1 Introduction

The ¹²C+¹²C fusion reaction is famous because of its complicated molecular resonances and its importance in nuclear astrophysics [1-3]. The mechanism of these strong resonant structures has been studied and discussed by many experimental and theoretical works [1, 2, 4-17]. The ¹²C+¹²C fusion reaction at low energies plays important roles in nucleosynthesis during the stellar evolution of massive stars and is considered to ignite a carbon-oxygen white dwarf into a type Ia supernova explosion [18, 19]. The effective energy of carbon burning is approximately from 1 to 3 MeV (Gamow window) [18], at which the cross-section varies from 10^-21 to 10^-7b. 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 cannot reliably extrapolate the cross-sections down to the unmeasured stellar energies. Despite more than five decades of studies [1, 2, 4-16], the ¹²C+¹²C fusion cross-sections at stellar energies are still highly uncertain. Measurements that are more precise are urgently needed, especially at stellar energies, to understand the resonance-like structure and provide more reliable cross-section data for 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 μg/cm² are usually used to measure the resonant structure and cross-sections of the ¹²C+¹²C fusion reaction at relatively high energies. However, this type of 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. In addition, the small cross-sections at stellar energies demand a high-intensity ¹²C beam (gt;10 pμA). 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 approximately 50 keV or less. To precisely map the resonant structures, fine energy steps (e.g., Δ<100 keV in the lab frame) are required. The yield difference between two adjacent energy points Y(E) and Y(E-Δ) is calculated to determine the dY/dE. Because the thick target yields only slightly changes within a fine energy step Δ, reasonably high statistics is required for each yield to obtain a reliable derivative for determination of the cross-section.

In the present study, 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. By contrast, conventional methods require more than 10 energy points with fine steps to accomplish such a task. This 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 for 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,p₁)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 discussed.

2 The ¹²C(¹²C,p)²³Na 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 of up to 1 pμA 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-μm-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,α)²⁰Ne. Each detector was covered with a 12.7-μm-thick Al foil in the front to completely stop the α particles emitted from the ¹²C(¹²C,α)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 approximately 1.8°. The detectors were calibrated using an Am-Cd mixed α source. The energy resolution for an individual strip was approximately 40 keV (FWHM) for 5.486-MeV α particles. The total solid angle of silicon detectors was determined to be 2.59% of 4π. A ¹²C beam with an intensity of ∼0.5-1 pμA was used to bombard a natural graphite target having a thickness of 1 mm.

Fig. 1.Full-screen View

The setup for the present experiment is shown. Two wedge-shaped silicon strip detectors cover the angles from 113.5° to 163.5° in the lab frame.

3 The principle of the thick target method

Consider a reaction,

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 can be determined by

where Ma, Mb,and MB are the masses in amu of the beam, ejectile, and residual particles, respectively; Ea is the beam energy and Eb is the energy of the ejectile particle b; θ is the outgoing angle of b. The values of Eb can be measured by detectors.

The principle of the thick target method for ¹²C(¹²C,p)²³Na is shown in Fig. 2. An infinitely thick target (i.e., thickness much greater than the beam range inside the target material) is used in this method. The ¹²C beam particles, with incident energy E_beam, 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 μms under the front surface of the target. The ranges of the ¹²C beam in the ¹²C target are approximately 5.7 and 7.1 μm for E_beam= 8 and 10 MeV, respectively. When the ¹²C(¹²C,p)²³Na reaction occurs, 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 the 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. 2. Because the range of the ¹²C beam in the ¹²C target is quite small, the outgoing angles of protons (e.g., (180-θm), (180-θn)) were only determined from the strip numbers of detectors. If the Q-value is known for each individual proton group (Q= 2.24 MeV for p₀, Q=2.24-0.44=1.80 MeV for p₁, etc.), from the measured proton energies in the silicon (accounting for energy loss in the Al degrader foil) and outgoing angles (180-θm, or 180-θn), the actual reaction energy (${E^{\prime }}_m$, ${E^{\prime }}_n$) can be reconstructed by solving Eq. 2. Therefore, a range of reaction energies [E_beam-ΔE, E_beam] is scanned with a single, constant beam energy. The effective width of the scan, ΔE, usually spans from 500 to 800 keV, depending on the clear identification of the reaction for events from each channel. This will be discussed later using the measurement result for ¹²C(¹²C,p₁)²³Na.

Fig. 2.Full-screen View

The principle of the thick target method for the ¹²C(¹²C,p)²³Na reaction. For clarity, the dimensions in this figure are not drawn to scale.

The kinematic calculation of the ¹²C(¹²C,p)²³Na reaction is shown in Fig. 3. The emitted protons are labeled pi, corresponding to the i_th excited state (i= 0, 1, 2, 3 …) populated in the heavy residual ²³Na nucleus. For example, p₀ corresponds to ²³Na in its ground state, and p₁ to the first excited state, etc. Note that p₀ and p₁ possess significantly more energy than any of the other proton groups (e.g., p₂, p₃, p₄, p₅, etc.). The α-channel has similar rules, with emitted α particles and heavy residual ²⁰Ne nuclei.

Fig. 3.Full-screen View

The kinematic calculation of the ¹²C(¹²C,p)²³Na reaction at E_lab=8.2 MeV. The energies of protons at different outgoing angles are given. The protons (pi) associated with ²³Na in its ground and lowest excited states are identified.

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,pi)²³Na reaction is then calculated from the extracted dY/dE using the following equation

where ε is the detection efficiency, which is the geometric efficiency determined by an α 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(ρX) is the stopping power, calculated with the SRIM code [20].

For ease of comparison with other experimental data sets, the measured cross-sections from this experiment are converted into S^* factors [7], which are defined by the following equation

where σ is the cross-section and E_c.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^*=210¹⁶ MeV*b). In the simulation, all details introduced above were considered, including the geometry of the detectors, the aluminum degrader, and beam straggling. The reaction energy spectrum from the simulation of the p₁ group with an incident energy of 8.2 MeV is shown in Fig. 4.

Fig. 4.Full-screen View

The simulation results of a 8.2-MeV ¹²C beam bombarding a thick target. (a) Reconstructed reaction energy spectrum; (b) Derived cross-sections; (c) Derived S^* factors.

4 Obtaining the S^* factor of ¹²C(¹²C,p₁)²³Na with the thick target method

The measurement utilizing the thick target method was carried out in the energy range 3 MeV lt; E_c.m. lt; 5.3 MeV (6 MeV lt; E_beam lt; 10.6 MeV). The measured results at E_beam= 8.2 MeV are shown in Fig. 5.

Fig. 5.Full-screen View

The experimental spectra measured at E_beam= 8.2 MeV using the thick target method. (top) Energy vs. angle of detected protons; (bottom) The Q-value for the ¹²C(¹²C,p_0,1,2)²³Na reaction. The events between 2 and 2.4 MeV are from the p₀ channel. The events between 1 and 2 MeV mostly correspond to the p₁ channel, with minor contamination from the p₀ channel.

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 wider and more 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 p₁ and p₂ channels offers an excellent clear region for identification of the p₁ events. In the present work, we focused on analysis of the p₁ 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 p₁ channel events at each fixed angle extend toward lower values, as shown in the energy vs. angle plot (Fig. 5). The p₁ 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 p₀ channel also has a low-energy tail, similar to the p₁ channel. These low-energy events from the p₀ channel interfere with the high-energy events from the p₁ channel. As shown in Fig. 5, this is not a huge problem at E_c.m.= 4.1 MeV because the p₀ cross-section quickly decays with decreasing beam energy. However, the tail of the p₀ channel events may contribute to the events of the p₁ channel around 20% at E_c.m.= 10.6 MeV.

By following the procedure introduced above, the reaction energy can be computed for each event using the p₁ Q-value= 1.80 MeV. The reaction energy spectrum is shown in Fig. 6. Using this unique Q-value for p₁, only reaction energies of events from the p₁ 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 investigate the p₀ channel, Q-value= 2.24 MeV should be used instead of 1.80 MeV to properly reconstruct the reaction energy.

Fig. 6.Full-screen View

The reaction energy spectrum constructed from the p₁ Q-value= 1.80 MeV. The true p₁ events are distributed in the range from 7 to 8.2 MeV. All events corresponding to other channels appear with incorrect energy values.

The energy calibration is crucial for 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 p₁ channel, the maximum reaction energy is E_c.m.= 4.0 MeV. The smearing of the edge at 4.1 MeV is a result of the limited resolution of the detectors and the spread of the beam energy. The effective measurement of the S^* factor for the p₁ channel stops at approximately E_c.m.= 3.4 MeV, where the background events begin to take over. These background events lead to a quickly rising S^* factor below E_c.m.= 3.4 MeV.

Fig. 7.Full-screen View

The S^* factor of the p₁ channel. The effective range of E_c.m. is from 3.4 to 4.0 MeV.

5 Comparison with the conventional thin target measurement

The present S^* factor for the p₁ 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. [11]. The highest reaction energy obtained by the thick target method is E_c.m.= 4.0 MeV instead of E_c.m.= 4.1 MeV because of the smearing effect resulting from the limited energy and angular resolution. It is impressive that the complicated resonant structure of the p₁ channel through a wide energy range can be easily revealed with a single incident beam energy using the thick target method. A scan of approximately ten energy points is required if using the conventional thin target method (or with the differential thick target method). It is also clear that the S^* factor obtained with the thick target method is approximately 50% higher than Becker’s data at E_c.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. [11], a set of silicon detectors was used to provide a well-determined angular distribution.

Fig. 8.Full-screen View

The S^* factor of the p₁ channel of ¹²C(¹²C,p)²³Na. Present results are displayed in red, while the data from Ref. [11] are displayed in black.

6 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,pi)²³Na, both p₀ and p₁ are good candidates because of their high proton kinetic energies and large separation from the lower Q-value channels. The other proton channels, p_2,3,4,5,⋅s, are too close to each other. It is thus 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 [21].

The ¹²C(¹²C,α)²⁰Ne reaction also could be studied by the present thick target method. The dE-E telescope technique is required to identify the α particles from the protons and the ambient background of δ-electrons. The separation between α₀ and α₁, or between α₁ and α₂, is more significant than that in the p channels. This would provide a better identification of the reaction channel of each α event. However, the energies of these α particles are below 5 MeV at backward angles, and the energy loss of α 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 α particles, bringing more uncertainties into the result. The associated straggling is another limitation for the α particle. New techniques, such as a solenoid spectrometer [15], are helpful for α-detection.

A clean background is essential for 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) [22], would greatly reduce contributions from target contaminants to background [16, 23, 24]. Direct measurements in nuclear astrophysics benefit from an underground environment, which greatly reduces cosmic ray-induced background noise. The Jinping Underground laboratory for Nuclear Astrophysics (JUNA) [25] in China, which is being constructed and expected to deliver a beam in a few years, would be a suitable place to perform the ¹²C+¹²C measurement with the ultra-low background of the China Jinping Underground Laboratory (CJPL) [26].

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 for rare events at stellar energies. A snapshot of some particular channels can be obtained efficiently with a single constant incident energy. The snapshot provides 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.

7 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 [E_beam-ΔE, E_beam] using a single incident energy E_beam. The ¹²C+¹²C fusion reaction is one of the most important reactions in nuclear astrophysics. The efficient thick target method outlined in the present work will be useful in searching for potentially existing resonances of ¹²C+¹²C in the energy range 1 MeV lt;E_c.m. lt;3 MeV, where the cross-sections are extremely low.