oin-d itsrity aree on theenergy.

Nuclear Physics A 760 (2005) 3–18

Properties of the12C 10 MeV state determinedthroughβ-decay

C.Aa. Digeta, F.C. Barkerb, M.J.G. Borgec, J. Cederkälld,V.N. Fedosseevd, L.M. Frailed, B.R. Fultone, H.O.U. Fynboa,∗,

H.B. Jeppesena, B. Jonsonf, U. Kösterd, M. Meisterf, T. Nilssond,G. Nymanf, Y. Prezadoc, K. Riisagera, S. Rinta-Antilag,O. Tengbladc, M. Turrionc, K. Wilhelmsenf, J. Äystög,h

a Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmarkb Department of Theoretical Physics, Research School of Physical Sciences

and Engineering, The Australian National University, Canberra ACT 0200, Australiac Instituto Estructura de la Materia, CSIC, E-28006 Madrid, Spain

d ISOLDE-CERN, CH-1211 Genève 23, Switzerlande Department of Physics, University of York, Heslington, YO10 5DD, UK

f Fundamental Fysik, Chalmers Tekniska Högskola, S-412 96 Göteborg, Swedeng Department of Physics, University of Jyväskylä, FIN-40351 Jyväskylä, Finland

h Helsinki Institute of Physics, University of Helsinki, FIN-00014 Helsinki, Finland

Received 18 April 2005; received in revised form 18 May 2005; accepted 24 May 2005

Available online 4 June 2005

Abstract

Theβ-delayed triple-α particle decay of12B has been measured with a setup that favours ccidence detection. A broad state in12C, previously reported around 10 MeV, has been seen anproperties determined through R-matrix analysis of the excitation spectrum. The spin and pa0+. Interference between this state and the Hoyle state at 7.654 MeV has a marked influencspectrum. The coupling between the two states makes it difficult to determine the resonance 2005 Elsevier B.V. All rights reserved.

* Corresponding author.

E-mail address: fynbo@phys.au.dk (H.O.U. Fynbo).

0375-9474/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2005.05.159

4 C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18

rticle

rticularcitation

withed insical

ented

ofin thee give

irecthpro-nizationand

o.

PACS: 23.40.-s; 26.20.+f; 27.20.+n

Keywords: RADIOACTIVITY 12Be(β−)12B(β−3α) [produced in Ta(p,X)]; measuredα-particlecoincidences; R-matrix analysis; deduced spin and parity of levels in12C; double-sided Si strip detector

1. Introduction

Somewhat surprisingly, the properties of the excited states in12C a few MeV above thetriple-α particle threshold (positioned at 7.275 MeV, this is the lowest threshold for paemission) are not well established [1]. The interest in this region comes both fromα-clustermodels and from the important impact the states have in nuclear astrophysics. Of painterest here are the properties of the broad state seen earlier around 10 MeV exenergy. It was suggested already in 1966 [2] that this state could be a 2+ rotational state(built upon the 0+ Hoyle state at 7.654 MeV), an interpretation that is consistentlater results [3] of triple-alpha cluster calculations. Such a state is therefore includthe NACRE triple-alpha reaction-rate evaluation [4], which is widely used in astrophycommunities.

Tentative evidence for the 10 MeV state comes from the13C(p, d)12C [5] reaction,and more convincingly from inelastic scattering12C(x, x′)12C, with x = p, 3He, α and6Li [6–11].

The 10 MeV state has been populated throughβ-decay of12B in several earlier works[12–14]. TheQ-value is 13.370 MeV and the spin and parity of 1+ of 12B implies that 0+,1+ and 2+ states in12C can be populated in allowed decays. We also employβ-decay of12B, but as a major improvement over the earlier experiments we make use of segmSi detectors that allow triple-coincidences to be recorded.

A similar experiment using the decay of12N was performed a few years ago by someus [15,16]. A combined analysis of these two experiments gives improved accuracydetermination of the properties of the relevant states, as reported elsewhere [17]. Win the present paper a detailed account of the experimental method employed for12B andcomment briefly on the results of the combined analysis.

2. The experiment

The experiment was performed at the ISOLDE facility at CERN. There was no dmethod for producing an on-line beam of12B, but it could be produced indirectly througβ−-decay of12Be. The Be activity was produced in a Ta foil target by a pulsed 1 GeVton beam. The produced atoms were then extracted and ionized in the resonance iolaser ion source [18]. The Be+ ions were subsequently accelerated by a 40 kV voltagemass separated in the high-resolution magnetic separator. The12Be beam was guided tthe setup and stopped in a 40 µg/cm2 carbon foil tilted by 45◦ with respect to the beamThe average trigger rate in our setup was 62 s−1, corresponding to a yield of12Be of about

7× 103 per µC (based on a branching ratio to the 10 MeV state of 0.08% [1]).

C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18 5

placed

ckes ofthin

ip, andd alonge hitvent inentified

idenceSSDss [19].

n linese

wastectorust beal-ith no

ase forarbonect

Fig. 1. Top view of the experimental setup.

The detector setup consisted of two double-sided silicon strip detectors (DSSSD)opposite to each other as shown in Fig. 1. The DSSSDs are constructed from a 50×50 mm2

silicon wafer with a thin implantation layer on each side, doped to make front and bap-type andn-type, respectively. The implants are separated in 16 strips on both sidwidth 3.0 mm, where the front strips are vertical and the back strips are horizontal. Acontact grid of aluminium covering 2% of the area has been deposited on each strthe amplitudes of the resulting signals are transformed to energy signals and storewith parameters identifying in which strip the hit occured. Typically several strips arby different particles and the information describing those hits is stored as a single ethe data structure, the position and energy of the measured alpha particles being idin the offline analysis. Since the detectors each cover about one tenth of 4π and are placedopposite to each other we can measure two—or even three—of the alphas in coinc(double- or triple-coincidence events), as indicated in Fig. 1. A detailed test of the DShas been carried out recently and is published along with a description of the DSSSD

The detectors were calibrated withβ-delayedα-particles from the decay of20Na, pro-duced on-line in the same target. The calibration was performed using the two maifrom the decay and checked with the remaining lines and with a148Gd source. Using threlative intensity of the main line in the individual detector pixels, the beam positionextracted as in [20]. The correction for energy loss in the carbon foil and in the dedeadlayer was done using the SRIM2000 program package [21]; this correction mapplied individually for eachα particle. The final energy resolution (FWHM) was evuated at five energies between 2.1 MeV and 4.4 MeV and found to be 57 keV wsignificant energy dependence.

3. 12B decay data

The12B decay data was taken under basically the same conditions as was the cthe 20Na calibration data, the main difference being the implantation depth in the cfoil. The range of 37.5± 8.9 µg/cm2 for the 40 keV12Be beam was used to get the corr

energy loss corrections.

6 C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18

ofto reject

i-ratorye havearticleslysis ofr threectors is

minedath theis-

cleser elec-

itionalffect onase foreter-

Fig. 2. Triple-coincidence data, where| �psum| vs. Esum is projected for all identified events. The three typesevents where 3, 2 or 1 of the three particles were true alphas can be identified. A cut was chosen in orderfalse coincidences and is shown in the figure.

The recoil energy of the excited12C nucleus induced by theβ decay process is estmated to be of the order of 1 keV, and therefore to a good approximation, the labosystem and center of mass system of the three alpha particles coincide. Hence, wa complete kinematics measurement whenever more than one of the three alpha phave been detected. Such events will be the basis of the following analysis. The anathe coincidence data is divided into two parts corresponding to events where two oalphas were measured. In both cases the identification of the hit position in the detedone requiring the energy in the front and back strip to coincide within 80 keV.

3.1. Triple-coincidence reconstruction

In events where the energy deposited in three different pixels could be deteruniquely the total energyEsum and total momentum�psum are deduced by assuming thall three hits correspond toα particles. The Cartesian coordinate system is chosen witx-axis along the beam direction and thez-axis towards one DSSSD. This allows us to dtinguish between true triple-α coincidences and triples where one (or two) of the partiwere low-energy events as illustrated in Fig. 2. These events may be caused by eithtronic noise or aβ particle. The applied cut:| �psum| < 30 MeV/c andEsum> 0.85 MeVis also shown. The possibility of reconstructing the momentum vector has an addadvantage, since it is very sensitive to the assumed position of the beam spot. The ethepy distribution when the beam spot is chosen to be either the same as was the cthe 20Na beam spot or shifted 0.8 mm vertically is clearly visible and allows us to d

mine the beam position quite accurately. We shall return to this question in Section 4, but

C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18 7

e rightle-alpha

beamorrec-fents

plane.ctions- and

d inn aell as

d

ployeding thethe

Fig. 3. Left part shows the triple-coincidence data with the data restrictions described in the text. To this shown the projection on the sum-energy axis. The vertical axis shows the energy relative to the tripthreshold as well as relative to the12C ground state.

note that a shift of 0.8 mm is reasonable, since it would be impossible to optimize theposition better than roughly 1 mm. As a test of the whole procedure the energy loss ctions were performed with a wrong implantation depth (roughly 1/3 of the actual depth o37.5 µg/cm2). In this case it was impossible to optimize all three momentum componwith the necessary requirement that the implantation spot should be within the foilThis indicates both the importance of the energy loss corrections and that the correwere in fact done properly. The resulting beam-spot position is used in both tripledouble-coincidence analysis.

The resulting triple-coincidence data are shown in Fig. 3. In this plot, introduce[22], for each event(Eα,Esum) is plotted for all three alphas resulting in three dots ohorizontal line. Sum energies are shown relative to the triple-alpha threshold as wrelative to the12C ground state. For comparison tabulated energies of the populate12Cstates are shown.

The triple-coincidence data have been used to test the analysis procedure embelow for the double-coincidence events. We have two independent ways of extractenergy,E8Be, of the intermediate8Be system. The first relation involves the energy offirst-emittedα particle:

2

Eα1 = (E12C − E8Be)3

, (1)

8 C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18

st thisannelad ine can.

s weree- andtructednticallso see

ofot

Fig. 4. All identified double-coincidence events with theEsum, E8Be background cut-off. The8Be(g.s.) energyof 0.0919 MeV is indicated as well.

the second uses the relative momentum�p23 of the two alpha particles from the8Bebreakup:

E8Be = 2p23

2mα

2. (2)

The data are consistent with all decays proceeding through the8Be ground state; this ialready obvious from Fig. 3 where the line defined by Eq. (1) is drawn. (Note thaway of plotting the data in general allows one to immediately identify the decay chfrom the slope, the width and the intercept of the line with the vertical axis.) The spreE8Be is significantly smaller when the second relation is used. Using either method onunambiguously identify the two out of the threeα particles that formed the ground state

3.2. Double-coincidence reconstruction

The double-coincidence data set contains events where only two of the alphadetected, that is all triple-coincidence events are excluded. This makes the tripldouble-coincidence data sets statistically independent. The third alpha is reconsfrom conservation of momentum and all identification requirements were kept ideto what was used in the triple-coincidence analysis. In the total data set one can athe 12.7 MeV 1+ state of12C that decays through the broad 2+ exited state of8Be. Thebreakup of this state was analysed in detail in the earlier experiment on the decay12N[15]. We shall here focus on decays through the8Be ground state and will therefore n

consider these events further.

C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18 9

ed-there thedata

ouble-valuated, the

l the re-

onionn haveparticle

ng intoof theres that

.,now

ces ine beamof the

l ver-

he lefte summthe

m as

d.scaled

As in the triple-coincidence analysis the intermediate8Be energy may be reconstructusing Eq. (2). It is again assumed that the two alphas from the8Be breakup were the lowenergy alphas. The plot of8Be energy vs. sum energy is shown in Fig. 4 along withtwo-dimensional cut used to exclude the low-energy background. Only events whedecay proceeds through the8Be ground state are accepted using this cut. The resultingset was shown in [17].

4. Efficiency simulations

Before one can compare the data to theoretical expressions the triple- and dcoincidence efficiencies of the detector setup are needed. These efficiencies are ethrough Monte Carlo (MC) simulations that include the kinematics of the breakupspatial coverage of the detector system, energy thresholds in the detectors, and alconstruction cuts from the analysis.

The simulations performed consist of 2× 108 events with the12C energy uniformlydistributed over the interval 0.1919–7.0 MeV. The direction of the first emittedα particle isdrawn from a spherical distribution, as is the direction of the two lastα particles in the8Bec.m. system. These two directions cannot be correlated since the intermediate8Be(g.s.) hasspin 0. The position of the decaying12B nucleus in the foil is drawn from a parametrizatiof the range distribution of12Be given by a SRIM simulation. The parametrizing functis chosen such that the parametrization and the SRIM simulated range distributiosimilar shape and identical mean, spread and skewness. Whenever a simulatedwould hit a detector an event is stored in the same way as a raw-data event, takiaccount both the resolution of the strips and the calibration. Finally, an analysissimulated data is performed exactly as was done for the raw-data events. This ensuall cuts are identical for the data and the simulation.

In Section 3.1 it was mentioned that the components of the sum momentum (e.gpy )were very sensitive to the beam-spot position. Through the MC simulation we caninvestigate also the effect of a finite beam-spot size. (When reconstructing coincidenthe data events one cannot correct for the finite beam-spot size event by event and thspot is assumed to be a single point.) Both the finite beam size and the finite widthrange distribution are included.

For simplicity the beam-intensity profile is assumed to be Gaussian with identicatical and horizontal widths. Thus the beam profile has one free parameterσb which canbe optimized by comparison of the simulated and data triple-coincidence events. In tpart of Fig. 5 such a comparison is done for the distribution of the absolute value of thmomentum. It is evident that the simulation forσb = 0.8 mm peaks at a too low momentuvalue whereas theσb = 1.4 mm simulation peaks at a too high value and in additionhigh momentum tail is clearly too pronounced. On the other hand theσb = 1.0 mm simu-lation peaks atpsum= 10 MeV/c as does the data, and the conclusion is to take 1.0 mbest value forσb with 0.8 mm and 1.4 mm as lower and upper limits.

In the right half of Fig. 5 the three projectionsx, y, z of the sum momentum are plotteBoth data and simulation are shown, where the simulation distributions have been

down to fit the lower number of counts in the data. The agreement also for details in the

10 C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18

d three

e details

is thehod for

e-t high

triple-ergy,mewhatin thecom-Onent of

scribed

Fig. 5. To the left is shown the plot used for finding the correct beam-spot size for the simulation. Data andifferent simulations are shown. To the right theσb = 1.0 mm simulation is compared to data for thex, y, z

projections of the sum momentum. Note that agreement between data and simulation is seen even in thof the structure in thepx , py andpz plots.

momentum distribution is remarkable, in particular since the only parameter fittedbeam-spot size. This good agreement adds confidence to the use of the MC metdetermining the detector and data analysis efficiencies.

For the extracted beam-spot size ofσb = 1.0 mm the efficiencies for triple- and doublcoincidence events are shown in Fig. 6. The double-coincidence efficiency drops aenergy because it becomes less likely to detect just one of the two8Be α-particles in onedetector. This in turn causes the triple-coincidence efficiency to increase. For thecoincidence events the sensitivity is very low in the region below 1.5 MeV sum enwhereas for the double-coincidence events the measurement is sensitive even sobelow 1.0 MeV. This conclusion is independent of the size of the beam spot, withlimits given above: The difference in the triple-alpha coincidence efficiency whenparing the possible choices ofσb is 10% at most, largest in the high-energy region.the other hand the difference in the double-coincidence efficiency is almost independenergy and is about 2%. The effects of these differences were tested in the fitting de

below, and the changes in the parameter estimates were found to be negligible.

C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18 11

sity

ne

tion

ilities

For

d width

heial

Fig. 6. Triple- and double-coincidence efficiencies shown as function of the energy in12C above the 3α threshold.

5. Results of an R-matrix analysis

5.1. Separate analysis of 12B data

As mentioned earlier theβ-decay of the 1+ ground state of12B can only populate statein 12C with spin and parity 0+, 1+ or 2+. Moreover since we only study positive-par12C states that can decay through the 0+ ground state of8Be, the spin of the12C statesmust be either 0 or 2.

After the event selection described in Section 3 we have a total of 2.1 × 104 double-coincidence events and 3.7 × 103 triple-coincidence events. The data fitting will be dowithin R-matrix theory [23]. The specific general formulation applicable forβ-decays canbe found in [24] to which we refer for details. In our case it is a good approximato assume that the12C states will decay only through the ground state of8Be and it istherefore sufficient to use the one-channel formalism [25]. Apart from the 0+ Hoyle stateat 7.654 MeV we introduce one more state in the fit and shall consider the two possibfor its spin in turn. If the higher-lying state has spin and parity 0+ it will interfere with theHoyle state, in the case of 2+ the contributions from the two states add incoherently.the case of two 0+ levels we note that the intensity in the spectrum is proportional to

fβPc

∣∣∣∣∑2

λ=1 gλγλc/(Eλ − E)

1− (Sc −Bc + iPc)∑2

λ=1 γ 2λc/(Eλ − E)

∣∣∣∣2

. (3)

Here fβ is the beta-decay phase-space factor, the eigen energy and the reduceamplitude of the stateλ for decay into the channelc (here8Be(g.s.)+ α, denoted byα)are denotedEλ and γλc, respectively, andgλ is the beta-decay feeding amplitude. TpenetrabilityPc and the shift functionSc that contain the effect of the Coulomb potenton the outgoing particles both depend on energy. BothEλ andγλc will depend in a well-

determined manner [26] on the boundary parameterBc, and we shall exploit this below and

12 C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18

ectionextl

sts).issonor-

ienciesondingaden-ce

of thelly in-

nsistent

rmoreble-extendot bethe

s [17].the

Fig. 7. Triple- and double-coincidence fits for the region 0.93–4.15 MeV. Data, fit (for two 0+ levels) and fittinginterval are shown. The effect of the different coincidence efficiencies (Fig. 6) is clearly seen.

later discuss the relation to peak position and peak width. We here and in this subsuse a fixed channel radiusa of 6.71 fm (a value used previously [27]), but shall in the nsubsection look at the important dependence of the results ona. From the experimentaposition and width of the Hoyle state (λ = 1) [1] and choosingBα = Sα(E1) one findsE1 = 7.654 MeV andγ1α = 0.61 MeV1/2. Our final fit parameters are thenE2, γ2α and theratior2 = g2/g1 of Gamow–Teller feeding amplitudes for theβ-transitions to the two state(we do not have a normalization that allows us to extract the absolute matrix elemen

The fit to the experimental data is done employing the original data and the Pomaximum likelihood method making use of the MINUIT package [28]. All efficiency crections are applied to the theoretical spectrum, the statistical error on these efficbeing so small that it can be neglected. As mentioned above MC simulations correspto different experimental conditions gave negligible changes in the fit results. The broing induced by the detector resolution and theβ-decay recoil has no significant influenon the fit (the broadening of the individualα-particle energies will be larger than the12Crecoil energy, but we are insensitive to this because we measure the sum energythreeα-particles). Since the spectra for double- and triple-coincidences are statisticadependent we can do a combined fit to them (separate fits to the two spectra give coresults). The final fits when the upper state has spin and parity 0+ are shown in Fig. 7. Thebest fit hasχ2

λ = 222 for 181 degrees of freedom.The fit parameters depend somewhat on the chosen fit interval and are furthe

strongly correlated. This is shown explicitly for two different fit intervals and the doucoincidence data in Fig. 8. The large correlation occurs because the data do notto a sufficiently high excitation energy; a higher-lying and broader level can then nwell distinguished from a lower-lying narrower level. This problem is not present indata from theβ-decay of12N, and the joint analysis of data sets from12N and 12B inthe next subsection therefore allows to better pin down the exact parameter valueFrom an analysis of the12B data alone the correlations will give rather large errors onindividual parameters, the results areE2 = 12.7+1.2

−0.3 MeV, γ2α = 0.81+0.23−0.06 MeV1/2 and

r2 = −0.25+0.23−0.05. Regarding the values ofγ2α andr2 the sign of the product is of physical

C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18 13

to

sg

eterstionssumingividual

terfer-ementfit could

nt

Fig. 8. The value ofχ2λ is shown as a function ofE2 andγ2α with minimization being performed with respect

the remaining parameters. The grey scale traces the increase inχ2λ relative to its minimum (the “best” fit) and i

chosen such that the grey scale levels correspond to the 1,2,3, . . . σ levels. The left panel shows the fit includinthe low-energy region whereas in the right panel this region has been excluded.

Fig. 9. Calculated spectrum of12C energies relative to the triple-alpha threshold using the best-fit paramobtained for a broad 0+ state. The spectrum is divided by thefβ factor and has not been corrected by the detecefficiencies. Total spectrum and the individual level components are shown, as are the individual spectra aa single-level approximation for each level. See the text for an explanation of the two ways of giving the indcontributions.

importance since it determines which regions should have positive and negative inence. The sensitivity to this sign has been tested by fitting with the additional requirthat the product of these parameters should be positive in which case an acceptablenot be obtained.

The separate contributions from the two 0+ states are shown in Fig. 9. Two differe

ways of extracting the contribution from a given state have been used previously and are

14 C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18

curves

undary

ls aremarkedtate

ghostludedfor theup tod; thise struc-frome

.ve sig-fordata.

0–60%ainingly

ig. 10.

Fig. 10. Double coincidence data and best fit for a 2+ state atEλ = 12.0 MeV with a = 7.17 fm and the highenergy tail of the Hoyle state. The combined fit and the individual contributions are shown. The fittedinclude the detection efficiency.

both illustrated (one must remember that the contributions also depend on the boparameter, we here still useBα = Sα(E1)). In the first, theβ-feedinggλ of the other levelis put to zero while the other parameters remain at their fitted value (i.e., the levestill connected through a common decay channel). The corresponding curves are0+

1 and 0+2 in Fig. 9. In the second way of extracting individual contributions, each sis evaluated from a single-level formula; this corresponds to settingγλα = 0 for the otherstate. The corresponding curves are markedsingle level in Fig. 9. Note that the Hoylestate still gives a sizeable contribution in the single-level approximation, a so-calledpeak [13,29]. This contribution is modified considerably when the second level is inc(keepingg2 = 0); it has a marked influence on the spectral shape and is responsiblekink in the spectrum around 3.2 MeV. Note also that the two components do not addthe total spectrum (full line) since the interference term between them must be addecan be seen, e.g., at 4.1 MeV where the total is about twice as large as the sum. Thture of the spectrum is therefore completely different from what one would derivetwo individual contributions, but a fit making use of two interfering 0+ states describes thdata well (see Fig. 7).

Note that additional strength could be present if higher lying 0+ states also contributeTo do so, such states should have non-vanishing Gamow–Teller strength and hanificant coupling to theα + 8Be channel. There is no reliable theoretical guidanceincluding such contributions, and they are not required to achieve a good fit to theThe total observed Gamow–Teller strength can be estimated to be of the order of 4of the Gamow–Teller sum rule value depending on the level of quenching. The remstrength will go to higher lying 0+, 1+ and 2+ states that in general will couple not onto theα + 8Be channel.

We turn now to the fit for the alternative spin assignment of 2+. In this case it turnsout to be impossible to describe the spectrum in any reasonable way, as seen in F

The fits actually preferred infinitely high values ofE2 and γ2α so it was necessary to

C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18 15

e- andf theried the

e

e a

.

ntitieshe

egma”worthy

on thenless

rom then

theof the

esentedrticulara-decay

tatarformedbers

keep the energy fixed. These fits were performed by simultaneously fitting the doubltriple-coincidence data. Fig. 10 clearly shows that only the position and the width oobserved structure are reproduced, whereas the shape is wrong. We have here vachannel radius by fitting with botha = 6.71 fm anda = 7.17 fm. However, this changin a had a negligible effect on the goodness of the fits. The best fit hadχ2

λ = 496 for 182degrees of freedom compared toχ2

λ = 222 for the fit assuming a spin of 0+. Thus we canconclude thatif a 2+ state is present in12C in the energy range in question, it cannot bdominating state in the transition.

Our analysis therefore clearly points to a spin and parity of 0+ for the higher-lying stateThe same fit to the data can be obtained by changing the boundary parameterBα , the otherparameters will then change in a well-defined manner [26]. We go to primed quadefined byB′

α = Sα(E′2), whereE′

2 is the resonance energy [23] of the upper level. Tcorresponding observed width [23] of the level is given by

Γ o2 = 2Pα(E′

2)(γ′2α)2

(1+ (γ ′

2α)2 d

dESα(E)

)−1

E=E′2

.

This gives the valuesE′2 = 11.42+0.32

−0.12 MeV andΓ o2 = 2.9+2.1

−0.5 MeV. The error bars werextracted by doing the explicit transformation for the “central value plus/minus one sivalues keeping in mind the strong correlation between the two parameters. It is notethat the error bars on the resonance energy are significantly smaller than the oneseigen energyE2. The observed width is rather large and corresponds to a dimensioreduced width [23]

(θ ′2α)2 =

(h̄2

µαa2

)−1

(γ ′2α)2

of 1.14+0.75−0.17.

5.2. Combined analysis of 12B and 12N data

As mentioned above a combined analysis of the present data set and the data fdecay of12N [16] will yield a more accurate result. The12N data extend to higher excitatioenergy while the low-energy region (where both12C states contribute) is better seen inpresent data due to the lower detection cut-off. First results of the combined analysisdouble-coincidence data (that contain most of the statistics) have already been prelsewhere [17]. We give here a more complete account of the analysis results, in pathe dependence on the channel radius, and comment briefly on the extracted betobservables.

Energy values in the results will be quoted relative to the triple-α particle threshold a7.275 MeV. The level parameters of the12C states are the same in the fits to both dsets, but the beta-decay feeding amplitudes are fitted separately. Fits have been pefor a range of channel radiia and the results are collected in Table 1. The total numof data points in the fit region is 473. The minimum inχ2 is found for a channel radiuslightly below 6 fm, i.e., smaller than the value used above. The energyE′

2 and width

Γ o

2 are defined as above and are (for the same channel radius) consistent with the values

16 C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18

s, thectrum

q. (3)atrix, using

up inandenergywidth

eng 2r limitntheat the

ecent

r futureportant

y ofing R-t-offs

Table 1Results of a combined fit to12B and12N data

a χ2 E′2 Γ o

2 Em2 Γ m

2 log(f t)′2(fm) (MeV) (MeV) (MeV) (MeV) 12B 12N

5.6 617 5.06 10.34 3.49 1.44 5.40 6.575.8 568 5.16 10.70 3.43 1.71 5.06 5.696.0 571 4.66 5.82 3.47 1.73 4.90 5.376.5 589 4.15 3.09 3.48 1.72 4.69 5.077.0 621 3.90 2.40 3.45 1.74 4.59 4.96

reached using only12B data. The values depend rather sensitively on the channel radiufit uncertainties being much smaller than this variation. However, the calculated spewill not vary much. As an illustration of this we also quote the peak energyEm

2 and theFWHM Γ m

2 of the component due to the second level alone (defined as above from Edivided byfβ where one has putg′

1 = 0), these are almost constant. The beta-decay melements are estimated from the ratio of the components from each level separatelythe earlier determined log(f t) values of 4.13 and 4.34 for the decays of12B and12N to themain peak of the Hoyle state [1]. The rather substantial variations of the log(f t) values tothe second level are due to the decreasing contribution of the ghost asa increases.

As shown in [17] the combined analysis indicates that there is a level furtherenergy. Since no interference with the two 0+ states seems to occur, it must have spinparity 2+. It has been included in our analysis and turns out to have a resonance6.42 MeV above threshold almost independently of channel radius, with an observedthat varies from 1.7 MeV to 2.0 MeV.1 It is unlikely that this high-lying 2+ state is the samas the one predicted in cluster calculations [3]. We have therefore included a low-lyi+state at the theoretical energy (1.75 MeV above threshold) in the fit and find an uppeof the intensity to such a level corresponding to anf t-value at least 50 times larger thathat of the Hoyle state. Since this 2+ state is predicted to be built upon the Hoyle statebeta-decay feeding to it ought to be not too dissimilar; our data therefore indicates thlevel is lying higher in energy or is not present. There is conflicting evidence from r12C(α,α′) experiments for 2+ states at 9–10 MeV with width� 1 MeV [10] and near11.5 MeV with width� 0.5 MeV [9]. A detailed search for a low-lying 2+ state in a widerenergy range is unfortunately not possible with the present data sets, but we note fouse that the interference between such a state and the 6.42 MeV state could be imin searches for it.

6. Conclusion and outlook

We have detectedβ-delayed triple- and double-alpha coincidences from the deca12B with highly segmented detectors and shown how this data can be interpreted usmatrix theory, provided the data is properly corrected for detection efficiencies and cu

1 The value quoted in [17] was lower due to an error in the conversion to observed quantities.

C.Aa. Diget et al. / Nuclear Physics A 760 (2005) 3–18 17

fficien-from

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llider also

ensi-

ce is in

num-nder

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used in the analysis. Monte Carlo techniques were successfully used to find these ecies. We found that a significant contribution to the spectrum in the energy range1 MeV to 4 MeV above the triple-alpha threshold must come from a 0+ state interferingwith the 0+ Hoyle state, and not a 2+ rotational exitation of the Hoyle state. This regicorresponds to 8–11 MeV in12C exitation energy.

An R-matrix analysis of our data alone yields a resonance energy for the 0+ state clearlyabove 11 MeV. This is significanly higher than the corresponding value from the liter[14], 10.3 ± 0.3 MeV. However, since the interference effects were not included ininterpretation of the previous experiments, we cannot expect “the 0+

2 state energy” to havthe same meaning in the old and the present analysis. As we have shown expliccontribution from the ghost [13,29] of the Hoyle state is important and has to be inclIt is actually seen in Fig. 9 that the fitted distribution function peaks around an eEsum = 3.1 MeV corresponding toE12C = 10.4 MeV. This peak position is in perfecagreement with the previous measurements [12,14].

A combined analysis of the present data and the ones from an earlier decay expeemploying12N yields a resonance energy of the 0+ state around 12 MeV excitation energcorresponding to 4.5–5 MeV above the triple-α particle threshold, while the width of thstate is several MeV and not well determined. The reason for the rather large uncein the level properties is the strong influence of the Hoyle state in the region covereddata, which makes it practically impossible to isolate the new state. However, a calcuwhere feeding to the Hoyle state is turned off gives a spectrum that peaks at 10.73 ±0.03 MeV and has a full width at half maximum of 1.72± 0.02 MeV. We stress that in acircumstances where the properties of this new state are needed one needs to consthe Hoyle state and in general to take great care in the theoretical description.

The present setup was mainly sensitive to decays through the8Be ground state. Whave searched for a low-lying 2+ state, but have seen no indications for it. A more setive search could be done by using decays through the broad 2+ excited state of8Be. Anexperiment where this decay channel is also included in the experimental acceptanprogress.

Acknowledgements

This work has been supported by the Spanish CICYT Agency under Projectber FPA2002-04181-C04-02, by the EU-RI3 (Integrated Infrastructure Initiative) uContract number 506065, and by the European Union Fifth Framework Programmproving Human Potential, Access to Research Infrastructure”.

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