The effect of nuclear compressibility on the shell-model potential

HAL Id: jpa-00207129

https://hal.archives-ouvertes.fr/jpa-00207129

Submitted on 1 Jan 1971

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.

The effect of nuclear compressibility on the shell-model potential

M. Ismail

To cite this version:

M. Ismail. The effect of nuclear compressibility on the shell-model potential. Journal de Physique,

1971, 32 (10), pp.729-731. �10.1051/jphys:019710032010072900�. �jpa-00207129�

729

THE EFFECT OF NUCLEAR COMPRESSIBILITY ON THE SHELL-MODEL

POTENTIAL

M. ISMAIL

(*)

Service de

Physique Théorique,

^Centre d’Etudes Nucléaires de

Saclay,

B. P. n°

2, 91,

Gif-sur-Yvette

(Reçu

^{le 12 mars}

1971)

Résumé. 2014 L’étude de la densité des

énergies

^de

particules

^{et du}

potentiel

^{du 40Ca est faite}

avec une interaction et des coefficients de

compressibilité

^{obtenus à}

partir

de courbes de saturation de la matière nucléaire.

Abstract. ²⁰¹⁴ A model interaction is used which is determined

by

^a

given

^{nuclear matter saturation}

curve. This interaction is used in a Hartree-Fock calculation of 40Ca to

study

^the

density,

^the

single particle energies

and the shell-model

potential

for various

compressibility

coefficients derived from the nuclear matter saturation curves.

LE JOURNAL DE PHYSIQUE TOME 32, ^OCTOBRE 1971,

Classification

Physics

Abstracts : 12.10

1. Introduction. ^- One of the recent

developments

in the last few years has been the successful calculation of the effective interaction between nucleons in the nucleus in terms of the interactions between nucleons in free space. It ^{turns out} that the effective interaction is

density dependent [1].

Most of the nuclear models and hence the nuclear structure calculations need to be revised so as to include the

dynamic

effects of the

density dependence

of the effective interaction.

The use of the

density dependent

^{effective interac-}

tions was shown to

yield improved agreement

^with

experimental binding energies, single particle energies

and elastic electron

scattering

^{cross sections}

[2].

^The

rearrangement

^terms

arising

^{from the}

density depen-

dent interactions have

improved

^the shell-model

potential

^{and the}

corresponding charge

densities. This effect can be also understood in terms of the increase in the

compressibility

coefficient obtained with these forces.

In the

present work,

^{we use a}

simple

^model

density dependent

effective interaction to

study

^{the effect of}

compressibility

^{on the}

charge distribution,

the shell- model

potential

^{and the}

single particle energies

^{for the}

40Ca

nucleus. For this purpose the nuclear ^matter

binding

energy per

particle

^{is assumed to be the follow-}

ing

^{function of the}

density

^{p :}

For a

given

^{value of a, the}

parameters a

^{and C are} chosen-so as to

give

^the

binding

energy

(1)

^a

stationary

(*) ^On leave of absence from the Faculty ^of Science, ^Cairo University, Egypt.

and minimum value

(E/A)o

^{= -} 16 MeV for the

density

po ⁼ 0.18

fm-3.

The

parameter

^{a is then}

changed

^{to obtain}

binding

energy ^curves with different

compressibility

coefficients defined as :

The model interaction is taken to be zero

ranged

^and

its

density dependence

^{is chosen so as to}

yield

^the

binding

energy function

(1)

^{in nuclear matter. It is}

known that the finite _range of the effective interaction attenuates the shell-model fluctuations of the

density,

so that the

present

calculation is

only designed

^to

measure the effect of the

compressibility

^{on the}

charge

distribution.

II. The model interaction. ^- To show how the

compressibility

^{affects the}

charge densities,

^the

single particle energies

^{and the shell model}

potential,

^{we use}

the

simple

^model

density dependent

interaction :

where r ⁼ rl - r2 and R =

rl + r2

^are the relative and centre of mass

position

^vectors

respectively.

^The

total _energy of the nucleus or of nuclear matter may be written as :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019710032010072900

730

where

p2 j2 m

is the kinetic energy

operator

^{and where} the ^sums extend over the A

occupied

^orbits. For nuclear matter, ^the

density

^{has the constant value}

where

kf

^{is the fermi momentum.}

Substituting

the model interaction

(2)

^{into the} equa- tion

(3)

^{one obtains the}

following expression relating

the

density dependence f (p)

^{to the}

binding

energy per

particle

^{in nuclear matter :}

Substituting for E/A by

^its

expression (1),

^{one obtains}

the function

f (p)

^{in terms} of the coefficient a which determines the

compressibility

^{of nuclear matter.}

III. The Hartree-Fock solution of

4°Ca. -

With the zero range model interaction

(2),

the energy

(3)

^of

the nucleus is a function

E(p)

of the local

density p(r) only

and the Hartree-Fock

potential

^is

given by :

The orbits

4> ;.(r)

^{are obtained}

by solving

the Hartree- Fock

equation :

and the

density p(r)

^is

given by :

Equation (7)

^{was solved}

numerically by

^iteration.

Starting

^{with a}

density p(O) (r)

obtained from harmonic oscillator ^wave

functions, equation (7)

^{was solved}

yielding e03BB(1)

^and

0(’)(r)

^{and hence}

p(1)(r).

^{The second}

iteration was started with

p( 1 )(r)

and the process ^was

repeated

^{until self}

consistency

^{was obtained.}

IV. Results. ^- In the

present work,

calculations

were made of the

4°Ca

nucleus

using

six values of oc

corresponding

^to

compressibility

coefficients in the range between 30 and 700. It is found

that,

^{with the} effective interactions

commonly used,

^the

compressi- bility

coefficient of the nuclear matter is about 20-30 ^so that we have studied the Hartree-Fock solution for very

high compressibility

coefficients.

Figures

¹ and 2 show the

density

^{and the} shell-model

potential (6) ^{of 4°}

^{Ca. For} low values of the

compressi-

bility

coefficient it is seen that the shell-model

potential

is smooth and the

density

^{has the}

typical

shell-model fluctuations. As the value of the

compressibility

coefficient

increases,

^the

density

^{curve tends to smooth}

out and the fluctuations decrease

gradually

^{and at the}

same time fluctuations of

opposite sign

appear in the shell-model

potential.

^For

large

values of the compres-

sibility coefhcient,

^a small increase in the

density

^{at a}

given point

^causes the effective interaction to become very

repulsive

^{at that}

point

and the shell-model

potential

becomes much more

repulsive

^{at that}

point.

The Hartree-Fock solution _converges therefore _very

slowly

^{when the}

compressibility

^is

high

because of the

sensitivity

of the effective interaction and of the shell- model

potential

^to small variations in the

density.

^In

this case the usual iterative

procedure

^no

longer

converges. To reduce the

density change

^{from one}

iteration to the next, ^we started the

(n

⁺

1)th

^iteration

with the

density.

with

FIG. 1. ^- The density p(r) and the shell model potential U(r) of 40Ca obtained for various compressibility coefficients.

731

FIG. 2. ^- Same as Figure 1, ^for higher compressibility ^coeffi- cients.

Figure

3 shows the shell-model

potentials

^{and the}

energies

of the Hartree-Fock orbits for 3 values of the

compressibility

coefficients. It is noted that the energy levels are lower for small

compressibility

coefficients.

FIG. 3. ^- The shell model potential ^{and the} energies ^{of the} single particle ^orbits plotted for three compressibility ^coeffi-

cients. The orbits in the potential ^{well are the 1} s, 1 p, 1 d and 2 s states.

They

^are

pushed

up ^as the

compressibility

^increases

from b ⁼ 18 to b ⁼ 192. For

larger

values of the

compressibility

coefficient the energy levels ^{are not} much affected.

Acknowledgments.

^- The author wishes to thank G.

Ripka

^for

suggesting

^the

problem

^{and for}

helpful

discussions. He also thanks C. Bloch for his kind

hospitality during

^his

stay

^{at the}

Département

^de

Physique Théorique

^of

Saclay.

References

[1]

RIPKA, ^{Fast neutrons and the}

study

^of Nuclear Struc- ture, N. Cindro Editor

(Gordon

^and

Breach,

New

York, 1970).

[2]

^NEGELE

(J. W.), Phys.

Rev., 1970, ^C

1,

^1260.

VAUTHERIN

(D.)

^{and BRINK}

(D. M.), Phys.

Lett., 1970, 32B, 149.

VAUTHERIN

(D.),

^VENERONI

^(M.)

^{and BRINK}

(D. M.),

Phys. Lett., 1970,

³³ B, 381.