Classical description of low energy16O +16O collisions

Pramg.na, Vol. 20, No. 6, June 198, pp. 523-546. ~) Printed in tadia.

Classical description of low energy 160 21- 16 0 collisions

A N DIXIT, V S RAMAMURTHY* and Y R WAGHMARE** Department of Physics, Christ Church College, Kanpur 208 001, India *Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India **Department of Physics, Indian Institute of Technology, Kanpur 208 016, India

MS received 25 February 1983

Abstract. The aim of this paper is to study the scattering of the two ground state 160 nuclei, using classical microscopic approach. We have studied fusion cross-section (ae) for various incident energies, its energy variation with time, and also other aspects such as shape deformation in head-on collision, life-time of resonance-scattering etc. Our calculations indicate that for projectile energies of the order of 1-2 MoV per nucleon (low energies) classical microscopic calculations for heavy ion reactions seem to give satisfactory description. However, at very low energies (less than 1 MeV/ nucleon) appreciable deviations are seen which may indicate the breakdown of classical approximations.

Keywords. Fusion cross-section; resonance .~..attering; classical microscopic description; Lennard-Jones potential.

1. Introduction

In recent years there has been a good deal of interest in the study of the dynamics of heavy ion eoUisions (Hiifner et al 1975; Hiifner and Knoll 1977; Koonin 1977; De Guerra et a11977; Brink and Staneu 1975; Stringari eta11976; Canto 1977; Canto and Brink 1977; Hassan and Brink 1978; Sierk et al 1978; Rowley and Marty 1976; Gutbrod et al 1973; Austern 1975; Gritfm and Kan 1976; Cusson et al 1976; Bonche et al 1976; Broglia and Winther 1972; Broglia etal 1974; G6tz et al 1975). One can now reach those regions of nuclear physics which were not accessible in the past. However, all such studies require a priori the knowledge of the ion-ion potential which we do not have. Hence it is absolutely necessary to go to approximate calculations within a framework of some idealised model, which can simulate the gross properties of nucleus to some extent.

Keeping these facts in mind, physicists made various attempts to describe heavy ion reactions using quantum as well as classical mechanics. In quantum mechanics one has to make use of the time-dependent Hartrec-Fock approach (Bonche et al 1976, 1978, 1979; Flocard et al 1978; Koonin 1976; Koonin et al 1977). These calculations are extremely difficult to perform and involve various approximations. On the other hand in the classical macroscopic approach (Alonso 1974) the relevant equations of motion are easy to handle and the concept behind the approach is also simple to understand and hence has been widely used. Here it is assumed that there is a continuous distribution of nuclear matter and for the study of interactions

523

524 A N Dixit, V S Ramamurthy and Y R Waghmare

the same is subjected to a given set of forces. Although calculations based on this approach give a lot of information on the dynamics of heavy ion collisions, there are also a few limitations associated with these models. For example, assumptions of a nucleus as a charged liquid drop with continuous distribution of nuelear matter and having a sharp surface is justified in or near the dusters' equilibrium configuration, but these assumptions do not hold in the ease of highly-deformed nuelear shapes obtained during the processes of actual fusion or fission.

Another approach for the study of heavy ion reactions is the classical microscopic approach. Several groups (Bertini et al 1973, 1974, Bondorf et al 1976, Bodmer and Panos 1977; Ramamurthy and Kataria 1978) have reported the study of heavy ion interactions based on this approach. We have also adopted classical microscopic approach for the study of scattering phenomena for 180 + 160 interactions at low centre of mass energies of the colliding system ranging from 66.56 to 16.64 MeV. For our analysis the Ldnnard-Jones form for the nueleon-nueleon potential has been used and the parameters fitted to reproduce correct binding energy per particle and root mean square (RMS) radius comparable with the liquid drop model estimate. The same form of potential has been used earlier by Ramamurthy and Kataria (1978). Here nucleons within the colliding nuclei are considered as classical spinless particles interacting via a two body force. This involves solving of Newtonian equations of motion for the interacting system, which leads to a set of self-consistent equations. For these calculations, at first, ground state configuration of the colliding clusters is obtained. The two clusters so obtained are allowed to collide and the scattering phenomena for 160 + 160 at the centre of mass energies mentioned above are studied in detail. This study includes head-on collision trajectories in R-S plane, energy variations in head-on as well as peripheral collisions, life-time of resonance, angular momentum transfer from the initial orbital angular momentum of the clusters to the angular momentum of each of the two clusters about their respective centres of mass, fusion cross-section calculations, comparison of present results with those of other workers and finally discussions and conclusions arrived at from the earlier sections. The present calculations exhibit qualitatively all the essential features of low energy heavy ion collisions such as complete fusion, deep inelastic scattering and nucleon transfer.

2. Choice of interparticle potential and procedure for bringing cluster in ground-state configuration

In the classical microscopic approach, nucleons within nuelei are assumed to be spinless particles interacting only via a suitable two-body force. The choice of this force and, therefore, the corresponding potential is arbitrary to the extent that one can have many forms each of which reproduces the same experimental data. One usually chooses a convenient form of the potential and adjusts the parameters appropriately. But once the potential is chosen in this manner the configuration of minimum energy is essentially fixed except for random rigid rotations about its centre of mass.

For the present purpose, we have chosen the Ldnnard-Jones form for the N-N potential namely,

= - (I)

Classical description of 1~0 q- 180 collisions 525

In the ground-state configuration, parameters ~ and a may be fitted to reproduce the correct binding energy per partide and the RMS r~.dius comparable with the liquid drop model estimate. The ground state configuration is obtained as follows.

In the duster the initial number of particles chosen depends upon the particular nudei, ground-state configuration of which is to be obtained. Partides so chosen are assumed to be within a cube of side 2R, with their positions arbitrarily assigned, where R is approximately the size of the duster. The potential energy of the duster with respect to an arbitrarily chosen ith particle is calculated using equation (1). This particle is then moved from its initial position through a small step Ax firstly along the positive direction and tken along the negative direction of x-axis. After each of the two movements the potential energy of the cluster is calculated and compared with its initial value. The partide is then moved along that direction of x- axis in steps of Ax, corresponding to which the potential energy of the cluster starts decreasing. This process of movement of the partide in Ax steps is continued upto that step, beyond which the next step movement in the same direction starts increasing the potential energy of the duster. The partide is similarly moved along Y and then along Z axis and finally such a position of the particle is obtained that any further movement of the particle through the same step in any direction results in increase of the potential energy of the cluster. For the calculation of potential energy only those particles are considered, which lie within a distance of 20 fro, from the chosen particle. For each of the remaining particles within the duster, a similar position is obtained. The step movement is now gradually decreased and the calculations repeated. The final step movement is of 0.01 fm.

As the nucleus ~X N consists of Z protons and N = (A--Z) neutrons, the usual Coulomb term is also added to the interaction potential (equation (1)) at this stage. Thus the two-body interaction potential becomes

e 2 V~ = 4, [(cr/rij) le -- (~/r,j) e] -F - - . (2)

r i j

We also account for the fact that the potential between like partides is 20% weaker than that between unlike particles, while obtaining the values of E and or.

Now consider the ease of leO. All the sixteen particles (nudeons) are assumed to be at rest at positions obtained above. They arc then subjected to the field of potential (.equation (2)) assigning the first eight particles to have unit electrical charge. As such they start moving with acceleration along the direetion of the force and the motion is governed by the dynamical equations of motion. For obtaining position and velocity coordinates at successive small intervals of time, we take the help of the standard fourth order Runge-Kutta method (Kuo 1972). We also assume that the initial velocity coordinates for each of the successive intervals of time are zero. Due to this restriction particles are compelled to move in the direction corresponding to which the potential energy (PE) of the duster goes on decreasing. The potential and kinetic energies are also computed at the end of each step. To ensurt that the conservation laws of momentum, angular momentum and total energy to hold the time step is indeed very small such as 0.02 nuclear second (1 NS = 10 -32 see). These calculations are computed till all the particles occupy positions corresponding to which the Pr of the duster becomes a minimum.


3. Determination of the parameters ¢ and

The RMS radius of a cluster (r~) 1/2 in its ground-state configuration is essentially a function of the parameter or, and with different values of g, this radius in its ground- state configuration is calculated by using the relation

(r~)l/ ' = [27' (r' -- R)2] x/~" A (3)

In the above relation, r~ and R are the position vectors of ith particle and the centre of mass respectively with respect to the origin; A is the total number of particles in each cluster. From these calculations the ~ value chosen is that corresponding to which (r~) 1/~ is approximately in agreement with that of the liquid drop model and thus it is approximately equal to 1.2 A 1/a.

The other parameter e is essentially the scaling factor which decides the binding energy of the cluster. The binding energy of the cluster is calculated for different values of e. From these calculations, the appropriate c value, giving about 16 MeV as the binding energy per particle, is obtained.

Without the inclusion of the Coulomb part, the appropriate values of cr and e to be used in the potential (equation (1)) are 2 fm and 2.3 MeV respectively. With the potential (equation (1)) and the above mentioned values of ~ and e, the minimum potential energy per particles (PE/A) is calculated as a function of A -1Is.

From the plot between PE/A versus A -1/3 (figure 1) one observes a linear relation- ship of the form given below,

PE/A = ao - - a .4 -1/3. (4)

The a 0 and a values as obtained from the plot are 15.33 and 19.93 MeV (figure 1)

.5

-7

i

,13

-IL? olz. ~ I l

0.~ 0.6 ~.113

Figure 1. Potential energy per particle (e~/A) vs A -1Is. of particles within the duster.

A represents the number

Classical description of ~0 + ~0 collisions 527

respectively. This justifies the fact that the Len~tard.Jones form of potential is a good approximation even for short-range nuclear interactions, as far as their macroscopic properties are concerned.

With the modified form of the potential (equation (2)) appropriate values of ~ and are 1-8 fm and 2-2 MeV respectively. The potential (equation (2)) with ~ = 1.8 fin

and c = 2.2 MeV is actually used throughout the calculations in leO q- 1~O scattering. We now have a classical assembly of particles which interact via a nuclear as well as

appropriate Coulomb potential such that the total system is in its ground state. Such assemblies are used for studying the collision problems as described below.

4. Trajectory computations

For studying ~ 0 + 160 scattering, which includes head-on as well as peripheral collision with different initial velocities, we proceed as follows. The two identical clusters each with sixteen particles and half the particles charged are assumed to be in their ground-state configurations. The particle positions corresponding to their ground states are obtained by the method described in § 2. For the head-on collision, the clusters are displayed oppositely from the origin through a distance of 10 fm along Z direction (figure 2). From these positions, the clusters are allowed to approach each other with velocity v along Z direction. Trajectories traced by the centre of mass of each cluster as well as by all the individual particles within each cluster are computed as follows.

Let the initial radius vector of ith particle within the cluster with respect to the origin be h. We consider this particle to be in the field of potential V, defined by (2). The equation of motion of this particle is taken as

M d~-dT-~=ri -- Vl ~ V,. (5)

For obtaining position and velocity coordinates of this particle at successive small intervals of 0-02 NS we solve (5) by using the standard fourth order Runge-Kutta method (Kuo 1972) assuming initial position and velocity coordinates of the particle to be known. Similarly the position and velocity coordinates of all the remaining particles within the two colliding clusters are calculated after each of the successive intervals of 0.02 NS.

e v

© Cluster- Z

T ¥

- v

0 0 Q Z - - - ~ Clu l t e¢ - t

Figure 2. Starting positions of the two clusters in the head-on collision with initial velocity ~.


From the position and velocity coordinates of individual partides in each duster, the position and velocity coordinates of the centre of mass of each of them are also computed. The potential and kinetic energies of the individual dusters, internal angular momentum transfer and the variables

R = 2 I Z [, (6)

and S = 2[ Iz-- i - I (7)

are also computed after each successive interval of 0.02 Ns. For peripheral collision the two dusters are again displayed symmetrieaUy with

respect to the origin in YZ plane such that their centre of mass separation along Y and Z axes are b and 20 fm respectively, where b is the chosen value of the impact parameter say (4 fm). The two dusters are then allowed to approach eaek other along Z direction with the velocity v and impact parameter b (figure 3). Collision trajectories are computed for this as well as for various other impact parameters. Here also along with the trajectory computations, other calculations similar to head-on collision are made. For any given velocity (v) at a particular impact parameter, fusion starts, this impaot parameter is defined as critical impact parameter (be) and the same is determined from the computed collision trajectories.

5. Definition o f fusion cross-section

In a classical scattering the impact parameter b, scattering angle 0 in the centre of mass system, and the differential scattering cross-section (do/dn) are related as

d o __b db (8) dl~ sin 0 ~ '

since d[l = sin 0 dO de,, (9)

therefore do -- b dbsi n 0 d0d~ ---- b db d~. (10) sin 0 dO

l Cluster- I

. . . . . . i

I " 4

Figure 3. Starting positions of the two clusters in the peripheral collision with impact parameter b, initial velocity v and separation R.

Classical description of 160 + leO collisions 529

For obtaining the fusion cross-section cr F, (10) is doubly integrated between the

limits 0 to 2,r and 0 to b c. Thus

27r b c

OF= f d$ ~ b db= 27rb~/2. 0 0

(11)

Similarly the expression for the scattering cross-section (eSC) is as follows:

0O

OSC ----- 2rr f b db. b~

(12)

6. Random rotations

The distribution of nucleons within the cluster is not spherically symmetric, hence in the trajectory computations, it is necessary to take into account the situations wherein at each impact parameter the cluster trajectories are computed with different initial random relative orientations. The changes in the initial relative orientations do not alter the relative positions of the particles within the duster with respect to its centre of mass. Therefore the initial ground-state configuration of the cluster remains invariant with these changes.

The changes in the initial relative orientations of the clusters are made by the random rotations given to the clusters in space. For these rotations a rotation matrix is used.

For each initial velocity different initial random relative orientations are considered. At each orientation (designed as IQ) collision trajectories are computed at different impact parameters and from these the critical impact parameter (b,)is obtained, and these parameters are different for different orientations. The average of the critical impact parameters is therefore taken as the actual critical impact parameter (bAc)for each initial velocity.

7. R and S variables

In order to study the deformation of the clusters in the collision process, the follow- ing two macroscopic variables (Nix and Sierk 1974) are defined

R = 2 [ Z I, (13)

and S = 2[iz I - I ,, (14)

where 17 [ S Z d m (Z) 1 - f dm (Z ) - A I Z' [' (15)

i

and I fZ'dm(Z) 1EiZ~l ' (16) -- f dm(Z) --A

i


where the integration is between the limits corresponding to the half volume to the right of the mid-plane of the reflection symmetric shape (figures 4 and 5).

In the above equation the quantity dm represents the mass of an elementary circu- lar disc considered within the cluster at a distance Z from the origin 0 (figure 4). In the collision trajectory R represents the centre of mass separation between the two separated clusters. It is also the measure of elongation of the unseparated clusters during the state of fusion; S is the measure of deformation of one single cluster. A plot of S versus R is extremely useful in the study of shape deformation of the individual clusters during the entire interaction process.

In the head-on collision, we first calculate the values of R and S at the instant when the two colliding clusters are in contact with each other (figure 4) and these values are 2/7 and 2b/2.23 respectively. Hence we get S/R = 0.45. Thus in R -- S plane loeii of points representing touching spheroids will be lying on the line with the slope S/R = 0.45. This line is named as The line of tangent spheroids (LTS).

In order to study the shape deformation of the constituent clusters during the process of fusion in head-on collision, we consider one of the constituent clusters, represented by an ellipse (figure 5) and again calculate the values of R and S at any instant. These are 0.75b and 0.4873b respectively. Hence we get SIR = 0.6497. Thus in the case of fusion, just after the two clusters come in contact with each other the locii of points each representing single spheroid at different instants of time will be lying on the line with the slope S/R = 0.65. The line is named as the The line of spheroids.

Note that this interpretation of R and S is valid till the Z axis coincides with the line joining the centres of mass of the two clusters.

8. Head-on collision trajectory

For the head-on collision trajectory with initial velocity (v) equal to 1.5 fm/Ns the two clusters are displaced oppositely from the origin through a distance equal to 10 fm along Z direction (figure 2). From these positions they are allowed to approach each other with the velocity v----- 1-5 fm/Ns. In the beginning upto the fourth Ns from the start, both the clusters move oppositely along a straight line path. These clusters come in contact with each other at about the fifth NS. (figure 6). From now on the constituent clusters start compressing each other. As a result of this com- pression, within a period of 6 NS, the separation between their centres of mass along Z-direction decreases from 5.7 to 3.06 fro. While being compressed, both the clusters

l v': I ~ . *er-* Ct u l t t s r - 2 V

i

[" :

Figure 4. Two clusters just in contact with each other in the head.on collision.

r

Cluster ° I

Figure 5. Cluster-1 in the state of fusion in the head-on collisions

Classical description of x60 + ~60 collisions 531

oscillate along X and Y directions with very small amplitudes and their frequencies of oscillations along these two directions are not the same. After being compressed once along Z-direction, the constituent clusters oscillate along Z direction throughout the remaining period of their trajectories. During this period, small oscillations along X and Y directions also continue.

Figure 6 shows the positions of protons and neutrons within the clusters by Greek and Roman letters respectively, while they move in head-on trajectory. The relative positions of these nucleons in the clusters start changing with respect to their centre of mass from the instant the nucleons of each of the two clusters come within effective potential range of each other. The r~+te of change of relative positions of the nucleons increases with the decrease in separation between the two clusters.

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® • 32 °28 ~x~x 0 0

zS.,Vlll x ; . . 2 9 xxj.", v~,

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z;

Figure 6. a, b and e. Positions of protons (Greek letters), neutrons (Roman letters) Within the two colliding clusters at different timings (t) in head-on collision with initial velocity v (1.5 fm/Ns).

After fusion, changes in relative positions of the nucleons are mainly due to alter- nate compressions and expansions of the constituent dusters.

9. Head-on collision trajectory in R-S Plane

Figure 7 shows a plot of S versus R for the head-on collision discussed above. In the beginning the value of S, which is a measure of shape deformation, remains constant for about 5 Ns. This shows that till this time, there is no shape deformation. Just after this as clusters move further, they come in contact with each other. Finally these get fused and a composite system is formed, which remains in a bound state during the rest of the trajectory period. We find that just after coming in contact, R first decreases approximately from 5.7 to 3.49 fm and thereafter it increases and decreases alternately to different magnitudes (figure 7).

After the fifth NS the S value also starts changing along with the changes in the R value (figure 7). Initially it decreases slightly from its initial value of 2.53 fm and then increases and becomes equal to 2.60 fm at about the seventh NS. In the remaining period of the trajectory it oscillates about its initial value. The period of these oscillations also varies in a random fashion. Thus it is evident that the shape of the constituent clusters changes continuously. With the changes in the shape of the constituent clusters, the shape of the composite system also oscillates around the pro- late shape, i.e. the deformation becomes smaller and larger as a function of time.

Classical description of leO + leO collisions 533

J, f /

,6 Rlfm}

HEAD ON COLLISION

I 1~ 20

Figure 7. Head-on collision trajectory of one of the clusters in R - S plane with initial velocity v. (I. 5 fm/Ns).

Figure 7 also shows the line of spheroids and the line of tangent spheroids. On these lines lie the locii points representing single spheroids and touching spheroids respectively.

10. Energy variations in head-on collision

Figure 8 shows the plot of total kinetic energy of the colliding system (ET) , kinetic energy of the relative motion of the two centres of mass (EcM) and the kinetic energy of relative motion of nucleons (Ex) as a function of time in head-on collision. In the beginning, upto about the fourth Ns, the decrease in ECM (6 MeV) goes as pw of the system. The Coulomb interaction is mainly responsible for this nature of variation in ECM within this period. The clusters come in contact with each other in the fifth NS and form an excited composite system which remains bound permanently. The rate of decrease of ECM is maximum within about fifth and seventh NS and by the end of this period nearly the whole of ECM is lost, about 30 MeV of which goes as Ex and the remaining part (7 MeV) gets converted as 1,E of the system. The total kinetic energy (ET) is hereafter mainly E~, which varies in oscillatory manner about a mean value of 26 MeV. From the nature of energy variations it is evident that after the formation of the composite system, there is a continuous exchange of energy between E~ and PE of the system.

11. Peripheral collision trajectories

Figure 9 shows the trajectories of the centre of mass of one of the two clusters in YZ plane, with initial velocity v = 1.5 fm/Ns and impact parameters (b) ranging from 8 to 5.8 fro. In all cases, the clusters are initially displaced oppositely from the origin along Z and Y directions simultaneously through distances equal to 10 fm and half the value of the chosen impact parameter ~espectively (figure 3). From these posi-

534 A N Dixit, V $ Ramamurthy and Y R Waghmare

$e

30

2s

| 2Q

;. Is

10

S

0 0

f \Ex

.!]

S !0 15 20 t ( N.SJ

Figure 8. Total ~ (Er), r~ of the two centres of mass (EcM) and that of the relative motion of nucleons (E x) vs time (t) in head-on collision trajectory with initial velocity v (1"5 fm/NS).

b8 "0,%

b, S.O ,3 I !

| -...o

• v ~

b-8.3 / |

30 y 5 0 ~ ~E ---o

C Y~ r~

b~Q.O f .Z

0 1 0 / 5 C O* • !0 /

4,

o 'T ",_~

| . . . . .

Figure 9. Trajectories of the centre of mass of one of the clusters with initial velocity v (1.5 fm/Ns) and different impact parameters b.

tions they are allowed to approach each other along Z-direction. In what follows we shall now discuss three peripheral collision trajectories exemplifying three different scattering situations. For instance, for the value of b = 8 fm one has only the Coulomb scattering (figure 9A). For b = 6 fm nuclear forces show their presence (figure 9C) and for b = 5.8 fm one has the situation in which the two clusters fuse (figure 9I:)).

Classical description of 1~0 q- x60 collisions 535

Consider the trajectory with impact parameter equal to 8 fro. In the beginning upto 1 NS, the clusters move along a straight line path, without recognizing the presence of each other. As the clusters move further, they start experiencing Coulomb interaction and begin to deviate away from each other along Y-direction. As such the nature of the trajectory changes from straight line to curved (parabolic) path in YZ plane. Moving like this, the clusters cross each other. In the outward journey, the curved path motion continues till the clusters are within the 'range' of Coulomb interaction, but beyond it they again start moving along a straight line path.

Between the fifth and ninth Ns, both the clusters come within strong Coulomb force field of each other. It is this Coulomb interaction, which is responsible for the rapid changes in the value of S within this period.

Consider the trajectory with impact parameter equal to 6 fin (figure 9c). It shows that after the third Ns deviation of the clusters along Ydirection is of lesser magnitude as compared to the case with b = 8 fm. This is due to that fact that along with Coulomb interaction, nuclear interaction also starts playing its role just after the third NS. Moving like this the two clusters come to the positions where they combine to form a composite system. The composite system so formed is shortlived and the period of its existence is called its life-time. While the composite system rotates about the origin, the constituent clusters continue their motion along Z-direction. Finally a neck is formed and then the composite system splits into two fragments. In the present case, these are identical to initial clusters. The fragments so formed move away to infinity.

Now consider the trajectory with impact parameter equal to 5.8 fm. We find that in this case from the beginning upto 5 Ns, the nature of the trajectory is similar to that with b = 6 fm. With further motion, the two clusters come in contact with each other and thereafter they combine to form a composite system. The composite system so formed remains in bound state throughout the rest of the trajectory and rotates about the origin. In the present case, during fusion, loss of kinetic energy from the two centres of mass is very large and is responsible for the system to remain in bound state permanently.

Figures 10 to 12 show the positions of protons and neutrons within the clusters at different times, while they move along the above mentioned collision trajectories. Protons and neutrons have been shown by Greek and Roman letters respectively.

In each case very shortly after the start, relative positions of the nucleons with respect to the centres of mass, of each cluster, start changing and continue to change upto the end of the trajectory. The rate of change in the relative positions of the nucleons goes on increasing with decrease in the impact parameter. From the calculations, for ECM ---- 37.44 MeV, the criticalimpact parameter (at which fusion starts) is 5.9 fin. We have also calculated critical impact parameter (be) for collision trajectories with other initial velocities such as 2, 1.8, 1.3 and 1 fm/ss.

12. Energy variations in peripheral collisions

Figure 13 shows the plot of the kinetic energy of the centres of mass of the two clusters (EcM) as a function of time for the above mentioned three-collision trajectories. It shows that initially upto about 2 Ns ECM decreases with the same rate in all the three


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=5

I , 15

Figure 11. Positions of protons (Greek letters), neutrons (Roman letters) in a cluster in its collision trajectory with impact parameter b (6 fro) and initial velocity v (1.5

NS 22 fm/NS) at different timings (1 = 10- sec).

cases and thereafter it varies differently in different cases. Figure 14a shows the plot o f E r , ECM and Ex as a function of time for the collision trajectory with b = 8 fro.

We find that E r and ECM both decrease continuously till the clusters approach each

other, and thereafter start increasing continuously. It is observed that at the instant

Classical description of zsO + zeO collisions 537

t=O i

. -~0 )Nil 31 X~Ill

oXXIV •

~= ;9 xx., ;x

• % + % Vl 1=6

..!1 ° ' 0 S

;o 3"= . =+.m .

= 9 xxm xx

r6 0 ÷ 16 0

t=O

t ; ~ r ~ eeVHI .

• 11 V

~'.fi %"

,'o~ ; 10

t=13 v T x~x. t=9 ~ ~ Y l

x;,z =s z; ;

°11 I~ x~ ;= ;9 x;.~z vm ,; • xxlv l~,'vl lz. v~l ~.,,

~2 xv~ "z7 .3o 16. 1~ " 9 eXXI

I~ v'. ~ o ~ = o o • 15 m. x'v,, ¢o x'v.. ZV:xx~v.XV,j.xrx ~"

I'1 ..13 30 29. • XXIII

• (b) z~ ¢

160 + =~0

Y! 'l t = 21 16 eVtlt

• Xlll

10 • VI 13 II1 ~ ; g , • 30 ~'o I~ 9. ;,I ~;~ x~tv ,z

09 • o

00,~vl I "31 ~ =- XX~III 0 ,,IV ~7" e Vl o16 e30 • XVlll xv~ll XVll T3 V~

. (c)

t=17

Z

7 /

Figure 12. a, b and c. Positions of protons (Greek letters), neutrons (Roman letters) during fusion in a collision trajectory with impact parameter b (5.8 fm) and initial wlocity, v (I '5 fm/ss).

P.--7

538 A N Dixlt, V S Ramamurthy and Y R Waghmare

f,o- "0*%

zc

t5

1o I s

3C

z

" " b • 9,0 s I

b, 6.0

r I [ 1o ts 20

t (N ,S) F I $ .13

Figure 13. Kinetic energy of the two centres of mass ECM (E~M+ E~M) as a function of time for the collision trajectories with impact parameters(b) and initial velocity v (1.5 fm/Ns).

A )

" " 3 5 Z u

3 0 uJ

,.oL (a) t,.s o

• \ o ÷ o , f 3 5 - ~ _

3 0 . . . . . ~ "-- 46 ! t ]

.,~ (b) b= 6.0

! ' ,J. . , , /, . , . /

2 s - ~ ,,"" "-"ix \'-

i !! s s " " 2 0 - I ~ / ' E C ~

. . . . . . . I..,, " j ! x / ' / " I~ l , I o 0 § 10 15 20

t ( N . S . )

4Z

z~ 0

20

X

10 w

- 5

Figure 14. a and 5. Total v.~ (Er), KE of the two centres of mass (EcM) and that of the relative motion of nucleons (Ex) vs time (t) in the collision trajectories with impact parameters (b)0

Classical description of ~0 + leO collisions 539

/.0- (c) /

!

~ 3o

ZC- [

i 160 6

t

Figure 14 c. initial velocity v (1.5

r \V

b - 5.0

l ~ • , , l % •

, , , ",./zl 0 (N.$.)

fm/NS).

25

- 2 0

s< t~

10

when the two outgoing clusters are once again at an approximate separation of 20 fm (initial separation) from each other, E r and ECM are about 38 and 37 MeV respectively. From the comparison of these values of E T and ECM with the initial kinetic energy of the colliding system (37.44 MeV), we conclude that the scattering is approximately elastic. Clusters being many-body system, nucleons within them pick up internal motion even in pure Coulomb field. This explains why the present scattering is only approximately elastic. Thus in the present case the increase and decrease of energies along the collision trajectory are mainly due to Coulomb interaction between the clusters.

Figure I4b shows the variation~ of ET, ECM and E~ for the collision trajectory with b = 6 fm. Initially upto the fourth NS, ECM decreases by about 4 MeV and the whole of it gets converted as PE of the system. The Coulomb interaction is mainly responsible for this change. Between the fifth and seventh Ns Ex increases by 1 MeV along with a flight increase in ECM. From now onwards upto the eleventh Ns ECM decreases whereas E~, increases and the rate of change of both (EcM and E;,) is maximum. As such by the end of this period ECM and Ex become equal to 16 and 11 MeV respectively. After this ECM starts increasing and the rate of increase decreases with time whereas Ex oscillates about a mean value of 12 MeV. The collision trajectory and the nature of variation of energies show that between the fifth and eleventh NS short-range nuclear interaction is mainly effective and after this it is the Coulomb interaction which dominates.

Figure 140 shows the variations of E T, EeM and E:, with time for the collision trajectory with b = 5.8 fin. Within first 5 lqs ECM decreases by 5 MeV and the whole of it gets converted as the P~ of the system. Ex starts increasing from the sixth Ns and continues to increase upto the tenth NS. Within this period (sixth to tenth NS) the rate of increase of Ex is more as compared to the rate of decrease of ECM. This is due to the fact that during this period, along with ECM, the 1,~ of the system also


contributes to Ex. As a result by the end of the tenth Ns, E T becomes equal to 44

MeV, the corresponding values of ECM and Ex are 18 and 25 MeV respectively. In

the eleventh Ns E T and ECM both decrease and the rate of decrease of ECM is more than that of E T. As such E~ further increases and becomes equal to 27 MeV. From

now on, while the composite system rotates permanently about the origin ET, ECM and E~ all the three oscillate about their respective mean values of 41, 18 and 23 MeV.

13. Angular momentum transfer

We have also calculated the angular momentum of each of the two clusters about their centres of mass at successive intervals of 1 NS, in units of h. It has been verified from our calculations that initially in all collision trajectories it is equal to zero and it develops with time. Thus the calculated angular momentum at different instants of time actually represents the transfer of angular momentum from the initial orbital angular momentum of the clusters. Figure 16 shows the values of the angular momentum transfers Al 1 and Al~ for some of the collision trajectories each with initial velocity v ---- 1.5 fm/Ns and different impact parameters ranging from 8 to 6 fro. In each ease Alz and Al~ are calculated at the instant when the outgoing clusters are at a distance R from the origin (figure 15), where R is the initial separation between the centre of mass of each of the two clusters and the origin. It is observed that Al z and A12 increase with the decrease in impact parameter. We have found that the angular momentum transfer is also the function of the initial random relative orientations of the clusters in space. This is expected because in the ground-state configuration the particles within a cluster are not situated symmetrically with respect to its centre of mass. Calculations for the collision trajectories with other initial velocities give similar results.

II I 0 , A

t ~ , S O f m

i , .F' o

I *O+'*O i l - I o , 8 b c • (~i0 fm

Figure 15. Position o f the out going cluster at a distance (R) from the origin in its critical impact parameter trajectory at two different initial relative orientations 0o). Initial velocity v (= 1.5 fm]Ns).

Classical description o f xeO q- ~ 0 collisions 541

4' f3 I| 160 ÷ 160

<1 \ t~

f I r ! 6 7 8 - bffm) Figure 16. Angular momentum of nucleons within each cluster about its centre of mass (Al~ and Al:)vs impact parameter b.

14. Life-time of resonance

The life-time of resonance of the two colliding clusters is given as (figure 15).

T ---- (t -- t') q- [(R/v cos 0) -- (R/v' cos 0)], (17)

where R is the distance of the starting position of the centre of mass of the cluster from the origin, t is the actual time taken by the duster to move in the field of potential (2) through a distance corresponding to which once again it is at the same distance R from the origin (figure 15) in its collision trajectory, t' is the time which the duster would have taken to travel the distance 2R in the absence of any potential field along the direction passing through the origin, v is initial velocity and 0' is the velocity at the instant when it is at the distance R from the origin and moving away from it (origin). 0 is the angle between the initial direction of motion and the straight line joining the starting point with the origin. Similarly ~ is the angle between the direction of the motion of duster after the time t and the straight line joining this position of the duster with the origin.

The nuclear time is defined as the time taken by the duster to travel a distance equal to its root mean square radius ( r : ) 1/~ and is given as

TNuc! = (:)1/2/v (18)

with initial velocity v = 1.5 fm/NS and ( r : ) x / : = 2.24 fin approximately, TNuel

comes out to be 1.49 Ns. We have also calculated T for the collision trajectories. Table 1 gives calculated values of Tfor some of the collision trajectories. From table 1 it is evident that T is a function of initial random relative orientations. It also increases with the decrease in the impact parameter. The eases with T > TNucl are of special interest, representing the phenomenon of resonance scattering.


Table 1. Life-time of resonance of the two colliding clusters (T) as a function of initial random relative orientations 0o). The trajectories considered are with critical impact parameter (be).

be T I0 (i'm) (Ns) T/TNuel

A 6.0 3.22 2.16 B 6.0 7.40 4.96 C 6.1 2.97 1.99 D 5.7 0.12 0-08 E 6.0 2.43 1.63 F 5.7 2.39 1.60 G 5.8 0.30 0.20

Table 2. Calculated critical impact parameters and fusion cross, sections, with and without initial random relative orientations as a function of initial kinetic energy of the centre of mass of the system (EcM).

Calculated results

without v ECM 1/EcM rotations with rotations

fm/NS (MeV) (MeV) -x be ~F bAC CrAF

(fm) (mb) (fro) (rob)

2"0 66"56 0'0150 5"1 816"71 5 "lg+°q9 836"04 ~ v - 0 . 5 1

1"8 53"81 0"0186 5"3 882"03

1"5 37.44 0"0270 6"0 1130"50 5 '91+o°~ 1096"70

1"3 28'12 0"0360 6"3 1246'27

1"0 16.64 0'060 5'8 1056"30 5"81 +o.54 1059.94 - -0 .20

I t

15. Calculation of fusion cross-section

For different collision energies fusion cross-sections (aF)are calculated using(l l) for both the cases with and without the inclusion of different initial random relative orientations. Table 2 gives the values of the calculated critical impact parameters (be) and fusion cross-sections with and without initial random relative orientations. The error bars indicated stem from the spread of critical impact parameter values for various initial random relative orientations.

aFversus inverse of collision energy (EcM) curve is plotted in figure 17 in the form of a continuous line for the case when different relative orientations of the clusters are not considered in the determination of critical impact parameter. These results are compared with the experimental results of other workers shown in figure 17 (Spinka and Winkler 1974 (diamonds), Kolata et al 1977 (dots), Fernandez et al 1978 (triangles) and Tserruya et al I978 (open circles)). From figure 17 it is evident that although for higher energies the theoretical results agree with experiment, the deviation between the two increases with decrease in energy. Crosses in the figure show the three representative cases where fusion cross-sections have been calculated

Classical description ofZSO + JJO collisions 543

Figure 17.

1500

1250

1000

,10 E b~" 750

SO0

.o ,o il o__

o "Yo •

• o@

e@

°• '~

1 I I 1 0.02 0.0 4 0.06 0.08 0.10

E'cZom. ( l,~zV -z) Fusion cross-section (%.) vs E~[.

from the average value of the ¢riti~l impact parameter (bAc) for each energy. It is noticed that with different initial orientations the average value of the critical impact parameter for each energy and hence OAF are also in better agreement with the experimental results only for energies above 37.44 MeV.

16. Comparison with other calculations

The first application of TDHF to the collision reactions was made by Bonche et al 0976), who numerically studied the one-dimensional collision of slabs of nuclear matter. Koonin 0976) and Koonin et al (1977) studied x60 -I- xso and 4°Ca -~ 4°Ca reactions and these calculations were two-dimensional. Fully three-dimensional calculations were first made by Cusson et al 0976). Flooard et al (1978) made extensive studies of zeO + 160 reactions at Ela b = 105 MeV by using a technique for solving the TDHF equations in three dimensions. Bonche et al 0978) used the three-dimensional calculations in coordinate space for z ~ -t- zeO at various bombard- ing energies and impact parameters. These tesults of Bonche et al have been shown in figure 18.

Birkelund et al 0979) calculated fusion cross-section using classical dynamical models based on (a) modified nuclear proximity potential (Randrup 1978a) and Coulomb potential (b) standard nuclear proximity potential (Blocki et al 1977) and Bondorf Coulomb potential (Bondorf et al 1974) and (o) standard nuclear


1500

.o E 1000

500

0

- - Clauk:ol dF'czfl~l Pm~nt ¢al¢.

- ¥ °°'! Io

- Io

I I I I! 0 2 4 6 8 x 10 "a

EC M-t (M¢~t)

Figure 18. Fusion cross-section (o F) vs E ~ as calculated by other workers along with the present results.

proximity potential and point charge Coulomb potential. In conjunction with all three of the above models, they employed the nuclear one-body proximity friction as proposed by Randrup (1978b). Birkelund et a/found that the three different models give nearly identical results and the same have been shown in figure 18.

Figure 18 also shows the present results. We find that with the exception of ECM= 16.64 MeV, these results are in good agreement with the TDHF results. It is also observed that for ECM = 66-56, 53.81 and 37.44 MeV our results are also in good agreement with classical dynamical calculations. However, for ECM = 28.12 and 16.64 MeV our results are higher as compared to classical dynamical calculations.

17. Discussions and conclusions

The scattering phenomenon depends upon the impact parameter b of the collision trajectory. At large impact parameters it is of Coulomb nature. With the decrease in the impact parameter the nature of scattering changes from elastic to inelastic and in some cases even to deep inelastic with mass transfer. Sometimes a short lived composite system is formed. The life-time of composite system increases with decrease in the impact parameter. At the critical impact parameter (be) clusters fuse permanently.

From the study of the collision trajectories we find that the rate of loss of ECM increases with the decrease in the impact parameter. At critical impact parameter we find that by the time two clusters fuse together, the loss in ECM is so large that they are unable to part with each other just after fusion. With further decrease in impact parameter the rate of loss in ECM increases more and more; it becomes maximum in head-on collision. Especially in the head-on collision we find that after coming in contact with each other, the clusters come to a grinding halt within about 1 NS. This results in a great loss of ECM energy. The amount of energy lost from ECM is divided into two parts, partly in increasing the PE of the system and partly in internal kinetic energy of the system which results in exciting the nucleons. However, a major por- tion of the energy is used for the excitation of the nucleons. This is in agreement with experimental features observed in deep inelastic reactions of heavy ions.

Classical description of ~ 0 + leO collisions 545

From the head-on collision trajectory in R-S plane one finds that, just after fusion the surface of each of the constituent clusters (within the composite system) changes alternately to less and more prelate shapes. With the changes in the shape of the constituent clusters the shape of the composite system also oscillates around the pro- late chape. In peripheral collision cases it is observed that the trajectory in R-S plane is not the same in the entrance and exit channels. This shows that the entrance and exit channel deformations are not the same.

During the process of collision the angular momentum transfer from the orbital angular momentum of the (colliding system) clusters to the angular momentum of nucleons about their respective centres of mass takes place. This transfer increases with decrease in impact parameter. It is found that this transfer is also a function of the intial random relative orientation of the clusters.

The life-time of resonance of the colliding clusters (T), is the time during which clusters are held together at the centre due to nuclear interacions. This has also been calculated. It is found that T increases with decrease in the impact parameter and is a function of initial random relative orientation of the clusters. Trajectories with T/TNucl greater than one represent the phenomenon of resonance scattering.

In order to account for the fact that the distribution of nucleons within the cluster are not spherically symmetric we have included different initial random relative orientations.

It is observed that at any impact parameter the nature of scattering depends upon the initial orientations of the colliding clusters. Some times it changes from elastic to inelastic with change in the initial orientations of the clusters. The magnitude of angular momentum transfer from the initial orbital angular momentum of the clusters (colliding system) to the angular momentum of the nucleons about their respective centre of mass also changes with the change in the orientations of the cluster. But the contribution to o F from relative random orientations is only about 5-10%, the average value being around direct collisions i.e. without relative random orientation.

The impact parameter which barely fuses the colliding system, determines the fusion cross-section for the fusion process and is a function of initial energy of the centre of mass (EcM). Fusion cross-sections have been determined at different ECM values and have been compared with experimental as well as theoretical results of other workers. We find that with the exception of very low energies, the present calculations are in good agreement with the results of other workers.

The present results indicate that for projectile energies of the order of 1-2 MeV per nucleon (low energies), classical microscopic calculations for heavy ion reactions seem to give satisfactory description. However, at very low energies appreciable deviations are seen, which may indicate the breakdown of the classical approximations at very low energies.

Acknowledgement

The authors are thankful to Dr S K Kataria of BARe Bombay for many valuable discussions and suggestions.


References

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Classical description of low energy16O +16O collisions