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Analytical I-Dependent Folding Potential According to Alpha-Cluster Model -I.

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Annalen der Physik. 7. Folgo, Band 46, Heft 3, 1989, S. 207-224
VEB J. A. Barth, Leipzig
Analytical I-Dependent Folding Potential According
to Alpha-Cluster Model -I
B y M. Y. M. HASSAN,
M. N. H. COMSAN*)and I. M. A. TAGELDIN*)
Cairo University, Giza, Egypt
*) Nuclear Physics Department, Nuclear Research Center, Atomic Energy establishment^, Egypt
A b s t r a c t . The folding model is applied t o calculate the nucleon and alpha-nucleus interaction
for IzC and I60. The matter densities are obtained using the alpha-cluster model. AnaIytical t-dependent expressions for the folding potential as well as the deformation parameters are given. The effect
of applying different nucleon-nucleon forces is discussed.
Analytisches, l-abhangiges Faltenpotciitial eiitsprechend dern Alphaclustermodell 1
I n h a l t s u b e r s i c h t . Zur Berechnung der Nukleon- und Alpha-Nukleonwechselwirkung a m
und l 6 0 wird das Faltungsmodell angewandt. Aus dem Alphaclustermodell ergeben sich die
Xateriedichten. Analytische, I-abhlngige Ausdriicke des Faltungspotentials sowie des Deformationsparameters werden bostimmt. Die Auswirkung der Anwendung verschiedener Nukleon-Nukleonkrlfte
wird diskutiert.
1zC
I. Introduction
The study of optical potential is important not only for its intrinsic interest but also
for the extremely important role it plays in the description of nuclear reactions.
Greenless, Yyle and Tange [ L] developed a reformulation of the optical model in
which tlie real part of the optical potential was obtained using a folding model from
nuclear matter distribution and tlie effective nucleon-nucleon force.
Knyazkov and Hefter [a] have studied the nucleon and alpha-nucleus interactions
applying the folding model to light deformed target nuclei. The potential obtained
depends on the transferred momentrim through the assumption of an axially symmetric
distribution of the target.
The alpha-cluster model has proved to be successful to describe the nuclei 12C and
l 6 0 [3].
According to the alpha-cluster model, the matter distribution may deviate froin
axial symmetry. I n this work we apply the alpha-cluster model to calculate the matter
distribution of 12C and l60nuclei. These matter distributions are used to develop the
nucleon and alpha-nucleus folding potential. The effect of using different nucleonnucleon effective interactions is studied.
The nuclear densities of 12C and l60according to the alpha-cluster model are given
in See. 2. The nucleon-nucleus folding potential is given in Sec. 3. The deformation
parameters and the discussion of the nucleon-nucleus folding potential is presented in
Sec. 4. I n Sec. 5 the alpha-nucleus folding-potential is given. See. 6 contains the deformation parameters aid the discussion of the alpha-nucleus potential. Sec. 7 is a general
coiiclusion.
Ann. Physik Leipzig 46 (1989) 3
208
2. iluelear Density Distribution Becording to Alpha-Cluster Model
The nuclear density of the nucleus can be expressed as a sum of the alpha-cluster
densities eJ(r')
1
iv
I
N
where N is the number of alpha-particles inside the nucleus, the vectors r and rk represent the center of the nucleus and the kth alpha-cluster center respectively, and Rk is
the position vector of the center of the kti' cluster with respect to the nuclear center.
Both Q J r ' )and e ( r )will throughout be normalized to unity. We have considered that
12C-nucleusconsists of three alpha-clusters in an equilateral triangle configuration [4, 51
and l6O-nucleus consists of four alpha-clusters in a tetrahedral configuration [4, 51.
The density of the alpha-cluster was assunled to have a Gaussian distribution of the
following form [6]
where b is a parameter that characterizes the rate a t which the function falls off.
On the other hand, it is a size parameter associated with the alpha-particle. By
substituting equation (2) into equation ( I ) we obtain
where t , = 2rRk/b2and px:= c o s ( r . Rk/rRk).Making partial wave expansion of the exponent la1 e ' t @ h and using the well-linown addition theorem for Legendre function we
get
The values of 0, and pr are deduced from the above-mentioned relevant configurations.
The parameters Rkand b are taken from [7]. These parameters were chosen to give the
best fit to the r.m.5. radius and elastic electron scattering form factors (see Table 1).
Table 1 The values of the parameters Rk and 6
hTucleus
Rk
b
1
2
c
I .73
0
1.99
1.32
1.38
3. Nucleon-nucleus Folding Potential
The nucleon-nucleus folding potential can be calculated using the single folding
integral [a]
U n - N ( R ) = J e z l ( r Tv(
) IR
-
rT7
1) dr,,
(5)
M.T.&I.HASANct nl., Alpha-Clust~rXodel
209
where e T ( r T ,is) the density of the target nucleus and v( 1 R
nucleon force. The effective interaction has the forni
-
rT
1)
is the effective nucleon
The parameters up, up and d ( E ) of the effective nucleon-nucleon interaction are listed in
Table 2 [ 2 ] . The first set gives the direct part of an interaction used in cluster calculations. The interactions denoted by ( 2 ) and (3) are based on the “realistic” interactions
given in IS, 91. The Pauli principle was taken into account through the set (3) via the
pseudopotential approxiniation, nhile the sets (1)an(l ( 2 ) do not take care of I t . We use
for simplicity E / A = 1.
Table 2. The parameters U,, ug and d ( E ) of the three effective nucleon-nucleon forces. The value of
EIA is taken t o be equal t o one
E’l
E’,
E’,
-20.97
-553.18
-601.99
~
1781.4
2 256.4
1.47
0.80
0.80
-
0.5
0.5
-
-276
(1 - 0.05 E / A )
Substituting for e T ( r T )and o(r) from equations (4) and (6) into (5) we get
210
Snn. Physik Leipzig 46 (1989) 3
and
and
The above coefficients were drawn for different values of '2".
M. Y. If. HASSAX
et
nl.,
311
Alpha-Cluster Model
4. Deformation Parameters and Discussion of t,he Results for Xueleon-nucleus Potentbid
Both shape and potential deforniation parameters
the formula [lo]
and
Brmare calculated using
where
The ratio of
&J,$$
is introduced in Table 3.
Table 3. The ratio of the matter t o the potential deformation parameters for nucleon-nucleus interaction
I=2
Reaction
n
+ lZC
1.583
1.525
1.470
F,
F,
p3
n
+
160
1=3
1=3
1=4
1=4
3.140
2.660
2.250
-
Force
PI
-
Fz
-
F3
-
2.050
1.870
1.740
-
1.860
1.900
1.750
12
c
-
-
2.420
2.200
1.980
2.440
2.190
1.980
---
FORCE ( 1 1
FORCE ( 2 )
(1.0)
-.-
FORCE ( 3 1
6
1
R(frn1
-i
-60-
Fig. l a
/'
,/.
212
Ann. Physik Leipzig 46 (1989) 3
12
c
1.2
_---.-
Fig. 1b
FORCE ( 1 1
FORCE ( 2 )
FORCE (31
31. T.31.H a s s a ~et al., Alpha-Cluster Model
2 13
The forms of elastic form factors (1 = 0) ant1 inelastic form factors (1 = 2, 3 ancl 4)
are represented in Figs. 1 and 2.
I n the case of elastic form factors, the dependence on the nucleon-nucleon force is
strong. The differences between the Urn’s obtained by “B1”and “F;’ are rather small
in the physically important region with R > 2 fm. ,411 the ciirves nearly tend to zero
beyond R > 6 fni.
For inelastic form factors, we note a stronger influence of the nucleon-nucleon interact ion.
12
c
1.3
- --.-
Fig. l c
FORCE I 1 1
FORCE [ 2 1
FORCE ( 3 )
211
Ann. Physik Leipzig 46 (1989) 3
The ratio /?fm//?& nag calculated for the three sets of forces, we see from our results
that the differences between the matter and potential deformation parameters are sizeable and increase with increasing 1. We notice that this ratio for force F , is less than that
for F2 which, in turn is less than that of F,.
---.-
FJRCE ( 1 1
FORCE ( 2 1
FORCE ( 3 1
Fig. Id
Fig. 1. Nueleon-l*C folding potential for 2 = 0, 2 , 3 and 4 corresponding t o the three sets of forces
5. Alpha-Nucleus Folding Potential
To describe the interaction of alpha-particle with nuclei we use the double foltling
[a]
( R ) = JJ eArJ . @N(rAT) .
procedure given by the formula
v(/ +
r
R - RN 1) d r , d r N ,
(11)
where both e,(r,) and e H ( r x ) represent the density distribution which is the alphaparticle in this case and the target nucleus, respectively, R stands for the center-ofua-N
&I, Y. M. Hassm et, al., Alpha-Cluster Model
215
mass radius and V represents the density independent effective nucleon-nucleon force.
For the density of the projectile alpha-particle we use [ I l l
= Qo7 exp(--dA)
@,(Fa)
(12)
9
where eoa = 4 ( d , / ~ ) and
~ / ~ d, = 0.614 fm-2 which reproduces
fairly well.
Evaluating equation (ll),we get
( T ~ ) for
, ~ alpha-particle
~
(
Xexp( - x 2 P ) exp(--z~R:) I,, 1,2 2da d ( E )
b2g2
-T
I
Fig. 2a
10-
’)
FORCE ( 1
FORCE(2
FORCE( 3
Ann. Physik Leipzig 4G (1899) 3
316
16
0
Pig. 2b
I =3
M. Y. X. HASSAN
e t nl., Alpha-Cluster Model
-
I
10
217
\
-
5-
R(fm)
5t
.
/
'
Big. 7 c
Fig. 2. Nucleon-'60 folding potential for 1 = 0 , 3 and 4 corresponding t o the three sets of forces
218
Ann. Physik Leipzig 46 (1989) 3
and
UP
Jd, esp(-n2R2)
-7
2 l/Rh.l/R
fs y
exp(-h2B;)
p-l
6. Deformation Parameters and Discussion of the Results for Alpha-nucleus
Folding Potential
By using equation (lo), the ratio /?&,J@&,
in alpha-nucleus interaction was calculated
and presented in Table 4.
Both elastic and inelastic form factors are shown in Figs. 3 and 4.
M. Y.M. HASSANe t al., Alpha-Cluster Model
219
Table 4. The ratio of the matter t o the pot,ential deformation parameters for alpha-nucleus interaction
1='2
Reaction
Pn,o/s&
a
+ 'ZC
Fl
F,
E',
a
+
'00
1=3
1=3
1=4
1=4
Be3,alP:a
@,3/8&
B",,o/a:o
Be4,4/PZ*
5.17
4.63
4.40
-
Force
2.11
2.04
1.96
E',
-
3'2
-
F,
-
-
2.7.5
2.67
2.5'2
3.15
3.04
'2.86
-
---.-
Fig. 3a
4.12
3.93
3.66
FORCE ( 1 )
FORCE ( 2 )
FORCE ( 3 )
-
4.13
3.94
3.66
Ann. Physik Leipzig 46 (1989) 3
h
>
-FORCE
---
-_-
-10-
-20-
-30Fig. 3 b
Pig. 3 c
\ \
-1.1.
(1)
FORCE ( 2 )
FORCE ( 3 )
&I. Y.M. HASSAN
et al., Alpha-Cluster Model
“31
The elastic form factors have the same shape for the t v o nuclei under consideration,
the depth of the potential corresponding to tlie third set of the effective nucleonnucleon force is deeper than that of the first antl second sets of force at R = Ofitt,
i.e. U p s > UaIl > U I i l ZThe
.
values of the elastic form factors tend nearly to zero a t
R > 6.5 frri.
In the case of inelastic forin factors, lie note a similar influence of the effective nucleon-nucleon force as for nucleon-nucleus potential.
Our results shon that the difference between the niatter antl potential deformation
parameters are sizeable and increase with increasing orbital angular rnonient uni “I”.
The deformations of the alpha-particle potential prove to he smaller than that of the
nucleon potential. We notice the same behaviour of tlie effect of the three sets of nucleonnucleon force AS in the nucleon-nucleus potential on the ratio pf/py.
1% here
2
___
---
>
-.
a
I
-.-
FORCE ( 1 )
FORCE ( 2
FORCE ( 3 1
v
E
:
1
0
I
- 1
-2
Fig. 3d
Pig. 3. Alpha-**Cfolding potential for I
=
0, 2, 3 and 4 corresponding to the three sets of forces
Ann. Physik Leipzig 46 (1989) 3
16
0
I20
-
FORCE ( 1 )
-.-._
FORCE ( 3 1
--_
0
1
2
5
3
FORCE ( 2 )
6
7
I
R(fm 1
- 20-bO-
- 60-
- 80-
Fig. 4a
16
0
Fig. 4b
M. Y . X . Hassax et. al., Alpha-Cluster Model
>
I
u
5Y -2
3
-1
Fig. 4c
Fig. 4. Alpha-160 folding potential for I = 0 , 3 and 4 corresponding t o the three sets of forces
7. Conclusion
By using the alpha-particle model, an expansion of the density of nuclei lZCand lSO
in terms of the orbital angular momentum “1” was developed up to 1 = 4.
Using these densities and the effective nucleon-nucleon interaction as expressed in
terms of Gaussians, we developed the analytical 1-dependent folding potentials for
nucleon-nucleus and alpha-nucleus scattering. It has been shown that the form factors
r;TZ1;N(B)and Ut;N(R)are quite sensitive to the type of nucleon-nucleon interaction used.
Seemingly, the double folding procedure as applied to composite particles does lead
t o a slight reduction of this depenaence.
A difference between Uz;N(R)and U,4,N(R)is seen which is attributed to the averaging and smoothing effect of the second integration and the presence fo the density of
pa(.).
It is worth mentioning that this model has to be applied t o elastic and inelastic
scattering to justify its validity.
References
[l]
[2]
131
[4]
GREENLESS,
G. W.; PYLE,G. J.; TANG,Y. C.: Phys. Rev. 171 (1968) 1115.
KNYAZKOV,
0. M.; HEFTER,
E. F.: Z. Phys. 301 (1981) 277.
BAUHOFF,
W.; SCHULTHEIS,
H.; SCHULTHEIS,
R.: Phys. Rev. C 29 (1981) 1046.
INOPIN,
E.; TISCHEHENRA,
B. J.: JETP (Sov. Phys.) 11 (1960) 840.
224
Ann. Physik Leipzig 4G (1989) :1
[a] HARRINGTON,
D. R.: Phys. Rev. 147 (1966) 685.
[6] SAVITSKII,
G. A. ;AFANAS,
N. G., EV. ; GULLKIROV, I. E.; KHVASTUXOV,
V. 11.;KIIAMICH,
A. A. ;
SHERCHENKA,
N. G.: Sov. J. Nuel. Phys. 6 (19G8) 105.
[7] Noos, M. S. M.: M. Sc. Thesis, Cairo University 1979.
[ 8 ] SATCHLEIL,
G. R.; LOVE,W. G.: Phys. Lett. 65 B (1976)415.
[9] SATCHLER,
G. R.; LOVE,W. G . : Phys. Rep. 55C (1979) 183.
[lo] HNYAZKOV,
0. M.: Sov. J. Nucl. Phys. 33 (1981) 624.
[11] LE N~EILE,At.; TANG,y . c.; TaonIIwN, D. R.: Phys. Rev. c 1.2 (1976) 1 .
Bei der Redaktion eingegangen am 15. Mai 1987.
Anschr. d. Verf.: Dr. M. II'. M. HASSAN
Physics Department, Farulty of Science,
Cairo University, Giza, Egypt
Dr. M. N. H. Coim-m, Dr. I. &I. A. TAGELDIN
Kuclear Physics Department,, Nuclear Research
Center, Atomic Energy Establishment, Egypt
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