Analysis of ¹²C+ ¹²C scattering using different nuclear density distributions

Science & Technology Development Journal, 21(3):78- 83 Original Research Department of Nuclear Physics, Faculty of Physics and Engineering Physics, University of Science, VNU-HCM, Nguyen Van Cu Street, District 5, Ho Chi Minh City Correspondence Nguyen Dien Quoc Bao, Department of Nuclear Physics, Faculty of Physics and Engineering Physics, University of Science, VNU-HCM, Nguyen Van Cu Street, District 5, Ho Chi Minh City Email: ndqbao@hcmus.edu.vn History  Received: 26 July 2018  Accepted: 07 October 2018  Published: 16 October 2018 DOI : https://doi.org/10.32508/stdj.v21i3.431 Copyright © VNU-HCM Press. This is an open- access article distributed under the terms of the Creative Commons Attribution 4.0 International license. Analysis of 12C+12C scattering using different nuclear density distributions Nguyen Dien Quoc Bao, Le Hoang Chien, Trinh Hoa Lang, Chau Van Tao ABSTRACT Elastic 12C+12C angular distributions at three bombarding energies of 102.1, 112.0 and 126.1 MeV were analyzed in the framework of opticalmodel (OM) and compared to the experimental data. The reality of theOManalysis using the double foldingpotential depends on the chosen nuclear density distributions. In this work, we use two available models of nuclear density distributions obtained from the electron scattering experiments and the density functional theory (DFT). The OM results show that the former gives better description of the 12C nuclear density distribution than the latter. Therefore, the DFT should beworked on for improving the nuclear density description of 12C in the future. Key words: Density functional theory, DFT, Double folding potential, Nuclear density, Optical model, Scattering INTRODUCTION One of the approacheswhichwe have been utilizing to study nuclear properties is investigation of collisions of two particles, especially systems of two light heavy nuclei, such as 12C+12C 1. Because of the refractive effect, the scattering data of this system gives informa- tion of nuclear potential in a wider range in compari- son to what the heavy nuclei systems can give. In fact, when two heavy nuclei start overlapping each other, a strong absorption dominates at the surface, and leads to non-elastic processes. This phenomenon reduces the possibility of other effects which take place in the inner region of the nuclei and, thus, prevents us from getting any information about the nuclear potential in this region. Fortunately, the refractive effect, which happens in the inner region, can be observed in the data of large angles of elastic scattering of light heavy systems 2, enabling these systems to become promi- nent objects to study either theoretically or experi- mentally. One well-known model, which is able to handle the calculation of the scattering of two particles, is the op- tical model (OM). In this model, a complex poten- tial, a so-called optical potential (OP), is utilized to describe both elastic scattering and non-elastic pro- cesses. There are two main approaches which have been used to obtain the OP. One is the phenomeno- logical method in which parameters of the Woods- Saxon form are determined by experimental data. Another consists ofmicroscopicmodels which are de- rived from nucleon-nucleon (NN) interactions. The latter approach is able to give a physical interpretation to experimental data because of its basic physical in- gredients 2; therefore, microscopic models are an ap- pealing topic to study. One of such microscopic models is the double fold- ing model in which the NN interactions and nuclear density of two particles are two crucial inputs. In re- cent years, D. T. Khoa et al. have developed an en- ergy and density-dependent NN interaction, namely the effective CDM3Yn 3. For the nuclear density, the Fermi form, which is obtained from electron scat- tering experiments 4, is a classical distribution. Be- sides the studies of improving the density approxima- tions for the folding potential, approaches which have beendeveloped to studynuclear structure are also able to yield the nuclear density. Furthermore, the den- sity functional theory (DFT) for nuclear studies was developed by P. Ring et al. 5,6, the Green’s function Monte Carlo (GFMC) technique was investigated in the work of J. Carlson et al. 7, and the ab initio calcula- tion was studied byM. Gennari et al. 8. It is important to study nuclear reaction by applying these methods to obtain the density of particles, before putting it into the double folding potential to calculate cross section. The aim of this work was to compare the density dis- tributions which are calculated in the framework of DFT and the Fermi form by OM analysis of elastic 12C+12C scattering data. In particular, these densi- ties are put into the double folding potential to cal- culate angular cross sections of the 12C+12C system, before comparison with the experimental data. Cite this article : DienQuoc BaoN, HoangChien L, Hoa Lang T, Van TaoC.Analysis of 12C+12C scattering using different nuclear density distributions. Sci. Tech. Dev. J.; 21(3):78-83. 78 Science & Technology Development Journal, 21(3):78-83 METHODS Optical model In the scenario of OM, the nucleus is assumed as a cloudy ball that absorbs and scatters partially the in- coming particle flux in a similar way to the behavior of light. To describe this idea, the total potential (called the OP) is defined in term of a complex function, UOP (R) = VR(R) + iW (R) 1 + exp[ RR0 a ] + VC where the second term accounts for the non-elastic scattering channels. Three parameters of W, R0, and a correspond to the depth, radius and surface diffuse- ness parameters adjusted to obtain the best fit to the angular distribution data. VC is the Coulomb poten- tial. The real part of nuclear potential VR (depict- ing the elastic channel) is calculated within the double folding model 2,3,9 using the effective NN interaction as follows: VR = VD + VEX =P i2a; j2A [ + ] where ji > and jj > correspond to the single-particle wave functions of the nucleon i in the target and the nucleon j in the projectile. With the explicit treat- ment of these single-particle wave functions of ji > and jj > , we can obtain the direct term VD and ex- change term VEX , given as 9: VD( ! R;E) =R a( !r a)A(!r A)D(; E; s)d3rad3rA; VEX = ( ! R; E) =R a( !r a; !r a + !s )A (!r A;!r A !s )vEX (; E; s)exp  iK( ! R )s   d3rad 3rA: Here, vD and vEX are the direct and exchange terms of the effective NN interaction. s =!r A !r a + !R  is the relative distance between two interacting nucleons. Additionally, rA and ra are the nucleon coordinates with respect to target A and projectile a, respectively; R represents the nucleus- nucleus separation; and E and K correspond to the center of mass energy of the system and the relative momentum. In the framework of quantum scattering theory, the differential cross section for an elastic scattering pro- cess is defined as 10: d d = fN () 2k sin(/2 )eiLn(/2 )+20 2 where the nuclear scattering amplitude is expressed in terms of partial-wave  , fN () = 1 2ik 1X `=0 (2`+ 1)ei` [ei` 1]P`(cos) and  is the Sommerfeld parameter and k is the wave number of the incident nucleus. The nuclear and Coulomb phase shifts (i.e.,  and  ) are determined by the relative-motion wave function. Note, (R) in the Schrödinger equation and the knownUOP (R) are shown below 10: 2 2  d2 dR2 + E `(`+ 1) R2  UOP (R)  (R) = 0 Nuclear Density To perform theOM calculations with themicroscopic nuclear potential, the nuclear densities of colliding nuclei were required as the important inputs. In gen- eral, the nuclear density can be determined from the electron scattering experiment or DFT calculation. Thenuclear charge-density distribution or the nuclear root-mean-square (rms) charge radius is given by the form factor measurement in the electron scattering experiment 4. Generally, the nuclear charge-density distributions are parameterized in terms of the two- parameter Fermi functions as follows: a(A)(r) = 0a(A)  1 + exp  r ca(A) da(A) 1 (8) where the parameters ( 0a(A); ca(A); da(A) ) are chosen to reproduce correctly the nuclear rms charge radii. On the other hand, nuclear density distributions can also be calculated by the framework of relativistic self- consistent mean field using the relativistic Hartree- Bogoliubov (RHB) equations 5, hD m 

 hD +m+  ! un vn ! = En un vn ! (9) where un and vn are Hartree-Bogoliubov wave func- tions that are corresponding to energy level En. The single-nucleon Dirac Hamiltonian hD is defined as: hD = iN Z N + Z r+ M(!r ) + V (!r ) (10) The parameters , effective massM and vector po- tential V (!r ) are described in detail by the meson- exchange model 5. The pairing field4 reads
i1i 0 1 = 1 2 X i1i 0 1 D i1i 0 1 jV ppj i2i 0 2 E  i2i 0 2 79 (1) (2) (3) (4) (5) (6) (7) Science & Technology Development Journal, 21(3):78-83 The index i1; i 0 1; i2 and i 0 2 refer to the coordinates in space, spin and isospin. < i1i 0 1 jV ppj i2i 0 2 > are the matrix elements of two-body pairing interactions. The pp-correlation potentials are the pairing part of the Gorny force D1S 11. Because the vector potential in Eq. (10) depends on the nuclear density  and the pairing potential in Eq. (11), our formula relies on pairing tensor ; thus, it is crucial to define it. For the RHB ground state 5, it as follows: ii0 = P En>0 vinvi0n + P En<0 vinvi0n (11) ii0 = P En>0 vinui0n + P En<0 vinui0n (12) In order to determine the nuclear density, we begin by calculating theDiracHamiltonian hD and the pair- ing field4using parameters of the density-dependent meson-exchange relativistic energy functional DD- ME2 6 and of the Gorny force D1S 11. Then, the Hartree-Bogoliubov wave functions are obtained by solving Eq. (9), before applying to Eqs. (12-13) to ob- tain the nuclear density and pairing tensor. The pro- cedure is repeated until the nuclear density is conver- gent. RESULTS To begin with, nuclear density distributions were cal- culated by two methods: the electron scattering ex- periment and themicroscopicDFT calculation. In the former method, the two-parameter Fermi distribu- tion was chosen to describe the nuclear density with parameters adjusted to correctly give the experimen- tal value of nuclear rms matter radius (0 = 0.194 fm3, c = 2.214 fm, d = 0.425 fm) 9. In the lat- ter method, the nuclear density distribution was ob- tained by the self-consistent mean field calculation, using Eqs. (9)-(13). All the calculations of DFT were performed by the RDIHB program 5. In Figure 1, we show these 12C density distributions as the function of distance. The dashed line represents the nuclear density calculated from the DFT (called the DFT den- sity), while the solid line is the result of nuclear den- sity obtained from the electron scattering experiment (called the FER density). One can see that the DFT reproduces a tight nuclear density distribution in comparison with the Fermi function. As a result, to conserve the nucleon num- ber, the DFT density at the nuclear center is higher about 15% than the FER density. At the surface re- gion, the diffuseness of DFT density distribution is slightly larger than that of FER one, which leads the discrepancy of the nuclear rms matter radii obtained from two methods. In particular, the root-mean- square (rms) of nuclear matter radius was evaluated from DFT (about 2.44 fm) and is larger than the ex- perimental value (about 2.33 fm) 4. We will consider how these densities affect the nuclear potential. The calculation of the nuclear folding potential is per- formed using Eqs. (2)-(4) within a self-consistent procedure. In this work, the effective CDM3Y3 inter- action, proven to be successful in the OM analysis of elastic scattering data over a wide range of energies 3, is used as an input for the folding calculation. Both the DFT and FER density distributions were used in the folding procedure. Thenuclear 12C-12Cpotentials at the bombarding en- ergy of 102.1 MeV using two different density dis- tributions are shown in Figure 2. The dashed and solid lines describe the folding potentials using two different inputs of DFT and FER density distribu- tions, named P1 and P2, respectively. Both the P1 and P2 potentials have almost similar shapes and depths (about 280-290MeV).The effective CDM3Y3 interac- tion depends on the nuclearmedium surrounding the two interacting nucleons. Thus, this interaction feels a tight nuclear medium arising from the DFT density in comparison with that of the FER density in the sep- aration; R < 1.5 fm (as seen in Figure 1). Therefore, it is reasonable to obtain the P1 potential more shallow than the P2 potential in the deep region, i.e., R < 1.5 fm. Now we analyze the elastic 12C+12C scattering data 12 based on the OM with the real part of OP in (1) and sequentially replaced by the P1 and P2 potentials. In OM analysis, the Coulomb potential, the last term of (1), is calculated by folding two uniform charge distri- butions of 12C 2. TheWoods-Saxon parameters in (1) are taken from the global OP for the elastic 12C+12C scattering analysis 13. A comparison between the theoretical results and the experimental data of elastic 12C+12C angular distri- butions are shown in Figure 3. The theoretical eval- uation was obtained by OM calculation sequentially using P1 (dashed lines) and P2 (solid lines) poten- tials. In general, the P2 potential gives a better de- scription of the elastic 12C+12C scattering data than does the P1 potential. However, there are some points at which it has a large deviation from experimental data compared to results of the P1 potential, espe- cially at around 65 degree and 80 degree angles; Elab =102.1 MeV. At forward angles, although both poten- tials nearly give the similar results, the P1 tends to be better than the P2 at angles of around 30 degrees. At backward angles, the P1 potential almost describes a wrong shape of angular distributions. 80 Science & Technology Development Journal, 21(3):78-83 Figure 1: The nuclear density distributions of 12C nucleus obtained from the DFT calculation and the elec- tron scattering experiment. Figure 2: The nuclear potentials of 12C+ 12C system at the bombarding energy (Elab = 102.1 MeV) stage corresponding to the DFT and FER density distributions. 81 Science & Technology Development Journal, 21(3):78-83 Figure 3: The elastic angular distributions of 12C + 12C system at Elab =102.1, 112 and 121.6 MeV. The data are taken from Ref. 12 . DISCUSSION The results of this study provide some information about the DFT. The density of 12C which is calcu- lated by the DFT is quite similar to that from the electron scattering experiments at the surface. How- ever, the values in the inner region of both methods are different. This leads to differences in cross sec- tions at large angles. As can be seen in Figure 3, the DFT is inappropriate to give the angular cross sec- tions at backward angles, and thus, it gives a bad de- scription of the nuclear density of 12C in the inner re- gion. There are a few reasons to explain this. One is that the self-consistent mean field tends to be success- ful in describing the structural properties of medium- heavy and heavy nuclei rather than light ones. It is a rough approximation to consider the light nuclei as a mean field. Another reason is that the parameters of the effective interaction DD-ME2, which is utilized in the DIRHB program 5, were adjusted to reasonably reproduce the properties of nuclear matter, binding energies and charge radii. Some medium-heavy and heavy nuclei were obtained from the experiments (ex- cept 16O) 6. This leads to a poor description of the nuclear density of light nuclei, such as 12C, using the effective DD-ME2 interaction. CONCLUSIONS The OM analysis of elastic 12C+12C scattering data at medium energies has been performed. To illus- trate the difference between two nuclear density dis- tributions, the real part of OP was constructed in the 82 Science & Technology Development Journal, 21(3):78-83 framework of the double folding model without free parameters. Besides the chosen effective NN inter- action, two density distributions obtained from the elastic scattering experiment and the DFT calculation were used as the independent inputs for the nuclear folding procedure. The analysis shows that DFT gives a bad description of the nuclear density of 12C in the interior region, and from the results, it also de- scribes wrong shapes of elastic 12C+12C angular dis- tributions at backward angles in three considered en- ergies. Further studies include investigating the DFT to improve the density calculations for nuclear reac- tion uses. COMPETING INTERESTS The authors declare that they have no competing in- terests. AUTHORS’ CONTRIBUTIONS Nguyen Dien Quoc Bao and Le Hoang Chien devel- oped the theoretical formalism, performed the ana- lytic calculations and contributed to the manuscripts. TrinhHoa Lang and Chau Van Tao reviewed and pro- vided critical feedback. ABBREVIATIONS DFT: Density functional theory NN: Nucleon-nucleon OM: Optical model OP: Optical potential RHB: Relativistic Hartree-Bogoliubov rms: Root-mean-square REFERENCES 1. Feshbach H. Optical model and its justification. Annual Re- view of Nuclear Science. 1958;8:49–104. 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