Intermediate Energy Pion- 20 Ne Elastic Scattering in the α+ 16 O Model of 20 Ne

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1 Commun. Theor. Phys. 60 (2013) Vol. 60, No. 5, November 15, 2013 Intermediate Energy Pion- 20 Ne Elastic Scattering in the α+ 16 O Model of 20 Ne YANG Yong-Xu ( ), 1, Ong Piet-Tjing, 1 and LI Qing-Run (Óè ) 2 1 Department of Physics, Guangxi Normal University, Guilin , China 2 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing , China (Received April 7, 2013; revised manuscript received June 26, 2013) Abstract We present an analysis of π- 20 Ne elastic scattering at intermediate energy basing on the α+ 16 O model of the 20 Ne nucleus and in the framework of Glauber multiple scattering theory. Satisfactory agreement with the general features of the experimental data of pion elastic scattering on the neighboring 4N-type nuclei is obtained without any free parameters. Compared with the experimental angular distributions of pion elastic scattering on 12 C, 16 O, 24 Mg, and 28 Si nuclei, the diffractive patterns and the positions of the dips and peaks in the angular distributions of π- 20 Ne elastic scattering are reasonably predicted by the calculations. PACS numbers: Dj, Gx Key words: pion-nucleus elastic scattering, nuclear cluster model 1 Introduction The pion-nucleus elastic scattering has been studied for a long time. Considerable experimental and theoretical work on pion-nucleus elastic scattering has been done for various target nuclei. Among these, the pion elastic scattering on light 4N-type nuclei is an interesting subject, for which there have been a large amount of the experimental data. In Refs. [1 3], the experimental differential cross sections of π- 12 C elastic scattering in the energy range 60 to 280 MeV were presented. In Refs. [4 5], angular distributions for π- 16 O elastic scattering in the energy range 80 to 340 MeV were measured. For π- 24 Mg and π- 28 Si elastic scattering, there were the measured differential cross sections at 180 MeV and some other incident energies. [6 9] These data have been extensively analyzed from various approaches using diverse theoretical frameworks, and most of the theoretical works were based on the elementary π-n interaction and the target ground state nucleon density as the input information. From the point of view of the nuclear cluster structure, some light 4N-type nuclei such as 12 C, 16 O and 20 Ne are typical α structure nuclei. For 12 C and 16 O targets, by means of the intermediate energy pion scattering, the α structure models of 12 C and 16 O have been examined successfully. [10 12] In this work we present an analysis of π- 20 Ne elastic scattering at intermediate energy basing on the α+ 16 O model of 20 Ne. The α+ 16 O structure for 20 Ne has been studied for many years, and a recent study can be found in Ref. [13]. In our previous article, [14] we have proposed an α+ 16 O model of the 20 Ne nucleus. This model can very well reproduce the experimental charge form factor of 20 Ne and the elastic proton- 20 Ne scattering differential cross sections. [14] In the present work, we apply the α+ 16 O model of the 20 Ne nucleus to the analysis of π- 20 Ne elastic scattering to get further examination of this model. 2 Formulation The Glauber multiple scattering theory [15 16] has been considered as one of the most successful tools for analyzing hadron-nucleus elastic scattering at medium and high energies. In this work, we adopt the Glauber scattering theory for the description of π- 20 Ne elastic scattering. In the framework of the Glauber multiple scattering theory and based on the α+ 16 O model for 20 Ne, the π- 20 Ne scattering can be regarded as a multiple scattering process from the α particle and the 16 O nucleus as the scatterers. Then the π- 20 Ne elastic scattering amplitude can be written as F πa (q) = ik πa d 2 b e iq b Γ 00 (b), (1) 2π Γ 00 (b) = 0 1 [1 Γ πα (b + 4 )] 5 s [1 Γ πo (b 1 )] 5 s 0, (2) where q is the momentum transfer vector, b is the impact parameter, k πa is the momentum of the incident particle in the center-of-mass system, and s is the relative coordinate between the centers of mass of α and 16 O on the plane perpendicular to the incident direction. Γ πα and Γ πo are the π-α and π- 16 O profile functions respectively, which are related to the π-α amplitude f πα (q) and the Supported by National Natural Science Foundation of China under Grant No yyxu@mailbox.gxnu.edu.cn c 2013 Chinese Physical Society and IOP Publishing Ltd

2 No. 5 Communications in Theoretical Physics 589 π- 16 O amplitude f πo (q) by 1 Γ πα (b) = d 2 q e iq b f πα (q ), (3) 2πik πα 1 Γ πo (b) = d 2 q e iq b f πo (q ). (4) 2πik πo The wave function of the ground state of 20 Ne, in the framework of the α+ 16 O cluster structure model in our previous article, [14] has been given as φ 0 ( 20 Ne) = ϕ 0 (α)ψ 0 ( 16 O)χ 0 (r), (5) where ϕ 0 (α) and ψ 0 ( 16 O) are the wave functions of the ground state for the α particle and the 16 O nucleus respectively. The relative motion wave function can be written as χ 0 (r) = R 0 (r)y 00 (θ, φ), (6) where R 0 (r) is the radial wave function, and in Ref. [14] it has been given as R 0 (r) = sin β 2 O 10(r) + cos β 2 O 20(r). (7) Here O 10 (r) and O 20 (r) are the harmonic oscillator radial functions with the quantum numbers 1s and 2s respectively, that is O 10 (r) = 2(a 6 π) 1/4 e r2 /2a 2, (8) 8 O 20 (r) = 3 (a6 π) 1/4[ 3 ( r ) 2 ] 2 e r2 /2a 2. (9) a By fitting the experimental charge form factor of 20 Ne, the parameters a = 1.96 fm and β = have been given. This wave function can reasonably reduce the rms radius of the 20 Ne nucleus and the rms separation between 16 O and α. [14] This α+ 16 O model of the 20 Ne nucleus have been used to describe the proton- 20 Ne, alpha- 20 Ne, and 16 O- 20 Ne elastic scattering and satisfactory agreement with the measured angular distributions are obtained. [14,17 19] To perform calculation of the π- 20 Ne scattering amplitude equation (1), we need the π-α scattering amplitude f πα (q) and the π- 16 O scattering amplitude f πo (q). There have been the available π-α and π- 16 O amplitudes, i.e. the phenomenological parametrized π-α scattering amplitude obtained by Binon et al. [20] and π- 16 O amplitude by Fröhlich et al. [21] They are given as )( ) f πα (q) = f πα (0) 1 q2 e βαq2, t α1 t α2 (10) )( ) f πo (q) = f πo (0) 1 q2 t O1 t O2 ) e βoq2, (11) t O3 where f πα (0) and f πo (0) are respectively the forward π-α and π- 16 O scattering amplitudes, t αj and t Oj are complex parameters, and β α and β O are real parameters. The values of the parameters in the above amplitudes for the considered energies are given in Refs. [20 21] These parametrized amplitudes can reproduce the experimental π-α and π- 16 O scattering data very well. Using the above wave function of the ground state of 20 Ne and the parametrized π-α and π- 16 O scattering amplitudes, the π- 20 Ne elastic scattering amplitude of expression (1) can be conveniently obtained and expressed as an analytical function. 3 Results and Discussion We have calculated the differential cross sections of π- 20 Ne elastic scattering at incident energies E lab = 79, 114, 163, 180, and 240 MeV. The calculated results are shown by the curves in Figs. 1 and 2. It is a pity that we do not find the available experimental data of π- 20 Ne elastic scattering. Perhaps due to the absence of the experimental data, we do not find any theoretical calculation on π- 20 Ne elastic scattering either. However, from the available experimental data of pion elastic scattering on the neighboring 4N-type nuclei, [1 9] one can find that the experimental angular distributions for 12 C, 16 O, 24 Mg, and 28 Si targets display the features as follows: (i) for a same incident energy, the patterns of the experimental angular distributions for these targets are analogous, and the positions of the dips and peaks are somewhat forward located for heavier target; (ii) for a certain target, the positions of the dips and peaks are also forward located for higher incident energy, and the cross sections in the larger angles decrease when the incident energy increased. The above features for 12 C, 16 O, 24 Mg, and 28 Si targets can be used to justify the calculated results of π- 20 Ne elastic scattering for corresponding incident energies. In the calculations, we choose the incident energies E lab = 79, 114, 163, 180, and 240 MeV for the analysis of π- 20 Ne elastic scattering, because there are available experimental angular distributions at corresponding energies for 12 C, 16 O, 24 Mg, and 28 Si targets that can be used to do comparison. Also the corresponding parameters for the basic input amplitudes of f πα (q) and f πo (q) for theoretical calculations have been given in Refs. [20 21]. In Figs. l and 2, the experimental angular distributions for 12 C, 16 O, 24 Mg, and 28 Si targets at corresponding energies are shown for comparison. From Figs. 1 and 2, one can see that the theoretical results for π- 20 Ne elastic scattering (solid curves) show reasonable patterns similar as those for the neighboring 4N-type nuclei at the corresponding incident energy, and the diffractive minima and maxima are predicted in the appropriate positions. To see it clearer, the evaluated θ values related to the first dip and second peak for 12 C, 16 O, and 20 Ne targets at the corresponding incident energies are listed in Table 1 and shown in Fig. 3. For 12 C and 16 O targets, the θ values of the dips and peaks are evaluated from the experimental angular distributions, and for

3 590 Communications in Theoretical Physics Vol Ne the values are obtained from the theoretical prediction. One can see that the theoretical predictions exhibit similar behavior as that for 12 C and 16 O targets: the positions of the first dip and second peak are forward located for higher incident energy, and about the same shifted steps are obtained. In Table 2 and Fig. 4, the evaluated θ values of the dips and peaks for 12 C, 16 O, 20 Ne, 24 Mg, and 28 Si targets at incident energy of 180 MeV are listed and shown. Here, for the 16 O target at 180 MeV where no experimental data available, the θ values of the first dip and second peak are evaluated by extrapolating seen in Figs. 3 denoted by the open circle points. From Table 2 and Fig. 4, we see that the theoretical predictions agree rather well with the trend of the mass dependence of θ values of the dips and peaks. Table 1 The evaluated θ values related to the dips and peaks for pion elastic scattering on 12 C, 16 O, and 20 Ne at the energies denoted. For 12 C and 16 O targets, the values are evaluated from the experimental data, and for 20 Ne the values are obtained from the theoretical prediction. The first dips The second peaks E lab /MeV 12 C 16 O 20 Ne 12 C 16 O 20 Ne 79 (80 for 12 C) (120 for 12 C) (162 for 12 C) (170 for 16 O) (226 for 12 C) Table 2 The evaluated θ values related to the dips and peaks for pion elastic scattering on 12 C, 16 O, 20 Ne, 24 Mg, and 28 Si at 180 MeV. The values for 20 Ne are obtained from the theoretical prediction, and for the other targets the values are evaluated from the experimental data. The first dips The second peaks E lab /MeV 12 C 16 O 20 Ne 24 Mg 28 Si 12 C 16 O 20 Ne 24 Mg 28 Si Fig. 1 The angular distributions of pion elastic scattering on 12 C, 16 O, and 20 Ne at the energies denoted. The curves show the theoretical results for π- 20 Ne elastic scattering by the α+ 16 O model of 20 Ne. The experimental data for the 12 C and 16 O targets, shown for comparison, are taken from Refs. [1 5]. It should be pointed out that in our calculations there are no free parameters. The parameters in the wave function for the ground state of 20 Ne have been obtained by fitting the experimental charge form factor of 20 Ne in our previous article. [14] The parameters in the π- 16 O scattering amplitude f πo (q) have been given by Fröhlich et al. [21] by fitting the experimental π- 16 O scattering data. For the π-α scattering amplitude f πα (q), the parameters for inci-

4 No. 5 Communications in Theoretical Physics 591 dent energy of 51, 60, 68, 75, 110, 150, 180, 220, 260, and 280 MeV have been given by Binon et al. [20] by fitting the experimental π-α scattering data, and then we can obtain the parameters for the desired incident energies by interpolating from the available parameters. So the values of all the parameters used in the calculations of the π- 20 Ne elastic scattering are definite and no longer adjustable. Generally, the Glauber theory is applicable for higher incident energies. At lower energies, the various nuclear structure effects, e.g. the Fermi motion, the binding energy, the angle transformation, and true pion absorption, etc., become more significant. It had been shown by many calculations that the multiple scattering theory failed to reproduce the experimental data at lower energies. But, if various higher-order effects, in particular the true pion absorption, are taken into account, then the discrepancy between theory and experiment can be significantly reduced. This case comes about as the use of the π-n amplitude as the basic input for the nucleon structure model of nucleus, as consequence of the π-n amplitude does not contain the pion absorption and other nuclear effects. However in the present calculations based on the α+ 16 O structure model of 20 Ne nucleus, the basic input is the π-α and π- 16 O amplitudes obtained by fitting the experimental data, the various nuclear structure effects have been automatically included to some extent. So, it is expected that the calculated results should be reasonable. This is a significant advantage of using nuclear cluster model in the study of π-nucleus scattering. Fig. 2 The same as Fig. 1, but for 12 C, 16 O, 20 Ne, 24 Mg, and 28 Si. The experimental data shown for comparison are taken from Refs. [1,4,6 7]. Fig. 3 The energy dependence of the evaluated θ values related to the dips (a) and peaks (b) listed in Table 1. The circle points show the values for 12 C and 16 O targets evaluated from the data; the star points show the theoretical prediction for 20 Ne. The lines are drawn to guide the eye. The present work does not take into account the effects of Coulomb scattering. The experimental data of π + and π elastic scattering on 12 C, 16 O, 24 Mg, and 28 Si nuclei [1 9] have shown that there are only very slight Coulomb induced differences between the π + and π angular distributions. The Coulomb scattering mainly affects the depth of the dips. Therefore, the inclusion of

5 592 Communications in Theoretical Physics Vol. 60 the Coulomb effects would not change the main features of the calculated angular distributions. In summary, based on the α+ 16 O model of the 20 Ne nucleus and in the framework of Glauber multiple scattering theory, we have calculated the differential cross sections of π- 20 Ne elastic scattering at incident energies E lab = 79, 114, 163, 180, and 240 MeV. The calculated results are well in agreement with the general features of pion elastic scattering on the neighboring 4N-type nuclei. Comparing with the experimental data of pion elastic scattering on 12 C, 16 O, 24 Mg, and 28 Si nuclei, the diffractive patterns and the positions of the dips and peaks in the angular distributions of the π- 20 Ne elastic scattering are reasonably predicted by the calculations. This success provides a support to the α+ 16 O model of the 20 Ne nucleus. Fig. 4 The mass dependence of the evaluated θ values related to the dips (a) and peaks (b) listed in Table 2 for incident energy of 180 MeV. The circle points show the values for 12 C, 16 O, 24 Mg, and 28 Si targets evaluated from the data; the star points show the theoretical prediction for 20 Ne. The lines are drawn to guide the eye. References [1] F. Binon, et al., Nucl. Phys. B 17 (1970) 168. [2] J. Piffaretti, et al., Phys. Lett. B 67 (1977) 289; Phys. Lett. B 71 (1977) 324. [3] M. Blecher, et al., Phys. Rev. C 28 (1983) [4] M. Blecher, et al., Phys. Rev. C 10 (1974) [5] J.P. Albanese, et al., Phys. Lett. 73 (1978) 119; Nucl. Phys. A 350 (1980) 301. [6] C.A. Wiedner, et al., Phys. Lett. B 78 (1978) 26. [7] B.M. Preedom, et al., Nucl. Phys. A 326 (1979) 385. [8] G.S. Blanpied et al., Phys. Rev. C 41 (1990) [9] M.W. Rawool-Sullivan, et al., Phys. Rev. C 49 (1994) 627. [10] Q.R. Li, Nucl. Phys. A 415 (1984) 445; Phys. Rev. C 30 (1984) [11] J.F. Germond and C. Wilkin, Nucl. Phys. A 237 (1975) 477. [12] N. Bano and I. Ahmad, J. Phys. G 5 (1979) 39. [13] B. Zhou, et al., Phys. Rev. C 86 (2012) [14] P.T. Ong, Y.X. Yang, and Q.R. Li, Eur. Phys. J. A 41 (2009) 229. [15] R.J. Glauber, Lecture in Theoretical Physics, Vol. 1, eds. W.E. Brittin and L.G. Dunham, Interscience, New York (1959) p [16] V. Franco and R.J. Glauber, Phys. Rev. 142 (1966) [17] Y.X. Yang, H.L. Tan, and Q.R. Li, Phys. Rev. C 82 (2010) [18] Yang Yong-Xu, Tan Hai-Lan, and Li Qing-Run, Commun. Theor. Phys. 55 (2011) 655. [19] Y.X. Yang and Q.R. Li, Phys. Rev. C 84 (2011) [20] F. Binon, et al., Nucl. Phys. A 298 (1978) 499. [21] J. Frḧlich, H.G. Schlaile, L. Streit, and H. Zing, Z. Phys. A 302 (1981) 89.