Transfer induced by core excitation within an extended distorted-wave Born approximation method

PHYSICAL REVIEW C 92, 014613 (2015) Transfer induced by core excitation within an extended distorted-wave Born approximation method M. G´ omez-Ramos*and A. M. Moro† Departamento de FAMN, Facultad de F´ ısica, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain J. G´ omez-Camacho‡ Departamento de FAMN, Facultad de F´ ısica, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain and Centro Nacional de Aceleradores, Avda. Thomas A. Edison, E-41092, Sevilla, Spain I. J. Thompson§ Lawrence Livermore National Laboratory, L-414, Livermore, California 94551, USA (Received 17 February 2015; revised manuscript received 11 May 2015; published 13 July 2015) Background: Dynamic core-excitation effects have been found to be of importance in breakup reactions and may be of relevance when obtaining spectroscopic information from transfer reactions. Purpose: In this paper we extend the distorted-wave Born approximation (DWBA) formalism in order to allow for noncentral components in the core-core term appearing in the transition operator, which allows for dynamic core-excitation effects. Then we study these effects by applying the formalism to different (d,p) reactions. Methods: The expression of the nonlocal kernels required for the evaluation of the DWBA amplitudes has been extended so as to include noncentral parts in the core-core interaction. The DWBA scattering amplitude is then obtained by solving the corresponding inhomogeneous equation, with the new computed kernels, and the usual outgoing boundary conditions. A new DWBA code has been developed for this purpose. Results: For 10Be(d,p)11Be, core-excitation effects are found to be almost negligible (<3%). The importance of this effect has been found to depend to a large extent on the excitation energy of the core. This has been confirmed in the 30Ne(d,p)31Ne case, for which the excitation energy of the first 2+state is 0.8 MeV, and the effect of core excitation increases to ≈10%. Conclusions: We find dynamic core-excitation effects in transfer reactions to have small contributions to cross sections, in general. However, they should not be neglected, since they may modify the spectroscopic information obtained from these reactions and may become of importance in reactions with nuclei with a core with high deformation and low excitation energy. DOI: 10.1103/PhysRevC.92.014613 PACS number(s): 24.10.Eq,25.40.Hs,25.45.Hi,25.70.Hi I. INTRODUCTION Transfer reactions have for decades been one of the main sources of information on the structure of stable and, more recently, also of exotic nuclei. The angular distribution of the outgoing particles produced in these processes is very sensitive to the transferred orbital angular momentum between the projectile and target, whereas the magnitude depends on the product of the initial and final spectroscopic factors (the normalization of the overlap functions in the initial and final nuclei). Extraction of accurate structure information from these measurements relies on the comparison of the data (often angular distributions) with calculations using a suitable reaction formalism. Traditionally, the analysis of transfer reactions has been carried out in terms of the distorted-wave Born approximation (DWBA). Corrections and improvements of this method have also been used. For example, when the initial or final nuclei contain collective excited states, *[email protected] †[email protected] ‡[email protected] §[email protected] multistep processes involving excitations and deexcitations among these states can be suitably incorporated within the coupled-channels Born approximation (CCBA) method [1]. Moreover, when one the colliding nuclei is weakly bound, such as in the case of (d,p) reactions, coupling effects to breakup channels can also influence the transfer cross sections and hence the inclusion of these couplings become important. This has been done within the continuum-discretized coupled channels (CDCC) approximation [2–6] or, more simply, within the adiabatic approximation, such as in the adiabatic distorted wave approximation (ADWA) of Johnson and Tandy [7]. More recently, the Alt–Grasberger–Sandhas (AGS) formulation of the Faddeev equations [8,9] has also been successfully applied to transfer reactions (see Ref. [10] for a recent review of these methods). In a simple picture, a transfer reaction can be modelled in a three-body model, in which a nucleon, or group of nucleons, is transferred from one nucleus to another. For example, a stripping reaction of the form b(d,p)Bcan be viewed as a process in which the incident deuteron transfers a neutron to the target nucleus b, producing a composite nucleus B=b+n in a given state defined by the relative wave function of neutron and core b. However, this naive interpretation of the transfer process, which is the basis of the DWBA approximation, neglects possible effects derived from the excitation of the 0556-2813/2015/92(1)/014613(11) 014613-1 ©2015 American Physical Society G´ OMEZ-RAMOS, MORO, G ´ OMEZ-CAMACHO, AND THOMPSON PHYSICAL REVIEW C 92, 014613 (2015) subsystems. Indeed, it is well known that core excitations can affect the transfer process in several ways. First, the interaction between the transferred particle and the core bwill not be given by a simple spherical potential, but will contain noncentral terms that will give rise to core-excited admixtures in the states of the composite nucleus B. The weight of each component can be regarded as a spectroscopic factor. This is a structure effect, not related to the reaction mechanism, and will be referred to as static core excitation. In actual calculations, this effect can be included by using some particlecore model (particle-rotor, particle-vibrator, Nilsson, etc.) or, more commonly, by simply multiplying some single-particle wave functions by appropriate spectroscopic amplitudes. This is the usual procedure followed in the DWBA method. Another core-excitation effect that may influence this reaction arises from the initial-state interaction of the incoming deuteron with the target b. Before the neutron is transferred, the deuteron may induce excitations on the core b, thus altering the population probability of the different core states. These dynamic core-excitation effects can be conveniently treated within the CCBA method and are commonly referred to as multistep or coupled-channels effects. In presence of these additional couplings, the cross sections for the different final states will no longer be proportional to the corresponding spectroscopic factors. Finally, another way in which core excitation can affect transfer dynamics is through the interaction of the proton with the core b. In the DWBA and CCBA formalisms, this interaction appears in the so-called remnant term of the transition operator and gives rise to a core-recoil effect. Standard calculations consider only the central part of this core-core interaction. However, the presence of noncentral parts (e.g., inelastic couplings) in this interaction will give an additional contribution to the transfer cross section, even in the Born approximation. In our previous example, these noncentral terms could connect different states of the core b, thus affecting the cross sections leading to these states. We refer to this mechanism as transfer induced by prompt core excitation to distinguish it from the aforementioned dynamic (multistep) and static core-excitation effects. This problem of prompt core excitation in transfer reactions has been scarcely studied in the literature. In Ref. [11], the problem was studied within the zero-range DWBA approximation and for (d,p) reactions only. The contribution of transfer induced by prompt core excitation was found to be very small (∼6%) in most cases studied but the calculations were done with many approximations (the transfer amplitude was evaluated in zero range, a closed-shell model was used for the core nucleus, and the bound-state wave functions were approximated by harmonic-oscillator functions) and so the conclusions must be taken with some caution. In Ref. [12], Kozlowsky and de-Shalit proposed a zero-range core-excitation DWBA model for the (3He,d) case. The transfer was also treated in zero range and core excitation was modelled within the vibrational model. The angular distribution via core excitation was found to be very similar to that of ordinary stripping due to a single-particle mechanism but the calculated magnitude underpredicted the experimental data by about one order of magnitude. However, this magnitude was found to depend very critically on the radius of the particle-core potential, which is directly linked to the distance at which the stripping mechanism is assumed to occur. Recently, the problem has been addressed [13] within the AGS formulation of the Faddeev equations [8,9]. The method was applied to several (d,p) and (p,d) reactions involving the nuclei 11Be and 24Mg. Dynamic core-excitation effects were found to be important, often improving the description of the experimental data. Moreover, these effects could not be reproduced by a simple reduction of the cross section by the corresponding spectroscopic factor. It is worth noting, however, that these calculations include, in an nontrivial way, effects arising from the different sources of core excitation discussed above. In particular, dynamic multistep and prompt core-excitation effects will appear entangled in this approach, and so the assessment of their relative importance is not straightforward from the final cross sections. In view of these results, we believe that it is useful to investigate (and isolate) the phenomenon of prompt core excitation within the much simpler DWBA framework. To avoid the limitations of previous calculations of this kind [11,12], we aim at performing full-fledged DWBA calculations, including finite-range effects and a more realistic model of the core + valence system. With this premise in mind, in this work we present the theoretical formulation of the problem and apply this model to the 10Be(d,p)11Be and 30Ne(d,p)31Ne reactions. We finally note that this work follows a series of previous works aimed at understanding the effect of core excitation in the different reaction channels. In Refs. [14–16], the study was focused on the breakup channels, and for that purpose appropriate extensions of the DWBA and continuumdiscretized coupled-channels method were developed and applied to several physical cases. Dynamic core-excitation effects were found be small for the case of the scattering with heavy targets but, on the other hand, sizable effects were found for light targets, typically enhancing the breakup cross sections with respect to the inert-core case. The paper is structured as follows: In Sec. II, we present an extended DWBA formalism including prompt core-excitation effects. In Sec. III, the method is applied to the 10Be(p,d)11Be and 30Ne(p,d)31Ne reactions and compared with recent data in the former case. The effect of the excitation energy of the core is also studied by using a model with a fictitious 11Be nucleus with reduced core-excitation energy. In Sec. IV we summarize the main conclusions of this work. II. THEORETICAL FORMULATION Let us consider the transfer reaction A(a+v)+b→a+ B(b+v) in which a composite projectile Atransfers a particle vto the target nucleus bgiving rise to the ejectile aand the residual composite nucleus B. This is a many-body scattering problem which, under suitable approximations [10,17], can be reduced to an effective three-body problem. We will consider that nucleus bhas internal degrees of freedom (ξ) which are relevant in the reaction. The effective three-body Hamiltonian describing this problem can be expressed in two different forms, depending on whether one chooses the prior or post 014613-2 TRANSFER INDUCED BY CORE EXCITATION WITHIN AN . . . PHYSICAL REVIEW C 92, 014613 (2015) FIG. 1. Relevant coordinates involved in the calculation. aand b are the cores to which the valence particle vis bound before and after the transfer reaction. representations, i.e., Hprior =TR+HA(r)+Uab(rc,ξ)+Vvb(r,ξ),(1) Hpost =TR+HB(ξ,r)+Uab(rc,ξ)+Vva(r),(2) where TRand TRrepresent the kinetic-energy operators for the projectile-target relative motion before and after the transfer reaction, respectively. HAand HBare the internal Hamiltonians of the composite systems formed by the valence particle and the core to which it is bound. The real interactions Vva and Vvb are the binding potentials for the transferred particle in the initial and final nucleus, respectively. Finally, the potential Uab, complex in general, represents the effective interaction between the two cores. Note that possible excitations of aare not considered explicitly (this is indeed justified for the applications presented in the next section, in which a represents a proton). The relevant coordinates are shown in Fig. 1and obey the following relations: r=a R+b R, r=a R+b R,rc=ac R+bc R, νA=ma/mA,ν B=mb/mB,ω=1 1−νAνB , a=νBω, b =−ω, a=ω, b=−νAω, ac=−ω(1−νB),b c=−ω(1−νA).(3) Since bis allowed to be excited during the transfer reaction, its wave function as well as the interactions Uab and Vvb will depend on ξ.Usingthepost representation, the exact transition amplitude can be expressed as (see, for instance, Refs. [1,18]) Tβα =(−) β(r, R,ξ)|Uab(rc,ξ)+Vva(r) −Uβ(R)|(+) α(r,  R,ξ),(4) where (+) α(r,  R,ξ) corresponds to the full three-body wave function of the system and Uβ(R) is an arbitrary potential, which is usually chosen to reproduce the elastic scattering of the a+Bsystem. The final wave function (−) β(r, R,ξ)isan eigenstate of the Hamiltonian Hf=TR+HB+Haand can therefore be expressed in the factorized form: (−) β(r, R,ξ)=χ(−) β( R)ψB(r,ξ)ψa,(5) where ψBand ψacorrespond to the internal wave functions of Band a, and χ(−) βis the distorted wave with incoming boundary conditions obtained by solving the Schr¨ odinger equation with the potential Uβ. In the DWBA approximation, the total three-body wave function (+) αis approximated by an expression analogous to Eq. (5): (+) α(r,  R,ξ)≃χ(+) α( R)ψA(r)ψb(ξ),(6) where ψA,b are equivalent to ψB,a, and χ(+) αis the distorted wave with outgoing boundary conditions obtained by solving the Schr¨ odinger equation with an optical potential Uαbetween projectile and target in the incident channel. The DWBA transition amplitude results TDWBA βα =χ(−) β( R)ψB(r,ξ)ψa|Uab(rc,ξ)+Vva(r) −Uβ(R)|χ(+) α( R)ψA(r)ψb(ξ).(7) Expanding this expression in partial waves, for a certain total angular momentum JT, one obtains TDWBA βα,JTMT=(4π)2 kαkβ LM,J MJ LM,J M J iL−LY∗ LM (ˆ kα)YLM(ˆ kβ)ei(σL+σL) ×LJpMMp|JM JLJ pMM p|JM J ×JJ tMJMt|JTMTJJ tM JM t|JTMT ×dRdRχ(−) β,L,J (R)Kαβ (R,R)χ(+) α,L,J (R), (8) where σLare the Coulomb phase shifts, YLM denote the spherical harmonics, and χ(+) α,L,J and χ(−) β,L,J are the radial parts of the distorted waves introduced in Eqs. (5) and (6), respectively. Jpand Jtare the spin of the projectile and target, respectively, in the initial partition, and J pand J tare the spins of ejectile and residual nucleus in the final partition. Jand J are the result of coupling L⊗Jpand L⊗J p, respectively. Note that the S-matrix elements, from which the scattering observables can be computed, are related to the Tmatrix as Sβα,JTMT=−2iπTβα,JTMT. The quantities Kαβ (R,R), usually referred to as nonlocal kernels [1], are defined as Kαβ (R,R)=YL(ˆ R)⊗ψJ p aJ⊗ψJ t B∗ JTMT ×Uab +Vva −Uβ ×YL(ˆ R)⊗ψJp AJ⊗ψJt bJTMT dˆ Rdˆ Rdξ. (9) They include integration over the angular parts of  Rand  R and the internal degrees of freedom of the reacting particles, remaining functions of Rand Ronly (in modulus). These kernels encompass the transition potential, the internal wave functions of projectile and target in the incoming and outgoing channels and the angular part of the distorted waves in both channels. 014613-3 G´ OMEZ-RAMOS, MORO, G ´ OMEZ-CAMACHO, AND THOMPSON PHYSICAL REVIEW C 92, 014613 (2015) The wave function of the composite system A(a+v)is expanded as follows: ψJp A= l,j,sa ϕl,j,sa(r)[Yl⊗χs]j⊗ψa saJp,(10) where χsis the spin function of the valence particle, sais the intrinsic angular momentum of the core a, and lis the orbital angular momentum between valence particle and core. Likewise, for the composite nucleus B(b+v), ψJ t B= l,j,I  ϕ l,j,I (r)[Yl⊗χs]j⊗ψb I(ξ)J t.(11) The potential Uab depends on rcand ξ. Since it is a scalar, it can be expanded in multipoles, in a way analogous to Ref. [19]: Uab = Q ˆ QUQ ab(rc,ξ) q CQq (ˆ rc)T∗ Qq (ξ),(12) where CQq =ˆ QYQq /√4π,ˆ Q=√2Q+1, and TQq is a function with the same tensorial rank as CQq . Let us remark that, since the transition potential depends on ξand is not central in its dependence on rc, the transition between different states of the core bwith different angular momenta I,Iis possible during the reaction in the process which we call prompt core excitation. Inserting the multipole expansion (12) in Eq. (9) it is possible to express the complete kernels Kαβ (R,R) as a sum of terms with different multipoles Q: Kαβ (R,R)=K(Q=0) αβ (R,R)+K(Q>0) αβ (R,R).(13) Standard DWBA and CCBA calculations consider only the term with Q=0, and hence they exclude prompt coreexcitation effects. These effects are induced by the Q>0 terms, which are considered in detail in the following. Since the angular momentum of the core bis no longer conserved in the reaction and there are noncentral components of the potential Uab, it is necessary to recouple the relevant angular momenta in a somewhat more complex way for the kernels with Q= 0 than in the standard case [20] (see Appendix B). We now present the resulting form of the kernels and refer to Appendix Afor more details on their construction. We remark that the particle is transferred from the projectile to the target and the target is a deformed nucleus, with angular momentum Iin the incoming channel and Iin the exit channel. This calculation assumes all potentials but Uab to be central and therefore excludes the spin-orbit terms. Nevertheless, we believe that this simplification will not affect our conclusions regarding the importance of core excitation. With this simplification, the kernels can be expanded as Kαβ (R,R)= SsV PSsVssa γ,γF γ,γ(R,R),(14) where γrepresents the set of angular momenta {l,j,I,L}, with land jbeing the orbital and total angular momentum of the valence particle, respectively, Lbeing the orbital angular momentum between projectile and target, and Ibeing the state of the core nucleus b. The unprimed values correspond to the incoming channel and the primed values correspond to the outgoing channel. PSsVssa γ,γis a factor that appears due to the recoupling needed to separate the angular momenta which are spectators in the reaction (the spins of the valence particle, s, and of core a,sa) from those that actively participate in it (l, L, and I). All the other new angular momenta (,,V,Ss, and ) appear due to this recoupling. The radial form factors F γ,γ(R,R) include the actual integration over ˆ R,ˆ R, and ξ. The recoupling factors PSsVssa γ,γhave the following expression: PSsVssa γ,γ=(−)s+sp+Jp+L+I−JT++L+l+Ssˆ ˆ ˆ 2ˆ V2ˆ Ss 2ˆ jˆ jˆ Jˆ Jˆ Jpˆ Jtlsj spJpSs ×SslJ p LJ  SsJ IJ T⎧ ⎨ ⎩ lL ss pSs jJV⎫ ⎬ ⎭I SsJTVIjJ t JJTV,(15) whereas the form factors F γ,γhave the form F γ,γ(R,R)=(−)+I+l+T+L+qT γQγ (R,R)l NQ NCl Nˆ Q2ˆ T2ˆ Iˆ l2ˆ l2ˆ ˆ ˆ Lˆ Lˆ F2ˆ F2ˆ G2ˆ G2ˆ H2ˆ R2 ×(aR)N(acR)NC(aR)N(bR)l−N(bcR)Q−NC(bR)l−NNCTF 000  ×LFG 000 NN G 000 Q−NCTF  000 LF G 00 0  ×l−Nl −NG 000 Q  I I FF Q Q−NCNCTGF H QL FlH QL  ×ll R HL GGR HLF ⎧ ⎨ ⎩ lRl  NGN  l−NG l−N⎫ ⎬ ⎭IT∗ Q(ˆ ξ)I,(16) 014613-4 TRANSFER INDUCED BY CORE EXCITATION WITHIN AN . . . PHYSICAL REVIEW C 92, 014613 (2015) where the sum is extended over all indices but L,L,,, and , and the limits for the sums are given by the 3-nj symbols, except for N,N, and NC, which go from 0 to l,l, and Q, respectively. Here, a b=(2a)! (2b)![2(a−b)]!, and the coefficients a,a,ac,b,b, and bcare given by Eq. (3). The reduced matrix element IT∗ QIis defined with the convention of Brink and Satchler [21]. The quantities qT γQγ (R,R) include the radial dependence and are defined as follows: qT γQγ (R,R)=|b|3 21 −1 ϕ∗ l,j,I (r) rl UQ ab(rc) rQ c ϕl,j (r) rl ×PT(cos ϑ)d(cos ϑ),(17) where ϕ,ϕ, and UQ ab are defined in Eqs. (10)–(12), PTis the Legendre polynomial of degree T, and ϑis the angle between  Rand  R. Note that, for the Q=0 part of the transition potential, one must calculate also the analogous quantities qT γ0γ(R,R), but using the complete transition potential Vav(r)+U0 ab(rc)−Uβ(R) instead of UQ ab. The formulas of the kernels are more complicated than those obtained for the standard calculations (see, for instance, Ref. [20]) due to the coupling of the extra angular momenta Q(multipole of Uab) and I,I(spin of the core). For the calculations presented in the subsequent sections, the expressions derived above have been implemented in a new DWBA code [22]. The Q=0 limit has been tested against the FRESCO code [20]. III. CALCULATIONS A. The 10Be(d,p)11Be reaction As an application of the formalism presented in the previous section, we consider the stripping reaction of deuterium on 10Be leading to several states of the nucleus 11Be. Since the latter can be modelled as a halo nucleus with a deformed core (10Be), it seems an adequate nucleus to test the importance of these effects. To generate the states of 11Be, we use the particle-rotor model with the parameters given by the set named Be12b in Ref. [23], which were fit to give the correct separation energies of the ground and first-excited states and the position of the low-lying resonances 5/2+and 3/2+at 1.27 and 3.00 MeV above the neutron separation threshold. This model considers only two states of the core, the ground state (0+) and the first-excited state (2+) with an excitation energy of 3.368 MeV. To represent the resonant states, we use energy bins with a width of 0.8 MeV, obtained with the code FRESCO [20]. For further details on the calculation of multichannel bins we refer to Refs. [20,24]. The calculated radial parts of the bound states and bin wave functions for these states are shown in Fig. 2. We note that, in this model, the ground, excited, and first-resonant (5/2+) states have their main component (85%, 79%, and 67% of their wave functions, respectively) with the core in its ground state (0+). However, the main component -0.2 0 0.2 0.4 rϕ (fm-1/2) |0+⊗s1/2> S=0.847 |2+⊗d3/2> S=0.023 |2+⊗d5/2> S=0.130 -0.2 0 0.2 0.4 |0+⊗p1/2> S=0.786 |2+⊗p3/2> S=0.206 |2+⊗f5/2> S=0.008 02040 r ( fm ) -0.2 0 0.2 0.4 rϕ (fm-1/2) |0+⊗d5/2> S=0.673 |2+⊗s1/2> S=0.179 |2+⊗d3/2> S=0.009 |2+⊗d5/2> S=0.118 0204060 r ( fm ) -0.4 -0.2 0 0.2 |0+⊗d3/2> S=0.206 |2+⊗s1/2> S=0.693 |2+⊗d3/2> S=0.015 |2+⊗d5/2> S=0.071 11Be(1/2+) g.s. E=-0.5 MeV 11Be (1/2-) E=-0.18 MeV 11Be(5/2+) E=1.27 MeV 11Be(3/2+) E=3.00 MeV FIG. 2. (Color online) Radial parts of the wave functions for the different states of 11Be considered, obtained by integration of the Schr¨ odinger equation, using energy bins to represent unbound states. The main components (represented by black full lines) correspond to the 0+state of the core for the ground state, excited state and resonant 5/2+state, but the main component of the 3/2+resonant state has the core in its excited 2+state. (69% of the wave function) of the 3/2+resonant state has the 10Be core in its excited state. Therefore, we expect the prompt core-excitation effects to be most important for this state of 11Be. As a test of the quality of this particle-rotor model for 11Be, we compare in Table Ithe spectroscopic factors obtained with the model to those extracted from different experimental data for the 0+component of the ground and excited states of 11Be. Due to the difficulty of performing spectroscopy for resonant states, these states are excluded from the comparison. We find that our model tends to overestimate the contribution of the 0+component to both ground and excited states as compared to the spectroscopic factors obtained from the analysis of experimental data. We now calculate the reaction cross section for the transfer reaction 10Be(d,p)11Be at an incident energy of 21.4MeV.At this energy there are experimental data for this reaction for all the considered states of 11Be but the 3/2+resonance [25]. The transition amplitude is evaluated in the ADWA approximation, which is formally identical to the DWBA approximation, but with the deuteron optical potential replaced by an adiabatic potential. This potential is not meant to reproduce the elastic scattering but accounts for the coupling to the breakup channels, which are known to be important for weakly bound projectiles. In particular, we use the finite-range adiabatic potential of Johnson and Tandy [7], which is constructed TABLE I. Spectroscopic factors for the 0+component of ground and excited states of 11Be obtained from particle-rotor potential Be12b [23] and from the analysis of transfer data. 11Be state Stheor (Be12b) Sexpt [25]Sexpt [26] 1/2+0.85 0.74 ±0.06 0.73 ±0.06 1/2−0.79 0.65 ±0.05 0.63 ±0.15 014613-5 G´ OMEZ-RAMOS, MORO, G ´ OMEZ-CAMACHO, AND THOMPSON PHYSICAL REVIEW C 92, 014613 (2015) 0 5 10 15 20 dσ/dΩc.m. (mb/sr) No PCE PCE Schmitt (2012) 0 5 10 15 20 25 30 0 1020304050 θc.m.(deg) 0 20 40 60 80 100 dσ/dΩc.m. (mb/sr) 0 1020304050 60 θc.m.(deg) 0 5 10 15 20 25 10Be(d,p)11Be(1/2+) Ground state 10Be(d,p)11Be(1/2-) Ex=0.32MeV 10Be(d,p)11Be(3/2+) Ex=3.41MeV 10Be(d,p)11Be(5/2+) Ex=1.78 MeV FIG. 3. (Color online) Angular dependence of the cross section of 10Be(d,p)11Be to different states of 11Be with incoming energy Ed=21.4 MeV. Calculations without considering prompt coreexcitation effects are represented by a black dashed line, while calculations taking them into account are shown with a blue solid line. Experimental data are taken from Ref. [25]. based on the p-10Be and n-10Be potentials. In the calculations presented below, the potentials between p-10Be and n-10Be are obtained from the global CH89 parametrization [27], omitting spin-orbit terms. This parametrization is also used for the p-11Be potential in the exit channel. For the p-n interaction we have chosen the Gaussian parametrization of Ref. [6] which gives the correct binding energy and root-mean square (rms) of the deuteron. We perform two calculations. The first one considers a central p-10Be interaction, as done in standard ADWA or DWBA calculations. For the second one, we introduce a quadrupole deformation of this potential with the same quadrupole deformation length parameter, δ2, (see Appendix C)asthen-10Be potential. Note that this deformation is not considered in the calculation of the adiabatic d-10Be potential. The calculated cross sections are presented in Fig. 3.It can be seen that core-excitation effects are negligible for all states. In Table II we give the angle-integrated cross section for both calculations and the relative difference between them for each final state of 11Be. Although this small difference is to be expected for the ground, excited, and first resonant states, it is rather surprising for the 3/2+state, since its main component, with an excited 2+core, can only be populated in first order by a core-excitation mechanism. Remarkable is TABLE II. Cross sections for 10Be(d,p)11Be at Ed=21.4MeV, excluding and including prompt core-excitation (PCE) effects. 11Be state σNoPCE (mb) σPCE (mb) Diff. (%) 1/2+4.872 4.785 −1.8 1/2−9.800 10.009 2.1 5/2+54.10 55.97 3.5 3/2+7.479 7.717 3.2 also the poor agreement with the data for the 5/2+resonance. This discrepancy might be related to the inadequacy of the chosen optical-model potentials, to higher order (beyond the Born approximation) effects or to the limitations of the reaction formalism employed here to deal with the transfer to unbound states. We note that this is still an open problem [28]but, since our major point of interest is the relative importance of prompt core-excitation effects, we have not explored these issues further. Our results are of the same order of magnitude as those found by Levin [11]forseveral(d,p) reactions, within a zero-range DWBA model. On the other hand, we find smaller core-excitation effects than those reported in Ref. [13], where the same data were analyzed by using the more sophisticated Faddeev formalism. However, as noted in the introduction, those calculations include, in an intricate way, additional core-excitation effects, such as the multistep couplings. These effects were in fact found to be significant for the inverse reaction, 11Be(p,d)10Be, according to the CCBA calculations performed in Ref. [29]. We may suggest some possible explanations for the smallness of prompt core-excitation effects. First, from the expression of the qT γQγ functions (17), we see that a certain overlap is required between the wave functions of Aand B (deuteron and 11Be in our test case) and the multipole of the potential which is considered. Terms with Q=0, which include the potential Vpn, are expected to have a greater overlap with deuteron wave functions than terms with higher potential multipoles, which only include Up10Be. This could result in smaller contributions for larger values of Qand, therefore, small effects of core excitation. Another factor that may affect these results is the large excitation energy of the core (3.368 MeV). This means that, in our particle-rotor model, the effective separation energy of the valence neutron of 11Be is noticeably enhanced when the core is in its excited state, so the exponential decay of its wave function as a function of the neutron-core separation will be steeper (let us remark that, for all cases considered, the components of 11Be with excited core are effectively bound), as can be seen in Fig. 2. Since transfer reactions are peripheral processes, the resulting reduction of the effective radius of 11Be may be relevant to the results obtained. The combined effects discussed above will tend to make the Q>0 kernels more localized in configuration space as compared to their Q=0 counterparts. This is illustrated in Fig. 4for the channel L=6, J=5, L=4, J=7/2, JT=5 where the top and bottom panels correspond to the Q=0 and Q=2 contributions. As anticipated, the Q=0 kernels are of longer range, extending to relatively large distances along rpn =0 (dashed line in the plot). On the contrary, the Q=2 kernels are confined to smaller distances and are comparatively smaller in absolute magnitude, thus explaining the reduced effect on the transfer cross sections. Since one of the reasons for this smallness of the coreexcitation effects in the 10Be(d,p)11Be reaction is the relatively high excitation energy of the core, one may speculate that these effects will be enhanced in other systems, for which this excitation energy is smaller. This is quantitatively tested in the next section. 014613-6 TRANSFER INDUCED BY CORE EXCITATION WITHIN AN . . . PHYSICAL REVIEW C 92, 014613 (2015) FIG. 4. (Color online) Nonlocal kernels [see Eqs. (13)and(14)] (a) K(Q=0) αβ (R,R)and(b)K(Q=2) αβ (R,R) for channel L=6, J=5, L=4, J=7/2, JT=5inthereaction10Be(d,p)11Be at Ed=21.4 MeV. Notice that the independent variables are Rand D=R−R. B. Calculation with reduced core-excitation energy In order to study the influence of the excitation energy of the core on prompt core-excitation effects, we artificially reduced the excitation energy of 10Be to 0.10 MeV so as to obtain a bound 3/2+state at −0.16 MeV. The main component of this shifted state (88% of the wave function) has its core in its excited 2+state and it is bound, so the problems of transfer to the continuum should be avoided. The radial parts of the wave function corresponding to this state are shown in Fig. 5. We now perform calculations of the transfer reaction 10Be(d,p)11Be to this artificial state for different energies of the incoming projectile: 21.4, 30, 40, 60, and 100 MeV. The angle-integrated transfer cross sections are listed in Table III and the angular dependence and total-angular-momentum dependence of the transfer cross section for the projectile energies: 21.4, 40, and 100 MeV are shown in Fig. 6. We observe that the effects become much more important when transferring a neutron to this artificial state with reduced 0204060 R (fm) -0.4 -0.2 0 0.2 rϕ (fm-1/2) |0+⊕d3/2> S=0.031 |2+⊕s1/2> S=0.877 |2+⊕d3/2> S=0.014 |2+⊕d5/2> S=0.078 Toy model E=-0.16 MeV FIG. 5. (Color online) Radial parts of the wave function for the artificially bound state of 11Be built with an excitation energy for the core of 10Be of 0.1 MeV. The spectroscopic factors corresponding to each component are shown in the legend. core energy. From Fig. 6and Table III we see that the total transfer reaction cross section is reduced with increasing incoming energy. It is noticeable the weak dependence of the relative importance of prompt core-excitation effects with projectile energy, which seems to indicate that, in a range of 10–50 MeV per nucleon, these effects will contribute to the same proportion of the cross section. Therefore, their study at smaller energies, favored due to the larger cross sections, should be useful when predicting their relevance at higher energies. From this study we can conclude that prompt coreexcitation effects will be important when transferring particles between nuclei with cores with small excitation energies, and that their contributions are expected to vary slowly with energy. For example, very-neutron-rich Ne, Na, and Mg isotopes around the so-called island of inversion are characterized by small 2+excitation energies and hence are expected to be good candidates to exhibit non-negligible prompt core-excitation effects. An example of this is considered in the next section. C. 30Ne(d,p)31Ne As a final example, we consider the reaction 30Ne(d,p)31Ne at an incident energy of 30 MeV. The structure of the 31Ne nucleus is not completely known, but most studies suggest a halo-like structure with a weakly bound neutron with amixedp3/2/f7/2configuration. A combined analysis of TABLE III. Cross sections for 10Be(d,p)11Be, to the artificially bound 3/2+state described in the text, excluding and including prompt core-excitation (PCE) effects. Ed(MeV) σNoPCE (mb) σPCE (mb) Diff. (%) 21.42.72 3.09 13.5 30 1.68 2.01 19.5 40 1.05 1.28 22.0 60 0.458 0.564 23.0 100 0.133 0.158 18.6 014613-7 G´ OMEZ-RAMOS, MORO, G ´ OMEZ-CAMACHO, AND THOMPSON PHYSICAL REVIEW C 92, 014613 (2015) 0 1 2 3 4 5 No PCE PCE 0 0.5 1 1.5 0 0.5 1 1.5 dσ/dΩ(mb/sr) 0 0.1 0.2 0.3 σJ (mb) 0 1020304050 60 θc.m.(deg) 0 0.2 0.4 510 15 20 J 0 0.01 0.02 Ed=21.4 MeV Ed=40 MeV Ed=100 MeV FIG. 6. (Color online) Angular (left) and total-angular- momentum (right) dependence of 10Be(d,p)11Be reaction cross section to the artificially bound 3/2+state of 11Be defined in the text for energies of the incident deuteron of 21.4, 40, and 100 MeV. 1n-removal experiment of this nucleus on C and Pb targets performed at RIKEN, Japan, at 230 MeV/nucleon [30] gives Sn=0.15+0.16 −0.10 MeV, and a spin-parity of 3/2−.The first-excited state of the core (2+) is located at Ex≈800 keV, so core-excitation effects are expected to be significantly larger than in the 11Be case. The 31Ne nucleus was studied in the particle-rotor model by Urata et al. [31,32]. However, the validity of this model for this nucleus is still unclear so, for the transfer calculations presented in this work, we will rely on a phenomenological model of 31Ne with spectroscopic factors quoted in Ref. [30] [model SM(ii)], obtained from shell-model calculations using the modified monopole sd-pf cross-shell SDPF-M interaction. The spectroscopic factors are 0.21, 0.33, and 0.80 for the [0+ 1⊗2p3/2], [2+ 1⊗2p3/2], and [2+ 1⊗1f7/2] configurations, respectively. In combination with eikonal calculations, these spectroscopic factors give knockout cross sections in good agreement with the experimental data of Ref. [30]. The radial parts of the bound-state wave function and their relative signs are obtained from a particle-rotor calculation, similar to that of Refs. [31,32], but the normalization of each component was taken from the spectroscopic factors quoted above. The deformation parameter of the core-target potential was taken as β2=0.2[32]. As in our previous calculations, the potentials were calculated with the CH89 global parametrization [27], the finite-range potential of Johnson and Tandy [7], and the Gaussian deuteron potential of Ref. [6]. In Fig. 7we plot the calculated angular distributions, with and without core excitation. The difference between both calculations is about ∼11%, still a modest value, but significantly larger than in the 10Be(d,p) case, as anticipated. From these calculations it can be concluded that the effects of prompt core excitation are in general small but not negligible, with an influence on the cross section of the order of ∼2%−10%. Since these effects become more important when the excitation energy of the core is reduced, they may play a greater role in transfer reactions with more massive 0 1020304050 60 θc.m. (deg) 0 1 2 3 4 5 dσ/dΩ (mb/sr) No PCE PCE 0510 15 20 J 0 0.05 0.1 0.15 0.2 0.25 σJ (mb) FIG. 7. (Color online) (a) Angular distribution and (b) angularmomentum dependence for the 30Ne(d,p)31Ne reaction at 30 MeV, with the inclusion and omission of prompt core-excitation effects. The model used for 31Ne is described in the text. nuclei. Also, in reactions with less-halo-like nuclei, where the influence of the core might be more important, these effects may become more relevant and should be taken into account when obtaining spectroscopic factors from transfer cross sections. IV. SUMMARY AND CONCLUSIONS We developed a formalism to include prompt coreexcitation effects in transfer reactions within the DWBA approximation. The formalism accounts for the transition between different core states of the initial and final nuclei due to the presence of noncentral terms in the core-core interaction appearing in the transition operator. For the cases considered in this work, we adopted a collective model of the core-core interaction which has been described in terms of a deformed potential. We showed that the developed formalism reduces to the standard DWBA expression when a central potential is used for the core-core interaction (see Appendix B). We studied the influence of these effects on the 10Be(d,p)11Be and 30Ne(d,p)31Ne transfer reactions. In the former case, the effect of prompt core excitation is very small, with a contribution of ∼2%–4% to the transfer cross section. This is attributed in part to the relatively high excitation energy of the core, which leads to a large effective separation energy of the neutron in the configurations with an excited core, and hence to more localized nonlocal kernels. This reduces the effective radius of 11Be with an excited core, and its contribution to a peripheral reaction such as transfer. In the 30Ne(d,p)31Ne case, for which the excitation energy of the core is much smaller (∼800 keV) the core-excitation mechanism enhances the transfer cross section by 11%, and therefore the effect is no longer negligible. Although the calculations presented in this work are based on the DWBA approximation, the formalism could be extended to more sophisticated formalisms, such as CCBA. In the latter case, additional core-excitation effects would be present in the form of multistep couplings. 014613-8 TRANSFER INDUCED BY CORE EXCITATION WITHIN AN . . . PHYSICAL REVIEW C 92, 014613 (2015) ACKNOWLEDGMENTS We are thankful to Professor Hermann Wolter for his suggestions and critical reading of the drafts for this paper. This work has been partially supported under grant FPA2009- 07653, by the Spanish Consolider–Ingenio 2010 Programme CPAN (CSD2007-00042), by Junta de Andaluc´ ıa (FQM160, P07-FQM-02894), and by Spanish Ministerio de Econom´ ıa y Competitividad (FPA-2013-47327-C02-01-R) and under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52- 07NA27344. M.G.-R. acknowledges a research grant by the Spanish Ministerio de Educaci´ on, Cultura y Deporte. APPENDIX A: CONSTRUCTION OF THE KERNELS In this appendix, we give an overview on the steps taken in order to obtain the kernel from Eq. (14). The complete derivation is rather lengthy so we outline here only the main steps. As mentioned in Sec. II,thetermPSsV ssa γ,γresults from the recouplings |[[L,{(l,s)j,sa}Jp]J,Jt]JTMT |[[(L,l),I ],(s,sa)Ss]JTMT,(A1) |[[L,J  p]J,{(l,s)j,I }J t]JTMT |[[(L,l),I],(s,sa)Ss]JTMT.(A2) From the definition of the 6j symbol [21] and the one of the 12j symbol given by Jahn and Hope [33]aswellasits expansion in 6j and 9j symbols, the expression of PSsV ssa γ,γ can be obtained, bearing in mind that sand saare not modified in the reaction and that Jt=Iand J p=sa. As for F γ,γ, it involves the integration in the angular coordinates of  Rand  R. Therefore, all angular dependencies must be expressed in terms of these two coordinates. For this, we make use of the solid harmonics expansion [34]: rQCQq (ˆ r)= Nn Q N(aR)NbRQ−Nˆ Q(−1)Q+q ×Q−NN Q q−nn−qCNn(ˆ R)CQ−N,q−n(ˆ R), (A3) with r=a R+b Rand where the definitions of Sec. II apply. This formula is applied to the spherical harmonics that appear from ϕ γ( r), ϕγ(r), and CQq (rc) from expansion (12). In order to extract the dependence on ˆ Rand ˆ Rfrom the moduli of r,r, and rcwe perform an expansion on multipoles of the product of the radial part of the wave functions of Aand Band the multipole of the potential: |b|3ϕ∗ γ(r) rl UQ ab(rc) rQ c ϕγ(r) rl = Tt ˆ T2(−1)tqT γQγ (R,R)CTt(ˆ R)CT−t(ˆ R).(A4) From this expansion we obtain the radial qT γQγ functions from Eq. (17), which only depend on the moduli Rand R, and two extra spherical harmonics, one in ˆ Rand another in ˆ R. Considering also the angular part of the incoming and outgoing distorted waves, we have a total of five spherical harmonics dependent on ˆ Rand ˆ R. In order to perform the integration we reduce their number to three by using the property CAa(ˆ R)CBb(ˆ R)= Cc ˆ C2(−1)cCCc(ˆ R) ×ABC 000 AB C ab−c(A5) and finally integrate analytically by using dˆ RCAa(ˆ R)CBb(ˆ R)CCc(ˆ R) =4πABC 000 ABC abc .(A6) Theintegrationonξis made symbolically, through the reduced matrix element IT∗ QI, obtained with the Wigner– Eckart theorem and Brink and Satchler’s convention [21]: Ii|T∗ Qq (ˆ ξ)|Ii=(−1)I−iˆ IIQI −iqi IT∗ Q(ˆ ξ)I. (A7) After performing the integrals, the resulting 3j symbols (in our calculations 20 of them appear) can be added symbolically, to give a more compact result. For this, the graphical method of Yutsis, Levinson, and Vanagas [21,35,36] is very useful. This simplification is the last step to obtain the formula (16) for the radial form factors F γ,γ. APPENDIX B: Q=0 LIMIT If one considers a central Uab potential, then Q=0, and many simplifications apply to the expression (14). The reduced matrix element IT∗ QIbecomes δI,I , and therefore no prompt core excitation is allowed. Q=0 implies NC=0 and many 3j and 6j symbols can be reduced to Kronecker deltas, which allow summation over indices H,F, and F, and obtaining =. 014613-9