Semiclassical description of elastic heavy-ion scattering for very heavy systems

2.N

]

Nuclear Physics A307 (1978) 2 9 7 - 308 : (~) North-Holland Publishino Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher

SEMICLASSICAL DESCRIPTION OF ELASTIC HEAVY-ION SCATTERING FOR VERY HEAVY SYSTEMS P. FROBRICH and P. K A U F M A N N

Hahn-Meitner-btstitut /i'ir Kern/orsclum~! Berlin, Bereich Kern- uml Strahlelq~hvsil~, Bo'lilt-14"est, Germal o' Received 1 May 1978 (Revised 15 June 19781 Abstract: It is shown that due to the simple topology of the turning points in the complex r-plane the

semiclassical method of using complex trajectories leads to a quantitative and practicable description of the elastic heavy-ion scattering for very heavy systems. A number of experimental cross sections are analyzed with this method and a comparison with quantal optical model calculations is performed.

1. Introduction

The semiclassical method of using complex trajectories in describing elastic heavyion scattering has been widely investigated in recent years ~- 6). However, it has been applied only to s-particle 6) or light heavy-ion scattering t-s), where at most 150 partial waves have to be taken into account. In the present paper it is shown that the method can be extended to the description of elastic scattering of very heavy systems, where more than 1000 partial waves may be important. It turns out that very large numbers of partial waves can be handled without difficulty, and it is shown that the semiclassical theory leads to a practicable computational procedure which gives quantitative agreement with quantummechanically calculated differential cross sections over several orders of magnitude. In sect. 2 the theory of using complex trajectories for the description of elastic heavy-ion scattering is reviewed briefly, and it is discussed how the turning point structure changes in going from light to very heavy systems. It is the simplification in the topology of the turning points which is mainly due to the strong Coulomb repulsion, that leads to a very practicable procedure for calculating differential cross sections. In particular it is discussed how the proper complex paths have to be chosen in the case of very heavy systems. In sect. 3 a number of experimentally observed angular distributions for very heavy systems are analyzed with the present approach, and a comparison with quantal optical model calculations is performed. In sect. 4 a criterion is given for which heavy-ion systems the theory can be applied. A first result of our investigations has been reported in ref. 7). 297

P. FROBR[CH AND P. K A U F M A N N

298

2. The semiclassical theory and the turning point structure of very heavy systems For the investigation of elastic scattering of very heavy systems we apply the semiclassical theory essentially in the form as developed in ref. 2), where only light systems (160+ 58Ni) are considered. We start with the Hamiltonian H = p2 + 2m ~

12

+ Vcb(r) -

V° 1 + exp [(r -

RR)/%]

W° -- i 1 + exp [(r--

R.)/a.]"

(1)

Here Pr is the radial momentum, l the orbital angular momentum and Vcb the Coulomb potential. We restrict the discussion to nuclear potentials of the WoodsSaxon type. The scattering amplitude is calculated as the sum over all important partial waves, 1 /max f(0) = 2 ~ ~=o(21 + 1) exp (2ifiCb)(exp (2i6~) -- 1)P~(cos 0) +fcb(0), (2) do"

d@ -If(O)12 Here 6cb are the Coulomb phases, 6~ are the nuclear phase shifts and fcb(O) is the exact Coulomb scattering amplitude; lm,x ~ Ro(2mEc.m.)½. The phases are calculated as the classical action for given angular momentum l and are obtained by solving the classical equations of motion, 3 t = -rpr,

OH r - 0p,'

P~-

0H Or'

(3)

with the boundary conditions for t ~ - ~ . Re Pr = - (2mE) ½, Re r = R

Im Pr = 0, (large),

(4)

Im r = arbitrary (chosen according to the physical situation). All quantities in (3) are complex because of the presence of the imaginary part of the nuclear potential, and consequently all trajectories are complex. This is somewhat different from the usual situation with real potentials where complex trajectories enter the semiclassical description only when classically forbidden regions are reached. In calculating the trajectories the choice of the boundary conditions (4) is of greatest importance. In principle one could choose any boundary condition for Imr at large distances (Re r large) if one introduces a complex time; but then the problem is how to choose the proper complex path in the time plane. We found it most convenient, in the case of the very heavy systems, to restrict ourselves to real paths in the time. plane, and therefore to complex paths in coordinate space as other authors 2-6) did also in the discussion of light systems. In the case of light heavy-ion

SEMICLASS1CAL DESCRIPTION

299

systems one has to be careful in choosing the boundary condition (4) for Im r; it is inadmissable to assume Im r = 0 for all/-values 2). In order to understand this one has to discuss the topology of the turning points in the complex r-plane. For each trajectory characterized by an angular momentum 1 there exist in general three turning points in the complex r-plane; i.e. there are three branches of turning points (as functions of/). A typical example for light systems 6) at low energy, where one has a deep real potential and weak absorption, is shown in fig. 1. For a-particle scattering at low energy all turning point branches contribute to the classical action 6). Considering light heavy ions 2) where it usually is assumed that the potentials become flatter and the absorption stronger, the outermost turning point branch gives the dominant contribution. But this turning point branch is still close to the first pole of the Woods-Saxon potential in the lower half-plane. As has been mentioned one has to pay attention to the boundary conditions (4) in these cases. In particular it is not possible to choose Im r --- 0 far outside the interaction region for all/-values, because in that case the trajectories for some/-values run around a pole of the Woods-Saxon potential, thus producing erroneous results 2). In these cases the boundary conditions can be shifted into the complex plane because of analyticity in such a way that the trajectories circumvent the relevant turning points without surrounding a pole. This makes the automatization of a computer program difficult. In the case of very heavy systems, where the repulsion of the Coulomb interaction increases considerably, and where one usually assumes relatively flat potentials 8 - 11), one encounters some simplifying features as compared to the light-ion case. A typical turning point pattern for a very heavy system is shown in fig. 2 for the case of 13654X~..i. _~_ 2081~1~82_t.(Ela , b --- 1120 MeV). The change of the turning point structure in fig. 2 as compared to fig. 1 is mainly due to the much stronger Coulomb repulsion for the very heavy system. The motion of the turning points from fig. 1 to those of fig. 2 can be understood as a function of increasing Coulomb energy as follows. The turning point branch in the upper halfplane contracts gradually around the first pole of the Woods-Saxon potential. With increasing Coulomb energy the two lower turning point branches come close to one another, connect for a given energy at a certain/-value and then break up again into two branches in such a way that each new branch now consists of a piece of the two old branches. Finally one of the new branches shrinks around the first lower pole of the Woods-Saxon potential; the other new branch, which is the important one, remains relatively close to the real axis. This is the situation shown in fig. 2. In the following we explain the properties of those complex paths in r-space which lead to accurate values for the elastic scattering cross sections of the very heavy systems. In contrast to the case of light systems we found it possible in these cases to choose the initial values Im r = 0 at large Re r, i.e. we start the trajectories on the real axis for all paths. An example for such a complex path is shown in fig. 2 (trajectoryl) for the partial wave l = 350. The path starts far outside the interaction region on the real axis, then moves slowly into the complex plane, encircles counterclockwise

300

P. F R O B R 1 C H A N D P. K A U F M A N N lmR

~°Ca(~,~}Z°Ca (29.0 MeV)

[fm]

pole

0

3O

2

3

I

1

/o

~2.o

1/,/

~

9 I

I

I

20

Re R 25 X pate

Fig. 1. The three turning point branches for a light system at low energy [co + Ca, E.~b = 29 MeV, from ref. ")]

/ "

136

5/. xe

J

208ph

+

(3)

82--

/

ELob=1120MeV

o =oo / soo /

lm R [fm] I

/

/ 9 i

--0

-1

10

11

i

12 /

i

=

-

•

/

Re R [ fm]

13

I/

I

..t

(2)

~

_.i

lt,

~~ " ~

15 I

.

500

16 A

17 '-

600

18 I

19 _~

700

[3°~'" ~00 \important turning point branch 300 ~,u .... I. - - " . ((1) complex classical trajectory

200

D,

.400

0

Fig. 2. Turning point branches for a very heavy system ( X e + P b , Ea, b = 1120 MeV). The crosses are the first poles of the Woods-Saxon potential. Trajectory I is a proper classical path for I = 350 with the boundary condition lm r = 0 at large Re r. The trajectories can be deformed as long as they do not pass over a pole, e.g. trajectory 2. The classical trajectory 3 is a forbidden one because it encircles the first pole of the potential in the lower half-plane.

SEMICLASSICAL

DESCRIPTION

301

the relevant turning point in the interior of the nucleus and runs back inside the first poles of the Woods-Saxon potential. The trajectories for the other/-values show the same behaviour. N o trajectory runs around a pole. The possibility of choosing the same boundary condition for all trajectories facilitates the automatization of a computer program. Instead of using the procedure outlined above, one can, by analytical continuation, deform the trajectories in the complex r-plane as long as they do not pass over a singularity. For instance one does not have to use the solutions of the classical equations of motion for the trajectories, but can instead choose the s a m e straight-line path for all/-values (trajectory 2 in fig. 2) which encircles all important turning points counterclockwise without surrounding a pole. This has also been discussed for the case of light systems in ref. 5). In fig. 2 we further show the example of a forbidden path (trajectory 3) for l = 350, which is a solution of the classical equations of motion with a non-vanishing value of Im r (Im r = 3.4 fm) at large Re r. This trajectory encircles the first pole of the Woods-Saxon potential in the lower half-plane, and therefore leads to erroneous results. The results of the next section show that it is sufficient to take into account the contribution from the turning point branch closest to the real axis; the contributions from the turning point branches which are concentrated around the poles of the Woods-Saxon potential can be neglected. For all the examples in the next section it is sufficient to choose for all /-values classical trajectories with Im r = 0 for large Re r. For not too heavy systems at relatively low energy [cf. example (iii) in the next section] as well as for the case of light systems 2.5), where the turning point branches are not so well separated as in fig. 2, we found it advantageous not to use Im r = 0 as boundary condition for all trajectories. Instead, for large Re r the imaginary part of each trajectory is set equal to the imaginary part of the turning point of the preceding (as a function of l) trajectory: Im rt(t --~ - - oO) = I m r~U_r~ing~a°int. This guarantees that w e stay in the complex plane inside the first poles of the Wood'sSaxon potential. We should add some technical details. In solving eqs. (2) numerically we found it most convenient to subtract the Coulomb trajectories, i.e. the equations for ? = r - rc, fi = P - P c and ~- = f i - t i c (C = Coulomb) are solved and finally the exact results for r o Pc and 6c at the end of the integration are added. This can be easily performed by parametrizing the Coulomb trajectories in the form r c = a(1 + e c o s h co),

12

t = - (e sinh co + e~),

(5) Pc - d t

'

t$c = -

rcdp o

e=

1 -t- #]2) '

Here a is half the distance of closest approach,/Ji the velocity of the incoming particle and ~/the Sommerfeld parameter. The divergent part of the Coulomb phase can be

P. F R O B R I C H A N D P. K A U F M A N N

302

neglected because it is independent of l and therefore enters only as an overall phase in the S-matrix. Another simplifying feature is that all quantities are smooth functions of the orbital angular momentum l. Therefore it is not necessary to calculate trajectories for all partial waves, but it is sufficient to interpolate all quantities between reasonably choosen/-intervals. This saves a lot of computing time so that our semi-

~L Reflection

i~r

m

coefficient

L-os

Oquantal • semiclassicol

J v

v

v

100

v

~

w

200

300

/.00 L_

500

500

700

Fig. 3. Semiclassical and quantal reflection coefficients for 136Xe+ 2°sPb at Ej. b = 1120 MeV. o/o R

=10

-

~

~

'01 quentol and se miciassical -0,01 136Xe÷ 208pb 1120MeV -0001 El experiment I

20

I

30

I

i

40 50 ecru.

I

60

70

Fig. 4. Comparison of the semiclassical and the quantal angular distributions with the experimental data of ref. 8) for the same case as in fig. 3.

SEMICLASSICAL DESCRIPTION

303

classical code is generally faster than e.g. the optical model code of Vandenbosch 5), in particular when a large number of partial waves have to be used.

3. Comparison with quantal calculation and experiment We applied the procedure outlined above to a number of experimentally observed cross sections of very heavy systems for which also quantal optical model calculations have been done. As a first, typical example we choose the system 136Xe+Z°8pb ( E l a b = 1120 MeV), whose turning point structure has been studied in the preceding section. This system has been measured and analyzed with an optical model code by Vandenbosch et al. 8). Whereas the semiclassically calculated reflection coefficient shows small deviations from the quantal values, as can be seen in fig. 3 and table 1, TABLE 1 The quantal and the semiclassical reflection coefficients for 136Xe +-'°SPb at El. ~ = 1120 MeV Angular m o m e n t u m / 400 420 430 440 450 460 470 480 490 500 520 550 600

Reflection coefficient q(/) quantal

semiclassical

0.0111 0.0996 0.2068 0.3483 0.4993 0.6364 0.7459 0.8275 0.8850 0.9242 0.9678 0.9913 0.9991

0.0096 0.0924 0.1955 0.3354 0.4871 0.6258 0.7377 0.8214 0.8807 0.9213 0.9662 0.9907 0.9990

TABLE 2 C o m p a r i s o n of the quantal and the semiclassical cross sections for t~6Xe+ 2°SPb at Etab = 1120 MeV

a(O)/odO)

Oc,m. (deg) 40 45 50 52 54 56 58 60 62

quantal

semiclassical

1.0025 1.0089 1.0202 0.9136 0.5347 0.1895 0.0433 0.0068 0.0008

1.0015 1.0090 1.0187 0.9149 0.5376 0.1909 0.0436 0.0069 0.0008

304

P. F R O B R I C H A N D P. K A U F M A N N

the agreement for the differential cross section is very good over four orders of magnitude, covering the whole range o f the experimental data (see fig. 4). In tables 1 and 2 the values of the reflection coefficients and the differential cross sections of the semiclassical calculations are compared with the quantal results in detail, because in the logarithmic plot of the angular distribution one cannot distinguish the semiclassical and the quantal curves. Similar agreement between the quantal and the semiclassical calculations as for the X e + P b case is obtained for the following reactions, which also have been observed experimentally: (i) S4Kr+2°spb, Elab = 494 MeV, ref. 9). (ii) 1 3 6 X e + 2 ° 9 B i , Ela b = 1130 MeV, ref. lo). (iii) 4°Ar+2°9Bi, Elab = 340 MeV, ref. 11). (iv) 84Kr+ 2°9Bi, Elab = 712 MeV, ref. 11). In table 3 the parameters of the Woods-Saxon potentials for all investigated cases TASt.E 3 Parameters of the W o o d s - S a x o n potentials of the investigated systems System

Ref.

J 36Xe + 2O8pb 84Kr + 2OSpb 136Xe _1_2OOBi • OAr+ 2O~Bi 84Kr + .,OgBi 8 4 K r + 2O9Bi

8) 9) 1o) i J) 11) it)

El. b (MeV) 1120 494 I 130 340 712 600 j

Vo

aR

rR

Wo

(tI

rI

30 50 8.47 68

0.6 1. I 0.44 0.54

1.195 I. 178 1.3 I. 167

25 2.5 6.95 83.9

0.4 0.33 0.44 0.54

1.235 1.282 1.3 1.167

7.68

0.217

1.344

10

0.217

1.344

are given. The results of the calculations of the angular distributions are shown in fig. 5; the semiclassical and the quantal calculations fall on the same curves. We should mention that in the references above 9- ~1) the cross sections are also analyzed with a conventional Fresnel formula which does not give as good results as the semiclassical method. Meanwhile Frahn has modified the Fresnel formula for application to very heavy systems and also obtains very good fits to the observed angular distributions 12). In order to demonstrate that our method can handle even a very large number of partial waves, we consider the cross sections for the 84Kr+Z°9Bi system up to energies of ten times the Coulomb barrier in fig. 6. The agreement between the quantal and the semiclassical calculations is again very good for E~ab = 600 MeV [the experimental data are from ref. ~)] and also for E l a b • 2El~b (the Coulomb barrier is E~,b ~ 533 MeV; no experiment available). For E~ab = 5E~,b and 10Ec~,b the optical model code available to us did not work because too high partial waves are involved. On the other hand, the present method has no difficulties with calculating angular distributions at very high energies as is shown for 5Eli,b and 10El~,b in fig. 6. In particular one observes as expected the

SEMICLASSICAL DESCRIPTION i

ole) oriel

i

i

T

8t.Kr. 208pb' E

I

L

I

a

I

b

I

305 I

I

~

4°Ar. 2°9Bi, EL=340MeV

+ experiment

10"

10-2

I

I

I

I

I t

5O

I

I

I

I lOO

ec.m.(degrees)

Fig. 5. Comparison of calculated and experimental angular distributions. Quantal and semiclassicalcross sections fall on the same curve. appearance of Fraunhofer-like structures at these energies. These oscillations are real because they remain unchanged when 3000 partial waves are used instead of 2000. Moreover, the justification for the semiclassical procedure should be the better the higher the partial waves are.

4. Discussion and conclusion The cases investigated in the present paper as well as our experience in a n u m b e r of other examples indicate that the semiclassical method can always be applied without difficulties (no surrounding of poles, only one turning point is important) if there are no pockets in the real part of the effective potential Veff =

V c b + I/real" nuclear "-L"Vcentrifugal

at the projectile energy and in the important range of the angular momenta, and

P. FROBRICH A N D P. K A U F M A N N

306

o(o1

i

~

~

~

r

~

r

i

i

-("

1

"

~

i

i ///el

r

i.zf"l

~

T

65l

IS

oR(O) I

10-I

10-2

I0-3

10-I.

5

10

3

2

35 J 3I, 8 " 5~ 5

I

Oc.m

Fig. 6. Elastic cross sections for 8'~Kr+Z°'~Bi (Coulomb barrier Et~h = 533 MeV) for the energies E~ab = 600 MeV, 2E~.b,~ab5 E ~ and 10E~ b. For the two highest energies only the semiclassical calculation is available. The same potential parameters as for Eh,b = 600 MeV (see table 3) are used for all energies.

provided that the imaginary po'tential is of the kind considered in sect. 3, i.e. not too strong and with a diffuseness that is large enough. Then the turning point structure is essentially that of fig. 2. The relevant turning point branch is far away from the nearest pole of the Woods-Saxon potential and consequently the use of the boundary condition Im r = 0 is generally adequate. To avoid a possible running around a pole for cases where the topology is not as clear as in fig. 2, for each trajectory the imaginary part of the turning point of the preceding (as a function of/) trajectory can be used as boundary condition. A rough estimate for which systems can be treated in the single turning point approximation is obtained as follows. The absence of pockets in the real part of the effective potential means that d Veff/dr --- 0 has no solution. This condition leads to the relation

V0 4a R

<

ZIZ2 e2 R2

+

/2min mR 3

.

(6)

Here r ,,~ R 0 is assumed and lml. is the partial wave where the reflection coefficient

SEMICLASSICAL DESCRIPTION

307

starts to change rapidly away from zero. It can be expressed roughly a s / m i n = I v - A, where the grazing angular momentum lgr = Ro(2mEc.m.) ~ and A is something like twice the diffuseness if the reflection coefficient is approximated by a Fermi function. Inequality (6) can be written in the form 4a~

" "

l g J / R o'

(7)

where Ecb is the Coulomb barrier. This relation is fulfilled for all very heavy systems considered in sect. 3, because there the potentials are flat and the Coulomb repulsion is very strong. More generally the present theory should work for very heavy systems for all projectile energies, whenever the potentials are assumed to be sufficiently flat. A large diffuseness also helps to satisfy inequality (7) and therefore supports the validity of our approach. This can be also understood from the topological structure of the turning points (fig. 2), because the poles move away from the real axis with increasing diffuseness a, thus reducing the probability for a trajectory to run around a pole. From (7) it can be also understood that in the case of light systems, where the Coulomb barrier is small and the potentials are assumed to be rather deep, the turning point structure is still the same as for very heavy systems if the projectile energy is high enough. This means that the present procedure is also applicable for light heavy-ion systems at high energy. The change of the turning point structure of e-particle scattering with increasing energy and the use of a one turning point approximation for this case have been discussed already in ref. 6). We conclude that the treatment of the elastic scattering of very heavy systems outlined above is practicable and can be used with advantage for analyzing experimental data. It is superior to the usual quantum-mechanical codes when very high partial waves have to be taken into account. In our opinion the present method is also preferable to that of Knoll and Schaeffer 1) from a computational point of view if differential cross sections have to be calculated for all scattering angles. This is because in ref. 1) a stationary phase equation has to be solved separately for each scattering angle, which is numerically inconvenient, although it has the advantage that it shows which trajectories run to a definite scattering angle. We are indebted to R. Vandenbosch for providing us with his optical model code for very heavy systems, to K. M6hring for valuable discussions and to R. Lipperheide for commenting on the manuscript.

308

P. FR~)BRICH AND P. KAUFMANN

References 1) 2) 3) 4) 5) 6) 7)

8) 9) 10) 11) 12)

J. Knoll and R. Schaeffer, Phys. Lett. 52B (1974) 131: Ann. of Phys. 97 (1976) 307 T. Koeling and R. Malfliet, Phys. Reports 22C (1975) 183 N. Rowley and C. Marty, J. of Phys. 62 (1976) 217 Y. Avishai and J. Knoll, Z. Phys. A279 (1976) 415 S. Landowne, C. H. Dasso, B. S. Nilsson, R. A. Broglia and A. Winther, Nucl. Phys. A259 (1976) 99 D. M. Brink and N. Takigawa, Nucl. Phys. A279 (1977) 159; D. M. Brink, J. de Phys. 37 (1976) C5-47 P. Fr6brich, P. Kaufmann and K. M6hring, Proc. Int. Conf. on nuclear structure, Tokyo, 1977, p. 578; Proc. Int. Symp. on nuclear reaction models, Balatonf~ired, 1977, ed. L. P. Csernai, Budapest 1978, p. 331 R. Vandenbosch, M. P. Webb, T. D. Thomas and M. S. Zisman, Nucl. Phys. A269 (1976) 210; R, Vandenbosch, Optical model code HOP-TWO for heavy ions, private communication R. Vandenbosch, M. P. Webb, T. D. Thomas, S. W. Yates and A. M. Friedman, Phys. Rev. C13 (1976) 1893 W. U. Schr6der, J. R. Birkelund, J. R. Huizenga, K. L. Wolf, J. P. Unik and V. E. Viola, Phys. Rev. Lett. 36 (1976) 514 J. R. Birkelund, J. R. Huizenga, H. Freiesleben, K. L. Wolf, J. P. Unik and V. E. Viola, Phys, Rev. CI3 (1976) 133 W. E. Frahn, Proc. Int. Conf. on nuclear structure, Tokyo, 1977, p. 576; Nucl. Phys. A302 (1978) 267, 281