Direct Nuclear Reactions in Lithium-Lithium Systems:7Li+7Li at Elab = 2 - 16 MeV

P. Rosenthal, H. Freiesleben?, B. Gehrmann, I. Gotzhein, K. W. PotthastInstitut fur Experimentalphysik I, Ruhr-Universitat-Bochum,

D-44780 Bochum, Germanyand

B. Kamys, Z. RudyInstitute of Physics, Jagellonian University, PL-30059 Cracow, Poland

? Present address: Institut fur Kern- und Teilchenphysik,Technische Universitat Dresden, D-01062 Dresden, Germany

August 3, 2018

Abstract

Angular distributions of 7Li(7Li,t), (7Li,α) and (7Li,6He) reactions were mea-sured for laboratory energies from 2 - 16 MeV. Exact finite range DWBA anal-yses were performed with the aim to identify contributions of direct processesand to investigate the applicability of DWBA to such few nucleon systems. Itturned out that DWBA can be successfully applied to estimate differential andtotal cross sections of direct transfer processes in 7Li+7Li interaction. The di-rect mechanism was found to play a dominant role in most of these reactionsbut significant contributions of other, strongly energy dependent processes werealso established. It is suggested that these processes might be due to isolatedresonances superimposed on the backround of statistical fluctuations arisingfrom interference of compound nucleus and direct transfer contributions.

Nuclear Reactions: 7Li+7Li at 2≤ Elab ≤ 16 MeV; measured angular distribu-tions and excitation functions of p, d, t, α, and 6He-channels; deduced directmechanism contribution to t, α and 6He-channels by DWBA analysis; reactionmechanism inferred.

PACS: 25.70.-x; 25.70.Hi

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1 Introduction

Prominent clusterization of weakly boundlithium nuclei may influence in various ways themechanism of reactions in which they take part.A high probability of direct cluster-transfer re-actions should be a likely consequence of clus-terization, indicated by both large spectroscopicamplitudes and small separation energies of theclusters. They might play a dominant role in thesystem of two interacting lithium nuclei. How-ever, other reaction mechanisms cannot be ex-cluded since the small binding energy of the en-trance channel nuclei leads to a compound nu-cleus with high excitation energy, i.e. with highdensity of states. Hence, large compound nu-cleus cross sections might occur. Furthermore,due to the clusterization, states with a specific,simple structure may appear as isolated reso-nances [1, 2] superimposed on the statisticalbackground of compound nucleus decay [3].

The aim of the present work is twofold.First, to study the contribution of direct reac-tions in the 7Li+7Li system and, in turn, to es-timate the magnitude of other possible mech-anisms. If direct processes were to contributesignificantly we have an opportunity to test thequality of DWBA predictions by straightforwardcomparison with experimental data and, thus,to investigate the applicability of direct reactiontheory in the extreme case of a nuclear systemconsisting of few nucleons. It is a priori not ob-vious whether the standard DWBA method canbe used for such a system since this approachexplicitely assumes transfer reactions to be onlya perturbation to elastic scattering. This condi-tion may be fulfilled for transfer reaction crosssections because they are typically smaller by anorder of magnitude than those of elastic scatter-ing. However, each transfer, even that of a singlenucleon, modifies the mass of target and pro-jectile in this light nuclear system to an extentwhich can hardly be considered as perturbation.

The second intriguing aspect of the7Li+7Li system is the possibility to test inDWBA calculations various optical model (OM)

potentials which give an equivalent descriptionof elastic and inelastic scattering data. In a re-cent extensive investigation of elastic and inelas-tic scattering in the 7Li+7Li system [4] no po-tential could be singled out on the basis of thequality with which experimental data were de-scribed. One may hope that cross sections oftransfer reactions will be more sensitive to opti-cal model potentials used to generate distortedwaves for DWBA calculations than elastic or in-elastic scattering cross sections.

It seems likely that both alpha particletransfer 7Li(7Li,t)11B and the triton transfer7Li(7Li,4He)10Be are the best candidates for di-rect - transfer reactions due to the very smallseparation energy of 7Li→4He + t. These trans-fers, however, significantly change the mass oftarget and projectile during the collision (40 -60%). Thus, as pointed out above, the appli-cability of a perturbation theory must be ques-tioned. In this respect, nucleon transfer reac-tions, namely proton transfer 7Li(7Li,6He)8Beand neutron transfer 7Li(7Li,6Li)8Li, are con-sidered most adequate. Unfortunately, the neu-tron transfer reaction appears to have a negativeQ-value and therefore its cross section is verysmall in the energy range studied here. Thusonly alpha particle, triton, and proton trans-fers were selected for the present investigation.They correspond to triton, alpha particle and6He exit channels, respectively. The proton andthe deuteron exit channels also measured in thepresent experiment were used to estimate the to-tal fusion cross section and compound nucleuscontribution to the reactions under investigation[5].

Studies of 7Li(7Li,t) and 7Li(7Li,4He) re-actions are reported in the literature mainly forlow energies (Elab= 2 - 6 MeV) [6, 7, 8] withexception of the (7Li,4He) reaction for which aforward angle excitation function was measuredin the range of Elab= 2 - 21 MeV [9]. The7Li(7Li,4He)10Be reaction was also studied at 26and 30 MeV with the aim to look for resonantstates of 10Be [10], while the 7Li(7Li,11B)t reac-tion was investigated at 79.6 MeV as part of a

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study of reactions leading to multi - neutron fi-nal states [11]. All these investigations were notconcerned with the questions asked here.

The 7Li(7Li,6He) reaction was reported inthe literature at very low energies (Elab= 3 MeV[12], Elab= 3-3.8 MeV [13]) where only spectraat small reaction angles were measured, and atenergies above our energy range (Bochkarev etal. [14], Elab=22 MeV). Angular distributions oftransitions to the ground and the first excitedstate of 8Be were found in the latter experimentto show pronounced oscillations and to be rathersteep. The authors suggested a direct reactionmechanism since DWBA calculations agreed rea-sonably well with the experimental data. More-over, estimation of the compound nucleus contri-bution made by means of the Hauser-Feshbachmodel indicated [14] that the compound nucleusmechanism is responsible for only a small part(approximately 10%) of the experimental crosssection. Thus the 7Li(7Li,6He)8Be reaction maybe used to study the applicability of the DWBAformalism for this few - nucleon system and fortesting various OM potentials in the 7Li+7Li sys-tem.

In the present work angular distributionsof light ejectiles (tritons and 4He) are measuredin the energy range from Elab= 2 - 16 MeV whilethose for 6He are restricted to 8 - 16 MeV. Theexperimental procedure and results are given inthe next chapter. The analysis of the data inthe frame of the DWBA formalism is presentedin the third chapter while a summary with con-clusions is provided in the last one.

2 Experimental procedure

The experiments were carried out at the4 MV Dynamitron Tandem accelerator at theRuhr-Universitat Bochum. 7Li-beams were pro-duced with a deflection sputter source; for theseexperiments the beam currents (max. 2.5 µALi− ions) were limited to 30 nA up to 100 nA(electric) in order not to destroy the targets and

backings. The beam was focused onto the tar-gets via two collimators (� =1.5 mm) 40 and60 cm upstream of the target. Targets of metal-lic lithium in natural isotopic abundance (92.5%7Li, 7.5% 6Li) were prepared by evaporation ontodifferent backing materials adapted to each ex-perimental setup.

The standard technique to identifycharged particles in low energy nuclear reactionsis the ∆E-E-discrimination. This techniquewas employed with triple telescopes of surfacebarrier detectors in order to cover the broaddynamical range of the p, d, t, α and 6Heexit channels in one measurement. These exitchannels were investigated in an energy rangefrom Elab = 2 - 16 MeV in steps of 0.5 MeVby measuring the differential cross sectionsfrom θlab = 0◦ - 80◦. Some typical spectra aredepicted in Fig. 1.

Due to the identity of projectile and tar-get angular distributions are symmetrical withrespect to 90◦ in the center - of - mass system.Hence, it is sufficient to measure up to 80◦ in thelaboratory system in order to obtain the wholeangular distribution.

To determine absolute cross sections forthese exit channels three separate experimentalsetups were utilized:

1. With the first differential cross sectionswere measured from θlab = 10◦ - 80◦ in steps of10◦. These measurements were carried out withtwo triple telescopes. The first telescope with anaperture of 0.311 msr and total energy resolutionof 280 keV covered the forward angles from θlab

= 10◦ up to 40◦. The second telescope (aper-ture: 0.179 msr; total energy resolution: 240keV) was used to cover the more backward an-gles from θlab = 50◦ up to 80◦. In front of eachtelescope aluminium foils were positioned to ab-sorb 7Li ions elastically scattered from the Ni-backing. A thickness from 30 up to 55 µm wassufficient depending on beam energy and labo-ratory angle of the telescopes was sufficient. Forthis setup transmission targets were used con-sisting of metallic 7Li with an area density of 55µg / cm2 evaporated onto a Ni-backing with an

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Figure 1: Experimental spectra of tritons (θlab =30◦, Elab=9 MeV), α-particles (θlab = 10◦,Elab=8 MeV) and 6He ejectiles (θlab = 10◦,Elab=16 MeV)

area density of 90 µg / cm2.2. The differential cross section of the

7Li(7Li, α0,1)10Be reaction at 0◦ and 5◦ was mea-sured separately with a special target. It con-sisted of 55 µg / cm2 metallic 7Li evaporatedonto Ni-foils; their thickness of 5.0 µm up to35.4 µm, depending on beam energy, sufficed tofully stop the 7Li-beam, rendering possible mea-surements at 0◦. One triple telescope with anaperture of 0.314 msr was utilized. Both mea-surements were carried out independently on ab-solute scale. The purpose of this second experi-ment was twofold. First, to extend the angulardistributions to 0◦, a region which is very sen-sitive to transfers with orbital angular momen-tum of l=0. Second, to verify the experimentaldata of Wyborny and Carlson [9] who found veryprominent structures in the 0◦ - excitation func-tion (cf. Fig. 4).

3. For either setup the target thicknesswas determined via Rutherford scattering of58Ni4+ off 7Li in a well defined geometry. Exper-imental details of target preparation and thick-ness determination may be found elsewhere [15].

It can be inferred from Fig. 1 that crosssections for triton exit channels are readily deter-minable for transitions to the ground state andthe first three excited states in 11B. In case of theα-particle exit channel only transitions to theground and first excited state were evaluated,and for the 6He-exit channel only the groundstate transition was selected for the analysisbeause the higher lying excited states in eithercase are residing on a continuous backgroundof three particle decays, namely 14C? →9Be +n + α and 14C? → α+10Be→6He+α, respec-tively, the shape of which is unknown, renderingrather difficult a reliable evaluation. In Fig. 2 wepresent an overview of angular distributions forthe three ground state transitions investigated.They are represented by a least square fit of aLegendre polynomial expansion to the data. Forreason of legibility experimental data are onlyincluded for the 6He-channel. Experimental un-certainties are of symbol size, if not shown ex-plicitely.

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Figure 2: Overview of angular distributions of ground state transitions 7Li(7Li,t0), 7Li(7Li,α0) and7Li(7Li,6He0) represented by result of a least square fit of series of Legendre polynomials to thedata. For reason of legibility experimental data are only included for the 6He channel.

Fig. 3 contains the excitation functionsof angle integrated cross sections for the triton

channel (four lowest states of 11B), the α-channel(two lowest states of 10Be) and the 6He-channel

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Figure 3: Excitation functions of angle in-tegrated cross sections for 7Li(7Li,t)11B tothe four lowest states of 11B; (g.s.;3/2−),(2.125 MeV;1/2−), (4.445 MeV;5/2−) and (5.020MeV;3/2−); excitation functions of angle inte-grated cross sections for 7Li(7Li,4He)10Be to thetwo lowest states of 10Be; (g.s.;0+) and (3.368MeV;2+) and excitation function of angle inte-grated cross section for 7Li(7Li,6He)8Beg.s..

(ground state of 8Be).

Fig. 4 shows the excitation function for7Li(7Li,α1) measured at 0◦ together with thedata of Wyborny and Carlson [9]. A very goodagreement can be stated.

3 DWBA analysis

The calculations were performed using the ex-act finite range DWBA computer code Jupiter-5[16] in both representations (prior and post) ofthe transition potentials. In the ideal case ofthe exact knowledge of transition potentials e.g.from some microscopic model, DWBA calcula-tions should lead to the same result in both rep-resentations [17]. Therefore the quality of agree-ment between results of calculations made withineither representations may be taken as indicationof both the applicability of the DWBA formal-ism in its standard form and the proper choiceof potentials.

Figure 4: Excitation function for 7Li(7Li,α1)measured at θlab = 0◦ (full dots) together withdata of Wyborny and Carlson (open dots) [9]

We followed the standard prescription ofDWBA for choosing the strong interaction po-tential responsible for the transfer. This is dis-cussed in the following using the proton transferreaction 7Li(7Li,6He)8Be as an example. Eitherthe potential which binds the proton to 7Li (post)or that which binds the proton to 6He (prior rep-resentation) was taken as nuclear transition po-

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Table 1: Optical model potentials used in the DWBA calculations.System Family U rU aU W rW aW rC Ref.

MeV fm fm MeV fm fm fm7Li+7Li S-1 3 1.456 1.416 9.8 1.369 0.441 1.25 [4]

S-2 21 0.802 1.279 13.8 1.354 0.409 1.25 [4]S-3 49 0.800 1.036 18.2 1.354 0.387 1.25 [4]S-4 69 0.940 0.860 21.2 1.381 0.345 1.25 [4]S-5 101 0.965 0.781 23.3 1.397 0.320 1.25 [4]S-6 136 0.994 0.721 26.2 1.411 0.296 1.25 [4]

10Be+4He S 136 0.673 0.792 35.6 0.924 0.137 0.795 [20]11B + t V 133 0.923 0.570 19.5 1.090 0.220 0.923 [21]

Real parts of all potentials have the Woods-Saxon form with the following parametrization of radii :

R=r0*(A1/31 + A

1/32 ). The imaginary potentials of the ”V” families have the volume shape of Woods-Saxon

form while those of the ”S” families use the surface shape of derivative of Woods-Saxon form.

tential.

Thus it was assumed that perfect cancel-lation occurs of the so called ”indirect transitionpotentials” i.e. the core-core (7Li-6He) interac-tion potential is equivalent to either the opticalmodel (OM) potential for the 7Li-7Li channel(prior representation) or the optical model po-tential for the 6He-8Be channel (post represen-tation). Such an assumption seems to be jus-tified because, in contradistinction to a rathergood knowledge of entrance/exit channel opti-cal model potentials, the core-core potential isnot known and it is necessary to make assump-tions concerning this potential. The best approx-imation should be an optical model potentialfor scattering of the core - core nuclear system.However, this potential is usually not known,and moreover, it is not obvious at which energyof the relative motion of the core - core systemsuch potential should be taken. Thus the stan-dard prescription for choosing the nuclear tran-sition potential is to approximate the core - corepotential by the optical model potential of eitherthe entrance or the exit channel.

The binding potentials were taken inWoods-Saxon form. Their geometrical parame-ters were arbitrarily fixed at the following values:the reduced radius r0 = 0.97 fm (for symmetri-

cal parametrization i.e. R=r0∗(A1/3core + A1/3

cluster)and the diffuseness parameter a= 0.65 fm. Suchvalues were successfully used when describingthe alpha particle transfer in 14N(d,6Li)10B reac-tions [18]. The depth of the potentials was fittedto reproduce the appropriate binding energy.

The Coulomb ”indirect transition poten-tials” were all taken into account in the presentanalysis and they were used in the standard formof uniformly charged spheres.

The Jupiter-5 computer code evaluates thereaction amplitudes for one orbital of the clus-ter in the donor and one orbital in the acceptornucleus. For some reactions e.g. alpha parti-cle transfers to 11Bg.s., 11B4.45 and 11B5.02 tworeaction amplitudes must be calculated becauseof the two different orbitals of an alpha par-ticle cluster in the boron nucleus. The coher-ent superposition was performed by means of aseparate computer code SQSYM [19] which alsotook care of the antisymmetrization of ampli-tudes which is required by the identity of pro-jectile and target.

The introduction of free parameters wasavoided by the following procedure: The transi-tion potentials were fixed according to the stan-dard prescription described above.

Parameters of optical model potentials

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Table 2: Cluster spectroscopic amplitudes used in the DWBA calculations.Nucleus Core Cluster n l j C*A Ref.

7Lig.s.(3/2−) t 4He 1 1 1 1.084 [24, 25]4He t 1 1 3/2 1.084 [24, 25]6He p 0 1 3/2 -0.831 [22]

8Beg.s.(0+) 7Li p 0 1 3/2 1.287 [22]10Beg.s.(0+) 7Li t 1 1 3/2 0.556 [25]10Be3.37(2+) 7Li t 1 1 1/2 0.568 [25]

1 1 3/2 0.040 [25]0 3 5/2 0.604 [25]0 3 7/2 -0.299 [25]

11Bg.s.(3/2−) 7Li 4He 2 0 0 -0.509 [27]1 2 2 0.629 [27]

11B2.13(1/2−) 7Li 4He 1 2 2 -0.585 [27]11B4.45(5/2−) 7Li 4He 1 2 2 0.725 [26]

0 4 4 0.018 [26]11B5.02(3/2−) 7Li 4He 2 0 0 -0.292 [26]

1 2 2 -0.322 [26]

”n” corresponds to the number of nodes of the bound state radial wave function (exluding r=0 and infinity),

”l” is the orbital angular momentum of relative motion of the cluster and the core,

”j” is the total angular momentum of the orbital,

”C*A” denotes the product of the isospin Clebsch-Gordan coefficient and the spectroscopic amplitude.

were taken from the literature, whenever it waspossible i.e. for 7Li+7Li [4], 10Be+α [20] and11B+t [21] systems, or they were approximatedby 7Li+7Li potentials for the unaccessible elas-tic 8Be+6He channel. All potential parametersare listed in Table 1. The spectroscopic ampli-tudes were taken from shell model calculationspublished in the literature for protons [22], tri-tons [23, 24], and for α - particles [25, 26, 27].Their values are given in Table 2. In that senseall calculations of cross sections were performedwithout any free parameter.

3.1 The proton transfer reaction -7Li(7Li,6He)8Be

A qualitative inspection of the experimental datasuggests the (7Li,6He) reaction to be dominatedby a direct reaction mechanism. The differen-tial cross section of this reaction is significantly

smaller (at least two orders of magnitude) thanthe elastic scattering cross section in the 7Li+7Lisystem [4] compared at corresponding scatter-ing angles. Therefore one is allowed to treat theproton transfer reaction (7Li,6He) as a perturba-tion to the elastic scattering and hence to applythe distorted wave Born approximation. Fur-thermore, the transfer of one nucleon results ina relatively small rearrangement between the in-teracting lithium nuclei and may thus, also fromthis point of view, be considered as a perturba-tion. These arguments led us to consider the7Li(7Li,6He)8Be reaction as the best candidateamong the reactions under investigation for test-ing the applicability of DWBA to such light nu-clear system.

Calculations of angular distributions forproton transfer were performed for 8, 10, 12,14 and 16 MeV laboratory energy for which ex-perimental data were measured in the present

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work. Additional calculations were performedat an energy of 22 MeV for the reaction leadingto both the ground and the first excited stateof 8Be rendering possible a comparison with theexperimental data of Bochkarev et al. [14].

Figure 5: Experimental angular distributionof 7Li(7Li,6He)8Beg.s. reaction at Elab=16 MeV(full dots). Solid and dashed curves representresults of DWBA calculations in post and priorrepresentation, respectively. Six OM potentialswere applied which describe elastic scattering inthe 7Li+7Li system equally well [4]. The ”fam-ily” of the OM potential parameters is specifiedby quoting the depth of its real part. The sameOM potentials were used in entrance and exitchannel.

The quality of reproduction of the exper-imental data, the similarity of results in priorand post representations, and the dependence of

the results on the optical model potentials wasfound to be almost the same for the energies un-der investigation. Therefore, we present in Figs.5, 6 and 7 only results for one bombarding en-ergy,i.e., 16 MeV.

Figure 6: Same as Fig. 5, but the entrance chan-nel OM potential is fixed (family #4, i.e. ”69MeV” of [4]) while all six equivalent potentialsof ref. [4] are used for the exit channel.

Identical optical model potentials used forboth entrance and exit channel lead to a perfectequivalence of the prior and post representations.This can be inferred from Fig. 5 where angu-lar distributions evaluated in prior (dashed lines)and in post representation (full lines) almost co-incide for six different ”families” of parametersof OM potentials listed in Table 1. Thus we con-clude, that DWBA is applicable for this reaction

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under the stated conditions.

Figure 7: Same as Fig. 5, but the exit channelOM potential is fixed (family #1, i.e. ”3 MeV”of [4] while all six equivalent potentials of ref. [4]are used for the exit channel. Note, that for themost shallow potential in the entrance channelresults of calculations in prior and post repre-sentations are almost indistinguishable.

It remains an open question whether the7Li(7Li,6He)8Be reaction is realized in natureonly via a direct mechanism. It may be an-swered by comparing the experimental angulardistributions with the theoretical ones. It is vis-ible in Fig. 5 that theoretical cross sections sur-pass the experimental data if the calculations areperformed with the shallow optical model poten-tials. This may hint at the possibility to selectsome potentials among those which reproduce

equally well the experimental elastic scatteringdata. However, in order to do so one has tovary the entrance and exit channel OM poten-tials independently since there is no obvious rea-son to assume them to be equal. The calcula-tions were thus repeated for all possible pairs ofentrance/exit channel potentials listed in Table1. Figs. 6 and 7 illustrate part of the resultsobtained.

In Fig. 6 results of calculations are shownobtained with a rather deep potential in the en-trance channel (”69 MeV” family) and with vari-ous potentials for the exit channel. A reasonablereproduction of the data is possible only for avery shallow potential (”3 MeV” family) in theexit channel. It indicates that in spite of thesmall difference in mass partition (7Li+7Li vs.6He+8Be) completely different optical model po-tentials seem to be responsible for scattering inthese two channels. They may reflect an adia-batic nucleon motion during the 7Li(7Li,6He)8Betransfer reaction in which the deep entrancechannel potential is modified in shape and depthto become the shallow one in the exit channel.

In Fig. 7 the most shallow one from equiv-alent 7Li+7Li OM potentials was used in theexit channel (”3 MeV” family), but different po-tentials were applied in the entrance channel.Again, as in Fig. 5 the shallow potentials inthe entrance channel overestimate the cross sec-tions. The potentials deeper than 60 MeV re-produce the experimental angular distributionsquite well. Moreover, it can be seen that theagreement of prior and post representations isbetter assured by deep entrance channel poten-tials than by shallow ones (with the exception ofidentical entrance and exit channel potentials;see above). In summary we conclude that the7Li(7Li,6He)8Be reaction favors rather deep OMpotentials in the entrance and shallow ones forthe exit channel.

In Fig. 8 we present results of the calcula-tions which were performed for several bombard-ing energies between 8 MeV (lab.) and 22 MeVwith this selected pair of OM potentials. Thereproduction of the experimental angular distri-

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butions may be judged as very good in particu-lar in view of the fact that the calculations werecarried out without any free parameters and thatcluster spectroscopic factors are known only withsome (model dependent) accuracy.

Figure 8: Experimental angular distributionsof the 7Li(7Li,6He)8Beg.s. reaction for severalbombarding energies (full dots, present work)together with results of calculations performedwith selected pairs of OM potentials: deep po-tential (family #4, i.e. ”69 MeV” of [4]) in theentrance channel and a very shallow one (family#1, i.e. ”3 MeV” of [4]) in the exit channel. Fulllines correspond to post, dashed lines to priorrepresentations. The data at Elab=22 MeV weretaken from ref. [14].

This good agreement is also seen in theenergy dependence of angle integrated cross sec-

tions which are shown in Fig. 9. The dotsrepresent the experimental data of the presentwork, lines show results of DWBA calculationsfor prior (upper part of the figure) and post rep-resentation (lower part of the figure)

Figure 9: Experimental angle integrated crosssections for the 7Li(7Li,6He)8Beg.s.(0+) reactionas a function of projectile energy (solid dots) andresults of DWBA calculations performed in prior(upper part of the figure) and in post represen-tations (lower part). Different lines correspondto the use of different OM potentials (ref. [4])for the entrance channel and the same OM po-tential (family #1, i.e. ”3 MeV” of [4]) for theexit channel.

calculated with different OM potentials forthe entrance 7Li+7Li channel and with the same,

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shallow potential (”3 MeV” family) in the exitchannel.

Figure 10: Experimental angular distributionfor the 7Li(7Li,6He)8Be3.04 reaction at 22 MeV(lab.) [14] (full dots) and results of DWBA cal-culations performed with a very shallow (family#1, i.e. ”3 MeV” of ref. [4]) 6He-8Be OM po-tential and with various 7Li-7Li OM potentialstaken also from ref. [4].

The analysis of the reaction leading to thefirst excited state of 8Be may yield an addi-tional test of the proton transfer mechanism.In our experiment we were not able to evalu-ate properly the differential cross sections fromthe experimental spectra but there are dataavailable of Bochkarev et al. [14] measuredfor the 7Li(7Li,6He)8Be3.04 reaction at 22 MeV(lab.). Calculations were performed along the

lines which yielded the results shown in Figs. 5- 8. A very similar picture arises. Therefore wepresent in Fig. 10 only results of calculationsobtained with very shallow 6He-8Be OM poten-tial (”3 MeV” family) and with various 7Li-7LiOM potentials. Again only rather deep entrancechannel potentials can well describe the experi-mental data and the quality of this descriptionis quite satisfactory.

From these results we conclude the directproton transfer to dominate in 7Li(7Li,6He)8Bereactions and the distorted wave Born approx-imation to be able to properly describe experi-mental cross sections in the studied range of en-ergies. The somewhat poorer reproduction of 22MeV angular distributions may indicate a needfor some energy dependence of the optical modelpotentials (which were fitted to elastic scatter-ing data independently of energy in a restrictedrange of 7Li projectile energies i.e. 8 - 17 MeVin the lab. system [4]).

3.2 The alpha particle transfer reac-tion - 7Li(7Li,t)11B

This reaction seems to be less suited for DWBAin comparison with the proton transfer reactionsince the rearrangement of nucleons during alphaparticle transfer is more drastic than in protontransfer. Thus it is not clear whether it is appro-priate to treat alpha particle transfer in such alight nuclear system as a perturbation. Further-more, cross sections of alpha particle transfer aretypically larger by an order of magnitude thanthose for proton transfer as can be seen in theFig.3. They represent approximately 10% of theelastic scattering cross section, a fact which maydisqualify the perturbation approach.

The DWBA analysis of the alpha parti-cle transfer was performed along the same linesdelineated for proton transfer. Again, no freeparameters were allowed for optical model po-tentials, spectroscopic amplitudes and transitionpotentials. It was found that results of the cal-culations are only weakly sensitive to the exitchannel t+11B optical model potential. Thus,

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in the systematic calculations only one OM po-tential, taken from the literature (ref. [21]), wasused for generating distorted waves in the exitchannel.

Figure 11: Experimental angular distributionfor the 7Li(7Li,t)11Bg.s.(3/2−) reaction at 14MeV projectile energy (solid dots), and the re-sults of DWBA calculations performed in prior(dashed lines) and in post representation (fulllines). Different frames in the figure correspondto calculations performed with the same 11B-tOM potential (ref. [21]) but with different 7Li-7Li OM potentials of ref. [4]. The depth of thereal part of these potentials is given in the cor-responding frames.

However, it was found that optical modelpotentials of the entrance channel which equallywell reproduce elastic scattering produce quite

different results when applied in the DWBA.Hence, for comparison, the calculations were per-formed with the same six OM potentials of the7Li+7Li channel which were already used in theanalysis of the proton transfer reaction.

Figure 12: Same as Fig. 11, but forthe 7Li(7Li,t)11B2.13(1/2−) reaction at Elab=14MeV.

One may conjecture by inspecting Fig.3 that contributions of mechanisms which arecharacterized by a strong energy dependence oftheir cross section e.g. statistical fluctuationsand/or excitation of individual resonances, arepresent in this reaction but least important atthe highest energy. Therefore we need to com-pare both angular distributions and excitationfunctions as given by DWBA with experimentaldata.

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Figure 13: Same as Fig. 11, but forthe 7Li(7Li,t)11B4.45(5/2−) reaction at Elab=14MeV.

Results of the calculations performed atElab=14 MeV are presented in Fig. 11 for theground state transfer 7Li(7Li,t)11Bg.s. and inFigs. 12, 13 and 14 for the the transfer to thefirst (2.13 MeV; 1/2−), second (4.45 MeV; 5/2−),and third (5.02 MeV; 3/2−) excited state, re-spectively.

The description of the experimental angu-lar distributions by DWBA is acceptable but sig-nificantly poorer than for proton transfer. Fur-thermore, the prior - post equivalence is notwell established (especially for deep OM poten-tials) pointing at the limits of accuracy of theDWBA approach for alpha particle transfer inthe 7Li+7Li system.

Figure 14: Same as Fig. 11, but forthe 7Li(7Li,t)11B5.02(3/2−) reaction at Elab=14MeV.

In Figs. 15 - 18 the angle integrated crosssections are depicted versus beam energy. Blackdots represent experimental cross sections andthe lines correspond to DWBA calculations per-formed with different 7Li-7Li OM potentials (cf.Table 1) in prior (upper part of the figure) andpost (lower part of the figure) representations.It may be concluded that results obtained inpost representation are in general more consis-tent than those in the prior one, i.e. the angleintegrated cross sections for all OM potentialsyield similar values.

Moreover, the ”post - prior” equivalencewhich was reasonably well fulfilled at Elab=14MeV for shallow potentials remains to be ful-

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Figure 15: Experimental angle integrated crosssection for the 7Li(7Li,t)11Bg.s.(3/2−) reaction asa function of projectile energy (solid dots) andresults of DWBA calculations performed in prior(upper part of the figure) and in post represen-tations (lower part). Different lines correspondto the use of different OM potentials for the en-trance channel from ref. [4] and the same OMpotential (ref. [21]) for the exit channel.

filled for these potentials in the whole energyrange. Thus, one has either to use the post rep-resentation or to choose the shallow potentialsfor calculations in the prior representation. Inthese cases the shape of the energy dependenceof experimental cross sections is reasonably wellreproduced by DWBA calculations at least forhigher energies. The size of the cross section,however, is predicted too small, clearly show-

Figure 16: Same as Fig. 15, but for the7Li(7Li,t)11B2.13(1/2−) reaction.

ing the presence of other reaction mechanisms.DWBA gives an almost constant value of the an-gle integrated cross section for the ground statetransition while the experimental one varies verystrongly with energy and has a maximum aboutElab= 8 MeV.

Resonances are likely to contribute to thetriton channel in the low energy region. Theyexhaust the major part of the experimentalcross section. Only at the highest energy ofElab=14 MeV direct reactions become importantwith contributions of approximately 50% for theground state transition, and 20 % , 50% and 60 %for the transitions leading to the first, the secondand the third excited states of 11B, respectively.

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Figure 17: Same as Fig. 15, but for the7Li(7Li,t)11B4.45(5/2−) reaction.

3.3 The triton transfer reaction -7Li(7Li,4He)10Be

It is apparent from examining Figs. 3 and 4 thata strong variation of the experimental angle in-tegrated cross sections versus energy is presentfor the 7Li(7Li,4He) reactions. Therefore thischannel is the least suited among all reactionsunder investigation for the application of the di-rect reaction formalism. Led by results for thealpha particle transfer reactions we can expectDWBA to be appropriate at the highest energy(Elab=14 MeV). DWBA calculations were per-formed at this energy for transitions to both theground state of 10Be and the first excited state10Be3.37(2+).

Figure 18: Same as Fig. 15, but for the7Li(7Li,t)11B5.02(3/2−) reaction.

It turned out that in these cases the resultsof DWBA calculations depend only weakly onthe parameters of the exit channel (4He+10Be)optical model potential. Hence, only one OMpotential was applied for evaluation of distortedwaves in the exit channel [20]. All six OM poten-tials (cf. Table 1) used previously in the analysisof the proton and the alpha particle transfer wereexploited for generation of distorted waves in theentrance channel.

The experimental angular distributions(full dots) are shown in Figs. 19 and 20 togetherwith results of DWBA calculations performedusing the prior (dashed lines) and the post repre-sentation (full lines) for 7Li(7Li,4He)10Beg.s.(0+)

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Figure 19: Experimental angular distribu-tion for the 7Li(7Li,4He)10Beg.s.(0+) reaction atElab=14 MeV (solid dots) and the results ofDWBA calculations performed in prior (dashedlines) and in post representation (full lines). Dif-ferent frames in the figure correspond to calcu-lations performed with the same 10Be-4He OMpotential (ref. [20]) but with different 7Li-7LiOM potentials from ref. [4]. The depth of thereal part of these potentials is given in the cor-responding frames.

and 7Li(7Li,4He)10Be3.37(2+) reactions, respec-tively. The calculations in both prior and postrepresentation result in angular distributionssimilar in shape as well as in magnitude. DWBAangular distributions are in either case smoothwith small oscillations reproducing qualitativelythe shape of the experimental angular distribu-

Figure 20: Same as Fig. 19, but for thethe 7Li(7Li,4He)10Be3.37(2+) reaction at Elab=14MeV.

tions. The magnitude of the theoretical crosssections is, however, in either case smaller thanthat for the experimental data. In case of theground state transition the theoretical cross sec-tion is smaller by a factor 40 - 50 than the exper-imental cross section and for alpha transfer lead-ing to the first excited state of 10Be the DWBAcross section exhausts approximately 40 - 50 %of the experimental data.

To estimate the average contribution of adirect mechanism to triton transfer the energydependence of the angle integrated DWBA crosssection was calculated and compared with theenergy dependence of the experimental angle in-tegrated cross section.

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Figure 21: Experimental, angle integrated crosssections for the 7Li(7Li,4He)10Beg.s.(0+) reactionas a function of projectile energy (solid dots),and results of DWBA calculations performed inprior (upper part of the figure) and in post rep-resentation (lower part). Different lines corre-spond to the use of different OM potentials (ref.[4]) for the entrance channel and the same OMpotential ([20]) for the exit channel.

This is illustrated by Figs. 21 and 22 forthe ground state transition and for the transitionto the first excited state of 10Be, respectively.The theoretical cross section for the ground statetransition varies smoothly versus energy in theinvestigated energy range. It is, on average,smaller by a factor 20 - 40 than the experimen-tal cross section. Note that the theoretical crosssection shown in Fig. 21 is multiplied by a factor

Figure 22: Same as Fig. 21, but for the7Li(7Li,4He)10Be3.37(2+) reaction.

of 10 for better representation. In contradistinc-tion to the DWBA predictions the experimentaldata vary rapidly with energy (the experimen-tal uncertainties are smaller than the dot size inthe figure). This is also true for the transitionto the first excited state shown in Fig. 22. Inthis case, however, the theoretical cross sectionestablishes approximately 40 - 50 % of the ex-perimental cross section.

It is interesting to note that the remainingparts of the experimental cross sections whichcannot be ascribed to a direct reaction mech-anism fulfill a simple ”2J + 1” relationship fortransitions to both the ground and the first ex-cited state. Such a relationship is indicative forthe compound nucleus mechanism. The strongenergy dependence of the cross sections seems to

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confirm this conjecture.

4 Summary and conclusions

In the present work experimental data arepresented from a measurement of angular dis-tributions of 7Li(7Li,t)11B, 7Li(7Li,4He)10Be and7Li(7Li,6He)8Be reactions at several energies be-tween 8 and 16 MeV in the laboratory system.Transitions to the ground states as well as tosome low lying excited states (three in the tri-ton channel and one in the 4He channel) werestudied. Already by inspection of these data weare led to the conclusion that the (7Li,6He) re-action proceeds predominantly as a direct pro-cess while various mechanisms may contributeto (7Li,t) and (7Li,α) reactions. The latter re-action seems to proceed predominantly throughisolated resonances of the 14C compound system.

A DWBA analysis was performed for allchannels, the results confirm the qualitative con-clusions derived from inspection of the experi-mental data. The (7Li,6He) reaction is - withinthe accuracy of DWBA calculations - completelydescribed by direct proton transfer. The othertwo reactions proceed partially by direct mecha-nisms: in average 20 - 60 % for the α-particletransfers (7Li,t) and 40 - 50 % for the tritontransfer (7Li,4He) reaction to the first excitedstate of 10Be and only approximately 3 - 5 % fortransition to the ground state.

Estimations based on the Hauser-Feshbachmodel [5] indicate a rather small contributionof compound nucleus reactions (approximately12-20% for the triton channels and even less forα particle and 6He channels). Thus, processesdifferent than pure direct and pure compoundnucleus mechanisms are present in the investi-gated energy range. The interference betweendirect and compound nucleus reaction - ampli-tudes may lead to fluctuations of the cross sec-tion which, however, are expected to be narrower(typical width approx. 0.6 MeV) than the struc-tures observed here. Indeed, the presence of

strong peaks in the excitation functions of theangle integrated cross section for α1, correlatedwith the structures visible in the excitation func-tion measured at forward angles suggests a con-tribution of isolated resonances superimposed onthe background from both direct and statisticalcompound nucleus reactions.

The good reproduction of both shape andmagnitude of the experimental angular distri-butions for the 7Li(7Li,6He)8Be reaction by theDWBA calculations as well as the reproductionof the energy dependence of the cross section in-dicates that the methods of direct reaction the-ory can be successfully applied for such a systemof few nucleons. The calculations within bothprior and post representations lead to equivalentresults and thus manifest the adequacy of theDWBA approach to this reaction. The use ofdifferent optical model potentials in the DWBAcalculations allowed us to select from potentialswhich describe elastic and inelastic scatteringequally well those which are appropriate, namelya rather deep (appr. 70 MeV) OM potential forthe entrance (7Li+7Li) and a very shallow po-tential (approx. 3 MeV) for the exit (6He+8Be)channel. Only this combination of OM poten-tials produces results which agree with the ex-perimental data of the 7Li(7Li,6He)8Be reaction.

In spite of rather big cross sections forthe α particle and triton channels (approxi-mately 10% of the elastic cross section) andthe relatively strong rearrangement processes ofthe 7Li+7Li system after triton or α particletransfer the method of DWBA can be success-fully applied to evaluate the contribution of thedirect mechanism to the 7Li(7Li,4He)10Be and7Li(7Li,t)11B reactions. Results of DWBA cal-culations turned out to be insensitive to vari-ation of the exit channel optical model poten-tials in both cases but some caution is in or-der when applying the prior or post representa-tions together with different OM potentials inthe entrance channel. In general the post repre-sentation is superior and may be applied with-out further selection of the optical model po-tentials. The calculations in the prior repre-

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sentation lead to equivalent results, as it is de-manded by the DWBA formalism, only if opticalmodel potentials of the 7Li+7Li system are cho-sen which are rather shallow (< 60 MeV ). Aselection of 7Li+7Li OM potentials on the basisof DWBA calculations applied to triton and αparticle transfers yields results contrary to thoseobtained for the proton transfer. This may in-dicate that either the proton transfer reaction issensitive to different parts of the OM potentialthan the cluster transfers or the prior represen-tation is not well suited for these reactions e.g.due to poor cancellation of ”indirect transition”potentials.

It should be emphasized that the good de-scription of the experimental data by DWBAwas achieved without introducing any free pa-rameters. This strongly supports the applicabil-ity of DWBA for the 7Li+7Li system although,at first sight, the methods of direct reaction the-ory seem hardly adequate for a system consistingof such a small number of nucleons.

Acknowledgements: We are grateful toDr. E. Kwasniewicz for supplying spectroscopicamplitudes.

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