Coupled reaction channel study of the 12C(alpha,8Be) reaction, and the 8Be+8Be optical potential
Do Cong Cuong, Pierre Descouvemont, Dao T Khoa, Nguyen Hoang Phuc
aa r X i v : . [ nu c l - t h ] A ug Coupled reaction channel study of the C( α , Be) reaction, andthe Be+ Be optical potential
Do Cong Cuong , Pierre Descouvemont , Dao T. Khoa , and Nguyen Hoang Phuc Institute for Nuclear Science and Technology,VINATOM, 100000 Hanoi, Vietnam. Physique Nucl´eaire Th´eorique et Physique Math´ematique, C.P. 229,Universit´e Libre de Bruxelles (ULB), B-1050 Brussels, Belgium
Abstract
Background:
Given the established 2 α structure of Be, a realistic model of 4 interacting α clusters must be used to obtain a Be+ Be interaction potential. Such a four-body problem posesa challenge for the determination of the Be+ Be optical potential (OP) that is still unknown dueto the lack of the elastic Be+ Be scattering data.
Purpose:
To probe the complex Be+ Be optical potential in the coupled reaction channel (CRC)study of the α transfer C( α, Be) reaction measured at E α = 65 MeV, and to obtain the spectro-scopic information on the α + Be cluster configuration of C. Method:
The 3- and 4-body Continuum-Discretized Coupled Channel (CDCC) methods are usedto calculate the elastic α + Be and Be+ Be scattering at the energy around 16 MeV/nucleon, withthe breakup effect taken into account explicitly. Based on the elastic cross section predicted by theCDCC calculation, the local equivalent OP’s for these systems are deduced for the CRC study ofthe C( α, Be) reaction.
Results:
Using the CDCC-based OPs and α spectroscopic factors given by the cluster modelcalculation, a good CRC description of the α transfer data for both the Be+ Be and Be+ Be ∗ + exit channels is obtained without any adjustment of the (complex) potential strength. Conclusion:
The α + Be and Be+ Be interaction potential can be described by the 3- and 4-body CDCC methods, respectively, starting from a realistic α + α interaction. The α transfer C( α, Be) reaction should be further investigated not only to probe of the 4 α interaction but alsothe cluster structure of C. , 024622 (2020) I. INTRODUCTION
The α -cluster structure established for different excited states in several light nuclei like C or O has inspired numerous experimental and theoretical studies, especially, the directnuclear reactions measured with C as projectile and/or target [1]. Given the cluster statesabove the α -decay threshold of C, some direct reactions with C were shown to produceboth the free α particle and unstable Be in the exit channel [2–4]. Consequently, theknowledge of the α + Be and Be+ Be interaction potentials should be important for thestudies of such reactions within the distorted wave Born approximation (DWBA) or coupledreaction channel (CRC) formalism.Given a well established 2 α -cluster structure of the unbound Be nucleus, the Be+ Beinteraction potential poses a four-body problem which is a challenge for the determination ofthe Be+ Be optical potential (OP) that cannot be deduced from a standard optical model(OM) analysis because of the lack of the elastic Be+ Be scattering data. The knowledgeabout the α + Be and Be-nucleus OP’s should be also important for the studies of thosedirect reaction processes that produce Be fragments in the exit channel [5–7]. Althoughthe α -nucleus and nucleus-nucleus OP’s are proven to be well described by the double-folding model (DFM) using the accurate ground-state densities of the colliding nuclei anda realistic density dependent nucleon-nucleon (NN) interaction (see, e.g., Refs. [8–12]), theDFM cannot be used to calculate the α + Be and Be+ Be OP’s because of a stronglydeformed, extended two-center density distribution of Be. In general, one could think ofthe triple- and quadruple folding models for the α + Be and Be+ Be potentials, respectively,but these will surely be complicated and involve much more tedious calculation in comparisonwith the standard DFM method. Although some phenomenological OP’s are available in theliterature for Li and Be, two nuclei neighboring Be, a strongly (two-center) deformationof the unstable Be nucleus casts doubt on the extrapolated use of these potentials for the Be+ Be and Be-nucleus systems.Given a very loose (unbound) Be nucleus that breaks up promptly into 2 α particles, wedetermine in the present work the Be+ Be OP using the Continuum-Discretized CoupledChannel (CDCC) method which was developed to take into account explicitly the breakup2f the projectile and/or target. A textbook example is a direct reaction induced by deuteron,which is loosely bound and can be, therefore, easily broken up into a pair of free protonand neutron. Originally, the deuteron breakup states were included in terms of a discretizedcontinuum by the CDCC method (see, e.g., Refs. [13–15] for reviews). In the recent versionof the CDCC theory, the continuum of deuteron is approximated by the square-integrablefunctions corresponding to positive energies. As a result, this approach can be well extendedto study elastic scattering of exotic nuclei that have rather low breakup energies (typicalexamples are He and Be).The first developments of the CDCC method were done in the framework of a three-body system where the projectile is seen as a two-body nucleus and target is assumedto be structureless, being in its ground state. More recently, four-body calculations weredeveloped, for either a three-body projectile on a structureless target [16], or a two-bodyprojectile and a two-body target [17]. The latter approach is highly time consuming, but wassuccessfully applied to study Be + d scattering in terms of Be = Be+ n and d = p + n .The goal of the present study is to determine the Be+ Be OP based on the elastic scatteringmatrix predicted by the 4-body CDCC calculation of four interacting α clusters. While sucha 4 α model is not appropriate for the spectroscopy of O [17], the derived OP for elastic Be+ Be scattering is expected to be reliable. The only input for the present 4 α CDCCcalculation is a realistic α + α interaction potential.Although Be is particle-unstable, its half life around 10 − s is long enough for the Be-nucleus OP to contribute significantly to a direct reaction that produces Be in the exitchannel, like the α transfer C( α , Be) reaction. This particular reaction was shown to be agood tool for the study of the high-lying or resonance states of O [18, 19] and to determinethe α cluster configurations of this nucleus [18, 20, 21]. Because of the unbound structureof Be, the direct reaction reactions A ( α , Be) B usually have a very low cross section (of afew tens microbarn), but they are extremely helpful for the study of the α -cluster structureof the target nuclei [22]. In particular, the α spectroscopic factors of different cluster stateswere deduced from these measurements at the α incident energies of 65 to 72.5 MeV.In the present work, the α + Be and Be+ Be optical potentials deduced from the scat-tering wave functions given by the 3- and 4 α CDCC calculations are used as the core-coreand the exit OP, respectively, in the CRC study of C( α , Be) reaction measured at 65 MeV[22]. The OP of the entrance channel is calculated in the DFM using the density dependent3DM3Y6 interaction that was well tested in the mean-field studies of nuclear matter as wellas in the OM studies of the elastic α -nucleus scattering [9, 11], and it accounts well for theelastic α + C scattering data measured at 65 MeV [23, 24]. The α spectroscopic factors ofthe α + Be cluster configurations of C are taken from the results of the complex scalingmethod (CSM) by Kurokawa and Kato [25].
II. THREE- AND FOUR-BODY CDCC METHODS
We discuss here the CDCC method used to determine the α + Be and Be+ Be opticalpotentials, where the α particles are treated as structureless and interacting with each otherthrough a (real) potential v αα ( r ). The Hamiltonian of the α + α system is given by H αα ( r ) = T r + v αα ( r ) , (1)where T r is the relative kinetic energy. There are two versions of the α + α potential[26, 27] parametrized in terms of Gaussians amenable for the present CDCC calculation.In the present work, we have chosen the α + α potential suggested by Ali and Bodmer [26](referred to hereafter as AB potential). The AB potential simulates the Pauli blocking effectby a repulsive core that makes this potential much shallower than the deep α + α potentialsuggested by Buck et al. [27]. Both potentials reproduce equally well the α + α phase shifts,and they were shown by Baye [28] to be linked by a supersymmetric transformation. TheBuck potential, however, contains some deeply-bound states (two states with ℓ = 0 and onewith ℓ = 2), which do not have physical meaning but simulate the so-called Pauli forbiddenstates [29] in the microscopic α + α model. As long as the two-body α + α system is considered,the choice of either AB or Buck potential is not crucial. However, when dealing with morethan two α clusters, the forbidden states have to be removed as they produce spurious statesin a multi-cluster system like α + Be or Be+ Be. There are two methods to remove theforbidden states in the multi-cluster systems: either to apply the pseudostate method [30]or to use the supersymmetric transformation [28]. These two techniques, however, give riseto a strong angular-momentum dependence of the α + α potential that cannot be used inmost of the multi-cluster models. Among the 3 α models, only the hypersperical methodand Faddeev method are able to use the deep α + α potential with an exact removal ofthe α + α forbidden states. This is why other studies of the 3 α and 4 α systems [17, 31–43] have used only the ℓ -independent AB potential. In the present work, we perform theCDCC calculation of the α + Be and Be+ Be optical potentials using the AB potential ofthe α + α interaction, so that the spurious effects arising from the Pauli forbidden statescan be avoided.The present CDCC method is based on the eigenstates Φ ℓmλ ( r ) of the Hamiltonian (1)which can be written as Φ ℓmλ ( r ) = 1 r u ℓλ ( r ) Y ℓm ( ˆ r ) , where ℓ is the relative orbital momentum of the α + α system. The radial wave functions u ℓλ ( r ) of the two- α state λ are expanded over a basis of N orthonormal functions ϕ i ( r ) u ℓλ ( r ) = N X i =1 f ℓλ,i ϕ i ( r ) , (2)where f ℓλ,i are determined by diagonalizing the eigenvalue problem X j f ℓλ,j (cid:0) h ϕ i | H αα | ϕ j i − E ℓλ δ ij (cid:1) = 0 . (3)The eigenvalues with E ℓλ < E ℓλ > Be. Note that there is only a small number of physical statesin a CDCC calculation (often one for exotic nuclei). Although Be is unbound, its energyis very close to the α + α threshold, and the lifetime is long enough to use a quasi-boundapproximation for the ground state (g.s.). Equations (2) and (3) are general for any choice ofthe basis functions ϕ i ( r ). We use here a Lagrange-mesh basis [34] derived from the Legendrepolynomials, and the calculation of matrix elements in Eq. (3) is fast and accurate. We referthe reader to Ref. [34] for more details and the application of the Lagrange-mesh basis.The α + Be and Be+ Be systems are described by the 3 α and 4 α Hamiltonians, respec-tively, as H = H αα ( rrr ) + T R + X i =1 v αα ( S i ) H = H αα ( rrr ) + H αα ( rrr ) + T R + X i,j =1 v αα ( | SSS i − SSS j | ) , (4)5 (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:7) a a a (cid:1)(cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:7) a a (cid:4) (cid:5) (cid:6)(cid:7) (cid:2) (cid:8) aa (cid:9)(cid:10)(cid:11)(cid:9)(cid:12)(cid:11) FIG. 1. Alpha configurations and coordinates used for α + Be (a) and for Be+ Be (b). where R is the projectile-target coordinate and ( r , r ) are the internal coordinates of the Be nuclei, as illustrated in Fig. 1. Coordinates
SSS i and SSS j are expressed as a function of( RRR, rrr ) and of ( RRR, rrr ), respectively.In the CDCC approximation, the α + Be wave function for a total angular moment J and parity π is written asΨ JMπ ( R , r ) = X λℓL g JπλℓL ( R ) (cid:2) Φ ℓλ ( r ) ⊗ Y L ( ˆ R ) (cid:3) JM , (5)where the summation over the pseudostates λ is truncated at a given energy E max . The Be+ Be wave functions involve 4 α clusters and can be expressed asΨ JMπ ( R , r , r ) = X λ ℓ X λ ℓ X IL g Jπλ ℓ λ ℓ IL ( R ) × (cid:20)(cid:2) Φ ℓ λ ( r ) ⊗ Φ ℓ λ ( r ) (cid:3) I ⊗ Y L ( ˆ R ) (cid:21) JM , (6)where the parity conservation imposes ( − L = ( − I . There are two parameters definingthe CDCC basis: the maximum Be angular momentum ℓ max and maximum pseudostateenergy E max . The physical quantities obtained from the CDCC calculation (scattering ma-trix, elastic cross section, and local equivalent OP) must be converged with respect to theseparameters. In practice, the 4 α calculations involve many channels and are, therefore, highlytime consuming [17]. 6he relative radial wave functions χ Jπc ( R ), with the indices c = ( λℓL ) for α + Be and c = ( λ ℓ λ ℓ IL ) for Be+ Be , are obtained from the solutions of the following coupled-channel equations (cid:20) − ~ µ (cid:18) d dR − L ( L + 1) R (cid:19) + E c + E c − E c . m . (cid:21) χ Jπc ( R )+ X c ′ V Jπcc ′ ( R ) χ Jπc ′ ( R ) = 0 , (7)where µ is the reduced mass, E c . m . is the center-of-mass energy, E c and E c are the excitationenergies of the two interacting nuclei, separated by the distance R as shown in Fig. 1. Thecoupling potentials V Jπcc ′ ( R ) are determined by the method explained in Refs. [17, 35]. Thesystem of the coupled channel equations (7) is solved using the R -matrix method whichprovides explicitly the scattering matrix and the associated wave functions [36, 37]. Althoughthe AB potential [26] is real, it consistently reproduces the experimental α + α phase shiftsup to about 20 MeV. In the present CDCC approach, the loss of flux from the elasticscattering channel is due entirely to the breakup channels, and the local equivalent α + Beand Be+ Be optical potentials are therefore complex. Owing to the strong 2 α structure of Be it is likely that these breakup channels represent the main source of the absorption.
III. RESULTS AND DISCUSSIONA. Local equivalent OP for the α + Be and Be+ Be systems
The main goal of our study is to determine the local ( J -independent) equivalent opticalpotential U for the α + Be and Be+ Be systems at the considered energies, based on thescattering wave functions given by the solutions of the CDCC equations (7). The mainrequirement for this procedure is that the solutions χ Jπ of the one-channel OM equationwith the optical potential U (cid:2) E c . m . − T R − U ( R ) (cid:3) χ Jπ ( R ) = 0 (8)give the cross section of the elastic α + Be or Be+ Be scattering close to that given bythe 3 α or 4 α CDCC calculation (7), especially, the cross section at forward angles which issensitive to the surface part of the 3 α or 4 α interaction potential. We briefly discuss thetwo approaches used in the present work for this purpose.7 Be+ Be optical potentialE c.m. = 41.3 MeV U ( M e V ) R (fm) Re U
LEP
Im U
LEP
Re U
WSD
Im U
WSD
FIG. 2. Complex OP for the elastic Be+ Be scattering at E c . m . = 41 . (i) The quantum-mechanically consistent method using the matrix inversion was sug-gested in Refs. [17, 38] to derive a local equivalent potential (LEP) that exactly reproducesthe elastic cross section given by the CDCC calculation (7). However, this LEP has twomajor drawbacks that prevent its further use in the direct nuclear reaction calculation.Namely, the derived LEP strongly depends on the total angular momentum J , and its ra-dial dependence has the singularities caused by the nodes of the scattering wave functions.These problems can be handled by the method proposed in Ref. [38] which averages theobtained LEP over the angular momenta to obtain a smooth J -independent OP withoutdiscontinuity that approximately reproduces the CDCC elastic scattering cross section. Therecent 4-body CDCC calculation [17] has shown that such an averaging method to a fairly8ood approximation determines the local J -independent OP. The complex OP derived usingthis approach is denoted hereafter as U LEP , with its imaginary part W LEP originating fromthe breakup channels included in the CDCC calculation (7). We have first performed theCDCC calculation (7) for the elastic α + Be and Be+ Be scattering at E c . m . = 43 . α + α interaction. The maximumangular momentum of the α + α system is ℓ max = 2, and the maximum pseudostate energyis E max = 10 MeV. These cutoff values were well tested to ensure the convergence of boththe elastic cross section and U LEP . The complex U LEP for the α + Be and Be+ Be systemswere obtained first in a Lagrange mesh [38], and then interpolated into the smooth shapesfor use as the external input of the complex OP in the CRC calculation.(ii) The standard OM method can also be used to determine a phenomenological J -independent OP in the conventional Woods-Saxon (WS) form, with its parameters adjustedto obtain a good OM fit to the elastic cross section given by the 3 α or 4 α CDCC calculation.We have assumed in the present work the following (volume+surface) WS form for thecomplex OP of the α + Be and Be+ Be systems at the energies under study, denotedhereafter as U WSD − U WSD ( R ) = V v f v ( R ) − V d a d df d ( R ) dR + i (cid:20) W v f w ( R ) − W d a s df s ( R ) dR (cid:21) , where f x ( R ) = 11 + exp[( R − R x ) /a x ] , x = v, d, w, s. (9)The elastic Be+ Be and α + Be cross sections at E c . m . = 41 . α and 3 α CDCC calculations (7), respectively, have been used in the method (ii) as the“experimental data” with the uniform 10% uncertainties for the OM analysis to determinethe WS parameters of U WSD . We found quite a shallow WS potential that gives a goodagreement of the OM result with the CDCC elastic cross section (see the OP parameters inTable I).The radial shapes of both U LEP and U WSD potentials for the Be+ Be system at E c . m . =41 . α + α interaction isshown to result on quite a shallow potential U LEP . A moderate oscillation of U LEP is seen atsmall radii that might originate from the J -dependence of the exact LEP discussed above.The best-fit WS complex OP determined by the method (ii) has the strength of Re U WSD enhanced slightly at the surface ( R ≈ U WSD . One can seein Table I that the volume integrals of Re U WSD and Im U WSD are close to those of Re U LEP -3 -2 -1 -2 -1 U WSD U LEP
CDCC Be+ Be elastic scattering E c.m. = 41.3 MeV+ Be elastic scattering E c.m. = 43.3 MeV c.m. (deg) d / d R FIG. 3. The CDCC prediction for the elastic Be+ Be (upper panel) and α + Be (lower panel)scattering at E c . m . = 41 . U LEP and U WSD determined by the methods (i) and (ii), respectively. and Im U LEP , which indicates that the OP’s given by both methods belong to about thesame potential family. The results of the CDCC calculation (7) for the elastic Be+ Be and α + Be scattering at E c . m . = 41 . U LEP and U WSD . The OM results given by bothOP’s agree fairly good with the CDCC prediction at forward angles, while at medium and10
ABLE I. WS parameters (9) of U WSD given by the method (ii) based on the OM fit to the elastic Be+ Be and α + Be cross sections at E c . m . = 41 . α and 3 α CDCC calculations (7), respectively. J V and J W are the volume integrals per interacting nucleonpair of Re U WSD and Im U WSD , respectively. J V LEP and J W LEP are those of U LEP given by themethod (i). V v ( W v ) R v ( w ) a v ( w ) V d ( W d ) R d ( s ) a d ( s ) − J V ( J W ) − J V LEP ( J W LEP )(MeV) (fm) (fm) (MeV) (fm) (fm) (MeV fm ) (MeV fm ) Be+ Be, E c . m . = 41 . He+ Be, E c . m . = 43 . large angles the phenomenological U WSD determined by the method (ii) better reproducesthe CDCC cross sections. The agreement with the CDCC results becomes worse at largeangles, and it might be due to the nonlocality effects.We note further that the method (i) fails to derive a smooth ℓ -independent U LEP basedon the CDCC results obtained with the deep Buck α + α potential. Namely, the obtained ℓ -independent U LEP turns out to be deeper but strongly oscillatory, and it gives the elasticcross section substantially different from that given by the CDCC calculation. Such afailure of the ℓ -independent U LEP based on the Buck potential is presumably caused by thePauli forbidden states, and this remains an unsolved problem for the present 4 α CDCCmethod. Therefore, we deem hereafter reliable only the CRC results obtained with α + Beand Be+ Be OP’s derived based on the CDCC elastic cross section obtained with the ABpotential of the α + α interaction. 11 . CRC study of the C( α, Be) reaction
The α + Be and Be+ Be optical potentials determined by the methods (i) and (ii) havebeen further used as the potential inputs for the CRC study of the α transfer C( α , Be)reaction measured at E α = 65 MeV [22]. We briefly recall the multichannel CRC formalism,to illustrate how the OP’s of the α + Be and Be+ Be systems enter the CRC calculationof the α transfer cross section. In general, the CRC equation for the initial channel β of thetransfer reaction can be written as [39, 40] (cid:2) E β − T β − U β ( R ) (cid:3) χ β ( R ) = X β ′ = β (cid:8) h β | W | β ′ i + h β | β ′ i (cid:2) T β ′ + U β ′ ( R ′ ) − E β ′ (cid:3)(cid:9) χ β ′ ( R ′ ) . (10)Without coupling to the inelatic scattering channels, the indices β and β ′ in Eq. (10) stand Be r r’ R’ R R cc C Be FIG. 4. Cluster configurations in the entrance and exit channels of the C( α, Be) reaction, andthe corresponding coordinates used for the inputs of the potentials in the CRC calculation. for the initial α + C and final Be+ Be partitions of the α transfer reaction, respectively,as shown in Fig. 4. In the present CRC analysis, the index β ′ is used to identify both the Be+ Be g . s . and Be+ Be ∗ + exit channels of the final partition. The distorted waves χ β and χ β ′ are given by the optical potentials U β and U β ′ of the α + C and Be+ Be systems,12espectively. The α transfer proceeds via the transfer interaction W which is determined inthe post form [39, 40] as W = V α − C ( r ) + (cid:2) U α + Be ( R cc ) − U Be+ Be ( R ′ ) (cid:3) , (11)with the radii of the potentials illustrated in Fig. 4. Here V α − C ( r ) is the potential bindingthe α cluster to the Be core in the g.s. of C. The difference between the core-core OPand that of the final partition, U α + Be ( R cc ) − U Be+ Be ( R ′ ), is the complex remnant termof W . The CRC equations (10) are solved iteratively using the code FRESCO written byThompson [41], with the complex (nonlocal) remnant term and boson symmetry of theidentical Be+ Be system properly taken into account. One can see that the Be+ Be OPenters the CRC calculation of the α transfer C( α , Be) reaction as the input of both theremnant term and the OP of the final partition. Therefore, it can be tested indirectly basedon the CRC description of the α transfer data.The OP of the initial partition U α + C has its real part given by the double-folding modelusing the density dependent CDM3Y6 interaction [11], and imaginary part chosen in the WSshape, with the parameters fine tuned to the best CRC fit of the elastic α + C scattering datameasured at E α = 65 MeV [23, 24]. A reasonably good CRC description of the elastic α + Cscattering data at E α = 65 MeV (see Fig. 5) is achieved without renormalizing the strengthof the real OP. In principle, we could also think of using the 4 α CDCC method to predict the α + C OP at the considered energy. However, Suzuki et al. [42] have shown that the use ofa local α + α potential (that properly reproduces the experimental α + α phase shifts) cannotprovide a proper 3 α description of both the g.s. and 0 +2 excitation (known as Hoyle state) of C. This problem could only be solved by introducing a microscopically founded nonlocal α + α force that mimics the interchange of three α clusters in the phase space allowed bythe Pauli principle [42]. The use a nonlocal α + α interaction remains beyond the scope ofthe present 4-body CDCC method [17]. On the other hand, the elastic α + C scattering atenergies above 10 MeV/nucleon is proven to be strongly refractive [9, 11, 43], with a far-sidedominant elastic cross section at large angles typical for the nuclear rainbow, which can bewell described by the deep (mean-field type) real OP predicted by the double-folding model[11, 43].For the α transfer reaction, the initial (internal) state of the α cluster bound in C isassumed to be 1 s state. Then, the relative-motion wave function Φ NL ( r ) of the α + Be13
20 40 60 80 100 120 14010 -2 -1 Elastic He+ C scatteringE = 65 MeV d / d R c.m. (deg) Real folded + Imag. WS Data by Goncharov et al. Data by Yasue et al.
FIG. 5. CRC description of the elastic α + C scattering at E α = 65 MeV given by the (unrenor-malized) real folded OP and imaginary OP chosen in the WS form [11], in comparison with thedata taken from Refs. [23, 24] configuration in the C target ( L -wave state) has the number of radial nodes N determinedby the Wildermuth condition [39], so that the total number of the oscillator quanta N isconserved N = 2( N −
1) + L = X i =1 n i −
1) + l i , (12)where n i and l i are the principal quantum number and orbital momentum of each constituentnucleon in the α cluster. Φ NL ( r ) is obtained in the potential model using V α − C ( r ) chosenin the WS shape, with its radius and diffuseness fixed as R = 3 .
767 fm and a = 0 .
65 fm,and the WS depth ( V = 51 . α separation energy of C.Because the ground state of Be is unbound by 92 keV, we have used in the present CRCcalculation a quasi-bound approximation for Be similar to that used for the g.s. of Be inthe CDCC calculation as discussed in Sec. II, to describe the formation of Be on the exit14hannel of the α transfer reaction. For this purpose, the repulsive core of the AB potentialwas slightly weakened to give the (1 s ) state Φ α ( r ′ ) of the α cluster in Be a quasi-bindingenergy of 0.01 MeV. The cluster wave functions Φ NL ( r ) and Φ α ( r ′ ) are used explicitly inthe calculation of the complex nonlocal α transfer form factor h β ′ | W | β i ∼ h [Φ α ( r ′ ) ⊗ Y L β ′ ( ˆ R ′ )] J β ′ | W | [Φ NL ( r ) ⊗ Y L β ( ˆ R )] J β i , (13)where L β and L β ′ are the relative orbital momenta of the initial and final partitions. Thewave functions of Be core in the initial and He core in the final partitions are omitted in(13) because they are spectators and do not contribute to the transfer [40].The CRC calculation of the α transfer C( α , Be) reaction requires the input of thespectroscopic factor of the α cluster in Be which is naturally assumed to be unity, and thatof the cluster configuration α + Be in C. The latter is determined as S α = | A NL | , wherethe spectroscopic amplitude A NL is given by the dinuclear overlap h Be | C i = A NL Φ NL ( r ) . (14)Because two different exit channels of the α transfer C( α , Be) reaction were identified,with the emitting Be being in the g.s. and excited 2 + state [22], one needs to evaluate (14)for the two configurations α + Be g . s . and α + Be ∗ + , which are associated with Φ N =3 ,L =0 ( r )( S -wave) and Φ N =2 ,L =2 ( r ) ( D -wave). In general, one can treat these two S α values as freeparameters to be adjusted by the best DWBA or CRC fit to the α transfer data. Insteadof this procedure, we have adopted in the present work the S α values predicted for theseconfigurations by Kurokawa and Kato using the CSM method [25]. Namely, S α (g . s . ) ≈ . S α (2 + ) ≈ .
38, which are rather close to the spectroscopic factors predicted recently byother cluster models [44, 45]. Note that the S α values used in our CRC calculation are alsoclose to those extracted from the DWBA analysis of the Be transfer reaction Mg( α , C) O[46]. The same Be+ Be OP has been used for both Be+ Be g . s . and Be+ Be ∗ + exit channelsof the final partition (see more discussion below).The CRC results for the α transfer C( He, Be) reaction at E α = 65 MeV to the groundstate of Be are compared with the measured data [22] in Fig. 6. One can see that theCDCC-based optical potentials of the Be+ Be partition ( U LEP and U WSD determined bythe methods (i) and (ii), respectively) give a good CRC description of the α transfer datawithout any adjustment of its strength, using S α (g . s . ) = | A | ≈ .
36 taken from the results15
10 20 30 40 50 60 70 8010 -1 E = 65 MeV c.m. (deg) d / d ( b / s r) C( , Be) Be g.s. U WSD U LPE U Int WS
FIG. 6. CRC description of the α transfer reaction C( α , Be) Be g . s . measured at E α = 65 MeV[22], using the α spectroscopic factor S α (g . s . ) ≈ .
36 taken from the CSM calculation [25]. The CRCresults obtained with the CDCC-based optical potentials U WSD and U LEP for the Be+ Be partitionare shown as the solid and dash-dotted lines, respectively. The dashed line is the CRC resultobtained with the WS potential U IntWS , interpolated from the OP’s adopted for the , Be+ Besystems at the nearby energies [5]. of the CSM calculation [25]. With a better OM description of the CDCC elastic cross sectiongiven by the U WSD potential (see Fig. 3), the CRC cross section given by U WSD also agreesslightly better the measured α transfer data.Because the Be+ Be OP is unknown so far, a practical assumption is to estimate itfrom the phenomenological OP’s adopted for the neighboring , Be isotopes. For example,the proton transfer reaction Li( B, Be) Be was measured by Romanyshyn et al. [5], anda deep WS potential was deduced for the real OP of the Be+ Be system at E c . m . = 31 . Be+ Be OP is quite close to that adopted earlier for the Be+ Be system (see16ig. 10 of Ref. [5]). Therefore, one might expect the Be+ Be OP to be close to the WSoptical potentials adopted for the , Be+ Be systems. To explore the reliability of thispractical approach, we have interpolated the Be+ Be OP from those of the , Be+ Besystems adopted in Ref. [5] and denoted it as U IntWS , with V v = 155 . R v = 3 . a v = 0 .
768 fm; and W v = 13 . R w = 5 . a w = 0 .
768 fm. The use of U IntWS in the CRC calculation of the C( α , Be) reaction completely fails to account for the data(see dashed lines in Fig. 6). In fact, the spectroscopic factor S α (g . s . ) taken from Ref. [25]must be scaled by a factor of 25, so that the CRC cross section obtained with U IntWS can becomparable with the measured α transfer data. -1 d / d ( b / s r) c.m. (deg) U WSD U LEP U Int WS C( , Be) Be + E = 65 MeV
FIG. 7. The same as Fig. 6 but for the α transfer reaction with one emitting Be nucleus beingin its 2 + state ( E x ≈ .
94 MeV), using the α spectroscopic factor S α (2 + ) ≈ .
38 taken from theCSM calculation [25]
We have also performed the CRC calculation of the C( α, Be) Be ∗ reaction at E α = 65MeV with one emitting Be nucleus being in its 2 + state ( E x ≈ .
94 MeV). Although the17 + state of Be is a broad resonance, its 2 α -cluster structure remains similar to that of theground state, and the α transfer cross section measured for the 2 + state is of about the samestrength as that measured for the g.s. as shown in Figs. 6 and 7. The α spectroscopic factors S α predicted by the CSM calculation [25] are also close for both the ground- and 2 + states.It is, therefore, reasonable to use the same CDCC-based Be+ Be OP for the partition Be+ Be ∗ + in the exit channel. The results of the CRC calculation are compared with thedata [22] in Fig. 7, and one can see that the (unrenormalized) CDCC-based Be+ Be OPalso delivers a good description of the α transfer data using S α (2 + ) = | A | ≈ .
38 givenby the CSM calculation [25]. The use of U IntWS for the Be+ Be OP in the CRC calculationalso strongly underestimates the α transfer data (see dashed line in Fig. 7).In conclusion, a good CRC description of the measured C( α, Be) data has been obtainedwith the Be+ Be OP’s determined by the methods (i) and (ii) from the elastic Be+ Becross section given by the 4 α CDCC calculation, and with the α spectroscopic factors givenby the CSM calculation [25]. The fact that no adjustment of the potential strength of U LEP and U WSD was necessary suggests that the 4 α CDCC method is a reliable approach to studythe Be+ Be system. These results also show that, despite the short life-time of Be, the α transfer C( He, Be) reaction is very sensitive to the OP of the Be+ Be partition. Themeasured α transfer data clearly prefer the shallow OP based on the result of the 4 α CDCCcalculation using the AB potential of the α + α interaction [26], over the deep WS potentialinterpolated from those adopted for the , Be+ Be systems [5].
IV. SUMMARY
The 3-body and recently suggested 4-body CDCC methods [17] have been used to predictthe elastic α + Be and Be+ Be scattering at E c . m . = 43 . α + α interaction suggested by Ali and Bodmer [26] that well reproduces the experimental α + α phase shifts. The elastic cross sections predicted by the CDCC calculation were usedto determine the local OPs of the α + Be and Be+ Be systems.The CDCC-based α + Be and Be+ Be OP’s are further used as the inputs of the core-core OP and that of the final partition, respectively, in the CRC study of the α transfer C( α, Be) reaction measured at E α = 65 MeV [22], with the emitting Be being in both theg.s. and 2 + state. These α transfer data are well reproduced by the CRC results obtained18ith the CDCC-based OPs and α spectroscopic factors of the Be+ Be g . s . and Be+ Be ∗ + configurations in C taken from the CSM cluster calculation [25].As alternative to the shallow (surface-type) Be+ Be OP determined from the elastic Be+ Be cross section predicted by the 4 α CDCC calculation, a deep WS potential withparameters interpolated from the OPs adopted for the , Be+ Be systems at the nearbyenergies [5] has been used in the CRC calculation, and it grossly underestimates the α transfer data. This might be due to the fact that Be and Be are well bound nuclei, andthe breakup effect is therefore much weaker than that of the unbound Be.We conclude that the α + Be and Be+ Be optical potentials can be determined from theelastic scattering cross section predicted, respectively, by the 3 α and 4 α CDCC calculationsusing the realistic α + α interaction that properly reproduces the experimental α + α phaseshifts [26]. The present CRC study should motivate further theoretical and experimentalstudies of the α transfer C( α, Be) reaction as a probe of the 4 α interaction and the α -cluster structure of C. ACKNOWLEDGEMENT
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