I. INTRODUCTION

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. [1-7] 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 100A–400A MeV ¹²C+¹²C 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 ¹²C+¹²C 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]. However, so far there are not many theoretical and experimental studies for heavy ions at higher energies than those of deuteron [8]. Therefore, the angular distribution of ¹²C on ¹²C elastic scattering needs to be experimentally measured in the incident energy range of 100A–400A 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

Firstly, the 100A MeV ¹²C+¹²C 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]. In order to make full use of the magnetic spectrometer, the beam was transported under the achromatic focusing along the WS beamline [10]. During the experiment we used a 1.181 mg/cm²-thickness natural carbon target and a 11.40 mg/cm²-thickness (CH₂)_n target. The TOF-ΔE signals, which were obtained from focal plane detectors, were used for particle identification. The details of focal plane detectors are described in Ref. [11]. After particle identification, the two dimensional plots of outgoing ¹²C 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. 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 ¹²C+¹²C scattering. However, another tilted band crossing with the horizontal bands exists in Fig. 1. It increases the number of ¹²C+¹²C scattering events, especially at small angles. The contribution of this part cannot be neglected. In comparison with those from the excited states of ¹²C on ¹²C 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 ¹²C on ¹²C scattering can be affected. In the paper, we introduce a method to subtract this disturbance and obtain the content on the experimental target.

Fig. 1.Full-screen View

(Color online) The two dimensional plot of outgoing ¹²C particles excitation energies with laboratory angles when the central angle of the magnetic spectrometer was set to 2.0 for 100A MeV ¹²C+¹²C scattering.

III. RESULTS AND DISCUSSION

Firstly, we compared the results of Fig. 1 on the natural target with that on the (CH₂)_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 ¹²C 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),

where Ka, KA, Kb, and KB denote the kinetic energies of the projectile, the target, the scattered particles, and the recoiled particles, respectively. Here, KA=0. According to the momentum conservation

where pa, pb, and pB, 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

where mi 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

where

Through Eq. (4), for the ground state of ¹²C+p and the excited states of ¹²C+¹²C at 100A MeV incident energy, it is shown that the first excited state 4.4 MeV ($2_1^{+}$) of ¹²C+¹²C scattering and the ground state of ¹²C+p cross at 1.015. The 9.64 MeV ($3_1^{-}$) state of ¹²C+¹²C and the ground state of ¹²C+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, 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

For easy selection of ¹²C+p events, we convert the energy spectrum of ¹²C+¹²C, shown in Fig. 1 to that of ¹²C+p system, which is shown in Fig. 2.

Fig. 2.Full-screen View

(Color online) The two dimensional plot of outgoing ¹²C particle excitation energies with laboratory angles of ¹²C+p elastic scattering at 100A MeV.

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):

$\begin{array}{ll}{Q}^{\prime }=0.0275{\theta }^4\hfill & +0.0333{\theta }^3+2.9658{\theta }^2+3.4918\theta \hfill \\ \hfill & -2.2786+Q\mathrm{,}\hfill \end{array}$ (6)

where Q’ and Q represent the outgoing ¹²C excitation energies of the ¹²C+p and ¹²C+¹²C systems, respectively. The uncrossed region of 1.7–2.2 is selected, as it is mainly caused by ¹²C+p elastic scattering. Fig. 3 shows the outgoing ¹²C 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.

Fig.Full-screen View

3. The selection of the excitation energy spectrum at 1.8 in the uncrossed region of Fig. 2 and the fitting results. The square points and the solid line denote the experimental data and the fitting results, respectively.

The differential cross sections of 96 MeV p+¹²C elastic scattering in Ref. [12] in the center of mass frame was converted to that of ¹²C+p elastic scattering in the laboratory frame with the incident energy 95.3A MeV as shown in 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. 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 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.

Fig. 4.Full-screen View

The angular distribution of 95.3A MeV ¹²C+p elastic scattering in laboratory system.

Fig. 5.Full-screen View

(Color online) The comparison between the normalized counts (circle) and real counts (square).

Fig. 6.Full-screen View

(Color online) The test of the subtraction of hydrogen disturbance at 1.8. The original data (square), the effect of ¹²C+p elastic scattering (circle) and the result after subtraction (triangle).

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 ¹²C+p elastic scattering. These deduced counts are acted as the Gaussian distribution in the excitation energy spectra of the ¹²C+¹²C system. The values of the center and the width of the fitted Gaussian distribution in Fig. 3 were transformed to those of ¹²C+¹²C system by using Eq. (6). So the values of the width and the center of the Gaussian distribution in the ¹²C+¹²C system are determined. The width is usually set as 3σ. The results of four examples are shown in Fig. 7. It’s obvious that the ¹²C+p elastic scattering contribution is large in comparison with that of ¹²C+¹²C in Fig. 7(c).

Fig. 7.Full-screen View

(Color online) The results of hydrogen subtraction at 1.0, 1.1, 1.4 and 1.5.

IV. CONCLUSION

In this paper, we used two-body kinematics to calculate the cross angles of the ground state of 95.3A MeV ¹²C+p elastic scattering and the excited states of 100A MeV ¹²C+¹²C, including the 4.4 MeV ($2_1^{+}$) state and the 9.64 MeV ($3_1^{-}$) 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 ¹²C 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)%. At the same time, the method used in present paper can help us improve the future experiment. The experiment using the (CH₂)_n target can calibrate some parameters, such as the efficiencies of the faraday cup, detectors, and so on. It provides a method for nuclear physics experiments.