Almost medium-free measurement of the Hoyle state direct-decay component with a TPC
J. Bishop, G.V. Rogachev, S. Ahn, E. Aboud, M. Barbui, A. Bosh, C. Hunt, H. Jayatissa, E. Koshchiy, R. Malecek, S.T. Marley, E.C. Pollacco, C.D. Pruitt, B.T. Roeder, A. Saastamoinen, L.G. Sobotka, S. Upadhyayula
AAlmost medium-free measurement of the Hoyle state direct-decay component with aTPC
J. Bishop,
1, 2
G.V. Rogachev,
1, 2, 3
S. Ahn, E. Aboud,
1, 2
M. Barbui, A. Bosh,
1, 2
C. Hunt,
1, 2
H. Jayatissa, E. Koshchiy, R. Malecek, S.T. Marley, E.C. Pollacco, C.D. Pruitt, B.T. Roeder, A. Saastamoinen, L.G. Sobotka, and S. Upadhyayula
1, 2 Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA Department of Physics & Astronomy, Texas A&M University, College Station, TX 77843, USA Nuclear Solutions Institute, Texas A&M University, College Station, TX 77843, USA Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA IRFU, CEA, Universite Paris-Saclay, Gif-Sur-Ivette, France Department of Chemistry, Washington University, St. Louis, MO 63130, USA (Dated: December 16, 2020)
Background:
The structure of the Hoyle state, a highly α -clustered state at 7.65 MeV in C, has long been thesubject of debate. Understanding if the system comprises of three weakly-interacting α -particles in the 0s orbital,known as an α -condensate state, is possible by studying the decay branches of the Hoyle state. Purpose:
The direct decay of the Hoyle state into three α -particles, rather than through the Be ground state,can be identified by studying the energy partition of the 3 α -particles arising from the decay. This paper providesdetails on the break-up mechanism of the Hoyle stating using a new experimental technique. Method:
By using beta-delayed charged-particle spectroscopy of N using the TexAT (Texas Active Target)TPC, a high-sensitivity measurement of the direct 3 α decay ratio can be performed without contributions frompile-up events. Results:
A Bayesian approach to understanding the contribution of the direct components via a likelihoodfunction shows that the direct component is < . − is most likely. Conclusion:
The measurement of the non-sequential component of the Hoyle state decay is performed in analmost medium-free reaction for the first time. The derived upper-limit is in agreement with previous studiesand demonstrates sensitivity to the absolute branching ratio. Further experimental studies would need to becombined with robust microscopic theoretical understanding of the decay dynamics to provide additional insightinto the idea of the Hoyle state as an α -condensate. I. Introduction.
Near-threshold states in C have alarge effect on the formation of elements. Through thetriple-alpha process, the synthesis bottleneck associatedwith the instabilities of the A=5 and 8 isobars is over-come. This reaction is enhanced by several orders ofmagnitude by the existence of a 0 + state just above the3 α threshold known as the Hoyle state. The structureof the state has been an area of interest since its discov-ery [1]. While the fact that the Hoyle state is a highly-clustered 3 α structure is common knowledge, the exactnature of the clustering is a subject of debate to thisdate, and has ramifications for other light few-body sys-tems involved in nucleosynthesis. It has been suggestedthat the Hoyle state may be the manifestation of a newstate of matter known as an α -condensate [2]. Whenthe average nuclear density drops below 1/3 of its nor-mal value, the lowest-energy state is a bosonic clusterof α -particles, a state with some properties similar toa Bose-Einstein condensate. Such a hypothesis has re-ceived extensive study in the past decade theoretically,but experimental observables for such an exotic state areextremely difficult to obtain [3]. One relevant observableis the direct 3-body decay of the Hoyle state, i.e. bypass-ing the Be(g.s) intermediate. If an α -condensate were to exist, this branching ratio can be predicted using asimple formulation of the α -condensate wavefunction inconjunction with two- and three-body tunneling calcula-tions. This value is very small and is estimated at 0.06% [4, 5] although extracting a value is highly model de-pendent. The latest high-sensitivity experiments [4, 6, 7]can only provide upper limits for this value and the bestlimit currently lies at 0.019% [7]. Beyond this point, onereaches the limitations of background associated with theuse of silicon detector arrays [8]. Recent indirect meth-ods predict a branching ratio of 0.00057% [9], a factor of45 lower than the current limit, indicating the magnitudeof the improvement likely needed to directly measure thedirect decay channel.One may differentiate between the decay mechanisms,sequential and direct, by the energy-partition of the 3 α -particles. The sequential decay mechanism restricts theenergy of one of the α -particles (in the center-of-massframe) to roughly 50% due to the well-constrained mo-mentum and energy conservation associated with C → α + Be. Direct decays have no such energy restric-tion and can occupy the full available phase space forthree-body decays. The most-likely direct decay compo-nents correspond to when all α -particles share the energy a r X i v : . [ nu c l - e x ] D ec equally. II. Experimental setup
To study the role of the di-rect decay to the Hoyle state decays, excited statesin C were populated using the β -delayed charged-particle spectroscopy technique [10] using the TexAT(Texas Active Target) TPC (Time Projection Chamber)[11]. A N beam was produced using the K500 cy-clotron at the Cyclotron Institute at Texas A&M Uni-versity. This beam was created via the interaction ofan 11 MeV/nucleon B primary beam undergoing a He( B , N) n reaction in a gas cell. The beam of inter-est was then selected using MARS (Momentum Achro-matic Recoil Spectrometer) [12] and delivered into theTexAT. TexAT is a general-purpose TPC using Mi-cromegas (MICRO MEsh GASeous) + THick Gas Elec-tron Multipliers (THGEM) amplification and segmenta-tion. The signals induced on the Micromegas are dig-itized at 10 MHz by the GET (General Electronics forTPCs) [13] and written to disk. This experiment was per-formed in the ‘2p-mode’ made available by GET wherebytwo half-events are taken to disk. The first half ofthe event corresponds to the implanting of the N intoTexAT. The second half corresponds to the decay of Cinto 3 α -particles. For decays that proceed via the Cground-state or first-excited state, the second half of theevent is absent but the partial half event is still taken todisk. As discussed in further detail for this experimentalsetup, this allows for one-at-a-time implant and decayspectroscopy using 20 Torr CO . Details of the exper-imental setup and analysis of the data are provided indepth in Ref. [10]. III. Almost medium-free branching ratio measurementsof the Hoyle state decay.
Unlike observables extractedfrom heavy-ion reactions, the use of β decay to pop-ulate the Hoyle state provides direct access to an al-most medium-free determination of a direct 3-body de-cay. Furthermore, this route takes maximal advantage ofthe characteristics of TPCs to remove the contributionsfrom pile-up events and other effects that contribute tothe limit currently achieved using solid-state arrays. Theintrinsic limitations for identifying different decays insidea TPC correspond to low-energy scattering of the parti-cles in the fill gas and limited segmentation/thresholdswhich influence the accuracy of track reconstruction ofthe decay-particles. In order to identify any rare directdecays in the data set, each track was fitted with threearms, one for each decay α -particle. The initial parame-ters for these arms were seeded by a Hough transform [14]and the decay vertex was identified by a combination ofusing the stopping point of the implanting N beam andthe highest energy deposition point of the decay tracks.Due to scattering effects in the gas, these tracks maydeviate from their original momentum vector introduc-ing an uncertainty in the measured final momentum vec-tor. In order to minimize this uncertainty, we employeda technique that ensures exact momentum conservationbetween the three α -particles [10, 15]. As a consequence,the uncertainty in the length of the longest track was re- FIG. 1. Definition of θ and θ as the angle between thelongest α -particle track and the second and third-longest re-spectively. duced and the ability to identify direct decay improved.Our procedure to identify direct decays makes use of twoexperimental parameters, one using two extracted anglesand the other making use of standard Dalitz plots. Theseare described, and shown below. a. Angular decay information The angles between themost-energetic α -particle and the two others (as shownin Fig. 1), are determined by a fitting procedure to the3 α -particle tracks. The results (after kinematic fitting)from the data are shown in Fig. 2, overlaid with the lo-cus for sequential and direct decay. The events for directdecay would be centered on (120 ◦ , ◦ ) for an equal-energy partition. While the dominance of sequential de-cay is clear, additional information is required for clearidentificiation of any direct decays. b. Dalitz formulation The Dalitz plot affords a conve-nient way to show the population of the available phasespace in three-body decays. By taking linear combina-tions of the partial α -particle energies in the center-of-mass frame, ε i , such that (cid:80) i ε i = 1, the energy partitionof the 3 α -particles can be represented on a 2-dimensionalplot. Figure 3 shows how the linear combinations of thesethree parameters can differentiate sequential and directdecay. Here, the direct decay component is simulatedusing the DDP (Direct Decay Phase space + Penetra-bility) model [5] that weights an otherwise uniform pop-ulation of phase space by the three-body penetrabilities.The Dalitz population within this model reconstructedwith TexAT is shown in Fig. 3c. This preferentially popu-lates the center of the Dalitz plot where all the α -particleshave similar energies. It is therefore practically identicalto the DDE (Direct Decay Energy-sharing) model wherethe α -particles have identical energies, smeared only bythe uncertainty principle [3, 5]. The experimental data,shown in Fig. 3 demonstrate the dominance of the se-quential decay (as per Fig. 3b). c. Branching ratio measurement Taking the angularinformation and location on the Dalitz plot, the χ wasevaluated for each event for both sequential and direct de-cay (incorporating the varying contribution of the directcomponent across the Dalitz plot). This was formulated
100 110 120 130 140 150 160 170 180 (deg) q ( deg ) q FIG. 2. Reconstructed θ against θ after kinematic fitting.The locus for sequential decay is shown with the dashed ma-genta line. The region occupied by direct decays is shown by adash-dotted red line and is focused mainly around 120 ◦ , ◦ for both θ , θ . as follows: χ θ = min { ( θ − θ theory ) + ( θ − θ theory ) } σ θ , (1)where θ i theory is determined for either the direct or se-quential case and σ θ is the experimental error determinedvia the Monte Carlo simulation (5 ◦ ). The kinematics forsequential decay constrain θ , θ to the locus as shownin Fig. 2. For direct decay, the equal α -particle energyconstraint is slightly relaxed so that the highest-energy α -particle fractional energy cannot exceed ε = 0 .
35 whichgenerates a small region around θ , θ = 120 ◦ , ◦ ,shown in Fig. 2 by the dash-dotted red line. As withthe sequential case, the shortest distance to this locus isfound. This χ θ value was also combined with χ D fromthe Dalitz plot measurement, defined as: χ D = (cid:40) ( y − y seq σ D ) , for sequential( x + y σ D ) , for direct (2)where x and y are the Dalitz plot co-ordinates for eachevent, y seq is the expected Dalitz co-ordinate for sequen-tial decay and σ D (=0.059) is the experimental error de-termined via the width of the projection of the experi-mental data shown in Fig. 3d. The expected y seq is ∼ however the experimentally-observed value is slightly off-set at 0.2 which is used for the χ formulation. This offsetis also replicated in the GEANT4 simulations (Fig. 3e)and is attributed to a combination of energy-loss uncer-tainties at the low energies causing a slight systematic ) / e - e - e ( (a) Exp. data ) / e - e - e ( (b) Seq. Simulated e - e (2300.20.4 ) / e - e - e ( (c) Dir. simulated (d) Exp. data projection(e) Seq. simulation projection (f) Dir. simulation projection FIG. 3. Dalitz plots for (a) experimental data, (b) simulatedfully-sequential decays and (c) simulated fully-direct (DDP )decays. The projections of these Dalitz plots onto the y-axisare shown in (c), (d) and (e) for the experimental data, sim-ulated sequential and simulated direct decays respectively. shift as well as threshold effects inside the TPC whichcannot be fully corrected for. This effect is dominant forsmall α -particle energies and, as such, does not greatlyaffect the center of the Dalitz plot where the α -particleshave a sufficiently large energy ( ∼
130 keV each). Theglobal χ is then defined as the sum of χ θ and χ D .A total of 19019 Hoyle decay events were taken as thecleanest unbiased subset of data whereby the implantedbeam stops sufficiently centrally in the TexAT sensitivearea such that no α -particle may escape and such thatthe beam stops in the central region of the Micromegaswhere the detector has no multiplexing [11]. Therefore,the decay vertex can be much more confidently identified,thereby improving the fitting and energy-partition deter-mination. By selecting the most direct-looking events(where χ dir < χ seq ) and manually checking these - - - - - - Z d i s t an c e ( mm ) · (a) - - - - - - - - Z d i s t an c e ( mm ) · (b) FIG. 4. (a) An example of a direct-looking event lookingat a side-on (XZ) projection where the beam is implantedalong the y-axis (into the plane of the plot) and the z-axiscorresponds to the drift axis. The three α -particle arms havevery similar lengths/energies ( ε = 0 . , . , .
28) and the θ angles between the α -particles are 119 ◦ and 130 ◦ . The com-bination of these parameters favors the classification of thisevent as direct-looking and has a p-value for direct/sequentialdecay of 0.1 and 1 . × − respectively. (b) An example ofa sequential event looking at a side-on (YZ) projection. Thep-values for direct/sequential decay are 1 . × − and 0 .
224 events had the decay vertex and corresponding de-cay arms correctly identified, a double-check was pos-sible to ensure that sequential events were not erro-neously misidentified as direct decays. A small subset ofevents (9), after manual checks, still had χ -values thatindicated that the event was more direct-looking thansequential-looking and corresponded to good Hoyle-statedecays. An example event of which is shown in Fig. 4a - - - - a BR 300.10.20.30.40.50.60.70.80.91 I n t eg r a l / R e l a t i v e li k e li hood LikelihoodIntegral
FIG. 5. (Solid magenta) Likelihood function for different val-ues of the direct 3 α branching ratio ( δ ) for our data using theformulation in Eq. 5. (Dashed red) Integral of the relativelikelihood function. At the 95% C.L., the branching ratio is < . in contrast to an example sequential event in Fig. 4b.Due to the finite resolution afforded by small-angle scat-tering and longitudinal straggling effects, it may still bethat these events are statistically outlying sequential de-cays rather than direct decays. To determine the relativeprobabilities, the χ values were converted into p-values, p χ . These describes the probabilities that if the event waseither direct or sequential, it would produce the observedvalues. For direct decays, the intensity distribution of theDPP is applied at this point.The probability that an event is sequential is given via: P seq = p χ seq (1 − δ ) , (3)where δ is the direct 3 α branching ratio. Similarly, theprobability this event is direct is given by: P dir = p χ dir δ. (4)The event-by-event probabilities were then used to cre-ate a log-likelihood distribution as a function of the 3 α branching ratio: L ( δ ) = (cid:88) n log( p χ seq (1 − δ ) + p χ dir δ ) , (5)which is representative of the product of the probabilityof each event being direct or sequential. This was thenused to generate the likelihood function and the 95% C.L.can be set from the integral of this likelihood function.These plots are shown in Fig. 5. At the 95% C.L., onecan determine that the 3 α branching ratio is < . > . λ = 0 . ∼ α decays, being a combi-nation of real direct-decays and contributions from theso-called ‘ghost peak’ in Be [16]. This ghost peak ap-pears when one has a near-threshold resonance when thepenetrability factor rises faster than the steeply-droppingbut still long-tailed form (i.e. Breit-Wigner) of resonanceline-shapes [17]. A non-zero branching ratio was also pre-dicted in previous studies [3, 4] although the ability todetermine these as direct-looking rather than pileup on an event-by-event basis was not possible.
IV. Conclusion
An almost medium-free measurementof the Hoyle direct decay to three α -particles has beenperformed with a TPC. With 95% C.L., the directbranching ratio is < . Be ghost peak [16] and analyzing theseevents show that they are genuine 3 α -decays and are notpileup events as experienced with previous experimentsthat measured a similar upper-limit. The preferential10 − branching ratio seen here is in agreement with pre-dictions from Faddeev calculations [18]. The strengthof this work relies on the use of a TPC and removal ofuncertainties related to pile-up, a problem that plaguedall previous measurements. More sensitive experimentalstudies of the direct component of the Hoyle state decaywill also require a better theoretical understanding of the3 α -particle dynamics at the microscopic level in generaland the contribution of the sequential decays to the Beghost state in particular.
V. Acknowledgments
This work was supported bythe U.S. Department of Energy, Office of Science, Of-fice of Nuclear Science, under award no. DE-FG02-93ER40773 and by National Nuclear Security Admin-istration through the Center for Excellence in NuclearTraining and University Based Research (CENTAUR)under grant number DE-NA0003841. G.V.R. also ac-knowledges the support of the Nuclear Solutions Insti-tute. [1] D. N. F. Dunbar, R. E. Pixley, W. A. Wenzel, andW. Whaling, Phys. Rev. , 649 (1953).[2] A. Tohsaki, H. Horiuchi, P. Schuck, and G. R¨opke, Phys.Rev. Lett. , 192501 (2001).[3] R. Smith, J. Bishop, J. Hirst, T. Kokalova, and C. Whel-don, Few-Body Systems , 14 (2020).[4] R. Smith, Tz. Kokalova, C. Wheldon, J. E. Bishop,M. Freer, N. Curtis, and D. J. Parker, Phys. Rev. Lett. , 132502 (2017).[5] R. Smith, J. Bishop, C. Wheldon, and Tz. Kokalova,Journal of Physics: Conference Series , 012021(2019).[6] D. Dell’Aquila, I. Lombardo, G. Verde, M. Vigilante,L. Acosta, C. Agodi, F. Cappuzzello, D. Carbone,M. Cavallaro, S. Cherubini, A. Cvetinovic, G. D’Agata,L. Francalanza, G. L. Guardo, M. Gulino, I. Indelicato,M. La Cognata, L. Lamia, A. Ordine, R. G. Pizzone,S. M. R. Puglia, G. G. Rapisarda, S. Romano, G. Santa-gati, R. Spart`a, G. Spadaccini, C. Spitaleri, and A. Tu-mino, Phys. Rev. Lett. , 132501 (2017).[7] T. Rana, S. Bhattacharya, C. Bhattacharya, S. Manna,S. Kundu, K. Banerjee, R. Pandey, P. Roy, A. Dhal,G. Mukherjee, V. Srivastava, A. Dey, A. Chaudhuri,T. Ghosh, A. Sen, M. Asgar, T. Roy, J. Sahoo, J. Meena,A. Saha, R. Saha, M. Sinha, and A. Roy, Physics LettersB , 130 (2019).[8] R. Smith, Experimental measurements of break-up re-actions to study alpha clustering in carbon-12 and beryllium-9 , Ph.D. thesis, University of Birmingham(2017).[9] R. Smith, M. Gai, M. W. Ahmed, M. Freer, H. O. U.Fynbo, D. Schweitzer, and S. R. Stern, Phys. Rev. C , 021302(R) (2020).[10] J. Bishop, G. V. Rogachev, S. Ahn, E. Aboud, M. Bar-bui, P. Baron, A. Bosh, E. Delagnes, J. Hooker, C. Hunt,H. Jayatissa, E. Koshchiy, R. Malecek, S. Marley,R. O’Dwyer, E. Pollacco, C. Pruitt, B. Roeder, A. Saas-tamoinen, L. Sobotka, and S. Upadhyayula, Nuclear In-struments and Methods in Physics Research Section A:Accelerators, Spectrometers, Detectors and AssociatedEquipment , 163773 (2020).[11] E. Koshchiy, G. V. Rogachev, E. Pollacco, S. Ahn,E. Uberseder, J. Hooker, J. Bishop, E. Aboud, M. Bar-bui, V. Z. Goldberg, C. Hunt, H. Jayatissa, C. Magana,R. O’Dwyer, B. Roeder, A. Saastamoinen, and S. Upad-hyayula, Nuclear Instruments and Methods in PhysicsResearch Section A: Accelerators, Spectrometers, Detec-tors and Associated Equipment , 163398 (2020).[12] R. Tribble, R. Burch, and C. Gagliardi, Nuclear In-struments and Methods in Physics Research Section A:Accelerators, Spectrometers, Detectors and AssociatedEquipment , 441 (1989).[13] E. Pollacco, G. Grinyer, F. Abu-Nimeh, T. Ahn, S. An-var, A. Arokiaraj, Y. Ayyad, H. Baba, M. Babo,P. Baron, D. Bazin, S. Beceiro-Novo, C. Belkhiria,M. Blaizot, B. Blank, J. Bradt, G. Cardella, L. Car- penter, S. Ceruti, E. D. Filippo, E. Delagnes, S. D.Luca, H. D. Witte, F. Druillole, B. Duclos, F. Favela,A. Fritsch, J. Giovinazzo, C. Gueye, T. Isobe, P. Hell-muth, C. Huss, B. Lachacinski, A. Laffoley, G. Lebertre,L. Legeard, W. Lynch, T. Marchi, L. Martina,C. Maugeais, W. Mittig, L. Nalpas, E. Pagano, J. Pancin,O. Poleshchuk, J. Pedroza, J. Pibernat, S. Primault,R. Raabe, B. Raine, A. Rebii, M. Renaud, T. Roger,P. Roussel-Chomaz, P. Russotto, G. Sacc`a, F. Saillant,P. Sizun, D. Suzuki, J. Swartz, A. Tizon, N. Usher,G. Wittwer, and J. Yang, Nuclear Instruments and Methods in Physics Research Section A: Accelerators,Spectrometers, Detectors and Associated Equipment , 81 (2018).[14] R. O. Duda and P. E. Hart, Commun. ACM , 11(1972).[15] R. Smith and J. Bishop, Physics , 375 (2019).[16] J. Refsgaard, H. O. U. Fynbo, O. S. Kirsebom, andK. Riisager, Physics Letters B , 414 (2018).[17] F. Barker and P. Treacy, Nuclear Physics , 33 (1962).[18] S. Ishikawa, Phys. Rev. C90