Introduction

¹²C+¹²C fusion reactions play a crucial role in various stellar burning scenarios [1-3], particularly during the final stages of massive star evolution, Type Ia supernovae events [4, 5], and superbursts [6, 7]. Hence, many direct measurements, primarily employing charged particles [8-10] and gamma ray spectroscopy [11-13], have been conducted to study nuclear astrophysical reaction rates [14]. Because the Coulomb barrier height for the ¹²C+¹²C system, at approximately 7.5 MeV, is significantly higher than the Gamow window energy (E_c.m. = 1.5 ± 0.3 MeV), the cross-section decreases rapidly to below one nanobarn in the energy region of interest. This makes it extremely difficult to measure the ¹²C+¹²C fusion reaction directly within the Gamow energy region. A series of direct measurement data [15, 9, 13, 12] suggests the existence of sub-barrier resonances in the ¹²C+¹²C fusion reaction, leading to an increase in the S-factor as the energy decreases below 2.5 MeV in the frame of the center of mass. Therefore, simple low-energy extrapolation based on high-energy data cannot accurately describe the reaction cross-section within the Gamow energy region.

The ¹²C+¹²C fusion reaction rate is mainly contributed by ¹²C(¹²C, p)²³Na, ¹²C(¹²C, α)²⁰Ne, and ¹²C(¹²C, n)²³Mg. The contribution of the neutron emission channel is marginal because of its negative Q value [16]. In 2018, the Trojan horse method (THM) was employed to measure the S factor of the ¹²C+¹²C fusion reaction at E_c.m. < 2.7 MeV [17]. To date, this is the only measurement that has entered the Gamow window. The astrophysical S* factor derived from the experiment is much larger than the values of the compilation [18] and various phenomenological and microscopic models [19], such as wave-packet dynamics (TDWP) [20], and coupled channel calculations, such as CC-M3Y+Rep [21, 22]. By contrast, Jiang et al. [23] proposed a hindrance model [24, 25] that showed the opposite trend in the energy region of interest. Despite extensive experimental and theoretical efforts, the exact behavior of the ¹²C+¹²C fusion reaction remains unclear on the existence of resonances or not, particularly in the Gamow window.

Thick-target inverse kinematics (TTIK) is a novel experimental method that has been widely used in radioactive ion beam measurements [26, 27] over the last two decades. In the TTIK measurement, the excitation function is obtained in a one-shot experiment with a single-beam energy. Despite the simplicity of its experimental setup, it has been proven by many measurements [28-35] that a satisfactory resolution of the excitation function can be obtained owing to the inverse kinematics enhancement.

Proton resonance scattering induced by radioactive secondary beams has been investigated [36] with the TTIK method at CIAE since 2005. A series of measurements were conducted for ¹²C+p, ¹³N+p, ¹⁷F+p, and ²²Na+p using stable and radioactive ion beams [37-42]. In this study, we extend the conventional TTIK method to complex exit channels for the first time using γ-charged particle coincidence spectroscopy and demonstrate its applicability to simultaneously extracting the excitation functions of different reaction channels.

Experiment Setup

The experiment was performed in the HI-13 tandem accelerator laboratory [43-47] at the China Institute of Atomic Energy (CIAE) in Beijing. Figure 1 illustrates the experimental setup for the ²³Na+p thick-target experiments. A beam of 110 MeV ²³Na⁹⁺ ions, with a current of approximately 0.2 enA, was directed onto a (CH₂)_n target with a thickness of 5.8 mg/cm², resulting in an energy loss of approximately 66 MeV. Finally, the ²³Na ions were fully stopped in a 15.7 mg/cm²-thick carbon target. Because a high-energy Na beam can easily undergo fusion-evaporation reactions with the carbon atoms in the target, a thick carbon target was used to measure the background.

Fig. 1Full-screen View

Experimental setup

A silicon telescope system was placed at 0° along the beamline to measure the protons and α particles emitted from the compound nucleus ²⁴Mg. The silicon telescope system consisted of a 70 μm double-sided silicon strip detector (DSSD), a 1.5 mm multi-guard silicon quadrant (MSQ), and a 1 mm MSQ. The DSSD has 16 strips on each side, dividing the entire Si surface into 256 pixels. The small detection unit of the DSSD provided high-precision angular resolution during the kinematic reconstruction of the charged particles. The charged particles of interest originating from the ²³Na+p reaction primarily deposited energy in the first MSQ, and the signals from the second MSQ were used as anti-coincidence signals to reduce the background of high-energy protons and α particles. The wall of the target chamber was made of organic plastic to minimize absorption of the emitted γ rays. Six 3-inch lanthanum bromide detectors were uniformly arranged around the target chamber and placed on sliding rails to allow movement along the vertical beam direction to measure the characteristic γ rays from the residual nuclei ²³Na and ²⁰Ne.

γ-particle coincidence technique

3.1

LaBr₃ detector array calibration

A three-component γ source of ⁶⁰Co, ¹³⁷Cs, and ²⁴¹Am was used to calibrate the LaBr₃ array. The Geant4 package [48] was also applied to simulate the efficiency curves of the LaBr₃ detectors. The simulation of γ-ray energy spectrum of ⁶⁰Co was compared with the measured data, as shown in Fig. 2. Using this set of simulation parameters, the efficiency of γ detectors at different energies was obtained. The efficiency curves of the LaBr₃ detectors were fitted using Eq. (1) [49], considering several points in the simulation, as shown in Fig. 3. The systematic error in the simulation process was approximately 3%. $\ln \eta \left(E\right)={\displaystyle \sum _{i=0}^n{a}_i}{\left[\ln E\right]}^i\mathrm{,}$ (1) For the residual nuclei ²³Na and ²⁰Ne, the full-energy peak efficiencies at the characteristic energies of 440 and 1634 keV were 28.2% and 12.6%, respectively.

Fig. 2Full-screen View

(Color online) Geant4 simulation for ⁶⁰Co γ source

Fig. 3Full-screen View

Efficiency curve of the LaBr₃ array. The black points are the full-energy peak efficiency of the LaBr₃ array for γ rays of different energies obtained from Geant4 simulations. The red line represents the fitting result

3.2

γ-charged particle coincidence

The γ background originates mainly from the fusion-evaporation reaction of the high-energy ²³Na beam with a carbon target nucleus. For the exit channel p₁ of the compound nucleus, its residual nucleus ²³Na emits 440 keV characteristic γ rays, that is, corresponding to the first excited state of ²³Na. Because of the large number of γ-high-energy proton accidental coincidence events, it is difficult to quantitatively analyze the p₁ exit channel based on purely γ-charged particle coincidence. The black and red lines shown in Fig. 4 represent the γ energy spectra associated with the proton emissions. After subtracting the C-induced background and correcting for a γ efficiency of 440 keV, the absolute counts of the p₁ channel were obtained.

Fig. 4Full-screen View

(Color online) γ ray single energy spectrum measured with the LaBr₃ detector array in coincidence with protons. The γ spectrum obtained with (CH₂)_n-target and carbon-target are presented as black and red lines, respectively. The purple line is the net count spectrum obtained by subtracting the carbon background spectrum after beam normalization

Two-body kinematics reconstruction

The energy spectrum of charged particles obtained by a silicon detector is a mixture of a series of excitation function effects. Because γ-charged-particle coincidence introduces a large number of interfering events originating from the carbon-induced background, it is necessary to subtract these effects from the energy spectrum. In the energy range of the measurement, the exit channels of the compound ²⁴Mg consist of p₀, p₁, α₀, and α₁. Taking the p₀ exit channel as an example, the detailed processing steps for the two-body kinematic reconstruction are outlined below.

Step 1: Extract the energy spectrum of the proton exit channels from the total proton energy spectrum obtained by the silicon detectors after particle identification and energy calibration. The energy spectrum of the p₁ exit channel can be obtained from the charged particle spectrum of the silicon detector via ${\gamma }_1$-proton coincidence. After subtracting the p₁ energy spectrum corrected by γ efficiency from the total proton energy spectrum, the energy spectrum of p₀ can be obtained. At this stage, both the p₀ and p₁ exit-channel energy spectra contain C-induced backgrounds. Therefore, the proton energy spectrum obtained from the carbon target must be analyzed in a similar manner.

Step 2: Perform an event-by-event two-body kinematic reconstruction based on where the reaction ${}^{23}\text{Na}{\left(\text{p}\mathrm{,}\text{p}{\gamma }_1\right)}^{23}\text{Na}$ occurs within the thick target and calculate the relationship between E_c.m. and the energy deposited in the silicon detectors. Through energy-loss calculations and reaction kinematics, the energy deposition in the silicon detectors E_Si of protons is attributed to E_c.m. of the ${}^{23}\text{Na}{\left(\text{p}\mathrm{,}\text{p}{\gamma }_1\right)}^{23}\text{Na}$ reaction. As shown in Fig. 5, the red and black dots represent the correspondence between E_Si and E_c.m. for the ${}^{23}\text{Na}{\left(\text{p}\mathrm{,}\text{p}{\gamma }_1\right)}^{23}\text{Na}$ reaction on the (CH₂)_n and carbon target, respectively.

Fig. 5Full-screen View

(Color online) Relationship between E_c.m. and the deposited energy in silicon for the p₁ exit channel

The correspondence between E_Si and E_c.m. can be fitted using the least-squares method with a linear function, as expressed in Eq. (2). As shown in Fig. 5, the red and black dots approximately follow straight lines. $E_{\text{c}\text{.m}\text{.}}\left(x\right)=E_{\text{Si}}\left(x\right)\cdot k+b\mathrm{,}$ (2)

Fig. 6Full-screen View

(Color online) Relationship between the bin target thickness and center-of-mass energy

Step 3: Subtract the background events from the energy spectrum in the frame of the center of mass. Corrections for the beam particle and ¹²C atom numbers are required for the carbon-induced background. The (CH₂)_n and carbon targets have different energy stopping powers for ²³Na. The correction factor f_n for the ¹²C atom numbers is the ratio of the number of C atoms corresponding to 1 keV of energy deposited in the (CH₂)_n target and the carbon target for the same energy of the ²³Na beam. This relationship was calculated using LISE++ and fitted to a linear function (Fig. 6). $\text{d}{m}_{\text{c}}/\text{d}{E}_{\text{c}\text{.m}\text{.}}=E_{\text{c}\text{.m}\text{.}}\cdot k_{\text{c}}+b_{\text{c}}\mathrm{,}$ (3) $\text{d}{m}_{{\left({\text{CH}}_{\text{2}}\right)}_{\text{n}}}/\text{d}{E}_{\text{c}\text{.m}\text{.}}=E_{\text{c}\text{.m}\text{.}}\cdot k_{{\left({\text{CH}}_{\text{2}}\right)}_{\text{n}}}+b_{{\left({\text{CH}}_{\text{2}}\right)}_{\text{n}}}\mathrm{,}$ (4) $f_{\text{n}}=\frac{E_{\text{c}\text{.m}\text{.}}\cdot k_{{\left({\text{CH}}_{\text{2}}\right)}_{\text{n}}}+b_{{\left({\text{CH}}_{\text{2}}\right)}_{\text{n}}}}{E_{\text{c}\text{.m}\text{.}}\cdot k_{\text{c}}+b_{\text{c}}}$ (5) The number of ¹²C atoms corresponding to each bin of the carbon target and the (CH₂)_n target energy spectra is given by Eq. (3) and Eq. (4). The correction factor f_n is given by Eq. (5). Subsequently, the carbon background can be subtracted from the energy spectrum of $p_{\text{0}}$. As shown in Fig. 7, the black line represents the energy spectrum of the (CH₂)_n target and the red line represents the energy spectrum of the carbon target. The net energy spectrum of the ${}^1\text{H}{\left({}^{23}\text{Na}\mathrm{,}\text{p}\right)}^{23}\text{Na}$ channel can be obtained by subtracting the normalized carbon background.

Fig. 7Full-screen View

(Color online) Yields from ¹H(²³Na, p₀)²³Na elastic scattering channel. The background from the pure carbon target is flat without any resonance structure

Step 4: Calculate the excitation function of ${}^1\text{H}{\left({}^{23}\text{Na}\mathrm{,}\text{p}\right)}^{23}\text{Na}$, based on the net yield of p₀ exit channel. Because each bin in the p₀ energy spectrum corresponds to different H atom numbers of the (CH₂)_n target, the differential cross-section needs to be calculated separately for each energy point. The excitation functions of the proton and α channels are shown in Fig. 8 and Fig. 9, respectively, according to Eq. (6). $\text{d}\sigma /\text{d}\Omega =\frac{N_{\text{count}}}{N_{\text{H}}\cdot N_{\text{I}}\cdot \Omega }$ (6) where N_H and N_I represent the numbers of H atoms in the target and incident ²³Na⁹⁺ beam ions, respectively. Ω denotes the solid angle of the Si detector.

Fig. 8Full-screen View

(Color online) Excitation function for the proton exit channels. p₀ and p₁ represent ¹H(²³Na, p₀)²³Na and ${}^1\text{H}{\left({}^{23}\text{Na},p_1\right)}^{23}{\text{Na}}_{440}^{*}$, respectively

Fig. 9Full-screen View

(Color online) Excitation function for the α exit channels. α₀ and α₁ represent ¹H(²³Na, α₀)²⁰Ne and ${}^1\text{H}{\left({}^{23}\text{Na},{\alpha }_1\right)}^{20}{\text{Ne}}_{1634}^{*}$, respectively

Throughout the data analysis process, each step resulted in an error that was ultimately contained in the individual data points of the excitation function. These errors mainly include γ-charged-particle coincidence, statistical, carbon background deduction, and p₁(α₁) deduction errors in the p₀(α₀) calculations. For γ-particle coincidence, the coincidence efficiency mainly depends on the γ detection efficiency, silicon efficiency, and accidental coincidence events. The errors of these factors were independent. The γ detection efficiency error evaluated through the Geant4 simulation process was less than 3%, whereas the silicon detection efficiency error was negligible. Accidental coincidence events were uniformly distributed in the coincidence time spectrum and were evaluated as having an error of less than 2%. Before the carbon background deduction, the energy spectrum must be divided into individual bins, with statistical errors assigned to each bin. For energy points with low counts, the statistical error was relatively large, close to 2%. The error after carbon background deduction depends on statistical errors and errors introduced during the subtraction process. During carbon background deduction, the ratio of carbon background counts to total counts positively correlated with the error introduced by the subtraction process. As shown in Fig. 7, for the p₀ channel, the error after carbon background subtraction at 3.8 MeV was 5%. By comparison, for most energy points with lower background contributions, the error after background subtraction was approximately 3.5%. For the p₀(α₀) channel, similar errors caused by data processing were introduced when subtracting the p₁(α₁) channel. The γ-efficiency error must be considered separately when calculating the final error in the p₁(α₁) channel. Overall, the two subtraction processes were the main sources of excitation function errors. The total errors for p₁ and α₁ were less than 8% and 10%, respectively. p₀ and α₀ errors were correspondingly larger, that is, less than 10% and 17% over the entire energy range.

The R-matrix theory [50, 51] is a parametric framework that describes compound nuclear reactions and theoretically describes the resonance phenomena in nuclear reactions. For excitation functions that include resonance states, R-matrix analysis can decompose the differential cross section into three overlapping components: hard-sphere scattering background, independent resonances, and interference of multiple resonances. This study provided the exit-channel excitation functions of the compound nucleus ²⁴Mg populated by the ²³Na+p entrance channel. The Azure package was used to extract a series of exit channel resonance parameters related to the ¹²C+¹²C fusion reaction. The fitting results are indicated by the red lines in Fig. 8 and Fig. 9. The fitting results indicate that nearly 50 ²⁴Mg resonances are included to reproduce the excitation functions, not only the 0⁺ and 2⁺ resonance levels relevant to the ¹²C+¹²C fusion reaction, but also other levels such as 1^- and 3^-. The ²⁴Mg resonances observed in the ²³Na+p entrance channel are very complex and discrete, but densely packed in the energy region of ¹²C+¹²C Gamow window. However, the resonance parameters of the proton and α decay channels are useful for the further study of the ¹²C+¹²C fusion reaction. The detailed results of the R-matrix analysis and its impact on the ¹²C+¹²C astrophysical S factor will be presented in a forthcoming paper.

Summary

The thick-target inverse kinematics method is widely used to measure the excitation functions of p(α) elastic and inelastic scattering induced by radioactive ion beams. In this study, the conventional TTIK method was extended to include complex reaction channels for the first time, which enables the simultaneous extraction of the excitation functions of different reaction channels. The high energy resolution and high detection efficiency γ-charged particle coincidence are essential for promoting this novel method for radioactive ion beam-induced reactions. Although the ²⁴Mg resonances observed in this study were useful, they were largely beyond the relevance of the ¹²C+¹²C fusion reaction. A similar measurement utilizing ²⁰Ne+α can be performed to significantly reduce ²⁴Mg resonances in the excitation functions.