Two-nucleon stripping process with effective interactions

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Two-nucleon stripping process with effective interactions

M. Ismail

To cite this version:

M. Ismail. Two-nucleon stripping process with effective interactions. Journal de Physique, 1977, 38

(8), pp.897-903. �10.1051/jphys:01977003808089700�. �jpa-00208656�

TWO-NUCLEON STRIPPING PROCESS

WITH EFFECTIVE INTERACTIONS

M. ISMAIL

(*)

Faculty

^of

Science,

^Cairo

University, Egypt

(Reçu

^{le 15 décembre}

1976,

^{révisé le 14 mars}

1977, accepté

^{le 29 avril}

1977)

Résumé. ²⁰¹⁴ On étudie le facteur de forme du processus d’arrachement

(stripping)

de deux nucléons

4oCa(t, p)42Ca

^{en utilisant deux} types d’interactions effectives, ^{celles de}

Skyrme

^{et de} Brink-Boeker.

On compare les résultats ^{avec ceux} obtenus avec le

potentiel

habituel de portée ^{nulle. On trouve} que

l’emploi

des interactions effectives réduit l’importance ^{de la} région nucléaire interne. De

plus

^les

facteurs de forme calculés avec ces interactions effectives diffèrent du facteur de forme obtenu pour

une portée nulle à la fois dans la

région

de surface et dans la région

asymptotique.

Abstract. ²⁰¹⁴ The form factor of the two-nucleon

stripping

process

40Ca(t, p)42Ca

is studied

using

two types ^{of effective} interactions, namely ^the Skyrme and Brink-Boeker interactions. The results ^are compared with those of the

ordinary

zero-range

potential.

^{It is found that the effective} interactions reduce the

importance

of the nuclear internal region. ^{In addition the form} factors calculated with effective interactions differ from the zero-range form factor in both the surface and asymptotic

regions.

Classification

Physics ^Abstracts

25.50

1. Introduction. ^-

Steady

progress has been made in recent years towards a

microscopic understanding

of nuclear

reactions,

^{that is to} say, ^a

description

ⁱⁿ

terms of detailed nuclear wave functions and an effec- tive interaction between nucleons which has some

realistic basis

[1].

The two-nucleon transfer reactions have taken on great

importance,

^since

Glendenning [2]

has shown these reactions to be _very sensitive to

spectroscopy.

The DWBA is a useful method for

describing

the two-nucleon transfer process. For definiteness we consider the reaction

A(t, p)B.

^{The DW}

calculation for this reaction consists of two inde-

pendent

parts. ^{The first} part

depends

^{on the}

overlap

of initial and final nuclear states and the effective interaction between the

outgoing proton

^{and the two}

captured

neutrons, referred ^{to as} the form factor. The

second part of the calculation is standard and can be carried out with well known computer programs.

In references

[3]

^and

[4],

^the effects of

configuration mixing

and the choice of

single particle

^{wave functions}

on the two-nucleon

stripping

^{form factor were studied}

in zero range DWBA. The present work studies the effect of effective interactions on the two-nucleon transfer form factor. For this _purpose the reaction

40ca(t., p)42Ca(g.s.)

^{at an} incident triton _energy of 12 MeV is considered. The

12 Ca

nucleus is taken to consist of two neutrons in a 1

f7/2

^level

^plus

^the

^4°Ca

closed shell. The

stripping

form factor is calculated

using

^two

types

of effective nucleon-nucleon interac-

tions, namely

^the Brink-Boeker

[5]

^and

Skyrme [6]

interactions.

2. The two-nucleon

stripping

^{form factor with}

Skyrme potential.

^{- The} two-nucleon

stripping

^transition

amplitude

for the reaction

A(t, p)B

may be written in DWBA as

The ’1’(-) and ’1’(+) are the usual distorted waves while the _cp functions describe the internal functions of the

particles

^involved. _Xp ^{is the}

spin

^wave function of the

outgoing

proton.

(*) Present address : Science & Mathematics Centre, ^{P.O. Box} 4577, Riyadh, Kingdom of Saudi Arabia.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003808089700

898

Using

^{the classical}

decomposition

^of

QJ JBMB(ÇA"

^rl,

r2)

^{one gets}

If the

interacting potentials

^are

spin independent,

^the differential cross-section _may be

given by

where

The coordinates are defined in

figure

^{1. We now evaluate}

integral (4) by making

^three

hypotheses.

2.1 The incident triton is assumed to be in a pure S-state. This is not a crude

approximation

since it is known that the

ground

^{state of a triton is at least 94}

%

^S-wave

[7].

The internal wave function of the triton is chosen as a Gaussian

where

2.2 The interaction

potential V;p

^{is taken as a}

density-dependent

effective interaction of the

Skyrme

type

[6],

^{and the sum of the two}

potentials (V1p

⁺

V2p)

^is

approximated by

where the

operators V2

and

V2

^{act on} the left and on the

right

of the matrix element.

p(R )

^{is the}

density

^{of the}

residual nucleus.

2.3 The

transferred-particle

^{wave functions are assumed to} be those of harmonic oscillators. The Talmi-

Moshimsky

transformation

gives

the result

[8]

We use the

Taylor expansion

^{of the wave functions}

and

to get

Substituting

eq.

(5), (6), (7)

^and

(8)

^into eq.

(4)

^and

setting Q

⁼

cxvf - ’ 3 Vr

^{we obtain}

The operators

V2n

^and

V>2n

^{now act on} the functions

e1J.Q

and

e-4"õ,,2 respectively,

and their values can be

easily

obtained. The double

integration

^over

r and n

^{can be}

performed

and its value is

where

Green’s theorem

implies that, if O2

^{is the}

Laplacian acting

^on

pl+),

^{we can}

replace

^the

operator Q 2

ⁱⁿ eq.

(10) by

Use of the

Schrodinger equation

allows each operator ^{to be}

replaced by

^its

eigenvalue

where

mi*

denotes the reduced mass of the

particle

^i.

Substituting

^{for the}

eigenvalue

^of

Q 2

^from eq.

(11)

^one gets the

following expression

^for

I L

where

The factor

Fn (Rt)

^is

complex

because of the existence of

optical-model potentials

^{in both}

kf

^and

kf .

^{The real}

part

of Fn (Rt)

goes ^{to a constant} value when the real

potentials

ⁱⁿ

equation (13)

^{become zero. The}

imaginary

part ^of

Fn (Rt)

is determined

directly by

^the

optical-model imaginary potentials

of the incident triton and the

outgoing

proton. ^{For the}

ordinary

^zero range

potential, Vo d(n).,

^{the factor}

Fn(Rt)

^{tends to a real and constant}

value,

900

The two-nucleon

stripping

form factor

corresponding

^{to the}

Skyrme potential

^is

given by

where

QNL(Rt) is

the radial part ^{of the wave function}

describing

^{the center of mass motion of the two}

captured

neutrons. For the

ordinary

^zero range

potential

^the two-nucleon

stripping

^{form factor is}

given by

3. The two-nucleon

stripping

^form factor with the Brink-Boeker

potential.

^{- The}

finite-range

^effective

interaction of the Brink Boeker type may be written as

where the first term is attractive while the second is

repulsive [5].

Let us consider first the attractive term

only.

^{In terms} of the coordinate vectors

r and n) (defined

ⁱⁿ

figure 1),

Eq. ( 17)

^{can be written as}

Substituting

eq.

(5), (7), (8)

^and

(18)

^into eq.

(4),

^one gets

where

If one

expands

^the

plane

^waves

on a

spherical

harmonic basis and assumes that A _{= p} ⁼

0,

the double

integral

^over

r and q

^{can be}

performed

and _eq.

(19)

^{reduces to}

where

As in the

preceding

^{section we}

replace

^the operator

Q 2 by

and the value of

can be found

by using

^{the LEA. This}

approximation simply replaces

^each

Laplacian by

^its

eigenvalue

^obtained

from _eq.

(11).

The

repulsive

^term of the Brink-Boeker

potential gives

^{a factor}

FBR(Rr)

^{which is}

exactly

^{the same as that}

given by

eq.

(21 ) except

^that

V 1 and P

^are

replaced by Y2 and B’ respectively.

The two-nucleon

stripping

^form

factor

corresponding

^to the Brink-Boeker

potential

^is

given by

where

4. Numerical calculations. ^- In order to show how the effective interactions affect the radial form factor of the two-nucleon

stripping

^process, the reaction

40Ca(t, p)42Ca(g.s.)

^{at an} incident triton _energy of 12 MeV was considered. The factor

Fn(R)

^{was cal-}

culated for this reaction

using

the Brink-Boeker and

Skyrme

^effective

interactiQns

and the results are

represented

^on

figures

^{2 and 3.}

The difference in

shape

between form factors calculated

using

^zero range and effective interactions

can be seen from

figure ^4,

which shows the real

parts

of the form factors for both the

Skyrme

^{and the}

Brink-Boeker

potentials together

^{with the zero} range form factor. In each case the form factor is obtained

by multiplying

the factor

Fn.(Rt,) by

^{the wave func-}

tion _TNL as indicated

by

eq.

(14), (15)

^and

(22).

^The

three form factors shown on

figure

^{4 are calculated}

for the case J ⁼ L ⁼ 0 and N ⁼ 4.

FIG. 2. - The real part ^of F1(Rt) calculated using Skyrme ^and

Brink-Boeker effective potentials. They ^are multiplied by the factors 1.50 x 10-3 and 3.23 ^x 10-3 respectively. - - - - Skyrme; ^20132013

Brink-Boeker.

FIG. 3. ^- The imaginary part ^of F1(Rt) calculated using Skyrme

and Brink-Boeker potentials. - - - - Skyrme; ^- Brink-Boeker.

The size

parameter "0

of the triton wave function

was chosen such that the rms radius of the triton is 1.68 fm. The oscillator

parameter

^{v was defined so} that the

single

^{nucleon wave functions are} propor- tional to

This

parameter

^is

typically

^about

^{A - 113}

^{fm -}

2,

^which

corresponds

^{to an} oscillator

spacing

nw = 41 A -1/3 MeV.

The

density p(R )

^{was assumed to be}

(9)

902

FIG. 4. ^- Plot of the real parts of the two-nucleon stripping ^form

factors obtained from Skyrme, Brink-Boeker and zero range inter- actions. All three curves are normalised at the point ^{R = 4.5 fm.}

zero range ; - - - - Skyrme; ^{+ + + +} Brink-Boeker.

where _po ⁼

0. 172 fm - ’, a

⁼

0.55fmandro

^{= 1.12 fm.}

The

optical-model

parameters ^were taken from refe

rence

[10].

The values

of to, t, and t2

^{were taken as}

[6]

to

^{= 3 298}

MeV . fm3 ,

t 1 ^{= - 59}

MeV . fm5

and

The wave function

TNL(2

^v,

Rt)

^was

replaced by

^the

spherical

Hankel function

(- i)L hi1)(ikR)

^{at a radius}

where the

following equality

^holds

where

EB is the

separation

energy of the

pair

from the nucleus and M* is the reduced nucleon mass.

5. Discussion. ^- For the

ordinary

^zero range

approximation

^{the factor}

F,(,z-’-)

is real and does not

depend

^on

Rt.

^{Thus the zero} range two-nucleon

stripping

form factor is

proportional

^{to the wave}

function

TNL(Rt).

^The

Skyrme potential

results in a

complex

^factor

F:(Rt)

^whose

imaginary

part ^{is due to}

the _presence of the

V2

term in the

potential.

^This

imaginary

part ^{is small}

(about

^2.3

%

of the real part

at

Rt

^{= 4.5}

fm)

because the

V2 coefficient,

_t1., ^{is small}

compared

with the values of the other two coefficients to ^and t2. ^{The real} part ^of

Fs(R,)

reduces the

impor-

tance of the nuclear interior

by

^{about 54}

%

^{as shown in}

figure

^2. This is because the effective nucleon-nucleon interaction is small in the inner

region

and increases as

the

density p(R )

decreases. The

increasing

^behaviour

of

Fn (Rt) ^beyond

the nuclear surface

yields

^{a form}

factor which falls off less

rapidly

^{in the}

asymptotic region

^{than the zero} range form factor

does,

^{as is}

shown in

figure

^4. Iano and Pinkston

[3]

^{found that}

slight

differences in form factors in the

region

^from

4-7 fm have

important

^{effects on both the}

shapes

^and

the

magnitudes

of the differential cross sections. Thus

we expect ^that the small difference between the

Skyrme

^{and the zero} range form

factors,

^{shown in}

figure 4, yields significant

^{effects on} the two-nucleon

stripping

differential cross-section.

The Brink-Boeker

potential gives

^a form factor which differs from that of the

Skyrme

interaction in both the

asymptotic

^{and inner}

regions.

^It falls off less

rapidly beyond

the nuclear surface and

yields

^less

reduction in the

importance

of the inner

region compared

^{with the} form factor

resulting

^{from the}

Skyrme potential. Moreover,

^the

imaginary

part of

FnBB(R )

^is

large (about

⁷⁵

%

of the real part ^at R ⁼ 4.5

fm)

^and

peaks

^{at about 5 fm. This}

imaginary

part increases the contribution of the surface

region

to the cross-section.

The value of the two-nucleon

stripping

cross-section

depends

^on both the value and the

shape

of the form factor in the

region

^{from 4-7 fm. The} greater ^the extension of the form factor in the

asymptotic region,

the

larger

the differential cross-section

[3].

^{The inner}

region

^{has a} small effect on the cross-section because

nonlocality

effects and the

absorption parts

^{of the}

optical-model potentials strongly

reduce the contri- butions to the cross-section from this

region.

^{In the}

present ^{work it was} found that the form factors calculated with effective interactions extend farther in the

asymptotic region

^{than the zero} range form factor.

Ibarra

[11]

found that if the nuclear

overlap

^which

enters in two

particle

transfer DWBA

theory

^is

calculated

using

^{a realistic}

method,

^{the zero} range form factor will increase in the surface

region by

^{about 43}

%.

Thus if we use a realistic nuclear

overlap together

^with

an effective

interaction,

^{we will} get ^a form factor which differs from the zero range form factor in both the surface and the

asymptotic regions.

Iano and Pinkston

[3]

^{have used a Gaussian}

potential

to calculate the form factor

assuming [12]

where

They

found that the calculated cross-section is smaller than the

experimental

^data

by

^{a factor of 14.6.}

In the present ^{work we} found that the _average value of

is about 1.9 Moreover the _average value

of I F,,s(R,) I2

is about 4.8 times that of the Brink-Boeker

potential.

Thus the

Skyrme

interaction

predicts

^an absolute value for the two-nucleon

stripping

cross-section which is 0.63 times that found

experimentally.

This factor will increase if one introduces a realistic nuclear

overlap together

^{with the}

Skyrme

interaction.

References

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ISMAIL, M., ^J. Physique ³⁴ (1973) ^369.

[2] GLENDENNING, ^N. K., Phys. ^{Rev. B 137} (1965) ^102.

[3] IANO, P. J. and PINKSTON, ^W. T., ^Nucl. Phys. ^{A 237} (1975) ^189.

[4] DRISKO, ^{R. M. and} RIJBICKI, F., Phys. Rev. Lett. 16 (1966) ^275.

[5] BRINK, D. M. and BOEKER, E.,Nucl. Phys. ^{A 91} (1967) ^1.

[6] VAUTHERIN, ^{D. and} BRINK, ^D. M., Phys. ^{Rev. C 5} (1972) ^626.

[7] PASCUAL, R., Phys. ^{Lett. 19} (1965) ^221.

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