Towards the pair spectroscopy of the Hoyle state in 12 C

The triple–alpha process leading to the formation of stable carbon in the Universe is one of the most important nuclear astrophysical processes. The radiative width of the so–called Hoyle state, involving the 7.654 MeV E0 and the 3.2148 MeV E2 transitions, is known with 10–12% accuracy. A novel, more direct approach to determining the width is proposed here, based on the measurement of the E0 and the E2 internal pair conversion intensities. We report on the development of a new magnetic pair spectrometer with high sensitivity for electron– positron pairs and with excellent energy resolution.


Introduction
Carbon, the fourth most abundant element in the Universe, is produced through the triple-alpha process.In the stellar environment, two α particles ( 4 He nuclei) fuse to form the highly unstable nucleus 8 Be, which has a halflife of only 6.7 × 10 −17 s for decay back to two α particles.Occasionally, a third α particle combines with the 8 Be before the decay takes place, forming a cluster of three α particles.At the right energy, these α-particles form a 7.654 MeV resonant state, the Hoyle state, in the stable nucleus 12 C (see Fig. 1).However 99.96% of the time the resonant state decays back to 8 Be by alpha emission, producing no stable carbon nuclei.The remaining 0.04% of the time it decays to one of two lower energy states in 12 C.This rare decay is the only path for the formation of carbon in the universe.Since the original prediction of the existence of a second 0 + state in 12 C by Fred Hoyle [1] in 1953, this state has been of great interest, not only in nuclear synthesis, but also in nuclear structure and reactions.Its existence was experimentally confirmed soon afterward [2].However, the structure of the Hoyle state is still not understood in detail.
The decay pathways of the Hoyle state are illustrated in Fig. 1.Because γ-ray emission is forbidden for transitions between two 0 + states, this pathway (E0) can only take place via internal conversion and/or pair production.Calculations show that for a 7.654 MeV E0 transition, the a e-mail: Tibor.Kibedi@anu.edu.au

C
Fig. 1. 3α-process and the formation of 12  The total rate of 12 C producing decays from the Hoyle state depends directly on the radiative width Γ rad , which includes the width for γ emission (Γ γ ), for internal conversion (Γ CE ), and for pair production (Γ π ); i.e.
The rate r 3α for the triple-alpha reaction [3] can be written as where Q 3α is the energy released in the α decay of the Hoyle state and T is the temperature.In previous studies Γ rad has been determined as a product of three independently measured quantities: The total width of the Hoyle state, Γ is defined as the sum of the radiative and the alpha decay widths: The currently adopted Γ rad /Γ, Γ E0 π /Γ and Γ E0 π values are summarized in Table 1.The largest contribution to the uncertainty in the triple α rate is from Γ E0 π /Γ (8.9%) followed by Γ E0 π (8.8%).The later one is dominated by a large, 6×σ difference between the two most recent measurements of Γ E0 π by Crannell et al. [4] and Chernykh et al. [5].Further studies are required to resolve the discrepancy between these two measurements, which use electron scattering methods.

Design of the new pair spectrometer
The Hoyle state will be populated in the laboratory using the 12 C(p, p ′ ) 12 C(7.654MeV) reaction at 10.5 MeV, a resonant bombarding energy [27].Electron-positron pairs will be recorded using the ANU Super-e electron spectrometer [24] augmented with an array of Si(Li) detectors as shown in Fig. 2. The 2.1 Tesla superconductive solenoid is mounted perpendicular to beam of the 14UD Heavy Ion accelerator.The target is rotated at 45 degrees to the beam direction allowing the beam to pass through and the transportation of electrons and/or positrons (referred here as "particles") from the rear of the target.For a given magnetic field the two axial baffles and the diaphragm (Fig. 2) define an energy range of particles which can reach the detector.A key element of the new pair spectrometer is a Si(Li) array, consisting of six pie-shaped detectors, located at 35 cm distance from the target, where most of the particles will complete two and a half loops.In this arrangement, a valid pair event is defined as one in which any pair of the six detectors has fired and the summed energy of the two particles, E + and E − satisfies the relation: where ω is the transition energy and m • c 2 is the electron rest mass.Due to the selectivity of the magnetic transporter, both particles, the electron and positron, can only reach the detector if they have similar kinetic energies.The pair spectrometer will sample a well defined energy window centered around (ω Electrons and positrons share the available kinetic energy and are ejected with a separation angle, θ s .The double differential of the emission probability, d 2 P π /dE + dθ s depends on the atomic number, transition energy, multipolarity as well as E + and θ s .The relevant emission rates for the 7.654 MeV E0 and 3.2148 MeV E2 transitions have been evaluated [28] using the Born approximation, which is a sufficiently accurate approach for a relatively low Z value and for cases when E + ≈ E − .For electric monopole transitions the pair conversion probability, P π (Z, ω), is The monopole transition strength ρ 2 is a dimensionless parameter for an E0 transition between the initial (0 + i ) and final (0 + f ) states and depends only on the nuclear structure [29].On the other hand, the so-called electronic factor, Ω π (Z, ω) [30] does not depend on nuclear properties.Its second derivative, d 2 Ω π /dE + dθ s can be calculated using the formulae given by Oppenheimer [31]: Heavy Ion Accelerator Symposium 2012 where p + (p − ) and W + (W − ) are the positron (electron) momenta and total energy.To take into account the effect of the nuclear Coulomb field a small correction [32] was also applied.The distribution of the theoretical d 2 Ω π /dE + dθ s values over the full positron energy and separation angle parameter space is shown for the 7.654 MeV E0 transition in the upper panel of Fig. 4. It peaks for equal energy sharing and for a separation angle of ∼ 60 • .For multipole transitions with L > 0, the differential pair conversion coefficient, d 2 α π /dE + d cos θ is defined as: where d 2 P π (Z, ω, τL)/dE d cos θ s is the double differential probability of the pair emission.P γ (Z, ω, τL) is the photon emission probability.The lower frame of Fig. 4 shows the probability distribution for the 3.2148 MeV E2 transition, as was evaluated using the Born approximation [33] with a correction [6].For the E2 transition the distribution is a much sharper function of the separation angle than the one is for the E0 transition.It peaks at θ s ∼ 30 • .On the other hand, it is less sensitive to the positron energy (E + ), but approximately equal energy sharing is still favoured.There is a slight tendency for E + > E − due to the Coulomb field.
In the absence of detailed tabulations of the double differential Ω π (E0) and α π (E2) values, Eqns 7 and 8 will be used to evaluate the pair conversion efficiency of the spectrometer.By comparing the dα π (E2)/dE + (single differential) values with those from ref. [34,35], we found that for Z = 6 and for cases when E − ≈ E + the above approximations are accurate to better than 1%.
Central to the development of the new pair spectrometer (Fig. 2) was the evaluation of the transportation of the electron-positron pairs.A modified version of the original Monte Carlo code, developed for conversion electron measurements [24], was used.A new emission module was added, which samples the probability distributions (Eqns.6 and 8) using randomly selected E + and θ s values.The initial take-off angles defined in Fig. 3 (θ − , θ + , φ − , φ + ) are randomly selected and the trajectories for both particles are evaluated in the axially symmetric magnetic field using the Runge-Kutta method.Typical particle trajectories (coloured lines) as well as the envelope of all trajectories for a fixed magnetic field ("cloud") are shown in Fig. 2. The initial experiments demonstrated [26], that the pair spectrometer works and that the main source of the background is the high energy photons from the target itself.This is dominated by the 4.439 MeV E2 transition from the first excited state in 12 C.Using extensive simulations, a new absorber system has subsequently been designed.The shape of the two axial absorbers and diaphragm has been set to maximise the electron-positron pair efficiency and the material between the target and the Si(Li) detectors.The absorbers are made from Heavymet (a non-magnetic tungsten alloy) with a minimum thickness of 8.2 cm to shield high energy photons.To minimise the scattering of electrons and positrons, as well as to reduce the production of photoelectrons, a thin (1-2 mm) layer of low Z material (epoxy resin, Torr Seal) was added to the absorbers and to the diaphragm.

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EPJ Web of Conferences in Fig. 4 and Table 2.These are based on 500,000 simulated events in each case.The distribution of electronpositron pairs, when both particles hit different detector segments, are superimposed on the theoretical distribution of pair emission in Fig. 4 and labelled as "detected pairs".For clarity, these distributions were constructed from the initial coordinates of the pairs.The figure illustrates the well defined window centered at the energy of [ω − (2 × m • c 2 )]/2.The distributions are much wider along the separation angle, θ s , reaching a maximum value just lower than the optimum separation angle of ∼ 60 • for E0 and ∼ 30 • for E2.There is also a finite probability that both particles will hit the same segment, which will be registered as a single event at the energy of ω − 2 × m • c 2 .

Conclusion
A new magnetic pair spectrometer is being developed for the spectroscopy of the electromagnetic transitions from the Hoyle state in 12 C.It combines a homogenous magnetic lens with a new array of six Si(Li) detectors mounted in a well shielded location.The design of the spectrometer is expected to be completed in 2012 and a series of experiments radioactive sources and well known reactions are planned to test the new spectrometer.

Fig. 4 .
Fig. 4. (Color online) Calculated double differential pair emission rates for (a) the 7.654 MeV E0 and (b) the 3.2148 MeV E2 transitions in12 C. The distribution of simulated events of electronpositron pairs detected with the magnetic pair spectrometer is also shown ("detected pairs").

Table 1 .
Adopted experimental values required to determine Γ rad from Eqn. 3.

Table 2 .
Selected spectrometer parameters for the pair spectroscopy of the Hoyle state. 1 Million electron-positron pair events (for each case) were used to determine optimum parameters for the 3.2148 MeV E2 and 7.654 MeV E0 in12C.