An efficient method for mapping the 12C+12C molecular resonances at low energies

TL;DR

This paper introduces a new efficient thick target method for measuring low-energy 12C+12C fusion resonances, enabling rapid cross section scans and aiding in the search for astrophysically relevant resonances.

Contribution

A novel thick target approach for measuring 12C+12C resonances at low energies, allowing comprehensive scans with a single beam energy and improving resonance detection efficiency.

Findings

Successful measurement at Ec.m.= 4.1 MeV

Method enables wide energy range scans with one beam setting

Potential to identify resonances between 1 and 3 MeV

Abstract

The 12C+12C fusion reaction is famous for its complication of molecular resonances, and plays an important role in both nuclear structure and astrophysics. It is extremely difficult to measure the cross sections of 12C+12C fusions at energies of astrophysical relevance due to very low reaction yields. To measure the complicated resonant structure existing in this important reaction, an efficient thick target method has been developed and applied for the first time at energies Ec.m.<5.3 MeV. A scan of the cross sections over a relatively wide range of energies can be carried out using only a single beam energy. The result of measurement at Ec.m.= 4.1 MeV is compared with other results from previous work. This method would be useful for searching potentially existing resonances of 12C+12C in the energy range 1 MeV<Ec.m.<3 MeV.

Equations

$$A+a\to B+b+Q$$

$$Q=(\frac{M_a}{M_B}-1)E_a+(\frac{M_b}{M_B}+1)E_b-2\frac{\sqrt{M_a{M}_b{E}_a{E}_b}}{M_B}cos(\theta )$$

$$\sigma (E)=\frac{1}{\epsilon }\frac{M_T}{f{N}_A}\frac{dY}{dE}\frac{dE}{d(\rho X)}$$

$$S^{\ast }=\sigma E_{c.m.}exp(\frac{87.21}{\sqrt{E_{c.m.}}}+0.46E_{c.m.})$$

Full text

12C+12C, Molecular resonance, Thick target method, 12C(12C,$p$)23Na

I Introduction

The 12C+12C is famous for its complicated molecular resonances and its importance in nuclear astrophysics b1 ; b2 ; nsac2015 . The mechanism of these strong resonant structures have been studied and discussed by many experimental and theoretical works b1 ; b2 ; b3 ; b4 ; b5 ; b6 ; b7 ; b8 ; b9 ; b10 ; b11 ; b12 ; b13 ; b14 ; b15 . The 12C+12C fusion reaction at low energies plays important roles in the nucleosynthesis during stellar evolution of massive stars, and is considered to ignite a carbon-oxygen white dwarf into a type Ia supernova explosion b16 ; b17 . The effective energy of carbon burning is approximately from 1 to 3 MeV (Gamow window) b16 at which the cross section varies from 10*-21* to 10*-7*b. Therefore, it is extremely difficult to directly measure the 12C+12C fusion cross sections at stellar energies. Lacking a clear understanding of the complicated resonances in the 12C+12C fusion cross section, one can not reliably extrapolate the cross sections down to the unmeasured stellar energies. Despite more than five decades of studies b1 ; b2 ; b3 ; b4 ; b5 ; b6 ; b7 ; b8 ; b9 ; b10 ; b11 ; b12 ; b13 ; b14 ; b15 , the 12C+12C fusion cross sections at stellar energies are still highly uncertain. More precise measurements are urgently needed especially at stellar energies in order to understand the resonance-like structure and provide more reliable cross section data for the astrophysical applications.

Both thin and thick targets have been used in experiments measuring the 12C+12C fusion cross sections. Thin carbon foils with thicknesses of a few tens of $\mu$g/cm2 are usually used to measure the resonant structure and cross sections of the 12C+12C fusion reaction at relatively higher energies. However, this target suffers from carbon build-up on its surface, which increases the target thickness continuously during an experiment and brings significant discrepancies into the results. Besides that, the small cross sections at stellar energies demand a high-intensity 12C beam ($>10p\mu A$). The thin carbon foils are easily damaged by such high-current beams. In the thick target approach, the beam is fully stopped inside the target, and the thick target yield is measured. The cross section is then obtained by calculating the derivative (dY/dE) from the measured thick target yield. The typical resonance width in the 12C+12C excitation function is about 50 keV or less. To precisely map the resonant structures, fine energy steps (e.g. $\Delta\textless$ 100 keV in the lab frame) are required. The yield difference between two adjacent energy points Y(E) and Y(E-$\Delta$) is calculated to determine the dY/dE. Since the thick target yield only slightly changes within a fine energy step $\Delta$, reasonably high statistics is required for each yield to get a reliable derivative for the cross section determination.

In the present article, a new thick-target approach is developed based on an analysis of the 12C(12C,$p$)23Na reaction. A scan of the cross sections over a relatively wide range of energies can be carried out using a single, constant beam energy. On the other hand, conventional methods require more than 10 energy points with fine steps to accomplish such a task. The new approach is much more efficient at mapping the 12C+12C resonance structure and is extremely useful in searching for new resonances at stellar energies.

The paper is organized as follows. First, we introduce a thick target experiment of 12C+12C. Second, the principle of the new thick target method is explained, and validated with a detailed Monte Carlo simulation by Geant4. Third, we apply this method to analyze the 12C(12C,$p_{1}$)23Na reaction. Fourth, the experimental results obtained with this thick target method are compared with a measurement using the traditional thin target. Finally, the strengths and weaknesses of the thick target method are also discussed.

II The 12C(12C,$p$)23N$\sc{a}$ Experiment

The 12C(12C,$p$)23Na reactions were measured by experiments in the center of mass energy range of 3 MeV to 5.3 MeV using thick targets. A 12C beam with an intensity up to 1 p$\mu$A was provided by the 10 MV FN Tandem accelerator at the University of Notre Dame. A gas stripper system was used to enhance the intensity of the 2+ charge state. The beam energies were determined by measuring the magnetic field of an analyzing magnet after the accelerator. The magnet was calibrated using the 27Al($p$,$n$) and 12C($p$,$p$) reactions.

The setup for the present experiment is shown in Fig. 1. Two 500 $\mu$m thick YY1-type silicon detectors from Micron Semiconductor Ltd were placed at backward angles from 113.5∘ to 163.5∘ in the lab frame. For the 12C+12C fusion reaction at energies below the Coulomb barrier, the most important two reaction channels are 12C(12C,$p$)23Na and 12C(12C,$\alpha$)20Ne. Each detector was covered with a 12.7 $\mu m$-thick Al foil in the front to completely stop the $\alpha$ particles emitted from the 12C(12C,$\alpha$)20Ne reaction. One detector surface was perpendicular to the beam direction covering the angular range 143.5∘ to 163.5∘, and the other detector surface held a 54.4∘ angle with respect to the beam direction covering from 113.5∘ to 143.5∘. Each wedge-shaped YY1 detector was segmented into 16 strips on the front side. Thus the angular resolution for charged particles was about 1.8∘. The detectors were calibrated using a Am-Cd mixed $\alpha$ source. The energy resolution for an individual strip was about 40 keV (FWHM) for 5.486 MeV $\alpha$ particles. The total solid angle of silicon detectors was determined to be 2.59% of 4$\pi$. A 12C beam with an intensity of $\sim$0.5-1 p$\mu$A was used to bombard a natural graphite target having a thickness of 1 mm.

The kinematic calculation of the 12C(12C,$p$)23Na reaction is shown in Fig. 2. The emitted protons are labeled as $p_{i}$ corresponding to the $i_{th}$ excited state ($i$= 0, 1, 2, 3 $\dots$) populated in the heavy residual 23Na nucleus. For example, $p_{0}$ corresponds to 23Na in its ground state, and $p_{1}$ for the first excited state, etc. Note that $p_{0}$ and $p_{1}$ possess significantly more energy than any of the other proton groups (e.g. $p_{2}$, $p_{3}$, $p_{4}$, $p_{5}$, etc.). The $\alpha$-channel has similar rules with emitted $\alpha$ particles and heavy residual 20Ne nuclei.

III The principle of the thick target method

Considering a reaction,

[TABLE]

where Q is the reaction energy which means the energy produced or absorbed by this reaction. The Q-value for reaction $A$($a$,$b$)$B$ is the total kinetic energy difference between the initial and final states. It could be determined by

[TABLE]

where Ma,Mb and MB are the masses in $amu$ of the beam, projectile, and residual particles, respectively; Ea is the beam energy, and Eb is the energy of the projectile particle $b$; $\theta$ is the outgoing angle of $b$. Values of Eb could be measured by detectors.

The principle of the thick target method for 12C(12C,$p$)23Na is shown in Fig. 3. An infinitely thick target (i.e. thickness much greater than beam range inside the target material) is used in the method. The 12C beam particles with incident energy Ebeam bombard the target. As they collide with the target nuclei, they continuously lose energy until they either react with a target nucleus or stop within a distance of a few $\mu$ms under the front surface of the target. The range of the 12C beam in the 12C target is about 5.7 and 7.1 $\mu$m for Ebeam= 8 and 10 MeV, respectively. When the 12C(12C,$p$)23Na reaction happens, the actual energy of the 12C beam is unknown. Protons produced at backward angles can easily penetrate through the target surface with an insignificant energy loss, and reach silicon strip detectors. The energies and outgoing angles of these protons are recorded by detectors. Two examples of protons, $m$ and $n$, are shown in Fig. 3. Since the range of 12C beam in the 12C target was really small, the outgoing angles of protons (e.g. (180-$\theta_{m}$), (180-$\theta_{n}$)) were only determined by strip numbers of detectors. The Q-value is known for each individual proton group (Q= 2.24 MeV for $p_{0}$, Q=2.24-0.44=1.80 MeV for $p_{1}$, etc.), from the measured proton energies in the silicon (accounting for energy loss in the Al degrader foil) and outgoing angles (180-$\theta_{m}$, or 180-$\theta_{n}$), the actual reaction energy (E${}^{\prime}_{m}$, E${}^{\prime}_{n}$) can be reconstructed by solving Eq. 2. Therefore, a range of reaction energies [Ebeam-$\Delta$E, Ebeam] is scanned with a single, constant beam energy. The effective width of the scan, $\Delta$E, usually spans from 500 to 800 keV depending on the clear identification of the reaction for events from each channel. It will be discussed later using the measurement result of 12C(12C,$p_{1}$)23Na.

The target yield derivative dY/dE is computed for each reaction energy bin after normalizing the count by the total number of incident 12C particles. The cross section for the 12C(12C,$p_{i}$)23Na reaction is then calculated from the extracted $dY/dE$ using the following equation

[TABLE]

where $\varepsilon$ is the detection efficiency which is the geometric efficiency determined by an $\alpha$ source, ${M}_{T}$ is the molecular weight of the target nucleus, $f$ is the molecular fraction of the target nucleus, ${N}_{A}$ is Avogadro’s number, and ${dE}/{d(\rho X)}$ is the stopping power, calculated with the SRIM code SRIM .

For ease of comparison to other experimental data sets, the measured cross sections from above are converted into S*∗* factors which are defined by the following equation

[TABLE]

where $\sigma$ is the cross section and Ec.m. is the energy in the center of mass frame.

To validate the proposed thick target method, a Geant4 simulation was performed to generate a reaction energy spectrum from a constant S*∗* factor input (S*∗=2$\times 10^{16}$ MeVb). In the simulation, all details introduced above were considered, including geometry of the detectors, the aluminum degrader, beam straggling, etc. The reaction energy spectrum from the simulation of the $p_{1}$ group with an incident energy of 8.2 MeV is shown in Fig. 4.

IV Obtaining the S*∗* factor of 12C(12C,$p_{1}$)23N$\sc{a}$ with the thick target method

The measurement utilizing the thick target method was carried out in the energy range 3 MeV$\textless$E${}_{\rm c.m.}\textless$ 5.3 MeV (6 MeV$\textless$E${}_{beam}\textless$ 10.6 MeV). The measured results at Ebeam= 8.2 MeV are shown in Fig. 5.

The reaction Q-value spectrum is computed from Eq.2 with a constant incident energy of Ea= 8.2 MeV. Because the 12C beam loses its energy as it passes through the target medium, the actual beam energy varies from the initial incident beam energy (8.2 MeV) down to 0. As a result, the shape of the Q-value spectrum becomes much more wider and complicated than the simple sharp Gaussian shape obtained from measurement with a thin target.

The corresponding Q-values of the 12C(12C,$p_{0,1,2}$)23Na channels are 2.24, 1.80, and 0.164 MeV, respectively. The large Q-value difference between the $p_{1}$ and $p_{2}$ channels offers an excellent clear region for the identification of the $p_{1}$ events. In the present work, we focused on the analysis of the $p_{1}$ channel. The Q-value spectrum obtained with a thin target is expected to be narrowly 1.80 MeV. However, in the thick target method, as the reaction energy decreases in the target, the energies of the $p_{1}$ channel events at each fixed angle extend towards lower values as shown in the energy vs. angle plot (Fig.  5). The $p_{1}$ channel events obtained with the thick target form a wide Q-value spectrum when we compute the Q-value using a fixed beam energy, Ea= 8.2 MeV.

The Q-value of the $p_{0}$ channel also has a lower energy tail similar to the $p_{1}$ channel. These low-energy events from the $p_{0}$ channel interfere with the high-energy events from the $p_{1}$ channel. As shown in Fig. 5, this is not a big problem at Ec.m.= 4.1 MeV, because the $p_{0}$ cross section quickly decays with the decreasing beam energy. However, the tail of the $p_{0}$ channel events may contribute to the events of the $p_{1}$ channel around 20% at Ec.m.= 10.6 MeV.

By following the procedure introduced above, the reaction energy is computed for each event using the $p_{1}$ Q-value= 1.80 MeV. The reaction energy spectrum is shown in Fig. 6. Using this unique Q-value for $p_{1}$, only reaction energies of events from the $p_{1}$ channel are correctly constructed. These events are distributed in the range of the actual reaction energies from 7.0 to 8.2 MeV. Meanwhile, all the events from other channels are placed at the ′wrong′ energies. If we wanted to take a look at the $p_{0}$ channel, Q-value= 2.24 MeV would be used instead of 1.80 MeV to properly reconstruct the reaction energy.

The energy calibration is crucial for the determination of the actual reaction energy in this approach. Considering the uncertainties in the detector energy calibration and the degrader thickness, a minor tuning of the energy shift was applied to the reconstructed reaction energy obtained from the experiment to match the S*∗* factors at the edge of the high-energy sides between the experimental and simulated spectra.

The present S*∗* factor measurement using the thick target method is shown in Fig. 7. For the $p_{1}$ channel, the maximum reaction energy is Ec.m.= 4.0 MeV. The smearing of the edge at 4.1 MeV is a result of limited resolution of detectors and the spread of beam energy. The effective measurement of the S*∗* factor for the $p_{1}$ channel stops around Ec.m.= 3.4 MeV where the background events begin to take over. These background events lead to a quickly rising S*∗* factor below Ec.m.= 3.4 MeV.

V Comparison with the conventional thin target measurement

The present S*∗* factor of the $p_{1}$ channel at energies from 3.4 MeV to 4.0 MeV is shown in Fig. 8, compared with an earlier measurement using the conventional thin target method by Becker $et$ $al$. b10 . The highest reaction energy obtained by the thick target method is Ec.m.= 4.0 MeV instead of Ec.m.= 4.1 MeV because of the smearing effect resulting from the limited energy and angular resolutions. It is impressive that the complicated resonant structure of the $p_{1}$ channel through a wide energy range can be easily revealed with a single incident beam energy using the thick target method. A scan of about ten energy points is required if using the conventional thin target method (or with the differential thick target method). It is also noticed that the S*∗* factor obtained with the thick target method is about 50% higher than Becker’s data at Ec.m.= 3.78 MeV. This is possibly an effect of the angular distribution. In the thick target experiment, we assumed a simple isotropic angular distribution because of the limited detector angular coverage. In the conventional thin target measurement of Becker $et$ $al$. b10 , a set of silicon detectors was used to provide a well-determined angular distribution.

VI Discussion

The thick target method established in the present work requires a clear identification of the reaction channel of each candidate event. For the 12C(12C,$p_{i}$)23Na, both $p_{0}$ and $p_{1}$ are good candidates because of their highest proton kinetic energies and large separation from the lower Q-value channels. The other proton channels, $p_{2,3,4,5,\dots}$, are too close to each other. It is rather difficult to achieve clear identifications of each channel using only their Q-values. Therefore, the particle-gamma coincidence technique is required for identification of the Q-value stella .

The 12C(12C,$\alpha$)20Ne reaction also could be studied by the present thick target method. The dE-E telescope technique is required to identify the $\alpha$ particles from the protons and the ambient background of $\delta$-electrons. The separation between $\alpha_{0}$ and $\alpha_{1}$, or $\alpha_{1}$ and $\alpha_{2}$ are more significant than that in the $p$ channels. This would provide a better identification of the reaction channel of each $\alpha$ event. However, the energies of these $\alpha$ particles are below 5 MeV at the backward angles, and the energy loss of $\alpha$ particles is significant inside the carbon target and Al degrader, the latter is often used in front of the detector to shield scattered 12C particles. A more careful correction is needed for the detected energies and angles of the $\alpha$ particles, bringing more uncertainties into the result. The associated straggling is another limitation for the $\alpha$ particle. New techniques, such as the solenoid spectrometer b14 , are helpful for the $\alpha$-detection.

A clean background is essential for the clear identification of the reaction channel for each event. The graphite target contains the impurity D2O which produces a proton background. A clean carbon target, e.g. highly ordered pyrolytic graphite (HOPG) HOPG , would greatly reduce contributions from target contaminants to background b15 ; bucher2015 ; fang2017 . Direct measurements in nuclear astrophysics benefit from an underground environment, which greatly reduces cosmic ray induced background. The Jinping Underground laboratory for Nuclear Astrophysics (JUNA) JUNA in China, which is being constructed and expected to deliver beam in a few years, would be a proper place to do the 12C+12C measurement with the ultra-low background of the China Jinping Underground Laboratory (CJPL) CJPL .

The present thick target method provides an efficient way to map the resonant structure of the 12C+12C fusion reactions. An intense beam can be used with such a target to search rare events at stellar energies. A snapshot of some particular channels can be obtained efficiently with a single constant incident energy. The snapshot provides an important guidance for the following detailed energy scan using the thin target method or the differential thick target method which may reveal more details of the resonances in other reaction channels.

VII Summary

In summary, an efficient thick target method has been applied for the first time to measure the complicated resonant structure existing in 12C(12C,$p$)23Na. It can provide cross sections within a range of [Ebeam-$\Delta$E, Ebeam] using a single incident energy Ebeam. The 12C+12C fusion reaction is one of the most important reactions in nuclear astrophysics. The efficient thick target method of the present work will be useful in searching for potentially existing resonances of 12C+12C in the energy range 1 MeV$\textless$E${}_{\rm c.m.}\textless$3 MeV, where the cross sections are extremely low.