Analysis of 12C12C scattering using different nuclear density distributions

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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 be worked on for improving the nuclear density description of 12C in the future.

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Nội dung Text: Analysis of 12C12C scattering using different nuclear density distributions

Science & Technology Development Journal, 21(3):78- 83 Original Research 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 12 C+12 C angular distributions at three bombarding energies of 102.1, 112.0 and 126.1 MeV were analyzed in the framework of optical model (OM) and compared to the experimental data. The reality of the OM analysis using the double folding potential 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 12 C nuclear density distribution than the latter. Therefore, the DFT should be worked on for improving the nuclear density description of 12 C in the future. Key words: Density functional theory, DFT, Double folding potential, Nuclear density, Optical model, Scattering INTRODUCTION latter approach is able to give a physical interpretation to experimental data because of its basic physical in- One of the approaches which we have been utilizing to study nuclear properties is investigation of collisions gredients 2 ; therefore, microscopic models are an ap- of two particles, especially systems of two light heavy pealing topic to study. nuclei, such as 12 C+12 C 1 . Because of the refractive One of such microscopic models is the double fold- Department of Nuclear Physics, Faculty ing model in which the NN interactions and nuclear of Physics and Engineering Physics, effect, the scattering data of this system gives informa- University of Science, VNU-HCM, tion of nuclear potential in a wider range in compari- density of two particles are two crucial inputs. In re- Nguyen Van Cu Street, District 5, Ho son to what the heavy nuclei systems can give. In fact, cent years, D. T. Khoa et al. have developed an en- Chi Minh City ergy and density-dependent NN interaction, namely when two heavy nuclei start overlapping each other, a Correspondence strong absorption dominates at the surface, and leads the effective CDM3Yn 3 . For the nuclear density, the Nguyen Dien Quoc Bao, Department of to non-elastic processes. This phenomenon reduces Fermi form, which is obtained from electron scat- Nuclear Physics, Faculty of Physics and the possibility of other effects which take place in the tering experiments 4 , is a classical distribution. Be- Engineering Physics, University of sides the studies of improving the density approxima- Science, VNU-HCM, Nguyen Van Cu inner region of the nuclei and, thus, prevents us from Street, District 5, Ho Chi Minh City getting any information about the nuclear potential in tions for the folding potential, approaches which have Email: ndqbao@hcmus.edu.vn this region. Fortunately, the refractive effect, which been developed to study nuclear structure are also able happens in the inner region, can be observed in the to yield the nuclear density. Furthermore, the den- History • Received: 26 July 2018 data of large angles of elastic scattering of light heavy sity functional theory (DFT) for nuclear studies was • Accepted: 07 October 2018 systems 2 , enabling these systems to become promi- developed by P. Ring et al. 5,6 , the Green’s function • Published: 16 October 2018 Monte Carlo (GFMC) technique was investigated in nent objects to study either theoretically or experi- DOI : mentally. the work of J. Carlson et al. 7 , and the ab initio calcula- https://doi.org/10.32508/stdj.v21i3.431 One well-known model, which is able to handle the tion was studied by M. Gennari et al. 8 . It is important calculation of the scattering of two particles, is the op- to study nuclear reaction by applying these methods tical model (OM). In this model, a complex poten- to obtain the density of particles, before putting it into tial, a so-called optical potential (OP), is utilized to the double folding potential to calculate cross section. describe both elastic scattering and non-elastic pro- The aim of this work was to compare the density dis- Copyright cesses. There are two main approaches which have tributions which are calculated in the framework of © VNU-HCM Press. This is an open- access article distributed under the been used to obtain the OP. One is the phenomeno- DFT and the Fermi form by OM analysis of elastic 12 terms of the Creative Commons logical method in which parameters of the Woods- C+12 C scattering data. In particular, these densi- Attribution 4.0 International license. Saxon form are determined by experimental data. ties are put into the double folding potential to cal- Another consists of microscopic models which are de- culate angular cross sections of the 12 C+12 C system, rived from nucleon-nucleon (NN) interactions. The before comparison with the experimental data. Cite this article : Dien Quoc Bao N, Hoang Chien L, Hoa Lang T, Van Tao C. Analysis of 12 C+12 C scattering using different nuclear density distributions. Sci. Tech. Dev. J.; 21(3):78-83. 78
Science & Technology Development Journal, 21(3):78-83 METHODS where the nuclear scattering amplitude is expressed in terms of partial-wave ι , Optical model 1 ∑ ∞ In the scenario of OM, the nucleus is assumed as a fN (θ) = (2ℓ + 1)eiσℓ [eiδℓ − 1]Pℓ (cosθ) (6) cloudy ball that absorbs and scatters partially the in- 2ik ℓ=0 coming particle flux in a similar way to the behavior of light. To describe this idea, the total potential (called and η is the Sommerfeld parameter and k is the wave the OP) is defined in term of a complex function, number of the incident nucleus. The nuclear and Coulomb phase shifts (i.e., δι and σι ) are determined by the relative-motion wave function. Note, χ(R) in iW (R) UOP (R) = VR (R) + + VC (1) the Schrödinger equation and the known UOP (R) are R − R0 1 + exp[ ] shown below 10 : a { 2 [ 2 ] } d ℓ(ℓ + 1) where the second term accounts for the non-elastic +E− − UOP (R) χ(R) = 0 2µ dR2 R2 scattering channels. Three parameters of W, R0 , and (7) a correspond to the depth, radius and surface diffuse- Nuclear Density ness parameters adjusted to obtain the best fit to the To perform the OM calculations with the microscopic angular distribution data. VC is the Coulomb poten- nuclear potential, the nuclear densities of colliding tial. The real part of nuclear potential VR (depict- nuclei were required as the important inputs. In gen- ing the elastic channel) is calculated within the double 2,3,9 eral, the nuclear density can be determined from the folding model using the effective NN interaction as follows: electron scattering experiment or DFT calculation. The nuclear charge-density distribution or the nuclear VR = VD + VEX = ∑ (2) root-mean-square (rms) charge radius is given by the i∈a, j∈A [< ij |VD | ij > + < ij |VEX | ji >] form factor measurement in the electron scattering 4 where |i > and |j > correspond to the single-particle experiment . Generally, the nuclear charge-density wave functions of the nucleon i in the target and the distributions are parameterized in terms of the two- nucleon j in the projectile. With the explicit treat- parameter Fermi functions as follows: ment of these single-particle wave functions of |i > { [ ]}−1 r − ca(A) and |j > , we can obtain the direct term VD and ex- ρa(A) (r) = ρ0a(A) 1 + exp (8) da(A) change term VEX , given as 9 : − → where the parameters ( ρ0a(A) , ca(A) , da(A) ) are VD ( R , E) = ∫ chosen to reproduce correctly the nuclear rms charge ρa ( − → r a )ρA (− → r A )νD (ρ, E, s)d3 ra d3 rA , (3) radii. On the other hand, nuclear density distributions can − → VEX = ( R , E) = also be calculated by the framework of relativistic self- ∫ ρa (− → r a, − → ra + −→s )ρA (− →r A, − → rA − − → s )vEX consistent mean field using the relativistic Hartree- [ − → ] (ρ, E, s)exp iK( R )s 3 3 d ra d rA . (4) Bogoliubov (RHB) equations 5 , µ ( )( ) ( ) hD − m − λ ∆ un un Here, vD and vEX are the direct and exchange = E n (9) − ∆∗ − h∗D + m + λ vn vn
terms →
of the effective − NN interaction. s =
−
→rA − − →r a + R
is the relative distance between where un and vn are Hartree-Bogoliubov wave func- two interacting nucleons. Additionally, rA and ra are tions that are corresponding to energy level En . The the nucleon coordinates with respect to target A and single-nucleon Dirac Hamiltonian hD is defined as: projectile a, respectively; R represents the nucleus- N −Z nucleus separation; and E and K correspond to the hD = −i r ) + V (− ∇ + βM ∗ (− → → r) (10) center of mass energy of the system and the relative N +Z momentum. The parameters β, effective mass M ∗ and vector po- In the framework of quantum scattering theory, the tential V (− → r ) are described in detail by the meson- differential cross section for an elastic scattering pro- exchange model 5 . The pairing field △ reads cess is defined as 10 :
2 1∑ ⟨ ′ ′ ⟩ dσ
η
∆i1 i′ = i1 i1 |V pp | i2 i2 κi2 i′ =
fN (θ) − e−iηLn(θ/2 )+2σ0
(5) 1 2 ′ 2 dΩ 2k sin(θ/2 ) i1 i1 79
Science & Technology Development Journal, 21(3):78-83 ′ ′ The index i1 , i1 , i2 and i2 refer to the coordinates from two methods. In particular, the root-mean- ′ ′ in space, spin and isospin. < i1 i1 |V pp | i2 i2 > are square (rms) of nuclear matter radius was evaluated the matrix elements of two-body pairing interactions. from DFT (about 2.44 fm) and is larger than the ex- The pp-correlation potentials are the pairing part of perimental value (about 2.33 fm) 4 . We will consider the Gorny force D1S 11 . Because the vector potential how these densities affect the nuclear potential. in Eq. (10) depends on the nuclear density ρ and the The calculation of the nuclear folding potential is per- pairing potential in Eq. (11), our formula relies on formed using Eqs. (2)-(4) within a self-consistent pairing tensor κ; thus, it is crucial to define it. For the procedure. In this work, the effective CDM3Y3 inter- RHB ground state 5 , it as follows: action, proven to be successful in the OM analysis of ∑ ∗ ∑ ∗ elastic scattering data over a wide range of energies 3 , ρii′ = vin vi′ n + vin vi′ n (11) En >0 En 0 En
Science & Technology Development Journal, 21(3):78-83 Figure 1: The nuclear density distributions of 12 C nucleus obtained from the DFT calculation and the elec- tron scattering experiment. Figure 2: The nuclear potentials of 12 C+ 12 C 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 12 C + 12 C system at Elab =102.1, 112 and 121.6 MeV. The data are taken from Ref. 12 . DISCUSSION rough approximation to consider the light nuclei as a mean field. Another reason is that the parameters The results of this study provide some information of the effective interaction DD-ME2, which is utilized about the DFT. The density of 12 C which is calcu- in the DIRHB program 5 , were adjusted to reasonably lated by the DFT is quite similar to that from the reproduce the properties of nuclear matter, binding electron scattering experiments at the surface. How- energies and charge radii. Some medium-heavy and ever, the values in the inner region of both methods heavy nuclei were obtained from the experiments (ex- are different. This leads to differences in cross sec- cept 16 O) 6 . This leads to a poor description of the tions at large angles. As can be seen in Figure 3, the nuclear density of light nuclei, such as 12 C, using the DFT is inappropriate to give the angular cross sec- effective DD-ME2 interaction. tions at backward angles, and thus, it gives a bad de- scription of the nuclear density of 12 C in the inner re- CONCLUSIONS gion. There are a few reasons to explain this. One is The OM analysis of elastic 12 C+12 C scattering data that the self-consistent mean field tends to be success- at medium energies has been performed. To illus- ful in describing the structural properties of medium- trate the difference between two nuclear density dis- heavy and heavy nuclei rather than light ones. It is a tributions, the real part of OP was constructed in the 82
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