Role of 6Li non-sphericity in nuclear reactions below the Coulomb barrier

DOI 10.1393/ncc/i2022-22123-7 Communications: SIF Congress 2021 IL NUOVO CIMENTO 45 C (2022) 123 Role of 6Li non-sphericity in nuclear reactions below the Coulomb barrier S. S. Perrotta(1)(2)(3)(4), L. Fortunato(4)(5),J.A.Lay(2)(6) and M. Colonna(3)(1) (1)Universit`a degli Studi di Catania - Catania, Italy (2)Universidad de Sevilla - Seville, Spain (3)Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud - Catania, Italy (4)Universit`a degli Studi di Padova - Padua, Italy (5)Istituto Nazionale di Fisica Nucleare, Sezione di Padova - Padua, Italy (6)Instituto Interuniversitario Carlos I de F´ısica Te´orica y Computacional - Seville, Spain received 31 January 2022 Summary. — We investigate the impact of the 6Li ground-state deformation on Coulomb-barrier penetrability in nuclear reactions between 6Li and a structureless projectile. The 6Li ground state is described through a quantum di-cluster model, including a quadrupolar component which induces a tensor term in the projectiletarget interaction. The corresponding ground-state form factor, for each possible 6Li orientation, is employed as a potential barrier in a Wentzel-Kramers-Brillouin- Jeffreys (WKB) radial penetrability calculation. The formalism was applied to the 6Li–p scattering. Throughout the sub-Coulomb energy range, we find no significant influence of the tensor interaction on the overall penetrability. 1. – Introduction In a nuclear reaction between charged particles at energies below the Coulomb barrier, the process of overcoming the reactants electrostatic repulsion through quantum tunnelling plays a paramount role, hence the need to study it in detail. In heavy-ion fusion reactions, correlations between the reactants internal degrees of freedom are known to increase the sub-barrier cross sections [1]. In light-ion reactions, the role of clustering in the barrier penetration process has been studied through semi-classical models [2], in the attempt to explain a set of experimental anomalies known as the electron screening problem, which is also important for its possible astrophysical implications. The goal of this work is to analyse the effects of the quadrupolar deformation of the 6Li ground state on the Coulomb barrier seen by another nucleus, which is a proton in the present application. In particular, a 6Li wave-function including both sand dcomponents is constructed phenomenologically as in ref. [3] (sect. 5.3), then employed to generate a form factor for the 6Li–p interaction, in analogy with the procedure in refs. [3,4]. Finally, the transmission coefficient for barrier penetration is computed in the WKB approximation as in ref. [2]. The model and the performed calculations are covered in greater detail in ref. [5]. Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0) 1 2S. S. PERROTTA et al. 2. – Construction of the 6Li di-cluster-model deformed state The 6Li nucleus is here modelled as a bound state of two inert clusters, an α-particle and a deuteron, each found in the ground state of the corresponding isolated nuclide, with spin-parity 0+and 1+respectively. Let |Ψ1,σbe a state for the internal motion of both clusters, in which the deuteron has definite spin projection equal to σ. Similarly, let |Li1,M bethegroundstateof6Li with spin-parity 1+and spin projection M.To correctly couple both angular momentum and parity, the orbital-angular-momentum modulus quantum number for the inter-cluster motion, L, must be 0 or 2. The 6Li state, projected on a specific inter-cluster displacement r, expressed in spherical coordinates as r=(r, θ, φ), can thus be written as (1) r|Li1,M = L,m (L, m),(1,M −m)|1,McLφL(r)YL,m(θ,φ)|Ψ1,M−m where YL,m is a spherical harmonic, (j1,m 1),(j2,m 2)|J, Mis a Clebsch-Gordan coefficient, the cLare a set of weights, and each φL(r) is a radial wave-function normalised to 1. Since only bound states are considered, all φLand cLare taken to be real, further setting that each φL(r) is non-negative at infinity and c0=1−|c2|2. The radial wavefunctions are the same employed in ref. [3] (sect. 5.3). c2can be adjusted with respect to the di-cluster-model predictions for the charge electric quadrupole moment [3] (eq. (5.4)) or the magnetic dipole moment (deduced from ref. [6], eq. (A.9)), finding respectively c2=−0.0909 or −0.257, using experimental data in [7]. The agreement may possibly be improved using a more microscopical construction of the radial wave-functions (as in ref. [4]). Within the present model, the conclusions of the study do not change by adjusting c2on either observable. The figures shown here refer to c2=−0.257. 3. – Construction of the projectile-target potential Let V1pand V2pbe phenomenological potentials, taken as central for simplicity, for the interaction between the structureless projectile, a proton, and each cluster composing the 6Li target. The potentials can be expressed in terms of the α–d displacement, r, and the 6Li–p distance, R. The complete projectile-target potential, Vtp(r,R), is then V1p(r,R)+V2p(r,R). To focus on the impact of ground-state deformations, all couplings to excited states are neglected. Then, the eigenvalue problem for the α–d motion can be decoupled from the 6Li–p scattering problem. To solve the latter, it is sufficient to consider the form factor Li1M|Vtp|Li1M, denoted as V(M,M,R). Performing the integration over r,itisV(M,M,R)=1,M|˜ Vtp|1,M,where|J, Mis a state describing only the 6Li spin degree of freedom, and ˜ Vtp acts only on such spin and on R. Vtp can be decomposed in multipoles of both coordinates (rand R). In a classical cluster model, as in ref. [2], all multipoles would be retained. Here, the form factor selects only those compatible with the state under study (|Li1Min eq. (1)), namely the monopole and the quadrupole. These multipoles of Vtp generate, respectively, a central and a tensor component in ˜ Vtp, which can thus be written as (2) ˜ Vtp =UC(R)+UT(R)1 2 Jt·R/R2 −1 3 J2 t ROLE OF 6Li NON-SPHERICITY IN REACTIONS BELOW THE COULOMB BARRIER 3 where  Jtis the 6Li spin vector operator and R=|R|. The central functions UC(R) and UT(R) were computed as in ref. [4] (eqs. (9), (11)), but without truncating to first order in c2. A sample of the radial profiles of the form factor computed in this manner isshowninfig.1. Theadoptedα–p interaction is the central part of the potential in ref. [8] (eq. (4.3)), while the d–p potential was obtained from the Fr2in code [9]. 4. – The 6Li + p barrier penetrability In analogy with the approach followed in ref. [2], consider a fixed centre-of-mass collision energy E, spin projection Mof 6Li, and orientation of R. The corresponding s-wave radial transmission coefficient, TM,θ(E), was calculated in WKB approximation [10] (sect. VI.10), adopting a central potential barrier VM,θ(R) equal to the form factor V(M,M,R). Treating the complete non-central problem phenomenologically as a set of many central problems (as a function of the direction) greatly simplifies the computations, but may destroy some features of the system which are relevant for the process of interest. This appears to be the most severe approximation performed in the calculation discussed here. Let mbe the reactants reduced mass and Zthe product of their charge numbers. The angle-integrated cross-section for barrier penetration in s-wave, σ(E), can be expressed as σ(E)=2π/(2mE)T(E). Furthermore, let S(E) be the corresponding astrophysical S-factor, defined as S(E)=Eexp(2πη)σ(E), where η=αeZmc2/(2E), with αebeing the fine-structure constant and cthe speed of light in vacuum. Figure 2 displays, as orange points, the average over all orientations and spin projections of the computed astrophysical factor. The black solid line is instead the astrophysical factor calculated using the average of VM,θ (shown as the black solid line in fig. 1) as potential barrier. The difference between the two calculations is seen to be negligible. Note that taking the average of the deformed potential is slightly different than considering a spherical 6Li, but the deviation is comparable with the uncertainty introduced by the phenomenological radial wave-functions employed in eq. (1). Fig. 1. – Radial profiles of the form factor V(M,M,R), for different 6Li spin projections, M and M, and angles θbetween projectile-target momentum and 6Li quantization axis. The black solid line is the average over all orientations (i.e.,justUCin eq. (2)). 4S. S. PERROTTA et al. Fig. 2. – Preliminary astrophysical factor for 6Li–p WKB radial barrier penetration in arbitrary units. Orange points refer to the average of the penetrability over all angles and spin projections. The black solid line was computed using the potential shown as the black solid line in fig. 1. See text for details. 5. – Conclusions The role of the ground-state quadrupole deformation of 6Li on Coulomb barrier penetrability in reactions with a proton was explored by modelling this system as a quantum bound state of αand d, expanding the semi-classical model proposed in ref. [2]. 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