An excitation function for the fusion reaction Si28 + Si30 (Q=14.3MeV) has been measured down to 40μb. Deviations from the behavior predicted by the optical model and standard coupled-channels calculations have been observed in this system. The fusion cross sections can be reproduced by a shallow potential model well, which was originally developed to explain the hindrance of heavy-ion fusion for systems with negative Q-values

Back, B.B.

Collon, P.

Esbensen, H.

Greene, J.P.

Henderson, D.J.

Janssens, R.V.F.

Jiang, C.L.

Lee, H.Y.

Lister, C.J.

Mişicu, S.

Notani, M.

Pardo, R.C.

Patel, N.

Rehm, K.E.

Seweryniak, D.

Shumard, B.

Tang, X.D.

Wang, X.

Zhu, S.

Carolina Digital Repository

PHYSICAL REVIEW C 78, 017601 (2008)Fusion hindrance for a positive Q-value systemC. L. Jiang,1 B. B. Back,1 H. Esbensen,1 J. P. Greene,1 R. V. F. Janssens,1 D. J. Henderson,1 H. Y. Lee,1 C. J. Lister,1M. Notani,1 R. C. Pardo,1 N. Patel,1 K. E. Rehm,1 D. Seweryniak,1 B. Shumard,1 X. Wang,1 S. Zhu,1 S. Mişicu,2P. Collon,3 and X. D. Tang31Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA2National Institute for Nuclear Physics, Bucharest, P.O. Box MG6, Romania3University of Notre Dame, Notre Dame, Indiana 46556, USA(Received 7 May 2008; published 8 July 2008)An excitation function for the fusion reaction 28Si + 30Si (Q = 14.3 MeV) has been measured down to 40 µb.Deviations from the behavior predicted by the optical model and standard coupled-channels calculations havebeen observed in this system. The fusion cross sections can be reproduced by a shallow potential model well,which was originally developed to explain the hindrance of heavy-ion fusion for systems with negative Q-values.DOI: 10.1103/PhysRevC.78.017601 PACS number(s): 25.70.Jj, 25.60.Pj, 26.50.+x, 26.30.−kHindrance in heavy-ion fusion at extreme sub-barrierenergies was discovered several years ago [1]. The phe-nomenon was first observed in medium-mass systems, butlater measurements and analyses [2–7] showed that this mightrepresent a general behavior of heavy-ion fusion at extremesub-barrier energies. A systematics has been established for awide mass region, showing that the hindrance is closely relatedto the entrance channel properties [5–7]. This hindrance couldalso affect fusion reactions that are of importance in nuclearastrophysics, such as 12C + 12C [6,8]. There is, however, animportant difference between the systems studied earlier andthe astrophysically important fusion reactions coming fromthe reaction Q-values: these are positive for lighter nucleiand negative for the medium-mass systems studied previously.In this experiment we have, therefore, measured a fusionexcitation function in the system 28Si + 30Si, which has aQ-value of 14.3 MeV, very close to that of the 12C + 12Creaction (Q = 13.9 MeV).The fusion of 28Si + 30Si was studied previously by Garyand Volent [9] down to cross sections of about 17 mb, i.e., toan energy region much higher than the energy where fusionhindrance might occur. The new experiment was performedat the superconducting linear accelerator ATLAS at ArgonneNational Laboratory. The 28Si beams delivered were in theenergy range of 48.5–71 MeV. The isotopic abundance of the30Si target was 96.02%, with the remainder coming from 29Si(0.33%) and 28Si (3.43%). Thin targets with thicknesses ofabout 20 µg/cm2 SiO2 (evaporated on 40 µg/cm2 carbonfoils) were used in order to minimize the target thicknesscorrections in the energy regime where a steep fall-off in theexcitation function occurs. The contaminations from lighter Siisotopes did not interfere with the actual measurements sincethe associated Coulomb barriers are higher than the one for30Si. Two surface-barrier Si detectors, located at ±45◦ withrespect to the beam direction, were used for beam particlenormalization.The experimental procedure was similar to that used inearlier measurements of the fusion-evaporation excitationfunctions for the systems 28Si + 64Ni (Q = −1.78 MeV)and 64Ni + 100Mo (Q = −92.29 MeV) [1]. The evaporationresidues were measured with the Fragment Mass Analyzer(FMA) [10], placed at 0◦ with respect to the beam direction.This instrument has been upgraded with the installation of asplit-anode in the first electric dipole, resulting in a suppressionof the background originating mostly from beam particlesscattered off the first anode. The evaporation residues weredetected and identified behind the FMA with detectors ofthe configuration PGAC1-TIC1-PGAC2-TIC2-PGAC3-IC (seeRef. [11] for further details). Here, the symbol PGAC standsfor an x-y position-sensitive, parallel-grid avalanche counter,TIC for a transmission ionization chamber, and IC for a largevolume multi-anode ionization chamber. The first PGAC1counter was mounted at the focal-plane of the FMA, wherethe evaporation residues are dispersed according to theirmass-to-charge ratio m/q. Full charge state distributions weremeasured for four energies, while four to five charge stateswere recorded for five intermediate points. At the four lowestenergies, only one or two charge states were measured becauseonly one charge state could be measured at the time for eachFMA setting. These data proved sufficient to determine thecharge state fractions of the detected evaporation residues withthe required accuracy.In order to determine the FMA transmission it is necessaryto characterize the angular distribution of the evaporationresidues. In the present experiment this was achieved by usingthe statistical model code PACE [12]. It was found that the totalangular distribution changes very little over the range of beamenergies measured, thus making the efficiency correctionsquite simple.The system measured here is much lighter than the ones wehave measured in our previous fusion hindrance studies [1].The large differences in velocity (time of flight) and energydeposited in the detectors between the beam particles (orreaction particles from the contaminants in the target) andthe evaporation residues observed in the previous experimentsallowed us to measure the excitation function down to thetens of nb region. In the 28Si + 30Si system the separationbetween the evaporation residues and others was still sufficientat the higher incident energies. At the lowest energy of48.5 MeV (corresponding to a cross section of about 40 µb) the0556-2813/2008/78(1)/017601(4) 017601-1 ©2008 The American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW C 78, 017601 (2008)FIG. 1. (Color online) Comparisons of experimental evaporationresidue cross sections with various calculations for the system28Si + 30Si. Green solid: optical model, black: CC with shallowpotential. Light-blue dash-dotted: CC with Woods-Saxon potential.Magenta dotted: an extrapolation corresponding to the curve: L(E) =A0 + B0/E1.5 in Fig. 3(a).background from scattered beam particles and other reactionparticles increased strongly (to ∼30%) and prevented us fromextending the excitation function to lower energies.The experimental cross sections (red circles) are presentedin Fig. 1, as a function of center-of-mass energy E. Whilethe normalization of the data in the most interesting energyregion below 31 MeV is straightforward, one has to considerthat for four data points at higher energies (E = 31.7–36.4 MeV), the elastic scattering at the monitor angles of 45◦ isnot pure Rutherford in character. In this energy range we have,therefore, used optical model calculations with the potentialparameters from Ref. [13] (V = 100 MeV, W = 10 MeV,r0 = 1.20 fm, r0i = 1.28 fm, a0 = 0.46 fm and ai = 0.39 fm),obtained from a fit to the elastic scattering of 28Si + 30Si atenergies of 33.6 and 36.2 MeV. The fusion cross sectionsobtained with this potential are shown by the green curvein Fig. 1. At the higher energies, the calculations are ingood agreement with both the present and previous [9] crosssections. At the lowest energies, however, the optical modelcalculations overpredict the fusion cross sections by a factorof more than 10. A similar behavior has been observed formany lighter systems where a standard average potential (V =50 MeV, W = 10 MeV, r0 = r0i = 1.28 fm and a0 = ai = 0.4fm, [14]) gives good agreement at the higher energies, butoverpredicts the fusion cross sections at very low energies[6,15]. It should be noted that the calculated fusion crosssections obtained by these two sets of potential parametersare nearly identical over the whole energy range underinvestigation (23–37 MeV).TABLE I. States and corresponding parameters used in the CCcalculations for nuclei 28Si and 30Si (from Refs. [18,19]).Nucleus λπ Ex(MeV)B(Eλ)(W.U.)βCλ βNλ2+ 1.78 13.2 −0.41 −0.4128Si 2PH(2)+ 4.69 8.8 −0.24 −0.243− 6.88 13.9 0.42 0.422+ 2.24 7.4 0.31 0.3130Si 2PH(2)+ 4.31 5.2 0.18 0.183− 5.49 6.1 [19] 0.27 0.27Standard coupled-channels calculations (CC) with aWoods-Saxon (WS) nuclear potential from Akyüz and Winther[16] are shown by the light-blue dash-dotted curve. The excitedstates of 28Si and 30Si included in the CC calculations are listedin Table I along with their excitation energies, and the values ofthe reduced electric quadrupole transition probability, B(Eλ)and the deformation parameter, β. While they give in generala better agreement, they still overpredict the cross sections atthe lowest energies (see insert in Fig. 1).In Ref. [17] a model was suggested to explain the fusionhindrance by including the saturation property of nuclearmatter. This model introduces a repulsive core, resulting in ashallow potential, and has reproduced the hindrance behaviorobserved in many systems with negative Q-values, such as 64Ni+ 64Ni [17]. The same recipe is used here. The potential used isthe M3Y potential added to a repulsive core. The calculationsof the repulsive core are based on a nuclear incompressibilityof K= 234 MeV and a diffuseness arep of the hard core densityvarying within the 0.410–0.425 fm range. These potentialsare compared in Fig. 2 with the standard Woods-Saxon (WS)potential. It is evident that the M3Y+repulsive potentials leadto a shallower pocket as compared to the WS potential. Thebest fit to the present data is achieved with arep = 0.415 fm.The resulting cross sections are given in Fig. 1 by the blackcurve. This CC calculation with a shallow potential reproducesthe experimental data much better. The magenta curve is an 40 35 30 25 20 15 10 5 0-5 12 11 10 9 8 7 6 5U(r)+VC(r) (MeV)r (fm)28Si + 30SiM3Y arep=0.42M3Y arep=0.415M3Y arep=0.41Woods-Saxonpure M3YFIG. 2. (Color online) Comparison of potentials used in the CCcalculations. For Ref. [17], different diffusenesses arep of the hardcore density are used (arep = 0.42, 0.415, 0.41 fm, respectively). Theblack double-dotted curve is the pure M3Y potential.017601-2BRIEF REPORTS PHYSICAL REVIEW C 78, 017601 (2008)FIG. 3. (Color online) (a) The logarithmic derivative L(E) =d(ln σE)/dE plotted as a function of the center-of-mass energy E.The solid circles were derived from the data by least-squares fits tothree consecutive data points while the open circles were obtainedwith the two-points method. The magenta dotted curve is a fit to thelow energy part of the data with a formula L(E) = A0 + B0/E1.5.(b) The S factor versus E plot. The magenta dotted curve is anextrapolated one. See text for details.extrapolation obtained with the recipe developed in Ref. [6](see the insert), which will be discussed in the next paragraph.The conversions of the fusion cross sections to eitherlogarithmic derivatives, L(E) = d ln(σE)/dE or an S factorare given in Fig. 3. The solid circles for the L(E) values werederived from the data by least-squares fits to three consecutivedata points, while the open circles were obtained with thetwo-points method. The dashed curve in Fig. 3(a) correspondsto the constant S factor function Lcs(E) = πηE , where η isthe Sommerfeld parameter [4]. The dotted magenta curve inFig. 3(a) corresponds to a fit to the low energy part of theexperimental data with the formula suggested in Ref. [6]:L(E) = A0 + B0/E1.5 MeV−1. (1)The dotted magenta curve in Fig. 3(b) is in turn the extrapo-lation corresponding to the magenta curve in Fig. 3(a), whichis also developed in Ref. [6]. In Fig. 3, the calculations withthe optical model, standard coupled-channels and coupled-channels with a shallow potential are shown with the samecolors and definitions as in Fig. 1. Figure 3(b) clearly demon-strates that the optical model and the standard CC calculationsoverpredict the data by a large amount, while the CC calcula-tions with a shallow potential represent the data rather well.From the definition of the constant S factor function,the intersection point (Ls and Es) of the lines L(E) andLcs(E) corresponds to the location of the maximum ofthe S factor. From previous studies, we concluded that theobservation of a maximum signals the onset of sub-barrierfusion hindrance. A maximum of the S factor cannot bereproduced either by standard coupled-channels calculationsfor negative Q-value systems, or by optical model calculationsfor positive Q-value systems. Since the lowest energy pointin the measured excitation function is at E = 24.67 MeV, theintersection between L(E) and Lcs(E) can be inferred onlyfrom a single point derived with the two-points method in thepresent experiment. As a result, no maximum in the S factor(Fig. 3(b)) has been observed with certainty. Depending onthe extrapolation, this maximum might occur at an energybelow 24.7 MeV. There is a systematics study about theradius-of-curvature, ρ, of the S factor maximum in Ref. [5].From this study, the maximum of the S factor in the system28Si + 30Si could be a broad one, since one obtains a largepredicted ρ value from Eq. (8) of Ref. [5].While no conclusion about a maximum for the S factorcan be drawn from the data, it is obvious from Figs. 1and 3, however, that optical model calculations, which havebeen extensively used in the past in extrapolations of fusioncross sections to lower energies especially for systems ofastrophysical interest, do not result in a satisfactory descriptionof the data. They predict a continuous increase of the Sfactor toward lower energies, which is at variance with thedata by more than an order of magnitude. On the otherhand, the calculations with a shallow potential model doreproduce the present data. Thus, it appears that the presentexperimental results support the conclusion given in theprevious systematical study [5,6] that the fusion hindranceoccurs also in systems with positive Q-values.From the definition of L(E) = d ln(σE)/dE it is clear thatL(E) → ∞ for E → 0 in systems with Q > 0, whereas forQ < 0, L(E) → ∞ for E → −Q, because the cross sectionmust vanish at the energy corresponding to the ground state ofthe fusing system. This implies that the slope of the logarithmicdependence of σE as a function of E becomes steeper with adecrease in energy, as demonstrated experimentally in Ref. [5]from an analysis of many systems.For a negative Q-value system, S(E) → 0 when E → −Q.This leads to the conclusion that for negative Q-value systems,a maximum in the S factor must occur at low energies,which cannot be reproduced by the standard coupled channelscalculations. For a positive Q-value system, however, thereis no restriction on S(E) when E → 0. There are manyindications, however, that the general tendency of the measuredS factor is to exhibit a decrease at the lowest energies, whichcan also not be reproduced by optical model calculations[6].Recently, coupled channels calculations have been per-formed for lighter heavy-ion systems, such as 16O + 16O [20].With the same parameters obtained in the present calcula-tions for the system 28Si + 30Si (K = 234 MeV and ares =0.415 fm), the shallow potential model can also reproducethe low energy data of 16O + 16O, and even the extrapolationsobtained with the fusion hindrance recipe [6]. Since this model,which was developed to explain the hindrance for negativeQ-value systems, can also reproduce the low energy data forthe positive Q-value systems 28Si + 30Si and 16O + 16O, itappears likely that the saturation property of nuclear matterinfluences the fusion of these positive Q-value systems aswell, and results in the hindrance of the fusion process.017601-3BRIEF REPORTS PHYSICAL REVIEW C 78, 017601 (2008)In conclusion, the first measurement studying heavy-ionfusion hindrance in a light system with a positive Q-value,28Si + 30Si, has been reported. While the excitation functioncould not be measured down to sufficiently low cross sectionsto exhibit a clear maximum in the S factor, and thusindicate firmly the occurrence of fusion hindrance in systemswith positive Q-values, strong deviations from the opticalmodel predictions and the standard CC calculations havebeen observed. The present results appear to support thatfusion hindrance occurs for positive Q-value systems. Sinceoptical model calculations (and other potential penetrationmodels, like the ones used in Ref. [21,22]) are frequentlyused to extrapolate experimental cross sections toward lower,astrophysically interesting energies, these results will alsoinfluence the field of nuclear astrophysics, e.g., supernovaeand pycnonuclear burnings, where heavy-ion fusion plays animportant role.This work was supported by the U.S. Department of Energy,Office of Nuclear Physics, under Contract No. DE-AC02-06CH11357.[1] C. L. Jiang et al., Phys. Rev. Lett. 89, 052701 (2002); 93, 012701(2004); Phys. Rev. C 71, 044613 (2005); Phys. Lett. B640, 18(2006).[2] M. Dasgupta et al., Phys. Rev. Lett. 99, 192701 (2007).[3] M. Trotta et al., Nucl. Phys. A787, 134c (2007).[4] C. L. Jiang, H. Esbensen, B. B. Back, R. V. F. Janssens, andK. E. Rehm, Phys. Rev. C 69, 014604 (2004).[5] C. L. Jiang, B. B. Back, H. Esbensen, R. V. F. Janssens, andK. E. Rehm, Phys. Rev. C 73, 014613 (2006).[6] C. L. Jiang, K. E. Rehm, B. B. Back, and R. V. F. Janssens, Phys.Rev. C 75, 015803 (2007).[7] C. L. Jiang, B. B. Back, R. V. F. Janssens, and K. E. Rehm, Phys.Rev. C 75, 057604 (2007).[8] L. R. Gasques et al., Phys. Rev. C 76, 035802 (2007).[9] S. Gary and C. Volant, Phys. Rev. C 25, 1877 (1982).[10] C. N. Davids and J. D. Larson, Nucl. Instrum. Methods Phys.Res. B 40, 1224 (1989); C. N. Davids et al., ibid. 70, 358 (1992).[11] C. L. Jiang et al., Nucl. Instrum. Methods A 554, 500 (2005).[12] A. Gavron, Phys. Rev. C 21, 230 (1980).[13] K. D. Hildenbrand et al, Nucl. Phys. A234, 167 (1974).[14] G. J. Michaud and E. W. Vogt, Phys. Rev. C 5, 350 (1972); G. J.Michaud, ibid. 8, 525 (1973).[15] R. G. Stokstad et al., Phys. Rev. Lett. 37, 888 (1976).[16] Ö. Akyüz and A. Winther, in Proceedings of the Enrico FermiSchool of Physics, 1979, Course on Nuclear Structure andHeavy-Ion Reactions, edited by R. A. Broglia, C. H. Dasso andR. Ricci (North Holland, Amsterdam, 1981).[17] S. Mişicu and H. Esbensen, Phys. Rev. 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Fusion hindrance for a positive Q-value system

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Fusion hindrance for a positive Q-value system

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