CHIN.PHYS.LETT.
Vol. 25, No. 8 (2008) 2834
Surface Diffuseness Anomaly in
16
O+208 Pb Quasi-elastic Scattering at Backward Angle ∗
JIA Hui-Ming(_¬²)∗∗ , LIN Cheng-Jian(«
), ZHANG Huan-Qiao(Üz), LIU Zu-Hua(4yu), YANG Feng( ¸), JIA Fei(_), ZHANG Chun-Lei(ÜSX), AN Guang-Peng(S2*), WU Zhen-Dong(ÇÀ), XU Xin-Xing(M#(), BAI Chun-Lin(xS), YU Ning(w) China Institute of Atomic Energy, Beijing 102413
(Received 23 April 2008) The quasi-elastic scattering excitation function of the doubly magic 16 O+208 Pb system at a backward angle is measured at sub-barrier energies with high precision. The diffuseness parameters extracted from both the singlechannel and the coupled-channels calculations give almost the same value a = 0.76 ± 0.04 fm. The results show that the coupling effect is negligible for the spherical system. The obtained value is smaller than the extracted value from the fusion excitation function, but larger than the value of a = 0.63 fm, which is from the systematic analysis of elastic scattering data.
PACS: 25. 70. −z, 25. 70 .Bc, 24. 50. +g For heavy-ion reactions, the Woods–Saxon (WS) form of nuclear potential, which is characterized by the depth V0 , radius parameter r0 and diffuseness parameter a, has been widely used. Recently, the diffuseness anomaly extracted from the above-barrier fusion excitation functions with the WS form of the nuclear potential have been reported.[1−4] Newton et al.[3] found that the diffuseness parameters of the Woods-Saxon potential increase with the increasing reaction charge products ZP ZT by analysing 46 high precise fusion excitation functions above the barrier energies. For the systems of ZP ZT < 1600, the diffuseness parameters increase from 0.75 to 1.5 fm. The result indicates that the diffuseness parameters are obviously greater than the commonly accepted value of 0.63 fm from the systematic analysis of elastic scattering data.[5−8] Up to now, the origins responsible for the discrepancy in the surface diffuseness parameter between the scattering and fusion processes have not been found yet. In order to check the reasons of the anomaly, Hagino et al.[9] proposed to extract the diffuseness parameter from the quasi-elastic scattering excitation function theoretically. For a single channel problem, the ratio of the elastic scattering cross section σel to the Rutherford scattering cross sections σRu at the backward angle θ is given by[9−11] √ dσel VN (rc ) 2aπkη (Ec.m. , θ) ∼ 1 + , (1) dσRu ka Ec.m. at energies well below the Coulomb barrier, where the tunnelling probability exponentially decreases. Here √ Ec.m. is the centre-of-mass energy, k = 2µEc.m. /¯h2 the wave number, µ the reduced mass of the system,
and η is the Sommerfeld parameter. This formula is obtained with the semi-classical perturbation theory, which is assumed that the nuclear potential VN (rc ) is proportional to exp(−r/a) around√the distance of the closest approach, i.e. rc = (η 2 + η 2 + λ2c )/k, where λc = η cot(θ/2) is the classical angular momentum for the Rutherford scattering. The deviation of the ratio of the cross section from unity is therefore sensitive only to the surface property of nuclear potential and provides a relatively model independent way to study the anomalous effect of surface diffuseness parameter. The reflection coefficient R0 (Ec.m. ) for angular momentum l = 0 is given by the ratio of Eq. (1) at 180◦ . Due to the difficulty of detecting scattered particles at θlab = 180◦ , the detectors were setup at angles as close to 180◦ as possible. In order to compare the value at 180◦ , the energy of the former was reduced by the centrifugal energy Ecent [12] Ecent = 2Ec.m.
sin(θ/2) . 1 + sin(θ/2)
(2)
In this study we present precise excitation function for the quasi-elastic scattering of 16 O impinging on 208 Pb at a backward angle. Generally, the quasi-elastic cross section σqel is defined as the sum of the elastic scattering and all other peripheral reaction processes. Here σel can be considered as σqel to a good approximation at deep subbarrier energies. Up to now there have been some experimental results by means of this method.[9,13,16] Washiyama et al.[13] measured the quasi-elastic scattering excitation function of 16 O+208 Pb system at backward angles. The best fit to the experimental data was obtained with a diffuseness parameter
∗ Supported by the National Natural Science Foundation of China under Grant Nos 10575135, 10575134, 10675169 and 10735100, and the Major State Basic Research Programme of China under Grant No 2007CB815003. ∗∗ Email:
[email protected] c 2008 Chinese Physical Society and IOP Publishing Ltd °
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JIA Hui-Ming et al.
a = 0.59 fm using a single-channel model. However, the quasi-elastic scattering data are not suitable for this purpose due to the large uncertainty and scarce low energy data. So the available result from the data is not conclusive. However Gasques et al.[16] measured the quasi-elastic excitation function of 32 S+208 Pb with high precision, which was extended to deeper sub-barrier energies. The extracted diffuseness value is a = 0.75 fm. The experiment was carried out at the HI-13 tandem accelerator of the China Institute of Atomic Energy. A collimated 16 O beam with incident energies from 40.5 to 80.25 MeV bombarded on the target. The energies were varied in steps of 3 MeV at the very low energies and in steps of 0.68 MeV at the relatively high energies. The 208 Pb target in thickness 140 µg/cm2 with a diameter 5 mm was evaporated onto a 20 µg/cm2 carbon backing. The energies were increased monotonically in order to reduce the magnetic hysteresis. Some of the energies were measured repeatedly to make sure that the detector response did not degrade. Two collimators were mounted in the entrance and exit tubes about 110 cm apart from each other. The scattered particles were detected by four Si (Au) surface-barrier detectors located at 175◦ with respect to the beam direction. Four Si (Au) monitor detectors (upper and lower, left and right), located at the angle 41◦ with respect to the beam are used to detect the elastic scattering for normalization, as well as to check the beam quality and angle. The relative angles subtended by the semiconductors were determined using a 241 Am alpha source. The low energy data whose values are close to unity were used to get a better normalization.
Fig. 1. Quasi-elastic excitation functions for the 16 O+208 Pb system at 175◦ with the same diffuseness parameter 0.63 fm calculated by using a single-channel model. The numbers in the figure represent the depth V0 of the Woods-Saxon potential.
In general, the region of “deep sub-barrier energies” is defined as around 3 MeV below the lowest
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barrier height or smaller, where the collective inelastic excitations and transfer reactions do not strongly coupled to the relative motion. According to the CCFULL calculation,[12] the definition in this way corresponds to the region where the experimental value of the ratio of the quasi-elastic to the Rutherford cross sections is larger than around 0.94. We therefore include only those experimental data which satisfy dσqel /dσRu ≥ 0.94 in the χ2 fitting. The single channel and coupled-channels calculations were preformed using a nuclear potential of WS form. In the calculations imaginary potential with W = 30 MeV, aW = 0.4 fm, and rW = 1.0 fm represents the rather small absorption from barrier penetration. The values assumed for the imaginary part of the potential result in negligible strength in the surface region. As long as the imaginary part of the potential is confined inside the Coulomb barrier, the quasi-elastic scattering cross section predictions are insensitive to the variation of the parameters. At the surface region the effect of variation in V0 and r0 on the Coulomb barrier height compensates for each other, so we can determine surface diffuseness parameter unambiguously. The parameters of the WS real potential were searched for the best fit to the quasi-elastic data, with the important constraint that the expected average fusion barrier energy had to be reproduced. The uncertainty of the diffuseness parameter was evaluated from the value at which the total chi-squared increases from the minimum by the minimum chi-squared per degree of freedom. Displayed in Fig. 1 are the cross section ratio trends of 16 O+208 Pb system with different depths and the corresponding radius parameters, which are calculated by a single channel model with the same diffuseness parameter. The average barrier of 74.52 MeV were extracted from fusion excitation function.[17] The trends of the curves are very similar except the curve calculated with the depth of potential 10 MeV. As long as the parameters of the real part of the potential are in the reasonable range, the trends of curves are almost the same at the deep sub-barrier energies. In the following analysis keeping the depth parameter V0 fixed at 100 MeV, we vary r0 and a to reproduce the barrier energy. The experimental quasi-elastic excitation function for the 16 O+208 Pb system measured at 175◦ is shown in Fig. 2. The energies have been corrected for the target thickness and converted to the centre-of-mass system. The relative errors of all the cross sections are about 0.4%. The solid curve in the figure indicates the result of the best-fitting by using singlechannel nuclear potential with diffuseness parameter a = 0.76 ± 0.04 fm. This value is larger than the diffuseness parameter value reported in Ref. [13] but is in good agreement with the parameter for the 32 S+208 Pb
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JIA Hui-Ming et al.
reaction.[16]
Fig. 2. Ratio of the quasi-elastic scattering to the Rutherford cross section measured at θ = 175◦ for the 16 O+208 Pb reaction. The solid squares are the present data. The solid curve represents the results obtained from a single-channel analysis, whereas the dashed curve (almost coinciding with the solid line) corresponds to the coupled-channel calculation (CC). The diffuseness parameter was obtained by performing a least-square fit to the data.
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pected, couplings play a negligible role in quasi-elastic reactions involving spherical nuclei at deep sub-barrier energies, and their effects can be almost ignored. It turns out that changing the average barrier height by ±1 MeV essentially changes the best fitted diffuseness parameter by ±0.04 fm. If we change the normalization of the data by ±0.4% (the typical statistical error), the corresponding change of diffuseness parameter is ±0.03 fm. Figure 3 shows the effects of varying the diffuseness of the nuclear potential for the 16 O+208 Pb system. We can conclude from the figure that the diffuseness parameters remarkably affect the trends of the quasi-elastic excitation function. In summary, we have measured a precise quasielastic excitation function for the 16 O+208 Pb system at the deep sub-barrier energies and at a backward angle. Both single-channel and coupled-channels calculations fit the data well with the same diffuseness parameter a = 0.76 ± 0.04 fm for the spherical system. The value is smaller than the diffuseness parameter a = 1.11±0.02 fm extracted from fusion experiment.[3] Now the suggested reasons for the large value of a extracted from fusion reaction have been related to the departure from the Woods–Saxon potential at closer distance or the dynamical fusion process. Combining the elastic scattering, quasi-elastic scattering and fusion reactions, possible reason is that more dynamical processes participate in the reactions with the approaching of the two colliding nuclei. The larger surface diffuseness parameter may be partly an artifact of the dynamical potential. Therefore, it appears that further theoretical work is needed in order to reach an understanding of the underlying physical process.
References Fig. 3. The same as Fig. 2, but the solid curves represent single-channel analysis with different diffuseness parameters. The numbers in the figure represent the diffuseness parameter a of the Woods–Saxon potential.
The experimental quasi-elastic scattering excitation function was also compared with the coupledchannels calculations. The projectile 16 O has been treated as inert and couplings to 3− , 5− states in 208 Pb were included in the calculations. The coupling strengths were obtained from Ref. [18]. The 3− and 5− single-phonon states of 208 Pb have excitation energies ε2 = 2.615 and ε3 = 3.198 MeV, respectively. The diffuseness parameter giving the best fit to the experimental data is almost identical to the one obtained by using a single-channel potential. The result is indicated by the dashed curve in Fig. 2. As is ex-
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