Further limit on 3α decay of Hoyle state

The quantitative contributions of various direct three alpha decay mechanisms in the decay of the famous Hoyle state, the 02 + resonant excited state of 12 C at excitation energy of 7.65 MeV, has been estimated using 60 MeV α inelastic scattering on 12 C target in complete kinematical measurement. Simultaneous optimisation of three different distributions (the relative energy of 8 Be like pairs, the root mean square energy deviation and the radial projection of symmetric Dalitz plot) derived from the experimental data with those generated from the Monte Carlo simulated event sets, have been done to arrive at a consistent estimation of the contributions of various direct decay modes.


Introduction
The Hoyle state, second 0 2 + resonant excited state of 12 C at an excitation energy of 7.654 MeV, plays an important role to understand a variety of problems of nuclear astrophysics such as elemental abundance as well as the stellar nucleosynthesis process as a whole [1,2]. The triple alpha reaction rate calculation assumed implicitly that the decay mechanism of Hoyle state in 12 C is exclusively through sequential two-step process, i.e. through the intermediate ground state of 8 Be nucleus [3]. However, the structure of this state has unusual in nature, as from nuclear structure point of view, there are many unanswered questions regarding the configuration of this state; from the cluster model, it has a linear chain like structure of three alpha particles [4] and at the same time from inelastic scattering it was found that this state has an abnormally larger radius compare to the ground state of 12 C [5] and also found possesses a gas like structure i.e., loosely bound 3α [6,7]. All these unusual properties of this state may change the decay mode of 12 C, from which reaction rates for carbon as well as other heavy elements have been calculated [8]. Deviation from the two-step process has been predicted to modify the relative abundance of 12 C, and thereby affect the future evolution of stars in the universe [9]. So, it is crucial to determine the quantitative contributions of all direct processes other than sequential decay to get a clear picture of the Hoyle state decay.
In recent years, several experiments have been performed to investigate and quantify the roles of various decay modes i.e. sequential decay (SD) and 3α direct decay (DD) [10][11][12][13]. In addition, attempts have also been made to identify and differentiate between various types of direct decays, such as decay into equal energies (DDE), decay in linear chain (DDL) and direct decay in phase space (DDΦ), and quantify their contributions in the exotic structure of the Hoyle  ing 60 MeV ded Si-strip orthogonal lar range of ided silicon (DSSD) in ly scattered nematically rticles from n of data on true Hoyle gnal as well rgy of 12 C n shown in collected in en analysed

Results and discussion
In order to estimate the various direct decay modes of Hoyle state (other than the most dominant mode, sequential decay, of the Hoyle state), an event-by-event Monte Carlo simulation has been performed for each three body decay scenario. In these simulations, we have taken into account, in addition to the sequential decay, three types of direct decay mechanisms (DDE -representing the decay mode where equal energy sharing occurs among three α-particles, DDL -representing the decay of a linear chain like configuration, leading to one α-particle at rest and the other two moving with equal but opposite velocities, and, DDΦ -characterising uniform sampling of phase space by the decay products, all in the rest frame of 12 C). In all these simulations, we have considered all experimental effects, such as, geometrical coverage, dead area, angular and energy resolutions of the strip detectors, event rejection due to multiple hit in single strip, etc. To determine the contributions of different decay modes of the Hoyle state, three different methods have been used as described below.

The distribution relative energy of 8 Be like pairs (E rel )
This is the distribution of the lowest relative energy between any two α-particles in each 3α decay event of Hoyle state [10,11]. So, all SD events decaying through 8 Be ground state will contribute to the peak at relative energy of 92 keV i. e. the ground state breakup energy of 8 Be. On the other hand, for DD mode of decay, it should be around the tail region of the peak. The relative energy distribution of the 8 Be like pairs in the experimental data (filled circles) is displayed in Fig. 2a.

The distribution of root mean square (rms) energy deviation (E rms )
The root mean square energy deviation, E rms , has been defined as [11,13], Where, E  are the energies of the α-particles from the decay of Hoyle state, and the average is over the energies of the 3α-particles of each event. So, E rms is the root mean square deviation of the energies of the α-particles of each Hoyle state decay event in 12 C rest frame. Fig. 2b displays the distribution of E rms for the fully detected events (experimental data, filled circles). It is clear from Eq. 1 that, in this case DDE should contribute prominently in the neighbourhood of E rms ≃ 0 in the distribution, subject to finite broadening due to the total instrumental resolution.

The distribution of radial projection of symmetric Dalitz plot
The radial projection of Dalitz plot is very useful; in particular for this case of isotropic decay of a state (spin zero state) into three equal mass particles, to gain deeper insight into the decay mechanism [12,14]. The radial coordinate of the symmetric Dalitz plot, ρ is given by where, ϵ i,j,k = E i,j,k /(E i +E j +E k ) are the normalized α particle energies in the 12 C frame and E i > E j > E k . Fig. 2c displays the distribution of radial coordinate of symmetric Dalitz plot for the fully detected events (experimental data, filled circles).
The experimental results have been compared with those obtained using simulated data sets containing contributions of different decay processes in varied proportions. Simultaneous optimisation of three different distributions (the relative energy of 8 Be like pairs, the root mean square energy deviation and the radial projection of symmetric Dalitz plot) derived from the experimental data with those generated from a simulated event set, have been done with χ 2 minimization to arrive at a consistent estimate of the contributions of various direct decay modes. The optimisation procedure was further repeated for a large number of times with different sets of simulated data sampled randomly from a much larger pool of simulated events to extract the complete range and distribution of the best fit values for each mode. Finally, these distributions have been used to obtain the optimum values of the contributions of various direct decay modes [15]. In figure 2, solid circle are the experimental data, dotted line (black), solid line (red) and the dash line (blue) are the SD, best fit and the upper limit with 99.75 % confidence limit, respectively. Hence, from the present analysis, we conclude that, at 99.75 % confidence limit, the upper limits of the DDL is 0.1 % and the estimated value for fractional contributions of other two direct decays mode are 0.60 ± 0.09 % for DDΦ, and 0.3 ± 0.1 % for DDE.