Recent applications of four-body continuum-discretized coupled-channels calculations to 6He reactions

AIP Conference Proceedings 1351, 29 (2011); https://doi.org/10.1063/1.3608932 1351, 29 © 2011 American Institute of Physics. Recent applications of four-body Continuum-Discretized Coupled-Channels calculations to reactions Cite as: AIP Conference Proceedings 1351, 29 (2011); https://doi.org/10.1063/1.3608932 Published Online: 10 August 2011 Manuela Rodríguez-Gallardo, and Antonio M. Moro ARTICLES YOU MAY BE INTERESTED IN Comparison of the effects of couplings to breakup channels in reactions induced by 6Li and 6He on the same 64Zn target AIP Conference Proceedings 1681, 060006 (2015); https://doi.org/10.1063/1.4932292 Elastic scattering of and on a target AIP Conference Proceedings 1231, 173 (2010); https://doi.org/10.1063/1.3428911 Pairing interaction and reaction mechanism for one- and two-particle transfer reactions: A simple model in one dimension AIP Conference Proceedings 1681, 060001 (2015); https://doi.org/10.1063/1.4932287 Recent applications of four-body Continuum-Discretized Coupled-Channels calculations to 6He reactions Manuela Rodríguez-Gallardo∗,† and Antonio M. Moro† ∗IEM, CSIC, Serrano 123, 28006 Madrid, Spain †Depto. FAMN, Universidad de Sevilla, Apto. 1065, 41080 Sevilla, Spain Abstract. The four-body continuum-discretized coupled-channels approach using a continuum-bins scheme of discretization for three-body projectiles, that has been recently developed, is presented. The formalism is discussed and applied to reactions induced by the Borromean nucleus 6He on different targets (27Al, 64Zn, 120Sn, and 208Pb), with special emphasis on the role of the Coulomb couplings. Keywords: Few-body systems. Direct reactions. Reactions induced by unstable nuclei. Coupledchannel and distorted-wave models. Halo nuclei. PACS: 21.45.-v,24.50.+g,25.60.-t,24.10.Eq,27.20.+n INTRODUCTION Exotic nuclei have been object of study in recent years due to their special properties, very different from the nuclei of the stability valley. Among the exotic nuclei, an interesting case are the so-called halo nuclei. These are weakly-bound nuclei comprising a core and one or two valence neutrons orbiting at distances larger than the typical nuclear radii. In recent years, large accelerators have been built around the worldto study this kind of exotic nuclei. Simultaneously, appropriate reaction theories have been also developed to extract reliable information from the experimental data on exotic nuclei. The continuum-discretized coupled-channels (CDCC) framework has been wide and successfully used for years to analyze experimental data on two-body weakly-bound nuclei using a three-body formalism (that is, two-body projectile plus target). Recently, the method has been extended to four-body problems (three-body projectile plus target), what allows to study reactions with three-body weakly-bound nuclei, in particular, Borromean nuclei. These nuclei has the special characteristic that none of the three binary subsystems is bound. This is the case of 6He(4He+n+n) or 11Li(9Li+n+n). The CDCC method, for a three-body scattering problem, was developed in the 80s. The basic idea of the method, explained in detail in Refs. [1, 2], consists in expanding the states of the whole system (projectile+target) into the states of the projectile. For weakly-bound projectiles, the coupling to unbound states (that form a continuum) is relevantwithin the reaction dynamics. Then, we need to treat properly the continuum of the projectile. For this purpose, in the standard three-body CDCC formulation, the continuum spectrum of the projectile is truncated at a maximum excitation energy ε max and the XXXIII Brazilian Workshop on Nuclear Physics AIP Conf. Proc. 1351, 29-38 (2011); doi: 10.1063/1.3608932 © 2011 American Institute of Physics 978-0-7354-0908-8/$30.00 29 model space, from the breakup threshold to ε max, is then divided into intervals, where the number and positions of the intervals can depend on the properties (e.g. resonant or non-resonant) of the continuum of the system. For each such interval, or energy bin, a representative square-integrable state is constructed as a linear superposition of the twobody scattering states in the interval. The method has been enormously successful in the description of elastic and breakup observables in reactions involving weakly-bound two-body projectiles [3, 4, 5] and has been recently extended to include core excitation [6]. An alternative to treat the continuum of the projectile is to use a Pseudo-State (PS) basis. This procedure consists in diagonalizing the Hamiltonian of the system in a truncated discrete basis and taking the positiveenergy eigenstates as representativeof the continuum. There are many PS methods in the literature. Within the CDCC framework, different bases have been used like complex-range Gaussian [7], Transformed Harmonic Oscillator (THO) [8], and Lagrange [9]. In the past decade the CDCC framework has been extended to four-body problems. Due to the difficulties in calculating the true continuum of a three-body projectile and making bins, the CDCC was first extended using PS bases like complex-range Gaussian [10, 11] and THO [12, 13]. Most recently, the binning procedure has been developed [14], providing more stable results for heavy targets. In this work we present some recent calculations for elastic and breakup observables, for different reactions induced by the Borromean nucleus 6He, obtained using the fourbody CDCC formalism with the binning procedure. In Section II we explain the formal aspects of the method. In Section III we show the observables obtained for the different reactions considered. Section IV is the summary and conclusions. FOUR-BODY CDCC FORMALISM To describe the three-body ground and excited continuum states of the projectile we make use of a hyperspherical harmonics (HH) expansion basis [15]. This involves use of the one radial and five angular hyperspherical coordinates, ρ , α ,bx,by, obtained from the normalized Jacobi coordinates~x,~y(see Fig. 1) of the three bodies [15, 13]. Note that there are three different Jacobi sets. The quantum number set, β , that defines each threebody channel [13] are the hypermomentum K, the orbital angular momenta lxand lyin coordinates~xand~y, their total~ l=~ lx+~ ly, the total spin Sxof the particles associated with coordinate~x, and the intermediate summed angular momentum ~ jab =~ l+~ Sx. If the spin of the third particle, I, is assumed fixed then the total angular momentum is ~ j=~ jab +~ I with projections µ . Note that, for each continuum energy ε and total angular momentum j, there will be as many independent solutions of the three-body scattering problem, as the number of outgoing channels β considered. These solutions can be chosen as the incoming channels β ′, but any orthogonal combination of these could be equally valid. Based on these total angular momentum eigenstates, Y β j µ (Ω)[13], where Ω≡ ( α ,bx,by), the bin wave functions are defined as φ bin nj µ (~x,~y) = ∑ β Rbin n β j( ρ )Y β j µ (Ω),(1) 30 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 x r yr r R 2 1 3target projectile FIGURE 1. (Color online) Relevant coordinates for the scattering of a three-body projectile by a structureless target. where the label nincludes reference to the energy interval of the bin [ κ 1, κ 2], as well as to the set of quantum numbers β ′. The functions Rbin n β j( ρ )in Eq. (1) are the associated hyperradial wave functions, Rbin n β j( ρ )≡Rbin [ κ 1, κ 2] β ′ β j( ρ ) = 2 p π N β ′jZ κ 2 κ 1 d κ e−i δβ ′j( κ )f β ′j( κ )R ββ ′j( κ , ρ ),(2) where κ =p2m| ε |/¯ his the momentum associated to the continuum energy ε , R ββ ′j( κρ )are the continuum hyperradial wave functions with δβ ′j( κ )their scattering phase shift, and f β ′j( κ )is a weight function with N β ′jits normalization constant. Note that for each jand three-body energy bin we must construct wave functions for all allowed incoming channels β ′. Further, as is explicit in Eq. (1), for each nwe must also construct Rbin n β j( ρ )for all allowed outgoing channels β . It follows that to include a large number of β ′channels is a severe computational challenge, and that it is desirable to establish a hierarchy of the continuum states according to their importance to the reaction dynamics. In so doing, we may be able to describe scattering observables using only a selected set of states, the number of them depending on the reaction under study. To this end, we make use of the eigenstates of the multi-channel three-body S-matrix [16], or eigenchannels (EC), as follows. (i) For each jand continuum energy ε , the S-matrix in the β basis is diagonalized to obtain its EC, enumerated by γ , their corresponding eigenvalues exp[2i δγ j( κ )] and eigenphases δγ j( κ ). (ii) The magnitudes of these eigenphases are used to order the EC. It has been shown [14] that those EC with largest phase shifts are the most strongly coupled in the reaction dynamics, and thus a hierarchy of states can be established by such an ordering. This leads to the possibility of a truncation in the number nec of EC included and testing of the convergence with respect to this number. The bin states given in Eq. (1) are a discrete representation of the states of the threebody projectile. From them, the four-body wavefunction of the projectile-target system, schematically depicted in Fig. 1, is formed as ΨJM(~ R,~x,~y) = ∑ nj µ LML φ bin nj µ (~x,~y)hLMLj µ |JMiiLYLML(b R)1 RfJ Lnj(R),(3) where ~ Ris the coordinate from the target to the center of mass of the projectile, Lis the orbital angular momentum of the projectile-target relative motion and Jis the total 31 angular momentum, ~ J=~ L+~ j. The radial functions fJ Lnj(R)satisfythe system of coupled equations −¯ h2 2mrd2 dR2−L(L+1) R2+ ε nj −EfJ Lnj(R)+ ∑ L′n′j′ iL′−LVJ Lnj,L′n′j′(R)fJ L′n′j′(R) = 0, (4) where mris the reduced mass of the projectile-target system. The coupling potentials VJ Lnj,L′n′j′(R)are then VJ Lnj,L′n′j′(R) = hLnjJM|b Vpt(~r1,~r2,~r3)|L′n′j′JMi,(5) where the ket |LnjJMidenotes the function ΦJM Lnj(b R,~x,~y)given by ΦJM Lnj(b R,~x,~y) = ∑ µ ML φ bin nj µ (~x,~y)hLMLj µ |JMiYLML(b R).(6) To calculate these coupling potentials, a multipole expansion of the projectile-target interaction is developed. The procedure is analogous to that for a three-body problem reported in Ref. [4]. We assume that the projectile-target interaction is the sum of the interactions of each particle of the projectile with the target, Vkt(~rk)with k=1,2,3. For each pair potential, an appropriate Jacobi set is chosen so that the corresponding coordinate~rkdepends only on the vectors ~ Rand ~yk. Assuming that the potentials are central, the coefficients of the multipole expansion are generated as Vk Q(R,yk) = 1 2Z+1 −1Vk(rk)PQ(zk)dzk,(7) where PQ(zk)is a Legendre polynomial, Qis the multipole order and zk=b ~yk·b ~ Ris the cosine of the angle between~ykand ~ R. So, the coupling potential can be expressed as VJ Lnj,L′n′j′(R) = ∑ Q (−1)J−jˆ Lˆ L′L Q L′ 0 0 0 W(LL′jj′,QJ)FQ nj,n′j′(R),(8) where the radial form factor FQ nj,n′j′(R)is FQ nj,n′j′(R) = (−1)Q+2j−j′ˆ jˆ j′(2Q+1)∑ ββ ′ 3 ∑ k=1∑ β k β ′ k N ββ kN β ′ β ′ k(9) ×(−1)lxk+Sxk+j′ abk−jabk−Ik δ lxkl′ xk δ SxkS′ xk ˆ lyk ˆ l′ yk ˆ lkˆ l′ kˆ jabk ˆ j′ abk lyk Q l′ yk 0 0 0  ×W(lkl′ klykl′ yk;Qlxk)W(jabk j′ abklkl′ k;QSxk)W(jj′jabk j′ abk;QIk) ×Z Z(sin α k)2(cos α k)2d α k ρ 5d ρ Rbin n β j( ρ ) ϕ lxklyk Kk( α k)Vk Q(R,yk) ϕ lxkl′ yk K′ k( α k)Rbin n′ β ′j′( ρ ), 32 with β kbeing the set of quantum numbers in the k’th Jacobi system where the potential does not depend on xk, and β being the set in the Jacobi system in which the states of the projectile are calculated. The matrix elements N ββ ktransform the hyperangular, angular and spin parts of the wave functions from one Jacobi set to another. Their explicit expression as a function of the Raynal-Revai coefficients is developed in Ref. [17]. Note that Eqs. (8) and (9) are completely general, and do not depend on the nature of the basis. APPLICATION TO 6HE+TARGET REACTIONS We apply the formalism of the preceding section to several reactions induced by the Borromean halo nucleus 6He. The reactions considered, for which elastic experimental data exist, are 6He+27Al at 11 MeV, 6He+64Zn at 13.6 MeV, 6He+120Sn at 17.4 MeV, and 6He+208Pb at 22 MeV. Note that each target has a mass approximately twice that of previous target and the energies in the laboratory frame are near the Coulomb barrier. So, a study of this set of reactions can give us a insight on the role of the Coulomb interaction as the mass of the target increases. Here we use the same structure model for the three-body system 6He, as in Refs. [12, 13, 14]. The nucleus is treated as a three-body system of an inert α particle core and two valence neutrons. A notable property of 6He is that none of its binary sub-systems bind, while the three-body system has a single bound state with binding energy of 0.973 MeV and total angular momentum j π =0+. Its low-lying continuum spectrum is dominated by a narrow j π =2+resonance, 0.825 MeV above threshold. The Hamiltonian includes two-body potentials plus an effective three-body potential. The 6He ground-state and continuum wavefunctions are generated using the codes FACE [17] and STURMXX [18]. The maximum hypermomentum used was Kmax =8. The parameters of the threebody interaction are adjusted to reproduce the ground-state separation energy and matter radius and the resonance energy (for j=1−and 2+states). The calculated ground state energy was 0.953 MeV and the root mean squared (rms) radius was 2.46 fm (assuming a rms radius of 1.47 fm for the α particle). The coupled-channels equations were solved using the code FRESCO [19], that reads the coupling potentials externally. We included in the calculation the states with angular momentum j=0+, 1−, 2+and the projectile-target interaction multipole couplings with order Q=0,1,2. The fragment-target interactions were represented by optical potentials which reproduce the elastic scattering at the appropriate energy. The potentials used are from the global parametrization of Koning and Delaroche [20] for the n+target subsystems. For the 4He+target subsystems the choice depends on the target, being from [21] for 27Al, [22] for 64Zn, [23] for 120Sn, and [24] for 208Pb. For all these reactions we have found convergence including only the first fourth EC and discretizing the continuum into 6 and 9 bins, for each EC, for j=0+,2+and j=1−, respectively. The maximum energy of 6He included varies from 6 MeV for 6He+27Al and 6He+64Zn to 7 MeV for 6He+120Sn, and 8 MeV for 6He+208Pb. 33 0 10 20 30 40 50 60 θc.m. (degrees) 0 0.2 0.4 0.6 0.8 1 σ/σRuth Experimental data No continuum 4b-CDCC: Nuclear+Coulomb 4b-CDCC: Nuclear Al 0 20 40 60 80 100 120 θc.m.(degrees) 0 0.2 0.4 0.6 0.8 1 σ/σRuth Experimental data No continuum 4b-CDCC: Nuclear+Coulomb 4b-CDCC: Nuclear Sn 0 30 60 90 120 150 180 θc.m. (degrees) 0 0.2 0.4 0.6 0.8 1 σ/σRuth Experimental data No continuum 4b-CDCC: Nuclear+Coulomb 4b-CDCC: Nuclear Zn 0 30 60 90 120 150 180 θc.m. (degrees) 0 0.2 0.4 0.6 0.8 1 σ/σRuth Experimental data No continuum 4b-CDCC: Nuclear+Coulomb 4b-CDCC: Nuclear Pb FIGURE 2. (Color online) Elastic differential cross section (ratio to Rutherford) in the center of mass frame for 6He+27Al at 11 MeV (top left), 6He+64Zn at 13.6 MeV (top right), 6He+120Sn at 17.4 MeV (bottom left), and 6He+208Pb at 22 MeV (bottom right). The experimental data are taken from Refs. [25], [22], [26], and [27] for 6He+27Al, 6He+64Zn, 6He+120Sn, and 6He+208Pb, respectively. Elastic scattering In Fig. 2 we show the elastic differential cross section distributions for the reactions considered. The yellow circles are the existing experimental data from Refs. [25, 22, 26, 27]. The dotted blue lines are the calculations without including the continuum. The black solid lines are the full four-body CDCC calculations including both nuclear and Coulomb interactions. Finally the dashed red lines are the full four-body CDCC calculations including only nuclear couplings (keeping the central Coulomb interaction between the colliding nuclei). We can see that in all the reactions the four-body CDCC calculations, with Coulomb and nuclear interactions, reproduces fairly well the experimental data. Only for the 6He+208Pb system, the experimental data at backward angles is somewhat underestimated by the calculation. We can also see the relevance of including the continuum to reproduce the experimental data. Comparing with the nuclear calculation we see that the role of the Coulomb interaction is practically negligible for the lightest target 27Al whereas is crucial for the heaviest target 208Pb. Breakup In Fig. 3 we present the breakup differential distributions as a function of the center- of-mass scattering angle. The black solid lines are the full four-body CDCC calculations 34 0 10 20 30 40 50 60 θ (degrees) 1 10 100 dσbu/dΩ (mb/sr) 4b-CDCC: Nuclear+Coulomb 4b-CDCC: Nuclear Al 0 20 40 60 80 100 120 θc.m. (degrees) 1 10 100 dσbu/dΩ (mb/sr) 4b-CDCC: Nuclear+Coulomb 4b-CDCC: Nuclear Sn 0 30 60 90 120 150 180 θc.m. (degrees) 1 10 100 dσbu/dΩ (mb/sr) 4b-CDCC: Nuclear+Coulomb 4b-CDCC: Nuclear Zn 0 30 60 90 120 150 180 θc.m. (degrees) 1 10 100 dσbu/dΩ (mb/sr) 4b-CDCC: Nuclear+Coulomb 4b-CDCC: Nuclear Pb FIGURE 3. (Color online) Breakup differential cross section distribution in the center of mass frame for 6He+27Al at 11 MeV (top left), 6He+64Zn at 13.6 MeV (top right), 6He+120Sn at 17.4 MeV (bottom left), and 6He+208Pb at 22 MeV (bottom right). TABLE 1. Total breakup and reaction cross sections. 6He+27Al 6He+64Zn 6He+120Sn 6He+208Pb σ nuc+coul reac (mb) 1661.7 1442.9 1495.9 1389.8 σ nuc+coul bu (mb) 187.09 97.06 161.85 266.28 σ nuc bu (mb) 166.71 56.15 67.44 70.84 including both nuclear and Coulomb interactions and the dashed red lines are the full four-body CDCC calculations including only nuclear couplings. Here we see that the Coulomb breakup is not relevant for the lightest target, whereas gets dramatically important as the mass of the target increases. We can see this fact more clearly in Table 1 where the total reaction and breakup cross sections together with the total nuclear breakup cross section are shown. It is noticeable the large decrease of the nuclear breakup cross section with respect to the nuclear plus Coulomb breakup cross section in the case of the heaviest target, whereas both breakup cross sections are very close for the lightest target.1 1Note that the total reaction and breakup is larger for the lightest target since the energy of the reaction is well above the Coulomb barrier with respect the other three reactions. 35 Trivial local polarization potential We have also extracted from the four-body CDCC calculations the so-called trivial local polarization (TLP) potential [28]. This is a local and L-independent potential which represents the overall effect of the breakup channels on the elastic scattering. This potential is constructed in such a way that the one-channel calculation performed with the potential Ubare(r) +UTLP(r)gives approximately the same elastic scattering as the full CDCC calculation. The bare potential, Ubare(r)is just the sum of the fragmenttarget interactions convoluted with the ground state density of the 6He nucleus. Figure 4 shows these polarization potentials, due to Coulomb and nuclear interaction with solid black lines and nuclear with dashed lines, calculated for the different systems at different energies. The real part of the TLP potentials is repulsive (except at very short distances where the details of this potential are probably not meaningful). This repulsive component is mainly due to nuclear couplings [29, 30, 31]. However, as the target mass increases the real part starts to exhibit a long-range negative tail at distances larger than the strong absorption radius. The effect of this long-range attractive tail is to reduce the the height of the barrier of the bare potential leading to an enhancement of the absorption [32]. This tail is known to arise from dipole Coulomb couplings and is consistent with qualitative features found in recent optical model fits of the 6He+208Pb data [29]. The imaginary parts of the TLP potentials are mostly absorptive. Both the real and imaginary parts extend to large distances, well beyond the strong absorption radius. These features are consistent with the findings of Mackintosh and Keeley [33] and Rusek [34]. The TLP extracted from the calculation with only nuclear couplings does not show the attractive tail for the real part, evidencing the Coulomb origin of this tail. Besides, the imaginary parts for all the targets have a longer range than in the calculation including the Coulomb couplings. This effect is larger for the heaviest target and very small for the lightest target. SUMMARY AND CONCLUSIONS We have presented the four-body CDCC formalism to study the reactions induced by three-body projectiles like the Borromean nucleus 6He. To treat the three-body continuum of the projectile we use a continuum-bins scheme that has been recently developed. The formalism has been applied to several reactions of 6He with increasing mass targets: 6He+27Al at 11 MeV, 6He+64Zn at 13.6 MeV, 6He+120Sn at 17.4 MeV, and 6He+208Pb at 22 MeV. We have compared the elastic cross section distributions calculated with the fourbody CDCC with the existing experimental data. In general, we have found a good agreement. We have compared the calculated distributions including Coulomb plus nuclear couplings with those performed including only nuclear couplings and we have found that the role of the Coulomb breakup grows drastically from the lightest to the heaviest target. The same effect is present in the breakup angular distributions. We have also presented the TLP potentials extracted from the four-body CDCC calculations, with and without Coulomb breakup. Inclusion of Coulomb couplings has 36