Fusion Reactions. V` (r) = V N (r) + V c (r) + }2` (` + 1) 2r 2 (1)

Transcription

1 Genera Considerations 1 Fusion Reactions 1 Genera Considerations The fusion of two nucei to form a heavier compound nuceon, in a very important reaction. Besides its a way to produce nucei with very high spins and accordingy to study the rapid rotation phenomena of nucei. At very ow energies nucear fusion is of paramount importance for stear energy production and nucear synthesis as originay proposed by H. Bethe [1]. Further, they are very sensitive to nucear structure when measured beow-the-couomb-barrier energies (to be expained in what foows). We sha emphasize the fusion of hao nucei, where break up e ects are important and aso the production, through fusion, of other type of exotic nucei, the super heavy eement (SHE). As a background to both subjects, we present beow a detaied account of a simpe picture of the fusion reactions namey, the tunneing through a oca one-dimensiona rea potentia barrier formed from the addition of an attractive nucear and a repusive Couomb potentias. We further assume that absorption into the fusion channes (compound nuceon) ensues in the inside region to the eft of the barrier after the tunneing through the barrier (from the right) has occurred. The one-dimensiona potentia for the ` + 1 partia wave is wave V` (r) = V N (r) + V c (r) + }` (` + 1) r (1) where V N (r) is the attractive nucear potentia takes to be e.g. of a Woods-Saxon form V o V N (r) = 1 + exp [(R R 0 ) =a], () V c (r) is the Couomb potentia given by Z1 Z V c (r) = e =r r R c Z 1 Z e =R c 3 (r=rc ) (3) r R c and the centrifuga potentia is given as usua by } ` (` + 1) =r : In gure 1(a) we show the e ective potentias for 64 Ni+ 64 Ni for three vaues of `. The Couomb barrier (peak) is ceary seen in the owest two curves and it becomes ess conspicuous for ` = 150 as the centrifuga repusion starts dominating. If the c.m. energy is 160 MeV, then a partia wave up to about ` = 100 wi contribute. In the region beow ` = 95 the tunneing probabiity is 1, for ` > 160 it is zero. Then one expects a form for this tunneing probabiity (fusion transmission coe cient) to be cose to a Fermi function (being unity for sma ` and going graduay to zero at arger ` vaues).

2 Genera Considerations Figure 1 (a) One dimensiona potentia of Eq. 1 for the 64 Ni+ 64 Ni system for servera ` vaues. The owest barrier is for ` = 0 (the bare barrier). The midde and top barriers are for ` = 100 and ` = 150, respectivey. (b) Cassica (on the eft) and quantum-mechanica (on the right) transmission probabiities for a one-dimensiona potentia barrier. The fusion cross-section is just the sum of a the T` is weighted with the factor ` + 1, i.e., ` 1X 1X F (E) = (` + 1) T` (E) = F (`; E) (4) k `=0 where F (`; E) is the partia fusion cross-section. The cacuation of T` (E) must rey on soving the appropriate Schrödinger equation with incoming wave boundary condition (IWBC). Here IWBC enforces the condition that one the system is in the pocket, it does not return An aternative approximate way of cacuating T` is through the use of the semicassica approximation [] T`E = f1 + expg [S` (E)] 1, (5) where the WKB integra S` (E) is given by S` (E) = r } Z r (`) r 1 (`) dr " `=0 V o (r) + } (` + 1=) r here r 1 (`) and r (`) are the outermost cassica turning points determined by setting V o + } (` + 1=) r (`) E # 1 (6) = E (7)

3 Genera Considerations 3 The width of the barrier is just r (`) r 1 (`). For energies that correspond to a trajectory cose, but beow, the top of the barrier, the barrier ooks ike an inverted paraboa. Then one expands V N +V c V o (r) 1 V o (r) = V B! (r R B ) (8) A more precise statement concerning Eq. 8 shoud invove the centrifuga potentia. However the incusion of this atter potentia ony changes V Bo to V B` V Bo + } ` + 1 =R B. The change in the curvature of the barrier, measured by!, is very sma and is thus negected. With the paraboic approximation for V o +} (` + 1=) =r 1 = V B`! (r RB ), the integra S` (E) can be performed straight forwardy and when inserted in T`, gives the we-known Hi-Wherer formua with T` = exp (V B` E) =}!, (9) V B` = V Bo + } (` + 1=) (10) RB The above Hi-Wherer expression for T`, Eq. 9, is vaid ony in the proximity of the barrier (E sighty above and sighty beow V B`). For energies that do not satisfy this condition the paraboic approximation is a bad one. One has to evauate S` (E) exacty. With the form of T` (E) determined (at east in the vicinity of the height of the Couomb barrier), one can now evauate the fusion cross-section F (E) = X 1 (` + 1) k 1 + exp V Bo + } (` + 1=) =RB E (11) `=0 Since many partia waves contribute to the sum, one may repace the sum by an integra. Ca (` + 1=) =, and write The integra yieds F (E) = k Z 1=4 1 d 1 + exp [(V Bo E) + } =RB ] (1) F (E) = }!R B E `n 1 + exp }! (E V Bo) The above expression for F (E) is referred to as the Wong formua [3]. At high energies, E V Bo, the exponentia dominates over unity and one nds the geometrica for F (E) F (E) = RB V Bo 1 (14) E (13)

4 Genera Considerations 4 The above form can aso be obtained in the cassica imit where T` (E) is a step function and accordingy by using! = 0. Before we concentrate our discussion on F at sub-barrier energies, we induge a bit on fusion at energies above the barrier. Here it is seen that after an initia rise, F (E) then reaches a maximum foowed by drop. The drop in F (E) is attributed to the competing deep ineastic processes. To account for such a structure in F, one can mutipy T` by the fusion probabiity [4]. Thus T` takes into account the tunneing into the strong absorption radius, R sa, which is situated sighty to the eft of the barrier. Part of the ux that reaches R sa woud fuse whie the other part is ost to deep ineastic processes. Accordingy F (E) = k 1 X `=0 (` + 1) T` (E) P (F ) ` (E) (15) Since T` (E) contains reference to the barrier radius R Bo ony, P (F ) ` (E) must contain reference to R sa R critica = R c. The sum over ` now extends to `c (R c ), the critica anguar momentum associated with R c. The cacuation of F (E) using the integra form, Eq. 1, with an upper imit in the integra c (`c + 1=), can be done easiy and one obtains the Gas-Mose formua [4] F = }!R B E `n 1 + exp }! (V Bo E) (16) " exp " }! } (`c+1=) R c!# # 9 1 = + exp }! (V Bo E) ; (17) Caing E V Bo = } `B + 1 =R B, the above expression can be reduced further into `n ; (18) F = ~!R B E ~! ~ R B exp (`B ) + 1 exp ( (`B `c)) + 1 Ceary, if F of Eq. 13 is considered very cose to the tota reaction crosssection, since periphera processes have much smaer cross-sections than F at ower enegies, Eq. 18, shows ceary that F < R. It has become common to ca the region where F R as Region I whie that where F < R as Region II. In this atter region, deep ineastic coisions DIC (where a arge amount of reative energy is converted into interna excitation of the two fragments) constitute a considerabe part of R. Figure??(a) shows the data of systems exhibiting Regions I and II. (19)

5 Genera Considerations 5 Figure (a) Fusion data for severa systems exhibiting regions I and II. See text for detais. (b) The maximum vaues of F (in barns) for severa systems. The theoretica cacuation are within the statistica yrast ine mode of reference 8. A possibe physica interpretation for `cr, besides the one reated to the competition between fusion and DIC, may reside in the compound nuceus formed in the process. The yrast ine mode states that the anguar momentum imparted to the compound nuceus (CN) can be associated with the yrast ine which is the maximum anguar momentum for a given excitation energy. Since the excitation energy of the CN is E c:m: + Q where Q is the Q-vaue of the fusion A 1 + A! (A 1 + A ), then the yrast ine is E c:m: + Q = ~ J y (J y + 1) (0) = where = is the moment of inertia of the CN. For a rigid spherica body, = = MRCM =5: Then J y = =E c:m: ~ 1 + Q E c:m: One then makes the identi cation `cr = J y. at high energies, F ' ~ `cr = = 1 + Q E c:m: E c:m: A variance of the CN mode is the statistica Yrast Line Mode of Lee at a. [5], which says that part of the excitation energy E c:m: + Q is statistica (therma) in (1) ()

6 Sub-Barrier Fusion 6 nature. Caing this energy Q one has ~ J 0 y J 0 y + 1 and thus = F = = + Q = E c:m: + Q (3) 1 + Q Q E c:m: The above expression can account for a wide range of experimenta data if Q = 10MeV and r o = 1:fm, where r o is the radius parameter that enters in the cacuation of the moment of inertia =. An exampe of how the above formua works is shown in Figure??(b) which exhibits the maximum vaue of F as a function of the system (Q-vaue), obtained with Q = 5 MeV. Sub-Barrier Fusion At energies cose or beow the ` = 0 barrier (the Couomb barrier), the coision time become onger and speci c e ects of the structure of the participating nucei in the fusion process become important. In such cases the one-dimensiona barrier penetration mode does not work anymore and one has to resort to couped channes treatments. A case which iustrates this e ect is suppied by the fusion of ony 16 O with the we deformed 154 Sm target [6]. Here, since the rotationa band is we formed in 154 Sm, one may freeze the rotation axis and perform a one-dimensiona cacuation of the type described before. An average over a orientations woud then suppy F, Z F (E c:m: ) = d F (E c:m: ; ) (5) where represents the soid ange that describes the direction of the rotation axis of the assumed rigid rotor 154 Sm. One can easiy convince onesef that the equivaent sphere F of Eq. 4, is arger than F cacuated for a spherica 154 Sm: The sub-barrier enhancement of F is found to occur in many systems. The cacuation of F according to Eq. 4 assumes an in nite moment of inertia of the rotor (degenerate or sudden imit). A more reaistic way of cacuating F which takes into account the energy oss (Q-vaue) must rey on couped channes theory. Another simpe mode that can be evauated anayticay is that of a twoeve system. Ignoring the anguar momentum of the excited state, and its excitation energy one has the two-couped equations ~ d E dr + V (r) 1 = F 1 ~ d E dr + V (r) = F 1 (6) (4)

7 Sub-Barrier Fusion 7 If the couping F 1 = F 1 = F is taken to be constant, one can sove for F, by diagonaizing the above two equations F = 1 ( F (V B + F ) + F (B B F )), (7) which is aways arge than F (V B ). The incusion of the non-zero excitation energy can be done without great di cuty. The resut of diagonaization gives for F the foowing F = A A (8) where F A = 4F + Q Q p 4F + Q = F (V + ) = 1 Q p Q + 4F when Q = 0, one recovers Eq. 7. The above resuts can be extended to severa channes. One introduces the eigen channes jci that diagonaize the many-channes Schrödinger equation. The eigen vaues are denoted by c. Then (9) F = X C jhcj0ij T C (E; V + c ) (30) where j0i is the entrance channe. For more detais on sub-barrier enhancement of fusion see, e.g., Ref. [7]. An interesting observation was made in Ref.. [8] and ater eaborated upon in Ref. [9] concerning the extraction of the barrier or eigen barriers directy from the data. It can be seen from the form of T` given by Hi-Wheeer, Eq. 9, that T` (E) may be written as T o (E `` + 1)~ = R ). Therefore it is not surprising that the approximation T` (E) ' T o (E) ` (` + 1~ ) = R (E), with R (E) being a sowy varying function of E, may tunr out to be reasonaby accurate. Then, with ` + 1= = p` (` + 1) and E E = R (E) F = k 1 X `=0 = R (E) E (` + 1) T` (E) ' k Z E 1 Z 1 0 d T (; E) de0 T o (E0) : (31) Therefore the second derivative d (E F (E)) =de can be directy reated to the rst derivative dt o (E) =de, up to corrections coming from the energy dependence

8 Sub-Barrier Fusion 8 Figure 3 (a) Cassica (on the eft) and quantum-mechanica (on the right) transmission probabiities for a two-channe couping. V 0 is the height of the one-dimensiona potentia barrier couped to these channes. (b) Fusion cross section and barrier distribution for the 16 O+ 154 Sm system by Leiht et a. [11]. of R (E) d T o (E) de 1 d dr R (E) de [E F (E)] + de In gure 3(a) we show an exampe of dt o =de for the one-dimensiona and the two-channe barrier tunneing probem. The actua data, e.g. 16 O Sm, show a cear deviation from the one-dimensiona case whendt =de is examined, g. 3(b). Contrary to the sub-barrier fusion enhancement found in ight heavy systems, the fusion of very heavy systems of the type empoyed in the production of superheavy eements (SHE), shows hindrance when compared to the simpe one-dimensiona barrier penetration mode. The incident energy has to be much higher than the fusion barrier which accounts for fusion of ighter systems, in order for heavy eement production to proceed. One needs an extra push, as proposed by Bjornhom and Swiatechi [10]. This extra energy needed for fusion to occur comes about from the fact that the ssion barrier for massive systems is ocated inside the potentia barrier in the entrance channe. Ref. [10] (BS) introduced the extra push concept; the energy needed to overcome the sadde point in the potentia energy surface under the constraint of mass asymmetry, and the extra extra push which is the energy needed to carry the system beyond (inside) the ssion sadde point. (3)

9 Fusion of Hao Nucei 9 3 Fusion of Hao Nucei We have seen that at near or at sub-barrier energies the fusion cross section for stabe nucei exhibits enhancement when compared to the one-dimensiona barrier penetration mode. Here we extend the discussion to neutron-rich and proton-rich nucei. As has been aready emphasized, the case of hao nucei is one that observes a specia attention. In these oosey bound systems, one has two competing e ects. The presence of the hao, namey an extended matter density, tends to ower the Couomb barrier then enhances the sub-barrier fusion. On the other hand the ow Q-vaue for break-up impies a strong couping to this channe, which by its nature, woud resut in a reduction of sub-barrier fusion. In the foowing we give an account of the work done on these systems and mention the scarce avaiabe data. In cases where the hao nuceus has a bound existed state, the enhancement shoud increase. The overa e ect on F can ony be assessed with a compete couped channes cacuation. A way of discussing fusion of oosey bound systems subject to break-up is a couped channe mode invoving, at east, the entrance channe and the break-up channe. The break-up channe here invoves eastic break-up ony, since ineastic break-up is a ux that is di cut to recover in a dynamic poarization framework. The recent avaiabiity of radioactive beams has made possibe to study fusion reaction reactions invoving unstabe nucei. Such reactions are important in processes of astrophysica interest, as we as in the search for superheavy eements. The main new ingredient in fusion reactions induced by unstabe projecties is the strong in uence of the breakup channe. In the case of not too unstabe projecties, the e ect of this channe in the fusion cross section at ow energies is, as in the case of stabe beams, to enhance it. At high energies, however, the situation is quaitativey di erent from the case where ony stabe nucei are invoved. The contribution from the breakup channe to the fusion reaction is strongy in uenced by the ow probabiity that a fragments are captured. Thus, in this case, the fusion cross section is partitioned into a compete and one or more incompete fusion contributions [1]. The introduction of the breakup channe into a couped channes cacuation is by no means trivia. The di cuty ies in the fact that this channe ies in the continuum, and invoves at east, three body system. This probem has been addressed by severa authors, using di erent approaches. Severa recent experiments invoving fusion of neutron rich 6 He and proton rich 17 F with heavy targets have been performed with the purpose of exporing these theoretica proposas [13, 14, 15]. A of these measurments, however, exihibit data for the summed fusion pus incompete fusion cross sections. As, such, the theoretica modes proposed for compete fusion can not be presenty tested in comparison with the data. To get an idea of the type of data now avaiabe,we exhibit in gures 4(a and b), and 5(a) the resuts obtained in Refs. [13], [14] and [15] for the systems 6 He 38 U, 6 He+ 09 Bi and 17 F+ 08 Pb, respectivey. In Refs. [1, 16, 17, 18, 19], the couped-channe probem is simpi ed by the introduction of the poarization potentias arising from the couping with the breakup channe [0, 1]. In Refs. [, 3, 4], the couped-channe probem is soved directy

10 Couped channe equations and poarization potentias 10 Figure 4 (a) Fusion pus incompete fusion cross section for the system 6 He+ 38 U from reference [13]. (b) Same as gure 7 for the system 6 He+ 09 Bi from reference [14]. within di erent approximation, ranging from the schematic mode of Ref. [] to the huge numerica cacuation Ref. [4], performed through continuum discretization. The poarization potentia approach of Refs. [1, 16, 17, 18, 19] has the advantage of eading to simpe expressions, which can easiy be used in data anaysis[5]. However, it empoys severa approximations which were not thoroughy tested. These approximations can be grouped in two cathegories. In the rst are those used in the derivation of the poarization potentias. In the second are the semicassica approximations for fusion and breakup coe cients, used in cacuations of the cross sections. These coe cients are written in terms of barrier penetration factors and surviva probabiities, which are evauated within the WKB approximation. In the next section we ascertain the quaity of the approximations for the transmission coe cients. Approximations in the derivation of the poarization potentia wi be aso be presented. For our purposes, we consider a case where a compete quantum mechanica cacuation is feasibe and compare exact and approximated cross sections. We study the 11 Li + 1 C coision, using typica optica and poarization potentias. For simpicity, our poarization potentia has no anguar momentum or energy dependence and the range is given by the 11 Li breakup threshod energy. The strength is consistent with that found in Ref. [0] for the most reevant partia waves in near-barrier fusion. 4 Couped channe equations and poarization potentias In a standard couped channes cacuation, the system is described through the distance between centers of projectie and target, r, and a set of intrinsic coordinates,

11 Couped channe equations and poarization potentias 11 Figure 5 (a) Fusion cross section for the 17 F+ 08 Pb system from reference [15]. (b) Hi-Wheeer and WKB approximations to the fusion cross section. The vertica arrow indicates the position of the Couomb barrier. See text for further detais., that describe the interna degrees of freedom of one of the nucei, e.g. the target. These coordinates are associated to an intrinsic Hamitonian h and its eigenfunction set, h () = () ; (33) where Z The system Hamitonian may then be written as () ()d = ; : (34) H = T + U opt + h + v(r; ) : (35) Above, T is the kinetic energy of the reative motion, U opt is the optica potentia and v(r; ) is the interaction couping intrinsic and coision degrees of freedom. The optica potentia, which is diagona in channe space, accounts for the average interaction between projectie and target. Usuay the soution of Schrödinger s equation H (r; ) = E (r; ) ; (36) where E is the coision energy in the center of mass frame, is expanded as (r; ) = X (r) () ; (37)

12 Couped channe equations and poarization potentias 1 where (r) describes the reative motion in channe. Substituting this expansion in Eq. (36) we obtain the couped channes equations (see e.g. Ref. [6]), (E H ) (r) = X V (r) (r) : (38) Above, E = E and H = T + U opt (r); where U opt V opt i W opt (39) is the optica potentia in channe. The imaginary parts have the purpose of accounting for the ux ost to channes negected in the expansion of Eq.(37). The channe couping potentias, in Eq.(38) are given by Z V (r) = d () v(r; ) () : (40) A consequence of the non-hermitian nature of H (see Eq.(39)) is that the continuity equation breaks down. This can be checked foowing the usua procedure to derive the continuity equation. For each ; we evauate (r) [Eq:(38)] [Eq:(38)] (r) and then sum the resuts. Assuming that V is hermitian, we obtain r X j = X W opt (r) j ~ (r)j 6= 0 Integrating the above equation inside a arge sphere with radius arger than the interaction range and using the de nition of the absorption cross section, we obtain the usefu reation[7] a = k X h E j W k i : (41) 4.1 Poarization potentias In some couped channe probems, it occurs that one is ony interested in the eastic wave function. One exampe is the study of compete fusion in coisions invoving nucei far from stabiity, where the breakup threshod is very ow. An extreme exampe is 11 Li, which has no bound excited state. In such cases, the couped channe probem invoves ony the eastic and the breakup channes. Since the breakup channes contain at east three fragments, their contribution to compete fusion is expected to be negigibe. Therefore, ony the eastic wave function is required for the cacuation of the compete fusion and breakup cross sections. In such cases, the poarization potentia approach becomes very convenient. It consists of repacing the couped channe equations by a singe Schrödinger equation for the eastic state. This equation contains a poarization term, U po, added to the optica potentia and its soution is identica to the eastic wave function obtained from the couped channe equations. According to Feshbach [8], the poarization potentia

13 Couped channe equations and poarization potentias 13 is obtained through eimination of the couped channe equations for excited states and it is given by U po = ( 0 jp V QG (+) QQ QV P j 0) : (4) Above, P = j 0 )( 0 j is the projector on the eastic channe, Q = 1 P = P 6=0 j )( j, and the propagator G (+) QQ is de ned as The wave function is then obtained by soving G (+) QQ = 1 E QH 0 Q + i : (43) (E H 0 U po )j 0 i = 0 ; (44) which, in the position representation is written E T U opt (r) Z (r) U po (r; r 0 ) (r 0 )d 3 r 0 = 0 ; (45) where U po (r; r 0 ) is the nonoca potentia U po (r; r 0 ) = X V 0 (r) G (+) (E ; r; r 0 ) V 0 (r 0 ) : (46) In principe, evauating the poarization potentia is neary as hard as soving the couped channe equations. However, for practica purposes it is repaced by triviay equivaent oca potentias, which are cacuated with approximations[0, 1]. 4. Fusion and breakup cross sections With the introduction of the poarization potentia, any ux going away from the eastic channe is treated as absorption. The sum in Eq. (41) is then reduced to a singe term, the one with = 0: The imaginary part of the potentia is (henceforth we drop the super uous index, since ony the eastic channe appears), the absorption cross section can be spit as W = W opt + W po ; (47) a = F + bu : (48) Above, F = k E Z d 3 r W opt (r) j (r)j (49) is identi ed with absorption through compete fusion and bu = k Z d 3 r W po (r) j (r)j (50) E

14 Couped channe equations and poarization potentias 14 corresponds to the oss of ux through the breakup channe. It incudes the breakup cross section and aso a cross section for absorption in the breakup channes, probaby incompete fusion. However, since for weaky bound nucei the range of W po is much arger than that of W opt ; we negect this contribution and use the notaton bu in Eq. (50). It is usefu to consider the expansion in partia waves of the wavefunction, = X ;m u (k; r) Y m (; ') ; (51) r where k = p E=~ and the u (k; r) are soutions of the radia equation, ~ d ( + 1) u dr r (k; r) + U opt (r) u (k; r) = E u (k; r) ; (5) normaized such that u (k; r! 1) = i h i H ( ) (kr) S H (+) (kr) : (53) Using the partia wave expansion in Eq.(49), the fusion cross section may be rewritten as F = X ( + 1)T F k ; (54) where the transmission coe cient is given by T F = 1 js j = 4k E Z 1 Proceeding simiary with Eq. (50), we get with 4.3 Approximations T bu 0 dr W opt (r)ku (k; r)j :r (55) bu = X ( + 1)T bu k ; (56) = 4k E Z 1 0 dr W po (r)ju (k; r)j : (57) In what foows, we study di erent approximations to the coe cients T F and T bu : In order to x ideas, we consider a 11 Li beam incident on a 1 C target, using an optica potentia U opt = V opt i W opt parameterized in the standard way: V opt (r) = V N (r) + V C (r) ; (58)

15 Couped channe equations and poarization potentias 15 with the nucear part given by and the Couomb one by V opt (r) = V opt exp [(r R r ) =a r ] ; (59) V Cou (r) = Z p Z t e =r; for r > R C (60) r = Z p Z t e =R C 3 ; for r R C : Above, Z p ; A p (Z t ; A t ) are the atomic and mass numbers of projectie (target), e the eectron charge, R C the radius of the nucear charge distribution, and R r;i = rr;i 0 Ap 1=3 + A 1=3 t : The imaginary part is simiary parameterized as W opt (r) = We take the foowing parameter vaues: R C W opt exp [(r R i ) =a i ] : (61) V opt 0 = 60 MeV; r 0 r = 1:5 fm; a 0 r = 0:60 fm ; (6) W opt 0 = 60 MeV; r 0 i = 1:00 fm; a 0 i = 0:60 fm : (63) Note that, since W opt corresponds excusivey to short range fusion absorption, ri 0 is appreciaby smaer than rr: 0 In order to review the standard approximations in the optica potentia cacuations, we initiay disconsider the breakup channes. In the absence of breakup, the imaginary part of the nucear potentia has a short range, and therefore fusion may be approximatey described through an in nitey absorbing imaginary potentia with a we de ned radius R F. In this case T F may be estimated by T, the transmission coe cient through the e ective potentia V (r) = V opt (r) + ~ ( + 1) : (64) r If one approximates the region around the maximum of V by a paraboa, then one obtains the Hi-Wheeer expression for T F [] T F 1 T T HW B E = 1 + exp ; (65) ~!

16 Couped channe equations and poarization potentias 16 Figure 6 (a) Fusion cross section with di erent contributions of the poarization potentia. See text for detais. (b) Fusion cross sections obtained with di erent approximations empoyed in previous pubications. The soid ine indicates exact quantum mechanica cacuations and the remaining ones are obtained with surviva probabiity approximation (Eq. (75)). The soid circes were obtained with Rutherford trajectories whiethe starts were obtained with cassica trajectories taking into account both the Couomb and the nucear potentias. where R B is the position of this maximum, B its vaue, and! the curvature of V at r = R B,! 1= ~ d V (r) ~! = : (66) dr R B In Fig. 5(b), we show an exampe of a cross section cacuated within the Hi- Wheeer approximation (dashed ine) compared with the exact quantum mechanica cacuation (fu circes). One notices that the approximation is exceent at energies above the Couomb barrier, E > V B B =0, but worsens rapidy for E << V B. The probem at ow energies may be improved using the WKB approximation. The transmission factor is then given by T exp( ) (67) where Z rout = Im k(r)dr : (68) r in

17 Couped channe equations and poarization potentias 17 Above, k(r) = 1 ~ p [E V (r)] (69) and r in e r out are the inner and outer cassica turning points for the potentia V, determined through the condition V (r in(out) ) = E : However, this approximation is not good at energies E B ; for E = B it yieds T = 1 instead of the correct vaue T = 1=, and even worse, it does not predict re ections above the barrier. Improvement is obtained by substituting the approximation of Eq. (67) by Kembe s expression [9] beow the barrier whie keeping Hi-Wheeer s approximation above it, T = (1 + e ) 1 (E < B ); T = T HW (E B ): (70) This appproximation has been empoyed in Ref. [1]; it is equivaent to empoying the Hi-Wheeer formua for a energies, abeit with the modi cation ~! (B E)! ; for E > B : The cross section obtained within this approximation is depicted in Fig. 5(b) (soid ine). We see that it reproduces the fu quantum cacuations for a coision energies. Let us now consider the incusion of the breakup channes. As we have seen, this may be done through the introduction of an appropriate poarization potentia. Such potentias were studied in Refs. [17, 18], for pure nucear couping, and in [19, 1] for the eectromagnetic couping. In [17, 18] ony the imaginary part of the poarization potentia was cacuated. Since the rea part of the poarization potentia reduces the height of the potentia barrier, this e ect was simuated by a shift in the coision energy in the cacuation of T. Namey, T (E)! T (E + E); E = V po (R B ) : As we wi see, the rea part of the poarization potentia pays a very important roe at energies beow the Couomb barrier. In the case of 11 Li + 1 C, the breakup process is dominated by the nucear couping. Therefore we write W po (r) = W po 0 (; E CM ) 1 + exp [(r R po ) =] ; (71) where R po may be approximated by the optica potentia radius, and the di useness is given in terms of the breakup threshod energy B bu as 1= bu B bu = : (7) ~ Above, bu is the reduced mass of the fragments produced in the breakup process. In the case of 11 Li, B bu = 0: MeV and thus = 6:6 fm.

18 Couped channe equations and poarization potentias 18 The strength of the poarization potentia varies with and E CM, and, for the partia waves reevant to the fusion process, is of the order of 1 MeV in the region around r R po. Since we are not concerned with its derivation, but with the approximations empoyed in the determination of the cross section, et us adopt the constant vaue W po 0 (; E) W po 0 = :0 MeV : (73) Since the rea part of the poarization pays a very important roe at energies beow the Couomb barrier, we sha incude it here. In the cacuations of Ref. [1] the rea and imaginary parts of the poarization potentia have quaitativey the same strengths. For simpicity we then take them to be equa, i.e. V po 0 (; E) V po 0 = :0 MeV : (74) The e ect of the rea and imaginary parts of the poarization potentia are shown in Fig. 6(a). As it coud be expected, the rea part eads to a substantia increase in the fusion cross section, most evident at energies beow the Couomb barrier. On the other hand, the imaginary part reduces the cross section both above and beow the barrier. When both the rea and imaginary parts are incuded, there is a competition between the e ects of the rea and imaginary parts. With the poarization strength vaues considered above, suppression dominates above the barrier and enhancement beow it. This situation was aso encountered in the couped channes cacuations of Refs. [3] and [4]. The presence of a ong-ranged absorption requires the introduction of modi - cations in the approximations to T F. Now the ux that reaches the strong absorption region is attenuated not ony because of the re ection at the barrier, but aso because of its absorption into the breakup channe. In Ref. [16] it was proposed the approximation T F T (E + E) P surv ; (75) where T (E +E) is the WKB transmission factor (Eq. (67)) evauated at the energy E + E and P surv is the breakup surviva probabiity. Within the WKB approximation we may take Z W P surv po (r) = exp ~ v (r) dr ; (76) where v (r) is the oca radia veocity aong a cassica trajectory with anguar momentum ~. A more forma justi cation for Eq (75), based on a WKB cacuation with three turning points was presented in Ref. [18]. This approximation is consistent with the resuts of Fig..6(a) The enhancement due to V po is incuded in T whie the suppression arising from W po is contained in P surv : In order to estimate P surv one needs to de ne the cassica trajectories to be empoyed in the cacuation. In Ref. [16] pure Rutherford trajectories were considered, negecting the nucear potentia di ractive e ects. These trajectories present a singe

19 Couped channe equations and poarization potentias 19 turning point. The corresponding fusion cross section is shown in Fig. 6(b) as a thin ine with soid circes. This gure aso depicts the fu quantum mechanica resuts (thick sodid ine). We see that athough the approximation obtained with the Rutherford trajectory is reasonabe at high energies, it breaks down at energies cose and beow the Couomb barrier (V B = :67 MeV). The incusion of the nucear potentia in the trajectory cacuations improves consideraby the resuts (thin ine with stars). In this case we may have, depending on the partia wave and coision energy, one or three turning points. The treatment with three turning points is not accurate in the region around the Couomb barrier, and that is the reason why there are arge deviations in the approximated fusion cross section. Later we wi show how one may improve this approximation, but et us rst brie y consider the breakup cross section. In Ref. [16] the breakup was cacuated by considering that T bu = 1 P surv : (77) This approximation is based on the notion that T bu corresponds to the probabiity of non-surviva to the breakup process. The resuts depend strongy on the cassica trajectory considered. In Fig. 7(a) we compare the exact quantum mechanica breakup cross section to the ones obtained using Eq. (77) with di erent trajectories. The resuts are far from satisfactory. In particuar, when the nucear potentia is incuded in the trajectory cacuations the ow energy breakup cross section has a competey wrong behavior. The reason for this discrepancy has been discussed by Ref. [18] and wi be considered in further detai ater in this section. Let us now deveop an improved WKB approximations for T F and T bu. In coe cients in a di erent order to expain them, it wi be usefu to rewrite the T F way. In the WKB approximation, the radia wave funcions with incoming ( outgoing (+) boundary conditions are given by u () (r) = p A Z exp i k(r) ) and dr k(r) ; (78) where k(r) = 1 ~ s E U opt (r) ~ ( + 1) r U po (r) : (79) The vaue of T F is given by the ratio between the probabiity density current that reaches the strong absorption region, j ( ) (r = R F ), to the incident one j ( ) (r = 1), where the radia currents are j () (r) = ~ i " u () (r)! (r) dr du () u () (r)! # (r) : (80) dr du ()

20 Couped channe equations and poarization potentias 0 Figure 7 (a) Exact breakup cross section (soid ine) and and cross sections approximated by Eq. (77). The soid circes were obtained with pure Rutherford trajectories whie the stars takes into account nucear potentia e ects on the trajectory. (b) Exact compete fusion cross section (soid circes) compared to the od approximation, depicted aso in Fig. 6(b) (stars), and with the improved WKB approximation (soid ine). See text for more detais. From Eqs.(78) to (80), we obtain T F = j( ) (r = R F ) j ( ) (r = 1) exp ; (81) where Z 1 = Im dr k(r) : (8) R F If one does not incude the poarization potentia, the integrand in the equation above is rea on the whoe cassicay aowed region (note that W opt (r > R F ) = 0). In this way, ony the cassicay forbidden region contributes to attenuate the current that reaches the fusion region (r < R F ), i.e.! = Z rout r in dr k(r) ; (83) where r in and r out are the inner and outer turning points. In this case T F reduces to the expression given in Eq. (67). However, if there is ong-ranged absorption as a resut of the couping to the breakup channes, the integrand in Eq. (8) becomes compex in a the integration region. The contributions to the integra that de nes from the cassicay aowed

21 Couped channe equations and poarization potentias 1 and forbidden regions may be cacuated separatey. In this case, T F is written as the product of factors resuting from each of them. Disregarding the imaginary part of U po (r) in the cassicay forbidden region, the corresponding factor reduces to the WKB tunneing probabiity T. On the other hand, in the cassicay aowed regions k(r) can be cacuated in an approximate way. Assuming that the imaginary part of U po (r) is sma in comparison to the remaining terms in the square root appearing in Eq. (79), we may take a series expansion to the owest order, where k 0 (r) = 1 ~ s and v(r) is the oca veocity, k(r) ' k 0 (r) + i W po (r) ~ v(r) ; (84) E U opt (r) v(r) = ~k 0(r) ~ ( + 1) r V po (r) (85) : (86) Since k 0 (r) does not attenuate the incident probabiity current, we obtain the same factor P surv as before. In our procedure we do not expicity distinguish between cassicay aowed and forbidden regions, and cacuate directy from Eq. (8), without any of the additiona approximations mentioned in the previous paragraph. Fig. 7(b) shows the fusion cross section obtained within this approximation, compared with the exact resuts and with the od approximation. We see that the present approximation yieds exceent resuts in a energy regions, incuding the one around the Couomb barrier where the od approximation totay faied. As noted in Ref. [18], the reationship between T bu and P surv that appears in Eq. (77) is not actuay correct. The reason for this is that when we cacuate the surviva probabiity we consider ony the incident branch of the trajectory, aong which the system approaches the strong absorption region. However, the breakup process may take pace both on the entrance or exit branches. Let us rst consider the cacuation of Ref. [16], which determines P surv aong a Rutherford trajectory. The surviva probabiity associated with T bu is the one cacuated aong the whoe trajectory, i.e. aong both branches A and B in Fig. 8(I-a), and not just aong branch A, as it was done in the cacuation of P surv. Since the contribution from both branches to the integra that de nes (Eq. (8)) are equa, the breakup probabiity ampitude may be written as T bu = 1 (P surv ) : (87) If we now take into account the e ect of the nucear potentia on the cassica trajectory, the situation changes very much. For ow partia waves, where E > B,

22 Couped channe equations and poarization potentias Figure 8 (I) Branches (A, incoming; B, outgoing) of the coision trajectory that contribute to the breakup process, (a) pure Rutherford, and (b) incuding nucear potentia e ects. In this ater case, the incoming branch has an additiona segment (C). (II) Exact cacuations of the breakup cross section (soid circes) compared to WKB cacuations taking into account a branches of the cassica trajectory (A, B, C of eft gure(b) and Eq. (91)) (soid ine) and taking ony branches A and B in the Rutherford trajectory (eft gure - a) and Eq. (87)) (dashed ine). the in nite absorption condition in the strong absorption region aows for ony an ingoing branch. On the other side, for partia waves for which E < B we may have two cassica turning points, as iustrated in Fig.8(I-a). In that case a segments A, B, C do contribute to the breakup cross section. In this case the ampitude T bu is given by T bu = h i j ( ) (1) j ( ) (r out ) + h i j ( ) (r in ) j ( ) (R F ) j ( ) (1) + h i j (+) (r out ) j (+) (1) (88) The rst term in the numerator corresponds to the contribution to the breakup channe aong incoming branch A in Fig. 8(I-b). The second term corresponds to the other incoming segment, C, whie the third one is the contribution associated to the exit :

23 Couped channe equations and poarization potentias 3 branch B. The currents in this equation are given by j ( ) (r out ) = e 1 j ( ) (1) j ( ) (r in ) = T j ( ) (r out ) j ( ) (R F ) = e j ( ) (r in ) j (+) (r out ) = (1 T ) j ( ) (r out ) j (+) (1) = e 1 j ( ) (r out ) ; (89) where 1 e are given by Z 1 1 = Im r out Z rout dr k(r) ; = Im r in dr k(r) : (90) Substituting the density currents in Eq. (88), we obtain T bu = 1 e 1 + e 1 T 1 e + (1 T ) 1 e 1 : (91) The breakup cross section cacuated using Eqs. (87) (dashed ine) and (91) (fu ine) are shown in Fig. 8(II), where they are compared to exact resuts (soid circes). We notice that the two approximations ead to simiar resuts, and both are reasonaby cose to the exact vaues. Comparing the two curves we reach two important concusions. One is that the inaccuracy in the resuts in Fig. 7(a) is due to the omission of the exit branch in the trajectories. The other is that in the present case nucear e ects on the trajectory are not very reevant. This is because the most important contributions to the breakup cross section arise from the high- partia waves. Whie for the energy range considered the fusion cross section converges for = 10, the breakup one requires the incusion of partia waves as high as 80. In this way, for most partia waves reevant for the breakup cacuation the externa turning point is paced outside the nucear potentia range. The situation changes somewhat when a more reaistic potentia is considered. In that case, its intensity decreases at high vaues, and the breakup cross section becomes more sensitive to ow partia waves. 1. H. Bethe, Phys. Rev. 55 (1939) D.L. Hi and J.A. Wheeer, Phys. Rev. 89 (1953) C.Y. Wong, Phys. Rev. Lett. 31 (1973) U. Mose, Comm. Nuc. Part. Phys. 9 (1981) S.M. Lee, T. Matsuse and A. Arima, Phys. Rev. Lett. 45 (1980) R.G. Stokstad, Y. Eisen, S,. Kaanis, D. Pete, U. Smianski and I. Tserruya, Phys. Rev. Lett. 41 (1978) A.B. Baantekin and M. Takigawa, Rev. Mod. Phys. 70 (1998) A.B. Baantekin, S.E. Koonin and J.W. Negee, Phys. Rev. C38 (1983) 1565.

24 Introduction 4 9. N. Rowey, G.R. Satcher and P.H. Steson, Phys. Lett. B54 (1991) S. Björnhom and W.J. Swiatecki, Nuc. Phys. A391 (198) J.R. Leigh et a. Phys. Rev. C5 (1995) L.F. Canto, R. Donangeo, Lia M. Matos, M.S. Hussein and P. Lotti, Phys. Rev. C58 (1998) M. Trotta et a. Phys. Rev. Lett. 84 (000) J.J. Koata et a. Phys. Rev. Lett. 81 (1998) K.E. Rehm et a. Phys. Rev. Lett. 81 (1998) M.S. Hussein, M.P. Pato, L.F. Canto and R. Donangeo, Phys. Rev. C46 (199) M.S. Hussein, M.P. Pato, L.F. Canto and R. Donangeo, Phys. Rev. C47 (1993) N. Takigawa, M. Kuratani and H. Sagawa, Phys. Rev. C47 (1993) R L.F. Canto, R. Donangeo, P. Lotti and M.S. Hussein, Phys. Rev. C5 (1995) R L.F. Canto and R. Donangeo, M.S. Hussein and M.P. Pato, Nuc. Phys. A54 (199) M.V. Andrés, J. Gómes-Camacho and N.A. Nagarajan, Nuc. Phys. A579 (1994) C.H. Dasso and A. Vitturi, Phys. Rev. C50 (1994) R1. 3. A.M.S. Breitschaft, V.C. Barbosa, L.F. Canto, M.S. Hussein, E.J. Moniz, J. Christey and I.J. Thomson, Ann. of Phys. 43 (1995) K. Hagino, A. Vitturi, C.H. Dasso and S. Lenzi, Phys. Rev. C61 (000) J. Takahashi et a., Phys. Rev. Letters 78 (1997) G.R. Satcher, Direct Nucear Reactions, Oxford University Press, G.R. Satcher, Phys. Rev. C3 (1985) H. Feshbach, Ann. Phys. 19 (196) E.C. Kembe, Phys. Rev. 48 (195) 549. Fusion of Very Heavy Nucei 5 Introduction Contrary to the sub-barrier fusion enhancement found in ight + heavy systems, the fusion of very heavy systems of the type empoyed in the production of superheavy eements (SHE), shows hindrance when compared to the simpe one-dimensiona barrier penetration mode. The incident energy has be much higher than the fusion barrier which accounts for fusion of ighter systems, in order for heavy eement production to proceed. One needs an extra push, as proposed by Björnhom and Swiatecki [1]. This extra energy needed for fusion to occur comes about from the fact that the ssion barrier for massive systems is ocated inside the potentia barrier in the entrance channe. Björnhom and Swiatecki (BS) introduced the term extra-push; the

25 The extra push and the extra-extra push 5 Figure 9 Schematic iustration of the reation between three critica energies, four types of nucear reactions, and two kinds of extra push. This gure is appropriate when the three miestone con gurations discussed in the text exist and are distinct. In some situations the critica energies I, II, III may merge (pairwise or a together) squeezing out the regimes corresponding to dinuceus and/or mononuceus reactions. In other situations one or both of the upper boundaries (II and III) may dissove, making the adjoining regions merge into continuousy graduated reaction types. energy needed to overcome the sadde point in the potentia energy surface under the constraint of mass asymmetry, and the extra-extra-push which is the energy needed to carry the system beyond (inside) the ssion sadde point. We sha discuss the BS mode. The hinderance of the fusion cross section for very heavy systems can be accounted for using a formua simiar to the Gas-Mose mode. A very nice account of such a cacuation has been recenty given by Smoanczuk []. The discussion in these sections is important for the genra account of Super Heavy Eements (SHE). 6 The extra push and the extra-extra push The schematic mode of Ref. [3] iustrated the expectation that there shoud often be three con guration of specia importance three miestones in the dynamica evauation of a nuceus-nuceus coision. These three miestone con gurations, which de ne three associated threshod energies, are as foows: I - The contact con guration, where the two nucei come into contact and the growth of a neck between them becomes energeticay favourabe. This type I con guration is usuay cose to the top of the interaction barrier in a onedimensiona pot of the potentia energy of two approaching nucei, whose den-

26 The extra push and the extra-extra push 6 sities are assumed to be frozen. (It woud coincide exacty with the top of the barrier if the nucear surfaces were sharp and the range of nucear forces were negected). However, for systems with su cient eectric charge and/or anguar momentum, the maximum in the (e ective) interaction may disappear, but the contact con guration is expected to retain its miestone signi cance associated with the rather sudden unfreezing of the neck degree of freedom around contact. II - III - The con guration of conditiona equiibrium (a sadde -point pass) in a mutidimensiona pot of the potentia energy at frozen mass asymmetry. (The equiibrium is conditiona because its energy is stationary ony on condition that the asymmetry be hed xed). The physica signi cance of this type II miestone is proportiona to the degree of inhibition of the mass-asymmetry degree of freedom, which in turn is reated to the severity of the constriction (the smaness of the neck) in the conditiona equiibrium shape. When the constriction is not severe, in particuar when the shape is convex everywhere, the type II con guration oses its physica signi cance as a miestone con guration. The con guration of unconditiona equiibrium (the ssion sadde-point shape). The associated ssion barrier ensure the existence of a compound nuceus and guards it against disintegration. For a system with su cient eectric charge and/or anguar momentum, the ssion barrier disappears and a compound nuceus ceases to exist. The type II and type III con gurations are identica for mass-symmetric systems. They become substantiay di erent ony for su cienty asymmetric systems. In cases when a three miestone con gurations exist, are distinct and physicay signi cant, the three associated threshod energies suggest a division of nuceusnuceus reactions into four more or ess distinct categories (see Fig. 9): (a) Reactions whose dynamica trajectories in con guration space do not overcome the type I threshod (i.e. trajectories that do not bring the nucei into contact) ead to binary reactions (eastic and quasi-eastic scattering). (b) Trajectories that overcome threshod I but not threshods II and III correspond to dinuceus (deep-ineastic) reactions. (c) Trajectories that overcome threshods I and II but not III correspond to mononuceus (fast- ssion) reactions. (d) Trajectories that overcome threshods I, II and III [and are trapped inside barrier III] correspond to compound-nuceus reactions. We may note that in the ideaization where the nucear surfaces are assumed to be sharp and the range of nucear forces is disregarded, there woud be a cearcut distinction between deep-ineastic and eastic reactions (ooking apart, that is,

27 The extra push and the extra-extra push 7 Figure 10 (a) Contour ines of the estimated extra push (in MeV) over the interaction barrier, needed to overcome the conditiona sadde (i.e. the sadde at frozen asymmetry). For an asymmetric system and an injection energy beyond about the 10 MeV contour the conditiona sadde begins to ose its physica signi cance because of the unfreezing of the asymmetry. The gure refers to the schematic mode and must not be used for actua estimates. (b) Contour ines (in MeV) of the estimated extra-extra push over the interaction barrier needed to make a compound nuceus (or, at east, to form a spherica composite nuceus) out of two nucei with atomic numbers Z 1 and Z. (Schematic mode, not to be compared with experiment). (c) Contour ines of equa extra-extra push E, using an anaytic scaing formua tted to the schematic mode.

28 The extra push and the extra-extra push 8 from ineasticities induced by eectromagnetic interactions). This is iustrated in the mode of Ref. [3] where, in their Fig. 6, trajectories that have resuted in contact are discontinuousy di erent from those that have not. [The atter correspond to motion back and forth aong the -axis in Fig. 6 of Ref. [3]]. This eaves a bank space between the ast eastic trajectory and the rst deep-ineastic trajectory. With the di useness of the nucear surfaces taken into account, this bank space woud be ed out with trajectories that vary from binary (eastic) to dinuceus (deep-ineastic) reactions in a way that washes out the origina discontinuity. The degree of washing out is, however, proportiona to the di useness of the surfaces, and it shoud be possibe to maintain an approximate but usefu distinction between eastic and deep-ineastic reactions to the extent that the eptodermous (thinskin) approach to nucear processes is approximatey vaid. The quaitative consequences of the existence of the three miestone con gurations were iustrated in the mode of Ref. [3]. The physica ingredients of that mode were: (a) Conservative driving forces derived from a eptodermous (iquid-drop) potentia energy: (b) Dissipative forces derived from the chaotic-regime, one-body dissipation function in the form of the wa or wa-and-window formua; (c) A schematic (reduced-mass) inertia force in the approach degree of freedom. The mode was further simpi ed by assuming the nucear shapes to be parame trized as two spheres connected by a portion of a cone and by reaying heaviy on the sma-neck approximation and centra coisions. One consequence of the above mode, which is obvious from quaitative considerations and had aready been studied for symmetric systems in Ref. [4], is that for reativey ight reacting nucei the overcoming of the threshod I is su cient to over-come barrier II, but that for heavier systems, or for systems with su cient anguar momentum, an extra push [an extra radia injection veocity over the threshod condition] is necessary. The upper part of Fig. 10 [taken from Ref. [5]] iustrates the dependence of the extra push (in MeV and for head-on coisions) on the atomic numbers Z 1, Z of the reacting nucei. In the schematic mode used to construct the gure there emerges an approximate sma-neck scaing parameter, the e ective ssiity parameter x eff, proportiona to the e ective ssiity (Z =A) eff, and given by eff = Z =A = eff Z =A ; (9) crit where Z =A 4Z 1 Z ; (93) eff A 1=3 1 A 1=3 A 1=3 1 + A 1=3