Modelling inclusive breakup: application to incomplete fusion

Journal of Physics: Conference Series PAPER • OPEN ACCESS Modelling inclusive breakup: application to incomplete fusion To cite this article: Antonio M. Moro et al 2022 J. Phys.: Conf. Ser. 2340 012034 View the article online for updates and enhancements. You may also like Effect of projectile structure on evaporation residue yields in incomplete fusion reactions K Surendra Babu, R Tripathi, K Sudarshan et al. - Nuclear Structure Effects in the Quasi- Continuum Deexcitation Process of 152Dy and 156Dy Nuclei Produced in Complete and Incomplete Fusion Reactions D C J M Hageman, R V F Janssens, J Lukasiak et al. - Reaction dynamics studies for the system 7Be+58Ni D Torresi, M. Mazzocco, L Acosta et al. - This content was downloaded from IP address 150.214.182.235 on 18/05/2023 at 12:34 Content from this work may be used under the terms of theCreativeCommonsAttribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd XLIV Brazilian Workshop on Nuclear Physics Journal of Physics: Conference Series 2340 (2022) 012034 IOP Publishing doi:10.1088/1742-6596/2340/1/012034 1 Modelling inclusive breakup: application to incomplete fusion Antonio M. Moro1,2, Jin Lei3, Edward C. Simpson4 1Departmento de FAMN, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain 2Instituto Interuniversitario Carlos I de F´ısica Te´orica y Computacional (iC1), Apdo. 1065, E-41080 Sevilla, Spain 3School of Physics Science and Engineering, Tongji University, Shanghai 200092, China. 4Department of Nuclear Physics and Accelerator Applications, Research School of Physics, The Australian National University, Canberra, ACT 2601, Australia E-mail: [email protected] Abstract. We propose an extension of the inclusive breakup model of Ichimura, Austern and Vincent [Phys. Rev. C 32, 431 (1985)] for the evaluation of incomplete fusion (ICF) cross sections in nuclear reactions induced by two-body projectiles. The main idea, adopted in other methods, consists in the separation of the participant-target optical potential into its direct reaction and compound reaction components, the latter being responsible for the ICF contribution of the total nonelastic breakup cross section. Preliminary comparison with experimental data shows encouraging results. 1. Introduction Fusion of neutron- and proton-rich nuclei is a hot topic with numerous applications, ranging from nuclear astrophysics [1, 2] to the production of superheavy elements [3]. A proper quantitative description of fusion dynamics in reactions involving these exotic species has challenged reaction theory for several decades. Part of the difficulty arises due to the influence of the other reaction channels on the fusion cross sections, such as inelastic excitations of the colliding nuclei, transfer and breakup. There has been also a long-standing debate regarding the interplay and competition between the so-called complete (CF) and incomplete fusion (ICF) mechanisms. Experimentally, they are operationally defined as the fusion processes involving either the total capture (CF) or the partial capture (ICF) of the projectile charge by the target nucleus. There are many experimental evidences indicating that, in reactions induced by light, weakly-bound nuclei, such as 9Be or 6,7,8Li the CF cross sections are suppressed by 20-30% with respect to the prediction of single-barrier penetration models or even coupled-channels calculations accounting for collective excitations of the interacting partners [4–10]. The phenomenon has been attributed to the flux going to either the elastic breakup of the projectile or to ICF. A quantitative assessment of this hypothesis has been hampered by the lack of reliable and predictive models for ICF, a situation that has triggered a significant theoretical and experimental activity in recent years. In addition to its inherent interest for the understanding fusion dynamics, ICF cross sections are also needed for several practical applications, such as the so-called surrogate method, an indirect method for evaluating compound-nucleus cross sections in reactions for which the direct XLIV Brazilian Workshop on Nuclear Physics Journal of Physics: Conference Series 2340 (2022) 012034 IOP Publishing doi:10.1088/1742-6596/2340/1/012034 2 measurement is difficult or even not possible. An example is the extraction of neutron-induced cross sections of the form (n, χ), where χis a given decay channel product (γ, fission fragment, etc) from a surrogate reaction giving rise to the same compound nucleus, such as a (d, p) reaction. In this contribution, we present a practical approach to evaluate ICF cross sections in reactions induced a two-body projectile nucleus. The method is based on the inclusive breakup model developed by Ichimura, Austern and Vincent (IAV) in the middle 1980s [11]. The IAV model provides total inclusive breakup cross section for a generic reaction of the form A(a, b)B where a projectile a, after colliding with target A, dissociates into fragments band x, but only the former is observed experimentally. The method proposed here tries to isolate the ICF contribution from other breakup contributions provided by the IAV model. This is accomplished by replacing the x+Aoptical potential by a short-range imaginary potential, which simulates the fusion process. The model is particularly suited for projectiles with a developed two-body structure, such as deuterons, 6Li or 7Li. In this contribution, we consider the 7Li case as an illustrative example and present calculations for the 7Li+209Bi reaction, for which experimental data exist. The document is organized as follows. In Sec. 2.1 we present a brief remainder of the IAV DWBA formula for nonelastic breakup. Then, in Sec. 2.2 we present the proposed modification of the IAV formula to extract the ICF content of the NEB cross section. In Sec. 3 the model is applied to the reaction 7Li+209Bi and compared with experimental data for this reaction. Finally, in Sec. 4 we summarize the main conclusions of this work and anticipate some future extensions. 2. Theoretical considerations 2.1. Reminder of the IAV formula for inclusive breakup The IAV model has been presented in several recent works [12–15] so we give here only the main formulas. The goal of this model was to provide a practical way of evaluating the singles cross section for a inclusive breakup reaction of the form A(a, b)B(following the same notation used in the introduction) in which on the fragment bis detected and xcan undergo any possible interaction with the target nucleus. Thus, the observed byield includes any possible final state of the B=x+Asystem. The model separates these contributions into two groups: (i) the elastic breakup part (EBU), in which the fragment xescapes after the collision whereas the target Aremains in its ground state, and (ii) the nonelastic breakup part (NEB), which includes any nonelastic interaction of xwith A. The ICF is one of the channels contributing to NEB, namely, that in which xis captured (fused) by the target, forming some compound nucleus. Using the Feshbach projection formalism, the model provides compact expressions for the EBU and NEB parts. Since the focus of this contribution is on ICF, we give only the expression for the NEB part, which can be written as follows: d2σ dEbdΩb      IAV NEB =−2 ¯hvi ρb(Eb)⟨φ(+) x|Wx|φ(+) x⟩,(1) where Ebis the center of mass (c.m.) energy in the b+Bchannel, Ωbthe c.m. solid angle for b ejectile, viis the projectile-target relative velocity in the entrance channel, ρb(Eb) is the density of states of particle b,Wxis the imaginary part of the x+Aoptical potential Uxand φ(+) xis x-channel wavefunction which, in DWBA post-form representation, is obtained as a solution of the following inhomogeneous equation (Ex−Tx− Ux)φ(+) x( kb,rx) = ⟨rxχ(−) b|Vpost|χ(+) aϕa⟩,(2) where χ(+) aand χ(−) bare distorted waves describing the relative motion of a+Aand b+B, respectively, Vpost =Vbx +UbA −UaA is the post form transition operator, with UaA the distorted XLIV Brazilian Workshop on Nuclear Physics Journal of Physics: Conference Series 2340 (2022) 012034 IOP Publishing doi:10.1088/1742-6596/2340/1/012034 3 potential generating the distorted wave χ(+) a,ϕa(rbx) is the initial bound state wavefunction of the b+xsystem, Exis the relative energy of the x+Asystem in the final state and Txthe kinetic energy operator for this system. The relevant coordinates are depicted in Fig. 1. Note that, among these relevant coordinates, are Jacobi coordinates connecting the center of mass of each binary sub-system to the third particle. bA rx ra rb rbx x rbA Figure 1. Relevant coordinates for the description of the A(a, b)B reaction. Note that the energy Exappearing in (2) is linked to Ebas Ex=Ea−Sbx −Eb, where Sbx is the projectile separation energy into b+xand Eais the initial c.m. energy. Note also that, above a certain final energy of the bfragment, the energy Exbecomes negative and Eq. (2) corresponds to the transfer of the xfragment to bound states of the x+Asystem. Therefore, the model accounts simultaneously for both the population of bound and unbound states of the x+Asystem. For that, it becomes apparent that the Uxpotential must be energy dependent. Ideally, one should use a dispersive potential, as those introduced by Mahoux and Sartor [16]. In practice, dispersive potentials are only available for specific systems so other global, nondispersive potentials, are commonly adopted in actual calculations. The IAV model has been recently revisited by several groups [12–14] and its accuracy assessed against experimental data with rather encouraging results [15,17]. 2.2. Incomplete fusion within the IAV model As discussed in the previous section, the IAV model is intended to provide the total inclusive cross section corresponding to the detection of the bfragment in reactions of the form A(a, b)B. This results from the fact that the imaginary part appearing in the expectation value of Eq. (1) accounts in principle for all processes in which the participant fragment xinteracts non-elastically with the target nucleus. This will include the ICF cross section, but also other NEB processes not associated with the formation of a compound nucleus of the x+Asystem, such as target excitation or nucleons exchange between xand A. Keeping with the fact that compound nucleus formation occurs at small inter-cluster separations, when the nuclei have overcome their mutual barrier, it is plausible to associate the ICF with the absorption due to a short range x+Aimaginary potential. This idea has been exploited by other authors in alternative approaches [18–21] with reasonable success. With this simple prescription, the ICF cross section is evaluated using the IAV expression, Eq. (1), in which the imaginary part of Uxis replaced by a short-range potential. Note that, for a meaningful and predictive application of this idea, the calculated cross section should be independent of the choice of this potential, provided the latter is sufficiently short-ranged and deep, so that it can simulate the usual ingoing boundary condition for fusion. 3. Applications To illustrate the method outlined in the previous section, we consider the reaction 7Li+209Bi, for which complete and incomplete fusion data have been measured at the ANU facility at Canberra [6,7,22]. XLIV Brazilian Workshop on Nuclear Physics Journal of Physics: Conference Series 2340 (2022) 012034 IOP Publishing doi:10.1088/1742-6596/2340/1/012034 4 The following potentials were used in the IAV calculations: for the entrance channel (7Li+209Bi) we used the 7Li+208Pb potential from Ref. [23]. For potentials for α+209Bi and α+211Po* were taken from [24] and the triton optical potentials from [25]. The 7Li bound state wavefuntion was obtained with the potential model of Buck and Merchant [26]. In the left panel of Fig. 2 we compare the measured ICF data for this reaction with the present calculations. As explained in [6,7], ICF was identified from the decay of the At and Po products, presumably stemming from the partial fusion of the projectile. For the calculations we show the individual contributions to the ICF cross section, namely, α-ICF (i.e., αabsorbed) and t-ICF (t-absorbed) as well as their sum. To compute the α-ICF (t-ICF), the imaginary part of the α+209Bi (t+209Bi) system was replaced by a short-range imaginary potential of Woods-Saxon form and parameters W0= 50 MeV, Ri= 1.0×2091/3fm, ai= 0.2 fm. The calculated ICF agrees well with the data for incident energies up to ∼35 MeV, but overestimate them at higher energies. Interestingly, the results are very similar to those reported in Ref. [20], in which the authors employed a very different approach based on the CDCC method. 25 30 35 40 45 50 55 Elab (MeV) 0 200 400 600 800 σICF (mb) Exp.: ICF (ΣAt + ΣPo) t ICF α ICF Total ICF 7Li+209Bi 0 0 10 10 20 20 30 30 40 40 Ex in 213At (MeV) 0 10 20 30 dσ/dEx (mb/MeV) 0 10 20 30 40 Ex in 212Po (MeV) 0 20 40 60 dσ/dEx (mb/MeV) 209Bi(7Li,t)213At* @ Elab=40 MeV Sα 209Bi(7Li,α)212Po* @ Elab=40 MeV St Figure 2. Left: ICF cross sections for 7Li+209Bi as a function of the incident energy. Experimental data from [6, 7] are compared with the calculations based on the IAV model. Right: Excitation energy distributions for the α-ICF (top) and t-ICF (bottom) processes at an incident energy of 40 MeV. The arrows indicate the αand triton separation energies in the respective residual nuclei. A virtue of the present method is the possibility of computing, in addition to angle and energy integrated cross sections, more detailed observables, such as angular and/or energy distributions of the detected fragments. As an example, we show in the right panel of Fig. 2 the excitation energy distribution of the residual nucleus formed after the capture of an αparticle (upper panel) or a triton (lower panel) for a projectile incident energy of 40 MeV. The arrows indicate the respective αand triton separation energies of the residual nuclei, namely, 213At and 212Po, respectively. It can be seen that most of the α-ICF (t-ICF) corresponds to the α(triton) capture to very high excited states of the target, forming a highly excited state of the 213At (212Po) nucleus that will eventually decay mostly by neutron emission. The calculations provide also the angular momentum distribution of the populated states. This is shown in Fig. 3 for the t-ICF and α-ICF processes. In these calculations, the intrinsic spins are ignored so the angular momentum of the compound nucleus stems from the relative orbital angular momentum of the captured fragment and the target. It is seen that states with very high angular momentum are populated (ℓ∼15) for both the tand αcapture. We note that XLIV Brazilian Workshop on Nuclear Physics Journal of Physics: Conference Series 2340 (2022) 012034 IOP Publishing doi:10.1088/1742-6596/2340/1/012034 5 0 5 10 15 20 Angular momentum of residual nucleus 0 20 40 60 σ (mb) t-ICF α-ICF 7Li+209Bi @ Elab=40 MeV Figure 3. Angular momentum distribution of populated states in the tand αICF for the reaction 7Li+209Bi at 40 MeV. Note that the intrinsic spins of tand 209Bi are ignored. the combined excitation energy and angular momentum distributions could be used as input of Hauser-Feshbach calculations to study the subsequent decay of the residual nucleus, a possibility that would be worth exploring in future works. It is also worth noting that a more realistic spin distribution could be obtained incorporating the intrinsic spins of the colliding partners and adding, when appropriate, some spin-dependent terms between the fragments. For example, the inclusion of a spin-orbit interaction in the x−Asystem would produce the usual j=ℓ±1/2 splitting of the cross section. We believe however that the adopted model without intrinsic spins is realistic enough for the present illustrative purposes. A more detailed analysis of this and similar reactions, including comparison with available data, will serve to better evaluate the validity and predictability of these calculations. 4. Summary and conclusions We have presented a model to evaluate incomplete-fusion cross sections in the collision of a twobody composite nucleus with a target. The model uses a modified version of the IAV model for inclusive breakup in which, to extract the ICF component of the NEB cross section, the potential of the captured fragment with the target is modeled by a short-range absorptive potential. The model has been compared with data for the 7Li+209Bi reaction with encouraging results. Comparison with data from other reactions will be necessary to assess the validity of this prescription. For example, the same model could be applied to reactions induced by 6Li, described as α+d. Work in this direction is in progress and will be presented elsewhere. Acknowledgments J.L. is supported by National Natural Science Foundation of China (Grants No.12105204) and by the Fundamental Research Funds for the Central Universities. A.M.M. is supported by the I+D+i project PID2020-114687GB-I00 funded by MCIN/AEI/10.13039/501100011033, by the grant Group FQM-160 and by project P20 01247, funded by the Consejer´ıa de Econom´ıa, Conocimiento, Empresas y Universidad, Junta de Andaluc´ıa (Spain) and by “ERDF A way of making Europe”. 5. References [1] Chamel N and Haensel P 2008 Living Rev. Rel. 11 10 XLIV Brazilian Workshop on Nuclear Physics Journal of Physics: Conference Series 2340 (2022) 012034 IOP Publishing doi:10.1088/1742-6596/2340/1/012034 6 [2] Steiner A W 2012 Phys. Rev. C 85 055804 [3] Loveland W 2007 Phys. Rev. C 76 014612 [4] Dasgupta M, Hinde D J, Butt R D, Anjos R M, Berriman A C, Carlin N, Gomes P R S, Morton C R, Newton J O, Szanto de Toledo A and Hagino K 1999 Phys. Rev. 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