The excitation function for the fusion-evaporation reaction 28Si + 64Ni has been measured down to a cross section of 25 nb. This is the first observation of fusion hindrance at extreme sub-barrier energies for a system with a small, negative Q-value (- 1.78 MeV). This result is further proof that heavy-ion fusion hindrance, reported earlier only for systems with large, negative Q-values, is a general phenomenon. The measured behavior can be reproduced by coupled-channels calculations with a modified ion-ion potential incorporating the effects of nuclear incompressibility

Back, B.B.

Collon, P.

Davids, C.N.

Esbensen, H.

Greene, J.

Henderson, D.J.

Janssens, R.V.F.

Jiang, C.L.

Jisonna, L.

Kurtz, S.

Lister, C.J.

Mişicu, S.

Notani, M.

Pardo, R.

Paul, M.

Peterson, D.

Rehm, K.E.

Seweryniak, D.

Shumard, B.

Tang, X.D.

Tanihata, I.

Wang, X.

Zhu, S.

Carolina Digital Repository

Physics Letters B 640 (2006) 18–22www.elsevier.com/locate/physletbFirst evidence of fusion hindrance for a small Q-value systemC.L. Jiang a,∗, B.B. Back a, H. Esbensen a, R.V.F. Janssens a, Ş. Mişicu a, K.E. Rehm a, P. Collon b,C.N. Davids a, J. Greene a, D.J. Henderson a, L. Jisonna a,c, S. Kurtz b, C.J. Lister a, M. Notani a,d,M. Paul e, R. Pardo a, D. Peterson a, D. Seweryniak a, B. Shumard a, X.D. Tang a, I. Tanihata a,1,X. Wang a, S. Zhu aa Argonne National Laboratory, Argonne, IL 60439, USAb University of Notre Dame, Notre Dame, IN 46556, USAc Northwestern University, Evanston, IL 60208, USAd Michigan State University, East Lansing, MI 48824, USAe Hebrew University, Jerusalem 91904, IsraelReceived 1 June 2006; accepted 3 July 2006Available online 14 July 2006Editor: V. MetagAbstractThe excitation function for the fusion–evaporation reaction 28Si + 64Ni has been measured down to a cross section of 25 nb. This is thefirst observation of fusion hindrance at extreme sub-barrier energies for a system with a small, negative Q-value (−1.78 MeV). This result isfurther proof that heavy-ion fusion hindrance, reported earlier only for systems with large, negative Q-values, is a general phenomenon. Themeasured behavior can be reproduced by coupled-channels calculations with a modified ion–ion potential incorporating the effects of nuclearincompressibility.© 2006 Published by Elsevier B.V.PACS: 25.70.Jj; 24.10.Eq; 26.30.+kThe phenomenon of sub-barrier fusion enhancement inheavy-ion induced reactions at energies in the vicinity of theCoulomb barrier has been well documented for many systems[1–7]. In contrast, the unexpected hindrance of heavy-ion fu-sion at extreme sub-barrier energies has only been discoveredmore recently [8–12]. Indeed, it has been shown that the ex-citation functions of such reactions exhibit an abrupt decreasein cross section at the lowest beam energies and, as a result,a maximum appears in the associated astrophysical S factor atenergy Es [9]. This behavior cannot be described by conven-tional coupled-channels calculations. A new mechanism wasproposed in Ref. [13], which explains heavy-ion fusion hin-* Corresponding author.E-mail address: jiang@phy.anl.gov (C.L. Jiang).1 Present address: TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T2A3,Canada.0370-2693/$ – see front matter © 2006 Published by Elsevier B.V.doi:10.1016/j.physletb.2006.07.007drance quite well for the fusion reaction 64Ni + 64Ni. In thismodel, the steep falloff of fusion is related to the saturationproperties of nuclear matter which inhibit the large overlap ofthe reaction partners, cause a change of the nuclear potentialinside the barrier and result in a hindrance of the quantum tun-neling.Thus far, the study of the sub-barrier hindrance phenom-enon has focused solely on colliding systems with very negativefusion Q-values. As a result, the S factor must always reacha maximum at sufficiently low beam energies (correspondingto an abrupt decrease in the measured cross sections) since thefinal state phase space disappears when the center-of-mass en-ergy, E, approaches −Q [9]. The systematic study of Ref. [14]suggested, on the basis of a phenomenological analysis, that thehindrance behavior might be a general phenomenon in extremesub-barrier fusion reactions, but this has not yet been demon-strated convincingly for a system with either a small negativeQ-value, or even a positive one.C.L. Jiang et al. / Physics Letters B 640 (2006) 18–22 19In order to explore this issue, the fusion–evaporation exci-tation function for the 28Si + 64Ni system with a Q-value ofonly −1.78 MeV has been measured down to 25 nb. This rep-resents a qualitative difference from the well studied systemsmentioned above, where the Q-values ranged from −36.6 MeVfor 32S + 89Y [15] to −157.4 MeV for 90Zr + 90Zr [16]. Fur-thermore, the present study also represents an opportunity to in-vestigate whether the ion–ion potential used in Ref. [13] is ableto describe fusion systems much lighter than the one discussedin the earlier work. It should be noted that this 28Si + 64Ni sys-tem had been measured previously down to a cross section of41 µb [17], i.e., down to an energy insufficient to investigate theissue under discussion here.The experiment was performed with 28Si beams in the en-ergy range of 63–95 MeV delivered by ATLAS, the supercon-ducting linear accelerator at Argonne National Laboratory. Themaximum beam current used was ∼100 pnA. This relativelyhigh beam intensity does not represent a problem in terms oftarget damage because of the high melting point of the highly-enriched, metallic nickel material (which was evaporated on a40 µg/cm2 carbon foil). Thin targets with thicknesses of about15 µg/cm2 were used in order to reduce the correction for tar-get thickness in the energy regime where the steep fall-off inthe excitation function occurs. The isotopic abundance of 64Niwas 98.02%, with the remainder coming from 58Ni (0.97%),60Ni (0.57%), 61Ni (0.05%) and 62Ni (0.39%). Small conta-minations from lighter isotopes did not interfere with the ac-tual measurements since the associated barriers are all higherthan for 64Ni. Two surface-barrier Si detectors, located at ±43◦with respect to the beam direction, served as monitors. The ab-solute cross sections for fusion–evaporation were determinedby using elastic scattering measured with these two coun-ters.The experimental procedure was similar to that used in ear-lier measurements of the fusion–evaporation excitation func-tion for the system 64Ni + 100Mo [11,18]. Briefly, the evapo-ration residues were measured with the Fragment Mass Ana-lyzer (FMA) [19] placed at 0◦ with respect to the beam direc-tion. The instrument has been upgraded with the installation ofa split-anode in the first electric-dipole [20]. The background,originating mostly from the beam scattered off the first an-ode, is greatly suppressed as a result. The evaporation residueswere detected and identified behind the FMA with detectors ofthe configuration PGAC1–TIC1–PGAC2–TIC2–PGAC3–IC de-scribed in detail in Ref. [18]. 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 PGAC1 wasmounted at the horizontal (x-direction) focal-plane of the FMA,where the evaporation residues are dispersed according to theirmass-to-charge ratio m/q .Only one charge state at the time could be measured at eachFMA setting. As a result, full charge state distributions wereobtained at five energies, while three to five charge states wererecorded for five intermediate points. Only one charge state wasmeasured for the six lowest energies because of the low crosssections involved. These data proved sufficient to determine theFig. 1. m/q spectra obtained at three of the lowest bombarding energies for28Si + 64Ni fusion evaporation residues. The corresponding fusion evaporationcross sections were 25 nb, 150 nb and 8.3 µb, respectively. The peaks observedin (a) are for m/q = 89/14 (low channel number) and 88/14, respectively.charge state fractions of the detected evaporation residues withthe desired accuracy.In order to establish the FMA transmission, it is neces-sary to characterize the angular distribution of the residues. Inthe present experiment this was achieved by determining theirvelocity distribution. This method was developed at Argonnepreviously [21] and relies on an accurate determination of thetime of flight of the reaction products on an event-by-event ba-sis. For this purpose the flight time from the target to PGAC1,and the flight time between PGAC1 and PGAC3 were used.The inferred angular distributions agreed well within the de-sired uncertainty with that measured at Elab = 84.5 MeV inRef. [17], as well as with those calculated with the statisticalcode PACE [22].The separation between the evaporation residues and thebackground was very good even at the lowest energy mea-sured. Three position-spectra are given for the measurementscarried out at the lowest, 2nd-lowest and 4th-lowest energies inFig. 1. The peaks observed in Fig. 1(a) correspond to residueswith m/q = 89/14 (lower channel number) and 88/14, re-spectively. At the lowest-energy, two evaporation residues withm/q = 89/14 were observed. The fusion evaporation cross sec-tions measured at these three energies were 25 nb, 150 nb and8.3 µb, respectively.The experimental cross sections, covering more than 7 or-ders of magnitude, are presented in Fig. 2, as an excitation func-tion in the center-of-mass system. Comparisons with the previ-ous data [17] are provided as well. In the near-barrier region,the two measurements agree within experimental uncertainties.At lower energies, however, an energy shift Elab ∼ 1.2 MeVbetween the two data sets appears; the situation is similar to thatfound earlier for fusion in the 64Ni+64Ni system. In this case as20 C.L. Jiang et al. / Physics Letters B 640 (2006) 18–22Fig. 2. Excitation function for the fusion of 28Si+64Ni. The solid circles are thepresent results, while the open ones are taken from Ref. [17]. The dot-dashedand solid curves are the results of coupled-channels calculations with a usualWoods–Saxon potential and a shallow potential incorporating the effects of nu-clear incompressibility, respectively (see text for details). The dotted curve isthe result of a one-dimensional, no-coupling calculation.well, an energy shift Elab ∼1.5 MeV was observed betweenthe cross sections measured at Argonne [10] and those reportedin Ref. [23].Detailed coupled-channels calculations have been carriedout in order to account for the data. As a starting point, theytook advantage of the fact that the earlier data of Ref. [17] hadbeen analyzed in the coupled-channels approach [24]. Specifi-cally, sensitivity to transfer reactions was found, in particularto 2-neutron transfer, a process with a positive ground-stateQ-value of 2.59 MeV. The present calculations are similar tothose of Ref. [24] for the coupled-channels part. Thus, theyinclude the 2+ and 3− excitations in both 28Si and 64Ni, themutual excitations below 7 MeV and the two-phonon quadru-pole excitations. All together, ten channels were coupled. Twotypes of potentials were considered. The first was a conven-tional Woods–Saxon ion–ion potential with a diffuseness para-meter a = 0.63 fm. In order to reproduce the present data in thecross section range above 0.2 mb, the radius parameter R =0.20 fm used in Ref. [17] had to be adjusted to 0.31 fm. Thecoupled-channels calculations, together with a one-dimensionalpenetration calculation (without couplings) can be seen in Fig. 2as dot-dashed and dotted curves, respectively. As in previousstudies [8–12], the conventional coupled-channels calculations(with a Woods–Saxon potential) reproduce the data well forall cross sections above ∼0.1 mb, but overpredict the measure-ments at extreme sub-barrier energies.As stated above, a new mechanism was proposed to explainfusion hindrance at extreme sub-barrier energies by consider-ing explicitly the incompressibility of the overlapping reactionpartners and estimating the ion–ion potential in a double fold-ing model [13]. The inclusion of this compressional energyresults in a shallower potential and thicker barriers, that is ulti-mately responsible for the fast fall-off of the excitation functionFig. 3. Spherical ion–ion potentials for 28Si + 64Ni; the solid curve is the po-tential employed in the present study. The dashed curve is the Akyüz–Wintherpotential. The horizontal solid and dotted lines are schematical plots of the tra-jectories for incident ions at an energy E = Es .at the lowest beam energies. The same approach was adoptedto analyze the present experiment. The potential proposed inRef. [13] (M3Y + repulsion) is compared with a more con-ventional Woods–Saxon one (Akyüz–Winther [25]) in Fig. 3,where the schematic trajectories of particles of energy E = Esis also indicated.The coupled-channels calculation results with this shallowpotential are given by the solid curve in Fig. 2. The agree-ment between the model and the data is satisfactory and of thesame order as in the 64Ni + 64Ni case. Thus, a consistent inter-pretation is achieved for two systems with strikingly differentQ-values, as the rather small negative Q-value, −1.78 MeV,for the fusion reaction 28Si + 64Ni does not appear to affect theability to reproduce fusion hindrance at extreme sub-barrier en-ergies.The logarithmic derivatives L(E) and the S factor have alsobeen derived using the expressions given in Ref. [9]. Thesetwo quantities are shown in Figs. 4(a) and 4(b), respectively.In Fig. 4(a) the star symbols are obtained from a least squaresfit to three neighboring entries for the cross sections. The heavysolid line in the figure is the so-called constant S factor ex-pression, Lcs(E) = πηE , derived in Ref. [9], while the dottedline corresponds to a linear fit to the low-energy part of L(E).The crossing point of these two lines determines the loca-tion of the maximum in the S factor curve, which occurs atEs = 45.6 MeV and Ls = 2.78 MeV−1 for the 28Si + 64Ni sys-tem. It is worth noting that, from the previous data (Ref. [17]),a value of Es = 47.3 MeV is obtained with the extrapolationmethod described in [9]. The difference in the values extractedfrom the two measurements is due to the combination of the en-ergy shift in the excitation functions mentioned above and theprecise knowledge of the fall-off of the cross section that is nowavailable. The data for L(E) and S(E) deduced from Ref. [17]are given as open circles in Figs. 4(a) and 4(b) for comparison.Obviously, there is a significant, but broad maximum in theS factor curve as a function of energy (see Fig. 4(b)). This be-havior cannot be reproduced by the calculations based on theWoods–Saxon potential (dot-dashed curves). On the other hand,C.L. Jiang et al. / Physics Letters B 640 (2006) 18–22 21Fig. 4. (a) The logarithmic derivative L(E) = d(lnσE)/dE, plotted as a func-tion of the center-of-mass energy E for the fusion of 28Si + 64Ni. The starsare obtained from least squares fits to three data points. (b) The S factorS(E) = σE exp(2πη), plotted as a function of the center-of-mass energy E.The solid circles are the experimental results. The L(E) and S(E) values ob-tained from the data of Ref. [17] are shown as open circles. The heavy solidline corresponds to a constant S factor. See text for details.the new potential by using the recipe of Ref. [13] (solid curves)follows the patterns exhibited by the data.The extracted values for the parameters Ls and Es in thereaction 28Si + 64Ni are described well by the empirical ex-pressions given in Ref. [14]:(1)Lemps = 2.33 + 500/(Z1Z2√μ) (MeV−1),and(2)Eemps =(0.495Z1Z2√μ/Lemps)2/3(MeV),within the desired accuracy. The derivatives (dL/dE)exp and(dL/dE)cs obtained at Es are −0.39 and −0.09 MeV−2, re-spectively. The general trend exhibited by the ratio of these twoderivatives is also described well by another empirical equationproposed in Ref. [14],(3)Ratioemp = 1 + 10−5(Z1Z2√μ)5/3,where μ is the reduced mass number A1A2/(A1 + A2).All these observations confirm the conclusion proposed inRef. [12], that the systematic behavior of the hindrance phe-nomenon is closely related to the properties of the entrancechannel of the colliding system.As mentioned above the maximum of the S factor of thefusion reaction 28Si + 64Ni is broad. Since(4)d2 ln(S(E))dE2= dL(E)dE− d(πηE)dE,the radius of curvature ρ of ln(S(E)) vs. E can be expressedas,(5)1 = dL(E) − d(Lcs(E)) .ρ dE dEThus, ρ is determined by the difference between the derivativeof L(E) and that associated with a constant S factor (dL/dE)csat E = Es . Expanding to second order around the S factor max-imum we thus find(6)S(E) = S(Es) × e(E−Es)2/2ρ,as shown by the dashed curve in Fig. 4(b). It describes theexperimental S(E) values at energies E ∼ Es rather well. Itshould be mentioned that in comparison with the medium-massfusion systems where Q-values are large and negative, the ab-solute value of dL(E)dE− d(Lcs (E))dEfor the 28Si + 64Ni reactionis smaller, and the absolute value of ρ is larger, resulting ina S(E) factor with a broader maximum. This observation alsoholds for other systems with small negative Q-value, althoughtheir S(E) maxima have not yet been explored as thoroughly asis the case for the reaction under study here. Nevertheless thisbehavior can already be seen by using the extrapolation methoddeveloped in Ref. [10].This work has reported on the first observation of sub-barrierfusion hindrance, and a clear maximum in the S factor as a func-tion of energy, in a colliding system with a near zero Q-value.This result confirms the assumption made previously that thesub-barrier hindrance is a general phenomenon in heavy-ionreactions. Furthermore, the success of the comparison of thenew data with the model proposed in Ref. [13] can be viewedas further evidence that the hindrance is a consequence of themodification of the ion–ion potential inside the barrier origi-nating from the incompressibility for total overlap between thereaction partners. Because this phenomenon manifests itself atthe lowest energies, it is expected to impact stellar processessignificantly and should be incorporated in network calcula-tions involving fusion of light heavy-ion reactions which areof interest in astrophysics [26].AcknowledgementsThis work was supported by the US Department of Energy,Office of Nuclear Physics, under contract No. W-31-109-ENG-38. One of the authors (Ş. Mişicu) is grateful to the FulbrightCommission for financial support.References[1] M. Beckerman, Phys. Rep. 129 (1985) 145.[2] M. Beckerman, Rep. Prog. Phys. 51 (1988) 1047.[3] K. Hagino, N. Takigawa, M. Dasgupta, D.J. Hinde, J.R. Leigh, Phys. Rev.C 55 (1997) 276.[4] A.B. Balantekin, N. Takigawa, Rev. Mod. Phys. 70 (1998) 77.[5] R. Vandenbosch, Annu. Rev. Nucl. Part. Sci. 42 (1992) 447.[6] N. Rowley, G.R. Satchler, P.H. Stelson, Phys. Lett. B 254 (1991) 25.[7] M. Dasgupta, D.J. Hinde, N. Rowley, A.M. Stefanini, Annu. Rev. Nucl.Part. Sci. 48 (1998) 401.[8] C.L. 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First evidence of fusion hindrance for a small Q-value system

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First evidence of fusion hindrance for a small Q-value system

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