Z. Phys. A - Atomic Nuclei 333, 57-69 (1989) Zeitschrift ffir Physik A

Atomic Nuclei �9 Springer-Verlag 1989

Light Particles Emission and Incomplete Fusion in Heavy Ion Collisions

V.E. Bunakov* and V.I. Zagrebaev** Max-Planck-Institut fiir Kernphysik, Heidelberg, Federal Republic of Germany

Received February 12, 1988; revised version December 14, 1988

Light particles emission in heavy-ion collisions is considered in terms of direct-reaction mechanisms. The extreme high-energy part of the inclusive spectrum of light particles is shown to arise essentially from the stripping mechanism while the usually observed maximum of the spectrum is formed by a superposition of several mechanisms. The significance is emphasized of the heavy-fragment stripping in the synthesis of the exotic superheavy nuclei.

PACS: 25.70.Cd; 25.70.Jj; 25.70.Np

1. Introduction

Light particle emission and incomplete fusion reac- tions in heavy-ion collisions are attracting nowadays a good deal of interest which is caused essentially by two reasons.

The investigation of heavy-ion reactions revealed a new type of reaction mechanism, namely deep-in- elastic reactions (DIR) for the nuclei, which fills the gap between the two idealized processes: the com- pound-nucleus formation (i.e. the complete fusion of the colliding nuclei leading to the thermodynamical equilibrium of the compound system and its sequen- tial decay) and the direct processes. The recent experi- mental and theoretical studies lead to the suggestion that these intermediate DIR play in fact the dominant role not only in the cross section of some exotic pro- cesses but also in the total cross section of the heavy- ion reaction. The complicated nature of the DIR mechanisms together with our lack of knowledge about the properties of the nuclear highly-excited states immensely complicate the construction of the adequate theory. Alongside with the statistical pro- cesses a large role in the heavy-ion collisions is played by the coherent nonstatistical collective phenomena. The emission of light particles (in particular a-parti- cles) often accompanied by the fusion of heavy frag-

* On leave of absence from Leningrad Nuclear Physics Institute, Gatchina, Leningrad Distr. 188350, USSR ** Chuvash State University, Cheboksary, USSR

ment with the target nucleus is an important example of these phenomena. The cross section of the light particle yield is comparable with the total reaction cross section. For example the e-particle formation cross section in the reactions tStTa+ZZNe of EN~ = 178 MeV [1] or 93Nb+ ~4N at EN=208 MeV [-2] is about 30% of the total cross section. There are also indications [-3] that the contribution of the com- pound-nucleus evaporation in these reactions is about 1% even for the isotropic part of the c~-particle spec- tra. These facts together with the large experience ac- cumulated by now in the theory of direct interactions provide the good opportunity for the approach to DIR mechanisms from the direct process side.

Another reason attracting the interest to the col- lective coherent phenomena in heavy-ion collisions comes from the possibility to use these processes (in- cluding the incomplete fusion reactions) in order to obtain the exotic nuclear states: the cold rapidly ro- tating nuclei, nuclei with anomalous neutron to pro- ton ratio, superheavy nuclei with low excitation ener- gies, etc.

In spite of the possibility to single out few degrees of freedom the theoretical analysis of the heavy-ion direct interactions is quite cumbersome. For instance the direct computation of the DWBA amplitudes for heavy-ion reactions with large transfers of mass, ener- gy and angular momentum is complicated by the ne- cessity to take into account hundreds of partial waves in the exit and entrance channels. Additional difficul-

58 V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission

ties arise in the description of multistep processes. The large number of channels, the complexity of inter- action coupling the relative motion with the internal degrees of freedom, the unknown nature of the highly- excited nuclear states practically invalidate the tradi- tional coupled-channels approach in this case. The situation becomes even more complicated since the experimental data exist as a rule only for the inclusive cross sections of the different product yields. This caused the appearance of lots of phenomenological approaches to the description of heavy-ion collisions which sometimes produce nice fits of experimental data but have no predictive power, do not clarify the reaction mechanism and thus do not allow to study such interesting aspects of nuclear structure as yrast states, multinucleon clustering, surface vibrations etc.

The aim of the present paper is to investigate in the framework of DWBA theory the contribution of different direct mechanisms to the inclusive spectrum of light particles emitted in heavy-ion collisions. We also discuss the mass distribution of these particles and the possible role of the multistep processes. Spe- cial attention is paid to the direct process of incom- plete fusion in the vicinity of Coulomb barrier in the entrance channel as a possible way to obtain the cold superheavy nuclei.

2. Direct Mechanisms of the Fast Light Particle Formation

As mentioned above the heavy ion collisions even at comparatively low energies of few MeV/nucleon are accompanied by the large yields of light charged particles (most frequently alphas) which are character- ized by the following properties.

1. The maximum yield lies at the energy of the particles defined by the incident beam velocity.

2. Their energy spectrum decreases with increasing energy much slower than the evaporation spectrum of the corresponding compound-nucleus and practi- cally reaches its two-body kinematic limit (corre- sponding to the "cold" fusion of the heavy fragment with the target).

3. Throughout the whole energy spectrum the for- ward peaking in angular distribution is observed which is even more pronounced for the high energies.

4. The correlation experiments (e.g. 7-multiplicity measurements) demonstrate that the emission of fast light particles is accompanied by the large angular momentum transfer, which reveals the peripheric character of the reaction.

All these properties indicate an important contri- bution of direct mechanisms to the formation of fast light particles. First arguments on these lines were given in [4, 5].

2.1. Stripping, Break-up and Inelastic Break-up

Consider first the direct formation processes of the light particle 1 in the case of ion I incident on the target A:

I ( = l + 2 ) + A - - 1 + 2 + A (g.s.) (1.1)

1 + (2, A)a (1.2)

I + B ( = 2 + A ) . (1.3)

These 3 physically different processes correspond to the "elastic break-up" (1.1) of the incident ion in the field of the target, "the inelastic break up" (1.2) in which the heavy fragment 2 interacts inelastieally with the target (without the fusion) giving rise to all the possible states of the system 2+ A and the strip- ping of heavy fragment 2 (also named the "incomplete fusion reaction" or the "massive transfer") (1.3) in which the fragment 2 and the target A form the com- pound nucleus B. The (1.1) channel can be distin- guished experimentally by the coincidences of parti- cles 1 and 2 at a fixed Q-value corresponding to the target ground state. One might also distinguish the (1.3) channel by measuring particle 1 in coincidence with the B nucleus characteristic 7-transitions. How- ever the major and most reliable part of the experi- mental data gives only the inclusive spectrum of parti- cles 1.

The transition amplitude for any of the processes (1.1)-(1.3) is defined in the '"post" representation

T~(k~, kl ; ki) = ~-) V121~k~(+) ) (~/~, kB, kl [ (2.1)

or in the "prior" representation

T~(k,,kl;ki)=(O~7)ka,kl]V1A + V2A + UiI4(k+)). (2.2)

Here k s denotes the asymptotic momenta in channel ft. For the sake of convenience we use the same index fi to denote the decay channel of the compound sys- tem B while fi = 0 (ko- k2) corresponds to the elastic break-up channel, qS~-) is the asymptotic state of the Hamiltonian (H--Vt2) and qS~ +) is the asymptotic state of the Hamiltonian ( H - VIA- g2a Dr- UI). The to- tal Hamiltonian in our model is

H = HA (0 + t l (r l) + t2 (r2) + V1A (r l, ~)

+ V2A(r2, ~)+ V12 ( r=r l --r2),

while UI (R=(ml rl +m2 r2)/mi) is the optical model potential in the system I+A. Thus, we neglect for the time being the interval structure of particles 1 and 2.

Now we make two main assumptions essential for our further considerations.

V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission 59

1. Assume that the processes (1) are initiated by the direct mechanism�9 This assumption allows to sub- stitute the exact wave function ,/,(+) in (2.1) by t/'k~ qSo(0 (pi(r)2(k+)(R), ~b0, ~oi and 2k, standing for the target ground state, the relative motion of 1 and 2 in I and the incident ion distorted wave respectively.

2. We also assume that the light particle 1 interac- tion with the target A is of elastic character and sub- stitute the V1a(r, ~) interaction in the total Hamilto- nian by the optical model potential U1A(ra) of the particle 1. This means physically that the particle 1 leaves the region of interaction without substantially exciting the target.

Then the amplitude (2.1) can be written as

T B (k,, k ~, k,) = ~ d 3 r ~ d ~ r2 2*, (r ~) ~b~, ks (r2)

�9 V~ 2 (r) ~0,(r) 2(k+) (R) (3)

where

/,/,(-) t r , qSa,ks(r2)= \v'~,k~t 2 r IqSo(~)> - (4)

Here (-) ~r is the wave function of a system 2 + A with the boundary conditions corresponding to the fi channels, ~k, and ~k, are the distorted waves of the incident ion and the detected particle 1. Then the total cross section for the emission of particle 1 is defined as

d2a _ 2 ( 2 n ) 4 m~h~6M r k~ k~ df2a dE1 ~ kr ~1T~lEdf2us" (5)

We assumed in (5) the two-body character of de- cay in fl channel (with the relative momenta k~ and reduced mass #~) only for the sake of simplicity�9 This is quite inessential since we sum and integrate over all the observed states ft.

In order to simplify (5) even further we shall use the method suggested in [6] which employs the com- pleteness of states in the (2 + A) channel and allows to carry the summation over in the peripheric direct reaction�9 For the case of deuteron break up this pro- cedure was also performed in [7] in order to separate the contribution to the inclusion proton spectrum arising from the inelastic break-up�9 Contrary to [7] we shall also try to separate the contribution arising from the channels of fragment 2 stripping into reso- nance states of compound nucleus B using the formal- ism suggested in [8]. One can easily see that the func- tion q~, us satisfies the relation

~bfl, ks (r2) -- Zk2 (r2) 6~ o + ~ ta 0 (kt~, k) G(2 +)(E 2 ; k, k2) d 3 k. (6)

Here tao is the transition matrix in the system 2 + A, G(2 +) is the Green's function for this system, E2

= E~ - E~ -- ea z is the fragment 2 energy in the'elastic ' channel fi = 0, in which q~B,ks coincides with the elastic scattering wave function of the particle 2 on the target A:

4~=o,~s=~2(r2)=0 (-~*~2(r9.

For f i # 0 qSp,ks contains in its asymptotics the outgo- ing wave�9 Expanding the Green function in terms of partial waves we get

~bt~,0,ks(r2)=~ / ~Yzu(r2)]~ f t~(kl~, k)

Gz(E2 ;k, r2) k2dk. (7)

The exact integration over k is possible only for the large values of r 2 when the Green function becomes separable in its variables�9 Since the major contribu- tion to (3) comes essentially from the large values of rl and r2 due to the strong absorption in the en- trance and exit channels and taking into account the pole character of the Green function we approximate the integral over k by the following expression

St~ Gi +) k2dk

tpo(kp, k2) Stgo(k2 ' k2) G(~+)(E2 ;k, r2)k2dk t~o(k:, k~) 4# t~ o (k~, k2)

,~ t~o(k2 ' k2 ) [t~(~+)(k2 r2)-Fx(k2 r2)] (8)

which is exact in the limit of large r 2. Here ~(z +~ is the partial wave of the elastic scattering wave func- tion, F x is the Coulomb function regular at the origin.

Now we separate the direct and resonance pro- cesses in the system 2 + A which lead into the same channel fi: tBo = t-~o + t~e~. This separation is essential since the formfactors ~bB, k~(r2) for these two cases are quite different. If the wave function r in (8) is calculated in the optical model approach it can not be used to model the compound resonances of the nucleus B = 2 + A. Therefore we shall use the follow- ing approximation for the function

~b~, kr (r2) ~bp, k2 (r2)

=2.. t z~tr2)l/ --~- ~ [O~�9176 r2)] x~ ! / ~ L ~ot~C2)

~_(m2 k2~�89 ei~ ]'0 ei~'6 ~�89 yff~(~fl)h(+,(k 2 r2)~" (9) \#~ kp] (E2-E~)+i2 J

Here 6~ and F o are the partial phase shift and the width in the elastic channel, h(~+)=G~+iF~, is the ' Coulomb' Hankel function taken at the point of res-

60 V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission

onance maximum, when the subbarrier behaviour of h~ +) is practically defined only by the irregular func- tion G~. Therefore the second term in (9) stands for the resonance contribution to the formfactor. The possibility of the resonance energy factorization in the case of stripping to resonances was discussed in [9]. Now taking into account that

~[~(k~, ka)[Z dOk, _ rn 2 k 2 ~ o (10) It~o(k2)l 2 #a kp ~ o

and the fact that energy averaging cancels the interfer- ence terms between the direct and resonance pro- cesses we can perform the summation over fl in (5). Thus we get the expression for the inclusive cross section of the particle 1 yield:

d zo- =(2n )4mlm2Mx k l k 2v ~ lT ( 1 ) ~k . /,)1 ~. t 1,kI)l 2 dE21 dE1 h 6 k1 zv

+ ]1 - Sal ~ [ T~(~)(kl' kz)12

§ 2 ~ I,.~l I T~,3)(k~ k,)l 2 } ( ) (11)

where

r}~2 (k~, k~)= ~2~)*(r~)1/2 ~0~")(k~ r2) Y~.(f2)

. V12 (r) q)i(r) -(+) 3 Zk~ (R)d r 1 d3rz (12)

(p(i) = I/](+)0M(k2 r2) ~012) = I/](2+)0M (k 2 rz)-Fx(k 2 r2) 0(3) - - ~,(+ )cb

z - - r~& t ~ , 2 F 2 ) .

Here Sz is the partial S-matrix for the elastic scatter- ing of particle 2 on nucleus A. It is defined in the optical potential which does not take into account the resonance contribution. The 3 terms in (1 l) corre- spond to the contribution of 3 different mechanisms into the process of particle 1 emission, namely, the elastic break-up of the incident ion, the "inelastic" break-up followed by the target excitation and var- ious direct exchange processes between the fragment 2 and the target, and, finally, the heavy-fragment strip- ping into the compound resonance of the final nucle- us B. The ratio of the average resonance width in the elastic channel to the resonance spacing is defined in the black nucleus case by the barrier penetrability P(2) or the S~ e~ calculated for the optical potential which takes into account only the resonance absorp- tion:

rc / "0(2) 1 - 1 S ~ l 2 D(2) 4

- - ~ P ( 2 ) .

A similar investigation of the light particle formation processes was carried out in a series of works by T.

Udagana, T. Tamura and collaborators. In their first publications [5, 10] they simply wrote the inclusive cross section for the emission of particle 1 as a prod- uct of the squared modulus of the elastic break-up partial amplitude (T~(, ~) in our notation) and the factor P(s 2. This factor was used to simulate the fusion of fragment 2 with the target A. Such a structure of the cross section defined their use of the somewhat misleading name 'breakup fusion' attributed to the whole mechanism of the light particle production. The above analysis shows that while at any rate the three different direct mechanisms contribute to the inclusive spectra of light particles only one of them, namely stripping to resonances leads to the direct fusion of the heavy fragment 2 with the target. Moreover, the experimental lack of coincidences between the outgo- ing particles 1 and 2 does not necessarily imply that the fusion of 2 + A took place since the existence of the inelastic breakup allows all softs of inelastic inter- actions between 2 and A including the mass exchange between them. Another drawback of the inclusive cross section expressions in [5, 8] consists in the lack of motivation for the appearance of the above factor P(2)/IS~I 2 which is at the same time catastrophically sensitive to the value of the elastic scattering S-matrix for small 2. This is especially evident in the natural limit of the absolutely absorbing nucleus A when IS~l=0 for 2

V.E. Bunakov and V.L Zagrebaev: Light Particles Emission 61

critical ,t value which cannot be exceeded in fusion of 2 + A), Our present knowledge of 2~rit is quite inade- quate but the decrease of the breakup amplitudes Ta(ut) and T~} z) due to the absorption at small 2 is so rapid that any reasonable choice of 2~t (say 2~,~ ) ~ t . The consequence of this cutoff will be described be- low.

The computation of 6-dimensonal integraIs (12) is still quite complicated. Therefore we use the quasi- classical approximation for the three-dimensional wave functions

~ t ) ( m = a , ( m e ~ ~ '~) ,

)~(k7 )* (r) = 2(_~, = A~a-)(r ,) e iss- '('*)

and expand the action functions S~ and S~ in series (see, e.g., [13]) at a point r2 assuming that the ampli- tudes A~+~ vary more slowly than exp(iS) and intro- ducing local momenta k(r):

. m 1

~ + ) ( R ) ~" tk,~/+)rrt 2I ~ e~k'('~} MWr (15.1)

~,~ )* (r 1) ~)~7)* (r2) e -ik'r . (15.2)

Then the transition amplitudes (12) can be written as follows:

/ 3 - Tz(~) (ks, k,)---5A,(r2) A]-)(r2) e ~ts~{'~)+ ss- )~,ng ~

-(,O(n) (k2 r2) Yz~ (~2) Di(q(r2)) d 3 re,

{ h2q2\ (P,z(q)). D, (q)=(qlV~21~ot)=~e,2-~]

(16)

(17)

We shall see below that the processes of interest to us are localized in a rather narrow range around r2 ~ r'~ ~ Rff(I), where Rff(I) is the position of the Cou- lomb barrier maximum for the entrance channel. In this range the Dr(q) changes with the change of r 2 much slower than the rest of the functions in the inte- gral (16). Thus we can take this factor out of the integral. The function Iqb(q)l 2 defines the momentum distribution of particles 1 in the incident ion. In our further calculations we shall use for this function the expression

Const. exp ( - q2/2 G2),

where

0 -z = Cr z m l ( M r --mi) MI-- 1 '

ao = ]~5-5 ~0.51 fro-t,

which was obtained in [14] for the fragmentation ex- periments at high energies. The 3-dimensional integral in (16) was computed directly (without the partial wave expansion) using the quasi-classical methods suggested in [15]. The final expression for the transi- tion amplitude is

~ 5 Ta(~ ) (ks, k,) = D, (q (r~')) ~ V ~ @n)(k 2 r2)

�9 I),u(kt, k t ; r2) r~ dr2 (1.8)

where Ix~ is the integral over the angular variables in (16) calculated with the aid of the stationary phase method. The radial integration of (18) was practically carried out only along the so-called 'transition lines' (see [15]) r(2) where the angular momentum conser- vation takes place: [r x k I (r) ] - [r x k~ (r) ] = 2.

The spectroscopic factor (or more generally the effective number) of particles I in the incident ion is the only unknown constant which enters the theory via the got(q). But even without this absolute normal- ization of the theoretical curves we can still define the character of the inclusive spectrum of particles l, the relative contributions of the three different mecha- nisms and of the different angular momenta 2 trans- ferred in the reaction, etc.

Figure 1 shows the inclusive spectrum of e-parti- cles emitted at 0 = 0 ~ in the reaction 22No + l SaTa for Elab 178 MeV [16] in comparison with the theoreti- N e ~ - cal calculations according to the (11), (12) and (18).

Figure 2 shows the contribution of different angu- lar momenta for the system 1SO+ lS*Ta to the high- energy part of the e-spectrum where the stripping (in- complete fusion) mechanism dominates.

Figure 3 demonstrates the dependence of the aver- age transferred angular momentum for the system

80 + 181Ta on the energy of the emitted e-particles. Figure 4 demonstrates the transition lines r(2) for

different 2 and two values of E~ together with the shadow regions in the entrance and exit channels (in the regions to the right and to the left of the corre- sponding dashed lines A1 < 0. I and A 1 < 0.2). This fig- ure illustrates how the optimal 2 values of Figs. 2 and 3 are formed_ For example, with 2 decreasing the stripping formfactor rapidly increases in the sub-

62 V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission

barrier region as a result of the diminishing centrifu- gal barr ier

"~exp [ 9 / 2 ] / 2 , 2(,;(+1)�9 ] --~! V--~-[V2a(r)-E~]+~r~or/" (19) 1

However the cor respond ing transi t ion line r(2) simul- taneously goes deeper into the shadow for the outgo- ing ~-particle A~-)(r~, 0 a) rapidly decreases with in- creasing angle 0 ~ (r~ and 0;" are the spherical coordi- nates of the cor responding transi t ion line). The joint effect of these two factors allows to define the opt imal value of 2 for a given reaction.

Our calculat ions allow to make the following con- clusions.

I. While all the 3 mechanisms contr ibute to the inclusive ~-spectrum its high-energy par t is domi- nated by the str ipping (incomplete fusion) contr ibu- t ion accompan ied by the fo rmat ion of the residual nucleus B = 2 + A. The cont r ibu t ion of elastic and in- elastic b reakup is negligible for Ea lower than the C o u l o m b barrier V~(2 + A). This follows f rom the dif- ference in the subbarr ier behaviour of the str ipping and b reakup formfactors ~o~.

d~/df~adEcx rnb/Sr N~eV

t0t o o

" R . O

, \~ I I

t6~ J

I I I I J [ ~/ 20 40 60 80 t00 t20 E~x Me

Fig. 1. The inclusive ~-particle spectrum [16] from t8~Ta(22Ne, ~) at lab Er~ = 178 MeV and 0~ = 0 ~ Dashed line denotes the contr ibut ion of tim eIastic and inelastic breakup, dast>dotted line - fl~e stripping contribution

2 A

12

8 4

I

arbltrar~ units

Ec( 90 MeV

0 20 40 60

Fig. 2. The partial stripping cross sections for 0~ = 0 ~ and different a-particle energies. The arrows indicate lye,,, values for the corre- sponing ~ 99Tl excitation energies

I

Eo( 106 M~V

E(x 125 MeV

I

8 o A. (~)

/~otY

8O

60

40

%

~gcx

40 80 120 ~AeV

Fig. 3. The average values of the angular momentum transfer in 181Ta(22Ne, ~) 199T1

5o . . . . . . . . : 70 ..... 3 0 / ~ ~

-t . "T Ecx = 40 ~eV I~ = 400MeV

Fig. 4. Transition Iines r(A) in lSlTa(22Ne, ~) reaction for the two energies of a-particles emitted at 0~ = 0 ~

V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission 63

2. The calculations show that the slope of the e- spectrum is essentially defined not by the momentum distribution ]~0~(q)[ 2 but rather by the dynamics of the process. In our case the momentum distribution in the spectrum range up to E,,~ 120 MeV falls down only by ,-~ 2 orders of magnitude. The rest 5-6 orders arise from the energy dependence of the integrals in (18). Therefore in the phenomenological approaches (see, e.g., [17]) neglecting this dynamics the attempts to describe the whole spectrum with the aid of the only factor ]q~I(q)[ 2 cause the necessity to choose the ao parameter as a function of the incident ion energy (smaller o- o for smaller E,) which contradicts the phys- ical meaning of the parameter.

3. All the processes considered including stripping are of a peripherical nature and are strongly localized in space. The shaded areas in Fig. 4 show the regions where the dominant contribution to the stripping am- plitude with optimal 2(E~) values are formed. All the transitions for other 2 values take place in the similar narrow straps around their transition lines r(2). Just this spatial localization allows to introduce the value r~(2) and take the Dx(q(r2) ) factor out of the integral (16).

4. Our quantitative predictions [18] of the break- down in the slope of the high-energy part of the ~- spectrum are confirmed in the theoretical calculations and in experiment [-16]. This breakdown arises from the limitations on the angular momentum which can be "absorbed" by the almost unexcited compound system B. Figure 2 demonstrates how this breakdown is formed. The dashed part of the lower curve in Fig. 2 corresponds to the values of 2 that exceed the maxi- mal allowed value lyra~t of the B nucleus spin for a given excitation energy and therefore do not contrib- ute to the cross section of this nucleus formation. The dashed part of the curve in Fig. 3 also corre- sponds to the yrast-line of B nucleus. This figure shows that the average 2 decreases with increasing E~ as one should expect from angular momentum conservation laws. This behaviour of 2 ..... go is con- firmed by the experimental dependence of average 7- multiplicity as a function of E, [19].

5. The low-energy part of the spectrum around the maximum is dominated by the breakup (with al- most equal contributions of elastic and inelastic mechanisms). We have already mentioned the necessi- ty to introduce the cutoff of the stripping mechanism contribution for 2 > 2crit. Since the present estimates of 2er~t vary in a wide range we did not introduce this cutoff in our present calculations. One can easily see from Figs. 2 and 3 that the introduction of the cutoff value of say )oe~t ~ 70 should greatly damp the stripping contribution to the spectrum for E~ ~< 60 MeV.

One should not take the quantitative agreement between the theory and experiment in Fig. 1 too liter- ally since our present knowledge of the heavy-ion op- tical potentials is sufficient only for the order of mag- nitude estimates. This is especially true in the low- energy part of the spectrum where different multi-step processes might contribute considerably.

2.2. The Role of the Multistep Processes

A considerable contribution to heavy-ion collisions with large excitations of the final systems comes from the multistep processes. It is quite difficult to take these processes into account in the coupled-channels approach even for nucleon-nucleus collisions. The number of coupled channels in the heavy-ion col- lisions is extremely large, the interactions and the na- ture of the highly excited states involved are quite complicated which makes the coupled-channels ap- proach practically useless. However the high density of the excited states allows to treat their contribution to the cross section incoherently and to introduce the phenomenlogical friction forces to describe the coupling of the relative motion with the intrinsic de- grees of freedom. This allows us to make a qualitative estimates of the changes in the relative contributions of the three different mechanisms (1.1)-(1.3) arising from the multistep processes

(1,2)+A--(1,2)'+A*(e~,LAMA)--I+2+A (1.1)'

1+ {2, A}~ (1.2)' 1 + B. (1.3)'

There is an experimental evidence that the excitation energy is shared between the colliding heavy ions pro- portionally to their masses. Since in our case M r < M A we shall further on consider for the sake of simplicity only the target excitation in the initial channel. The conclusions below might be easily generalized to in- clude the projectile excitation (sequential breaking) but the expressions loose their transparency. In order to consider the role of the multistep processes we shall use the approach suggested by one of the authors [20] and expand the total incoming wave function in the amplitude (2.1) in terms of the target states:

(+)-- r (a~ ~(+) Okr --~0I( )Z (~))~kI, v(R)" (201 v

The 'elastic' component ~o~ q~o )~+) of (20) describes the above one-step processes (1). The wave functions of relative motion in the channel v for sufficiently

64 V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission

small energy transfer (e~ < El) can be expressed in the form

)~k, ~(R)= C,(R) .4i (R) e isr(m. (21)

Here AI(R) is the same amplitude factor as in the elastic channel but obtained with the absorption (Im U~(R)) caused by all the channels which are not taken into account in (20) (i.e. mass redistribution and complete fusion of I and A nuclei); S~(R) is the action function for the incident ion moving along the 'elas- tic' trajectory in the field U~(R) with the initial mo- mentum

./2#,

C~ (R) is the probability to excite the target into the state 4~,(~) with the energy e, while the incident ion

reaches the point R, C~ ~-t= 1 �9 When the state

density PA(eJ of the target is sufficiently high (e,+~ - e , ~ E,) it is possible [20] to simplify the approach and introduce, instead of C~(R), the new values C(e, R)= ~ C, (R) which are normalized

QO

I c%, R) k,(~) o ~ d e = 1 (22)

and satisfy the transport equation

~c(~, R) 0 (23) VC(e,R)+F(R) Oe

with the boundary condition C 2 (e, k1 Ri ~ -- oo)= 6(e) and the phenomenlogical friction forces F(R).

Inserting (20) into the amplitude (2.1) and follow- ing the same way which led us to (11) we obtain the following expression for the inclusive cross section with the energy dissipation on the initial channel:

dZ~ =(2re)'* ml m2 M1 kl E~ d(21 dE~ - h6 k i 2 d 8 PA(G) k I ( e )

2gt

LA' yrast(e) ( 1 - - �9 y, tlTA1)(k,, ~; k3l 2 + I&( Ei)I2

LA, MA I 1-sx(g~)lZ

(2) } I TA. (ks, e; k,)l 2 + P~(E~) I T(a)e~ e; k,)[ 2 (11')

where

Tz(~) = ~A 1 (r 0 elSi - )(*0 ~ q~(~")(k2 (e) re) Y~(r2)

�9 1/12(1" ) ~oi(r) AI(R) C(~, R) eiSr(~'md3rl d3r2

E ~ = E 2 - e = E I - - E l - e 1 2 - e

E x = E 1 _ o _ g?(g.s. _ gT' u12 L'2 A z-~ 1 �9 (12')

The three terms in (11') correspond to the three pro- cesses (1.1') (1.Y).

Obviously the integration over the intermediate target states will not change considerably the one-step results for the high-energy part of the spectrum (i.e. for small excitation energies E~). For smaller values of Ea the range of integration over e becomes larger. This fact by itself will not much affect the single-step results due to the normalization condition (22). How- ever the major difference of the amplitudes (12') from those in (12) lies in the fact that the coefficients Al do not contain the absorption due to the energy dissi- pation in the initial channel and decrease with de- creasing R much slower than the corresponding values in (12). Thus the incident ion in the inelastic (v#0) channels comes much closer to the target A than in the elastic one down to the distances where the complete fusion takes place. Therefore the cross section of the fragment 2 stripping to the target in the excited state with e~>0 might be much larger than stripping to the ground state (direct single-step process). This is caused by the exponential increase of the formfaetor ~o~3 ) with the decreasing distance in the subbarrier region (see (19))�9 The multistep pro- cess influence on the breakup (elastic and inelastic) will be less pronounced since the formfactors q0(z ~' 2) decrease with decreasing distance. Therefore the mul- tistep processes with energy dissipation in the initial channel might lead to the dominance of stripping (in- complete fusion) over the breakup in the inclusive cross section (11'). Preliminary estimates confirm the dominant contribution of the multistep processes in (11'). However the characteristic features of the angu- lar and energy distributions as well as of the 2-distri- butions remain unaltered. The detailed analysis is be- ing carried out presently.

2.3. The Mass Distribution of the Emitted Particles

The problem of the mass distribution for the particles emitted in heavy-ion collisions is also of interest. The exponential dependence of the 'light particles' (ml < M3 yields on their relative binding energies in the entrance and exit channels was first pointed out in the works of Volkov et al. (see review [21])�9 Mind that the strong dependence of the cross section on the binding energy ~2 of the particles 1 and 2 in the projectile might be an evidence that the particles 1 are formed from the incident ion nucleons, i.e. the evidence that the mechanisms (1) dominate.

V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission 65

In our approach all the dependence on the projec- tile structure is defined by the factor Dlz(q(r~)) in the transition amplitude (16). However in obtaining (16) we completely neglected the structure of the frag- ments 1 and 2 and extended the integration over the relative coordinate r in (17) to the whole space. This was not quite correct because the strong absorption in the entrance and exit channels forms a shadow in the regions where As(R) and Al(r~) are small. In order to take these facts into account explicitly we shall consider another representation of the transition amplitude for the elastic breakup case. The results obtained might be generalized to the inelastic breakup as well.

Let the fragments 1 and 2 have the inner structure, (po(~) and q~O(~) being their ground state wave func- tions. Then the elastic breakup amplitude can be writ- ten as

To o (k,, k2; k i ) = ( (po Xk, (pO Zk2 ~o I VsA + (+)

V2 A - - V12 ] ~]/k~- ) . (24)

When we used V~a and VZA as distorting potentials in the exit channel we obtained the expression (2.1) (mind that Zk in (24) are the pain waves). But now we shall separate the relative motion of the 1 and 2 fragments in the exit channel with the momentum !112--(m2 k l - m l kz)/Mi and their center of mass mo- tion with the momentum k = k s +k2. Using now the V~ 2 (r, ~ , ~2) interaction as the distorting potential we get:

Too(P12, k; ks) = (-) r < ~hp,~ ( , ~,, ~2) Z~,(-)(R) q$o (~a) I Via

~- V2A - - u;I V'k,'/'(+)"/. (25)

Here ~,(-) is the wave function for the relative motion P12 of the 1 and 2 fragments in their ground states, Z~,(-)(R) is the wave function of their center of mass motion in the distorting potential UI(R). In the case of the one-step breakup (25) takes the following form:

TgoWaa(pl 2, k; k,)= < ~m:(r)Z(k-)(R) ] U1A(r,)

+ UZA (r2)-- U/(R)I ~o~(r). 2G +'(R)> (26)

where )) is the distorted wave function for the scattering of particle 1 on the fragment 2 which is defined by the optical potential UFpt(r); USA and U:A are the optical potentials for the interaction of particles 1 and 2 with the target. First attempts to use the amplitude (26) in the analysis of deuteron breakup [22] demonstrated that the agreement with the experiment is much worse than for the usual DWBA amplitude (2.1). However the

later investigations [23, 24J showed that with the proper choice of the U'(R) ~ U? pt the amplitude (26) is in good agreement with the experiment both for the inclusive proton spectra and for the correlation measurements.

In our case the two charged fragments 1 and 2 emerge in the breakup. Therefore we probably do not make a large error in considering only the long- range Coulomb distortion for Z(-)(R). The strong en- trance channel absorption together with the bound state (ps(r) behaviour leads to the fact that only the peripheric region contributes to the amplitude. In this region we can also leave only the Coulomb parts Z1 Z2 e 2 Z 2 Z A e 2 - - a n d - - i n the potentials UsA and U2A

r s r2 of (26). Then the amplitude can be expressed in the form

Tg T M (Pie, k; k,) = ~ C z (2 .72) I r E . (f)l m,(r)>

- (Z c ( - ) (R) l~ [2 (k~ +)(R)) (27)

where

4~ r l, , , , , ,v C~.--22+lZAeZ[z2t~ } --Zl\Ms/

and the summation starts with 2 = 1 (with 2 = 2 for the "symmetric" zs/ml = z2/m2 breakup).

Now the dependence on the projectile structure is defined by the factor a~u(p ) = (~(p-)[r'~y.~u[ (Ps) which is easily evaluated if one considers the rapid damping of the wave function ~(p ) in the inner region r R~2 , where R~2 is the radius of the absorptive potential for the elastic scattering of 1 and 2. Therefore the main contribution to a~,(p) comes from the peripheric region of the projectile I, where

g~ g0s (r) = S 1 (I) Yll~ ,,12(r~) - -

r

q~ -- S Vlz(r') r o

llz(112 + 1)] dr"( + J J"

(as)

Here r o is the inner classical turning point for the potential barrier which reaches its maximum Vf(12) at r=R~(12). The spectroscopic factor SI(I) can be crudely estimated as follows. The sum of the spectro- scopic factors over all the excited states of the frag- ments 1 and 2 (sum over all the decay channels) gives

us the effective number of particles 1 in I Neff(1 ) = (Zt ! NI !)/(Z, ! Z2 ! Ni ! N2 !). Taking all the channel contributions in this sum to be roughly equal we ob- tain S1 (I),-,Neff (1).

Denoting by r m the point where the function Av(r)ra+a~o12(r) reaches its maximum and approxi- mating this function in the vicinity of r,, by the Gauss- Jan of width A r we can estimate the integral

a~.(P12)~S~(I) qO12(rm) e - [ ' ~ ( r - ) ar]z (29)

"IJ

/ V, ( ] ~ Here p12(rM)=P12 ] / 1 ~2"r~'-is the local relative

g ER h 2

momentum, E12 = 2p1~ pzz and

B 7Z

Rc - t o /

exp - Y 12 (rm-- ro (30)

/ 2 / Z 1 2 y12= V - ~ - ( ~ B - e 1 2 ) and the r,, is easily proved

to be quite close to R~(12) which allows to make the second transition in (30).

The 2 first factors in (29) define the mass distribu- tion of particles 1 in the ion I breakup. The last factor in (29) leads to the maximum in the cross section for the minimal possible relative momentum P12(rm) and should be compared to the Iq~i(q)l z factor in (17). The angular and energy distributions of the particles are also defined by the matrix element

(~,c(-) Yau ~,(+),, (27) which is easily shown to ~-k ~ - A.k, / in

cause the strong forward peaking in the momentum distribution for the center of mass of the particles 1 and 2. This matrix element closely resembles the inte- grals in the breakup ampIitudes (18) which were al- ready shown above to define the energy slope of the inclusive spectrum. One should note that similar re- sults were obtained by Friedman [25] who simply postulated that the energy distribution is defined by the 'vertex' I~0~(q)l 2 with the cutoff due to the absorp- tion while the relative fragment yields are defined by the I~olz(rm)l 2. Now we see both from the numerical calculations of the amplitudes (18) and from the ana- lytical form (27) that the vertex I~pt(q)l 2 practically does not contribute to the formation of the spectrum slope. Figure 5 shows the relative yields of the differ- ent carbon and nitrogen isotopes in the 22Ne + 232Th

m C [] o N

m

0,t

66 V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission

1 t 4 5 ~ {'m -~

Fig. 5. Isotope yields of N and C isotopes emitted in reaction 22Ne + 232Th at E ~ = 174 MeV and 0 = 4 0 ~

reaction at ENe=174 MeV and 01=40 ~ of [21] as a function of the parameter Y12 in (30). The straight line shows the theoretical plot obtained from (29)-(30) with the reasonable parameter (rm-r0)= 1.9 fm. This together with the analogous plots of [25] is an indica- tion that the direct breakup mechanism dominates in these reactions. It is especially important since this exponential dependence on the ground state binding energies was usually considered (see, e.g., [21]) as a major argument in favour of the statistical nature of the heavy-ion deep inelastic reactions. This also seems to rule out the considerable contribution of the direct knock-out and knock-on processes which are charac- terized [17] by the dependence on the fragment 1 binding energy in the target.

3. The Incomplete Fusion Reaction in the Vicinity of the Barrier

The recent intensive studies of the complete fusion reactions demonstrated (see, e.g., [26]) that the ap- proach to the superheavy elements is complicated be- cause of the large excitation energies introduced into the residual nuclei in these processes. However we have seen above that the emitted light particle in the incomplete fusion direct reaction can carry away practically all the extra energy leaving the residual nucleus with excitations of about few MeV. Therefore the incomplete fusion reaction might be a good alter- native mechanism for the production of the super-

V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission 67

i0 ~

io -4

|cfi

,o"

:i: I

dO'A/clF..,a, at'b, unLts E 1',[r

~ 140 t30 120 ~ 1 0 0

~ 9 0

A I I I I ' '

I0 20 50

Fig. 6. Partial incomplete fusion cross sections in 181Ta(22Ne, a) reaction at different projectile energies and Ex = 5 MeV

dO-/d ~adEco arb. unLts l J L b

to "~ ~ - Id 2 ~ -

'07-, , / , , e, 7 0 ,.50 60 90-120 t50 180

Fig. 7. Angular distribution of a-particles

heavy elements. One anyway needs to study this mechanism in connection with the superheavy synthe- sis because as a rule the fact of formation of a super- heavy residual nucleus in the complete fusion is veri- fied by observing a sequence of decays which starts with one or several e-decays. Since such a fusion is very rare phenomenon it is very difficult to distinguish experimentally these e-particles from those emitted in the incomplete fusion reaction and therefore to be sure that the complete fusion really took place.

We have shown in Sect. 2.1 that the formation cross section of a 'cold ' final nucleus in the incomplete fusion process might be severely limited since the sys- tem with low excitation energy can not absorb the large transferred angular momentum ('yrast limita- tion'). With the lower incident energy the values of the transferred angular momentum diminish but in the vicinity of the Coulomb barrier and in the subbar- rier region the reaction cross section is also damped by the barrier penetrability in the incoming channel. Therefore it is of considerable interest to find the opti- mal incident energy which gives the maximal incom- plete fusion cross section for the formation of final nuclei with the excitation energies less than few MeV. It is also important to analyse the cross section depen- dence on the masses and charges of the target, the projectile and the transferred fragment as well as on the reaction Q-value.

Using the formalism of Sect. 2.1 we performed a quantitative analysis for the case of the 1 s 0 heavy- fragment stripping lSXTa(22Ne, e)~ogT1 in the inci- dent energy range lab ENo = 80-150 MeV. The results are shown in Figs. 6-8. All the curves correspond to the

101

t0 0

40-~

ld 2

dO-/d E~, arb.unLf.s Ex=5M(~V

/ /

!1// 90 410 150 150 L Ne,MeV

Fig. 8. Partial and total stripping cross section as a function of the projectile energy

final nucleus excitation energy Ex = 5 MeV and there- fore to the upper limit of the angular momentum for this nucleus lyrast(Ex)=30. The total cross section is

[ y r a s t

a sum over the partial cross sections d a = ~ do -a. , t , = 0

The cutoff becomes important (see Fig. 6) for Elab > 140 MeV.

N e

The results of Figs. (~8 can be summarized as fol- lows.

a) The formation cross section for a sufficiently ' cold' (Ex < 5 MeV) residual nucleus reaches its maxi- mum for the incident ion energy which is considerably

68 V.E. Bunakov and V.I. Zagrebaev: Light Particles Emission

it?optim higher than the Coulomb barrier. In our case "~Nr 130 MeV while the Coulomb barrier Bcn(Ne + Ta) 80 MeV. This behaviour is caused by the interplay

of the trajectory distributions, shadows due to the absorption and the transition lines.

b) The e-particle angular distribution for this op- timal energy is almost isotropic. For lower ener- gies the alphas are emitted predominantly in the backward direction. Only for higher energies (EN~ ~> 130 MeV) the angular distributions are becom- ing forwardly peaked (see Fig. 7).

c) The average angular momentum 2,v of the re- sidual nucleus is nonzero even in the subbarrier re- gion and gradually increases with the increasing inci- dent energy:

We have used again the Gaussian approximation for the integrand in (18) with the width AR around the point rm.

Now we see from (37), (38) that the incomplete fusion cross section falls down rapidly with the in- creasing mass and charge of the transferred frag- ment 2 because of the increasing transferred momen- tum q and Coulomb barrier V~B(2A) which essentially defines the V2A(r) in (37). We also observe from (37) that in order to make this cross section larger one should choose the particles A and 2 with the largest binding energies Eb.

4. Summary

)~:v=12 (EN~=80MeV), 2av=15 (ENo=90 MeV),

2,~=20 (ENd= 130 MeV).

In order to gain more insight into the mechanism of stripping we can try to obtain the analytic form for the corresponding amplitude (18). The values I~, in (11) can be expressed in the local momentum ap- proximation as [15] :

I~(EI, El, 01, r)~A,(Et, r(2)) A(~-)(EI, r(2)) eiq(r(a))r (36)

where q(r )=k1(r) -kl ( r ) is the local transferred mo- mentum and r(2) is a point on the transition line. The stripping formfactor qr for the residual nucleus with small excitation E x behaves in the peripheric re- gion as

( 0 ( 3 ) t • " -- ]~'g.s. -t- Ex, r) 2 I,~2-- a-~2A

- e x p { - - i /2#2A _E2] dr, } R0'2+A) V ~ [-V22At(Yt)

~ e x p { - ~ @ 2 2 A [VJa(r)-E~~i] ( r -R0)} . (37)

The ground-state separation energy E~'~/is expressed in terms of the binding energies Eb

EN~/= I Eb(2)l + IEb(a)l- IEb(B)l.

Since the amplitudes A1 and A1 are decreasing expo- nentially for small r we can again perform the proce- dure which lead us to (29) and obtain the expression for stripping amplitude:

Tz~(E~, El, 01) (3) E2 ' Az(Ez, rm(2)) A(1-)(E1, rm(2)) ~ox ( rm(2))

�9 e - [q (E2 ' rm (3")) AR]2. (38)

We have demonstrated that the main experimental characteristics of the light-particle emission in the heavy-ion collisions can be understood in terms of the direct mechanisms�9 Among these mechanisms stripping (or incomplete fusion) seems to be of utmost interest because of several reasons. First of all, strip- ping seems to be a unique way of obtaining the almost unexcited residual nuclei B which is extremely impor- tant in the synthesis of exotic new elements. Secondly, this mechanism gives a dominant contribution to the high-energy tail of the inclusive spectrum�9 Therefore a mere observation of a very fast outgoing particle 1 is an evidence that the corresponding cold residual nucleus B was formed. One should recommed to pro- ceed with the experimental investigatons of the ex- treme high-energy tail of this spectrum since this gives immediate insight into the properties of the exotic residual nuclei�9 On the other hand, we have seen that many different mechanisms including all sorts of breakup and perhaps the multistep stripping contrib- ute to the low energy part of the inclusive spectrum. Therefore in order to make the theoretical interpreta- tion here more meaningful one should perform the correlation experiments aimed to single out the con- tributions of the definite channels (at any rate to de- fine the charges, masses and angular momenta of the residual nuclei)�9 Otherwise the abundance of almost free parameters such as absorptive potentials, lerlt for incomplete fusion and phenomenological friction forces allows to say in advance that the maxima and the nearby slopes of the inclusive experimental spectra can be easily fitted in theory but this description will never be of much value.

We should point once more that with decreasing energy of e-particles many more complicated mecha- nisms start to contribute to the inclusive spectra (i.e. the above mentioned multistep processes, sequential decay of projectile-like fragments etc.). Our main aim here was to show that the analysis on the lines of

V.E. Bunakov and V.I. Zagrebaev: Light Particlez Emission 69

the peripherical direct and semi-direct processes might successfully explain the experimental data and therefore is worth of further examination and devel- oping.

One of the authors (V.B.) would like to 1hank Prof. H.A. Wei- denmiiller for the excellent opportunity of a sabbatical at Max- Planck-tnstilule in Heidelberg.

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VE. Bunakov Leningrad Nuclear Physics Institute Oatchina SU-188 350 Leningrad District USSR

V.I. Zagrebaev Chuvash State University Cheboksary USSR