Coupled-channels analyses for large-angle quasi-elastic scattering in massive systems
aa r X i v : . [ nu c l - t h ] D ec Coupled-channels analyses for large-angle quasi-elastic scattering in massive systems
Muhammad Zamrun F. and K. Hagino
Department of Physics, Tohoku University, Sendai 980-8578, Japan
S. Mitsuoka and H. Ikezoe
Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan (Dated: November 16, 2018)We discuss in detail the coupled-channels approach for the large-angle quasi-elastic scattering inmassive systems, where many degrees of freedom may be involved in the reaction. We especiallyinvestigate the effects of single, double and triple phonon excitations on the quasi-elastic scatteringfor Ti, Cr, Fe, Ni and Zn+
Pb systems, for which the experimental cross sections havebeen measured recently. We show that the present coupled-channels calculations well account forthe overall width of the experimental barrier distribution for these systems. In particular, it is shownthat the calculations taking into account single quadrupole phonon excitations in Ti and tripleoctupole phonon excitations in
Pb reasonably well reproduce the experimental quasi-elastic crosssection and barrier distribution for the Ti+
Pb reaction. On the other hand, Cr, Fe, Ni and Zn+
Pb systems seem to require the double quadrupole phonon excitations in the projectiles inorder to reproduce the experimental data.
PACS numbers: 24.10.Eq, 25.60.Pj, 25.70.Bc, 27.80.+w
I. INTRODUCTION
It is now well established that the internal structure ofcolliding nuclei strongly influences heavy-ion collisions atenergies around the Coulomb barrier. In particular, thecoupling to the collective excitations (rotation and vibra-tional states) in the target and projectile nuclei partic-ipating in the reaction significantly enhances the fusioncross sections for intermediate mass systems [1, 2]. Suchcouplings give rise to a distribution of the Coulomb bar-rier [1, 2, 3], which can most easily be visualized for re-actions involving a deformed nucleus. In this case, thenucleus-nucleus potential depends on the orientation an-gle of the deformed nucleus with respect to the beamdirection. Since the orientation angle distributes isotrop-ically at the initial stage of the reaction, so does the po-tential barrier. The concept of barrier distribution canbe extended also to systems with a non-deformed target[3], where the distribution originates from the couplingbetween the relative motion and vibrational excitationsin the colliding nuclei and/or transfer processes. Noticethat, although this concept is exact only when the exci-tation energy is zero, to a good approximation it holdsalso for systems with a non-zero excitation energy [4, 5].In Ref.[6], Rowley et al. have argued that the bar-rier distribution can be directly extracted from a mea-sured fusion cross section σ fus ( E ), by taking the secondderivative of the product Eσ fus ( E ) with respect to thecenter-of-mass energy E , that is , D fus = d ( Eσ fus ) /dE .This method has stimulated many high precision mea-surements of fusion excitation function for medium-heavymass systems [2, 7]. The extracted barrier distributionshave revealed that the concept indeed holds and the bar-rier distribution itself provides a powerful tool for in-vestigating the effects of channel coupling on heavy-ionfusion reactions at sub-barrier energies. It has also been shown recently that the concept of barrier distributionis still valid even for relatively heavy systems, such as Mo+
Mo [8].A similar barrier distribution can also be extractedfrom quasi-elastic scattering (a sum of elastic, inelas-tic and transfer processes) at backward angles [9, 10],that is a good counterpart of the fusion reaction [11].In this case, the barrier distribution is defined as thefirst derivative of the ratio of quasi-elastic to the Ruther-ford cross sections dσ qel /dσ R , with respect to E , i . e .,D qel = − d ( dσ qel /dσ R )/ dE . Since the fusion and thequasi-elastic scattering is related to each other becauseof the flux conservation, a similar information can be ob-tained from those processes and the similarity betweenthe two representations for barrier distribution has beenshown to hold for several intermediate mass systems[9, 10, 12].Recently, the quasi-elastic barrier distribution has beenexploited to investigate the entrance channel dynamicsfor fusion reactions to synthesize super-heavy elements[13, 14, 15, 16]. It has been shown that the conceptof barrier distribution remains valid even for such veryheavy systems once the deep-inelastic cross sections areproperly taken into account. As is expected, the strongchannel coupling effects on the barrier distribution havebeen observed.In this paper, we carry out a detailed coupled-channelsanalysis for large-angle quasi-elastic scattering data for Ti, Cr, Fe, Ni and Zn+
Pb systems leading tosuper-heavy elements Z = 104 , , , , and 112,respectively [15, 16]. We especially study the role ofmulti-phonon excitations of the target and projectile nu-clei, which has been shown to play an important role inquasi-elastic scattering for the Kr+
Pb system [14].The paper is organized as follows. We briefly explainthe coupled-channels formalism for quasi-elastic scatter-ing in Sec. II. We present the results of our systematicanalysis in Sec. III. We then summarize the paper in Sec.IV.
II. COUPLED-CHANNELS FORMALISM FORLARGE ANGLE QUASI-ELASTIC SCATTERING
In this section, we briefly describe the coupled-channelsformalism for large angle quasi-elastic scattering whichincludes the effects of the vibrational excitations of thecolliding nuclei. The total Hamiltonian of the system isassumed to be H = − ~ µ ∇ + V (0) N ( r ) + Z P Z T e r + H exct + V coup ( r , ξ P , ξ T ) , (1)where r is the coordinate of the relative motion betweenthe target and the projectile nuclei, µ is the reducedmass and ξ T and ξ P represent the coordinate of the vi-bration in the target and the projectile nuclei, respec-tively. Z P and Z T are the atomic number of the pro-jectile and the target, respectively, and V (0) N is the barenuclear potential, which we assume to have a Woods-Saxon shape. It consists of the real and imaginary parts, V (0) N ( r ) = V ( r ) + iW ( r ). H exct describes the excita-tion spectra of the target and projectile nuclei, while V coup ( r , ξ P , ξ T ) is the potential for the coupling betweenthe relative motion and the vibrational motions of thetarget and projectile nuclei.In the iso-centrifugal approximation [1, 17, 18], wherethe the angular momentum of the relative motion in eachchannel is replaced with the total angular momentum J (in the literature, this approximation is also referred toas the rotating frame approximation or the no-Coriolisapproximation), the coupled-channels equations derivedfrom the Hamiltonian (1) read (cid:20) − ~ µ d dr + J ( J + 1) ~ µr + V (0) N ( r ) + Z P Z T e r − E + ǫ n (cid:21) u n ( r ) + X n ′ V nn ′ ( r ) u n ′ ( r ) = 0 (2)where ǫ n is the eigen-value of the operator H exct for the n -th channel. V nn ′ ( r ) is the matrix elements for the cou-pling potential V coup .In the calculations presented below, we use the methodof the computer code CCFULL [17] and replace the vibra-tional coordinates ξ P and ξ T in the coupling potential V coup with the dynamical excitation operators ˆ O P andˆ O T . The coupling potential is then represented as V coup ( r, ˆ O P , ˆ O T ) = V C ( r, ˆ O P , ˆ O T ) + V N ( r, ˆ O P , ˆ O T ) , (3) V C ( r, ˆ O P , ˆ O T ) = R λ P P ˆ O P (2 λ P + 1) r λ P + 3 R λ T T ˆ O T (2 λ T + 1) r λ T ! × Z P Z T e r , (4) V N ( r, ˆ O P , ˆ O T ) = − V h (cid:16) [ r − R − ( R P ˆ O P + R T ˆ O T )] a (cid:17)i − V (0) N ( r ) . (5)Here, λ P and λ T denote the multipolarity of the vibra-tions in the projectile and the target nuclei, respectively.We have subtracted V (0) N ( r ) in Eq. (5) in order to avoidthe double counting.If we truncate the phonon space up to the triple phononstates (that is, n =0,1,2, and 3), the matrix elements ofthe excitation operator ˆ O in Eqs. (4) and (5) are givenby O nn ′ = 1 √ π β β √ β √ β √ β √ β (6)where β is the deformation parameter, that can be es-timated from a measured electric transition probabil-ity from the single phonon state ( n =1) to the groundstate ( n =0). We have assumed the harmonic oscillatormodel for the vibrations, where ǫ n in Eq. (2) is given by ǫ n = n ~ ω .The coupled-channels equations, Eq. (2), are solvedwith the scattering boundary condition for u n ( r ), u n ( r ) → i ( H ( − ) J ( k n r ) δ n,n i − r k i k n S Jn H (+) J ( k n r ) ) , ( r → ∞ ) (7)where S Jn is the nuclear S matrix. H ( − ) J ( kr ) and H ( − ) J ( kr ) are the incoming and the outgoing Coulombwave functions, respectively. The channel wave num-ber k n is given by p µ ( E − ǫ n ) / ~ , and k i = k n i = p µE/ ~ . The scattering angular distribution for thechannel n is then given by [18] dσ n d Ω = k n k i | f n ( θ ) | (8)with f n ( θ ) = X J e [ σ J ( E )+ σ J ( E − ǫ n )] r J + 14 π Y J ( θ ) × − iπ √ k i k n ( S Jn − δ n,n i ) + f C ( θ ) δ n,n i (9)where σ J ( E ) and f C ( θ ) are the the Coulomb phase shiftand the Coulomb scattering amplitude, respectively. Thedifferential quasi-elastic cross section is then calculatedto be dσ qel d Ω = X n dσ n d Ω (10)We will apply this formalism in the next sec-tion to analyze the quasi-elastic scattering data of Ti, Cr, Fe, Ni, and Zn+
Pb systems.
III. COMPARISON WITH EXPERIMENTALDATA : EFFECTS OF MULTI-PHONONEXCITATIONS
In this section, we present the results of our de-tailed coupled-channels analysis for quasi-elastic scatter-ing data of Ti, Cr, Fe, Ni, and Zn+
Pb systems[15, 16]. The calculations are performed with a version[19] of the coupled-channels code
CCFULL [17]. Noticethat the iso-centrifugal approximation employed in thiscode works well for quasi-elastic scattering at backwardangles [10]. In the code, the regular boundary conditionis imposed at the origin instead of the incoming waveboundary condition. We discuss the stability of the nu-merical calculations for the massive systems in AppendixA.The surface diffuseness of the real part of the nuclearpotential is taken to be a = 0 .
63 fm, as suggested byrecent studies on deep sub-barrier quasi-elastic and Mottscattering [20, 21, 22], while the radius parameter to be r = 1 .
22 fm for all the systems. Notice that a simi-lar value for a has been used also in the analysis of therecent experimental data for quasi-elastic scattering inthe Kr +
Pb system [14]. The depth parameter, V , is adjusted in order to reproduce the experimentalquasi-elastic cross sections for each system. The opti-mum values of the depth parameter and the resultantCoulomb barrier height are summarized in Table I. Asusually done, we use a short range imaginary potentialwith W = 30 MeV, r w = 1 . a w = 0 . R T = 1 . A / T and R P = 1 . A / P , respectively, inorder to be consistent with the deformation parameters[23, 24]. All the calculations shown below are performedat the scattering angle of θ c . m . = 170 ◦ . We plot thequasi-elastic cross sections and barrier distributions as afunction of the effective energy defined by [9, 10] E eff = 2 E sin( θ/ θ/ , (11) TABLE I: The depth parameter for the real part of the nu-clear potential for the Ti, Cr, Fe, Ni, and Zn+
Pbsystems. The radius and the diffuseness parameters are takento be r =1.22 fm and a =0.63 fm, respectively, for all the sys-tems. The resultant barrier height energy V B is also listed.System V (MeV) V B (MeV) Ti+
Pb 88.90 190.50 Cr+
Pb 91.70 205.50 Fe+
Pb 92.85 222.50 Ni+
Pb 95.10 236.25 Zn+
Pb 108.2 249.30TABLE II: The properties of the single phonon states includedin the present coupled-channels calculations. ~ ω and β are theexcitation energy and the dynamical deformation parameter,respectively.Nucleus I π ~ ω (MeV) β Pb 3 − .
614 0 . a Ti 2 + .
983 0 . b Cr 2 + .
834 0 . b Fe 2 + .
846 0 . b Ni 2 + .
346 0 . b Zn 2 + .
884 0 . ba taken from Ref. [23]. b taken from Ref. [24]. which takes into account the centrifugal energy. Wecalculate the quasi-elastic barrier distributions from thecross sections in a similar way as the one used to ob-tain the experimental barrier distributions [16]. Namely,we use the point difference formula with the energy stepof ∆ E =0.25 MeV and then smooth the resultant bar-rier distribution with the Gaussian function with the fullwidth at half maximum (FWHM) of 1.5 MeV. We havechecked that the shape of the barrier distribution doesnot change significantly even if we use a larger energystep for the point difference formula, e.g. ∆ E =0.5 MeV. A. Effect of double phonon excitations
Let us first discuss the effect of double octupole phononexcitations in the
Pb target. Such excitations havebeen shown to play a significant role in the sub-barrierfusion reaction between O and
Pb nuclei [25, 26].The dotted line in Fig. 1 shows the result of thecoupled-channels calculation for the Ti +
Pb systemobtained by taking into account the coupling to the singleoctupole phonon state in the target nucleus,
Pb, andthe single quadrupole phonon state in the projectile nu-cleus, Ti. The mutual excitations in the projectile andthe target nuclei are fully taken into account in this cal-culation as well as in all the other calculations presentedin this paper. Figs. 1(a) and 1(b) show the ratio of thequasi-elastic to the Rutherford cross sections, dσ qel /dσ R , d σ qe l / d σ R Ti+ Pb (a) Ti(1ph)-Pb(1ph)Ti(1ph)-Pb(2ph)Ti(2ph)-Pb(2ph)exp.0.000.050.100.15 175 180 185 190 195 200 205 D qe l [ M e V - ] E eff [MeV] (b) FIG. 1: Effects of multi-phonon excitations on the quasi-elastic scattering cross section (upper panel) and on the quasi-elastic barrier distribution (lower panel) for the Ti+
Pbsystem. The dotted line is the result of the coupled-channelscalculations including coupling to the one quadrupole phononstate in the projectile and the one octupole phonon states inthe target nucleus, while the solid line is obtained by includ-ing the coupling in addition to the two octupole phonon statein the target nucleus. The dashed line is the result of doublequadrupole phonon couplings in the projectile and the doubleoctupole phonon couplings in the target nucleus. The exper-imental data are taken from Ref. [16]. and the quasi-elastic barrier distribution, D qel , respec-tively. Although the overall width of the barrier distribu-tion is reproduced reasonably well with this calculation,the detailed structure is somewhat inconsistent with theexperimental data. The situation is similar even whenwe include the double quadrupole phonon state in theprojectile while keeping the single octupole phonon cou-pling in the target nucleus (not shown). We then in-vestigate the effect of the double octupole phonon cou-plings in the target nucleus. The solid and the dashedlines in Fig. 1 show the results with the single and thedouble quadrupole phonon excitations in the projectile,respectively, where as the double octupole phonon cou-pling in the target is included in both the calculations.The former calculation reproduces both the cross sections TABLE III: The value of χ for the quasi-elastic cross sec-tions for the Ti+
Pb system obtained with the coupled-channels calculations with various coupling schemes. The cou-pling schemes are denoted as [ n , n ], where n is the num-ber of quadrupole phonon excitation in the projectile nucleuswhile n the number of octupole phonon in the target nucleus.System [1,1] [1,2] [1,3] [2,2] Ti+
Pb 19.15 9.82 7.12 37.12 and the barrier distribution reasonably well, although thelatter calculation somehow worsens the agreement. Thisclearly suggests that the double octupole phonon exci-tations in the target nucleus is important in the quasi-elastic Ti+
Pb scattering. We summarize the χ value of our calculations in Table III.Since the coupling to the one quadrupole phonon statein the projectile and the two octupole phonon statesin the target reasonably well reproduce the experimen-tal quasi-elastic scattering data for the Ti+
Pb sys-tem, one may expect that the same coupling schemeaccounts for the experimental data for the other sys-tems, Cr, Fe, Ni and Zn+
Pb. The results of thecoupled-channels calculations with this coupling schemeis shown by the dashed line in Fig. 2. Figs. 2(a), (c), (e),and (g) are for the quasi-elastic cross sections for the Cr, Fe, Ni and Zn+
Pb systems, respectively,while Figs. 2(b), (d), (f), and (h) are for the quasi-elasticbarrier distributions. One can clearly see that these cal-culations underestimate the experimental cross sectionsat high energies, although the experimental barrier distri-butions themselves are reproduced reasonably well. Werepeat the same calculations by including the couplingup to the double quadrupole phonon states in the pro-jectile, in addition to the double octupole phonon statesin the target nucleus. These results are shown by thesolid in Fig. 2. The agreement with the experimentaldata is considerably improved, especially for the quasi-elastic cross sections. See Table IV for the χ values. It isthus evident that the coupling to the double quadrupolephonon states in the projectile is needed in order to ex-plain the experimental data for the Cr, Fe, Ni and Zn+
Pb reactions.The reason why the double quadrupole phonon cou-pling is not necessary for the Ti projectile while it isfor the heavier projectiles is not clear at this moment.This might reflect some ambiguity of the Monte Carloreaction simulation code
LINDA [27] which was used tosubtract the deep-inelastic component from the exper-imental yields at backward angles [16]. Clearly, a fur-ther investigation is still necessary concerning the effectof deep inelastic scattering on quasi-elastic scattering inmassive systems [13, 14, 15, 16]. d σ qe l / d σ R Cr+ Pb (a) Cr(1ph)-Pb(2ph)Cr(2ph)-Pb(2ph)exp.0.000.050.100.15 190 195 200 205 210 215 220 D qe l [ ( M e V - ] E eff [MeV] (b) Fe+ Pb (c) Fe(1ph)-Pb(2ph)Fe(2ph)-Pb(2ph)exp. 210 215 220 225 230 235 240E eff [MeV] (d)0.00.20.40.60.81.01.2 d σ qe l / d σ R Ni+ Pb (e) Ni(1ph)-Pb(2ph)Ni(2ph)-Pb(2ph)exp.0.000.050.100.15 220 225 230 235 240 245 250 255 D qe l [ M e V - ] E eff [MeV] (f) Zn+ Pb (g) Zn(1ph)-Pb(2ph)Zn(2ph)-Pb(2ph)exp. 235 240 245 250 255 260 265E eff [MeV] (h) FIG. 2: The quasi-elastic scattering cross sections ((a), (c), (e), and (g)) and the quasi-elastic barrier distributions ((b), (d),(f), and (h)) for the Cr, Fe, Ni and Zn+
Pb systems obtained with two coupling schemes as indicated in the insets. Thedashed line is obtained by including the one quadrupole phonon state in the projectile nuclei while the solid line is obtained withthe double phonon couplings. The double octupole phonon excitations in the target nucleus is included in all the calculations.The experimental data are taken from Ref. [16]. d σ qe l / d σ R Ti+ Pb (a) Ti(1ph)-Pb(2ph)Ti(1ph)-Pb(3ph)exp.0.000.050.100.15 175 180 185 190 195 200 205 D qe l [ M e V - ] E eff [MeV] (b) FIG. 3: Effects of triple phonon excitations on the quasi-elastic scattering cross section (upper panel) and on the quasi-elastic barrier distribution (lower panel) for the Ti+
Pbsystem. The dashed line is the result of the coupled-channelscalculations taking into account the coupling to the onephonon state in the projectile and the two phonon states inthe target nuclei. The solid line is obtained by including thecoupling to the one phonon state in the projectile and thethree phonon states in the target. The experimental data aretaken from Ref. [16].TABLE IV: Same as Table III, but for the Cr, Fe, Ni and Zn+
Pb systems.System [1,2] [1,3] [2,2] [2,3] Cr+
Pb 52.47 49.80 20.61 11.78 Fe+
Pb 28.46 28.36 10.44 10.28 Ni+
Pb 57.45 61.43 32.21 30.64 Zn+
Pb 26.52 24.81 11.36 6.87
B. Effect of triple phonon excitations
In the previous subsection, we have shown that thedouble octupole phonon excitations in the
Pb targetplay an important role in quasi-elastic scattering for thesystems considered in this paper. However, the calcu-lated quasi-elastic barrier distributions have a much more prominent peak than the experimental distribution athigh energies. Since it has been shown in Refs. [13, 14]that the triple octupole phonon excitations of the
Pbplay a significant role in the large-angle quasi-elastic scat-tering between Kr and
Pb nuclei, it is intriguing toinvestigate such effects in the present systems as well.The results of the coupled-channels calculations includ-ing the coupling to the triple octupole phonon states in
Pb for the Ti+
Pb reaction is presented in Fig. 3.The dashed line is the same as the solid line in Fig. 1, thatis the result of single phonon in Ti and double phononin
Pb. The solid line denotes the results of the triplephonon coupling in the target in addition to the singlephonon in the projectile. By including the triple octupolephonons in the target nucleus, the quasi-elastic cross sec-tions are improved slightly (see also Table III). On theother hand, one can see that the agreement for the bar-rier distribution with the experimental data is much moreimproved by the triple phonon coupling.The results for the other systems, Cr, Fe, Ni and Zn+
Pb reactions, are shown in Fig. 4. Figs. 4(a),(c), (e) and (g) are for the quasi-elastic cross sections,while Figs. 4(b), (d), (f) and (h) for the quasi-elasticbarrier distributions. Let us first discuss the calculationswith the single phonon excitation in the projectile. Thedotted line in the figures is obtained by taking the cou-pling to the single phonon state in the projectile and thetriple octupole phonon excitations in the target. This cal-culation underestimates the quasi-elastic cross sections athigh energies and the obtained barrier distribution is in-consistent with the experimental data. Therefore, theprevious results shown in Fig. 2 is not improved even ifthe triple phonon excitations in the target is taken intoaccount as long as only the single phonon excitation isconsidered for the projectile nucleus. The results withthe double phonon couplings in the projectile togetherwith the triple phonon excitations in the target are thenshown by the solid line in the figure. For comparison,we also show by the dashed line the results of the doublephonon excitations in both the projectile and the targetnuclei, which is the same as the solid line in Fig. 2.One can observe that the inclusion of the triple octupolephonon excitations in the
Pb somewhat improves theagreement between the calculations and the experimen-tal data for both the quasi-elastic cross sections and thebarrier distributions (see also Table IV).In Ref.[14], Ntshangase et al. reduced the couplingstrength of (3 − ) → (3 − ) states in Pb by a factorof (0 .
6) and that of (3 − ) → (3 − ) by (0 . in orderto explain the experimental barrier distribution for the Kr+
Pb reaction. In order to see whether such re-duction of the coupling strengths improves the agreementbetween the coupled-channels calculations and the exper-imental data for the present systems, we repeat the cal-culations by including those effects for the Zn+
Pbsystem. The results are shown in Fig. 5. The solid is ob-tained by reducing the coupling strengths as Ntshangase et al. did, while the dashed line is the same as the solid d σ qe l / d σ R Cr+ Pb (a) Cr(2ph)-Pb(2ph)Cr(1ph)-Pb(3ph)Cr(2ph)-Pb(3ph)exp.0.000.050.100.15 190 195 200 205 210 215 220 D qe l [ ( M e V - ] E eff [MeV] (b) Fe+ Pb (c) Fe(2ph)-Pb(2ph)Fe(1ph)-Pb(3ph)Fe(2ph)-Pb(3ph)exp. 210 215 220 225 230 235 240E eff [MeV] (d)0.00.20.40.60.81.01.2 d σ qe l / d σ R Ni+ Pb (e) Ni(2ph)-Pb(2ph)Ni(1ph)-Pb(3ph)Ni(2ph)-Pb(3ph)exp.0.000.050.100.15 220 225 230 235 240 245 250 255 D qe l [ M e V - ] E eff [MeV] (f) Zn+ Pb (g) Zn(2ph)-Pb(2ph)Zn(1ph)-Pb(3ph)Zn(2ph)-Pb(3ph)exp. 235 240 245 250 255 260 265E eff [MeV] (h) FIG. 4: Effects of triple phonon excitations on the quasi-elastic cross sections ((a), (c), (e), and (g)) and on the quasi-elasticbarrier distributions ((b), (d), (f), and (h)) for the Cr, Fe, Ni and Zn+
Pb systems. The dashed line is the same as thesolid line in Fig. 2 while the dotted line is the results of the calculations taking the coupling to the triple octupole phonon in thetarget and the one quadrupole phonon state in the projectile nucleus into account. The solid line is obtained by including thecoupling to the double quadrupole phonon states in the projectile and the triple otcupole phonon states in the target nucleus.The experimental data are taken from Ref. [16]. d σ qe l / d σ R Zn+ Pb (a) Zn(2ph)-Pb(3ph)Zn(2ph)-Pb(3ph-red.)exp.0.000.050.100.15 235 240 245 250 255 260 265 D qe l [ M e V - ] E eff [MeV] (b) FIG. 5: Effect of anharmonic octupole phonon excitationsin
Pb on (a) the quasi-elastic cross section and (b) thequasi-elastic barrier distribution for the Zn+
Pb reaction.The solid line is the coupled-channels calculation obtained byreducing the coupling strengths to multi-phonon states, whilethe dashed line denotes the results in the harmonic limit. Theexperimental data are taken from Ref. [16]. line in Figs. 4(g) and (h), that is obtained by assumingthe harmonic limit. In both cases, we take into accountthe coupling to the double quadrupole phonon excitationsin the projectile nucleus. One can see that the harmonicmodel leads to a better agreement with the experimen-tal data both for the cross sections and the barrier dis-tribution, as compared to the anharmonic calculation.The difference between Ref. [14] and the present calcula-tion concerning the role of anharmonicity may originatefrom the fact that Ref. [14] used a smaller value for R T (=1.06 A / T fm) and thus a larger value for β (=0.16). Inorder to clarify the role of anharmonicity of multi-phononexcitations in quasi-elastic scattering in massive systems,it would be required to take into account also the reori-entation terms [12, 28, 29]. It is beyond the scope of thispaper, and we will leave it for a future study. d σ qe l / d σ R Ni+ Pb (a) a = 0.63 fma = 1.0 fm a = 1.0 fm (shifted)exp.0.000.050.100.15 220 225 230 235 240 245 250 255 D qe l [ M e V - ] E eff [MeV] (b) FIG. 6: Comparison of the experimental data with thecoupled-channels calculations obtained using different val-ues of the surface diffuseness of the nuclear potential for Ni+
Pb reaction for (a) the quasi-elastic cross section and(b) the quasi-elastic barrier distribution. The solid and thedashed lines is obtained using the surface diffuseness of thenuclear potential a = 0 .
63 fm and a = 1 . .
40 MeV for the calculation using a = 1 . C. Surface diffuseness of the nuclear potential
We next discuss the dependence of the quasi-elasticscattering on the surface diffuseness parameter of the nu-clear potential. The standard value for the diffusenessparameter is around 0.63 fm [30, 31, 32]. Recently, sys-tematic studies on quasi-elastic scattering as well as Mottscattering at deep sub-barrier energies have revealed thatthe surface region of the nuclear potential is indeed con-sistent with the standard value of the surface diffusenessparameter [20, 21, 22]. On the other hand, it has beenknown for some time that the recent high precision dataof sub-barrier fusion cross sections require a larger valueof surface diffuseness parameter, ranging between 0.75and 1.5 fm [33]. Since the large-angle quasi-elastic scat-tering around the Coulomb barrier may probe both thesurface region and the inner part of the nuclear potential,it is interesting to study the sensitivity of quasi-elasticcross sections and barrier distributions to the surface dif-fuseness parameter.For this purpose, as an example, we repeat the coupled-channels calculation for the the Ni+
Pb reaction us-ing the nuclear potential with a =1.0 fm. We readjust thedepth and the radius parameters to be V = 160 .
70 MeVand r = 1 .
10 fm, respectively, so that the barrier heightremains the same as the one listed in Table I. We includethe coupling to the double phonon states in the projec-tile and the triple phonon states in the target. Fig. 6compares the results with a =0.63 fm (the solid line) tothe one with a =1.0 fm (the dashed line). One sees thatthe calculations with a =1.0 fm underestimate the quasi-elastic cross section, although the shape of barrier distri-bution itself is similar to the one obtained with a = 0 . a =1.0 fm as the one for thedashed line, but by changing the depth parameter V sothat the resultant barrier height is higher by 2.4 MeV.This calculation now reproduces the experimental quasi-elastic cross sections at energies larger than E c . m . = 230MeV reasonably well, but below this energy the crosssections are underestimated. Therefore, it seems dif-ficult to reproduce the experimental quasi-elastic crosssections with the diffuseness parameter of a =1.0 fm atenergies below and above the Coulomb barrier simulta-neously. We have checked that the situation is similar forthe other systems, Ti, Cr, Fe and Zn+
Pb. Thisresult clearly indicates that the standard value of surfacediffuseness, a = 0 .
63 fm, is preferred by the experimentalquasi-elastic data for the systems studied in this paper.
IV. SUMMARY
We have performed a detailed coupled-channels anal-ysis for large-angle quasi-elastic scattering of the Ti, Cr, Fe, Ni and Zn+
Pb systems, wheretheir experimental barrier distributions have been ex-tracted recently. Our coupled-channels calculations withmulti-phonon excitations in the colliding nuclei repro-duce the experimental quasi-elastic cross sections as wellas the barrier distributions, indicating clearly that thecoupled-channels approach still works even for massivesystems [8]. It was crucial to subtract properly the deep-inelastic components from the total backward-angle crosssections in order to reach these agreements between thecalculations and the experimental data.In more details, the calculation with the singlequadrupole phonon excitation in Ti and the triple oc-tupole phonon excitations in
Pb reproduces reason-ably well the experimental data for the Ti+
Pb sys-tem. On the other hand, for the Cr, Fe, Ni and Zn+
Pb systems, we found that the coupling to thedouble quadrupole phonon excisions in the projectile nu-cleus in addition to the coupling to the triple octupole phonon in the target nucleus seems to be needed to fit theexperimental data. These results suggest that the tripleoctupole phonon excitations in the
Pb nucleus playsan important role in describing the experimental data forthe quasi-elastic cross section and the quasi-elastic bar-rier distribution for the present massive systems. This isconsistent with the previous finding for the Kr +
Pbsystem [14].Although our calculations well reproduce the gross fea-tures of the experimental barrier distributions, higherprecision data are still required in order to study thedetailed structure of the barrier distributions, especiallythe role of multi octupole phonon states in
Pb. Fromthe theoretical side, a further detailed investigation willalso be necessary, taking into account the anharmonicityof the multi-phonon excitations.
Acknowledgments
This work was partly supported by The 21st CenturyCenter of Excellence Program “Exploring New Scienceby Bridging Particle-Matter Hierarchy” of Tohoku Uni-versity and partly by Monbukagakusho Scholarship andGrant-in-Aid for Scientific Research under the programnumber 19740115 from the Japanese Ministry of Educa-tion, Culture, Sports, Science and Technology.
APPENDIX A: NUMERICAL STABILIZATIONOF COUPLED-CHANNELS CALCULATION
In this Appendix, we discuss the problem of numer-ical instability of coupled-channels calculations and thestabilization methods which we employ in the presentcalculations.The coupled-channels equations (2) form a set of N second order coupled linear differential equations, where N is the dimension of the coupled-channels equations.These equations can be solved by generating N linearlyindependent solutions and taking a linear combinationof these N solutions so that the asymptotic boundarycondition, (7), as well as the regular boundary condi-tion at the origin, are satisfied. The linearly independentsolutions can be obtained by taking N different sets ofinitial conditions at r = 0. We denote these solutionsby φ nn i ( r ), where n refers to the channels while n i refersto a particular choice of the initial conditions. A simplechoice for the N initial conditions is to impose φ nm ( r ) → cr J +1 δ n,m , for r → , (A1)where c is an arbitrary number and J is the total an-gular momentum. With these boundary condition, the0coupled-channels equations for φ nm ( r ) given by (cid:20) − ~ µ d dr + J ( J + 1) ~ µr + V (0) N ( r ) + Z P Z T e r − E + ǫ n (cid:21) φ nm ( r ) + X n ′ V nn ′ ( r ) φ n ′ m ( r ) = 0 , (A2)are solved outwards up to a matching radius R max . Thewave functions u n ( r ) in Eq. (2) are then obtained as u n ( r ) = X m C m φ nm ( r ) , (A3)where the coefficients C m are determined so that theasymptotic boundary condition (7) is fulfilled.In the classical forbidden region, the scattering wavefunctions exponentially damp as the coordinate r de-creases. For the smaller energy, the damping is thestronger. Therefore, when the excitation energy ǫ n isfinite, the absolute value of the wave functions for eachchannel are different by order of magnitude in the clas-sical forbidden region, and thus the wave functions tendto be dominated by that of the channel which has thesmallest excitation energy. This easily destroys the lin-ear independence of the N numerical solutions φ nm , andcauses the numerical instability. This is a serious problemespecially when the coupling is strong, as in the massivesystems which we discuss in this paper.Several methods have been proposed in order to stabi-lize the numerical solution of coupled-channels equations[34, 35, 36, 37]. In the present calculations, we stabilizethe solutions by diagonalizing the wave function matrix φ nm at several points of r in order to recover the linearindependence. That is, at some radius r s , we computethe inverse of the matrix A nm = φ nm ( r s ), and define thenew set of wave functions,˜ φ nm ( r ) = X k φ nk ( r ) · ( A − ) km . (A4) The new wave functions ˜ φ obey similar coupled-channelsequations as Eq. (A2), with the boundary conditionsgiven by,˜ φ nm ( r s − h ) = X k φ nk ( r − h ) · ( A − ) km , (A5)˜ φ nm ( r s ) = δ n,m . (A6)Here, h is the step for the discretization of the radial co-ordinate, r . These coupled-channels equations are solvedoutwards from r s . The solutions φ can then be con-structed as φ = A · ˜ φ . We impose this stabilization proce-dure up to r = 15 fm with an interval of 1 fm. Althoughthis method is similar to those in Refs. [36, 37], ourmethod is much simpler to be implemented.This method is sufficient for intermediate heavy sys-tems, such as O +
Sm. For massive systems, how-ever, we still encounter a small numerical instability [38].In order to cure this problem, in addition to the stabi-lization method (A4), we also adopt two other methods,which are used in the computer code
FRESCO [39]. Thatis, we introduce two radii, R min and R cut . R min is the ra-dius from which the coupled-channels equations (A2) aresolved, i.e. , these equations are solved from r = R min in-stead of r = 0, by setting φ nm ( r ) = 0 for r ≤ R min . R cut is a cut-off radius for the coupling matrix, i.e. , the off-diagonal components of the coupling matrix V nn ′ ( r ) areset to be zero for r ≤ R cut . Both the procedures are jus-tified when the absorption is strong inside the Coulombbarrier, as in heavy-ion systems, and the results are insen-sitive to the particular choice of R min and R cut as long asthey are inside the Coulomb barrier. Typically, we take R min = 6 fm and R cut = 10 fm to obtain reasonable resultsfor the present systems (notice that the pocket and thebarrier appear at e.g., Ni+
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