fulltext

Calculated fusion cross-sections for neutron rich colliding nuclei

( 

)

RAJK. GUPTA, MANOJK. SHARMAand RAJEEVK. PURI

Centre of Advanced Study in Physics - Panjab University, Chandigarh 160014, India (ricevuto il 17 Luglio 1997; approvato il 15 Ottobre 1997)

Summary. — Fusion cross-sections are calculated by deriving the ion-ion interaction potential analytically from the Skyrme energy density formalism. For collisions be-tween neutron rich nuclei, the fusion cross-section is found to get enhanced with the increase of neutron number(s) in either (or both) of the reaction partners. Experimen-tal data are needed.

PACS 24.10 – Nuclear-reaction models and methods. PACS 21.60.Cs – Shell model.

PACS 25.70.Jj – Fusion and fusion-fission reactions. PACS 27.20 –6A19.

PACS 01.30.Cc – Conference proceedings.

1. – Introduction

It is now well known that fusion of nuclei is a low-density phenomenon. Two nuclei approach each other and fuse at/above the Coulomb barrier at a distance that is greater than the sum of radii of the two colliding partners. It is also clear that the fusion barrier cannot be explained by Coulomb barrier alone. The nucleon-nucleon interactions play very important and crucial role in deciding the fusion barriers and hence cross-sections. In other words, the barrier comes as an interplay between the Coulomb barrier and nu-clear interactions [1, 2]. A question that comes to mind is: can we describe this interplay and the contributions of nuclear and Coulomb interactions in terms of some simple, well-known quantities? If one neglects the spins of colliding nuclei, then the parameterization is quite simple. This has been shown to be acting like a proximity potential [1, 2]. But the job becomes quite tedious when one also includes the spins of colliding nuclei. We have addressed ourselves to this question in this paper. Another question of recent interest is the fusion of nuclei using neutron rich beams. Fusion using neutron rich beams is found to enhance the possibility of synthesis of new and superheavy elements [3]. In this paper, we study both the questions in terms of the well-known Skyrme energy density formalism. The nucleon-nucleon interaction is obtained analytically, as far as possible.

( 

)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and

Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.

2. – Skyrme energy density formalism

In the Skyrme energy density formalism (SEDF), the interaction potential

V

N (

R

)is

defined as

V

N

(

R

)=

E

(

R

),

E

(1)

;

(1)

where the two nuclei overlap at a distance

R

and are completely separated at infinity. The energy expectation value

E

is given by

E

= Z

H

(r)dr

;

(2)

with the Hamiltonian density

H

(r)of Vautherin and Brink [4] for an even-even spherical

nucleus,

H

(

;;

J)= 

h

2 2

m

+ 1 2

t

0[(1+ 1 2

x

0)



2 ,(

x

0+ 1 2 )(



2 n +



2 p )]+ 1 4 (

t

1+

t

2)



+ (3) + 1 8 (

t

2,

t

1)(



n



n +



p



p )+ 1 16 (

t

2,3

t

1)



r 2

_

+ + 1 32 (3

t

1+

t

2)(



n r 2

_

n +



p r 2

_

p )+ 1 4

t

3



n



p



, , 1 2

W

0(



r
J+



n r
J n +



p r
J p )

:

The terms involvingJ2 have been neglected in eq. (3). By using (1)-(3), the interaction

potential is

V

N (

R

)= Z f

H

(

;;

J),

H

1(



1

;

1

;

J1),

H

2(



2

;

2

;

J2)gdr

:

(4) Here,



i

;

i

;

J

i are the nucleon densities, kinetic energy densities and the spin densities

of the individual nuclei, respectively, and using sudden approximation, for a composite system



=



1 +



2,



=



1+



2 and J = J1 +J2. The parameters,

x

0,

t

0,

t

1,

t

2,

t

3 and

W

₀, appearing in eq. (3), have been fitted by several authors and in the present study, we use the force SIII whose parameters are

x

0 =0

:

45,

t

0 =,1128

:

75MeV
fm

3_,

t

1 =395

:

00MeV
fm 5_,

_t

₂ =,95

:

00MeV
fm 5_,

_t

₃ =14000

:

00MeV
fm 6_{, and}

_W

₀ =120.00 MeV
fm 5_.

Each term in the Hamiltonian (3) can be analysed separately. For our study, we divide eq. (4) into two parts, the spin-independent and the spin-dependent parts [1], as

V

N (

R

)=

V

P (

R

)+

V

J (

R

)

;

(5) with

V

P (

R

)= Z [

H

(

;

),

H

1(



1

;

1),

H

2(



2

;

2)]dr

;

(6) and

V

J (

R

)= Z [

H

(

;

J),

H

1(



1

;

J1),

H

2(



2

;

J2)]dr

:

(7)

Note that

V

P

(

R

) depends on the nucleon and kinetic energy densities, whereas

V

J

(

R

)

depends on the nucleon and spin densities.

Using the proximity force theorem treatment of Chattopadhyay and Gupta [5], we compute

V

P

(

R

)in (6) by using the sudden approximation of



=



1+



2and write it as

V

P (

R

)=2





R

(

s

)

;

(8) where

R

and

(

s

)are, respectively, the inverse of the root mean square radius of the

Gaussian curvature and the universal function which is independent of the geometery of the system.

R

is given by



R

=

C

1

C

2

C

1+

C

2

;

(9)

with the S¨ussman central radii

C

i =

R

i ,1

=R

iand the equivalent spherical radii

R

i = 1

:

28

A

1=3 i ,0

:

76+0

:

8

A

,1=3 i

. In eq. (8),

s

(=

R

,

C

1,

C

2)is the overlap distance between

colliding surfaces. This method of dividing the potential into a geometrical factor and the universal function is valid only if the nuclear surface is negligible compared to the central core, which means that this method is not good for lighter(

A <

16)reaction partners.

The spin-density potential from eqs. (3) and (7) reads as

V

J (

R

)=, 3 4

W

0 Z [



1(r
J2)+



2(r
J1)]dr

:

(10)

In terms of single particle orbitals that define a Slater determinant



i, the spin density

for

q

=

n

or

p

is given by J q (r)=(,

i

) X i;s;s 0



 i (r

;s;q

)[ r



i (r

;s

0

;q

)h

s

j



j

s

0 i]

:

(11)

Here the summation

i

runs over all the occupied single particle orbitals and

s

and

q

are, respectively, the spin and isospin indices. For a completely filled

j

-shell for

n

or

p

, eq. (11) reduces to J c q (r)= r 4

r

4 X  (2

j

 +1) 

j

 (

j

 +1),

l

 (

l

 +1), 3 4 

R

2  (r)

;

(12) with

R



(r)as the shell model radial wave functions. Note thatJ c q

(r) =0for completely

filled pairs of orbitals with

j

=

l

+12 and

j

=

l

,12.

For an even-even nucleus with valence particles (or holes) outside (inside) the closed

j

-shell, we divide the contribution toJ

q

(r)into two parts [1], one due to the core consisting

of closed shells and another due to the valence

n

vparticles (or holes), J q (r)=J c q (r)J nv q (r)

:

(13)

The(+)sign is for particles and(,)sign is for holes. The first term is the same as eq. (12)

and the second term reads as [1]

J nv q =

n

v r 4

r

4 

j

(

j

+1),

l

(

l

+1), 3 4 

R

2 l (r)

:

(14)

This model is applied to several hundreds of colliding partners involving light nuclei such as4He+4He with 0

s

+0

s

shell and heavy nuclei like150Hg+150Hg with 0

h

11

2 +

0

h

11 2

shell, and analytical formulas for

V

P

(

R

)and the spin-dependent part

V

J

(

R

)of the

interaction potential

V

N

(

R

)are obtained. Then, by adding Coulomb interaction potential

to

V

N

(

R

)we obtain the theoretical fusion barrier height

V

B and position

R

B. This allows

us to calculate the fusion cross-sections, using the sharp cut-off model,



fus=

R

2 B (1,

V

B

E

cm)

:

(15)

The quantity

E

_cmin eq.(15) is the centre of mass energy. This study is then extended to several neutron rich colliding nuclei.

3. – Calculations

The universal function(

s

)for some of the colliding nuclei from various shells is

plot-ted in fig. 1a). The general behaviour of(

s

)can be parametrized as [6]

(

s

)= ( ,0 exp  ,0

:

3325(

s

,

s

0)2 

;

for

s



s

0

;

,0+1

:

90(

s

,

s

0)2

;

for

s



s

0 (16)

with0 = 2

:

27and

s

0 = 0

:

2. Notice that the scattering of points for different reaction

partners is quite small from the solid line giving the behaviour of eq. (16). Similar plots have been made in refs. [1,2] for different force parameters. Then, the (spin-independent) ion-ion potential

V

P can be obtained by multiplying

(

s

)with the geometrical factor2





R

of colliding nuclei. Note that in parametrizing the universal function(

s

), no restriction

of shells is imposed on colliding pairs.

In contrast to the proximity part of the interaction potential, we cannot split the spin-dependent part of potential

V

Jinto any such geometrical factor and an universal function.

This is because the spin densityJ(

r

), appearing in

V

J (eq. (10)), depends on radial wave

functions

R

nl

(r)which vary from shell to shell [1]. Therefore, we have to restrict our

parametrization of spin-density potential

V

Jto colliding nuclei of the same shell.

Figure 1b) shows the spin-density part of ion-ion potential as a function of separation distance R for various reactions with nuclei coming from different shells (see different symbols for different reactions, marked ”exact”). We find that the spin-density part of the potential behaves opposite to the proximity potential. As we are trying to parameterize the potential within a shell, we assume that the0

f

and1

p

shells do not mix with each

other. In other words, the0

f

-shell, i.e. (0

f

7 2

,0

f

5 2

), is filled first and then the (1

p

3 2

,1

p

1 2

) shell is filled. This means that the0

f

5

2

shell is assumed to be lower in energy than the1

p

3 2

shell. Then, following our earlier work [1], the spin-density potential is characterized by the following four points (marked for one case in fig. 1b): i) the height of the repulsive maximum

V

JB, ii) the position

R

JB of

V

JB, iii) the position

R

J0, where the spin-density

potential changes its nature from repulsive to attractive and hence becomes zero, and iv) the limiting distance

R

JL, where

V

J

(

R

)goes to zero. In terms of these four points, we

Fig. 1. – a) Universal functionas a function of overlap distancesand b) spin-density potentialV J

as a function of relative separation distanceR.

by the following simple formula:

V

J (

R

)= 8 > > > < > > > :

V

JB exp " ln 

V

JL

V

JB 

R

,

R

JB

R

JL ,

R

JB  5 3 #

;

for

R



R

JB

V

JB ,

V

JB 

R

,

R

JB

R

J0 ,

R

JB 2

:

for

R



R

JB (17)

The parametrized expressions for four constants

V

JB,

R

JB,

R

J0and

R

JLare obtained in

the same way as reported in ref. [1]. In ref. [1], the variation of

V

JB was found to be very

smooth with a quantity called ”Particle-Strength”, defined as

P

s = X  (2

j

 +1) 4





j

 (

j

 +1),

l

 (

l

+1), 3 4  

n

v 4





j

(

j

+1),

l

(

l

+1), 3 4 

:

(18)

The spin-density potential barrier

V

JBcan be represented by

V

JB

=

CP

s

;

(19)

and the radii

R

JB,

R

J0and

R

JLvary as straight lines

R

i =

a

i +

b

i (

A

1

A

2)

; i

=

JB;J

0

;JL

(20)

with the constants(

C;a

i

;b

i

)for different shells obtained as listed in table I. These

con-stants reproduce the exact values of

V

JB,

R

JB,

R

J0and

R

JLquite nicely [6]. This is

Fig. 2. – Cross-sections as a function of center of mass energy for a)40

Ca+40

Ca and b)26

Mg+34

S. The experimental data in a) are from ref. [7-9] and in b) from ref. [11]. Our calculations marked ”Present” are comapred with those of other authors [2, 10].

solid line in fig. 1b). We find that our analytical expression (17) reproduces the exact spin-density potential (10) rather accurately.

An application of eqs. (17)-(20) is straightforward. Once we know the masses of col-liding nuclei, we can find

V

JB,

R

JB,

R

J0and

R

JLfrom eqs. (18)-(20) and table I. These

four parameters finally appear in eq. (17) to generate the spin-density potential

V

J (

R

).

Then, using eqs. (16) and (17), we can generate the ion-ion interaction potential

V

N (

R

)

analytically.

TABLEI. – Characteristic constants a, b and C for colliding nuclei belonging to different shells. Colliding nuclei V JB R JB R J0 R JL (shells) C aJB bJB aJ0 bJ0 aJ L bJL 0d+0d 1.5750 4.74 1.0710 ,3 3.68 8.8710 ,4 9.62 2.2610 ,3 0f+0f 1.1333 6.19 3.2210 ,4 4.77 2.8810 ,4 10.63 8.2910 ,4 1p+1p 1.3920 5.51 1.5110 ,4 4.71 1.2110 ,4 11.66 2.9310 ,4 0g+0g 0.8983 7.62 1.0410 ,4 5.96 8.2910 ,5 14.55 7.6210 ,5 1d+1d 0.0035 12.53 8.8510 ,5 12.19 7.4510 ,5 12.65 1.3710 ,4 0h+0h 0.7107 8.89 5.0010 ,5 6.84 4.4510 ,5 8.68 3.6910 ,4

As an illustration of our analytical procedure, fig. 2 shows the fusion cross-sections as a function of centre-of-mass energy for 40Ca+40Ca and 26Mg+34S reactions where experimental data are available. We find that formula (15) reproduces the results quite nicely. Since40Ca+40Ca is a spin-saturated(J=0)case, this provides a straightforward

Fig. 3. – Same as for fig. 2 but for neutron rich nuclei coming from different shells. No experimental data are available for these systems.

parameterization for spin-unsaturated(J 6= 0) colliding nuclei26Mg+34S. One can see

a good comparison between our analytical results and the experimental data. From the above comparisons, evidently our analytical formulation is able to reproduce the fusion cross-sections of both the spin-saturated and spin-unsaturated colliding nuclei.

In recent years, fusion of nuclei with radioactive nuclear beams has been reported [3]. It is found that with neutron rich beams (bombarded on an odd-even target) the height of fusion barrier is reduced and hence this leads to an enhancement in fusion cross-sections. In fig. 3 we show the same effect of addition of neutrons on the fusion cross-sections. We find that as neutrons are added to colliding(

N

=

Z

)nuclei, there is a gradual increase in

the fusion cross-sections.

4. – Conclusions

Summarizing, we have presented an analytical formulation of nucleon-nucleon inter-action potential. This analytical parameterization is shown to reproduce the experimental fusion cross-sections quite nicely. Furthermore, an addition of neutrons to colliding

N

=

Z

nuclei is shown to enhance the fusion cross-sections.

  

This work is supported in parts by the Department of Science and Technology, Min-istry of Science and Technology and the Department of Atomic Energy, Govt. of India.

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