A method of estimating and subtracting the hydrogen background in the natural carbon target used in the ¹²C + ¹²C experiment

A method of estimating and subtracting the hydrogen background in the natural carbon target used in the 12 C + 12 C experiment 1 2 first-author 1 2 corresp Corresponding author, zgl@buaa.edu.cn Corresponding author, zgl@buaa.edu.cn zgl@buaa.edu.cn 1 2 1 2 3 1 2 1 2 School of Physics and Nuclear Energy Engineering, Beihang University, Beijing100191, China School of Physics and Nuclear Energy Engineering, Beihang University, Beijing100191, China International Research Center for Nuclei and Particles in the Cosmos, Beihang University, Beijing100191, China International Research Center for Nuclei and Particles in the Cosmos, Beihang University, Beijing100191, China Research Center for Nuclear Physics, Osaka University, Osaka567-0047, Japan Research Center for Nuclear Physics, Osaka University, Osaka567-0047, Japan The experimental data of 100 A MeV 12 C + 12 C elastic scattering are checked by using two-body kinematic calculation and 12 C+p elastic scattering. It is shown that the measured data are true and reliable. In the paper, the transformation between the excited energy spectra of the 12 C+ 12 C system and the ground state energy spectra of the 12 C+p system is introduced. The method of subtraction of the hydrogen background in the natural carbon target used in the experiment is elaborately described and the results are discussed. It is indicated that this method of subtraction of hydrogen background is reasonable and can be used in the data analysis. Based on the elastic scattering cross section of the previous experiment of 12 C+p at 95.3 A MeV, the hydrogen content entered into the reaction is analyzed. The final hydrogen content in the natural carbon target is (2.73 ± 0.12)%. Elastic scattering Subtraction background Hydrogen content I. INTRODUCTION 1 The role of the three-body-force (TBF) in complex nuclear systems is one of the key issues not only in nuclear physics but also in nuclear astrophysics relevant to high-density nuclear matter, such as neutron stars and supernova explosions. In nuclear collisions, elastic scattering can provide important information on the nucleon-nucleon (NN) and nucleus-nucleus (AA) interaction. Recently in Ref. [ 1 1 - 7 7 1-7 ] a double folding model (DFM) with a new type of complex G-matrix interaction including the TBF, was developed. They applied it to calculate the optical potential of 100 A –400 A MeV 12 C+ 12 C elastic scattering. It was indicated that the real part of the optical potential changes its character from attraction to repulsion as the incident energy increases. The angular distribution of 12 C+ 12 C elastic scattering shows a different diffraction transition with and without the TBF effect. Through this evolution of optical potential and angular distribution of elastic scattering with an increase of incident energies, the contributions of TBF and tensor force can be extracted [ 5 5 ]. However, so far the re are not many theoretical and experimental studies for heavy ions at higher energies than those of deuteron [ 8 8 ]. Therefore, the angular distribution of 12 C on 12 C elastic scattering needs to be experimentally measured in the incident energy range of 100 A –400 A MeV. Through the precise measurement of elastic scattering, the repulsive nature of optical potential can be explained and the transition energy from attraction to repulsion of the real part in optical potential can be determined. It can provide important information about the TBF effect, the medium effect of high density nuclear matter, the energy dependence of TBF, and the role of tensor force. II. EXPERIMENTAL SECTION 1 Firstly, the 100 A MeV 12 C+ 12 C experiment was performed in Research Center for Nuclear Physics (RCNP) at Osaka University. The angular distribution was precisely obtained using the magnetic spectrometer, "Grand Raiden". This magnetic spectrometer has excellent ion-optical properties [ 9 9 ]. In order to make full use of the magnetic spectrometer, the beam was transported under the achromatic focusing along the WS beamline [ 10 10 ]. During the experiment we used a 1.181 mg/cm 2 -thickness natural carbon target and a 11.40 mg/cm 2 -thickness (CH 2 ) n target. The TOF- Δ E signals, which were obtained from focal plane detectors, were used for particle identification. The details of focal plane detectors are described in Ref. [ 11 11 ]. After particle identification, the two dimensional plots of outgoing 12 C particle excitation energies with laboratory angles were obtained when the central angle of the magnetic spectrometer was set to 2.0, as shown in Fig. 1 Fig. 1 . We can clearly observe three horizontal bands which correspond to the ground state, 4.44 MeV (2 + ) and 9.65 MeV (3 - ) excited state of 12 C+ 12 C scattering. However, another tilted band crossing with the horizontal bands exists in Fig. 1 Fig. 1 . It increases the number of 12 C+ 12 C scattering events, especially at small angles. The contribution of this part cannot be neglected. In comparison with those from the excited states of 12 C on 12 C scattering, it is much larger at the smaller angles. So if the contribution of the tilted band is not subtracted, the differential cross section of 12 C on 12 C scattering can be affected. In the paper, we introduce a method to subtract this disturbance and obtain the content on the experimental target. III. RESULTS AND DISCUSSION 1 Firstly, we compared the results of Fig. 1 Fig. 1 on the natural target with that on the (CH 2 ) n target. The results are similar. Moreover, there was no tilted band in other plots on the natural carbon target at larger angles. So it is concluded that the tilted band is from 12 C on hydrogen scattering. Second, we checked the experimental data on the basis of the relativistic kinematics calculation. For reaction A ( a , b ) B , a , A , b , and B represent the projectile, the target nucleus, the scattered particle, and the recoiled nucleus, respectively. According to the definition of Q (reaction energy), <math display="block" id="M1"><mrow><mi>Q</mi><mo>=</mo><msub><mi>K</mi><mi>B</mi></msub><mo>+</mo><msub><mi>K</mi><mi>b</mi></msub><mo>−</mo><msub><mi>K</mi><mi>a</mi></msub><mo>−</mo><msub><mi>K</mi><mi>A</mi></msub><mn>,</mn></mrow></math> where K a , K A , K b , and K B denote the kinetic energies of the projectile, the target, the scattered particles, and the recoiled particles, respectively. Here, K A =0. According to the momentum conservation <math display="block" id="M2"><mrow><msubsup><mi>p</mi><mi>B</mi><mn>2</mn></msubsup><mo>=</mo><msubsup><mi>p</mi><mi>a</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>p</mi><mi>b</mi><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>p</mi><mi>a</mi></msub><msub><mi>p</mi><mi>b</mi></msub><mi>cos</mi><mi>θ</mi><mn>,</mn></mrow></math> where p a , p b , and p B , respectively represent the momentum of the projectile, the scattered nucleus, and the recoiled nucleus. θ is the scattering angle in the laboratory frame. If the relativistic is considered, the relationship between momentum and kinetic energy can be obtained <math display="block" id="M3"><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>=</mo><msqrt><mrow><msubsup><mi>K</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>K</mi><mi>i</mi></msub><msub><mi>m</mi><mi>i</mi></msub></mrow></msqrt><mn>,</mn></mrow></math> where m i denotes the rest mass of nuclei. i = a , A , b , and B represent the projectile, the target nucleus, the scattered nucleus, and the recoiled nucleus, respectively. From Eqs. (1), (2), and (3), it can be obtained <math display="block" id="M4"><mrow><mi>Q</mi><mo>=</mo><msub><mi>K</mi><mi>b</mi></msub><mo>−</mo><msub><mi>K</mi><mi>a</mi></msub><mo>−</mo><msub><mi>m</mi><mi>B</mi></msub><mo>+</mo><msqrt><mrow><msubsup><mi>m</mi><mi>B</mi><mn>2</mn></msubsup><mo>+</mo><mi>Δ</mi></mrow></msqrt><mn>,</mn></mrow></math> where <math display="block" id="M5"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Δ</mi><mo>=</mo><msubsup><mi>K</mi><mi>a</mi><mn>2</mn></msubsup></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><mn>2</mn><msub><mi>K</mi><mi>a</mi></msub><msub><mi>m</mi><mi>a</mi></msub><mo>+</mo><msubsup><mi>K</mi><mi>b</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>K</mi><mi>b</mi></msub><msub><mi>m</mi><mi>b</mi></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn><msqrt><mrow><mo stretchy="false">(</mo><msubsup><mi>K</mi><mi>a</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>K</mi><mi>a</mi></msub><msub><mi>m</mi><mi>a</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>K</mi><mi>b</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>K</mi><mi>b</mi></msub><msub><mi>m</mi><mi>b</mi></msub><mo stretchy="false">)</mo></mrow></msqrt><mi>cos</mi><mi>θ</mi><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Through Eq. (4), for the ground state of 12 C+p and the excited states of 12 C+ 12 C at 100 A MeV incident energy, it is shown that the first excited state 4.4 MeV ( <math id="M6"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> ) of 12 C+ 12 C scattering and the ground state of 12 C+p cross at 1.015. The 9.64 MeV ( <math id="M7"><mrow><msubsup><mn>3</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> ) state of 12 C+ 12 C and the ground state of 12 C+p cross at 1.5. The influence of the beam uncertainty (including the beam angular and beam size shift, which is about 2%) can be neglected for the calculation of cross angles. From Fig. 1 Fig. 1 , we can clearly see that the crossing angles are about 1.0 and 1.5, which are consistent with the calculation. It proves the authenticity of the experimental data. A. Subtraction of hydrogen disturbance in target 2 For easy selection of 12 C+p events, we convert the energy spectrum of 12 C+ 12 C, shown in Fig. 1 Fig. 1 to that of 12 C+p system, which is shown in Fig. 2 Fig. 2 . During the conversion we used a polynomial function to rotate it. The rotation has to obey the rules. The excitation energy cannot change with the angles and the excitation energy of the ground state must stay zero. Finally we get the Eq. (6): <math display="block" id="M8"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>Q</mi><mo>′</mo></msup><mo>=</mo><mn>0.0275</mn><msup><mi>θ</mi><mn>4</mn></msup></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><mn>0.0333</mn><msup><mi>θ</mi><mn>3</mn></msup><mo>+</mo><mn>2.9658</mn><msup><mi>θ</mi><mn>2</mn></msup><mo>+</mo><mn>3.4918</mn><mi>θ</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2.2786</mn><mo>+</mo><mi>Q</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where Q’ and Q represent the outgoing 12 C excitation energies of the 12 C+p and 12 C+ 12 C systems, respectively. The uncrossed region of 1.7–2.2 is selected, as it is mainly caused by 12 C+p elastic scattering. Fig. 3 Fig. 3 shows the outgoing 12 C excitation energy spectrum at 1.8 and the fitting results. This method was applied to the other angles, we can obtain the real number of reactions with hydrogen and the width σ of the Gaussian distribution. The differential cross sections of 96 MeV p+ 12 C elastic scattering in Ref. [ 12 12 ] in the center of mass frame was converted to that of 12 C+p elastic scattering in the laboratory frame with the incident energy 95.3 A MeV as shown in Fig. 4 Fig. 4 . In order to obtain the relationship between the angles and the differential cross sections, the experimental data were fitted from 0.8 to 2.2. Then we used the fitting function to calculate the differential cross sections from 1.7 to 2.2, and obtained the ratios between the differential cross sections and the real counts at each angle. The least square method was used to optimize all the ratios. As a result, an optimized ratio, the normalization factor, was given. We then checked the reliability of the normalization factor through the comparison between the normalized counts and real counts, as shown in Fig. 5 Fig. 5 . It is obvious that all of values are in the range of error bars. So the obtained normalization factor is reasonable. We also checked the results of the uncrossed region of 1.7–2.2 after the subtraction of the background by using this normalization factor. Fig. 6 Fig. 6 shows the result at 1.8. It is shown that the peak can be subtracted within statistical fluctuation. After that, only the background is left. Therefore, this method for the subtraction of the hydrogen disturbance is reasonable. We applied this method for the crossed region of 0.9–1.6. The above normalization factor is used to calculate the real counts at the crossed regions of 0.9–1.6, along with the data from 12 C+p elastic scattering. These deduced counts are acted as the Gaussian distribution in the excitation energy spectra of the 12 C+ 12 C system. The values of the center and the width of the fitted Gaussian distribution in Fig. 3 Fig. 3 were transformed to those of 12 C+ 12 C system by using Eq. (6). So the values of the width and the center of the Gaussian distribution in the 12 C+ 12 C system are determined. The width is usually set as 3 σ . The results of four examples are shown in Fig. 7 Fig. 7 . It’s obvious that the 12 C+p elastic scattering contribution is large in comparison with that of 12 C+ 12 C in Fig. 7(c) Fig. 7(c) . B. Extraction of hydrogen content in the natural carbon target 2 The differential cross section is defined as <math display="block" id="M9"><mrow><mfrac><mrow><mtext>d</mtext><mi>σ</mi></mrow><mrow><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>N</mi><mo>/</mo><msub><mi>ε</mi><mi>d</mi></msub><msub><mi>ε</mi><mi>t</mi></msub></mrow><mrow><msub><mi>N</mi><mn>0</mn></msub><msub><mi>N</mi><mi>T</mi></msub><mtext>d</mtext><mi>Ω</mi></mrow></mfrac><mn>,</mn></mrow></math> where N , ε d , ε t , N 0 , N T , and d Ω denote the counts of reaction events, the detectors efficiencies, the trigger efficiency, the beam intensity, the target number per square centimeter, and the solid angle, respectively. Since we obtained the reacted counts from 12 C+p elastic scattering in the uncrossed region of the fitting in Fig. 3 Fig. 3 and the differential cross sections are given from Fig. 4 Fig. 4 , ε d , ε t , N 0 , and d Ω can be determined according to the experimental arrangement and measurement. Then we can obtain the hydrogen content for each angle. The results are listed in Table 1 Table 1 . Using the weighted mean method, the final hydrogen content in the natural carbon target is (2.73± 0.12)%. 1 1 Angles (deg.) 1 1 Real counts 1 1 H numbers (×10 18 /cm 2 ) 1 1 Relative errors (%) 1 1 1.7 1 1 598 1 1 1.58325 1 1 6.33 1 1 1.8 1 1 521 1 1 1.81728 1 1 6.94 1 1 1.9 1 1 354 1 1 1.64291 1 1 8.75 1 1 2.0 1 1 288 1 1 1.81152 1 1 12.18 1 1 2.1 1 1 159 1 1 1.34735 1 1 20.99 1 1 2.2 1 1 150 1 1 1.65899 1 1 20.76 IV. CONCLUSION 1 In this paper, we used two-body kinematics to calculate the cross angles of the ground state of 95.3 A MeV 12 C+p elastic scattering and the excited states of 100 A MeV 12 C+ 12 C, including the 4.4 MeV ( <math id="M10"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup></mrow></math> ) state and the 9.64 MeV ( <math id="M11"><mrow><msubsup><mn>3</mn><mn>1</mn><mo>−</mo></msubsup></mrow></math> ) state. By using this method, the experimental angles are calibrated and the experimental data are checked. Next, the method and the results of the subtraction of hydrogen contaminant are introduced and discussed, respectively. The elastic scattering of 12 C on hydrogen background from the experimental target has a large contribution at some angles, so in the data analysis the contribution of this part needs to be subtracted. Finally, the hydrogen content in the natural carbon target was extracted. The result is (2.73± 0.12)%. 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