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

An efficient method for mapping the 12 C+ 12 C molecular resonances at low energies 1 2 first-author 1 2 3 corresp Corresponding author, fangx26@mail.sysu.edu.cn Corresponding author, fangx26@mail.sysu.edu.cn fangx26@mail.sysu.edu.cn 4 5 5 5 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China Idaho National Laboratory, Idaho Falls, ID 83415, USA Idaho National Laboratory, Idaho Falls, ID 83415, USA Institute for Structure and Nuclear Astrophysics, University of Notre Dame, Notre Dame, IN 46556, USA Institute for Structure and Nuclear Astrophysics, University of Notre Dame, Notre Dame, IN 46556, USA The 12 C+ 12 C fusion reaction is famous because of its complication of molecular resonances, and it plays an important role in both nuclear structural research and astrophysics. It is extremely difficult to measure the cross-sections of 12 C+ 12 C fusions at energies of astrophysical relevance because of the very low reaction yields. To measure the complicated resonant structure that exists in this important reaction, an efficient thick target method has been developed and applied for the first time at energies E c.m. lt;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 E c.m. = 4.1 MeV is compared with results from previous work. This method will be useful for searching for potentially existing resonances of 12 C+ 12 C in the energy range 1 MeV lt; E c.m. lt;3 MeV. 12 C+ 12 C Molecular resonance Thick target method 12 C( 12 C,p) 23 Na 1 Introduction 1 The 12 C+ 12 C fusion reaction is famous because of its complicated molecular resonances and its importance in nuclear astrophysics [ 1 1 - 3 3 1-3 ]. The mechanism of these strong resonant structures has been studied and discussed by many experimental and theoretical works [ 1 1 , 2 2 , 4 4 - 17 17 4-17 ]. The 12 C+ 12 C 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 18 , 19 19 ]. The effective energy of carbon burning is approximately from 1 to 3 MeV (Gamow window) [ 18 18 ], at which the cross-section varies from 10 -21 to 10 -7 b. Therefore, it is extremely difficult to directly measure the 12 C+ 12 C fusion cross-sections at stellar energies. Lacking a clear understanding of the complicated resonances in the 12 C+ 12 C fusion cross-section, one cannot reliably extrapolate the cross-sections down to the unmeasured stellar energies. Despite more than five decades of studies [ 1 1 , 2 2 , 4 4 - 16 16 4-16 ], the 12 C+ 12 C 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 12 C+ 12 C fusion cross-sections. Thin carbon foils with thicknesses of a few tens of μ g/cm 2 are usually used to measure the resonant structure and cross-sections of the 12 C+ 12 C 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 12 C beam (gt;10  p μ 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 (d Y /d E ) from the measured thick target yield. The typical resonance width in the 12 C+ 12 C 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 d Y /d E . 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 12 C( 12 C,p) 23 Na 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 12 C+ 12 C 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 12 C+ 12 C. 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 12 C( 12 C, p 1 ) 23 Na 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 12 C( 12 C,p) 23 Na Experiment 1 The 12 C( 12 C,p) 23 Na reactions were measured by experiments in the center of mass energy range of 3 MeV to 5.3 MeV using thick targets. A 12 C beam with an intensity of up to 1 p μ 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 27 Al(p,n) and 12 C(p,p) reactions. The setup for the present experiment is shown in Fig. 1 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 12 C+ 12 C fusion reaction at energies below the Coulomb barrier, the most important two reaction channels are 12 C( 12 C,p) 23 Na and 12 C( 12 C, α ) 20 Ne. Each detector was covered with a 12.7-μm-thick Al foil in the front to completely stop the α particles emitted from the 12 C( 12 C, α ) 20 Ne 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 4 π . A 12 C beam with an intensity of ∼ 0.5-1 p μ A was used to bombard a natural graphite target having a thickness of 1 mm. 3 The principle of the thick target method 1 Consider a reaction, <math display="block" id="M1"><mrow><mi>A</mi><mo>+</mo><mi>a</mi><mo>→</mo><mi>B</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>Q</mi><mn>,</mn></mrow></math> 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 <math display="block" id="M2"><mrow><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>M</mi><mi>a</mi></msub></mrow><mrow><msub><mi>M</mi><mi>B</mi></msub></mrow></mfrac><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>E</mi><mi>a</mi></msub><mo>+</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>M</mi><mi>b</mi></msub></mrow><mrow><msub><mi>M</mi><mi>B</mi></msub></mrow></mfrac><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>E</mi><mi>b</mi></msub><mo>−</mo><mn>2</mn><mfrac><mrow><msqrt><mrow><msub><mi>M</mi><mi>a</mi></msub><msub><mi>M</mi><mi>b</mi></msub><msub><mi>E</mi><mi>a</mi></msub><msub><mi>E</mi><mi>b</mi></msub></mrow></msqrt></mrow><mrow><msub><mi>M</mi><mi>B</mi></msub></mrow></mfrac><mi>cos</mi><mi>θ</mi><mn>,</mn></mrow></math> where M a , M b ,and M B are the masses in amu of the beam, ejectile, and residual particles, respectively; E a is the beam energy and E b is the energy of the ejectile particle b ; θ is the outgoing angle of b . The values of E b can be measured by detectors. The principle of the thick target method for 12 C( 12 C,p) 23 Na is shown in Fig. 2 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 12 C 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 μ ms under the front surface of the target. The ranges of the 12 C beam in the 12 C target are approximately 5.7 and 7.1 μ m for E beam = 8 and 10 MeV, respectively. When the 12 C( 12 C,p) 23 Na reaction occurs, the actual energy of the 12 C 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 Fig. 2 . Because the range of the 12 C beam in the 12 C target is quite small, the outgoing angles of protons (e.g., (180- θ m ), (180- θ n )) 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 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- θ m , or 180- θ n ), the actual reaction energy ( <math id="M3"><mrow><msub><msup><mi>E</mi><mo>′</mo></msup><mi>m</mi></msub></mrow></math> , <math id="M4"><mrow><msub><msup><mi>E</mi><mo>′</mo></msup><mi>n</mi></msub></mrow></math> ) can be reconstructed by solving Eq. 2. Therefore, a range of reaction energies [ E beam -Δ E , E beam ] is scanned with a single, constant beam energy. The effective width of the scan, Δ E , 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 12 C( 12 C, p 1 ) 23 Na. The kinematic calculation of the 12 C( 12 C,p) 23 Na reaction is shown in Fig. 3 Fig. 3 . The emitted protons are labeled p i , corresponding to the i th excited state ( i = 0, 1, 2, 3 … ) populated in the heavy residual 23 Na nucleus. For example, p 0 corresponds to 23 Na in its ground state, and p 1 to 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 α -channel has similar rules, with emitted α particles and heavy residual 20 Ne nuclei. The target yield derivative d Y /d E is computed for each reaction energy bin after normalizing the count by the total number of incident 12 C particles. The cross-section for the 12 C( 12 C, p i ) 23 Na reaction is then calculated from the extracted dY/dE using the following equation <math display="block" id="M5"><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>ε</mi></mfrac><mfrac><mrow><msub><mi>M</mi><mtext>T</mtext></msub></mrow><mrow><mi>f</mi><msub><mi>N</mi><mtext>A</mtext></msub></mrow></mfrac><mfrac><mrow><mtext>d</mtext><mi>Y</mi></mrow><mrow><mtext>d</mtext><mi>E</mi></mrow></mfrac><mfrac><mrow><mtext>d</mtext><mi>E</mi></mrow><mrow><mtext>d</mtext><mo stretchy="false">(</mo><mi>ρ</mi><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> 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 d E /d( ρX ) is the stopping power, calculated with the SRIM code [ 20 20 ]. For ease of comparison with other experimental data sets, the measured cross-sections from this experiment are converted into S * factors [ 7 7 ], which are defined by the following equation <math display="block" id="M6"><mrow><msup><mi>S</mi><mo>*</mo></msup><mo>=</mo><mi>σ</mi><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mi>exp</mi><mo stretchy="false">(</mo><mfrac><mrow><mn>87.21</mn></mrow><mrow><msqrt><mrow><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub></mrow></msqrt></mrow></mfrac><mo>+</mo><mn>0.46</mn><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> 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 * =210 16 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 p 1 group with an incident energy of 8.2 MeV is shown in Fig. 4 Fig. 4 . 4 Obtaining the S * factor of 12 C( 12 C,p 1 ) 23 Na with the thick target method 1 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 . The reaction Q -value spectrum is computed from Eq.2 with a constant incident energy of E a = 8.2 MeV. Because the 12 C 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 12 C( 12 C, p 0,1,2 ) 23 Na 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 identification of the p 1 events. In the present work, we focused on 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 toward lower values, as shown in the energy vs. angle plot ( Fig. 5 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, E a = 8.2 MeV. The Q -value of the p 0 channel also has a low-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 Fig. 5 , this is not a huge problem at E c.m. = 4.1 MeV because the p 0 cross-section quickly decays with 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 E c.m. = 10.6 MeV. By following the procedure introduced above, the reaction energy can be computed for each event using the p 1 Q -value= 1.80 MeV. The reaction energy spectrum is shown in Fig. 6 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 investigate the p 0 channel, Q -value= 2.24 MeV should be used instead of 1.80 MeV to properly reconstruct the reaction energy. 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 Fig. 7 . For the p 1 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 p 1 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. 5 Comparison with the conventional thin target measurement 1 The present S * factor for the p 1 channel at energies from 3.4 MeV to 4.0 MeV is shown in Fig. 8 Fig. 8 , compared with an earlier measurement using the conventional thin target method by Becker et al. [ 11 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 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 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 11 ], a set of silicon detectors was used to provide a well-determined angular distribution. 6 Discussion 1 The thick target method established in the present work requires a clear identification of the reaction channel of each candidate event. For the 12 C( 12 C, p i ) 23 Na, both p 0 and p 1 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 21 ]. The 12 C( 12 C, α ) 20 Ne reaction also could be studied by the present thick target method. The d E - E telescope technique is required to identify the α particles from the protons and the ambient background of δ -electrons. The separation between α 0 and α 1 , or between α 1 and α 2 , 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 12 C 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 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 D 2 O, which produces a proton background. A clean carbon target, e.g., highly ordered pyrolytic graphite (HOPG) [ 22 22 ], would greatly reduce contributions from target contaminants to background [ 16 16 , 23 23 , 24 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 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 12 C+ 12 C measurement with the ultra-low background of the China Jinping Underground Laboratory (CJPL) [ 26 26 ]. The present thick target method provides an efficient way to map the resonant structure of the 12 C+ 12 C 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. 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Sci . 67 231 - 251 ( 2017 ). doi: 10.1146/annurev-nucl-102115-044842 http://doi.org/10.1146/annurev-nucl-102115-044842 The China Jinping Underground Laboratory and Its Early Science 10.1007/s41365-019-0652-9 This work was supported by the National Key R&D Program of China (No. 2016YFA0400500), the National Natural Science Foundation of China (Nos. 11805291, 11475228, 11490560, 11490564, 11875329), the National Science Foundation of the US (Nos. PHY-1068192, PHY-1419765), the US Department of Energy (No. DE-AC07-05ID14517), the Key Program (No. XDPB09-2) and the "Hundred Talents Program" of the Chinese Academy of Sciences, and the Fundamental Research Funds for the Central Universities (No. 18lgpy84). 2019-04-30 2019-05-17 2019-05-23 2019-07-13 2019-08-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2019 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-019-0652-9 http://dx.doi.org/10.1007/s41365-019-0652-9 . 2019 Nuclear Science and Techniques 08 30