Improved Eikonal Approach for charge exchange reactions at intermediate energies


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. doi: 10.1088/1674-1137/43/12/124102
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    沈阳化工大学材料科学与工程学院 沈阳 110142

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Improved Eikonal Approach for charge exchange reactions at intermediate energies

  • 1. School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
  • 2. School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China
  • 3. Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University, Beijing 100191, China

Abstract: To describe charge exchange reactions at intermediate energies, as the first step, we implemented the formulations within the Normal Eikonal Approach. The calculated differential cross sections based on this approach deviate significantly from those using the conventional DWBA code, for CE reactions at 140 MeV/nucleon. Therefore, improvements are made on the way of applying the eikonal approximation, so that to keep the strict three-dimensional form factor. The results obtained within this Improved Eikonal Approach are in good agreement with the DWBA calculations and also with the experimental data. Since this Improved Eikonal Approach can be formulated in a microscopic way, it is easy to be applied to CE reactions at higher energies, where the application of the phenomenological DWBA is of a priori difficulty due to the lack of the required phenomenological potentials in most cases.


    1.   Introduction
    • Charge exchange (CE) reactions, with hadronic probes such as (p, n), (3He, t), (12C, 12N) and so on, are used as one of powerful tools for nuclear structure studies. The related interesting topics include the Gamow-Teller(GT) transitions in excitation-energy regions inaccessible by $ \beta $-decay [14], the spin-dipole transitions [3], the isovector spin giant monopole resonances [2, 5], the symmetry energy [2, 6], the isospin symmetry breaking force in the asymmetric nuclear matter [68], the Gamow-Teller giant resonance and the Landau-Migdal parameter [8]. In particular, GT strengths are crucial for understanding various problems such as the late stellar evolution, neutrino nucleosynthesis and neutrinoless double-$ \beta $ decay [1, 3, 8]. It is realized that the heavy-ion probes allow a better extraction of the GT strength, compared with (p, n) reaction, due to the advantage of higher energy resolution as well as the stronger absorption at the surface of the target nucleus [14].

      Recently, more attention has been paid on experimental studies of CE reactions at intermediate energies ($ \geqslant $ $ 100$ MeV/nucleon) [2, 3, 8]. Systematic investigations indicate that the reaction mechanism at these energies is dominated by one-step process so that more precise extraction of the weak transition strengths or the nuclear structure information can be obtained, as long as the appropriate theoretical tools could be applied to describe the data [9]. However, so far few theoretical formulations and calculation tools exist at intermediate and high energies [4, 10, 11], although at lower energies (< 100 MeV/nucleon) CE reactions have been successfully reproduced by the conventional Distorted-Wave-Born-Approximation (DWBA) [1216]. It is difficult to apply DWBA calculation for CE reactions at intermediate energies, since the required phenomenological potential is rarely available there [4, 17]. Therefore, we try to follow the eikonal approximation which can be formulated microscopically by using the nucleon densities and the bare nucleon-nucleon (NN) interaction [18]. At the first step, we follow the Normal Eikonal Approach (NEA) [19], which means applying the eikonal approximation to both the relative motion and the structural form factor. The calculated results based on NEA show a considerable discrepancy compared with the conventional DWBA calculations for CE reactions at $ 140$ MeV/nucleon. It is found that the problem is caused by the approximate treatment of the form factor, which carries the sensitive nuclear structure information. At the second step, we developed the Improved Eikonal Approach (IEA), which means still applying the eikonal (straight-line) approximation to the relative motion part while removing this approximation from the structural part so that the form factor remains to be a strict three-dimensional function in coordinate space. Based on IEA, the calculated differential cross sections (DCS) are in good agreement with those using the conventional DWBA and also with experimental results, for CE reactions at $ 140$ MeV/nucleon.

    2.   Theoretical description
    • The DCS for the CE reaction A(a, b)B is usually expressed as [17, 20]:

      $ \frac{\rm{d}\sigma}{\rm{d}\Omega}(\theta) = \frac{1}{(2J_{\rm{A}}+1)(2J_{\rm{a}}+1)}\sum\limits_{\substack{M_{\rm{a}}M_{\rm{b}}\\M_{\rm{A}}M_{\rm{B}}}} \big|f(\theta)\big|^{2}, $


      where $ J_{{\rm i}} $ and $ M_{{\rm i}} $ are the spin and magnetic quantum number of the particle i (i$ \, $ = $ \, $a, b, A and B), respectively. At intermediate energies, the difference between the amplitudes of the initial and final momentum could be neglected, and thus the scattering amplitude $ f(\theta) $ in DWBA approach can be defined in terms of the interaction matrix element [17, 20]:

      $ f(\theta) = \frac{-\mu}{2\pi\hbar^{2}}\langle\chi^{(-)}_{{{k}}'}({{R}})\Phi_{\rm{b}}\Phi_{\rm{B}}\mid V\mid\Phi_{\rm{a}}\Phi_{\rm{A}}\chi^{(+)}_{{k}}({{R}})\rangle, $


      where $ \mu $ is the reduced mass of the reaction system, $ \chi^{(\pm)}({{R}}) $ the incoming (+)/outgoing (-) distorted wave function, $ {{k}}({{{k}}'}) $ the initial (final) relative momentum, and $ {{R}} $ the vector of the relative position between a and A or b and B as seen in Fig. 1 and Fig. 3. In Eq. (2), $ \Phi $ is the internal wave function of the corresponding nucleus, and V the effective interaction potential which results in the charge exchange.

      Figure 1.  Schematic shown of the coordinate system used in the text.

      Figure 3.  The coordinates used in the text. $R$ is the distance between the center of mass of the nuclei, and $r_{{\rm p}_{j}}$/$r_{{\rm t}_{i}}$ the distance between the interacting nucleons ${\rm p}_{j}$/${\rm t}_{i}$ and the center of mass of the projectile/target, respectively.

      Within the eikonal approximation, which is valid at relatively high incident energies, the incoming distorted wave $ \chi_{{k}}^{(+)} $ is assumed to have the form [17, 20]:

      $ \chi_{{k}}^{(+)}({{R}}) = e^{ikz}e^{i\chi({{R}})}, $


      where z axis is parallel to the direction of the incident particles. The outgoing distorted wave $ \chi^{(-)}_{{{k}}^{\prime}} $ is the time reversal of $ \chi_{{{k}}^{\prime}}^{(+)} $. In Eq. (3), $ e^{ikz} $ with large k is a rapidly oscillating plane wave along z axis, while $ e^{i\chi({{R}})} $ is characterized by the relatively slow oscillation. Here, one usually adopts the cylindrical coordinates so that $ {{R}} $ can be replaced by $ {{b}}+z{{{e}}_{{z}}} $ as shown in Fig. 1. The vector in the plane perpendicular to z-axis, $ {{b}} $, is usually called the impact parameter.

      Again according to eikonal approximation, the phase shift function $ \chi({{R}}) $ in Eq. (3) can be replaced by the two dimensional function $ \chi({{b}}) $ [17, 19, 20]:

      $ \chi({{b}})= -\frac{\mu}{\hbar^{2} k}\int^{+\infty}_{-\infty}U({{b}},z)\,{\rm d}z, $


      where U is the overall interaction potential between the initial particles a and A or the final particles b and B. Then the product $ \chi^{(-)\star}_{{{k}}'}\cdot\chi^{(+)}_{{k}} $ in the expression of $ f(\theta) $ becomes $ e^{-i{{q}}\cdot{{R}}}e^{i\chi({{b}})} $ [19]. Here, $ {{q}} $, the transferred momentum, is equal to $ {{{k}}'}-{{k}} $$ \, $ as illustrated in Fig. 2, with $ q\,\approx\,2k\sin\dfrac{\theta}{2} $ at relatively high energies.

      Figure 2.  Schematic definition of the scattering angle $\theta$: the angle between the final momentum ${{{k}}'}$ and the initial momentum ${{k}}$. ${{q}}$ is the transferred momentum.

      Generally the potential U in Eq. (4) includes the nuclear and Coulomb part. Correspondingly, $ \chi({{b}}) $ is also the sum of $ \chi_{\rm{N}} $ and $ \chi_{\rm{C}} $, for the nuclear and Coulomb phase shift functions, respectively.

      In order to make direct comparison with the results of DWBA calculations, we may simply take U as the phenomenological optical potential (OP), to be used in both CEX and DWBA codes. In the case of the unavailability of the phenomenological potential U, alternatively, a microscopic method, called “$ t_{\rho\rho} $” method, can be adopted to calculate $ \chi_{\rm{N}} $:

      $ \chi_{\rm{N}}(b) = \int^{\infty}_{0} {\rm d}q\, q \rho_{\rm{p}}(q)\rho_{\rm{t}}(q)f_{\rm{NN}}(q)J_{0}(qb), $


      where $ f_{\rm{NN}} $ is the NN scattering amplitude, $ \rho_{\rm{p}} $ and $ \rho_{\rm{t}} $ the nucleon densities of projectile and target, respectively. Using this method, the application of the eikonal approximation can be extended to higher incident energies ($ E\, $ up to $ \sim 1\,\rm{GeV} $) where the phenomenological potential U is rarely available [11].

      Based on the eikonal approximation, $ f(\theta) $ in Eq. (2) becomes:

      $ f(\theta) = \frac{-\mu}{2\pi\hbar^{2}}\int {\rm d}{{R}}\,e^{-i{{q}}\cdot{{R}}}\,e^{i\chi{({{b}})}}F({{R}}). $


      Here, $ F({{R}}) $, the form factor carrying the nuclear structure information, is defined by:

      $ F({{R}}) = \langle J_{\rm{B}}T_{\rm{B}}J_{\rm{b}}T_{\rm{b}} \mid V \mid J_{\rm{A}}T_{\rm{A}}J_{\rm{a}}T_{\rm{a}} \rangle, $


      where $ T_{{\rm i}} $ is the isospin of the nucleus i (i$ \, $ = $ \, $a, b, A and B). V in Eq. (7) is the residual interaction between the valence nucleons, being responsible for the CE reaction. More specifically, V is given by [19]:

      $\begin{split} {V = \sum\limits_{{\rm t}_{i}{\rm p}_{j}}\,V_{{\rm t}_{i}{\rm p}_{j}}} =&{ \sum\limits_{{\rm t}_{i}{\rm p}_{j}}\sum\limits_{s_{0}t_{0}K}\,A^{K}_{s_{0}}\,V^{K}_{s_{0}t_{0}}(r_{{\rm t}_{i}{\rm p}_{j}}) (\tau^{t_{0}}_{1}\cdot\tau^{t_{0}}_{2})}\\ &{\times[Y_{K}(\widehat {{{{r}}_{{{\bf{t}}_{{i}}}{{\bf{p}}_{{j}}}}}})\cdot(\sigma^{s_{0}}_{1}\otimes\sigma^{s_{0}}_{2})^{K}],} \end{split} $


      where $ {{{r}}_{{{\bf{t}}_{{i}}}{{\bf{p}}_{{j}}}}} $ is the spatial coordinate between the interacting target nucleons “${\rm t}_{i} $” and projectile nucleons “$ {\rm p}_{j}$” as indicated in Fig. 3. Here, $ s_{0} $, the spin change of the interacting nucleons, has two values 1 and 0, corresponding to the spin-flip and non-spin-flip processes, respectively. The change of isospin, $ t_{0} $, is 1 for CE reactions. In Eq. (8), $ K\, = \,0 $ and $ K\, = \,2 $ correspond to the central and tensor forces, respectively. The constants $ A^{K}_{s_{0}} $ have the values $ \sqrt{4\pi} $, $ -\sqrt{12\pi} $ and $ \sqrt{4\pi/5} $, for $ A^{0}_{0} $, $ A^{0}_{1} $ and $ A^{2}_{1} $, respectively [19]. In Eq. (8), the NN interaction strength functions $ V^{K}_{s_{0}t_{0}}(r) $ include both the central (K = 0) and the tensor (K = 2) parts. Their parameters are taken from [21, 22]. Given the exchange and medium effects, the modified NN interaction strength functions are given in [21], which were adopted in the present work.

      Combining Eq. (8) with Eq. (7), $ F({{R}}) $ can be decomposed into the partial form factor $ F^{JSL_{{\rm tr}}} ({{R}}) $ weighted by C-G coefficients:

      $\begin{split} F({{R}}) =&{ \sum\limits_{\substack{J,S,L_{\rm{tr}}\\M_{\rm{tr}}M_{J}M_{S}}}C^{J_{\rm{B}}M_{\rm{B}}}_{J_{\rm{A}}M_{\rm{A}}JM_{J}}C^{J_{\rm{b}}M_{\rm{b}}}_{J_{\rm{a}}M_{\rm{a}}SM_{S}}}\\ &{\times C^{L_{{\rm tr}}M_{\rm{tr}}}_{SM_{S}JM_{J}}F^{JSL_{{\rm tr}}} ({{R}}), } \end{split} $


      where $ J/S $ is the total spin transferred to the intrinsic motion of the target/projectile system, $ L_{\rm{tr}} $ the total transferred angular momentum, and $ M_{\rm{tr}} $ the associated magnetic quantum number. It is natural that the partial form factor (matrix element) $ F^{JSL_{{\rm tr}}}({R}) $ requires, as its inputs, the states and wave functions of the interacting (valence) nucleons. One possibility is to use the One Body Transition Densities (OBTD), including the configuration mixing, which can be obtained by shell model calculations using, for instance, OXBASH code [1, 23]. The exact expressions of OBTD, together with the corresponding single-particle wave functions, can be found in [1215, 24, 25]. The detailed expressions for OBTD is beyond the scope of this article and, hence, are omitted here.

      For simplicity, the numerical calculation is performed in momentum space. Therefore, $ F({{R}}) $ is expressed by the inverse Fourier transformation of the form factor in the momentum space $ F({{p}}) $:

      $ F({{R}}) = \int {\rm d}{{p}}\,e^{-i{{p}}\cdot{{R}}}F({{p}}). $


      Inserting Eq. (10) into Eq. (6), $ f(\theta) $ becomes:

      $ \begin{split} f(\theta)= &{-\frac{\mu}{2\pi\hbar^{2}} \int {\rm d}{{b}}\,{\rm d}z\,e^{-i{{q}}\cdot{{R}}}e^{i\chi({{b}})}}\\ &{\times\int{\rm d}{{p}}\,e^{-i{{p}}\cdot{{R}}}F({{p}}).} \end{split} $


      According to NEA, the integral over z on $ e^{-i({{q}}+{{p}})\cdot{{R}}} $ in Eq. (11) is performed and yields a delta function $ \delta(q_{{\rm z}}+p_{{\rm z}}) $. Further, by using $ q_{{\rm z}}\,\approx\,0 $, Eq. (11) in this approach becomes:

      $ \begin{split} f^{{\rm NEA}}(\theta) =&{ -\frac{\mu}{\hbar^{2}} \int {\rm d}{{b}}\,e^{-i{{q}}\cdot{{b}}} e^{i\chi({{b}})}}\\ &{\times\int{\rm d}{{p}}_{\perp}\,e^{-i{{p}}_{\perp}\cdot{{b}}} F({{p}}_{\perp}),} \end{split}$


      where $ {{p}}_{\perp} $ is a two-dimensional momentum in the plane perpendicular to z axis. In this way, $ F({{R}}) $ is quenched into a two-dimensional function $ F^{{\rm NEA}}({{b}})\!\! = \!\!\int{\rm d}{{p}}_{\perp}\,e^{-i{{p}}_{\perp}\cdot{{b}}} F({{p}}_{\perp}) $.

      For comparison, in IEA, firstly $ q_{{\rm z}}\,\approx\,0 $ is directly used on $ e^{-i{{q}}\cdot{{R}}} $, so that $ e^{-i{{q}}\cdot{{R}}} $ is replaced by $ e^{-i{{q}}\cdot{{b}}} $ and Eq. (11) becomes:

      $ \begin{split} f^{{\rm IEA}}(\theta) =&{ -\frac{\mu}{2\pi\hbar^{2}} \int {\rm d}{{b}}\,{\rm d}z\,e^{-i{{q}}\cdot{{b}}} e^{i\chi({{b}})} }\\ &{\times\int{\rm d}{{p}}\,e^{-i{{p}}\cdot{{R}}}F({{p}}) . } \end{split} $


      One can see that, in IEA, the form factor $ F({{R}}) $ is maintained to be a complete three-dimensional function. In other words, $ F^{{\rm IEA}}({{R}}) \, = \,\int {\rm d}{{p}}\,e^{-i{{p}}\cdot{{R}}}F({{p}}) $.

      The term $ e^{-i{{p}}\cdot{{R}}} $ above can be expanded into a series of three-dimensional partial terms:

      $ e^{-i{{p}}\cdot{{R}}} = 4\pi\sum\limits_{LM}i^{-L}j_{L}(pR)Y_{LM}(\hat{{{p}}})Y^{\ast}_{LM}(\hat{{{R}}}), $


      Correspondingly, DCS for CE reactions (Eq. (1)) can be decomposed in the form:

      $ \frac{\rm{d}\sigma}{\rm{d}\Omega}(\theta) = \frac{\mu^{2}}{\hbar^{4}}\frac{(2J_{\rm{B}}+1)(2J_{\rm{b}}+1)}{(2J_{\rm{A}}+1)(2J_{\rm{a}}+1)}\sum\limits_{\substack{JSL_{{\rm tr}}\\M_{\rm{tr}}}} \big|\beta(\theta)\big|^{2}. $


      The detailed expression for the partial amplitude $ \beta(\theta) $ can easily be deduced for NEA or IEA, based on the above derivations.

      Besides CE reactions, the eikonal approximation can also be applied to the elastic scattering cross sections. DCS for elastic scattering may be written as [11, 19]:

      $ \frac{{\rm d}\sigma}{{\rm d}\Omega}(\theta) = \big|f_{\rm{el}}(\theta)\big|^{2}, $

      where the elastic scattering amplitude, $ f_{{\rm el}}(\theta) $, is defined by [11]:

      $ f_{{\rm el}}(\theta) =-\frac{1}{4\pi}\int e^{-i{{{k}}'}\cdot{{R}}}U({{R}})\psi^{\prime}({{R}}){\rm d}{{R}}. $


      Within the eikonal approximation, $ f_{{\rm el}}(\theta) $ becomes [11]:

      $ \begin{split} f_{{\rm el}}(\theta) = &{f_{\rm{C}}(\theta)+ik\int_{0}^{\infty}{\rm d}b\, b J_{0}(qb) e^{[i\chi_{{\rm C}}(b)]}}\\ &{\times[1-e^{i\chi_{{\rm N}}(b)}], } \end{split}$

      where $ f_{{\rm C}}(\theta) $ is the scattering amplitude given by a point-charge Coulomb potential [20].

    3.   Results
    • Before going into the comprehensive treatment of CE reactions, we start with a test of the coding technique by evaluating a simple case, the elastic scattering calculation based on the eikonal approximation. As demonstrated in Fig. 4, the results are in perfect agreement with the conventional Optical Model (OM) calculations [26], for scattering at $ 140$ MeV/nucleon and at small angles ($ \theta< 20 $ deg). Some divergence appears at larger angles, which is reasonable since the eikonal approximation is valid only for small scattering angles and at relatively high energies [11, 17, 20]. It should be noted that, for a direct comparison, the same OP parameters from [1, 25] are used in both the eikonal approximation and OM calculations.

      Figure 4.  Comparison of the elastic scattering DSC between our calculation (red-solid lines) and the OM calculation (black-dashed lines). (a), (b), (c) and (d) correspond to the elastic scattering of 3He at 140 MeV/nucleon on 13C, 12C, 120Sn and 26Mg targets, respectively.

      For CE reactions, firstly we perform calculations on DCS within NEA. According to Eq. (15), DCS for CE reactions is contributed by all possible and independent $ JSL_{{\rm tr}}\,-\, $components. For GT-type CE reactions, there are 110 ($ J = 1,\,S = 1,\,L_{{\rm tr}} = 0 $) and 112-components. As can be seen in Fig. 5, The results show considerable discrepancies between NEA calculations and those using the conventional DWBA code (FOLD program), for the 112-component and the total DCS, although a good agreement on the 110-component is achieved (not shown). Three GT-type CE reactions, all at $ 140$ MeV/nucleon, are checked: 12C (3He, t) 12N, 26Mg (3He, t) 26Al, and 120Sn (3He, t) 120Sb, respectively. Again we note that all input parameters are the same for both calculations, including those for the overall optical potential U, the residual NN interaction potential V and OBTD [1, 21, 22, 25]. Therefore, the difference between the calculated results is relevant to the model approximations.

      Figure 5.  Comparisons between DWBA (black-dashed lines) and NEA (red-solid lines) calculations for the 112-component and the total DCS. Three GT-type CE reactions, all at 140 MeV/nucleon, are used: (a) 120Sn (3He, t) 120Sb; (b) 26Mg (3He, t) 26Al and (c) 12C (3He, t) 12N

      Therefore, calculations are performed based on the new approach, IEA. As shown in Fig. 6, the two calculations, again using exactly the same input parameters, are in perfect agreement to each other on both 112-component and the total DCS, for the three GT-type reactions at $ 140$ MeV/nucleon. The difference between NEA and IEA for the 112-component can further be understood by looking into the contributions related to each specific quantum number. As indicated above (Eq. (9)), 112-component means $ L_{\rm tr} = 2 $ and $ M_{\rm tr} $ = $ 0, \pm 2 $ (only even numbers are effective). It is found that, when applying the eikonal approximation to the form factor, such as in NEA, the important contributions from the $ M_{\rm tr} $ = ±2 terms vanish automatically. By using the strict three-dimensional form factor, such as in IEA, these terms are recovered and good results can be achieved. This finding is inline with the physics meaning of each term, according to the kinematics picture. Since only $ M_{\rm tr} $ = $ 0 $ term is allowed for the 110-component, it does not show any difference between NEA and IEA calculations. It is clear that the treatment in IEA is crucial for the applicability of the eikonal approximation for CE reactions at intermediate energies.

      Figure 6.  Comparisons between calculations using DWBA approach (black-dashed lines) and IEA (red-solid lines), for the 112-component and total DCS. The reactions used are the same as in Fig. 5.

      We also tried to apply the IEA to describe the experimental data for an mirror reaction 13C (13N,13C) 13N at $ 105 $ MeV/nucleon [27]. Here, $ V^{K}_{s_{0}t_{0}}(r) $ at $ 105 $ MeV/nucleon is deduced through interpolation on those at 50, 100, 140, 175, 210, 270, 325, 425, 515, 650, 725, 800 and $ 1000 $ MeV/nucleon [21, 22]. OP parameters are obtained from [28]. OBTD are produced using OXBASH code with pwt interaction [29] in p shell space. As shown in Fig. 7, the IEA calculation is comparable with the experimental results and with another calculation using the single-particle model (SPM) [28]. It would be worth noting that the experimental angular distribution has no oscillatory structure due to the very limited angular resolution of the detection system [27]. What's important here is the correct reproduction of the cross section at 0 degree, from which the GT transition strength can be extracted [1, 30]. The actual IEA prediction of $ \dfrac{{\rm d}\sigma}{{\rm d}\Omega}(0^{\circ}_{{\rm c.m.}}) $ gives a value of 50.43 mb/sr or 64.73 mb/sr by using pwt [29] or ckpot interactions [31], respectively, in the shell-model calculations. Both agree with the previously extracted value of 56$ \pm $10 mb/sr [27] within the error bar.

      Figure 7.  Comparisons of the angular distributions for 13C (13N, 13C) 13N at 105 MeV/nucleon among the IEA calculation (red-solid line), SPM calculation (red-dashed line) [28] and experimental data (black dots with error bars.) [27]

    4.   Summary
    • To describe CE reactions at intermediate energies, we began with the implementation of the NEA formulations. The test calculations show considerable discrepancies compared with the conventional DWBA calculations, for CE reactions at 140 MeV/nucleon. Therefore, the formulation is improved (IEA) by removing the influence of the eikonal approximation from the form factor so that to keep it as a strict three-dimensional function. Calculations based on IEA are in good agreement with the DWBA ones for various CE reactions at 140 MeV/nucleon and also with experimental data at 105 MeV/nucleon. Since the eikonal approximation can be easily formulated in a microscopic way, relying only on the nuclear matter density distribution and the bare nucleon-nucleon interaction (see Eq. (5)), it has the obvious advantage to be applied to CE reactions at higher energies, where the phenomenological OP, required by DWBA, are rarely available. Based on the current benchmark test, IEA approach is ready to be applied to describe experimental data and to extract the related physics quantities, although some improvements could further be made at relativistic energies.

      We would like to thank Prof. C.A. Bertulani for the initial ideas and useful discussions about the theoretical description of CE reaction at intermediate energies. He also provided part of the NEA formulas.

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