro- cess and C( α, γ ) O reaction compete for the available α particles, with the latter reaction also feeding on theavailable C nuclei. The carbon-to-oxygen abundanceratio is important for later stages of stellar evolution, andthe rates of production and consumption of C are there-fore important input parameters in astrophysical calcu-lations.A recent measurement [49] of the radiative branchingratio, Γ rad / Γ, suggests a value that is substantially higherthan the currently adopted ratio used in this work. Com-bining this recent result with the present measurement onΓ E π / Γ, results in a large increase of the radiative widthas compared to the adopted value. This increase wouldhave a significant impact on astrophysical calculations,and it is crucial to address the discrepancy observed forΓ rad / Γ.A new approach to determine the radiative widthfrom a direct measurement of the ratio of the pairtransitions de-exciting the Hoyle state, Γ E π / Γ E π , hasbeen developed [27]. However, the success of this newmethod requires a 20 times reduction in the backgroundcurrently observed in vicinity of the 3.22 MeV E rad . ACKNOWLEDGMENTS
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The radiative width ofthe Hoyle state from γ -ray spectroscopy-ray spectroscopy