16O-12C scattering : description of the gross structure features using an optical model

HAL Id: jpa-00208669

https://hal.archives-ouvertes.fr/jpa-00208669

Submitted on 1 Jan 1977

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

16O-12C scattering : description of the gross structure

features using an optical model

A.J. Cole, N. Longequeue, J.F. Cavaignac

To cite this version:

16O-12C

SCATTERING : DESCRIPTION OF

THE

GROSS STRUCTURE

FEATURES USING AN OPTICAL MODEL

A. J.

_COLE,

N.

_LONGEQUEUE

and J. F. CAVAIGNAC Institut des Sciences

_Nucléaires,

B.P.

257,

38044 Grenoble

_Cedex,

France

(Reçu

le 17 mars

_1977,

_accepté

le 16 mai

₁₉₇₇₎

Résumé. 2014 La _mesure de la diffusion de 16O sur 12C

a été faite dans une

_{région angulaire}

limitée dans les directions avant et arrière à des _énergies laboratoires de 16O de _{80, 85, 91, 93, 95,} 101 et

122 MeV. La fonction d’excitation du maximum de la distribution

_angulaire

situé _près de 156° _(c.m.)

a aussi été mesurée par pas de ~ _0,5 MeV entre 80 et 101 MeV _(lab.). Les résultats ont été _ajoutés à d’autres résultats existants et la gross structure de l’évolution de la diffusion avec

_l’énergie

a été

analysée en utilisant un modèle

_optique

_{dont le rayon imaginaire} croit avec

_{l’énergie.}

Abstract. - Measurements of the

scattering

of 16O on 12C have been made over a limited

_angular

range in the forward and backward

hemispheres

at

_laboratory

_(16O)

_energies

_{of 80,85,91,93,95,101,}

122 MeV. The excitation function of the maximum near 156° _(c.m.) has also been measured in _steps

of ~ 0.5 MeV between 80

_and

101 MeV (lab.). The data were combined with other

_existing

measure-ments _{and the gross} structure features _analyzed

_using

an

_optical

model whose _imaginary radius

increases with energy.

Classification

z

Physics Abstracts 24.50

1. Introduction. - The

12C_Ilo

system

has,

in

recent _years, been the

_subject

of intense

_experimental

and theoretical

_{investigations mainly}

due to the existence of

_{anomalously high}

backward

_angle

cross-sections

_exhibiting

_strong

oscillations

_{approximately}

described

_by

_[PLB(cos

_0)]2

where

_Lg

is the

_angular

momentum

_{corresponding}

to an

impact

_parameter

characteristic of the surface of the interaction

_{[1, 2, 3].}

In addition for

_energies

_EB

_Ec.m.

30 MeV

_(EB

is the Coulomb barrier ~ 12

MeV)

the backward

angle

cross-section fluctuates

_strongly

with _energy

giving

rise to the

_supposition

of the existence of

molecular _resonances,

_especially

at

_Er

= 19.7 MeV

[2]

and

_Ec...

= 22.5 MeV

[1]

where the influence of

resonance like effects in reaction channels have been

established

_{[3, 1].}

At low _energy

_EB

_Ec.m.

17 MeV

measurements of the total reaction cross-section

_[4]

were

_{explained using}

an

_optical

model which

_however,

possessed

the

_unphysical

characteristic of

_relatively

small ( ~

0.7)

transmission coefficients for low

_partial

waves.

One may

question

the wisdom of

_trying

to fit

fluctuations

which,

in this _system, have a width of

~ 0.5 MeV with a _gross structure model

_(optical

model).

Nevertheless

_{by using dependent optical}

models some success has been obtained in

_fitting

angular

distributions at backward

_{angles [5],}

_(1).

The

_{physical origin}

of the

_dependence

is however

not clear and a number of

explanations

related either

to the non formation of the

_compound

nucleus

_[6]

or to the

_angular

momentum mismatch in direct reaction channels

_[7]

have been

_proposed.

Yet another

possible origin

was

_{proposed by}

the authors of

refe-rence

_[5]

who

_supposed

that the

_{dependent imaginary}

potential

used in their

_analyses

of

’6p_12C

_scattering

at 65 and 80 MeV

_{(160 lab.)}

simulates coherent inter-ference between the elastic

_scattering

and the a

transfer reaction.

However,

it remains to be established that such interference can

reproduce

the

rapid

fluctua-tions observed in excitation function measurements

while

_preserving

the

_regular

_oscillatory

structure

e) Charles, P., Private communication.

1044

observed at backward

_angles.

_Certainly,

a

_single

direct mechanism _may be ruled out since such

ampli-’

tudes _vary too

_slowly

with _energy. This fact was

emphasized

by

Devries

_[8]

who limited his

_analysis

of

160_12C

_scattering

in terms of the a transfer

mechanism to the data at 80 MeV where

_rapid

fluc-tuations with _energy were

_supposed

to be less

impor-tant.

Above 80 MeV

_published

data

_{including angular}

distribution measurements is limited to forward

angle

distributions measured at 111 MeV

_[9]

and 168 MeV

_[10].

It was therefore considered useful to

extend the _range of the data

_by

_measuring

small

portions

of the forward and backward

_angle

distribu-tions between 80 and 120 MeV

_{(lab. 160) .}

The _energy resolution available

_{(dE/E N}

10-

₃₎

_precluded

a search

for details of fluctuations in the excitation function

although large

effects,

such as those observed at lower

energies

[2],

if

_present,

could be observed.

_Further,

a verification of the _presence of a backward

_angle

rise

in the

_angular

distributions and a test of the

_simple

[PLs(cos

0)]2

behavior observed at lower

_energies

was

considered useful.

_Using

the

_160(5+)

beam of the Grenoble ISN

_cyclotron

we therefore measured small

portions

of the forward and backward

_angle

distri-butions

for 160_12C

_scattering

at

_{laboratory energies}

of

80, 85, 91, 93, 95,

101 and 122 MeV. In

_addition,

the excitation function of the maximum situated near

1560 c.m. was measured between 80 and 101 MeV. 2.

_{Experimental procedures.}

- The

experiment

was

carried out at the ISN isochronous

_cyclotron

of Grenoble. The

_experimental

_arrangement

for the 80 to 101 MeV data is shown in

_figure

1. A collimated

FIG. 1. -

Experimental arrangement. Dimensions are _given in

’

millimetres. z

beam

_{of 160(5+)}

of _up to 15 nA

_(particle)

was directed

onto self

_{supporting 12C}

foils of thickness 100 ± 5

_{ug/cm2 .}

Two AE - E detector

_telescopes

T1

and

_T2

mounted

on a

_single

movable arm, and

separated by

20°

detected

_particles

on the left and

_right

sides of the beam. The

_angular

_aperture

of the detectors were 0.5°

and 0.7°. The

160

_particles

emitted in the forward direction and the

_recoiling

12C

_particles

(correspond-ing

to backward

_angle

160

in the c.m.

system)

were

identified on line

_by

a PDP9 data

handling

sys-tem

_[7].

The

_collimating

_system DD’ defined the beam direction to within 2° which was not

_sufficiently

precise

for the

_performance

of the

_experiment.

Thus before each

_angular

distribution run the

_angle

between the beam direction and the chamber axis was measured

using

a 100

J.lg/cm2

gold

_target.

This

_asymmetry

angle

was obtained when the ratio of the elastic counts in the left and

_right

detectors was

equal

to the ratio of their solid

_angles.

The _constancy of the beam direc-tion was then monitored

_{continuously using}

a

_large

gold

target

evaporated

in

a 12C

_backing

in a

secondary

chamber,

the ratio between the count rates in two

detectors

_(Ml

and

_M2)

at ± 25° with

respect

to this chamber axis

_indicating

if any

change

took

_place

in the beam direction. Data were

_accepted

_provided

that

the _asymmetry

_angle

remained constant to within 0.2°. As a further check the

_angular

distributions were measured on both sides of the beam and the

angular

shift _necessary for their

_{superposition}

was

compared

to the initial _asymmetry measured

_using

the

gold

target.

The

_agreement

was

always

better than

0.2° which was therefore

_assigned

as the

_angular

error

to each data

_point.

The

_quality

of the identification

spectrum did not

_permit

us to rule out the presence of small contributions of other

_isotopes

of,C

and 0. However because of

_negative

reaction

_Q

_values,

contamination of the elastic

_peaks

was due

only

to

the _presence of

the 13C

_target

_impurity

(1

%)

and was

therefore

_neglected.

A

_typical

_energy

_spectrum

is shown in

_figure

2. The measured solid

_angle

ratio between the two detection

_systems

was checked

_by

moving

the arm

_completely

to the left or to the

_right

side of the beam and cross

calibrating using

the

gold

target.

FIG. 2. -12C energy spectrum 81ab = 10.8°.

The beam _energy was measured in a _separate

magne-tic

_analyzing

_system to within 0.5 MeV. _{For energy}

changes

of _up to 5 MeV aluminium foils mounted on a

movable arm were used as

_degraders.

The

_experimental

energy resolution varied between 500 keV and 1 MeV

depending

on the

_degrader.

With no

_degrader

the

effects. The forward

angle

elastic

_scattering

measure-ment at 122 MeV was

_performed

in a

_larger

chamber

installed on the same beam line. For this

experiment,

the two detectors were mounted on

_separately

movable

arms. Otherwise the

_experimental

_arrangements

were

identical.

At forward

_angles

the

_high

count rate from the

16O

elastic could cause

pile

_up

problems

in the

12C

channel.

_Anti-pile

_up devices were therefore included

in the electronic set _up and scalers were used to

measure the raw AE count rates and the

_pile

_up rates.

In addition a

_generator

simulated

12C

_peak

was fed

through

each _system and was used to make dead time corrections.

_{Any gain changes}

in the electronics could thus also be detected.

3. Results and data

_analysis.

- In contrast to the

results at lower

_energies

our data exhibit a

_nearly

monotonic decrease in the

_magnitude

of the backward

angle

cross-section as the _energy increases. Fluctua-tions of width less than 500 keV if _present, were

obscured

_by

the

_experimental

_energy resolution. Further and rather

_{surprisingly,}

_although

the back .

angle

differential cross-sections exhibit

_deep

oscilla-tions the maxima and minima do not move

progres-sively

towards 1800 c.m. as would be

_expected

from

a

[PLg(cos 0)]2

_description.

The forward

_angle

data

present

no such

_anomaly.

Since no

_rapid

variations with energy were

_observed,

it was

_thought

reasonable

to

_try

and fit the

_positions

of the back

_angle

maxima with a

_potential

model.

_Initially,

a shallow

1-depen-dent

_potential

used

_{by Badawy et}

al.

_[11]

in an

_attempt

to fit the 1660 excitation function between 15 and 30 MeV

_(c.m.),

was tried but failed to fit the

anoma-lous back

_angle

behaviour.

_By

_adjusting

the

1-depen-dence _parameters at each _energy

_good

fits could be obtained

_but

no

_simple

_energy

_dependence

of the model was

apparenta

Recently,

von Oertzen

_[12],

has remarked on the

success of

_deep

potentials

given

for

_{example by}

the double

_folding

model

_[13]

in

_accounting

for the

systematics a of

heavy

ion transfer reactions., It was

therefore decided to

_try

a

_deep

7potential

of the

(conventional)

form :

The Coulomb

potential Vc(r, Rc)

was that due to a

charged sphere

of radius 2.86 fm. The value of the radius _parameter has been

_justified

in reference

_[14].

The

_depth

of the real _part was chosen

_following

Satchler

_[15]

as

roughly

half that

_given

by

the double

folding

model. The _{geometry parameters} were

deter-mined

_{by fitting}

the variation with energy of the

positions

of the backward

_angle

maxima. An excellent fit was obtained with the values

_given

in table I

TABLE I

Optical

model _parameters

(*) The potential radii are obtained _{by multiplying by}

A f/3

where A _{1 =} 12.

FIG. _{3 _l6å-l2C scattering.} Variations of the _positions of the back

angle maxima with energy. The solid lines extending up to N 100 MeV

correspond

_to calculations made with the _potential

parameters of table I. Above this energy the simple oscillatory

structure of the backward angle distributions is lost and the

evo-lution of a _particular maximum is thus impossible to follow.

and is

_displayed

in

_figure

3.

_Remarkably

these

para-meters

_imply

a mean _square

radius of

the real

poten-tial

r2

_>v

= 14.8

fm2

which

is

almost

exactly

that

given

by

the double

_folding

model

_prescription

were (

_{r2 ) pl}

and

_{r2 ) p2}

are the mean _square radii

of the densities of the

_{interacting particles and (}

_{r2 >d,}

that

of

a

phenomenological

nucleon-nucleon force.

Taking

the

mean

square

radii

of the densities from

electron

_scattering

measurements

_[16]

and that of the

two

_body

force from reference

_[17]

_gives

a value of

1046

The

_predictions

of

_figure

3 were shown to be sensi-tive to the values of a and

_Ro

_{independently.}

This

sensitivity

was not

_preserved

when fits to the

_positions

of the forward

_angle

minima were made. These

posi-tions

which,

as can be seen in

_figure

4,

were well

fitted

_by

the

_parameter

set

_given

in table I were

shown to be determined

_{uniquely by}

the value of

r2

_)v.

The broken lines in

_figure

4

_give

some idea of

the

_sensitivity

to this _parameter. It was verified that

variations in the

_{imaginary potential}

_parameters

have little effect on these conclusions. At _{any one}

energy variations in either the

depth

or the

_geometry

parameters

were shown to affect

_chiefly

the

_depths

of forward

_angle

minima and the

_magnitude

of the backward

_angle

differential cross-section.

FIG. 4. - l60_l2C _scattering. Variations of the _positions of forward

angle minima with _energy. The solid lines _correspond to calculations made with the _{potential parameters} of table I.

The determination of the

_{imaginary potential}

(table

_I)

was carried out as follows. Below 33 MeV c.m. a linear increase of the radius

_parameter

was found

by

comparing

the excitation function data of

_Badawy

et al.

_[11]

with the

_optical

model

_predictions.

However,

since this data exhibits

_rapid

fluctuations with _energy it was deemed _necessary to define an

experimental

energy

averaged

cross-section at each _energy

_Eo

through

the relation

with AE = 1.0 MeV. The linear

_part

of the function

rule)

given

in table I was thus obtained

_{by demanding}

a

good

overall fit to the

_magnitude

of this _energy

averaged

cross-section. As can be seen in

_figure

5 the energy

averaged

data reveal gross structure resonances

which are

_reproduced

_by

the calculation. This struc-ture was shown to be

_independent

of AE for

L wncr i

c.m.

FIG. 5. - l60_12C _{scattering. Energy averaged} excitation function

(solid line) for the maximum situated near 160° _(c.m.) calculated from the data of Badawy et al. [11] as described in the text. The

optical model _{prediction (dotted line)} was made using the

para-meters given in table I. Since the experimental data exist _only on the

range 15 to 34 MeV c.m., the averaged cross-section is incorrect in N 2 MeV wide _regions at the _beginning and end of this range.

FIG. 6. - l60.12C scattering. Variation of total reaction cross-section with energy [4] (full line) compared with the optical model

prediction made _using the _parameters of table I.

As a further check we

_compared

the

_prediction

made

using

these _parameters with the total reaction

cross-section data of Kuenher and

_Almquist

_[4].

_Figure

6 shows that excellent _agreement was obtained.

Above 30 MeV c.m. it was found _necessary to add

a non linear term to the function

_rule)

in order to

reproduce

the absolute value of the back

_angle

diffe-rential cross-section at 52.3 MeV c.m.

_(Fig.

11).

When this was done a

_good

fit to our excitation

func-tion measured between 34 and 41 MeV c.m. was

obtained

_(Fig.

_{7). Further,}

the maximum value of

rI(E)

(2.15 fm at 70 MeV

_c.m.)

_gives

an excellent fit

to the forward

_{angle scattering}

data of Hiebert and

Garvey

[10]

(Fig.

8).

At low

energies

the influence of the non linear term is

_negligible.

Since a smooth monotonic increase of the

imaginary

potential

radius

_parameter

_gives

a

_{good description}

of all data considered it would be

_tempting

to _try to

interpret

this

_comportment

in terms of some

_physical

model. However a number of

_{problems preclude}

such

speculation.

The most

_important

is the existence of

FIG. 7. - l60_12C scattering. Prediction of excitation function for the maximum situated near 1560 c.m. The data points are those

measured in the _present work.

FIG. 8. - l60_l2C

scattering. 168 MeV (lab.) data of ref. _[10]. The _{prediction (full line)} was made with the maximum value of

r1(E) as _explained in the text. Other predictions (dotted lines) are

shown to _give an idea of the _sensitivity to _rl(E). Other parameters

are as _given in table I.

parameter

with _{energy may} be

_{replaced by}

an

expo-nential increase in the

_{potential depth.}

Also a first

order

_{Taylor expansion}

indicates that the addition of a derivative Saxon Woods _term, whose

_depth

increases

_linearly

with _energy, to an _energy

indepen-dent volume Saxon Woods form is

_equivalent,

over

short _{energy ranges,} to a

_single

volume term with a

linearly increasing

radius

_parameter.

Calculations made with these forms over small _{energy ranges}

verified these

_equivalences

and thus underlined the

phenomenological ambiguity.

Even if no such

ambi-guity

existed a

_difficulty

would arise since one _expects

the

_optical

model to underestimate _{the average} of

a

rapidly fluctuating

cross-section

by

the _average

of the cross-section due to the

_fluctuating

_part

of the

scattering

wave function. Since the

origin

of the

observed fluctuations

_is,

at

_present,

unknown it is

impossible

to decide how much of the _energy

_averaged

cross-section should be allocated to

_potential

scatter-ing.

Finally

we remark on the fact that our _parameters

suffer from the same drawback as those of Kuenher

and

_Almquist

_[4]

in as much as at low

_energies

_they

imply

non zero values

of

_{1 1], for}

low

_partial

waves.

In order to

_investigate

this

_point

a

_{strongly absorbing}

core was added to our

_optical

_potential

the

_parameters

of which are

_given

in

figure

9. The small diffuseness was chosen to minimise the effect of the core in the

surface

_region

while

_avoiding

the introduction of

sharp

discontinuities in the

_potential.

As can be seen

in

_figure

9

_(inset),

_increasing

the

_depth

of this term

FIG. 9. -16O-12C _{scattering. Sensitivity} of the calculated 17.2 MeV

(c.m.) angular distribution to the addition of a _{strongly absorbing} core to the imaginary potential of table I

with _RCORE = _0.5.4 1/3 _{fm. (A,} = 12),

ACORE = _{0.3 fm. The effect}

of the core on the transmission coefficients _T, is shown as an inset.

has a

_large

effect on the transmission coefficients for

low

_partial

waves while

_producing

no

_significant

change

in the

_partial

waves scattered from the

poten-tial surface.

Further,

it is clear from the

_figure,

that the

_{large changes}

in the transmission coefficients for low

_partial

waves do not

_strongly

affect the back

angle

differential cross-sections. We thus conclude that the non

_physical

character of

_{the 111, values}

for low

_partial

waves is associated with the use of

the

Saxon Woods

_{imaginary potential}

form factor and that this form is

_probably

not

_appropriate

for use in

analysis

of

_{light heavy}

ion

_scattering.

Predictions for the back

_angle

differential

cross-sections are shown with the data in

figure

10

and,

for the more

_complete

data at 122 MeV in

_figure

11. No detailed

_fitting

was

_attempted

because of the

limited

_angular

_range of the data.

4. Discussion. -

Contrary

to the statement made in the introduction of reference

_[5]

it is

_possible

to fit both forward and backward

_{angle 16O-12C}

_scattering

1048

FIG. 10 - 16O-12C scattering. Back angle data between 80 and

100 MeV _{(lab.) together} with _optical model _predictions made _using the _{parameters given} in table I.

FIG. 11. - l60_12C scattering. Data at 122 MeV with the optical model _prediction made _using the _{parameters given} in table I.

or not the

_deep

_potential

found in the _present

_analysis

has

_{physical significance.}

The lack of

_sensitivity

of the

predictions

to the _presence of a

strongly absorbing

core

_(Fig.

₉₎

indicates that the interior _part of the

potential

at radii 1.5 fm is not well determined

by

the data considered. However at distances _greater than this value a

deep potential

would seem to be indicated.

Further,

our

_analysis

shows that the

geometry parameters of such a

_potential

are well

determined

_provided

that data at forward and back-ward

_angles

over a

_large

_{energy range} are considered. Theoretical calculations

_{[18, 19]}

of the ion-ion

potential

usually predict

a shallow

potential

such as

that used in the

_analysis

of the

12C_12C

data

_by

the Yale _group

_{[20]. However,}

a recent calculation of the

16O-16O

_potential

_[21]

_using

a

_generator

coordinate

method

_produced

a

_potential

rather like that used in

the

_present

_analysis.

Detailed

_comparison

is difficult

since the calculation of the interior and exterior

_parts

of the

_potential

were made in different limits of the

approximations employed.

We note however that the authors

_predict

a

_potential

of - 430 MeV central

depth

falling

to half of this value for radii between 2 and 4 fm.

_They

also conclude that the surface of the

potential

is well

_{approximated by}

the double

_folding

model.

From a

_{phenomenological point}

of view our results

agree with the

findings

of Kozab et al.

[22] who,

in

analyzing

the variation of

_complete

fusion

cross-sections with _energy for

2’Al

+

_160,

2’Al

+

2°Ne

and

2’Al

+

325,

were led to the conclusion that the real

_part

of the

_{optical potential}

was

_strongly

attrac-tive at radii

_considerably

smaller than the sum of the

radii of the

_{interacting particles. Analysis}

of syste-matics of transfer reactions

_by

von Oertzen

_[13]

also indicated the

_necessity

of a

_deep

real

_potential.

On the other hand there exist _many

phenomenolo-gical analyses

of ion-ion

_scattering

_using

shallow real

potentials

[24].

It is our

_opinion

that such

_analyses

are

_strongly

influenced

_by

the choice of radius

para-meter which is almost

_always

taken to be the sum of

the radii of the

_interacting

nuclei. A

_priori

of course

there is no reason for this choice

_which,

at least for

light

targets,

disagrees

with the

_supposition

that the

potential

surface is well

_{represented by}

the double

folding

model. For

_example

in the

16O-12C

case

the sum of the radii is

Even for zero diffuseness we obtain

which _may be

_compared

with the value of 14.8

fm2

given by

the double

_folding

model (section

3).

A somewhat

_surprising

result of our

_analysis

is the smallness of the

_{imaginary potential.}

Its main

justifi-cation comes from the

_agreement

between the

cal-culated and

_experimental

reaction cross-sections as

well as the overall _agreement with the

_magnitudes

of

the backward

_angle

differential cross-sections. Because of the

_ambiguities

detailed in the

_previous

section we

may conclude

only

that the

_{imaginary potential}

increases with _energy. This conclusion _may of course

be understood

_partly

in terms of the

_{increasing density}

of states in the 28Si

_compound

nucleus and

_partly

in terms of the

_availability

of an

_increasing

number

of

_direct

and semi-direct reaction channels with

increasing

energy.

Finally,

we should remark on

_possible

contribu-tions from the cx transfer mechanism. Our

_analysis

has

shown that all the _gross structure features of the data

are well described

_by

a

_single

direct _process (elastic

scattering).

The a transfer

reaction,

which _may be an

part of the

_{cross-section,}

would be

_expected

to be characterized

_by

a

[PLs(cos 0)]’

behaviour which

disagrees

with observation above 34 MeV

_(c.m.).

Below this _energy it is not clear whether the

interfe-rence of these mechanisms is

_capable

of

_producing

the observed fluctuations but this

_possibility

should

not be ruled out. Detailed

_{calculations,}

_clearly

necessary to test this

_point,

are in progress.

Acknowledgments.

- The

experiments

described in this _paper were carried out in collaboration with members of a _group of

_physicists

of C.E.N.

_Saclay.

We wish to thank them for their assistance. Our thanks are

_especially

due to Dr’s P. Charles and B. Fernandez for _many fruitful discussions and to

Professor J. P.

_{Longequeue (ISN)}

for

_experimental

assistance and for careful

_reading

of the

_manuscript.

References

[1] CHARLES, P., BADAWY, I., BERTHIER, B., DOST, M., FERNANDEZ, B., GASTEBOIS, J. and LEE, S. M., Proceedings of the International Conference on Nuclear Physics, Vol. 1,

edited _by J. de Boer and H. J. Mang (North-Holland, Amsterdam) 1973, p. 539.

[2] MALMIN, R. E., SIEMSSEN, R. H., SINK, D. A. and SINGH, P. _P.,

Phys. Rev. Lett. 28 ₍₁₉₇₂₎ 1590.

[3] STOKSTAD, R., SHAPIRA, D., CHUA, L., PARKER, D., SACHS,

M. W., WIELAND, R. and BROMLEY, D. _{A., Phys.} Rev. Lett.

28 (1972) 1523.

[4] KUENHER, J. A. and ALMQUIST, E., Phys. Rev. 134B (1964) 1229.

[5] GUTBROD, H. H., BOCK, R., VON OERTZEN, W. and

SCHLOT-THAUER-VOOS, U. C., Z. Phys. 262 ₍₁₉₇₃₎ 377.

[6] COHEN, S., PLASIL, F. and SWIATECKI, W. J., Ann. Phys. 82

(1974) 557.

[7] CHATWIN, R. A., ECK, J. S., ROBSON, D. and RITCHER, A., Phys. Rev. C1 ₍₁₉₇₀₎ 795.

[8] DEVRIES, R. M., Nucl. Phys. A 212 (1973) 207.

[9] WILLIAMS, D. J. and SIEGERT, F. E., Nucl. Phys. 30 (1962) 373.

[10] HIEBERT, J. C. and GARVEY, G. T., Phys. Rev. 135B ₍₁₉₇₄₎ 346.

[11] BADAWY, I., BERTHIER, B., CHARLES, P., FERNANDEZ, B. and

GASTEBOIS, J., Note CEA-N-1791 ₍₁₉₇₄₎ p. 14.

[12] VON OERTZEN, W., Preprint LBL 1985 _(1973).

[13] EISEN, Y., Phys. Lett. 37B (1971) 33.

[14] IWE, H., WIEBICKE, H. J., European Conference on Nuclear

Physics with Heavy Ions, Caen 1976, Communications

p. 20.

[15] SATCHLER, G. R., In International Conference on Reactions

Between Complex Nuclei, edited by R. L. Robinson,

F. K. McGowan, J. B. Ball and J. H. Hamilton

(North-Holland, Amsterdam) 1974, vol. 2, p. 171.

[16] HOFSTADER, R., Ann. Rev. Nucl. Sci. 7 (1957) 308.

[17] GREENLEES, G. W., PYLE, G. J. and TANG, Y. C., Phys. Rev.

171 (1968) 1115.

[18] BRINK, D. M. and STANCU, F. L., Nucl. Phys. A 243 ₍₁₉₇₅₎ 175.

[19] BRUECKNER, K. A., BUCHLER, J. R. and KELLY, M. M., Phys.

Rev. 173 (1968) 944.

[20] BROMLEY, D. A., In Nuclear Reaction induced _{by Heavy} Ions,

Proc. International Conference, Ed. R. Bock, W. R.

Hering (Heidelberg) 1969, p. 49.

[21] JAQAMAN, H. R. and MEKJIAN, A. Z., Nucl. _Phys. A 259 (1976)

157.

[22] KOZUB, R. _{L., Lu,} N. H., MILLER, J. M., LOGAN, D., DEBIAK,

T. W. and KOWALSKI, L., Phys. Rev. 11C (1975) 1497.

[23] See for example Heavy ion _{scattering. Proceedings} of the