The partial widths of the 16.1 MeV 2+ resonance in 12C
EEPJ manuscript No. (will be inserted by the editor)
The partial widths of the 16.1 MeV 2 + resonance in C Michael Munch and Hans Otto Uldall Fynbo Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, DenmarkReceived: date / Revised version: date
Abstract.
The 16 . + resonance in C situated slightly above the proton threshold can decay byproton-, α -, and γ emission. The partial width for proton emission cannot be directly measured due tothe low proton energy and the small branching ratio. Instead it must be indirectly derived from otherobservables. However, due to several inconsistent data the derived partial width varies by almost a factor 2dependent on the data used. Here we trace the majority of this inconsistency to different measurements ofthe ( p, α ) cross sections. We have remeasured this cross section using modern large area silicon strip detec-tors allowing to measure all final state particles, which circumvents a normalization issue affecting some ofthe previous measurements. Based on this we determine Γ p = 21 . . + resonance and for isospin symmetry in the A = 12 system.In addition, we conclude that the dataset currently used for the NACRE and NACRE II evaluation of the B( p, α ) reaction should be scaled by a factor of 2 /
3. This impacts the reaction rate accordingly.
PACS. α decay – 27.20.+n Properties of specific nuclei listed by mass ranges; 6 ≤ A ≤ Situated just above the proton threshold the 16 . + state in C has been the subject of numerous studies[1,2,3,4,5,6,7,8,9,10,11,12,13,14] with the most recentcompilation published in ref. [15] and a detailed review ofthe decay properties of the 16 . p + B reaction and it is known to decay via proton, α particle and γ ray emission as illustrated in fig. 1.In one of the first applications of the concept of isospin,the narrow width of only roughly 5 keV of this state sit-uated higher than 8 MeV in the 3 α continuum was ex-plained by Oppenheimer and Serber to be due to its T = 1nature [16]. Hence, its dominating α -decay mode is onlypossible due to admixtures of T = 0 in the state.In the narrow resonance limit the measured cross sec-tions, σ px , can be related to the partial widths, Γ x , of theresonance σ px = 4 π ¯ λ ω Γ p Γ x Γ , (1)where ω = J +1(2 j +1)(2 j +1) with J the resonance spin and j i the spin of the beam and target. ¯ λ = ¯ h/E is the reducedde Broglie wavelength with the center of mass energy, E . Γ p has a key role in this relation, but due to the low pro-ton energy and the fact that Γ p /Γ (cid:28)
1, it is not feasi-ble to measure it directly. Instead, Γ p must be inferredfrom measurements of both Γ x and σ px . This extraction Send offprint requests to : Hans Fynbo was performed in both refs. [12,15], however, the resultingproton widths differ by almost a factor of 2.The decay properties of T = 1 isobaric analog states inthe A = 12 system was analysed by Monahan et al. [17].This analysis was based on a comparison of the protonwidths in C with the neutron spectroscopic factors in B deduced from the B( d, p ) B reaction. Good agree-ment was found for most states with the notable excep-tion of the 16 . B( d, p ) B reaction was remeasured with a new methodwhich confirmed the spectroscopic factors deduced previ-ously within 25% [18]. The proton width recommended byref. [15] results in good agreement with this spectroscopicfactor, while that of ref. [12] does not. In the following wewill summarize the results of previous measurements andattempt to clarify the situation. γ ray emission predominantly occur to the ground state(GS), γ , and the first excited state, γ , in C. The crosssections for the ( p, γ ) reactions were most recently mea-sured by He et al. using a thin target for the first time [13].Here they confirmed the prior cross section measurements[1,4,8,9,14,19,20] yielding a combined result of σ pγ =5 . µ b and σ pγ = 139(12) µ b. Thus we consider the val-ues for these cross sections to be reliable. Complementaryto these measurements, Friebel et al. directly measured Γ γ = 0 . et al. have measured the relative yield of γ rays and charged particles; Γ γ /Γ α = 6 . × − and Γ γ /Γ α = 2 . × − [8]. a r X i v : . [ nu c l - e x ] J u l M. Munch and H. O. U. Fynbo: The partial widths of the 16.1 MeV 2 + resonance in C Table 1.
Prior measurements of the ( p, α ) channel.Measurement σ α, [mb] σ α, [mb] σ α [mb] σ α, /σ α, Γ [keV]Huus et al. [1] ∼ et al. [2] 0 . ± ± Segel et al. [3] 22(3)Anderson et al. [4] 41(3) . et al. [5] 54(6) 5 . +0 . − . Becker et al. [6] 2 . ±
5% 69 . ± . et al. [7] 19 . The authors note that the α particles were “barely detectable” [2]. This result will be disregarded. Assuming infinite target thickness and using the combined Γ of ref. [4,6] this should be rescaled from 38 . The authors note that their model did not reproduce the α data. The current understanding of the α particle decay mech-anism is a sequential decay proceeding either through the Be GS, α , or the first excited state, α [7]. The results ofprior investigations of the ( p, α ) channel are listed in ta-ble 1. There are multiple consistent measurements for theresonance width and combining the results from refs. [5,6]yields 5 . α /α branching ratio are consistentand, as the measurement by Laursen et al. was done withcoincident detection of multiple final state particles, wealso consider the branching ratio reliable. The measuredtotal cross sections for ( p, α ) generally show poor agree-ment. However, considering the ( p, α ) reaction yields adistinct high energy peak we expect the σ p,α measure-ment by Becker et al. to be accurate [6].By combining the various measurements for the α - and γ channels with eq. (1) and approximating Γ ≈ Γ α itis possible to derive several independent values for Γ p .These are listed in table 2. Interestingly, the values seem tocluster into two groups, with the measurements for ( p, α )split across the groups.All of the cross section measurements of the ( p, α )channel were performed with a small energy sensitive de-tector placed at various angles. The measured energy spec- g.s. +12 C4.44 + + α + α + α p + B 7.37 + α + Be10.4 + α α γ γ Fig. 1.
Illustration of the reaction scheme. The 16 .
11 MeV 2 + state C is populated with the p + B reaction. The state caneither decay via γ , α or proton emission. Energies and spinassignments are taken from ref. [15]. Energies are in MeV. trum was then extrapolated to 0 and integrated. refs. [4,5] performed a linear extrapolation, while ref. [6] used asequential decay model. Although Becker et al. note theirmodel performed poorly at this resonance, it does not ex-plain the discrepancy between the measurements. The keydifference is the choice of normalization for the α mea-surement where Becker et al. argue that their detector haseither detected the primary alpha particle α or the twosecondary α particles from the subsequent Be break-up.Thus, they divide their count number by 2. On the con-trary, refs. [4,5] argue they observe one out three finalstate α particles and divide by a factor of 3. The proba-bility for detecting both secondary α particles in a singledetector was discussed theoretically by Wheeler in 1941[21]. The probability depends on the opening angle be-tween the secondary α particles and the aperture of thedetector. The opening angle in turn depends on the en-ergy of the Be system with respect to the α threshold.This is small for the Be GS but quite significant for thefirst excited state. Based on information provided in ref.[6] we have estimated their maximum detector apertureto be of the order of 3 ◦ . In this case the probability for adouble hit is minuscule – even for the α channel. Thus,the α results by Becker et al. should most likely be scaledby a factor of 2 / B( p, α ) eval-uation, where they use the dataset of Becker et al. for itsrecommended value while using the dataset of refs. [5,23] Table 2.
Calculated values for Γ p . The values are calculatedusing eq. (1) with the quantities listed in the left column. Inall cases Γ = 5 . Γ α ≈ Γ was applied.Method Γ p [eV] σ pα [4] 20(2) σ pα [6] + Γ α /Γ a [3,7] 22(3) σ pα [5] 26(3) σ pα [6] 34(6) σ pγ [13] + Γ γ /Γ a [8] 35(3) σ pγ [13] + Γ γ /Γ a [8] 37(6) σ pγ [13] + Γ γ [10] 38(6). Munch and H. O. U. Fynbo: The partial widths of the 16.1 MeV 2 + resonance in C 3
Fig. 2.
Schematic drawing of the experimental setup. An arrowindicates the incoming proton beam. The enriched boron targetwas oriented 45 ◦ with respect to the beam axis. Two quadraticand two annular double sided silicon strip detector were usedto detect outgoing particles. Front and back segmentation isshown simultaneously for clarity. as a lower limit. The updated evaluation NACRE II [24]is less cautious and relies solely on Becker et al. The magnitude of the σ pα cross section has implica-tions beyond nuclear structure. For example the B( p, α )reaction is a candidate for a fusion reaction generating en-ergy without neutrons in the final state, see e.g. [25]. Therate of this reaction at the energies relevant for a fusionreactor is mainly determined by the 16 . + and thehigher lying 16 . − resonances. The proton width isrelated to the B( d, p ) B reaction by isospin symmetry.In turn, that reaction is used to deduce the B( n, γ ) Breaction cross section, which may play a role in the astro-physical r-process [18].The object of this paper is to remeasure σ pα in orderto clarify the situation. The measurement will circumventthe normalization ambiguity by observing all three parti-cles in coincidence using an array of large area segmentedsilicon detectors. In this paper we will only address thecross section of the 16 . A thin foil of B, oriented 45 ◦ with respect to the beamaxis, was bombarded with a beam of H molecules. Aresonance scan was conducted between 460 and 600 keV and afterwards data was collected for 30 hours at 525 keV.The beam was provided by the 5 MV Van de Graaff ac-celerator at Aarhus University and the beam spot wasdefined by a pair of 1 × p, α ) resonances in Aland the energy resolution was found to be better than afew keV for single charged beams. It should be noted thatthe energy stability is trifold improved for H .Upon impact with the target foil the H moleculeswill break up and additional electron stripping, neutral-ization and scattering might occur. This affected the inte-grated beam current, which was measured with a Faradaycup 1 m downstream of the target. The combined effectof this can be determined from the ratio of the observedcurrent both with and without a target foil in the beam.At the beginning of the experiment the effective chargestate was determined to be 1 . e . However, this ratiowas observed to change during the experiment. We at-tribute this to carbon build-up on the target. The changein charge state was 4 . × − µ C − . Correcting forthis, a total of 61(6) µ C was collected on the resonance.The target was produced at Aarhus University by evap-oration of 99 % enriched B onto a 4 µ g / cm carbon back-ing. The thickness was measured by bombarding the tar-get with 2 MeV α particles and the boron layer either fac-ing towards or away from the beam. Assuming a two layertarget the boron thickness can be inferred from the energyshift of particles scattered off the carbon layer using theprocedure of ref. [26], but including a correction for thechanged stopping power of the scattered particle. t = δEK ( θ ) S ( E b ) + S ( K ( θ ) E b ) / cos θ , (2)where E b the beam energy, S the stopping power, δE theenergy difference and K is the kinematic factor for thelaboraty scattering angle, θ , defined as K = m b cos θ + (cid:113) m t − m b sin θm b + m t , (3)where m t , m b is the mass of the target and beam ion re-spectively. The cosine factor in eq. (2) corrects for the in-creased path length for the scattered particle. The resultis 39(3) µ g / cm . The energy loss for a 525 / ◦ and 23 and 36 ◦ respec-tively. Each annular ring is approximately 1 mm wide withan approximate 2 ◦ resolution in polar angle. Additionally,two quadratic DSSDs (W1 from Micron Semiconductors)were placed 40 mm from the target center at an angle of90 ◦ with respect to the beam axis. These covered angles M. Munch and H. O. U. Fynbo: The partial widths of the 16.1 MeV 2 + resonance in C CM energy [keV] C o u n t s p e r k e V α secondaries α and secondaries α Fig. 3.
Full CM energy spectrum without any cuts. The highenergy peak corresponds to the primary α particle, α . between 60 ◦ and 120 ◦ . All pixels are 3 × ◦ . The analysis is structured in the following way. First a res-onance scan is shown for the α channel. This is followedby an extraction of the α angular distribution and crosssection from the multiplicity 1 data. Building upon thisfollows the analysis of the triple events i.e. events withexactly three alpha particles and afterwards a brief dis-cussion of how the detection efficiencies for the α and α channel have been determined. Assuming all ejectiles to be α particles the center-of-mass(CM) energy can be determined from the detected po-sition and energy. The full spectrum, without any cuts,is shown in fig. 3. The spectrum shows a clear peak at5 . . Be. The α particle givingrise to this peak will be referred to as the primary α par-ticle. Below the peak is a broad asymmetric distribution,which consists of secondary α particles and α particlesfrom the break-up via the first excited state of Be. Atlow energy the proton peak is visible. It is double peakedsince energy loss corrections are applied as if it was an α particle.The primary α particle is selected by requiring E CM > .
65 MeV. fig. 4 shows the α yield as a function of pro-ton energy. The yield is clearly resonant and peaks at ∼
175 keV. The curve shown in the figure is the best fitto the thick target yield for a Breit-Wigner shaped reso-
150 160 170 180 190 200 210
Proton energy [keV] Y i e l d [ / n C ] Fig. 4.
Scan of the 162 keV resonance. The individual datapoints corresponds to the α yield, while the solid curve is thebest fit to eq. (4). nance [28] Y = (cid:34) tan − E p − E r Γ lab / − tan − E p − E r − ∆EΓ lab / (cid:35) × Γ lab σ BW ( E = E r )2 (cid:15) η, (4)where Γ lab is the resonance width in the lab system, E p isthe beam energy, E r the resonance energy, ∆E the energyloss through the target, η the detection efficiency, σ BW the resonant Breit-Wigner cross section and (cid:15) = N dEdx ,where N is the number density of target nuclei and dEdx thestopping power. Using the factor outside the parenthesisas a arbitrary scaling factor, the best fit was achieved with ∆E = 24 . E r = 162 . Γ lab fixed to12 / · .
28 keV. The target thickness is consistent withthe result obtained from α -scattering and the resonanceenergy fits with the recommended literature value [20].The α angular distribution relative to the beam axiswas extracted for the long runs at E p = 175 keV. Theangular distribution, corrected for solid angle, can be seenin fig. 5. The solid line shows the best fit to the lowest fiveLegendre polynomials. W ( θ ) = A (cid:34) (cid:88) i =1 a i P i (cos θ ) (cid:35) . (5)The coefficients providing the best fit are a = − . a = 0 . a = − . a = − × − and A = 4 . × sr − . The lower panel of the figureshows the fit rescaled, W (90 ◦ ) = 1, along with the datafrom refs. [6,5], which have been rescaled to coincide withthe fit at 90 ◦ . Good agreement is observed with ref. [5]while the symmetric behavior seen by ref. [6] cannot bereproduced.The total number of counts is found by integration ofeq. (5) i.e. 4 πA . This can be related to the resonant Breit- . Munch and H. O. U. Fynbo: The partial widths of the 16.1 MeV 2 + resonance in C 5 − − d N / d Ω [ / s r ] FitThis datasetBecker et. al
Davidson et. al − − cos ( θ CM ) Fig. 5.
Angular distribution of this dataset corrected for solidangle along with the best fit to eq. (5). The datasets of ref. [6,5] have been rescaled to coincide with the fit at 90 ◦ . Wigner cross section using eq. (4) σ pα, = 2 . . (6)The main uncertainty is the variation in the effective chargestate. The end goal of this analysis step is to extract tuples ofparticles consistent with a decay of the 16 . C into three α particles. It applies the methods describedin ref. [29].The first and simplest requirement is that at least threeparticles must be detected in an event. This massively re-duces the data, since the majority of events consist of elas-tically scattered protons. Furthermore, it is required thatall three particles are detected within 30 ns of each other.This reduces the background from random coincidenceswith protons significantly while >
99 % of good eventssurvive. Additionally, it is required that the sum of CMangles between the CM position vectors must be largerthan 350 ◦ . All particles surviving these cuts are assumedto be α particles. From the detected energy and positionit is possible to calculate the four momentum of each par-ticle. From these, one can calculate the total momentumin the CM and the C excitation energy. This is shownin fig. 6, which has a distinct peak at 16 . . γ transitions in C as observed in ref. [12]. Projectingthe individual energy of the high momentum events it isclear that these correspond to events with one proton andtwo α particles. Hence, all events with p CM >
60 MeV / care removed.The classification of whether a tuple corresponds to adecay via the GS or first excited state, can be done based C Excitation energy [MeV] T o t a l m o m e n t u m [ M e V / c ] Fig. 6.
Total momentum of the three particles in the CM vs.the calculated excitation energy of C. The red line corre-sponds to the cut placed at 60 MeV / c. on whether the CM energy of the most energetic particlelies within the high energy peak in fig. 3. This is the samecut used in sect. 3.1. With this classification the countnumbers for the two channels are N = 3 . × (7) N = 4 . × , (8)where the uncertainties are due to counting statistics. α channel In order to relate the observed number of counts to a yieldit is necessary to determine the detection efficiency. Forthe ground state this is simple. A beam with a 1 × α was generated and emittedaccording to the observed angular distribution in fig. 3.The secondary particles were ejected isotropically accord-ing to conservation of angular momentum. These particleswere propagated out of the target and into the detectors.Energy loss was taken into account using the SRIM en-ergy loss tables [27]. The output of the simulation had astructure which was identical to the data and was thussubjected to the same analysis. From the survival ratio anacceptance was determined η = 7 . . (9)Correcting for the efficiency gives a cross section of σ pα, = 2 . , (10)which is consistent with the singles analysis. α channel The detection efficiency depends heavily on the Be ex-citation energy as it determines the opening angle be-tween the secondary α particles. Laursen et al. found that M. Munch and H. O. U. Fynbo: The partial widths of the 16.1 MeV 2 + resonance in C their sequential decay model fully described their data [7].Thus, events were generated using this model. Propaga-tion and energy loss was done with the same procedure asdescribed in the previous section. From the survival ratiothe acceptance was determined to be η = 0 . . (11)This yields a cross section to the excited channel of σ pα, = 38(5) mb . (12) Both values determined for the α cross section are consis-tent with the measurement by Becker et al. The weightedcross section is σ pα, = 2 . . (13)Comparing the angular distribution in fig. 5 with previousmeasurements, good agreement is observed for the regionaround 90 deg. However, while ref. [6] finds the distribu-tion to be nearly symmetric around 90 deg, this conclusionis not supported by the present measurement or ref. [5].Importantly, the integrated cross section is not very sensi-tive to the large angle behavior, which explains the goodagreement obtained nevertheless.In order to compare σ pα, between the different mea-surements, it is computed as σ pα, = σ pα − σ pα for refs.[4,5]. The result is shown in fig. 7 along with the present α cross section and that of ref. [6]. From the figure the ex-cellent argeement between the present measurement andref. [4] can be observed. Both values deviate more than 2 α from the measurement of ref. [5]. We suspect this is due toan overall normalization problem in ref. [5]. The originalmeasurement by ref. [6] differs significantly from all othermeasurements, but if rescaled by a factor 2 /
3, correspond-ing to the different normalization choice, the data pointis in agreement within the errors. However, in light of thesystematic problems reported by ref. [6] for the 16 . . Instead the recommended α cross section is basedon the results from the present experiment and ref. [4] σ pα, = 39(3) mb . (14)Similarly the recommended total α cross section is σ pα = 41(3) mb . (15)The the ratio of the two α channels from the presentmeasurement is 19(3), which is consistent with both pre-vious measurements. Combining all three measurementsyields σ pα, σ pα, = 19 . . (16) The short-comings of their model at the 16 . α permutations. The importance ofinterference was demonstrated in the work of refs. [30,31].
30 40 50 60 70 α cross section [mb] Anderson et. al
Davidson et. al
Becker et. al
Becker scaledPresent
Fig. 7.
Comparison of the present α cross section with themeasurement of Refs. [4,5,6]. Becker scaled is the data fromRef. [6] scaled by 2 /
3. The full line is the mean recommendedvalue while the dashed lines show the one sigma limit. See textfor details.
Using the present measurement of the ( p, α ) cross sectionthe partial proton width can be determined using eq. (1) Γ p = 19(3) eV , (17)while using the combined cross section yields Γ p = 19 . . (18)Both values are consistent, within the errors, with that ofthe latest compilation [15], but not with the value favoredin the recent review in ref. [12].Combining this proton width with the results of He etal. and Γ , the partial gamma widths can be determined Γ γ = 0 . Γ γ = 18(2) eV . (20)These values are consistent with the latest compilation[15], but inconsistent with a direct measurement usinginelastically scattered electrons, which measured Γ γ =0 . Γ γ is remeasured in a direct manner. The argument inthe recent review hinged on this value being correct [12].Combining the improved total α cross section with the γ cross sections measured by He et al. the branching ratiocan be determined Γ γ Γ α = 1 . × − (21) Γ γ Γ α = 3 . × − , (22) . Munch and H. O. U. Fynbo: The partial widths of the 16.1 MeV 2 + resonance in C 7 which is inconsistent with the measurement by Cecil et al.
Considering the general spread of the measured α crosssections, the most likely explanation for this discrepancyis that Cecil et al. have overestimated the α yield.Using the updated proton width the ratio between thereduced proton and neutron width can be computed. Theanalysis in ref. [17] was performed with Γ p = 69 eV andan updated value can be computed by scaling accordingly γ n γ p = 0 . . (23)This shows a similar degree of isopin symmetry as theother bound states analysed in the A = 12 system [17]. Using the p + B reaction, the break-up of the 16 . C into three α particles has been studied usingan array of large area segmented silicon detectors in closegeometry. The decay via the ground state of Be has beenstudied both with detection of single particles and coinci-dent detection of all three α particles. The derived crosssections are internally consistent and the combined resultis σ pα, = 2 . , (24)which is consistent with the result of ref. [6].Currently, there exists multiple incompatible measure-ments of the decay via the first excited state of Be. Thischannel was studied using coincident detection of all three α particles. The coincidence acceptance was determinedusing the decay model of ref. [7] yielding a model depen-dent cross section σ pα, = 38(5) mb . (25)which is, within the errors, consistent with ref. [4] but notrefs. [5,6].The inconsistency with ref. [6] is due to their claim ofhaving a substantial chance of detecting two out of threeparticles with a single detector. This was discussed basedon ref. [21]. The chance of this is minuscule and hencethe entire α dataset of ref. [6] should be rescaled by afactor 2 /
3. This has a significant impact on the recom-mended astrophysical reaction rate, as both NACRE [22]and NACRE II [24] have based their recommended valueson the dataset provided by ref. [6]. The recommended re-action rate should thus be scaled accordingly. In addition,this also has implacations for the expected yield from ananeutronic fusion reactor.Combining the present measurement of σ pα with thatof ref. [4] a refined partial proton width of Γ p = 19 . Γ γ = 0 . Γ γ = 18(2) eV,using the combined γ cross sections reported by ref. [13].The value for Γ γ differs by roughly a factor of 2 fromthe direct measurement of ref. [10]. Hence, we recommendthat Γ γ is remeasured. Based on these results, we can nolonger recommend the proton width deduced in ref. [12]. Additionally, improved γ - α branching ratios are de-rived. These are roughly a factor of 2 larger than themeasurements published by ref. [8]. We speculate that thisdiscrepancy is most likely due to ref. [8] having overesti-mated the α yield.The recommended value for the proton width can becompared to the spectroscopic factor for the analog statein B. By using the analysis presented in [17] the ratioof the corresponding reduced widths is 0.63, which showsa similar degree of isopin symmetry as the other boundstates analysed in the A = 12 system. A modern calcula-tion of the proton width would be highly interesting. We would like to thank Folmer Lyckegaard for manufac-turing the target. We also acknowledge financial supportfrom the European Research Council under ERC startinggrant LOBENA, No. 307447.
References
1. T. Huus, R.B. Day, Phys. Rev. , 599 (1953)2. O. Beckman, T. Huus, ˇC. Zupanˇciˇc, Phys. Rev. , 606(1953)3. R.E. Segel, M.J. Bina, Phys. Rev. , 814 (1961)4. B. Anderson, M. Dwarakanath, J. Schweitzer, A. Nero,Nucl. Phys. A , 286 (1974)5. J.M. Davidson, H.L. Berg, M.M. Lowry, M.R.Dwarakanath, A.J. Sierk, P. Batay-Csorba, Nucl. Physics,Sect. A , 253 (1979)6. H.W. Becker, C. Rolfs, H.P. Trautvetter, Zeitschrift f¨urPhys. A At. Nucl. , 341 (1987)7. K.L. Laursen, H.O.U. Fynbo, O.S. Kirsebom, K.O.Madsbøl, K. Riisager, K.S. Madsbøll, K. Riisager, Eur.Phys. J. A , 271 (2016),
8. F. Cecil, D. Ferg, H. Liu, J. Scorby, J. McNeil, P. Kunz,Nucl. Phys. A , 75 (1992)9. E.G. Adelberger, R.E. Marrs, K.A. Snover, J.E. Bussoletti,Phys. Rev. C , 484 (1977)10. A. Friebel, P. Manakos, A. Richter, E. Spamer, W. Stock,O. Titze, Nucl. Phys. A , 129 (1978)11. S. Stave, M. Ahmed, R. France, S. Henshaw, B. M¨uller,B. Perdue, R. Prior, M. Spraker, H. Weller, Phys. Lett. B , 26 (2011)12. K.L. Laursen, H.O.U. Fynbo, O.S. Kirsebom, K.S.Madsbøll, K. Riisager, Eur. Phys. J. A , 370 (2016),
13. J.J. He, B.L. Jia, S.W. Xu, S.Z. Chen, S.B. Ma, S.Q. Hou,J. Hu, L.Y. Zhang, X.Q. Yu, Phys. Rev. C - Nucl. Phys. , 1 (2016)14. D.S. Craig, W.G. Gross, R.G. Jarvis, Physical Review ,1414 (1956)15. J. Kelley, J. Purcell, C. Sheu, Nuclear Physics A , 71(2017)16. J.R. Oppenheimer, R. Serber, Physical Review , 636(1938)17. J.E. Monahan, H.T. Fortune, C.M. Vincent, R.E. Segel,Phys. Rev. C , 2192 (1971) M. Munch and H. O. U. Fynbo: The partial widths of the 16.1 MeV 2 + resonance in C18. H.Y. Lee, J.P. Greene, C.L. Jiang, R.C. Pardo, K.E.Rehm, J.P. Schiffer, A.H. Wuosmaa, N.J. Goodman, J.C.Lighthall, S.T. Marley et al., Physical Review C , 015802(2010)19. F. Ajzenberg, T. Lauritsen, Reviews of Modern Physics , 321 (1952)20. F. Ajzenberg-Selove, Nucl. Phys. A , 1 (1990)21. J.A. Wheeler, Phys. Rev. , 27 (1941)22. C. Angulo, M. Arnould, M. Rayet, P. Descouvemont,D. Baye, C. Leclercq-Willain, a. Coc, S. Barhoumi,P. Aguer, C. Rolfs et al., Nuclear Physics A , 3 (1999)23. R.E. Segel, S.S. Hanna, R.G. Allas, Physical Review ,B818 (1965)24. Y. Xu, K. Takahashi, S. Goriely, M. Arnould, M. Ohta,H. Utsunomiya, Nuclear Physics A , 61 (2013)25. D. Moreau, Nuclear Fusion , 13 (1977)26. M. Chiari, L. Giuntini, P. Mand`o, N. Taccetti, Nucl. In-struments Methods Phys. Res. Sect. B Beam Interact. withMater. Atoms , 309 (2001)27. J.F. Ziegler, M.D. Ziegler, J.P. Biersack, Nuclear Instru-ments and Methods in Physics Research B , 1818(2010)28. W.A. Fowler, C.C. Lauritsen, T. Lauritsen, Reviews ofModern Physics , 236 (1948)29. M. Alcorta, O. Kirsebom, M. Borge, H. Fynbo, K. Riis-ager, O. Tengblad, Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers,Detectors and Associated Equipment , 318 (2009)30. G.C. Phillips, Reviews of Modern Physics , 409 (1965)31. K. Sch¨afer, Nuclear Physics A140