Correction coulombienne du facteur de phase en théorie diffractionnelle de la diffusion subcoulombienne

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Correction coulombienne du facteur de phase en théorie diffractionnelle de la diffusion subcoulombienne

Ch. Leclercq-Willain

To cite this version:

Ch. Leclercq-Willain. Correction coulombienne du facteur de phase en théorie diffrac- tionnelle de la diffusion subcoulombienne. Journal de Physique, 1971, 32 (7), pp.475-482.

�10.1051/jphys:01971003207047500�. �jpa-00207101�

CORRECTION COULOMBIENNE DU FACTEUR

DE PHASE EN THÉORIE DIFFRACTIONNELLE

DE LA DIFFUSION SUBCOULOMBIENNE

Ch.

LECLERCQ-WILLAIN

Institut Interuniversitaire des Sciences

Nucléaires,

Université Libre de

Bruxelles, Bruxelles, Belgique (Reçu

^{le 28} janvier 1971)

Résumé. ²⁰¹⁴ La méthode

semi-classique

développée ^{en théorie} d’excitation coulombienne est utilisée _pour définir les sections efficaces différentielles

élastique

^et inélastique d’excitation d’un niveau nucléaire vibrationnel à un phonon 03BB, ^en diffusion subcoulombienne de particules ^{03B1. L’inter-} férence des interactions coulombienne et nucléaire modifie la structure diffractionnelle aux petits angles ^de diffusion. Les résultats théoriques présentés ^sont appliqués ^à l’étude des diffusions élas- tique ^Ni58 (03B1, 03B1) ^{Ni58 et} inélastique ^Ni58 (03B1, 03B1’) ^{Ni58 avec} excitation des niveaux 2+ et 3- du Ni⁵8.

Abstract. ²⁰¹⁴ A phase ^Coulomb correction is introduced in the usual diffractional model to define the differential cross sections for elastic and inelastic subcoulomb 03B1 scattering ^with excitation of a

vibrational one phonon 03BB nuclear level. The interference of Coulomb and nuclear interactions modifies the diffractional structure at small scattering angles.

Those theoretical results are used to study ^{the elastic} scattering ^Ni58 (03B1, 03B1) Ni58 and the inelastic

one with excitation of 2+ and 3- levels in Ni58.

Classification Physics Abstracts :

12.17, 12.37

1. Introduction. ^- One of the most

striking

^features

of the

scattering

^{of a}

particles,

^{with an energy E which}

exceeds the Coulomb

barrier,

^{is the}

persistence

^of

sharp, regularly spaced

oscillations in the

angular

distributions. Such oscillations are a characteristic feature of so-called surface reactions. ^{We assume} that the conditions

of

high

^incident energy and

for excitation of collective levels in an

absorbing

target nucleus, ^are fulfilled. In the

expressions (1)

^and

(2) ;

t1 ⁼ zz’

e21!iv

is the Coulomb

parameter, k

^{the wave}

number of the incident

particle, R

the radius of the nucleus and àE the _{energy of} the excited level.

Then,

in the process of

scattering,

^{the nucleus can be consi-}

dered immovable and the

change

in energy of the

particles

^{in inelastic}

scattering

^{can be}

ignored.

^The

adiabatic

approximation

^{is thus}

applicable.

^{The solu-}

tion of the

scattering problem

^{reduces to}

finding

^the

scattering amplitude fez2, ç) (3)

^of

particles

^{from a}

fixed nucleus

where ç

^{define the internal}

variables,

characteristic of the target

nucleus,

^{and Ç2 =}

(0, rp)

determine the direction of

scattering.

In this case the differential elastic

scattering

^cross-

section and the inelastic one for the excitation of ^a final

level IM

> in the target ^{nucleus are defined}

by

different

projections

^{of the same}

scattering amplitude f(Q, ç) (3) :

If condition

(1)

^is

fulfilled,

^{it is}

possible

^{to calculate}

the

scattering amplitude (3) using

^the diffraction

theory

^method

[1 ]-[4]

where k is the wave number of the

outgoing particle.

Integration

^{is defined in} the shadow

plane (p, cp) perpendicular

^{to the unit vector}

in a direction deflected

by 0/2

^{from the} incident beam.

Here,

Q(p)

^is the function which accounts for the characteristics of the nucleus and of the elastic and inelastic

scattering potentials [1 ]-[3].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01971003207047500

476

We find the differential cross-sections for elastic and inelastic

scattering

^with excitation of a vibrational one

phonon A

^nuclear level in the field of a nuclear and a

Coulomb

scattering potential.

2.

Sharp

cutoff-model. ^-

Assuming

^{that the}

scattering potential

^is

composed

^{of a} nuclear and a

Coulomb part

the function

Q(p)

^{can be}

split

up into a nuclear and a

Coulomb factor

2.1 THE

^{NUCLEAR FUNCTION. -} The function

Q’(p) ^accounts

^for the characteristics of a black nucleus :

Q’(p)

^{= 1 on all}

planes

^{Z = const. aside}

from the shadow of the nucleus in which the value is 0.

(9) Assuming

^{that the}

equation

^of the nuclear sur-

face is

the shadow line is defined

by

2.2 THE COULOMB FUNCTION. ^- The function

S2°b(p)

accounts for the

charge

^{effects of the} target ^nucleus

Assuming

^{that the}

charge density

^{function for the} target ^{nucleus is}

where

the Coulomb interaction

potential,

^{for small deforma-}

tion parameter aÂ, ^{in the}

expression (10),

^is

Then,

^the

explicit

^{form of}

Qc’(p)

^is

2. 3 SCATTERING AMPLITUDE AND CROSS-SECTIONS. ^-

With the

expressions (5), (9)

^and

(15)

^the

scattering amplitude f(Q, ç)

^is

To find the cross-sections for elastic and inelastic

scattering

^we

expand

^the

amplitude (16)

^{into a series}

for small deformation parameters ^{which is limited}

by

linear

approximation.

^We obtain the

following general

expression

^{for the}

scattering amplitude

^of

charged particles

^{from a fixed nucleus} whose surface has  vibrations

So the elastic

scattering amplitude

where the Coulomb

amplitude

^is

and the functions

Foo(kRo 0)

^{is defined in}

Appendice

^1.

Thus,

^{the total}

amplitude

^{of elastic}

scattering

^consists

of the

amplitude

of elastic

scattering

ⁱⁿ the electric field of the nucleus and a nuclear part

which,

^{for the Cou-}

lomb parameter il ⁼

0,

^is

exactly

^the

expression

^of

the Blair

sharp

cutoff model

[1].

So,

the elastic differential cross section

splits

^into

three terms :

The Coulomb cross section

a nuclear term in Coulomb field

and an interference term

with

With

regard

^{to the}

expression

^{of the total elastic}

cross-section divided

by

^the Rutherford cross-section

where

the condition _y ⁼ 1 defines a limit

scattering angle

For il

>

kRo > 1, 0.

^is

large

^{and the}

scattering

^fol-

lows the Rutherford law.

For

kRo > q

>

1,

^{the second term is dominant in} the

expression (22)

^{and the}

scattering

^{has a} diffractional structure. For

kRo -- il » 1,

^the

scattering

^{is « Cou-}

lomb » for 0

Oo

and « diffractional » for 0 >

00.

Those three

possibilities

^are summarized

by

^three

experimental

^{results in}

figures 1,

^{2 and 3.}

FIG. 1. ^- At incident energies ^{lower than the Coulomb} barrier, the angular distribution follows the Rutherford law.

The inelastic

scattering amplitude

^{for excitation}

of a one

phonon

vibrational level

is defined

by

^the

following expression :

where

478

FIG. 2. ^- At incident energies ^much higher ^than the Coulomb barrier,

a diffractional theory ^is applicable ^{to describe the} angular distribution.

FIG. 3. ^- At energies ^near the Coulomb barrier, ^the description of the scattering process must take both the Coulomb and

nuclear interactions into account.

So the differential

scattering

^cross section is :

and

splits

into three terms

(Appendice 2).

In the linear

approximation

for small deformation parameters ^{of the} target

nucleus,

^the

angular

^distri-

butions for elastic

scattering

^{on a}

spherical

^{or a non-}

spherical

^{nucleus are the same and the}

angular

^distri-

butions of inelastic

scattering

^with excitation of the first rotational level of a permanent ^{deformed nucleus}

are defined

by

^{the same}

expression (17)

^{with the}

dyna-

mic variable aA,

replaced by

(w

^{shows the} orientation of the axes of symmetry ^of the nucleus and _a is the permanent deformation

parameter)

^{and the}

expression (24)

^with

flz replaced by

The

general expressions (21), (25)

^and

(A 2)

^are

applied

to the excitation of a one

phonon A

⁼

2, 2+

^level

_{[5] [6]}

and a one

phonon A

⁼

3,

^{3- level}

[6].

2.4 RESULTS OF CALCULATION AND COMPARISON WITH EXPERIMENTAL DATA. ^-

According

^{to the} expres- sions of the

sharp

cutoff Blair model

[1] ]

^{both the}

elastic and inelastic cross-sections measured for

projectiles

^{of different wave} number k should follow

« universal » curves when divided

by ^k2

^and

plotted

versus x ⁼ 2

kRo

^sin

0/2.

In

figure 4,

^{a universal curve has been drawn to} summarize the oc elastic

scattering experimental

^data

on

Ni58

at incident

energies Eo

^{= 100 MeV}

[7],

FIG. 4. ^- Universal curve of the a elastic scattering experi-

mental data on Ni 5 8 at incident energies ^of 100, 41 and 33 MeV.

41 and 43 MeV

[8]

^{and 33 MeV}

[7].

^{Some elastic}

data

corresponding to

^the

highest energies (100-85 MeV)

are

displayed

^{at small}

angles.

We believe that this may be attributed to the

varying importance

^{of the}

Coulomb corrections to the scattered

amplitude

^at

small

angles.

^The theoretical universal curves calcu- ted

according

^{to formula}

(21) [narrow

^{solid curve for}

41,

³³

MeV,

^{dashed curve} for 100-85

MeV]

^{and to} formula of the

sharp

^{cutoff Blair model}

[1] [large

solid

curve]

^{have been drawn in}

figure

^{5. As the energy}

FIG. 5. ^- Theoretical universal curves of ce elastic scattering

on Ni 5 8 calculated according ^{to formula} (21) (narrow ^solid

curve for 41 and 33 MeV - dashed curve for 100 and 85 MeV) and to formula of the sharp cutoff B!air model (large ^solid curve).

FIG. 6. ^- Experimental and theoretical angular distributions in elastic a scattering ^{on Ni$g} at energy of 33 MeV. The results of the sharp ^{cutoff Blair model} and of this calculation are

compared.

480

is

increased,

^{the Coulomb}

complications occuring

^at

small

angles

^{should be} reduced and indeed it is seen

that the

high

energy data lie much closer to the Blair

curve than does the universal curve at lower

energies.

Similar

experimental

and theoretical results are

obtained in inelastic

scattering Ni58(OC, a’)

^{Ni58* 2+}

FIG. 7. ^- Angular distributions of inelastic ^ce scattering ^at

33 MeV with excitation of 2+ and 3- levels in Nis 8.

and 3 - levels. In

figure 6,

^the

experimental angular

distribution in elastic a

scattering

^on

^Nis8

^at

Eo

^{= 33 MeV is}

compared

^to the theoretical curves

calculated

according

^{to the}

sharp

cutoff Blair model

[1]

(dashed curve)

^{and to} the formula

(21)

with Coulomb and nuclear interference effects

(solid curve).

^{At this}

energy, in the

angular

distributions of inelastic scat-

tering

with excitation of 2+ and 3-

levels, (A

²

with ⁼ 2 and

3)

the interference of the Coulomb and nuclear interactions modifies the diffractional

structure at small

scattering angles (0 200) (see Fig. 7).

^A

sharp

^{minimum is defined at a}

scattering angle

^for which the Blair model result is near a

maximum.

3. Smooth Cutoff Model. ^- In the above

results,

it is seen that the

magnitudes

of the theoretical diffe- rential cross sections

drop

^{off much too}

slowly

^with

scattering angles.

^{Some authors}

[9], [10]

^have

already pointed

^{out the}

improvement

^{of the adiabatic nuclear}

theory by using

^a smooth cutoff

picture.

This rounded cutoff correction is introduced in

assuming

^{for the}

nuclear function

QN(P)

^{the form}

Some results are

developped

^{with an}

optical

^surface

nuclear

potential

By

^similar

calculations,

^{we can} define the _correspon-

ding scattering amplitude f(Q, aÂ,).

^{At the zero and}

first order

approximation

ⁱⁿ _az., ^{we have :}

where and

This

formalism has been

investigated by

^Drozdov

[12]

and

applied by

this author to calculate differential cross-sections

Nis8, Mg24 (p, p’)

^at 40 MeV with 2+

and 3 - excitation. The results seem to

depend largely

on the

imaginary

part ^of the nuclear

potential.

We must note that the

approximate expressions,

obtained

by

^Drozdov

display

^a difference from the formulas

(28), (29) ;

^{the last term of}

fi being

^absent

from the Drozdov

expression.

4. Conclusion. ^- Elastic and inelastic

scattering

of fast

charged particles

^is

investigated

ⁱⁿ the diffrac- tion

approximation

^{in which a Coulomb}

phase

^correc-

tion has been

partly

introduced.

Indeed,

^the

scattering

wave function

is an

approximate expression,

^{the z}

dependence being

characteristic of a

plane

^wave.

Perhaps,

^{it would be}

necessary ^to

develop

^a

theory

^{based on} the Coulomb

wave functions and ^on the Coulomb Green function.

However,

^{in this} case, it seems easier to use a distorted

wave method.

With the smooth cutoff

correction,

^{the results are} the same as those calculated

by

^{distorted wave}

theory [10], [11]

^{and the} agreement ^with

experiment

is

quite impressive.

^The D. W. B. A. is more

general

and

requires

^fewer

approximations. However,

^when Coulomb interactions have to be taken into account, many

partial

^{waves have to be} considered in the deve-

lopment

^{of the Coulomb} excitation terms ; then the distorted wave method

requires

^more

developed

numerical calculations on

big

computers.

So,

this Coulomb and nuclear smooth cutoff model

can be used

successfully

^and presents ^the

advantage

of

simplicity

^{on the} D. W. B. A. method.

Acknowledgements.

^{- We are}

grateful

^{to Profes-}

sor M. Demeur for many useful discussions and fo his constant and

unfailing help

^and encouragement

at all times.

We are indebted to the staff of the

Computer

^Center

of Brussels

University

for extensive use of their faci- lities.

APPENDICES 1. Function

We use the

multiplication

theorem of Bessel functions

So we obtain

By solving

^the

u-integration :

2.

Scattering

differential cross-sections for excitation of a one Â

phonon

^level.

482

where

is defined in

(A 1).

References

[1] ^BLAIR (J. S.), « Inelastic Excitation of Collective Levels ^» Lectures on Nuclear Interactions, Herceg- Novi, 1962, VII, 1.

[2] ^AKHIEZER (A. I.) ^{et SITENKO} (A. G.), Phys. Rev., 1957, 106, ^1236.

[3] ^GLAUBER (R. J.), « High-energy ^Collision Theory »:

Lectures in theoretical Physics, ^Boulder 1959, ed.

Interscience Publ. Inc., ^New York, 1959, p. 315.

[4] ^FESHBACH (H.), Intern. School of Physics ^{« E.} Fermi », Varenna Course 38-Academic Press, ^New York, p. 183.

[5] ^DROZDOV (S. I.), ^Soviet Physics JETP, 1959, 36, 1335.

[6] LECLERCQ-WILLAIN (C.), ^{« Etude} théorique de l’inter- férence des excitations coulombienne et nucléaire », Thèse Bruxelles.

[7] MERIWETHER (J. R.), ^{BUSSIERE DE NERCY} (A.) ^et

HARVEY (B. G.), Phys. Lett., 1964, 11, 299.

[8] ^BROEK (H. W.), ^YNTEMA (J. L.), BUCH (B.) ^{et SATCH-}

LER (G. R.), ^Nucl. Phys., 1965, 64, 259 ; Mc DANIELS (D. K.), ^BLAIR (J. S.), ^CHEW (S. W.) ^et

FARWELL (G. W.), ^Nucl. Phys., 1960, 17, ^614.

[9] ^BLAIR (J. S.), ^SHARP (D.) ^{et WILETS} (L.), Phys. Rev., 1962, 125, ^1625.

[10] ^RosT (E.), Phys. Rev., 1962, 128, ^2708.

[11] ^BASSEL (R. H.), ^SATCHLER (G. R.), ^DRISKO (R. M.) ^et

RosT (E.), Phys. Rev., 1962, 128, ^2693.

[12] ^DROZDOV (S. I.), Proc. of Conf. on « Direct Interac- tions and Reaction Mechanisms », ^Padua 1962, Gordon and Breach Science Publ., ^New York,

p. 799.