Absolute cross sections for a number of binary processes in the N14+Tb159 reaction have been determined by use of γ-particle coincidence techniques. The data are used to test a "sum-rule model" of complete- and incomplete-fusion reactions. The model assumes the reaction probabilities to be scaled by the available phase space and to be constrained additionally by the entrance-channel angular momentum limitations

GONGGRIJP, S

HAGEMAN, DCJM

JANSSENS, RVF

LUKASIAK, J

SIEMSSEN, RH

SIWEKWILCZYNSKA, K

VANDRIEL, J

WILCZYNSKI, J

NARCIS 

INCOMPLETE-FUSION REACTIONS IN THE N-14+TB-159 SYSTEM AND A SUM-RULE MODEL FOR FUSION AND INCOMPLETE-FUSION REACTIONS

University of Groningen

ARTS repository - University of Groningen

Wilczyński, J.

Siwek-Wilczyńska, K.

Van Driel, J.

Gonggrijp, S.

Hageman, D.C.J.M.

Janssens, R.V.F.

Łukasiak, J.

Siemssen, R.H.

Carolina Digital Repository

VOLUME 45, NUMBER 8 PHYSICAL REVIEW LETTERS 25 AU|-UsT 1980Incomplete-Fusion Reactions in the '"N+'~9Tb System and a "Sum-Rule Model"for Fusion and Incomplete-Fusion ReactionsJ„Wile zydskiInstitute of Nuclear E%ysics, 31 -342 Cracose, Poland, and Kemfysisch Versneller Instituut,9747-AA Gronin gen, The NetherlandsK. Siwek-%ilczy6skaInstitute of Experimental Physics, University of Warsacu, 00-68I Warsato, Poland, andKernfysisch Versneller Instituut, 9747-AA Groningen, The NethertandsJ. van Driel, S. Gonggrijp, D. C. J. M. Hageman, R. V. F. Janssens,J. Cukasiak, ' and R. H. SiemssenKernfysisch Versneller Instituut, 9747-AA Groningen, The Netherlands(Received 7 May 1.980)Absolute cross sections for a number of binary processes in the ' N+ "~Tb reactionhave been determined by use of y-particle coincidence techniques. The data are used totest a "sum-rule model" of complete- and incomplete-fusion reactions. The model as-sumes the reaction probabilities to be scaled by the available phase space and to be con-strained additionally by the entrance-channel angular momentum limitations.PACS numbers: 25.70.Hi, 25.70.8cAn investigation of the energy dependence ofthe cross sections for two main incomplete-fusionchannels in the "C+"Gd reaction" led us to aninterpretation of these reactions in terms of ageneralized concept of critical angular momen-tum. ' According to this concept incomplete-fu-sion reactions are localized in successive "lwindows" above the critical angular momentumfor a complete fusion, in a sequence beginning withthe capture of the heaviest fragment of a projec-tile, followed by the capture of lighter fragmentsat higher angular momenta. Predictions of thismodel have been confirmed by Geoffroy et al„'who measured y-multiplicity distributions incoincidence with the products of incomplete-fusion reactions.In this Letter we report on the measurementsof the absolute cross sections for a varj. ety ofbinary reactions in the "N+'"Tb collisions at140 MeV bombarding energy. It is shown that allthose binary reactions (with the exception ofsimple direct transfer) can be understood interms of the same mechanism that governs theincomplete-fusion processes. Moreover, itseems that this common reaction mechanismcomprises also complete fusion. A sum-rulemodel that enables one to predict absolute crosssections for complete fusion and all incomplete-fusion channels, as well as the distribution ofthese reactions in / space, is proposed.The "N+'"Tb system was chosen with the aimto observe a large number of reaction channels[contrary to the "C+"'Gd system studied pre-viously" for which the important reaction chan-nels were ("C, ct), ("C, 2o.), and ("C, 3n) only].Experiments were performed using a 140-MeV'4N beam from the variable-energy cyclotron ofthe Kernfysisch Versneller Institute bombardinga 1.4-mgicm'-thick target of metallic '"Tb.Selection of binary reactions was possible bymeasuring coincidences between y rays (used toidentify the target-residue nuclei) and chargedreaction products detected and identified with astandard ~-E telescope consisting of a 50- pmtransmission detector and a 5-mm E detector.Particle identification (Z and A) was achieved forall reaction products over the range 2 Z 8.Details of the experimental techniques used inour experiments- are contained in Ref. 2.Absolute cross sections for different binaryreaction channels were obtained by summing thecross sections for all subchannels with apprecia-ble yield:o[»gab(14N' &Z) ]0(14N AZxn)1 ts- A-x (72 Z)'The absolute cross sections for a number of bi-606 1980 The American Physical SocietyVOLUME 45, NUMBER 8 PHYSICAL REVIEW LETTERS 25 AUGUST 1980TABLE I. Cross sections for various reactions inthe Q + ' ~Tb system at 140 MeV determined fromparticle-y coincidence data, 0«p, , and the calculatedcross sections, 0(:~~(: (T = 3 MeV, 4~ = N, qc = 0.06fm ').where T is an effective temperature, and Qc isthe change of the Coulomb interaction energy dueto transfer of charge. Qc can be parametrizedasReactionFusion('4N, 4He)(14N 6He)('4N, 'Li)(14N 7L )('4N, 7Be)(14N 8Be)( N, Be)('4N, 'OBe)(14N 10B)(14N 11B)(14N' 12B)(14N 11C)(14N 12C)(14N 13')0«p~ (mb)~ ~ ~84+ 16& 1.6& 3.5&6& 1.536+ 85.6+ 1.15.8+ 1.35.3+ 2.211+43.5+ 1.22.4+ 0.736+ 522k 20'calc (m10301600.92.51.41.342.76.96.62.77.72.41.548.929.6Qc =qc(Z, ~Z, ~ —Z, Z, )e',where Z, , Z, and Zy Z2 are the atomic num-bers of the constituents of the dinuclear systembefore and after the transfer of charge, respec-tively, and qc is a parameter. It should bepointed out that we apply postulate (2) not only toall binary divisions of the dinuclear system, butalso to comPlete fusion of the system.Apart from the phase space, also the entmnce-channel angular momentum limitations constrainthe reaction probabilities We u.se the samerecipe as that proposed in Ref. 1: For each reac-tion channel i, the limiting angular momentum ismass of projectile/jim 2) = mass of captured fragmentnary reaction channels deduced from particle-ycoincidence data are listed in Table I. In a forth-coming paper we will discuss the results in de-tail, together with the data on inclusive crosssections, particle-particle coincidence data, andy-multiplicity measurements. In this Letter weconcentrate on a new interpretation of the incom-plete-fusion reactions using as a test of the pro-posed model the cross sections listed in Table I,and the previous data on the "C + "'Gd system.Our results on the ' N+'"Tb system show thatthe incomplete-fusion reactions characterized bythe capture of a massive fragment of a projectileand emission of "fast" light ejectiles, e.g. , nparticles, ' ' protons, deuterons, or tritons, ' be-long to a wide class of binary reactions in whichejectiles have the average kinetic energies closeto those required by the l-matching condition.On the other hand, it is known from inclusive ex-periments that for this class of reactions thecross sections depend exponentially on the ground-state Q value, Q«. ' This striking relation hasbeen interpreted by Bondorf et al. ' in terms of apartial statistical equilibrium of a strongly inter-acting dinuclear system. Following such phase-space arguments we postulate that the reactionprobabilities for all reactions i proceeding viathe partially equilibrated system are proportion-al to the exponential factor as proposed by Bon-dorf et al. ':p(i) -exp{[Q„(i)—Qc(i)]/T ), (2)(f„+—,')'p,(C, + C,)' C,C, Z,Z,e'Here C, and C, are the half-density radii, and yis the surface-tension coefficient. To make ourmodel more realistic we also assume a smoothcutoff in the distribution of the "transmission co-efficients" T, :T,(i) = 1+ expl —i „(i)&sCombining Eqs. (2) and (6), one can write thefollowing sum rule:(6)(7)provided all the reaction channels (including com-plete fusion) are taken explicitely into account.With the normalization factor N, calculated fromEq. (7) one obtains absolute cross sections foreach reaction channel:o(i) =era' Q (2E+1)N, T, (i)expl =0(8)We further postulate that the l,„ limiting thesum in Eq. (8) is the maximum l for which thex l„(target+captured fragment), (4)where l „is calculated from the balance of forces':607VOLUME 45, NUMBER 8 PHYSICAL REVIEW LETTERS 25 AUGUST 1980EJIOlCh100-CLlQI—50-60IEtab("maxi, Mev80 100 140 180 22010 20 30 40 50 60 70ANGULAR MOMENTUM, L (ti)&fu &e i &2(mb) (m1000-- 200500--10050 100 150 200BEAM ENERGY, MeVFIG. 1. (a) Reaction probabilities for the main reac-tion channels in the '2C+ '60Gd system (T = 3 MeV,q c = 0.06 fm ' ) . (b) Excitation functions forcomplete fusion and two main incomplete-fusion chan-nels in C+ ~6 Gd collisions calculated with the sameset of parameters. Experimental data are taken fromRef. 1.colliding system gets into the region where thetotal nucleus-nucleus potential is attractive and/or the distance of closest approach is smallerthan the sum of the half-density radii. Note thatthis criterion is identical with that proposed byGlass and Mosel" to determine the range of par-tial waves leading to complete fusion. Now wemake the assumption that this range of partialtopazes confines complete-fusion and incompletefusion reactions together.To calculate l,„one has to use a certain nu-cleus-nucleus potential. Results of the numer-ical calculations presented in this Letter wereobtained assuming the point-charge Coulombinteraction potential and the nuclear potential asproposed in Ref. 11, based on the boundary condi-tions following the liquid-drop model (adiabaticlimit for small relative distances, and suddenapproximation for proximity configurations). Thehalf-density radii [used, e.g. , in Eq. (5) and incalculation of t~,„]were parametrized as pro-posed by Myers. " Only for the lightest frag-ments (A (4) for which the approximation C =R—b'/R fails (R is the equivalent sharp radius,b=1 fm) the half-density radii were scaled pro-portionally to A' ' with the constant that matchesthe Myers parametrization at A =4.The only free parameters of the model outlinedabove are the effective temperature T, the dif-fuseness of the T, distributions b, , (assumed to bethe same for all reaction channels), and the Cou-lomb energy parameter qc. Contrary to standardstatistical-model calculations, the parameter Thas no clear interpretation. ' Therefore it shouldbe regarded rather as an empirical parameter,similarly as b, , and qc.Figure 1(a) shows the distributions of the reac-tion probabilities for the strongest channels in"C+"~Gd close-contact collisions. Reactionchannels involving complete fusion and reactionswith emission of n, ' 'H, "'He, "Li, ' ' Be,"B, and ""Cas ejectiles were taken explic-itly into the sum rule (7). Because of the largestvalue of Q —Q c, the complete-fusion channeltakes almost the whole geometrical cross sectionfor low l values. Only above the critical angularmomentum (l,„=42k) the probability for completefusion diminishes and thus makes room for othercompeting reaction channels. Other importantchannels next to complete fusion are ("C, n) and("C, 2n)—again because of the largest Q„—Qcvalues. The l1 values for these two channelsdiffer significantly ( 47li and--618, respectively).This is a reason why the reaction probability for("C, o.) decreases first while the probability forthe other reaction still increases until its ownangular -momentum limitation is reached. As itis seen from Fig. 1(a) the l windows for incom-plete-fusion channels postulated in Ref. 1 appearautomatically in the model proposed here. How-ever, with parameters that fit the experimentaldata, the windows are not sharply separated (asassumed in the model of Ref. 1) and the crosssections for particular incomplete-fusion chan-nels are significantly smaller than the geometri-cal cross sections within the / windows.Figure 1(b) shows a comparison of the modelpredictions with the excitation functions for the"Gd("C, n) and '"Gd("C, 2o.) reactions. ' Forsimplicity, the parameter T was assumed to beenergy independent. (With increasing bombard-ing energy the thermalized part of kinetic energyis probably shared between more and more de-grees of freedom. Therefore the effective tem-608VOLUME 45, NUMBER 8 PHYSICAL RKVIEW LETTERS 25 AUGUST 1980perature of a partially equilibrated system maynot tend to increase much. ) As it is seen fromFig. 1(b), the calculated excitation functions re-produce well the energy thresholds and absolutecross sections for both ("C, o) and ("C, 2o.)incomplete-fusion reactions.The proposed model can be checked more ex-tensively using our new data on the "N+ '"Tbsystem. The absolute cross sections for a num-ber of capture reactions (see Table I) are com-pared with the model predictions if we assumethe same set of parameters as that used for theC + Gd system. The calculated cross sectionsare given also for other channels for which wecould not determine the cross sections. [For in-stance, we were not able to measure the com-pound-residue cross section since the ("N, xn)channels in the "N+ '"Tb reaction at 140 MeVlead to unknown isotopes of Hf. ] There is a closecorrelation between the calculated and measuredcross sections. This result strongly indicatesthat both the Q„effects and the entrance-channellimitations built into the model influence thecross sections.The analysis of the experimental data on theC+'"Gd and '4N+ i59Tb systems gi~es strongarguments in support of the proposed model. Anintriguing question is whether the mass andcharge distributions of the products of stronglydamped reactions in collisions of very heavy sys-tems can also be described by this model. Theproposed model couples strongly the complete-fusion reactions with other reaction modes.[Actually, the sum rule (7) simulates in a sensea complete coupled-channels calculation in theno —imaginary-potential limit. ] Such an approachoffers, therefore, a possible explanation of thenuclear structure effects which show up in mag-nitudes of the fusion cross sections especiallyfor light systems. "' Permanent address: Institute of Nuclear Research,05-4()0 Swierk, Poland.'K. Siwek-Wilczynska, E. H. du Marchie van Voor-thuysen, J. van Popta, B. H. Siemssen, and J.Wil-czynski, Phys. Rev. Lett. 42, 1599 (1979).K. Siwek-Wilczynska, E. H. du Marchie van Voor-thuysen, J. van Popta, R. H. Siemssen, and J.Wil-czynski, Nucl. Phys. A330, 150 (1979).'K. A. Geoffroy, D. G. Sarantites, M. L. Halbert,D. C. Hensley, R. A. Dayras, and J. H. Barker, Phys.Rev. Lett. 43, 1303 (1979).T. Inamura, M. Ishihara, T. Fukuda, T. Shimoda,and H. Hiruta, Phys. Lett. 688, 51 (1977).5D. B. Zolnowski, H. Yamada, S. E. Cala, A. C.Kahler, and T. T. Sugihara, Phys. Rev. Lett. 41, 92(1978).'6H. Yamada, D. B. Zolnowski, S. E. Cala, A. C.Kahler, J. Pierce, and T. T. Sugihara, Phys. Rev.43 605 (1979)YA. G. Artukh, V. V. Avdeichikov, J. Er'o, G. F.Gridnev, V. L. Mikheev, V. V. Volkov, and J.Wil-czynski, Nucl. Phys. A160, 511 (1971).8J. P. Bondorf, F. Dickmann, D. H. E. Gross, andP. J. Siemens, J. Phys. (Paris), Colloq. 32, C6-145(1971).J.Wilczynski, Nucl. Phys. A216, 386 (1973).D. Glass and U. Mosel, Nucl. Phys. A237, 429 (1975)."J.Wilczynska and K. Siwek-Wilczynska, Phys. Lett.55B, 270 (1975).' W. D. Myers, Nucl. Phys. A204, 465 (1973).3P. Sperr, T. H. Braid, Y. Eisen, D. G. Kovar,F. W. Prosser, Jr., J. P. Schiffer, S. L. Tabor, andS. Vigdor, Phys. Bev. Lett. 37, 321 (1976).Spectra of Very Highly Charged Cu- and Zn-Like IonsJoseph Reader and Gabriel LutherNationa/ Bureau of Standards, washington, D. C. 20234(Received 5 June 1g80)The 4s-4p, 4P-4d, and 4d-4f transitions of ten copperlike and zinclike ions from Bato W + have been observed by means of a laser-produced plasma and a 2.2-m grazing-incidence spectrograph. The spectra are accompanied by a prominent continuum lyingjust below the 4p P~/2-4d D3/2 transitions in the copperlike ions. The results supportthe identification of the resonance lines of Xe + and Xe + in the Princeton UniversityST tokamak by Hinnov.PACS numbers: 32.30.-r, 52.70.-m, 52.50.JmThe spectra of ions having simple atomic struc-tures are currently of great interest because oftheir use in the diagnosis of hot plasmas found incontrolled-fusion devices. Of special importanceare ions in the copper and zinc isoelectronic se-quences. These ions contain one and two elec-Oc 1980 The American Physical Society 609

Incomplete-fusion reactions in the N14+Tb159 system and a "sum-rule model" for fusion and incomplete-fusion reactions

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INCOMPLETE-FUSION REACTIONS IN THE N-14+TB-159 SYSTEM AND A SUM-RULE MODEL FOR FUSION AND INCOMPLETE-FUSION REACTIONS

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