The effects of beam drifts on elastic scattering measured by the large solid-angle covered detector array

The effects of beam drifts on elastic scattering measured by the large solid-angle covered detector array 1 2 3 first-author https://orcid.org/0000-0003-1890-6102 3 corresp Corresponding author, cjlin@ciae.ac.cn Corresponding author, cjlin@ciae.ac.cn cjlin@ciae.ac.cn https://orcid.org/0000-0002-3988-6756 3 3 3 https://orcid.org/0000-0002-5524-1085 2 corresp Corresponding author, zgl@buaa.edu.cn Corresponding author, zgl@buaa.edu.cn zgl@buaa.edu.cn https://orcid.org/0000-0003-4615-3187 2 https://orcid.org/0000-0002-1112-3554 3 https://orcid.org/0000-0003-3489-0091 3 State Key Laboratory of Software Development Environment, Beihang University, Beijing 100191, China State Key Laboratory of Software Development Environment, Beihang University, Beijing 100191, China School of Physics, Beihang University, Beijing 100191, China School of Physics, Beihang University, Beijing 100191, China China Institute of Atomic Energy, Beijing 102413, China China Institute of Atomic Energy, Beijing 102413, China A new detector array characterized by compact structure and large solid-angle coverage was designed for radioactive ion beam (RIB) experiments and measuring multi-nucleon correlation. A Monte Carlo simulation was performed to explore the effects of beam drifts in different directions and distances on the angular distribution of the Rutherford scattering, as measured by the detector array. The results indicate that, when the beam drift distance is less than 2.0 mm, the symmetry of the detector array can maintain a count error of less than 5%. This confirms the property of the detector array for RIB experiments. Furthermore, the simulation was validated through the elastic scattering angular distributions of 6,7 Li measured by the detector array in 6,7 Li + 209 Bi experiments at different energies. Large solid-angle detector array beam drift Monte Carlo simulation elastic scattering 1 Introduction 1 With the development of accelerator facilities in many nuclear physics laboratories, the beams of weakly-bound nuclei are becoming increasingly available [ 1 1 ]. It is becoming critically important to understand the influence of weak binding on reaction dynamics, including elastic scattering [ 2 2 - 5 5 2-5 ]. Through the elastic scattering angular distributions of the weakly-bound nuclei, reaction mechanisms can be studied [ 6 6 , 7 7 ]. Therefore, a precise measurement of the elastic scattering angular distribution is necessary for the study of weakly-bound nuclei. A double-sided silicon strip detector (DSSD) is widely used for the measurement of charged particles because of its high position, energy resolution, and detection efficiency. Several well-known silicon detector arrays, such as the EXPADES detector array, MITA, and compact disc DSSD array [ 8 8 - 10 10 8-10 ], have been used for the measurement of elastic scattering angular distributions in experimental studies. A silicon detector array with a larger angle coverage than the previous arrays [ 8 8 - 15 15 8-15 ], both in scattering angles and azimuthal angles, was developed by the China Institute of Atomic Energy (CIAE). The detector array was characterized by a compact structure and small size. It contained eight independent telescopes, each of which consisted of two parts of silicon detectors: the first layer was 40 (or about 60) μ m DSSD, followed by one or two layers of quadrant silicon detectors (QSDs) with different thicknesses. The detector array was used in the experiment of 6,7 Li + 209 Bi at the HI-13 tandem accelerator in CIAE to obtain the elastic scattering angular distributions of 6,7 Li at different energies. However, the elastic scattering angular distributions measured by the array were sensitive to the position of the beam spot on the target, while beam drifts were common in the experiments. In this study, we conducted Monte Carlo simulations to explore the effect of beam drifts on the angular distribution of Rutherford scattering. Through this method, we obtained precise angular distributions, which are in agreement with the calculated results. In this paper, the detector array is described in Section 2. The experimental setup is presented in Sect. 3. In Sect. 4, we describe the Monte Carlo simulation method. In Sect. 5, the scattered events of 6,7 Li + 209 Bi are selected to obtain the elastic scattering angular distributions. Finally, we provide our conclusions. 2 Description of the detector array 1 The detector array used in the experiment is shown in Fig. 1. Fig. 1. The detector array should have consisted of ten detector telescopes; however, two telescopes were canceled due to the lack of a silicon detector with the appropriate thickness. Each telescopic unit contained one DSSD and one (or two) QSDs. A mylar foil with a thickness of 0.5 μ m was installed in front of the telescope to protect the first silicon detector from being hit by a low-energy electron. Following the foil, a series of silicon detectors were aligned in the same plane. Different units had different arrangements. A DSSD with a thickness of about 40 μ m and a QSD with a thickness of 1000 μ m were arranged in units 1, 2, 3, and 4. In unit 5, 7, and 8, in case the energy loss of light particles in first layer of telescope was insufficient, a DSSD with a thickness of about 60 μ m and a QSD with a thickness of 300 μ m, followed by a QSD with a thickness of 1000 μ m, were arranged. Due to the lack of 300 μ m thick QSD, a DSSD with a thickness of about 60 μ m and a QSD with a thickness of 1500 μ m were arranged in unit 6. The gap between the mylar foil and DSSD was 1 mm, and the distance from the DSSD to the first QSD was about 4 mm. The DSSDs were divided into 16 strips for each side, which had 50 × 50 mm 2 total active area, making up 256 pixels of 3 × 3 mm 2 . The size of each pixel determines the angular resolution of the detector array. Considering the distance from the target to the DSSDs, which was about 70 mm (or 82 mm), the angular resolution in the center region of each unit was at least about ± 1.23° (or ± 1.05°), which became better in the periphery area of the detector telescopes in the laboratory frame. The QSDs were divided into four squares, and each square had a 24 × 24 mm 2 active area. The eight Δ E - E telescopes surrounding the target were installed on a designed and fixed frame. In the array, the distance from the center of the DSSD to the target center was 70 mm in units 1, 2, 3, 4, 7, and 8, and 82 mm in units 5 and 6. The angle of the center of each unit in the beam direction, which is marked as θ , is listed in Table 2 Table 2 . Another angle, ϕ , which is listed in Table 1 Table 1 , is the angle of the center of each unit in the vertical plane. The arrangement of eight telescopes made the array cover a large area, both for the scattering angle and azimuthal angle. The angular coverage of each unit is shown in Fig. 2. Fig. 2. 1 1 Unit NO. 1 1 Thickness of DSSD ( μ m) 1 1 Thickness of front QSD ( μ m) 1 1 Thickness of back QSD ( μ m) 1 1 Distance between unit and target (mm) 1 1 θ (deg) 1 1 ϕ (deg) 1 1 1 1 1 40 1 1 1000 1 1 *** 1 1 70 1 1 132.07 1 1 139.23 1 1 2 1 1 40 1 1 1000 1 1 *** 1 1 70 1 1 132.07 1 1 40.77 1 1 3 1 1 40 1 1 1000 1 1 *** 1 1 70 1 1 132.07 1 1 220.77 1 1 4 1 1 40 1 1 1000 1 1 *** 1 1 70 1 1 132.07 1 1 319.23 1 1 5 1 1 60 1 1 300 1 1 1000 1 1 82 1 1 90 1 1 180 1 1 6 1 1 60 1 1 1500 1 1 *** 1 1 82 1 1 90 1 1 0 1 1 7 1 1 60 1 1 300 1 1 1000 1 1 70 1 1 47.93 1 1 139.23 1 1 8 1 1 60 1 1 300 1 1 1000 1 1 70 1 1 47.93 1 1 319.23 1 1 Projectile 1 1 E Lab (MeV) 1 1 V (MeV) 1 1 r V (fm) 1 1 a V (fm) 1 1 W (MeV) 1 1 r W (fm) 1 1 a W (fm) 1 1 σ R (mb) 1 1 6 Li 1 1 30.0 1 1 13.61 1 1 1.33 1 1 0.36 1 1 24.18 1 1 0.26 1 1 2.00 1 1 284.46 1 1   1 1 40.0 1 1 28.48 1 1 1.13 1 1 0.80 1 1 12.66 1 1 1.26 1 1 0.69 1 1 1062.20 1 1   1 1 47.0 1 1 44.26 1 1 1.22 1 1 0.50 1 1 13.43 1 1 0.95 1 1 1.43 1 1 1840.63 1 1 7 Li 1 1 30.0 1 1 6.72 1 1 1.39 1 1 0.40 1 1 22.09 1 1 0.49 1 1 1.54 1 1 114.44 1 1   1 1 40.0 1 1 32.52 1 1 1.15 1 1 0.77 1 1 13.60 1 1 1.26 1 1 0.58 1 1 1011.11 1 1   1 1 47.0 1 1 41.79 1 1 1.24 1 1 0.51 1 1 24.46 1 1 0.70 1 1 1.58 1 1 1855.36 To reduce the noise, the integrated preamplifiers designed in CIAE [ 16 16 ] were installed close to the detectors, so that the integrated preamplifiers were also arranged in the reaction chamber, which was evacuated in the experiment. A cooling system was used to decrease the temperature, to ensure that the integrated preamplifiers functioned stably. All integrated preamplifiers were installed on two hollow brass disks, which were fixed on the top and bottom of the array. During the experiment, the integrated preamplifiers transferred their heat to the disks, and the brass disks were cooled by the circulation of alcohol inside. 3 Experimental setup 1 The detector array was used to study the reaction mechanism in the 6,7 Li + 209 Bi systems at energies around and above the Coulomb barrier. In the experiment, the 6,7 Li beams were accelerated to energies of 30, 40, and 47 MeV, while the 6 Li + 209 Bi experiment was performed at an energy of 25 MeV. The schematic vertical view of the experimental setup for 6,7 Li + 209 Bi is shown in Fig. 3. Fig. 3. The self-hold target 209 Bi with a thickness of 210 μ g/cm 2 was located at the center of the detector array. To reduce the blind area on the detectors caused by the target frame, the target was tilted at 70° relative to the beam line, and aligned to the gap of the units. A collimator with a diameter of 3 mm was installed at 30 cm in front of the target along the beam line to limit the beam spot size and position. Four silicon detectors were installed at a distance of 250 mm from the center of the target to monitor the beam intensity and beam drifts. Two monitors with a diameter of 0.5 mm were installed symmetrically at a scattering angle of 12.5° in the horizontal plane, while the other two monitors with a radius of 0.5 mm were installed at a scattering angle of 25° in the vertical plane. 4 Simulation 1 It is common knowledge that the different reaction positions of the projectiles on the target represent different scattering angles for the particles measured by the same detector pixel. The beam spot size of the radioactive ion beam on the target was usually large (about 10 mm), which affected the accuracy of the scattering angle of the particles measured in the experiments. Therefore, several position-sensitive detectors, such as parallel plate avalanche counters, were often placed in front of the target to reconstruct the trajectories of beam particles and determine the reaction positions. For the beam generated by the HI-13 tandem accelerator, the beam spot size on the target was sufficiently small (at most 1 mm), so that the scattering angle of each particle could be directly obtained from the geometry relationship between the fired silicon pixel and beam spot. However, beam drifts, leading to changes in the reaction positions of the beam particles on the target, were common in the experiments, and the accuracy of the scattering angle of the particles measured in the experiments was often affected. To determine the influence of beam drifts on the measurement via the detector array, a Monte Carlo simulation was performed to simulate the angular distribution of the Rutherford scattering measured by the array [ 17 17 ]. In the simulation, the aforementioned geometry of the detector array including the eight telescopic units, was implemented, and the number of simulated events was 10 8 . The beam particles were simulated with the energies sampled from a Gaussian distribution with a large width to cover a wide range of energies. The distribution was centered (mean) at 25 MeV with an energy spread ( σ ) of 1 MeV. The ion distribution was uniform in circles with a radius of 0.5 mm, which was consistent with the beam spot size in the 6,7 Li + 209 Bi experiments. The elastic scattering occurred inside the target with the reaction positions sampled from a random distribution along the beam line, and the energy loss of the projectiles in the target was taken into consideration by the calculation with the distances 6 Li traversed in the target. The different beam drift distances in the horizontal and vertical directions were both chosen in the simulation. We used the ratios of the number of detected particles with different beam drift distances to those with no beam drift in the same pixels. By comparing the changes of ratios, the influence of beam drifts were obtained. To distinguish the directions of beam drifts, the distance was recorded as a positive value when the beam drifted to the right or down relative to the beam direction. The simulation results with different drift distances in the horizontal direction are shown in Fig. 4 Fig. 4 , while the results for beam drifts in the vertical direction are shown in Fig. 5. Fig. 5. To distinguish easily, the scattering angles were partitioned into four ° bins. By comparing Fig. 4 Fig. 4 and Fig. 5 Fig. 5 , it was easy to observe that the ratios were closer to 1 when all detectors were cooperating. As shown in Fig. 4(c) Fig. 4(c) and Fig. 4(d) Fig. 4(d) , when the beam drifted in the horizontal direction, the ratios were lower than 1.05 when the beam drift distance was larger than -2 mm, and higher than 0.95 when the distance was less than 3 mm. When the beam drifted in the vertical direction, as shown in Fig. 5(c) Fig. 5(c) and Fig. 5(d) Fig. 5(d) , the difference between the ratios and standard value (the dashed line) was less than 4% when the beam drift distance ranges from -5 mm to 5 mm. This indicates that the beam drifts in the horizontal direction had a greater effect on the measured elastic scattering angular distribution than those in the vertical direction. This may result from the tilted target and the asymmetry of the detectors in the horizontal–vertical direction. We concluded that the symmetrical placement of detector telescopes is beneficial for correcting the effect of beam drifts. The influence of beam drifts was small at distances from -2 mm to 4 mm in the horizontal direction or from -5 mm to 5 mm in the vertical direction, and the count error was less than 5%. When the beam drift distance was less than 1 mm, the influence was at most 2%. 5 Data analysis and results 1 To verify the above simulation, the elastic scattering data of 6,7 Li + 209 Bi, measured by the detector array, were analyzed. The energy calibrations of the DSSDs were performed utilizing the α particles decayed from the evaporation residues in both complete and incomplete fusion. In addition, two radioactive sources ( 239 Pu and 241 Am), providing α particles at energies of 5.156 MeV and 5.486 MeV, were employed to calibrate the energy spectrum. Then, the calibrations of the QSDs were performed by the energy deposited by charged particles in the QSDs, which were determined by subtracting the energy measured by the DSSDs from the expected energy of the particles calculated by LISE++ [ 18 18 ]. The typical particle identification spectrum in 6 Li + 209 Bi, which was obtained from one pixel of the unit 5 telescope at an energy of 40 MeV, is shown in Fig.6(a) Fig.6(a) , and the spectra measured by unit 4 at the same scattering angle are shown in Fig.6(b) Fig.6(b) and Fig.6(c) Fig.6(c) . As labeled correspondingly, the banana-like regions in Fig.6(a) Fig.6(a) , Fig.6(b) Fig.6(b) , and Fig.6(c) Fig.6(c) can be identified as 6 Li, protons, deuterons, tritons, and alphas. A typical energy spectrum of 6 Li + 209 Bi, which was measured by unit 4 at an energy of 40 MeV, is shown in Fig.7. Fig.7. The energy resolution was about 2.0%, which was sufficient to allow a clear separation between the elastic and inelastic scattering events (the first excited state energy of 6 Li is 2186 keV). Then, the pure elastic scattering events of 6 Li were chosen to calculate the elastic scattering angular distribution. This is the same for 7 Li + 209 Bi. Through the ratios of the elastic scattering events measured by the symmetrical monitors over a period of time, the average beam drift distance was obtained via calculation using the program, Mathematica [ 19 19 ]. Owing to the installed collimator, the beam drifts over the period of time were limited. Then, the influence of beam drifts was corrected via simulations using the determined beam drift distance. The angular distribution of elastic scattering is usually described as the ratio of the reaction cross section to the cross section of the Rutherford scattering, which is obtained using equation [ 11 11 ] <math display="block" id="M1"><mrow><mfrac><mrow><mtext>d</mtext><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><msub><mi>σ</mi><mrow><mtext>R</mtext><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mi>D</mi><mfrac><mrow><mi>N</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>N</mi><mrow><mtext>R</mtext><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where parameter D is a normalization constant that can be calculated by assuming that the elastic scattering is pure Rutherford scattering at small scattering angles. Subscript C refers to the center-of-mass frame. N ( θ C ) is the elastic scattering event at any scattering angle, and N RU ( θ C ) is the Rutherford scattering event with the same solid angles at the same scattering angle in the center-of-mass frame. We supposed the elastic scattering in 6 Li + 209 Bi at an energy of 25 MeV is pure Rutherford scattering because 25 MeV is below the Coulomb barrier of this system. Therefore, in the 6 Li + 209 Bi system, N ( θ C ) is the elastic scattering count measured in the experiment at designed energies, and N RU ( θ C ) is the experimental elastic scattering count of 6 Li + 209 Bi at 25 MeV. Then, the equation becomes <math display="block" id="M2"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>σ</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mi>L</mi><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><msub><mi>σ</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mi>L</mi><mi>i</mi><mn>,</mn><mi>R</mi><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>C</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><msub><mi>D</mi><mn>1</mn></msub><mfrac><mrow><msub><mi>N</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mi>L</mi><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>N</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mi>L</mi><mi>i</mi><mn>,25</mn><mi>M</mi><mi>e</mi><mi>V</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where θ L is the scattering angle of 6 Li in the laboratory frame, which corresponds to θ C in the center-of-mass frame, while D 1 is the new normalization constant. The influence of the detector efficiency and geometry of the detector array was reduced by taking the elastic scattering in 6 Li + 209 Bi at an energy of 25 MeV into calculations as Rutherford scattering. For the calculation of the angular distributions of elastic scattering in the 7 Li + 209 Bi system at different energies, a derivation is required. The number of Rutherford scattering events can be deduced from the elastic scattering events in 6 Li + 209 Bi at an energy of 25 MeV, and the transformation equation is as follows. <math display="block" id="M3"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>N</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mtext>1</mtext><mn>,</mn><mi>C</mi></mrow></msub><mo stretchy="false">)</mo><mi/></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mi>N</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo><mfrac><mrow><msub><mi>Ω</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mtext>1</mtext><mn>,</mn><mtext>C</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>Ω</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>×</mo><mfrac><mrow><mtext>d</mtext><msub><mi>σ</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mtext>1</mtext><mn>,</mn><mtext>C</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><msub><mi>Ω</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mtext>1</mtext><mn>,</mn><mtext>C</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mfrac><mrow><mtext>d</mtext><msub><mi>Ω</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><msub><mi>σ</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> where θ C and θ 1,C are the scattering angles of 6,7 Li, respectively, in the center-of-mass frames. <math id="M4"><mrow><msub><mi>Ω</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mtext>1</mtext><mn>,</mn><mtext>C</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></math> and <math id="M5"><mrow><msub><mi>Ω</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo></mrow></math> are the solid angles of 6,7 Li at the corresponding scattering angles. If θ C and θ 1,C correspond to the same scattering angle θ L in the frame of the laboratory system, the Rutherford scattering events of 7 Li at the scattering angle θ 1,C can be described as <math display="block" id="M6"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>N</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mtext>1</mtext><mn>,</mn><mtext>C</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>D</mi><mn>2</mn></msub><mi>P</mi><msub><mi>N</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>C</mtext></msub><mo stretchy="false">)</mo><mfrac><mrow><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mn>4</mn></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>L</mtext></msub><mo>+</mo><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>A</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mtext>Li</mtext></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msup><mrow/><mrow><mtext>209</mtext></mrow></msup><mtext>Bi</mtext></mrow></msub></mrow></mfrac><mi>sin</mi><msub><mi>θ</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mn>4</mn></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>L</mtext></msub><mo>+</mo><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>A</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li</mtext></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msup><mrow/><mrow><mtext>209</mtext></mrow></msup><mtext>Bi</mtext></mrow></msub></mrow></mfrac><mi>sin</mi><msub><mi>θ</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=8253706&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=8253702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=8253706&type= where D 2 is a constant related to the energies of 6,7 Li, and P is the ratio of the solid angle of 7 Li to 6 Li in the center-of-mass frame at the same scattering angle θ L . Then, the ratio to describe the angular distribution of elastic scattering of 7 Li with 209 Bi can be calculated by substituting Eq. (4) and Eq. (2) into Eq. (1). <math display="block" id="M7"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mi>d</mi><msub><mi>σ</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mtext>1</mtext><mn>,</mn><mtext>C</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><msub><mi>σ</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li,RU</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mrow><mtext>1,C</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><msub><mi>D</mi><mtext>3</mtext></msub></mrow><mi>P</mi></mfrac><mfrac><mrow><msub><mi>N</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>N</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mtext>Li,25MeV</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>×</mo><mfrac><mrow><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mn>4</mn></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>L</mtext></msub><mo>+</mo><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>A</mi><mrow><msup><mrow/><mtext>7</mtext></msup><mtext>Li</mtext></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msup><mrow/><mrow><mtext>209</mtext></mrow></msup><mtext>Bi</mtext></mrow></msub></mrow></mfrac><mi>sin</mi><msub><mi>θ</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mn>4</mn></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mtext>L</mtext></msub><mo>+</mo><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>sin</mi></mrow></mstyle><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>A</mi><mrow><msup><mrow/><mtext>6</mtext></msup><mtext>Li</mtext></mrow></msub></mrow><mrow><msub><mi>A</mi><mrow><msup><mrow/><mrow><mtext>209</mtext></mrow></msup><mtext>Bi</mtext></mrow></msub></mrow></mfrac><mi>sin</mi><msub><mi>θ</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> where parameter D 3 is the normalization constant for 7 Li. The normalization results of the elastic scattering for 6 Li on the 209 Bi target at energies of 30, 40, and 47 MeV are shown in Fig. 8(a) Fig. 8(a) , while those for 7 Li at the same energies are shown in Fig. 8(b) Fig. 8(b) ]. The results were fitted by the optical model calculation program “Ptolemy” citebib:20, and the potential parameters and total reaction cross-sections were extracted from the model. Here, the Woods–Saxon function was selected as the shape function of the real and imaginary parts of the nuclear potential. V , r V , a V , respectively, and W , r W , and a W , respectively, represent the amplitude, radius parameter, and diffusness parameter of the real and imaginary parts of the nuclear potential. The fitting results are shown in Fig. 8 Fig. 8 by the solid curves, and the potential parameters and the total reaction cross sections are listed in Table. 2 Table. 2 . In Fig. 8 Fig. 8 , we observed that the fit lines fitted the experimental data well at 30 MeV and 40 MeV, while the fit lines deviated the experimental data at large angles of 47 MeV, because the events measured at large angles of 47 MeV were too small and other reaction channels opened. Because the fitting results are in agreement with the experimental data, the results in Table 2 Table 2 provide the reference data for the nuclear potential and total cross-sections of 6,7 Li + 209 Bi. 6 Summary 1 In this study, we conducted a study introducing the elastic scattering measured by a large solid-angle-covered detector array, a silicon detector telescope array designed in CIAE, which covered the scattering angle from 24.31° to 155.69° and 300.96° in the azimuthal angle in the laboratory system. By simulating the angular distributions of Rutherford scattering measured by the detector array at different beam drift distances in the horizontal and vertical directions, we explored the influence of the beam drifts on the angular distribution of the Rutherford scattering measured by the detector array, obtaining the precise angular distribution of elastic scattering. The simulation results indicate that, benefiting from the symmetrical geometry of the detector array, the count error was less than 2% when the beam drift distance was less than 1 mm. The beam drifts had a weak influence on the measurements with the array. The elastic scattering angular distributions, measured by the detector array in the 6,7 Li experiments with the 209 Bi target at energies of 30, 40, and 47 MeV and performed at CIAE, were chosen to validate the simulation. After obtaining the elastic scattering angular distributions of 6,7 Li, the optical model calculation program “Ptolemy” was used to fit the results at different energies. The program fitted the experimental data well. The potential parameters V , r V , a V , W , r W , a W , and total reaction cross sections were extracted from the fitting. The agreement between the experimental results and theoretical calculations and the previous results[ 21 21 ] verifies the accuracy of the simulation. Furthermore, the method can be used in 8 B + 120 Sn and 7 Be + 209 Bi experiments to confirm the performance of the detector array for a deeper investigation of the nuclear reaction mechanism. References C. Signorini , A. Edifizi , M. Mazzocco et al . Exclusive breakup of 6Li by 208Pb at Coulomb barrier energies [J]. Phys. Rev. C 67 , 044607 ( 2003 ). doi: 10.1103/PhysRevC.67.044607 http://doi.org/10.1103/PhysRevC.67.044607 Exclusive breakup of 6Li by 208Pb at Coulomb barrier energies L. F. Canto , P. R. S. Gomes , R. Donangelo et al . Recent developments in fusion and direct reactions with weakly bound nuclei [J]. Physics Reports , 596 , 1 - 86 ( 2015 ). doi: 10.1016/j.physrep.2015.08.001 http://doi.org/10.1016/j.physrep.2015.08.001 Recent developments in fusion and direct reactions with weakly bound nuclei N. Keeley , R. Raabe , N. Alamanos , and J. L. Sida . Fusion and direct reactions of halo nuclei at energies around the Coulomb barrier [J]. Progress in Particle and Nuclear Physics , 59 , 579 - 630 ( 2007 ). doi: 10.1016/j.ppnp.2007.02.002 http://doi.org/10.1016/j.ppnp.2007.02.002 Fusion and direct reactions of halo nuclei at energies around the Coulomb barrier K. J. Cook , E. C. Simpson , L. T. Bezzina et al . Origins of Incomplete Fusion Products and the Suppression of Complete Fusion in Reactions of 7Li [J]. Phys. Rev. Lett 122 , 102501 ( 2019 ). doi: 10.1103/PhysRevLett.122.102501 http://doi.org/10.1103/PhysRevLett.122.102501 Origins of Incomplete Fusion Products and the Suppression of Complete Fusion in Reactions of 7Li M. Dasgupta , P. R. S. Gomes , D. J. Hinde et al . Effect of breakup on the fusion of 6Li, 7Li and 9Be with heavy nuclei [J]. Phys. Rev. C 70 , 024606 ( 2004 ). doi: 10.1103/PhysRevC.70.024606 http://doi.org/10.1103/PhysRevC.70.024606 Effect of breakup on the fusion of 6Li, 7Li and 9Be with heavy nuclei M. Mazzocco , D. Torresi , D. Pierroutsakou et al . Direct and compound-nucleus reaction mechanisms in the 7Be + 58Ni system at near-barrier energies [J]. Phys. Rev. C 92 , 024615 ( 2015 ). doi: 10.1103/PhysRevC.92.024615 http://doi.org/10.1103/PhysRevC.92.024615 Direct and compound-nucleus reaction mechanisms in the 7Be + 58Ni system at near-barrier energies E. F. Aguilera et al . Near-Barrier Fusion of the 8B + 58Ni Proton-Halo System [J]. Phys. Rev. Lett 107 , 092701 ( 2011 ). doi: 10.1103/PhysRevLett.107.092701 http://doi.org/10.1103/PhysRevLett.107.092701 Near-Barrier Fusion of the 8B + 58Ni Proton-Halo System E. Strano , A. Anastasio , M. Bettini et al . The high granularity and large solid angle detection array EXPADES [J]. Nucl. Instr. Meth. Phys. Res. B 317 , 657 - 660 ( 2013 ). doi: 10.1016/j.nimb.2013.06.035 http://doi.org/10.1016/j.nimb.2013.06.035 The high granularity and large solid angle detection array EXPADES D. Pierroutsakou , A. Boiano , C. Boiano et al . The experimental setup of the RIB in-flight facility EXOTIC [J]. Nucl. Instr. Meth. Phys. Res. A 837 , 46 - 70 ( 2016 ). doi: 10.1016/j.nima.2016.07.019 http://doi.org/10.1016/j.nima.2016.07.019 The experimental setup of the RIB in-flight facility EXOTIC N. R. Ma et al . MITA: A Multilayer Ionization-chamber Telescope Array forlow-energy reactions with exotic nuclei [J]. Eur. Phys. J. A 55 , 87 ( 2019 ). doi: 10.1140/epja/i2019-12765-7 http://doi.org/10.1140/epja/i2019-12765-7 MITA: A Multilayer Ionization-chamber Telescope Array forlow-energy reactions with exotic nuclei Gao-Long Zhang , Yong-Jin Yao , Guang-Xin Zhang et al . A detector setup for the measurement of angular distribution of heavy-ion elastic scattering with low energy on RIBLL [J]. Nucl. Sci. Tech . 28 , 104 ( 2017 ). doi: 10.1007/s41365-017-0249-0 http://doi.org/10.1007/s41365-017-0249-0 A detector setup for the measurement of angular distribution of heavy-ion elastic scattering with low energy on RIBLL K. J. Cook , E. C. Simpson , D. H. Luong et al . Importance of lifetime effects in breakup and suppression of complete fusion in reactions of weakly bound nuclei [J]. Phys. Rev. C 93 , 064604 ( 2016 ). doi: 10.1103/PhysRevC.93.064604 http://doi.org/10.1103/PhysRevC.93.064604 Importance of lifetime effects in breakup and suppression of complete fusion in reactions of weakly bound nuclei S. Kalkal , E. C. Simpson , D. H. Luong et al . Asymptotic and near-target direct breakup of 6Li and 7Li [J]. Phys. Rev. C 93 , 069904 ( 2016 ). doi: 10.1103/PhysRevC.93.069904 http://doi.org/10.1103/PhysRevC.93.069904 Asymptotic and near-target direct breakup of 6Li and 7Li D. H. Luong , M. Dasgupta , D. J. Hinde et al . Predominance of transfer in triggering breakup in sub-barrier reactions of 6,7Li with 144Sm, 207,208Pb and 209Bi [J]. Phys. Rev. C 88 , 034609 ( 2013 ). doi: 10.1103/PhysRevC.88.034609 http://doi.org/10.1103/PhysRevC.88.034609 Predominance of transfer in triggering breakup in sub-barrier reactions of 6,7Li with 144Sm, 207,208Pb and 209Bi D. H. Luong , M. Dasgupta , D. J. Hinde et al . Insights into the mechanisms and time-scales of breakup of 6,7Li [J]. Phys. Let. B , 695 , 105 - 109 ( 2011 ). doi: 10.1016/j.physletb.2010.11.007 http://doi.org/10.1016/j.physletb.2010.11.007 Insights into the mechanisms and time-scales of breakup of 6,7Li Dong-Xi Wang , Cheng-Jian Lin , Lei Yang et al . Compact 16-channel integrated charge-sensitive preamplifier module for silicon strip detectors [J]. Nucl. Sci. Tech . 31 , 48 ( 2020 ). doi: 10.1007/s41365-020-00755-0 http://doi.org/10.1007/s41365-020-00755-0 Compact 16-channel integrated charge-sensitive preamplifier module for silicon strip detectors G. X. Zhang , G. L. Zhang , C. J. Lin et al . The calibration of elastic scattering angular distribution at low energies on HIRFL-RIBLL [J]. Nucl. Instr. Meth. Phys. Res. A 846 , 23 - 28 ( 2017 ). doi: 10.1016/j.nima.2016.11.058 http://doi.org/10.1016/j.nima.2016.11.058 The calibration of elastic scattering angular distribution at low energies on HIRFL-RIBLL O. B. Tarasovab , D. Bazina . LISE++: design your own spectrometer [J]. Nucl. Phys. A 746 , 411 - 414 ( 2004 ). doi: 10.1016/j.nuclphysa.2004.09.063 http://doi.org/10.1016/j.nuclphysa.2004.09.063 LISE++: design your own spectrometer J. Mocaka , A. M. Bond . Use of MATHEMATICA software for theoretical analysis of linear sweep voltammograms [J]. J. Electroanal. Chem . 561 , 191 - 202 ( 2004 ). doi: 10.1016/j.jelechem.2003.08.004 http://doi.org/10.1016/j.jelechem.2003.08.004 Use of MATHEMATICA software for theoretical analysis of linear sweep voltammograms M. H. Macfarlane and S. C. PIEPER . PTOLEMY: A program for heavy-ion direction-reaction calculation [R]. US : Argonne National Laboratory , 1978 . PTOLEMY: A program for heavy-ion direction-reaction calculation S. Santra , S. Kailas , K. Ramachandran et al . Reaction mechanisms involving weakly bound 6Li and 209Bi at energies near the Coulomb barrier [J]. Phys. Rev. C 83 , 034616 ( 2011 ). doi: 10.1103/PhysRevC.83.034616 http://doi.org/10.1103/PhysRevC.83.034616 Reaction mechanisms involving weakly bound 6Li and 209Bi at energies near the Coulomb barrier All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Nan-Ru Ma, Dong-Xi Wang and Yong-Jin Yao. The first draft of the manuscript was written by Yong-Jin Yao and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-021-00854-6 This work was supported by the National Science Foundation of China (Nos. 11635015, U1832130, and 11975040), the State Key Laboratory of Software Development Environment (SKLSDE-2020ZX-16), the Continuous Basic Scientific Research Project (No. WDJC-2019-13), and the Leading Innovation Project (Nos. LC192209000701 and LC202309000201). 2020-11-18 2020-12-13 2020-12-20 2021-02-15 2021-02-01 2021-03-12 © 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. 2021 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/s41365-021-00854-6 http://dx.doi.org/10.1007/s41365-021-00854-6 . 2021 Nuclear Science and Techniques 02 32