Quan tum calculation of dip ole excitation in fusion reaction C
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Quan tum calculation of dip ole excitation in fusion reaction C

Simenel Ph Chomaz G de rance GANIL BP 55027F14076 Caen Ce dex 5 anc e Decem er 4 2000 The excitation of the gian dip ole resonance fusion is studied with NZ asymmetry in the en trance hannel The TDHF solution exhibits strong dip ole vibration whic c

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Quan tum calculation of dip ole excitation in fusion reaction C




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Quan tum calculation of dip ole excitation in fusion reaction C. Simenel, Ph. Chomaz, G. de rance G.A.N.I.L., B.P. 55027,F-14076 Caen Ce dex 5, anc e. (Decem er 4, 2000) The excitation of the gian dip ole resonance fusion is studied with N/Z asymmetry in the en trance hannel. The TDHF solution exhibits strong dip ole vibration whic can asso ciated with gian vibration along the main axis of uctuating prolate shap e. The consequences on the gamma-ra emission from hot comp ound uclei are discussed. Ordered collectiv motion are general prop ert of mesoscopic systems. In

metallic clusters, electron vibra- tions are plasmon excitations. In atomic uclei, oscilla- tions of protons against neutrons generate gian dip ole resonances [1,2]. The general to excite suc mo des is to use rapidly arying electromagnetic elds asso ciated with photons or generated fast electrically harged particles. The collectiv vibrations can also thermally excited as it as clearly demonstrated in the studies of the emission from hot uclei [3{ 6]. It has een re- cen tly prop osed that fusion reactions with N/Z asym- metric uclei ma lead to the excitation of dip ole mo de ecause of

the presence of net dip ole momen in the en- trance hannel [7,8]. The rst exp erimen tal indications on the ossible existence of suc new phenomenon ha een rep orted in [9] for fusion reactions and in [10] for deep inelastic collisions. Ho ev er, the real nature of suc vibration is still unclear oth from the exp erimen tal and from the theoretical oin of view. In particular only semiclassical approac hes or sc hematic mo dels ha een used to infer the prop erties of the generated dip ole mo de. In this letter, presen the rst quan tum cal- culation of suc mec hanism using the time

dep enden Hartree (TDHF) approac [11{ 14,16]. TDHF cor- resp onds to an indep enden propagation of eac single particle function in the mean eld generated the ensem ble of particles. It do es not incorp orate the dissipa- tion due to o-b dy in teraction [18{ 20], but tak es in to accoun one dy mec hanisms suc as Landau spreading and ev ap oration damping [21]. It fully tak es in to accoun the quan tal nature of the single particle dynamics whic is crucial at lo energy oth ecause of shell eects and of the dynamics. In the time dep enden Hartree-F (TDHF) ap- proac h, the ev

olution of the single particle densit matrix =1 is determined Liouville equa- tion, (1) where is the mean-eld Hamiltonian. ha used the co de built Bonc he and co ork ers with an eec- tiv Skyrme mean-eld and Ly parameters. FIG. 1. Densit plots, pro jected on the reaction plane, for the cen tral collision reaction 20 20 at MeV/u. Lines corre- sp onds to equidistan alues of the densit The eect of the isospin asymmmetry in the en trance hannel has een rst studied in the 20 20 fusion reactions at energies close to the Coulom barrier. Strong shell

eects are exp ected in these mirror-n uclei reactions leading to the N=Z 40 comp ound system. The densit plots obtained in the cen tral collisions at kinetic energy of eV er ucleon in the cen ter of mass frame is sho wn in Fig. 1. The system fuses and presen ts quadrup ole oscillation around slo wly damp ed deformation. Since 20 and 20 ha signican tly dieren er ratios (resp ectiv ely 1.5 and 0.67), the cen ter of mass of protons and neutrons do not coincide at the initial stage of the collision in con trast to what is obtained in the fusion of uclei with the same er

ratio. Let us rst start with head on head collisions along the axis. Adequate observ ables to study the collectiv mo- tion induced in this mec hanism are the dip ole momen and its conjugated quan tit is the net dis- tance et een protons and neutrons: where and are the proton and neutronís cen ters of mass co ordinates. Similarly where and are the momen ts of protons and neutrons. In Fig. is plotted as function of As time go es on, observ spiral in the collectiv phase space whic signals the presence of damp ed col- lectiv vibration. Indeed, it originates from oscillations in phase

quadrature of the conjugated dip ole ariables. In order to asso ciate the observ ed vibration with the gian dip ole resonance (GDR) in the comp osite ucleus ust study the collectiv vibration of 40 ucleus. The GDR asso ciated to the dip ole oscillation of 40 ground state has een excited an iso ector dip ole eld and follo ed using the TDHF approac h. The resulting
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momen is plotted as function of time in Fig. The erio of the observ ed oscillations is around 80 fm/c whic corresp onds to an energy 15 eV in go agreemen with skyrme RP calculations [20] and close to the exp

erimen tal alue 20 eV [15]. or the 40 formed fusion (Fig. it is around 150 fm/c. This large dierence can explained the deforma- tion of the fused system. Indeed, as can see in Fig. the comp ound ucleus relaxes its initial prolate elonga- tion along the axis of the collision with higher ypical time than the dip ole oscillationís erio d. The eraged alue of the observ ed quadrup ole deformation parameter dened 20 00 is around 23. FIG. 2. Time ev olution of the dip ole vibration. Dip ole momen Qd and its conjugate Pd are plotted in the phase space (a) and Pd is plotted in function

of time (b). Therefore, lo er mean energy is exp ected for this longitudinal collectiv motion GD as compared to the one sim ulated in spherical 40 GD GD (2) The energies of the GDR along the elongation axis, fullll this relation with 26 in go agreemen with the observ ed erage deformation. The dominan role of the deformation in the lo er- ing of the GDR energy can easily repro duced in the calculations constraining 40 to eep constan de- formation. mimic this situation ha erformed constrained Hartree calculation in an external quadrup ole eld. This external eld is ept

during all the dynamical ev olution so that the deformation do es not relax. time dip ole oscillation is induced dip ole ost. The resulting dip ole oscillations ha lo er oscillation erio d. Using the deformation param- eter 23 nd that this erio is close to the one observ ed in the fusion case and agrees with the phe- nomenological relation (2). FIG. 3. Time ev olution of the comp onen of the dip ole momen parallel to the deformation axis (a) and the comp onen erp endic- ular in the collision plane (b) for the reaction 20 20 with the impact parameter (solid line) and (dashed line). (c)

GDR in the 40 with spherical shap get deep er insigh in the dip ole oscillation ob- serv ed in fusion reactions ha analyzed the time ev o- lution of the erio d. rom eac oin on the collectiv tra jectory this quan tit can inferred from the time needed to reac the opp osite side of the observ ed spiral. The resulting ev olution is plotted in Fig. This erio presen ts oscillations to o. In other hand, calcu- lated the monop ole momen 00 and the quadrup ole mo- men 20 whic are presen ted in gure can see that those observ ables presen oscillations whic are al- most in phase with those in Fig.

This oin ts to ard ossible coupling et een the dip ole mo de and another mo de of vibration. The ev olutions of the monop ole and quadrup ole momen ts are ery similar. In particular, they presen the same oscillationís erio around 166 fm/c. Therefore, conclude that they originates from the same phenomenon, the vibration of the densit around prolate shap e. This oscillation mo dies the prop erties of the dip ole mo de in time dep enden In order to in estigate if this vibration is the origin of the observ ed eha vior of the dip ole erio can mo del the induced mo de coupling. consider

harmonic os- cillator with spring constan whic aries in time at frequency This is simple to tak in to accoun the fact that oscillations of the densit whic en ters in
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the mean eld oten tial of the TDHF equation, mo d- ify the dip ole restoring force. It is easy to understand that lo er densit as ell as in an elongated shap es induce eak er recall force. Th us, ariations of the den- sit proles in the TDHF equation can mo deled corresp onding ariation of the constan of rigidit in spring mo del. In suc mo del, the equation of motion ecomes the Mathieuís equation [1

cos )] (3) where is the pulsation without coupling and the pulsation of the densit yís oscillation while corresp onds to the magnitude of the induced frequency uctuations. ha computed the umeric solution of equation (3) with the ypical external frequency of the monop ole and quadrup ole oscillations. The bare frequency and the coupling strength ha een tuned in order to get the same oscillation frequency and ypical strength as in the full TDHF calculations. Indeed, ecause of the presence of the oscillating term, the observ ed frequency is dieren from the bare one. The umerical

solution presen ts oscil- lations with an oscillating erio whic ell repro duces our self consisten calculations with 15 and hanged factor 1.2 from the observ ed alue GD (see Fig. 4-a). rom this analysis it app ears that the observ ed dip ole motion corresp onds to gian vibration along the main axis of uctuating prolate shap e. The observ ed uctuation of the dip ole frequency is source of additionnal spreading of the resonance line shap e. FIG. 4. (a) Time eha viour of the dip ole erio (solid line) and its mo delisation with the Mathieuís equation (dashed line). (b) Time ev

olution of the monop ole (dashed line) and quadrup ole momen ts (solid line). The um er of excited phonons at the time of the fu- sion can estimated from the maximal elongation of the comp ound system whic is obtained when the fusing uclei are stopp ed for the rst time uclear forces. This maximal elongation max max is related to the um er of excited phonons the expres- sion max GD 2 (4) where GD 8MeV is the energy of the GDR along the axis and is the reduced mass of the neutron-proton system, eing the ucleon mass. It giv es 9. Ob viously this alue is an upp er limit ecause the mean

eld dynamics underestimate the con- sequences of the damping since dy excitation is not tak en in to accoun t. In the same as ref. [7,9,17,22], can compute the -deca probabilit er its mean alue in the equilibrium with the expression GD ev ap ev ap (5) where ev ap is the rate at whic the comp ound sys- tem deca (see ref. [17]), is the deca ying rate of the phonons and GD is the mean um er of excited phonons in the equilibrium. The latest can estimated GD GD where is the comp ound ucleus temp erature. or 40 at an excitation energy around 96 MeV, tak ev ap 76 eV eV and GD 6. It giv es

whic sho ws that the pre-equilibrium eects are only correction for this sys- tem while the alue for an en trance hannel quenc hing 0) of the GDR using the isospin symmetric reac- tion 20 20 ould lead to 9. It means the rst hance gamma emission will sligh tly enhanced ab out factor 20 %. This is in go agreemen with the alues rep orted in the litterature [9]. In order to compute the prop erties of the pre- equilibrium GDR in fusion shall not only consider the head on head collisions but also study non zero im- pact parameters. In particular the in terpla of dip ole vibration and

deformation can aected the rota- tion. dened new co ordinate system where is the deformation axis and is the rotation axis. or the head on head reaction studied efore, those frames are the same. In this case, for symmetry rea- sons the dip ole momen ts along the and axis are zero at ev ery time. If consider no nite the symmetry only forbids the vibration to ccur along ). In this case, the ampli- tude of the oscillations along sligh tly decreases with the impact parameter (Fig. dashed line) whereas an oscillation along app ears (Fig. dashed line). First can see that the

erio of this dip ole oscillation
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erp endicular to the deformation axis is lo er than one along the deformation axis. It is another consequence for the prolate deformation of the ucleus indeed the en- ergy of this excitation along should ob ey the relation GD GD (1 ). This relation giv es alue of the oscillationís erio of 67 fm/c whic is exactly the observ ed one on (Fig ). The oscillation along presen ts lo er amplitude than this along (appro x- imativ ely er cen in relativ in tensit y). This oscilla- tion along the axis results from eakly symmetry breaking due to the rotation

of the system. Indeed the hanging to the rotating frame leads to the Hamiltonian with non diagonal term J (6) where is the Hamiltonian in the rotating frame, is the angle et een the frames and is the generator of the rotations around ). The last term induces motion along the axis from dip ole vibration along ). This is indeed what is happ ening since initially the dip ole mo de is along the axis. If no lo ok in more details to the dep endance of the en trance hannel phonon excitation can see that the comp onen is sligh tly reduced and that the com- onen is of the order of few ercen ts

of the longitudinal mo de. These ariations comp ensate so that the net eect is almost no ariation of the phonon um er Therefore the conclusions reac hed for can ex- tended to the full range of impact parameters No ha to compare this with other energies. The lac of damping in TDHF do esnít enable us to go up in energy with this mass. So in estigate lo er energy 0.5 MeV er ucleon whic is the coulom barrier. In this case, the um er of exciting phonons is 1.0. This alue is lo er than one obtained at 1MeV/u ecause the maxim um dip ole momen decreases with decreasing en- ergy Therefore the

um er of pre-equilibrium phonons is lo er at lo er energy Ho ev er, at the same time due to the reduction of the temp erature the equilibrium um- er is also uc lo er so that the global pre-equilibrium con tribution to the sp ectrum is larger. ha also in estigated the role of the mass of the reactions partners to study if the pre-equilibrium eects surviv in hea vier comp ound systems. ha com- puted the 40 Ar 40 at MeV er ucleon in the cen- ter of mass system (sligh tly ab the fusion barrier). ha observ ed exactly the same phenomenology with strong excitation of pre-equilibrium GDR. The

mea- sured dip ole elongation is three times smaller than the one obtained for ligh ter uclei reaction. get 6. This comes from the fact that the reaction with small u- clei is less damp ed than the one in olving more ucleons. Therefore the pre-equilibrium GDR is also ery strong in the hea vier fusion reaction. In summary pre-equilibrium eects related to the N/Z asymmetry in the en trance hannel ha een for the rst time in estigated using the TDHF approac h. The rela- tion et een the mean oscillationís erio and the shap of the ucleus ha een established. Because of the de-

formation the erage dip ole frequency along the defor- mation axis is lo er than the usual GDR one ,but up- er in the erp endicular direction in the collision plane. In the last direction, the oscillation ccurs only for non zero impact parameter. The ev olution of the erio ha een mo deled Mathieuís spring mo del so that the link et een the observ ed phenomena and the gian dip ole resonance ha een established. This uctuation pla ys the role of an additionnal spreading of the GDR fre- quency The results sho ws that pre-equilibrium gamma emission are exp ected in the case of fusion

reaction with asymmetric N/Z uclei. an to thank aul Bonc he for pro viding his TDHF co de and Herv Moutarde for his help ab out the Ma- thieuís equation. Useful discussions with eter an Isac er and Denis Lacroix are ac kno wledged. [1] G. C. Baldwin and G. Klaib er, Ph ys. Rev. 71 (1947) [2] M. Goldhab er and E. eller, Ph ys. Rev. 74 1046 (1948) [3] D. M. Brink, Ph. D. thesis Univ ersit of Oxford (1955) Axel, ph ys. Rev. 126 671 (1962) [4] K. A. Sno er, Ann. Rev. Nucl. art. Sci. 36 545 (1986) [5] J. J. Gaardhoje, Nucl. Ph ys. A488 261c (1988) [6] A. Bracco et al, Ph ys. Rev. Lett. 62 2080

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(1995) [18] M. Gong, M. Tho ama and J. Randrup, Z. Ph ys. A335 331 (1990) [19] C. Y. ong and H. H. K. ang, Ph ys. Rev. Lett. 40 1070 (1978) [20] D. Lacroix, Chomaz and S. Ayik, Ph ys. Rev. C58 2154 (1998) [21] Ph. Chomaz, N. V. Giai and S. Stringari, Ph ys. Lett. B189 375 (1987) [22] F. Bortignon et al. Ph ys. Rev. Lett. 67 3360 (1991)