Fusion cross sections for 7Li + 12C system at near barrier energies

Nuclear Physics A 792 (2007) 187–200

Fusion cross sections for 7Li + 12C systemat near barrier energies

V.V. Parkar ∗, K. Mahata 1, S. Santra, S. Kailas, A. Shrivastava,K. Ramachandran, A. Chatterjee, V. Jha, P. Singh

Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India

Received 25 March 2007; received in revised form 3 May 2007; accepted 25 May 2007

Available online 12 June 2007

Abstract

Elastic scattering and alpha particle angular distributions for the reaction 7Li + 12C have been measuredat Elab = 7.5,9,12, and 15 MeV. Statistical and direct reaction model calculations have been performed todeduce the fusion and direct reaction cross sections. The fusion cross sections measured in the present workare found to be consistent with those obtained by measurement of characteristic γ -rays, which are verydifferent from those measured by direct detection of evaporation residues. The contribution from directreactions to the alpha particle cross section has been quantified by exact finite range transfer calculationsand found to be negligible (< 6%) at all the bombarding energies studied. The measured fusion crosssections at all the energies studied are nearly equal to the reaction cross sections which imply that the directreaction contributions are less and fusion is the dominant process.© 2007 Elsevier B.V. All rights reserved.

PACS: 25.70.Jj; 25.70.Mn; 25.60.Pj; 25.70.Hi

Keywords: Elastic scattering; Fusion cross sections; Charge particle detection; Statistical model analysis; Direct andcompound nuclear angular distributions

* Corresponding author.E-mail address: [email protected] (V.V. Parkar).

1 Present address: GSI, Planckstr. 1, D-64291 Darmstadt, Germany.

0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2007.05.013

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1. Introduction

In recent years, the study concerning the influence of breakup of weakly bound stable or ra-dioactive nuclei on fusion process at energies around the Coulomb barrier has generated a strongtheoretical as well as experimental interest [1]. Although sub-barrier fusion involving stable nu-clei is well understood, there are contradictory results and predictions about enhancement orsuppression of the fusion cross section (σFus) around the Coulomb barrier when one of the col-lision partners is a weakly bound nucleus. In particular, the small separation energies of weaklybound nuclei, e.g., 6Li (6Li → α +d, Sα/d = 1.474 MeV), 7Li (7Li → α + t, Sα/t = 2.467 MeV),9Be (9Be → α+α+n, S2α/n = 1.573 MeV), 6He (6He → α+2n, Sα/2n = 0.972 MeV), etc., in-creases the importance of the breakup channel. However, the role of breakup on fusion has beenstrongly debated both theoretically [2] and experimentally [3–17]. Recently, Hagino et al. [2]performed an improved coupled channels calculation which predicts enhancement of σFus atsub-barrier energies and reduction at above barrier energies.

For light mass systems like 6,7Li + 12,13C [10,11,14,15,17] and 6,7Li + 16O [12,13,16], thereexists a large discrepancy in the σFus measured using different techniques. The σFus obtainedfrom direct detection of evaporation residues (ER) [10–13] showed suppression compared to thereaction cross-sections, while those obtained from characteristic gamma-ray measurements [14–16] and detection of evaporation residues in inverse kinematics [17] agreed well with the reactioncross sections and showed no suppression. One thing to note here is that, from both the ER andγ -ray measurements, it is not possible to separate the direct and compound nuclear contributions.However, the same can be done from the angular distribution of outgoing charged particles likealpha.

In the present work, we have measured the σFus for 7Li + 12C using a different method, i.e., bydetection of charged (alpha) particles at four laboratory energies Elab = 7.5,9,12, and 15 MeV.Since the same final state of the residual nuclei can be populated from both fusion (evaporation)and direct reactions, it is necessary to separate these two contributions. This has been achievedby the analysis of the angular distribution of the charged particles with the statistical model andthe direct reaction model calculations.

2. Experimental details

The experiment was performed using 7Li beam at energies Elab = 9,12, and 15 MeV, fromthe 14UD BARC-TIFR Pelletron accelerator, Mumbai, and at Elab = 7.5 MeV from the FOTIAat BARC, Mumbai. The 12C targets were self supporting natural carbon films (98.9% enriched12C) and were prepared using vacuum evaporation technique. The target having thickness 52.2±0.8 µg/cm2 was used for 7.5, 9, and 15 MeV and the one with 96.4 ± 1.0 µg/cm2 thicknesswas used for 12 MeV. The target thicknesses were determined by the energy loss method with anAm–Pu α-source. Two telescopes (�E–E) of silicon surface barrier (SSB) detectors were placedon one of the movable arms inside a 1 m diameter scattering chamber. The detector thicknesseswere typically 10–25 µm for �E and 250–300 µm for E. A monitor (single SSB of thickness2000 µm) detector was mounted on the other arm at 20◦ with respect to the beam directionfor monitoring the beam quality and the variation in the target thickness when the target wasrotated. The solid angles of both the telescopes were measured accurately by measuring elastic(Rutherford) scattering from 209Bi target of known thickness. The beam intensity was measuredusing a precision current integrator. The angular distributions of charged particles (Li, α) weremeasured in the range θlab = 10◦–170◦.

V.V. Parkar et al. / Nuclear Physics A 792 (2007) 187–200 189

(a)

(b)

Fig. 1. (Colour online.) (a) A two-dimensional plot of �E–Etot obtained in the telescope at θlab = 31.6◦ . The differentreaction products were identified and labeled. (b) Projection of part of α band containing discrete states as marked by thedotted box in (a). Various discrete peaks of α particles corresponding to 15N∗ were identified and labeled.

A typical �E versus Etot spectrum at θlab = 31.6◦ for 9 MeV is shown in Fig. 1(a). Theprojection of a part of the alpha particle band is shown in Fig. 1(b). It shows discrete groups ofα particles which were identified and labeled. The discrete alpha groups corresponding to well-known states of 15N are clearly seen from the data. They are consistent with the measurement ofRef. [18]. The levels identified in 15N arise due to triton transfer mechanism as well as from decayof 19F compound nucleus. The hydrogen impurity in the target was identified via p(7Li, α)α

reaction as the rightmost peak in Fig. 1(b). Its contribution was subtracted out during analysis.

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Since �E detectors were 10–25 µm thick, the threshold energy for α particle detection was∼ 3–4.5 MeV. Absolute values of cross sections were obtained by making use of target thickness,beam intensity and solid angles of the detectors.

3. Results

3.1. Elastic scattering

The measured elastic scattering angular distributions are shown in Fig. 2. The optical modelcalculations were performed using the optical model code SNOOPY [19]. A standard four pa-rameter energy independent optical model potential with a Woods–Saxon form was taken fromRef. [20] for the calculations. These potentials are listed in Table 1. The reaction cross sections(σR) obtained from the analysis of the elastic scattering data were compared with the σFus asdiscussed later.

Fig. 2. Elastic scattering angular distributions for 7Li + 12C system at different laboratory energies. The solid linesrepresent optical model calculations.

Table 1Potential parameters used in HAUFES and FRESCO

V0 rR aR W0 WS rI aI

(MeV) (fm) (fm) (MeV) (fm) (fm) (fm)7Li + 12C Ref. [20] 134.0 1.66 0.65 – 22.0 1.66 0.65α + 15N Ref. [21] 183.0 1.40 0.57 15.0 – 1.50 0.60α + 12C Ref. [21] 177.0 1.30 0.57 12.0 – 1.50 0.60α + t Ref. [22] searched 1.80 0.70t + 12C Ref. [23] searched 1.25 0.65

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3.2. Competition between CN and direct reaction

The alpha particles that are detected can arise from direct (transfer, breakup) reaction andcompound nuclear (CN) decay. In order to separate these two contributions, angular distributionsof discrete groups of α particles were plotted. It was observed that, the weakly populated, neg-ative parity states show symmetry around 90◦, a characteristic feature of CN mechanism. Otherstrongly populated, positive parity states showed the forward angle peaking, a characteristic ofdirect reaction.

Angular distributions of discrete alpha groups from fusion (evaporation) were calculated usingthe Hauser–Feschbach code HAUFES [24]. The optical model potentials for the outgoing neu-tron and proton channels were taken from Ref. [25] while those for entrance and alpha outgoingchannels are listed in Table 1. Finite range DWBA (FRDWBA) calculations were performedusing the code FRESCO [26], to estimate the contribution from direct reaction to the alpha par-ticle cross section. The potential parameters used in FRESCO calculations are listed in Table 1.As in Ref. [27], the cluster configuration of (sd)3 [3p–4h] was assumed for the positive par-ity 15N states, while p1(sd)2 [2p–3h] was assumed for the negative parity states. All FRDWBAcalculations were carried out including the full complex remnant term and by using post formula-tion. Since the spectroscopic information needed for multinucleon transfer is not unique and thespectroscopic factor is dependent on the bombarding energy [28,29], the direct reaction angulardistribution calculations were normalised to the data, so that the addition of CN and direct reac-tion contributions explains the data. Fig. 4 shows normalisation constant (C2S) obtained for oneof the states (9.155 MeV, 5/2+ of 15N) plotted as a function of Elab, similar to that in Ref. [28].For higher energies Elab = 35 and 52.5 MeV, the data for the above mentioned state were takenfrom Refs. [30] and [27], respectively. The similar FRDWBA calculations were performed forthese two higher energies to get the normalisation constants. For Elab = 52.5 MeV, the imaginarypart of incoming channel potential was slightly modified to get the fit to elastic scattering dataat that energy, which is then used in FRDWBA calculations. The observed behaviour of energydependence of spectroscopic factor needs to be understood.

Angular distribution for the strongly populated state (9.155 MeV, 5/2+ of 15N) at all the ener-gies is shown in Fig. 3 along with CN and direct reaction calculations. The average direct reactioncontribution from all the discrete states was found to increase from 10% to 20% from lowest tohighest bombarding energy. Similar conclusions were drawn in Ref. [31] for 12C(7Li,p)18Odata measured at Elab = 16 and 18 MeV, where the outgoing protons showed dominant CNmechanism.

The continuum part of the alpha spectra was split into several energy bins of 1 MeV width. Theangular distribution for each of the energy bins was found to be dominantly symmetric around90◦. In Fig. 5, the data are shown for alphas corresponding to EX (X can be 15N or any otherresidue) = 12–13 MeV bin. The solid lines are 1 + cos2 θ (isotropic) fit to the data.

Further the contribution to alpha yield can also come from direct breakup (7Li, tα) and othertwo step processes like (7Li, 5He–nα), (7Li, 5Li–pα), (7Li, 6Li–dα), (7Li, 8Be–2α), etc. The en-ergy limits of alpha particles coming from these processes were calculated using three bodykinematics considering the reaction Q-values as in Ref. [32] and were found to fall in the con-tinuum region of alpha spectra. The FRDWBA calculation for the process (7Li, 5He) yielded asmall value of the cross section (1 mb).

As the continuum part showed purely CN character, the direct reaction contribution can comeonly from discrete states. The yield of discrete alpha groups is ∼ 30% of the total alpha yield,

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Fig. 3. Angular distribution of alpha particles corresponding to 9.155 MeV, 5/2+ state in 15N measured in the12C(7Li, α)15N∗ reaction. Here the data at backward angles was clearly explained by CN mechanism, while forwardangles show peaking which was explained by addition of direct and CN mechanisms.

which gave a direct reaction contribution to be only 3–6% (from lower to higher energy) of thetotal alpha yield.

3.3. Fusion cross sections

In order to determine the fusion cross section, the procedure of Refs. [33,34] was followed inthe present work. We determined the fusion cross section by measuring the alpha particle spectraat several angles, followed by a statistical model analysis of the data. The Monte Carlo statisticalmodel code PACE [35] was used for the analysis. At a given energy, the σFus value was assumed.The spin distribution following fusion was parametrized using the following Fermi distribution

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Fig. 4. Ratio of the experimental to the theoretical cross sections for alpha particles corresponding to 9.155 MeV, 5/2+state in 15N measured in the 12C(7Li, α)15N∗ reaction. See text for details.

σl = (2l + 1)(π/k2)[1 + exp

(l−lmax

Δ

)] . (1)

The diffuseness parameter Δ was obtained from the corresponding σl versus l distribution ofthe elastic scattering (reaction cross section) data. This value was consistent with the ones de-duced from σl versus l calculated using the CCFUS [36] fusion code. Using the above expressionfor σl and the relation, σFus = σl , the lmax value was determined.

The level density formalism used in the present analysis is similar to the recommendation inRIPL-2 [25], i.e., the constant temperature approximation was used for lower excitation energies(discrete states) and Fermi gas model at high excitation energies for all possible evaporationchannels. A value of A/7.4 for the level density parameter for the Fermi gas model, used in thehigher energy region, was obtained by matching of the level densities at Ux (excitation energy ata given x). The values of Ux and T (temperature) were taken from Ref. [25]. The transmissioncoefficients for the light particle emission (n,p,α) were determined using the optical modelpotentials of Refs. [37,38]. The alpha energy spectra and the angular distributions were fittedfollowing the method of Refs. [33,34] and the fusion cross section was optimised. Hence theexperimental σFus values were obtained from statistical model analysis of the measured alphadata. The statistical model parameters used for the above analysis could also reproduce all ERcross section data of Ref. [14]. The Mα (multiplicity of alpha particles) values were 0.53, 0.70,0.98, 1.12 at Elab = 7.5,9,12, and 15 MeV, respectively. Taking into account the uncertainty inthe measured σα and the variation in the corresponding calculated σα values due to the statisticalmodel parameters, it is estimated that the σFus determined will have an error of maximum 10%.

In Fig. 6, alpha particle energy spectra along with the PACE calculations at two angles at allthe measured energies are shown. The data are in good accord with the PACE calculations. ThePACE predictions to ER data along with predictions of Ref. [14] using the CASCADE code areshown in Fig. 7. The overall quality of the agreement with the data from the present as wellas from predictions of Ref. [14] is similar. We also found measurable contributions from 14Cchannel, not included in Ref. [14]. It should be mentioned that while the p,n evaporation cross

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Fig. 5. Angular distribution of the continuum part (for EX = 12–13 MeV bin) of the alpha particles measured in the7Li + 12C reaction. Here ‘X’ corresponds to 15N or any other residue. Solid lines show the 1 + cos2 θ (isotropic) fit tothe data.

sections are overpredicted by the calculations, the pn/d data are underpredicted. This may bepartly related to the non-inclusion of d and t evaporation channels [39,40].

The alpha particle energy spectra (d2σα/dEα dΩ versus θlab) were energy and angle integratedto get the total alpha particle cross section (σα). The energy integrated alpha particle angulardistributions along with PACE calculations are shown in Fig. 8. At very forward angles uptoθlab = 40◦, the cross sections were corrected for direct reaction contribution. As there is goodagreement between the statistical model calculations and the measured alpha yield, spectra andangular distribution, it is believed that the overall direct reaction contribution (in particular forthe continuum which constitutes nearly 70% of the total alpha yield) is not significant.

In Fig. 9, the σFus values determined from the present work are compared with the ones fromthe literature (obtained by other techniques). The present results agree with that of Refs. [14,17]

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Fig. 6. The alpha particle spectra from the reaction 7Li + 12C. The histograms represent the results from the PACEcalculations.

at all the measured energies. The reaction cross sections were calculated and plotted for Ec.m. =2.84 to 9.47 MeV, using the parameters given in Ref. [20].

4. Discussion

As stated earlier, the main purpose of the present measurement was to determine the fusioncross-sections and to estimate the direct reaction contribution to the total cross-section. Broadly,

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Fig. 7. Cross sections for different exit channels of the 7Li + 12C reaction along with CASCADE calculations [14] werecompared with the PACE predictions.

the direct reaction processes can be classified as arising due to inelastic scattering, break-upprocess, transfer reaction (both charged particle and neutron transfer). The inelastic scatteringis mainly decided by the structure of the nucleus (projectile/target). The break-up probability isfound to be considerably reduced with the decrease in target atomic number [41,42]. For haloprojectiles like 6He, large neutron transfer cross-section has been observed at near Coulombbarrier energies [43]. This feature is consistent with the neutron halo structure of 6He. Reactionswith weakly bound stable projectiles, e.g., 6,7Li, 9Be on heavy and medium mass targets showlarge direct reaction cross-sections [44,45]. Interestingly, in the case of 6,7Li + 28Si, Pakou et

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Fig. 8. Energy integrated angular distribution of alpha particles at various bombarding energies compared with the PACEcalculation. The hollow circles represent the data corrected for direct reaction contribution. For larger angles, the directreaction contributions are negligible.

al. [46,47] have observed larger transfer cross-sections relative to fusion at near barrier energies.Whereas, Hugi et al. [33] have reported a larger contribution of fusion compared to direct reactionfor 6Li + 28Si at above barrier energies. For weakly bound non-halo projectiles 6,7Li, 9Be, studieswith light targets like 16O, 11B, 9Be [48,49] have found fusion cross-sections comparable toreaction cross-sections, implying less contribution from the direct reaction process. In the presentwork, we have determined the cross-sections for the α-channel at energies E ∼ 1 VB–2 VB forthe system 7Li + 12C and found that the direct reaction contribution (discrete and continuumcontributions estimated separately) is negligible when compared to fusion.

5. Summary

The elastic scattering and alpha particle angular distributions for 7Li + 12C system have beenmeasured at energies Elab = 7.5,9,12, and 15 MeV. Optical model calculations have been per-formed to get σR by fitting the measured elastic scattering angular distributions. The angulardistribution of discrete groups of alpha particles have been explained by the addition of both CNand direct reaction contributions. As in the present work, alpha energy spectra, angular distrib-ution (for discrete and continuum) have been considered for the analysis, it has been possible todetermine reliably the direct contribution for this reaction. Direct reaction contribution has been

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Fig. 9. Fusion cross sections for the 7Li + 12C reaction. The solid curve represents the reaction cross sections.

estimated to be not more than 6% of the total alpha yield (mainly based on discrete alpha data).Fusion cross sections have been determined by the statistical model analysis of measured alphaparticle spectra.

The fusion cross sections for 7Li + 12C system obtained from present work are in good accordwith those obtained by Mukherjee et al. [14,17]. The σFus values are found to be in agreementwith the corresponding σR values, except at the highest bombarding energy, imply that the directreaction contributions are less and fusion is the dominant process.

Acknowledgements

The authors would like to thank the Pelletron and FOTIA crew for the smooth operationof the accelerator during the experiments and to Mr. Mahadakar for target preparations. Oneof the authors (V.V.P.) acknowledges the financial support of Department of Atomic Energy,Government of India, in carrying out these investigations.

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