Total Nuclear Reaction Cross Section Induced by Halo Nuclei and Stable Nuclei

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1 Commun. Theor. Phys. (Beijing, China) 40 (2003) pp c International Academic Publishers Vol. 40, No. 5, November 15, 2003 Total Nuclear Reaction Cross Section Induced by Halo Nuclei and Stable Nuclei GUO Wen-Jun, 1,2 JIANG Huan-Qing, 1,3 LIU Jian-Ye, 1,2, ZUO Wei, 1,2 REN Zhong-Zhou, 1,4 and LEE Xi-Guo 1,2 1 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou , China 2 Institute of Modern Physics, the Chinese Academy of Sciences, Lanzhou , China 3 Institute of High Energy Physics, the Chinese Academy of Sciences, Beijing , China 4 Physics Department, Nanjing University, Nanjing , China (Received March 21, 2003) Abstract We develop a method for calculation of the total reaction cross sections induced by the halo nuclei and stable nuclei. This approach is based on the Glauber theory, which is valid for nuclear reactions at high energies. It is extended for nuclear reactions at low energies and intermediate energies by including both the quantum correction and Coulomb correction under the assumption of the effective nuclear density distribution. The calculated results of the total reaction cross section induced by stable nuclei agree well with 30 experimental data within 10 percent accuracy.the comparison between the numerical results and 20 experimental data for the total nuclear reaction cross section induced by the neutron halo nuclei and the proton halo nuclei indicates a satisfactory agreement after considering the halo structure of these nuclei, which implies quite different mean fields for the nuclear reactions induced by halo nuclei and stable nuclei. The halo nucleon distributions and the root-mean-square radii of these nuclei can be extracted from the above comparison based on the improved Glauber model, which indicates clearly the halo structures of these nuclei. Especially, it is clear to see that the medium correction of the nucleon-nucleon collision has little effect on the total reaction cross sections induced by the halo nuclei due to the very weak binding and the very extended density distribution. PACS numbers: H, D Key words: total reaction cross section, improved Glauber theory, halo nuclei 1 Introduction The investigation of the structure of nuclei far from the β stability line is one of the most interesting subjects in nuclear physics. The use of the radioactive nuclear beam (RNB) offers a good opportunity to study the properties of nuclei far from the β stability line. Many new progresses have been made in recent years. The discovery of halo phenomena in exotic nuclei is one of the typical examples. In general, the halo nucleus has large neutron excess or proton excess where a few outside nucleons are very weakly bound. Such halo systems are well described by the fewbody models, which assumes that halo nuclei consist of a core and a few outside nucleons. [1 3] The small separation energy of the last nucleon, the extended density distribution, and the narrow momentum distribution due to the Pauli exclusion are important properties of halo nucleus. In this case the nuclear structure models based on the experimental data for stable nuclei should be improved and new theoretical models to describe the special structure of nuclei far from the β stability line are required. To probe the density distribution and structure properties of nuclei near the drip line, one method is to measure the total reaction cross sections induced by the radioactive nuclear beam on stable nuclei. The relations between the total reaction cross section and the nuclear density distribution are studied by the Glauber theory, Boltzmann Uehling Uhlenbeck (BUU) model, and the optical model. [4 9] Especially it is also very interesting to investigate the roles of the dynamical ingredients such as the mean field and two-body collisions in the dynamical collision process induced by the halo nucleus and stable nucleus. The Glauber model is based on the multiple scattering theory under the high-energy approximation conditions. [10,11] It was developed under the semi-classical approximation and is successful in describing the nucleonnucleus collisions at high energies. [10] However, to generalize the Glauber model to the nucleus-nucleus collisions in the low and intermediate energy regions one meets some difficulties. First of all, one has to justify the validity of the approximation conditions at low and intermediate energies. Secondly, to describe the nucleus-nucleus collision from the Glauber theory is lengthy and one has to find a simple and easy way for the calculation of the total reaction cross sections. One effort toward this direction has been done by Al-Khalili and Tostevin in the few-body The project supported by the State Key Basic Research Development Program of China under Contract No. G , the 100- Person Project of the Chinese Academy of Sciences, National Natural Science Foundation of China under Grant Nos , , and , and CAS Knowledge Innovation Project under Grant No. KJCX2-SW-N02 Corresponding author, liujy@ns.lzb.ac.cn

2 578 GUO Wen-Jun, JIANG Huan-Qing, LIU Jian-Ye, ZUO Wei, REN Zhong-Zhou, and LEE Xi-Guo Vol. 40 model and has received success. [12] Here we attack this problem in a different way. 2 Improved Glauber Theory In this paper we extend the Glauber theory for calculating the total reaction cross section of the nucleusnucleus collision at low and intermediate energies. The quantum correction and Coulomb correction are included in our calculations. The effective density distribution of the projectile nucleus and the effective reaction cross sections of the incident nucleon on the target nucleus are used. By using this improved Glauber theory we study the special structure of halo nuclei. The calculated results for the total reaction cross sections induced by stable nuclei agree with the experimental data satisfactorily. From the comparison between the theoretical results and the experimental data induced by the neutron halo nuclei 11 Li, 11 Be, 14 Be, 6 He, 8 He, and the proton halo nuclei 23 Al and 27 P, the root-mean-square (RMS) radii and the density distributions of these nuclei are extracted. It is found that the RMS radii are abnormally larger and the neutron or proton distribution tails are much longer than those of the neighboring nuclei. These properties indicate the special halo structure of the nuclei studied. The quantum correction and Coulomb correction are salient for the nuclear reactions induced by the stable nuclei in the low energy region, but for the nuclear reaction process induced by the halo nuclei in the same energy region except the quantum correction and Coulomb correction remain important, the in-medium effect of N-N collision on the total reaction cross section is not salient. We consider a nucleon with incident laboratory energy E 0 colliding with a nucleus. According to the Glauber theory [10,11] the total reaction cross section can be written as σ RT (E 0 ) = d 2 b(1 e σt (b) ). (1) Here T (b) is the thickness function of the target with density distribution ρ t (r), T (b) = + ρ t (b, z)dz = + ρ t ( b 2 + z 2 )dz, (2) and σ is the isospin-averaged N-N cross section. Equation (1) is a very good approximate formula for calculating the total reaction cross sections of the nucleon-nucleus at high energies. However, one has to improve Eq. (1) for describing the nucleon-nucleus scattering in the low and intermediate energy regions. First of all, the importance of the Coulomb effect increases with the incident energy decreasing. Secondly, because the expression for the total reaction cross section of nucleon-nucleus collisions based on the Glauber theory is semi-classical, [10,11] the position of the nucleon is undefined within a wavelength 1/k. At lower energies the wavelength of the nucleon becomes longer and therefore the quantum correction to the Glauber model calculation for the nucleon-nucleus cross section has to be considered. Finally due to the medium correction to the N-N cross section in free space the free N-N cross section σ has to be replaced by in-medium N-N cross section σ NN. In Ref. [13] H.C. Chiang and J. Hufner suggested a pragmatic approach for improving the Glauber model calculation of the total reaction cross section of the nucleon-nucleus collision at low energies as { } σ RT (E 0 ) = d 2 b(1 e σ NN T (b) ) {( ) 2 ( 1 V c(r t ) )}. (3) k 0 R t E 0 Here k0 2 = 2ME 0, and E 0 and M are the incident laboratory energy of the projectile nucleon and its mass, respectively. V c (R t ) is the Coulomb interaction energy between the proton and the target at the target radius R t. Calculations of the total reaction cross sections of nucleons on nuclei below 100 MeV agree with the experimental data satisfactorily. [13,14] Now we extend the above calculations to the nucleus-nucleus collision where the incident nuclei can be halo nuclei. In this case the projectile cannot be considered as a point particle. As a matter of fact, the dynamical process and outcome of the nucleus-nucleus collisions are based on the coherent sum of the dynamical behavior of all nucleon-nucleon collisions in the colliding system. To study the dynamical process of the nucleusnucleus collisions and calculate its total reaction cross section in low and intermediate energy regions we also have to trace the dynamical behavior of nucleon-nucleon collisions. In this case, it is similar to the nucleon-nucleus collisions and the quantum correction for such collisions has to be taken into account. Therefore, we postulate that the total reaction cross section of the nucleus-nucleus collisions with the quantum correction and the Coulomb correction can be expressed as { } σ R (E 0 ) = d 2 b(1 e σ RT T eff (b) ) {( ) 2 ( V (R) )} 1. (4) kr (A t + A p )E Here σ RT is an effective nucleon-nucleus reaction cross section calculated from Eq. (3) and R = r 0 (Ap 1/3 + A 1/3 t ) is the interaction radius of the projectile and target. E = 2mk 2 and V c (R)/(A t + A p ) are the incident nucleon laboratory energy and the Coulomb energy per nucleon at R. T eff (b) = + ρ eff(b, z)dz is an effective thickness function, which is calculated from the effective density

3 No. 5 Total Nuclear Reaction Cross Section Induced by Halo Nuclei and Stable Nuclei 579 distribution for the projectile, ρp (R r)ρ t (r)dr ρ eff (R) = ρt (r)dr with ρ p (R r) and ρ t (r) being the density of the projectile and the target, respectively. Here we have introduced the effective density distribution ρ eff, which is a folding of the projectile density distribution with target density distribution. At the same time the in-medium N-N cross section σ NN is replaced by the nucleon-target reaction cross section σ RT. So we calculate the total reaction cross sections of nucleus-nucleus collisions in two steps. First we use Eq. (3) to calculate the total reaction cross sections of nucleon-target collisions to get σ RT and then input the results Eq. (4) to calculate the total reaction cross section of nucleus-nucleus collisions. However, we know that the free neutron-proton cross-section σ np is about three times larger than that of the neutron-neutron (proton-proton) cross section σ nn (σ pp ) below about 400 MeV. [15,16] So the isospin effects of N-N collision should be considered in the low-medium energy region. In this case, σ RT T eff (b) in Eq. (4) is replaced by D(b), + [ D(b) = σ Rtn (E 0 )ρ eff n (b, z) (5) ] + σ Rtp (E 0 )ρ eff p (b, z) dz, (6) where σ Rtn and σ Rtp are the reaction cross sections of the neutron and proton from the projectile on the target, respectively, and ρ eff n and ρ eff p are the effective neutron and proton densities of the projectile, respectively. The total reaction cross section of a neutron with momentum k 0 in the projectile on the target is { }{( σ Rtn (E 0 ) = d 2 b(1 e Dn(b) ) ) 2 } (7) k 0 R t with D n (b) = + [σ nnρ tn (b, z) + σ np ρ tp (b, z)]dz. Similarly, we can write the total reaction cross section of a proton with momentum k 0 in the projectile { } σ Rtp (E 0 ) = d 2 b(1 e Dp(b) ) {( ) 2 ( 1 V c(r t ) )}, (8) k 0 R t E 0 where D p (b) = + [σ ppρ tp (b, z) + σ np ρ tn (b, z)]dz and V c = Z t Z p e 2 /R t with Z t and Z p being the numbers of protons in the target and projectile. The effective density distributions of neutrons and protons in the projectile are defined as ρnn (R r)ρ t (r)dr ρ eff n (R) =, ρt (r)dr ρpp (R r)ρ t (r)dr ρ eff p (R) =. (9) ρt (r)dr Here ρ tn, ρ tp, ρ pn, and ρ pp are the density distributions of neutrons and protons in the target and projectile, respectively. σ np, σ nn, and σ pp are the in-medium neutronproton, neutron-neutron, and proton-proton cross sections, respectively. 3 Numerical Results and Discussions We use the above method to calculate the total reaction cross sections of the radioactive nuclear beam on stable nuclei and to extract the neutron or proton distributions of the beam nuclei. To be sure that our approach is valid in the energy region studied, we first calculate the total reaction cross sections of the stable nuclei, where the density distributions of both the target and projectile are known. The density distributions of neutrons and protons in nuclei are calculated by the Relativistic Mean Field (RMF) model with parameter set NL2 [17] and compared with those extracted from the electron scattering data. The calculated results show that the total reaction cross sections by adopting both density distributions from the RMF theory and the experimental data are very close for stable nuclei. In our calculation the parametrized formula of inmedium nucleon-nucleon cross sections σ np, σ nn, and σ pp are taken from Ref. [18], σ nn = ( β β β 4 ) E0.06 lab ρ ρ 1.46, (10) σ np = ( β β β 4 ) E0.04 lab ρ ρ 1.90, (11) where β = /γ 2, γ = E lab / , σ is in mb, E in MeV, and ρ in fm 3. The calculated total reaction cross sections for stable nuclei are listed in Table 1 together with the experimental data σ exp. In Table 1, σ gl, σ med, and σ free indicate the results from the Glauber theory without any corrections, with full corrections, and with full corrections but the medium effect for the N-N cross sections, respectively. The corresponding relative differences are defined as D med = σ med σ exp σ exp, D free = σ free σ exp σ exp. (12) From Table 1, one can see that without any adjustable parameters our calculated results agree with the experimental σ exp within the accuracy of 10 are better than previous calculated results. We are confident that the improved Glauber theory is reliable and can be applied for calculating the total reaction cross sections at low and

4 580 GUO Wen-Jun, JIANG Huan-Qing, LIU Jian-Ye, ZUO Wei, REN Zhong-Zhou, and LEE Xi-Guo Vol. 40 intermediate energies by adopting the effective nuclear densities and the effective reaction cross section of the nucleon-nucleus σ RT replacing the NN cross section σ NN. From the calculation process one can see obviously that the quantum correction enhances the total reaction cross section in low energy region while the Coulomb correction reduces them. Table 1 The calculation results and experimental data for the reactions induced by stable nuclei. Reaction system E 0 σ gl σ med D med σ ree D free σ exp (NL2) (MeV/u) (mb) (mb) (%) (mb) (%) (mb) 12 C + 12 C ± C + 12 C ± C + 12 C ± C + 12 C ± C + 12 C ± N + 12 C ± N + 12 C ± N + 12 C ± N + 12 C ± O + 12 C ± O + 12 C ± O + 12 C ± C + 12 C ± C + 12 C ± C + 12 C ± C + 12 C ± 60 9 C + 12 C ± C + 12 C ± C + 12 C ± Be + 12 C ± N + 12 C ± O + 12 C ± O + 12 C ± O + 12 C ± B + 12 C ± Al + 12 C ± Al + 12 C ± Al + 12 C ± Al + 12 C ± Al + 12 C ± 121 The experimental data taken from Refs. [5], [19] [21] Now we calculate the nucleus-nucleus reaction cross sections induced by the neutron halo nuclei 11 Li, 11 Be, 14 Be, 6 He, 8 He, and the proton halo nuclei 23 Al and 27 P. There is few information about the density distributions of these nuclei. We follow the suggestion from Ref. [22] for the structures of the halo nuclei and separate the density distribution of the nucleus into a core density ρ c (r) and a halo distribution of neutrons or protons on the single particle orbit, ρ nl (r). Namely, ρ(r) = ρ c (r) + ρ nl (r). We make use of the RMF code with the parameters NL2 [17] to calculate the density distribution of the core ρ c (r) and the halo neutron or proton distribution ρ nl (r). For example, based on the lowest energy of colliding nucleus, one separates 11 Li into a core of 9 Li and two halo neutrons in the 1p 1/2 orbit. We assume that 11 Be contains a core of 10 Be and one neutron in the 1p 1/2 orbit and 14 Be is made

5 No. 5 Total Nuclear Reaction Cross Section Induced by Halo Nuclei and Stable Nuclei 581 of a core of 12 Be and two halo neutrons in the 1d 5/2 orbit. For the proton halo nuclei 23 Al it contains a core of 22 Mg and one halo proton in the 1d 5/2 orbit and 27 P is made of a core 26 Si and one halo proton in 2s 1/2 orbit. In Table 2 we show a comparison between the calculated reaction cross sections and experimental data induced by the halo nuclei. From Table 2 one can see that the numerical results agree with the experimental data for the neutron halo nuclei 11 Li, 11 Be, 14 Be, 6 He, 8 He, and the proton halo nuclei 23 Al and 27 P reasonably well after taking into account the halo structure of these nuclei. This implies that the mean fields are quite different for the reactions induced by the halo nuclei and the stable nuclei. One knows that the large contribution to the total reaction cross section comes from the surface region of the nucleus. For the halo nuclei the density distributions are quite extended and the densities in the surface region are quite dilute. In our calculations for the neutron nuclei 11 Li, 11 Be, 14 Be, 6 He, 8 He, and the proton halo nuclei 23 Al and 27 P we use both the free and in-medium N-N cross sections Eqs. (10) and (11) as input, respectively. From the comparison between D free and D med in the low energy region in Table 2, one can see that the agreement between the calculation results with the free N-N cross sections and the experimental data is better than those from the in-medium N N cross section due to quite extended and quite dilute densities for the halo nuclei. Since the incident energies of 6 He and 8 He beams are of big uncertainties, it is difficult to compare the calculated results with the experimental data directly. Table 2 The calculation results and the experimental data for the reactions induced by halo nuclei. Reaction system E 0 σ exp σ free D free σ med D med (NL2) (MeV/u) (mb) (mb) (%) (mb) (%) 14 Be+ 12 C ± Be+ 12 C ± Li+ 12 C ± Li+ 12 C ± Be+ 12 C ± Be+ 12 C ± Al+ 12 C ± P+ 12 C ± Li+ 12 C ± Li+ 12 C ± Li+ 12 C ± He+ 12 C ± He+ 12 C ± He+ 12 C ± He+ 12 C ± He+ 12 C ± He+ 12 C ± He+ 12 C ± He+ 12 C ± He+ 12 C ± The experimental data are taken from Refs. [21] and [23] [26] To see the importance of the isospin effect we calculate the total reaction cross section by using the isospin independent in-medium NN cross section for the 11 Be beam on the 12 C target at E = 33 MeV/nucleon and compare with the results by adopting the isospin dependent in-medium N N cross section. The isospin effect of in-medium N N cross sections enhances the total reaction cross section by about five percent. From the above comparison between the calculated results and experimental data we can find that without any adjustable parameter the experimental cross sections for halo nuclei are reproduced reasonably well by the present calculations using the density distributions from RMF model even though most of the calculated results are a little underestimated.

6 582 GUO Wen-Jun, JIANG Huan-Qing, LIU Jian-Ye, ZUO Wei, REN Zhong-Zhou, and LEE Xi-Guo Vol. 40 Using the density distribution from RMF model we can calculate the root mean square (RMS) radii of nuclei. The formula of the RMS radius is [ R rms = r 2 1/2 r 2 ρ(r)dr ] 1/2 =, (13) ρ(r)dr where ρ(r) = ρ c (r)+ρ nl (r). However, with assignment of the orbits of the halo nucleons the calculated RMS radii for neutron halo nuclei 11 Li, 11 Be, and 14 Be are not in agreement with that extracted from the experimental data [3] (see Table 3). We think that there are two reasons for this discrepancy. First the extraction of the RMS radii from the total reaction cross section is model-dependent. Secondly, here we use the RMF model without the pair correction, deformation effect, or any adjusted parameters to calculate the density distributions. Especially the RMF model has to be justified for the calculation of distributions of halo nucleons. In order to obtain some information for the special halo structure of these nuclei we perform a calculation for the total reaction cross section from the improved Glaber theory using a Yukawa form for the halo nucleon density distribution, ρ nl (r) = ρ 0 (α/ r) exp( r/α) with an adjusted parameter α. Here ρ nl (r) is normalized to the number of halo nucleons for determining ρ 0. The densities of the core nuclei ρ c (r) are calculated from the RMF model. The total density is ρ(r) = ρ c (r) + ρ nl (r). By adjusting α to fit experimental cross sections, we extract the density distribution. The extracted RMS radii of neutron halo nuclei 11 Li, 11 Be, 14 Be, 6 He, 8 He, and the proton halo nuclei 23 Al and 27 P are listed in Table 3. From Table 3 one can find that our calculated results for the RMS radii of neutron, proton, and total nuclear density distributions are in agreement with that previously extracted from the experimental data. Table 3 The RMS radii of neutron, proton, and total nuclear density distribution. EXP are results from Refs. [21] and [23]. RMF are calculations from RMF model. Yukawa are results from fitting the experimental reaction cross sections using a Yukawa form for the densities of halo nucleons. R n R p R t EXP ± ± ± Al Yukawa RMF EXP ± ± ± P Yukawa RMF EXP 3.53 ± Li Yukawa RMF EXP 2.90 ± Be Yukawa RMF EXP 3.20 ± Be Yukawa RMF The experimental data are taken from Refs. [21] and [23] [26] Fig. 1 The proton, neutron and total density distributions of 14 Be, 11 Li, 11 Be, 6 He, 8 He, 23 Al, and 27 P. The dotted line, dashed line, and solid line show the proton, neutron, and total density distributions, respectively.

7 No. 5 Total Nuclear Reaction Cross Section Induced by Halo Nuclei and Stable Nuclei 583 In Fig. 1 we plot the density distributions of 14 Be, 11 Li, 11 Be, 6 He, 8 He, 23 Al, and 27 P extracted from above comparisons. One finds clearly that the neutron distributions of neutron halo nuclei and those of proton halo nuclei are far extended compared with their proton distribution and neutron distributions. It indicates the neutron halo structure for 14 Be, 11 Li, 11 Be, 6 He, 8 He, and the proton halo structure for 23 Al and 27 P. Figure 2 shows the RMS radii extracted from the above comparisons as a function of the neutron number for the Li, Be, He, C, Al, and P isotopes. One can finds that the RMS radii of 14 Be, 11 Li, 11 Be, 23 Al, and 27 P are much larger than that of the neighborhood nuclei. This demonstrates again the neutron halo structure for 14 Be, 11 Li, 11 Be and the proton halo structure for 23 Al and 27 P. Fig. 2 The RMS radii for the Li, Be, He, C, Al, and P isotopes. 4 Summary Based on the improved Glauber model of the nucleusnucleus collisions at low and intermediate energies we calculate the total reaction cross section and get a satisfactory agreement between the calculated results and the experimental data in the low and intermediate energy region. The Coulomb correction, and especially quantum correction, are salient in the low energy region. However, for the nuclear reactions induced by halo nuclei 11 Li, 11 Be, 14 Be, 8 He, 6 He, 23 Al, and 27 P, it is found that besides the quantum correction and Coulomb correction, which remain important, one can get a better agreement between the calculation results and experimental data only after considering the halo structure of the exotic nuclei, which implies quite different mean fields for the reactions induced by the halo nuclei and the stable nuclei. The in-medium effect of the N-N collision on the total reaction cross section is not important in the low energy region due to the special halo structure, especially for the nuclear reactions induced by the halo nucleus. This is quite different from the reactions induced by the stable nuclei. From the above discussions it is clear to see that there are quite different roles of the dynamical ingredients, including one-body dissipation and two-body collision for the nuclear reactions induced by the halo nuclei and stable nuclei. The comparisons between the density distributions and RMS radii of the halo nuclei from the calculation by using the improved Glauber model and the experimental data indicate the special neutron halo structure for 11 Li, 11 Be, and 14 Be, and proton halo structure for 23 Al and 27 P. Acknowledgments We would like to thank Profs. Zhong-Yu Ma and Bao- Qiu Chen for their helpful discussions on the work and collaboration in the RMF code. One of the authors (Guo Wen-Jun) thanks Prof. Ma Wei-Xing and Dr. Peng Guang-Xiong for their instruction at IHEP. References [1] Y. Tosaka and Y. Suzuki, Nucl. Phys. A512 (1990) 46. [2] G.F. Bertsch and H. Esbensen, Ann. Phys. 209 (1991) 327. [3] D.Z. Ding, Y.S. Chen, and H.Q. Zhang, Prog. Nucl. Phys., Shang Hai Scientific & Technical Publishers (1997) pp [4] Y. Ogawa, K. Yabana, and Y. Suzuki, Nucl. Phys. A543 (1992) 722. [5] D.Q. Fang, W.Q. Shen, et al., Chin. Phys. Lett. 17 (2000) 655. [6] M. Zahar, M. Belbot, et al., Phys. Rev. C49 (1994) [7] Y.G. Ma, et al., Phys. Rev. C48 (1993) 850; Phys. Lett. B302 (1993) 386. [8] Cai Xiang-Zhou, Shen Wen-Qing, et al., Chin. Phys. Lett. 17 (2000) 565. [9] Y.L. Zhao, Z.Y. Ma, B.Q. Chen, et al., High Energy Phys. Nucl. Phys. 26 (2001) 506. [10] R.J. Glauber, et al., Phys. Rev. 100 (1955) 252; Lectures in Theoretical Physics, Boulder, Calorado (1958); Nucl. Phys. B21 (1970) 135. [11] J.M. Blatt and V.F. WeissKopf, Theoretical Nuclear Physics, Wiley, New York (1963). [12] J.S. Al-Khalili, J.A. Tostevin, et al., Phys. Rev. Lett. 76 (1996) 3903; Phys. Rev. C54 (1996) [13] H.C. Chiang and J. Hufner, Nucl. Phys. A349 (1980) 466. [14] M.H. Simbel, J. Hufner, and H.C. Chiang, Phys. Lett. B94 (1980) 11.

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