Nuclear deformation and sub-barrier fusion cross sections

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 605 (1996) 334-358

Nuclear deformation and sub-barrier fusion cross sections Akira Iwamoto, Peter M611er 1,2 Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Naka-gun, Ibaraki 319-11, Japan

Received 20 February 1996

Abstract We present calculations of sub-barrier fusion cross sections for spherical projectiles and deformed targets. For a given spherical-projectile and deformed-target combination we calculate exactly the sum of the Coulomb and nuclear potentials. There are no free parameters in the calculations, except for a simple energy shift of the calculated cross sections. The shapes of the target nuclei are taken from a global calculation of nuclear ground-state shapes that determined the ~2, e3, e4, and ~6 shape coordinates from a minimization of the nuclear potential energy. The cross sections are obtained by integrating the transmission coefficients over angle, which are obtained by calculating the barrier penetrability at each angular momentum by use of the WKB approximation. Some of the results, where high-precision experimental data are available, are analyzed in terms of "fusion-barrier distributions".

1. Introduction Sub-barrier fusion cross sections observed in experiments show substantial enhancement over cross sections obtained in calculations based on the assumption of spherical projectile and target shapes and a corresponding one-dimensional fusion barrier. This enhancement has been studied both experimentally and theoretically for about 20 years. Unless both the projectile and the target are very heavy, one can today in coupledchannel calculations normally obtain very good fits to the measured cross sections. I Permanent address: E Moiler Scientific Computing and Graphics, Inc. EO. Box 1440, Los Alamos, NM 87544, USA. 2E-mail: [email protected], 0375-9474/96/$15.00 Copyright@ 1996 ElsevierScience B.V. All rights reserved PH S0375-9474 (96) 00155-8

A. lwamoto, P MOUer/Nuclear Physics A 605 (1996) 334-358

335

These calculations have to account for, in some manner, the static and dynamic nuclear deformations of the projectile and target, and the associated "distribution" of fusion barriers. Normally these models contain several free parameters, in the most simple case for example the fusion-barrier height, the barrier frequency, the target deformation, and the fusion radius [ 1,2]. However, the values of these parameters are varied from reaction to reaction in order to obtain optimum agreement. It is also unsatisfactory that the value of the quadrupole deformation parameter r 2 obtained for one and the same target nucleus exhibits strong variations depending on the reaction studied and even on the model used to calculate the cross section. For example, for 154Sm/32 values ranging from 0.22 to 0.37 have been obtained [1,3]. It has been shown [3] that some of this variation has its origins in too simplistic model assumptions, but uncertainties in the value of/~2 remain. In contrast to the studies above, we calculate here for a series of X -4- 154Sm reactions fusion cross sections in a model that contains n o free parameters, except for a simple shift in energy of the calculated cross section. We also study in this model the reactions 160+186W and 160+23Su. Our model accounts for deformation, except that the used, calculated target deformation may differ from the actual deformation and that the projectile is always assumed to be spherical. It is not our aim to obtain a perfect fit to the data, but instead to study systematically what component of the fusion enhancement is due to the target deformation, by comparing the deformed results to calculations assuming a spherical projectile and a spherical target. We comment below on the effects of possible differences between the calculated and the actual projectile and target shapes. In a deformed treatment it is necessary, as is done here, to account for the relative orientation of the projectile and the target in the calculation of the fusion barrier. The fusion barrier is calculated in terms of a macroscopic potential energy, which is the sum of Coulomb and nuclear interaction terms [4]. The nuclear deformation parameters are obtained from a calculation of nuclear masses and ground-state shapes [5]. The variation of barrier height versus angle is directly obtained from the model without any new parameters. The transmission coefficients are determined by calculating the transmission through the barrier; no free parameters such as hto are introduced. The deviations that remain between the calculated cross sections and the measured values may then contain important information on additional effects that further enhance subbarrier fusion cross sections above the enhancement due to deformation.

2. Models

For heavy-ion collisions with spherical projectiles and targets the fusion-barrier penetration problem can easily be solved relatively exactly by use of the WKB method. When either the projectile or the target, or both, are deformed, the problem becomes substantially more complex. Previously, we have developed a model to calculate the macroscopic potential energy between two arbitrarily oriented, deformed heavy ions [4]. Thus, we can describe

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A. lwamoto, P. Mb'ller/Nuclear Physics A 605 (1996) 334-358

the static part of the barrier-penetration problem, namely the multi-dimensional fusion potential-energy surface, in terms of a fully deformed treatment. However, in the dynamic part of the deformed collision problem some approximations are always made. To illustrate some important features of our model for the deformed case we first present the normal formalism for the spherical-projectile/spherical-target configuration, then extend it to collisions involving spherical projectiles and deformed targets. Some earlier studies of sub-barrier fusion cross sections have utilized parameterized distributions of fusion barriers to describe the two-dimensional fusion potential-energy surface in collisions of spherical projectiles with deformed targets. In our description a twodimensional potential is an immediate and well-defined result of specifying the Coulomb and nuclear potentials in terms of four-dimensional integrals over the spherical projectile and the deformed target shapes. Because these integrals are calculated to desired accuracy by numerical integration, we do not introduce inaccuracies by resorting to multipole expansions. It has been shown [6] that fusion cross sections can be used to extract a "distribution of fusion barriers". It was further stated that an advantage of studying fusion-barrier distributions instead of the total fusion cross section, which is the primary quantity measured in experiments, is that the former show much more clear signatures of positiveand negative-hexadecapole target deformations. In Section 2.3 we review the expressions required to extract this distribution from the fusion cross section. In our cross-section model the fundamental properties of the fusing system are expressed directly in terms of the two-dimensional fusion potential-energy surface, so the fusion-barrier distributions extracted from the experimental cross sections cannot be simply related to any of our basic model features in a well-specified quantitative way. However, it is quite informative to relate the extracted experimental barrier distributions in a qualitative way to the saddle points and ridges that surround the target nucleus. The distribution of these calculated features actually constitute a type of barrier distribution. It is only because the twodimensional surface that arises in our approach is a more primary quantity than a barrier distribution that we prefer not to address how to extract barrier distributions directly from our calculated fusion potential-energy surfaces. However, we will extract a calculated distribution of fusion barriers in a somewhat indirect way by using the calculated fusion cross section instead of the calculated two-dimensional fusion potential-energy surface. Then we apply the same formalism to these calculated cross sections as is used to extract the experimental barrier distributions from the experimental cross sections. The distributions extracted from the calculated cross sections can then be compared in a quantitative way to the distributions extracted from the experimental cross sections.

2.1. Spherical projectile~spherical target

For configurations with spherical projectiles and targets the fusion cross section o-f is given by the sum of the partial fusion cross sections o-f(L):

A. lwamoto, P. MOiler~Nuclear Physics A 605 (1996) 334-358 Off =

~

o'f(L) .

337 (l)

L The partial cross sections o-f(L) are given by A2(2L + 1)TL 4rr

o-f(L) -

(2)

with a given by h Here h is Planck's constant, Ecm is the center-of-mass energy in the entrance channel, and the reduced mass/x is given by mlm2

/*-ml+m~"

(4)

The quantities ml and m2 are the projectile and target masses, respectively. The transmission probabilities TL are obtained in WKB theory as

T

B

L1 +

1

m

e KL '

(5)

where KL = 2

r~ { 2/,Z -]1/2 ~ - [Vp+c(r,L) - Ecm] ~ d r .

(6)

rl

Here Vp+c(r, L) is the potential-plus-centrifugal energy of the projectile-target configuration versus (1) the separation r between their centers of mass and (2) the angular momentum L. The barrier entry and exit points are given by rl and r2, respectively. We cannot calculate the fusion barrier in a unique way inside the point of touching, since the shape evolution of the fused system is not known. Unless the barrier exit point occurs outside the touching point we therefore assume that the exit from the fusion barrier occurs at the point of touching. This is equivalent to assuming that the fusion barrier is very low inside the touching point. For the light projectiles we consider here, we expect this to be a reasonable approximation. This approximation increases the calculated penetrabilities and cross sections relative to their actual values. The increase would be larger the further the energy of the reaction is decreased below the barrier. However, we will see below that the calculated cross sections far below the barrier are usually lower than the experimental data, and that the discrepancy increases as the energy is decreased, the opposite of the expected effect of our approximation. When Ecru is higher than the maximum BL of Vp+c(r, L), the WKB approximation in Eq. (6) cannot be used. To treat this case we use the parabolic approximation and estimate hwL at each incident energy from the average properties of the actual barrier in terms of the following model. We observe that in the case of a parabolic barrier the transmission probability TL is exactly

338

A. lwamoto, P. M611er/Nuclear Physics A 605 (1996) 3 3 4 - 3 5 8 1

TL = 1 + exp[2zr(BL -- Ecm)/hwL] '

(7)

where hWL is the characteristic eigenvalue separation of the inverted parabolic barrier. This expression for the transmission probability is valid for incident energies both above and below the barrier. We use this expression when Ecru > BL. We now also need a reasonable model for hZOL. Since the barrier is far from parabolic we proceed in the following way. For Ecru > BL we evaluate Eq. (6) for

Eeff = BL

-

[BL --

Ecru[

(8)

and obtain the result /eLff. We now obtain ru.o~et from hw~ff = 2¢r(BL -- Eeff) /~Lff

(9)

The potential-plus-centrifugal energy Vp+c(r, L) is given by Vp+c(r, L) = Vc(r) + VN(r) + VL(r) ,

(10)

where the sum Vc(r) + Vrq(r) of the Coulomb (Vc) and the nuclear (VN) interaction energies is the potential energy versus the distance r between mass centers [4]. The centrifugal barrier VL(r) is obtained from the usual expression

Vr(r) =

h2L(L + 1) 2#r2

(11)

2.2. Spherical projectile~deformed, axially and mass-symmetric target To treat the simplest non-spherical configuration, namely a spherical projectile colliding with a deformed, axially and mass-symmetric target we introduce the angle 0 between the target symmetry axis and the beam direction. In a simple extension of the spherical formalism, a generalization that contains several approximations, we write for the fusion cross section o-f

o'f = f sinOdOZo-r(L,O) o

(12)

L

with the partial cross section o-f(L, 0) for a particular angular momentum L and orientation 0 given by o-f(L, 0) =

AZ(2L + 1)TL(0) 4rr

(13)

The transmission probabilities TL(O) at various angles 0 are obtained in WKB theory as TL -

1

1 + e KL(O)

'

(14)

A. lwamoto, P. MSller/Nuclear Physics A 605 (1996) 334-358

339

where r2

KL(O) = 2

1/2

- ~ [Vp+c(r,O,L) -Ecm]

}

dr.

(15)

rl

Here Vp+c(r, 0, L) is the potential-plus-centrifugal energy of the projectile-target configuration as a function of the separation r between their centers of mass, the angle 0 between the target symmetry axis and the beam direction, and the angular momentum L. For incident energies above the barrier we obtain the transmission probability in complete analogy with the spherical case. The potential-plus-centrifugal energy Vp+c(r, 0, L) is given by Vp+c(r, 0, L) = Vc(r, O) + VN(r, O) + VL(r).

(16)

Thus, the centrifugal barrier VL(r) is treated here exactly as in the spherical case, a significant approximation. Another approximation is the treatment of the barrier-penetration problem as a superposition of transmission probabilities through one-dimensional barriers corresponding to different angles 0, instead of a treatment where the problem is solved exactly in a two-dimensional space. However, the total potential energy Vc(r,O) + VN(r,O) is obtained in a fully multi-dimensional treatment, in terms of our fusion-barrier model of deformed nuclei [4]. From the point of view of the coupled-channel model, the approximations leading to Eqs. ( 1 2 ) - ( 1 5 ) may be summarized as (i) An adiabatic model in which we assume that the typical splitting of the groundstate rotational-band members is much smaller than the characteristic eigenvalue splitting of the fusion barrier. (ii) The neglect of couplings between different L-values. For the well-deformed targets that we study here, these assumptions are thought to be valid.

2.3. Expressions for fusion-barrier distributions Because it has been clear for some time that a one-dimensional fusion-barrier concept is insufficient to describe many fusion excitation functions, multi-dimensional approaches are now often used. It has been proposed that the multi-dimensional structure of the fusion barrier can be extracted from the experimental fusion cross sections in terms of a distribution of fusion barriers [6]. This distribution is proportional to the curvature of the function Ecmo'f, that is to d 2 (Ecmo'f) dE2m

(17)

In practice these derivatives have to be obtained by numerical derivation. It is well known that numerical derivation is one of the most ill-behaved numerical problems that exists. The difficulty arises because numerical derivation involves taking the difference of a

340

A. lwamoto, P M6ller/Nuclear Physics A 605 (1996) 334-358

function in two close-lying points. Because the points are close-lying, the two numbers involved may be of similar magnitude so the loss of numerical significance can be quite substantial. When the function involved in the derivative is calculated theoretically on a computer it is sometimes possible to determine the function sufficiently accurately, often in double precision, that the derivatives can be evaluated to the required accuracy. This is normally not possible when the function is obtained in experimental measurements, because the associated statistical errors often makes it impossible to use close-lying data points in the difference formulas. However, recently high-precision data have been taken for a few reactions, for example for reactions with 160 ions incident on 1548m [7], 186W [7], and 238U [8] targets. These data allow the calculation of fusion-barrier distributions to a useful accuracy. There are also high-precision data available for some additional reactions, but we here limit our discussion of fusion-barrier distributions to these three representative reactions. The fusion-barrier distributions were extracted from the high-precision experimental data by use of the difference equation [ 8]

d 2 (Ecmo'f)

a ccm

( (Ecm°'f)3 -- (Ecmo'f) 2 =2\

E

(Ecmo'f)2-(Ecmo'f)l'~ (

l

)

E2 (18)

where (Ecmo'f) i are evaluated at energies E i. When experimental data points are used, there is an error contribution statistical error which with Eq. (18) and equidistant grid points is [8]

,,,(

+ 4

+

,

~stat due to the

9

where the (6o-0/ are the experimental statistical errors in the cross sections and AEcm is the distance between the grid points. However, the total error is larger than the error originating from the statistical uncertainties associated with the data points, because there occur additional error terms due to truncation and rounding [9] in the finite-difference estimate of the second derivative. The truncation error 8t~un corresponding to Eq. (18) is [9]

~trun -----O(Agc2m) •

(20)

Thus, the truncation error decreases when the distance between the grid points used is decreased. In the absence of statistical errors, rounding errors may become dominant for very small grid-point distances. Because of the presence of the statistical errors we here use sufficiently large distances so that the rounding errors can be neglected. Eqs. (19) and (20) show that the barrier distribution cannot be estimated to arbitrarily high accuracy from the experimental points. In the experimental barrier-distribution calculations [8] a step length of 2 MeV in the laboratory frame was usually used. For approximate compatibility we use in the evaluation of the theoretical barrier distributions a step length of 2 MeV in the center-of-mass frame. However, we also perform the evaluations for a 1 MeV step length to illustrate the uncertainty due to truncation.

A. lwamoto, P. MOiler~Nuclear Physics A 605 (1996) 334-358

341

32S 4-154S m 15 10 5 E

,.t,.,.. v

O...

0 -5 -10

-15 -20 -15 -10

-5

0 5 z (fm)

10

15

20

Fig. 1. Potential-energy surface for the reaction 32S+154Sm. The energy in the gray area, next to the target nucleus 154Sin in the center, was not calculated, because the points in this region are inside the touching configuration. Note the ridge with saddle points and peaks around the target nucleus. The shape of target nucleus 154Sm is the calculated ground-state shape, namely e2 = 0.250, e4 = - 0 . 0 6 7 , and e6 = 0.030.

It will turn out that these errors are well in excess of 15% in some regions of the barrier distribution. Similar truncation errors are associated with the experimental barrierdistribution points, but are not included in the experimental error bars.

3. Cross sections for two-dimensional fusion potential-energy surfaces We have previously presented a comprehensive selection of potential-energy surfaces for arbitrarily oriented, deformed, colliding heavy ions [4]. In our cross-section studies based on this general potential-energy model that we present here, we limit the calculations to spherical projectiles and deformed targets. Such systems correspond to two-dimensional fusion potential-energy surfaces.

Quadrupole and hexadecapole shape distortions cross sections 3.1.

-

fusion potentials and fusion

To illustrate how characteristic features of the fusion potential-energy surface and the fusion cross section depend on common target shapes we present in Figs. 1-5 calculated potential-energy surfaces and cross sections for the reaction 325+1545m for a spherical target shape and for targets with representative spheroidal and positive and negativehexadecapole shape distortions. Fig. 1 shows the potential energy when the shape of the target 1545mis taken from a recent calculation of ground-state masses and shapes, which

A. lwamoto, P MOiler~Nuclear Physics A 605 (1996) 334-358

342

32S +

t54Sm

15 10 5 E

0

o..

-5 -10 -15 -20 -15 -10

-5

0 5 z (fm)

10

15

20

Fig. 2. Potential-energy surface for the reaction 32S+1545m for a hypothetical, spherical target shape. The energy in the gray area, next to the target nucleus 154Sm in the center, was not calculated, because the points in this region are inside the touching configuration.

32S -F 154Sm

15 10 5 E "-

0

o..

-5 -10 -15 -20 -15 -10

-5

0 5 z (fm)

10

15

20

Fig. 3. Potential-energy surface for the reaction 32S+154Sm for a hypothetical, spheroidal target shape, namely E2 = 0.250, e4 = 0.000, and e6 = 0.000. The energy in the gray area, next to the target nucleus l ~ S m in the center, was not calculated, because the points in this region are inside the touching configuration.

A. lwamoto, P. MOiler~NuclearPhysics A 605 (1996) 334-358

343

32S + ~S4Sm 15 10 5 E v

0

Q..

-5 -10 -15

-20 -15 -10 -5

0 5 Z (fm)

10

15

20

Fig. 4. Potential-energy surface for the reaction 3 2 S + 1 5 4 S m for a hypothetical, negative-hexadecapoleshape, namely e2 = 0.250, e4 = 0.067, and ~6 = --0.030. The energy in the gray area, next to the target nucleus 154Smin the center, was not calculated, because the points in this region are inside the touching configuration. obtained e2 = 0.250, e4 = - 0 . 0 6 7 , and e6 = 0.030. In all our fusion potential-energy surfaces the projectile is shown touching one polar region of the target. However, the fusion potential energy is shown for any projectile location (z, p) with the target located as shown, in the origin and aligned with the z axis. Figs. 2 and 3 show hypothetical situations when the target shape is spherical and spheroidal. Finally, we show in Fig. 4 another hypothetical situation, in which the signs of e4 and E6 are reversed relative to their calculated values. The calculated fusion cross sections corresponding to these four situations are shown in Fig. 5. The figure shows that the cross sections for all three deformed cases are enhanced relative to the spherical case. Large positive values of e4, corresponding to negative-hexadecapole moments result in lower enhancements than pure spheroidal shapes with the same values of e2. The largest enhancements occur for large, negative values of e4, corresponding to positive-hexadecapole moments. 3.2. Fusion potentials and cross sections

In Figs. 6 - 2 0 we show calculated fusion potential-energy surfaces and related, calculated fusion cross sections compared to experimental data. The reactions studied are 4He, 12C, 160, 2SSi, 3es, and 4°Ar projectiles colliding with ]54Sm targets and 160 projectiles colliding with ]86W and 238U targets. For each reaction we show one two-dimensional fusion potential-energy surface followed by a figure data [ 1,2,7,10-14]. The calculated curves and arrows have been shifted in energy by the amount indicated in the figures. This energy shift is the only free parameter in our cross-section model, since the target

344

A. lwamoto, P. M#ller/Nuclear Physics A 605 (1996) 334-358

10 3

I

I

I

I

I

I

I

I

I

I I

I

I

I

I

I

I

I

I

I I

3284-154Sm~

d3

I

I

I

]

I

I

I

I

~ I

I

I

I

I

I

i

~ i~

-

,EE 102 tO101 (/3

,'/

£ 10° 0

¢0

t

/,! ]

i

-~ 10 -1 LL

/

10 -2 100

I

I

I

I

I

I

/ I

I~

110

Center-of-Mass

Calculated Calculated Calculated Calculated

.... - -

/

i

I

/

---I

I

I

I

I

120

I

I

I

I

I I

I

I

I

I

I

(~4 > (E4 = (E4 < (~2 = I

I

130

I

I

0) O) O) 0) I

I

I

140

E n e r g y Ecru ( M e V )

Fig. 5. Calculated fusion cross sections for the reaction 32S+154Sm. The solid curve corresponds to the fusion potential-energy surface shown in Fig. 1 where the shape of the target corresponds to the calculated ground-state shape. The remaining three curves correspond to hypothetical target shapes whose fusion potential-energy surfaces are shown in Figs. 2-4.

shapes are obtained from a global nuclear-mass calculation, and the projectile shapes are always assumed spherical. However, the projectiles in some of the reactions we study are known to be deformed [ 15], so below we discuss what deviations between the calculated and measured cross sections may arise due to the assumption of spherical projectiles. For the 4He+]54Sm reaction with its very light projectile, shown in Figs. 6 and 7, there is little difference between the cross section calculated for the actual, deformed shape of the target and the cross section for a hypothetical, spherical target shape. The calculated cross section agrees quite well with the experimental data both above the barrier and down to the energy corresponding to the Coulomb barrier in the polar direction, which is the lowest energy studied in this reaction. Also for the 12C+154Sm reaction, shown in Figs. 8 and 9, the calculated cross section agrees quite well with the experimental data both above the barrier and down to the energy corresponding to the Coulomb barrier in the polar direction. In the 1 6 0 + 1 5 4 S m reaction, shown in Figs. 10 and 11, the agreement between the calculated cross section and the measured fusion cross section is extremely good in the entire 6 MeV range from the highest points of the Coulomb barrier in the equatorial region to the lowest points in the polar regions. Except for a simple translation in energy by - 3 . 1 MeV of the calculated results, this agreement was achieved without any free parameters related to nuclear ground-state deformations, vibrational couplings, or barrier curvatures. The multi-dimensional fusion potential [4] is completely defined by its globally valid potential parameters and by the target ground-state deformation as obtained

A. lwamoto, P. MOiler~Nuclear Physics A 605 (1996) 334-358

345

4He+lS4Sm _,ll,,,Ul,,,,IV,,,l,,,,i,,,,l,,,,i,,,,i,,,,i,

15 10 5 v

0 -5 -10 -15

8

9

10

11

12

- 2 0 -15 -10 - 5

12

11

10

9

10

0 5 Z (fm)

8

15

20

Fig. 6. Potential-energy surface for the reaction 4He+154Sm. The energy in the gray area, next to the target nucleus 154Sm in the center, was not calculated, because the points in this region are inside the touching configuration. Note the ridge with saddle points and peaks around the target nucleus.

104

i

i

i

I

,

u

u

i

,

I

l

u

i

l

4He+154Sm E t~ 10 3 "6 ffl 0

c3

102

e-

fl'

•

kt.

--

101 10

i

issue

,,

Experiment Calculated (Deformed) Calculated (Spherical) i

I

I

1

I

15 20 Center-of-Mass Energy Ecru(MeV)

I

25

Fig. 7. Calculated fusion cross sections for the reaction 4He+154Sm, compared to experimental data [ 1 ]. The solid curve corresponds to the fusion potential-energy surface shown in Fig. 6 where the shape of the target corresponds to the calculated ground-state shape. The long-dashed curve is the cross section obtained for a hypothetical, spherical target. The arrows indicate the fusion-barrier heights in the polar (p) and equatorial (e) directions of the deformed target and the fusion-barrier height (s) corresponding to a hypothetical, spherical target. The two curves and the three arrows have been translated in energy by AEtran = --0.7 MeV.

A. lwamoto, P. M611er/Nuclear Physics A 605 (1996) 334-358

346

120 -I- 154Sm

15 10

5 E

0

v

Q..

-5 -10 -15 -20 -15 -10 -5

0 5 z (fm)

10

15

20

Fig. 8. Potential-energy surface for the reaction 12C+1545m. The energy in the gray area, next to the target nucleus 154Sm in the center, was not calculated, because the points in this region are inside the touching configuration. Note the ridge with saddle points and peaks around the target nucleus.

10 3

-

i

, 12

v

i

C

i

i

'

i

i

i

i

i

i

i

i

i

i

i

i

i

-

. 154t",__ ~

E

8

10 2

.[,.~

o

09 o9 o9 0

101 tO

/

IT !

, Mev

J. / / / / ~ l c u l /IP / she

IJ_

100 40

,

t,

,/

,

I~,~

i

• -a t i

i

Experiment . Calculated (Deformed) e d (Spherical) -I

i

t

i

i

I

,

I

45 50 55 Center-of-Mass Energy Eom (MeV)

,

,

60

Fig. 9. Calculated fusion cross sections for the reaction 12C+154Sm, compared to experimental data [ 1 ]. The solid curve corresponds to the fusion potential-energy surface shown in Fig. 8 where the shape of the target corresponds to the calculated ground-state shape. The long-dashed curve is the cross section obtained for a hypothetical, spherical target. The arrows indicate the fusion-barrier heights in the polar (p) and equatorial (e) directions of the deformed target and the fusion-barrier height (s) corresponding to a hypothetical, spherical target. The two curves and the three arrows have been translated in energy by dEtran = - 1 . 7 MeV.

A. lwamoto, P. M~ller/Nuclear Physics A 605 (1996) 334-358

347

160 + 154Sm

15 10

5 E

0

"v

o.

-5 -10 -15 -20 -15 -10

-5

0 5 z (fm)

10

15

20

Fig. 10. Potential-energy surface for the reaction 160+154Sm. The energy in the gray area, next to the target nucleus i54Sm in the center, was not calculated, because the points in this region are inside the touching configuration. Note the ridge with saddle points and peaks around the target nucleus.

103

-

I

I

I

I

I

I

I

1 6 0 + 154Sm

..13 v

I

I

I

I

I

I

I

I

i

i

i

i

:

~

-

-~ ~---

Calculated (deformed) Calculated (spherical) -

E

¢- 102

.o_ "5 CO o9 O ¢-

101

._o '-i ii

,j/ ~' ,

10 o 50

,

,p

J/ I

/ / ,

I

s , sj

e i{e . . . .

i

. . . .

55 60 65 Center-of-Mass Energy Ecru(MeV)

70

Fig. 11. Calculated fusion cross sections for the reaction 160+1548m, compared to experimental data [ 10]. The solid curve corresponds to the fusion potential-energy surface shown in Fig. 10 where the shape of the target corresponds to the calculated ground-state shape. The long-dashed curve is the cross section obtained for a hypothetical, spherical target. The arrows indicate the fusion-barrier heights in the polar (p) and equatorial (e) directions of the deformed target and the fusion-barrier height (s) corresponding to a hypothetical, spherical target. The two curves and the three arrows have been translated in energy by AEtra. = - 3 . 1 MeV.

348

A. lwamoto. P. M611er/Nuclear Physics A 605 (1996) 334-358

28Si + 154Sm 15 10 5 E •'v

0 -5 -10 -15 -20 -15 -10

-5

0 5 z (fm)

10

15

20

Fig. 12. Potential-energy surface for the reaction 28Si+154Sm. The energy in the gray area, next to the target nucleus ~54Sm in the center, was not calculated, because the points in this region are inside the touching configuration. Note the ridge with saddle points and peaks around the target nucleus. in a recent nuclear-mass calculation [5]. We observe the large difference between the curve calculated for the deformed target shape and the curve for a hypothetical, spherical shape. Clearly there is a substantial effect of nuclear deformation on the fusion cross section. In this reaction the deformation effect fully accounts for the "sub-barrier" enhancement of the fusion cross section. The remaining three reactions with 154Sm targets that we study are 28Si+154Sm shown in Figs. 12 and 13, 32S+154Sm shown in Figs. 1 and 14, and 4°Ar+154Sm shown in Figs. 15 and 16. The reactions with 28Si and 4°Ar projectiles agree extremely well with the data down to energies near the lowest points on the Coulomb barrier, which occurs in the polar directions. At energies below the polar-region barrier the deviations become increasingly larger for decreasing energy. In the 325+154Smreaction the deviations between the calculations and the data extend well above the barrier in the polar directions. Since the agreement with the data in the cross-section calculations with both the lighter 28Si projectile and the heavier 4°Ar projectile was quite good, the deviations in Fig. 14 are somewhat unexpected. All three projectiles are deformed with the magnitude of the 32S deformation in between the deformation of the other two projectiles [ 15]. We obtain very good agreement between our calculations and experimental data in most cases, although the projectile deformation is not taken into account. This is probably due to the small size of the projectile relative to the target. To investigate the discrepancies in the lowest energy regions and to study reactions with heavier projectiles it is a natural next step to account for the projectile deformation. Our computer code does not at this stage allow calculations with deformed projectiles. An extension of the model to deformed projectiles with retention of the axial symmetry of the projectile

A. lwamoto, P. MSller/Nuclear Physics A 605 (1996) 334-358

103

v

E 6 102

I IIIIIlllllllllll

I, I I I

28Si + lS4Sm

IIIIIlllllllll

II Ill

IIII

349 III-

f

!_

_~¢

-=

tO 0

co 101 ¢/)

£

0 ._ ] 0 °

• / [ / p

U_

10 -1 80

.......

I • Experiment _ = / ------Calculated (Deformed)= ~lculated (Spherical) , s e

, i p, ,,, i,

......

, .........

,

........

90 100 110 120 Center-of-Mass Energy Ecru (MeV)

130

Fig. 13. Calculated fusion cross sections for the reaction 28Si+154Sm, compared to experimental data [2]. The solid curve corresponds to the fusion potential-energy surface shown in Fig. 12 where the shape of the target corresponds to the calculated ground-state shape. The long-dashed curve is the cross section obtained for a hypothetical, spherical target. The arrows indicate the fusion-barrier heights in the polar (p) and equatorial (e) directions of the deformed target and the fusion-barrier height (s) corresponding to a hypothetical, spherical target. The two curves and the three arrows have been translated in energy by AEtran = --5.8 MeV.

and the target would lead to a four-dimensional potential instead of a two-dimensional one. The calculation of a cross-section curve corresponding to a two-dimensional fusion potential takes about 60 minutes on an HP-Apollo 9000-735 computer. An extension of the model to a four-dimensional fusion potential would increase the computing time for one cross-section curve to about one week. This is still a feasible calculation. Until this work is actually carried out, the origins of the deviations in the low-energy regions in the reactions involving 28Si, and 4°Ar and over the broader energy range in the reaction involving 32S cannot be completely identified. Even after the projectile deformation has been taken into account some deviations may remain. However, it is only natural to first include projectile deformation, before invoking additional mechanisms for the remaining deviations, such as vibrational enhancements. At this stage we consider the agreement between our calculations and the data very encouraging, considering that only one parameter is adjusted to data. In the final two reactions we consider here we investigate reactions between 160 projectiles and deformed targets other than 154Sm. Specifically we study the reactions 160-]-186W shown in Figs. 17 and 18 and 16Oq-238U shown in Figs. 19 and 20. In the first reaction, between oxygen and tungsten, the sign of the hexadecapole moment of the tungsten target is opposite to that of the 154Sm target studied in the previous reactions. Also with this negative-hexadecapole moment we obtain excellent agreement between the calculated fusion cross-section curve and the measured data points, from the highest

350

A. lwamoto, P. MOiler~Nuclear Physics A 605 (1996) 334-358

103 328 4-154Sm

..Q

E 102 cO ,= o 101 03 ¢/} 8 10 0

•

!

(.)

0

•g

N,~o = - 4.0 MeV

1 0 -t

Ii

--="

[ /

[ • / / ~ l c u l a t ,,,,,,,,/,L,,,i,, p s

Experiment . Calculated (Deformed) e d (Spherical) e , ......... ,,,,,

10-2 100

110

120

130

Center-of-Mass Energy Ecr. (MeV) Fig. 14. Calculated fusion cross sections for the reaction 32S-FI54Sm, compared to experimental data [ 1 1 ]. The solid curve corresponds to the fusion potential-energy surface shown in Fig. 1 where the shape of the target corresponds to the calculated ground-state shape. The long-dashed curve is the cross section obtained for a hypothetical, spherical target. The arrows indicate the fusion-barrier heights in the polar (p) and equatorial (e) directions of the deformed target and the fusion-barrier height (s) corresponding to a hypothetical, spherical target. The two curves and the three arrows have been translated in energy by AEu.an = --4.0 MeV.

40Ar + 154Sm

15 10

5 E

"v

0

(3_

-5 -10 -15 -20 -15 -10

-5

0 5 Z (fm)

10

15

20

Fig. 15. Potential-energy surface for the reaction 4°Ar+154Sm. The energy in the gray area, next to the target nucleus l ~ S m in the center, was not calculated, because the points in this region are inside the touching configuration. Note the ridge with saddle points and peaks around the target nucleus.

A. lwamoto, P. M611er/Nuclear Physics A 605 (1996) 334-358

103

III

I IIIIIII

Illlllllllllll

I IIIII

Illl

III

IIIIlll

351

II

40Ar+ 154Sm , ~ t . 4 ~ - ~ -Q E1 0 2 v

6 ¢-

O

.

101

/

•

I1)

!

•

co 100 t.o

!

•

£ 0 1 0 -1 t"-

._o o~

= 10 -a Ii

!

Etran= ? 5.8_MeV

•

/ ]

I

I i

/ p

10-3

/

i "

• ---"'-'s e

Experiment Calculated (Deformed) Calculated (Spherical)

,i,, ,/,, i,,, ......... , ......... ,,,, 110 120 130 140 150 Center-of-Mass Energy Ecru (MeV)

Fig. 16. Calculated fusion cross sections for the reaction 4°Ar+154Sm, compared to experimental data [ 12]. The solid curve corresponds to the fusion potential-energy surface shown in Fig. 15 where the shape of the target corresponds to the calculated ground-state shape. The long-dashed curve is the cross section obtained for a hypothetical, spherical target. The arrows indicate the fusion-barrier heights in the polar (p) and equatorial (e) directions of the deformed target and the fusion-barrier height (s) corresponding to a hypothetical, spherical target. The two curves and the three arrows have been translated in energy by zlEtran = - 5 . 8 MeV.

160 + 186W

15 10 5

--

CL

0 -5

-10 -15 -20 -15 -10

-5

0 5 z (fm)

10

15

20

Fig. 17. Potential-energy surface for the reaction 160-~-186W.The energy in the gray area, next to the target nucleus 186W in the center, was not calculated, because the points in this region are inside the touching configuration. Note the ridge with saddle points and peaks around the target nucleus.

352

A. lwamoto, P. M~ller/Nuclear Physics A 605 (1996) 334-358

103

f

I

I

I

I

I

160 + l

I

I

a

I

I

I

~

I

I

/

I

I

t

I

I

i

i

i

i

i

i

i

i

i

/

~

•

Et,a.=- 3.6 MeV Experiment Calculated (Deformed) Calculated (Spherical)

i

~

d~

E

v

102 tO

09 101 ~9

/

2

0

J / / ] / mln

~ 10 0 LL _

10-1

,

60

i

i,

S

e,

, ,1, ,1, I I

,

,

I

. . . .

' ' ' ' , I , , , ,

65 70 75 80 85 Center-of-Mass Energy Ecru (MeV)

90

Fig. 18. Calculated fusion cross sections for the reaction 160-J-186W, compared to experimental data [7]. The solid curve corresponds to the fusion potential-energy surface shown in Fig. 17 where the shape of the target corresponds to the calculated ground-state shape. The long-dashed curve is the cross section obtained for a hypothetical, spherical target. The arrows indicate the m i n i m u m fusion-barrier height (min), which for this target with its large negative-hexadecapole moment does not occur in the polar directions, the m a x i m u m fusion-barrier height located in the equatorial (e) direction of the deformed target, and the fusion-barrier height (s) corresponding to a hypothetical, spherical target. The two curves and the three arrows have been translated in energy by AEtran = --3.6 MeV.

energies down to energies corresponding to the lowest points on the Coulomb barrier. For the negative sign of the tungsten hexadecapole moment one should observe that the lowest points on the Coulomb barrier are not located in the polar regions but instead in between the polar and equatorial regions. This is perhaps even more evident in the color figure 1 of Ref. [4] for the fusion potential of 160+184W. In this color figure calculation was inadvertently carried out with e6 = +0.033 instead of the correct value e6 = -0.033 from the recent nuclear-mass calculation [5]. However, the calculated fusion potential corresponds to the shape configuration shown. In our calculations here the programs have been further developed so that the deformations may be automatically read in from the mass table, so that typing errors are avoided. Below the lowest points on the two-dimensional Coulomb barrier the deviations between the calculations and the measured cross sections increase with decreasing energy. In the reaction with oxygen on uranium we also obtain quite good agreement between the calculations and the measured cross sections from the highest energies to the lowest points on the Coulomb barrier in the polar regions. Also here there is a tendency towards increasing discrepancies between the calculations and the experimental results as the energy decreases below the lowest points on the Coulomb barrier.

A. lwamoto, P. Mi~ller/Nuclear Physics A 605 (1996) 334-358

353

160 + 238u

15 10 5 E

"-

0

EL

-5 -10 -15 -20 -15 -10

-5

0 5 z (fm)

10

15

20

Fig. 19. Potential-energy surface for the reaction ]60+238U. The energy in the gray area, next to the target nucleus 238U in the center, was not calculated, because the points in this region are inside the touching configuration. Note the ridge with saddle points and peaks around the target nucleus.

10 4

511111111'l'iIIll'''IIl~ll''''llll111'''l'llll''''

5

160 Jr238U

..Q E 103

~7 c

.o_ 02 CO

~ 101 f ~ment(Zhang e, al.) / I • Experiment(Shenet al.) ] / Calculated (Deformed) / ~lculated (Spherical)

0 E 0

O0

U-

,pl

10 -1 65

70

s

e

75 80 85 90 95 100 105 110 115 Center-of-Mass Energy Ecru (MeV)

Fig. 20. Calculated fusion cross sections for the reaction 16Oq-238U,compared to experimental data [ 13,14]. The solid curve corresponds to the fusion potential-energy surface shown in Fig. 19 where the shape of the target corresponds to the calculated ground-state shape. The long-dashed curve is the cross section obtained for a hypothetical, spherical target. The arrows indicate the fusion-barrier heights in the polar (p) and equatorial (e) directions of the deformed target and the fusion-barrier height (s) of a hypothetical, spherical target. The two curves and the three arrows have been translated in energy by /tEtran = - 4 . 5 MeV.

354

A. lwamoto, P. M611er/Nuclear Physics A 605 (1996) 334-358

' ' '+~s~ .... ~°0 Sm 800 .~ 600 1000

E

J ....

t

'

I

I

I

I

400

200

- 200 1 i e. ~iiei'~t( :eefVo 'iT1~ rmed) 400 50 55 60 65 Center-of-MassEnergyEem(MeV)

-

,

,

,

,

I

,

,

,

,

I

,

,

,

,

,

I

I

I

70

Fig. 21. Comparison of calculated and experimental [ 7] fusion-barrier distributions for the heavy-ion reaction 160-FI54Sm. The experimental points and the theoretical results given by the solid line are based on second derivatives calculated by use of the finite-difference equation (18) and a grid-point distance of 2 MeV. The arrows indicate the fusion-barrier heights in the polar (p) and equatorial (e) directions of the deformed target. The shaded region shows the change in the calculated curve if a 1 MeV grid-point distance is used in the finite-difference equation. These results can be compared to the calculated two-dimensional fusion potential-energy surface in Fig. 10, which should be shifted in energy by -3.1 MeV for the purpose of this comparison. 3.3. F u s i o n - b a r r i e r d i s t r i b u t i o n s

We discussed above in Section 2.3 that a quantity proportional to the "distribution o f fusion barriers" may be extracted as the second derivative with respect to the center-ofmass energy Eorn o f the product Ecmo'f. However, to calculate these second derivatives, high-precision experimental data are required. These are now available for several reactions. We present comparisons between experimental and calculated fusion-barrier distributions in Figs. 21-23 for J60 projectiles colliding with 154Sm, 186W and 238U targets. The experimental points in Figs. 21 and 22 are taken from Ref. [7]. In Fig. 23 the experimental points are from Ref. [8]. In the latter case we had to extract the values by measuring the location of the points in the published figure. Because o f the difficulty o f obtaining accurate information in this manner we did not extract the error bars in this case. However, they vary with energy in a similar manner as the error bars in Figs. 21 and 22, that is they rapidly become fairly large at energies beyond the peak of the distribution. We extract calculated fusion-barrier distributions in complete analogy with the method for obtaining the experimental fusion-barrier distributions. Thus, we take the second derivatives o f the product of the center-of-mass energy and the calculated fusion cross section. Just as in the extraction o f the experimental data we use, for compatibility a

A. lwamoto, P M611er/Nuclear Physics A 605 (1996) 334-358

1000 ---r'--'r---T"~ [ ~ ._. >

355

T r--'T--T---r-16O

80O

+ t86 W

.:#iiii~,

600

..Q

E

.

fi !ii:::/:~::~:

400

~E Lu°

200 E

o

v

%

•,,, Calculated ::;:r::e:t (Deformed)

-200 -

tt

--

400 6O

65 70 75 Center-of-Mass Energy Ecru(MeV)

80

Fig. 22. Comparison of calculated and experimental [ 7 ] fusion-barrier distributions for the heavy-ion reaction 160+186W. The experimental points and the theoretical results given by the solid line are based on second derivatives calculated by use of the finite-difference equation (18) and a grid-point distance of 2 MeV. The arrows indicate the minimum fusion-barrier height (min), which for this target with its large negative-hexadecapole moment does not occur in the polar directions and the maximum fusion-barrier height located in the equatorial (e) direction of the deformed target. The shaded region shows the change in the calculated curve if a 1 MeV grid-point distance is used in the finite-difference equation. These results can be compared to the calculated two-dimensional fusion potential-energy surface in Fig. 17, which should be shifted in energy by - 3 . 6 MeV for the purpose of this comparison.

2 MeV difference between the grid points in the finite-difference formula (18) for the second derivative. We mentioned in Section 2.3 that the experimental errors contribute increasingly to the error of the extracted second derivatives as the grid-point distance decreases. However, the numerical truncation errors are large for large grid-point distances. These truncation contributions to the error are not included in the experimental error bars. To give some indication of the magnitude of this error we show in Figs. 21-23 as gray areas the change in the second derivatives extracted from the calculated cross sections when the grid-point distance is deceased from 2 MeV to 1 MeV. Clearly the inaccuracies are the greatest in the peak regions. In general, the calculated fusion-barrier distributions agree extremely well with the experimentally obtained distributions. For the 1 6 0 + 1 5 4 5 m reaction there are hardly any significant differences between the calculated and experimental results. For the reactions 160"-}-186Wand 160+238U there are some differences in the peak regions. However, it is here that the finite-difference method as applied in these examples yields the largest uncertainties, so these differences may not be too significant. It is of interest to observe the characteristic differences between the fusion-barrier distributions corresponding to targets with positive and with negative-hexadecapole deformations. In the former case, for the Sm and U targets, the maximum in the fusion-barrier

356

A. lwamoto, P. M61ler/Nuclear Physics A 605 (1996) 334-358

1000

.-. > 0

800 600

E

v

~

ilillfrlllIjtililllltlli~lltl

-

160 + 238u

~

_-

~

o :ii::o ':~iiii•~•

•0

_-

400

~E ~o

200 E u~o

%

oil

o - 200

- 400

,I 70

"

/

raft, i i i I i i i , I , ,

75 80 85 90 95 Center-of-Mass Energy Eom (MeV)

Fig. 23. Comparisonof calculated and experimental [ 8] fusion-barrierdistributionsfor the heavy-ionreaction The experimentalpoints and the theoretical results given by the solid line are based on second derivatives calculated by use of the finite-differenceequation (18) and a grid-pointdistance of 2 MeV. The arrows indicate the fusion-barrier heights in the polar (p) and equatorial (e) directions of the deformed target. The shaded region shows the change in the calculated curve if a 1 MeV grid-pointdistance is used in the finite-differenceequation. These results can be compared to the calculated two-dimensionalfusion potential-energysurface in Fig. 19, which should be shifted in energy by -5.5 MeV for the purpose of this comparison.

I6oq-238U.

distribution occurs in the higher energy ranges. This corresponds to the fairly constant fusion barrier seen extending in Figs. 10 and 19 in both directions from the equatorial region towards the polar regions. Low barrier energies occur only in small regions near the poles. Therefore, the fusion-barrier distribution function is low at low energies for the Sm and U nuclei with their positive-hexadecapole moments. In contrast, for the W target with its negative-hexadecapole deformation the fusion-barrier distribution is maximum in its lower energy range. The fusion potential-energy surface in Fig. 17 corresponds to this structure. In this case the potential is almost constant from the polar regions to near the equatorial region where the Coulomb barrier rapidly increases to a maximum.

4. S u m m a r y We have calculated two-dimensional fusion potential-energy surfaces associated with collisions of spherical projectiles with deformed targets. The corresponding fusion cross sections were also calculated. The reactions studied were projectiles from 4He to 4°Ar colliding with 154Sm targets and 160 projectiles colliding with 154Sm, 186W,and

23Su

targets. An important feature of our cross-section model is that there are no free parameters that are adjusted to the experimental cross sections, except for a simple energy

A. lwamoto, P M61ler/Nuclear Physics A 605 (1996) 334-358

357

translation of the calculated cross sections. This free parameter is necessary, because there is always some deviation between the actual and the calculated heights of the Coulomb barrier. We obtain very good agreement between our calculated cross sections and the experimental data in almost all cases studied for energies above the lowest points on the Coulomb barrier. That is, we obtain good agreement in the upper part of what is often referred to as the "sub-barrier" region. The notable exception is the reaction with 32S projectiles. For some other projectiles small deviations start do develop for decreasing energies just above the lowest points on the Coulomb barrier. Below the lowest points on the Coulomb barrier these deviations increase further. They may be partly due to vibrational enhancements and, in some cases, due to our neglect of the projectile deformations. Alternatively, our calculated target deformations may not agree exactly with the actual nuclear deformation. Since the deformation parameters were not varied, some increased agreement would be expected if an optimization were undertaken. However, the aim of our studies here was to show the high degree of agreement that is obtained just by using the calculated shape parameters that were obtained in a recent nuclear-mass calculation [5]. We feel the agreement is outstanding, considering that only one parameter was adjusted to the data. This good agreement for energies above the lowest points on the Coulomb barrier, shows that the calculated deformations are in close agreement with the actual target deformations and that the model for the heavy-ion fusion potential is satisfactory. For the reactions involving 160 projectiles we also compared calculated results to experimental data in terms of a "fusion-barrier distribution". We found a high degree of agreement between the calculated fusion-barrier distributions and those extracted from experimental cross sections. Clear signatures of the sign of the target hexadecapole moments appear in both the calculated and in the experimental distributions. We conclude by again observing that for the projectiles and the well-deformed target nuclei studied here, the sub-barrier fusion enhancement is very well described as a pure static deformation effect, down to the lowest points on the Coulomb barrier. Whether this holds for somewhat heavier, deformed projectiles will have to await an extension of the model from spherical to deformed projectiles. Finally, we point out that the model discussed here will be inaccurate for nuclei in the transitional regions between spherical and deformed shapes. For such nuclei we expect that vibrational enhancements will be of considerable importance over and above the static deformation effects studied here.

References [ 1] S. Gil, R. Vandenbosch, A.J. Lazzarini, D.-K. Lock and A. Ray, Phys. Rev. C 31 (1985) 1752. [2] S. Gil, D. Abriola, D.E. DiGregorio, M. di Tada, M. Elgue, A. Etchegoyen, M.C. Etchegoyen, J. Fernfindez Niello, A.M.J. Ferrero, A.O. Macchiavelli, A.J. Pacheco, J.E. Testoni, P. Silveira Gomes, V.R. Vanin, A. Charlop, A. Garc[a, S. Kalias, S.J. Luke, E. Renshaw and R. Vandenbosch, Phys. Rev. Lett. 65 (1990) 3100. [3] R.J. Leigh, N. Rowley, R.C. Lemmon, D.J. Hinde, J.O. Newton, J.X. Wei, J.C. Mein, C.R. Morton, S. Kuyucak, and A.T. Kruppa, Phys. Rev. C 47 (1993) 47.

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A. lwamom. P. M611er/Nuclear Physics A 605 (1996) 334-358

P. M611er and A. lwamoto, Nucl. Phys. A 575 (1994) 381. E M~311er,J.R. Nix, W.D. Myers and W.J. Swiatecki, At. Data Nucl. Data Tables 59 (1995) 185. N. Rowley, G.R. Satchler and PH. Stelson, Phys. Lett. B 254 (1991) 25. R.J. Leigh, M. Dasgupta, D.J. Hinde, J.C. Mein, C.R. Morton, R.C. Lemmon, J.O. Newton, J.P. Lestone, J.O. Newton, H. Timmers, J.X. Wei and N. Rowley, Phys. Rev. C 52 (1995) 3151. [ 8 ] D.J. Hinde, M. Dasgupta, J.J. Leigh, J.P. Lestone, J.C. Mein, C.R. Morton, J.O. Newton and H. Timmers, Phys. Rev. Lett. 74 (1995) 1295. [9] M.J. Maron, Numerical Analysis: A Practical Approach (MacMillan, New York, 1982). [10] R.G. Stokstad, Y. Eisen, S. Kaplanis, D. Pelte, U. Smilansky, and I. Tserruya, Phys. Rev. C 21 (1980) 2427. I l l ] ER.S. Gomes, I.C. Charret, R. Wanis, G.M. Sigaud, V.R. Vanin, R. Liguori Neto, D. Abriola, O.A. Capurro, D.E. DiGregorio, M. di Tada, G. Duchene, M. Elgue, A. Etchegoyen, J.O. Fermlndez Niello, A.M.J. Ferrero, S. Gil, A.O. Macchiavelli, A.J. Pacheco and J.E. Testoni, Phys. Rev. C 49 (1994) 245. [12] W. Reisdorf, EP. Hessberger, K.D. Hildenbrandt, S. Hofmann, G. Miinzenberg, K.-H. Schmidt, J.H.R. Schneider, W.EW. Schneider, K. Siimmerer, G. Wirth, J.V. Kratz and K. Schlitt, Nucl. Phys. A 438 (1985) 212. [13] W.Q. Shen, J Albinski, A. Gobbi, S. Gralla, K.D. Hildenbrand, N. Herrmann, J. Kuzminski, W.EJ. Miiller, H. Stelzer, J. T6ke, B.B. Back, S. Bj~mholm and S.P. SCrensen, Phys. Rev. C 36 (1987) 115. [14] H. Zhang, Z. Liu, J. Xu, X. Qian, Y. Qiao, C. Lin and K. Xu, Phys. Rev. C 49 (1994) 49. [ 15] S. Raman, C.W. Nestor Jr., S. Kahane and K.H. Bhatt, At. Data Nucl. Data Tables 42 (1989) 1.