ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 7, с. 906-916
= ЯДРА
AN INVESTIGATION OF a-NUCLEUS ELASTIC SCATTERING
© 2014 Awad A. Ibraheem1),2)*, M. A. Hassanain2),3), S. M. Mokawesh2)
Received June 5, 2013; in final form, December 23, 2013
The a elastic scattering from various targets over a wide range of energy (120—400 MeV) has been analyzed in the framework of the double-folding (DF) optical model potential. The real DF optical potentials are generated based upon the M3Y and JLM effective nucleon—nucleon interactions. Density-dependent versions, DDM3Y, DDJLM, are also considered. The imaginary part of the optical potential is expressed either in a phenomenological Woods—Saxon form or microscopically using a normalization of the real DF potential. The angular distributions of the elastic scattering differential cross sections using the derived microscopic real potentials revealed successful description of nine sets of data all over the measured angular range. The extracted reaction cross sections have been also investigated.
DOI: 10.7868/S0044002714060087
1. INTRODUCTION
Much information can be learned about the structure of nuclei from the scattering of a particles by nuclei. Therefore, several analyses of differential cross sections for the elastic scattering of a particles by nuclei have been performed using both phenomeno-logical Woods—Saxon (WS) and microscopic potentials [1 — 18]. Although phenomenological approaches are consistent to explain the experimental results in some cases, however, considering a realistic approach such as the double-folding (DF) model is important to understand the details of reaction mechanism.
For the scattering of (a) particle from nuclei at energies above 100 MeV, the Fraunhofer diffraction in the angular distribution is usually observed followed by a smooth shoulder-like rainbow maximum which falls off exponentially at large angles. This observed refractive (or rainbow) pattern in the elastic scattering cross section is due to the weak-absorbing a—nucleus optical potential, which permits a deeper penetration of the a particle into the interior of the target nucleus. The study of refractive a—nucleus scattering, measured accurately up to large scattering angles, can be considered as an alternative method to determine the nuclear potential accurately at small separation distance [11]. Therefore, in the last decade, the elastic scattering cross sections of a scattering from 12C,16O, 28Si, 58Ni, and 124Sn nuclei, have been experimentally investigated [19—23].
!)Physics Department, Al-Azhar University, Assuit, Egypt.
2)Physics Department, King Khalid University, Abha, Saudia Arabia.
3)Sciences Department, New-Valley Faculty of Science, As-suit University, Egypt.
E-mail: awad_ah_eb@hotmail.com
Actually, most of the theoretical analyses of a-particle elastic scattering from several targets have been conducted by using the phenomenological WS potentials. The purpose of this paper is to investigate the validity of the microscopic DF potential model to explain the elastic scattering of the a particles from both light- and heavy-ion targets within the framework of the optical model. Therefore, the elastic scattering data over a wide range of energies (120— 400 MeV) are analyzed by performing a comparative study of the Michigan three-Yukawa terms M3Y (DDM3Y) and JLM (DDJLM) effective nucleon-nucleon (NN) interaction developed by Jeukenne, Lejeune, and Mahaux [9] within the framework of the DF model [9, 10, 24-29].
The basic inputs to calculate the a-nucleus potentials using the DF model are the nuclear densities of both projectile and target nuclei as well as an appropriate effective NN interaction. This model generates the first-order term of the microscopic optical potential that is derived from Feshbach's theory of nuclear reactions [11, 12]. This first-order term is the dominant part of the nucleon optical potential. Accordingly, this approach describes successfully the observed nucleon-nucleus elastic scattering data for many targets.
The most popular choices for the effective NN interaction used in the folding model, are the M3Y [2431] and JLM interactions [9]. The M3Y effective interaction was designed to reproduce the G-matrix elements of the Reid soft-core interaction [24-31], while JLM effective interaction was developed by using the Brueckner-Hartree-Fock (BHF) approximation from the Reid soft-core NN potential to be used in nucleon-nucleus calculations through
the single-folding (SF) approach [9]. It was parameterized through the energy and target-density dependence. This parameterization consists of the isoscalar, isovector and Coulomb components of the complex nucleon-nucleus optical model potential for energies from 0 up to 160 MeV. The JLM effective NN interactions has the advantage to independently generate both real and imaginary components of the DF potential.
The core of the present study is to investigate the recently measured 4He + 12C (at 240 and 386 MeV), 4He + 16O (at 240 and 400 MeV), 4He + 28Si (at 120 and 240 MeV), 4He + 58Ni (at 240 and 386 MeV), and 4He + 124Sn (at 386 MeV) [19-23]. We aim to study the effect of introducing different effective NN interactions through the DF procedure in describing the angular distributions of elastic scattering differential cross section. Additionally, it may be useful to compare the obtained results by those extracted using the density dependent versions of the considered M3Y and JLM interactions; namely, DDM3Y and DDJLM interactions, respectively. The present paper is organized as follows: in the next section, the theoretical formalism is described. The results and the discussion are presented in Section 3. Section 4 is devoted to summarizing the conclusions extracted from the present study.
2. THEORETICAL FORMALISM 2.1. The Folding Model
During the last three decades, the DF model has been extensively used to generate microscopically the real parts of both a—nucleus and heavy-ion (HI) optical potentials, where an effective NN interaction is folded with the mass distributions of both projectile and target nuclei. Thus, the DF potential can be expressed by the double-convolution integral
U(R) = Nr J p(ri)p(r2)v(s)dridr2, (1) s = |R - ri + r2\,
where Nr is the free normalization factor, p(ri) and p(r2) denote the nuclear matter density distributions of projectile and target nuclei, u(s) is the effective NN interaction, respectively. Using Fourier transforms [24], the DF integral equation (1) can be easily calculated.
Generally, the total effective a—nucleus interaction may be expressed as the summation of the attractive optical potential plus the repulsive Coulomb potential as
V(R) = -U(R) - iW(R) + VCoul(R), (2)
where U(R) and W(R) are the central real and imaginary nuclear parts of the optical potential, respectively. The repulsive Coulomb potential, VCoul(R), is considered due to the charge ZPe interacting with the charge ZTe distributed uniformly over a sphere of
the Coulomb radius Rcoui, where Rcoui = +
+ Ay3 ) [fm], where AP and AT are the projectile and target mass numbers, respectively. VCoul (R) is given by
VCoul (R) =
(3)
' ZpZTe2 2-Rcoul ZPZTe2 R
3 -
R2
W
Coul -
for R < Rcoul,
for R > Rcoul-
The imaginary part of the nuclear potential, W(R), is taken in the phenomenological WS form as
Wo
W (R) =
l+exp(^) Ri = n (a1/3 + at/3) ,
(4)
where W0, r/, and a/ are the depth, radius, and diffuseness parameters, respectively.
In the present work, the M3Y interaction has been used as an effective interaction in the folding procedure (see below Eq. (5)). The explicit radial strengths of the isoscalar components of the M3Y interaction based on the G matrix of the Reid NN potential are given in the following form
,M3Y
— 2134
(s) =
exp(—2.5s) 2ls
7999
exp(—4s) ~4i
(5)
— 276(1 — 0-005E/Ap )S(s),
where E is the projectile incident energy in laboratory system. The third term of Eq. (5) means that the knock-on exchange potential is treated approximately by adding zero-range pseudo-potential. Due to the attractive character of the M3Y forces, Eq. (5), the saturation condition for cold nuclear matter is not fulfilled and the nuclear matter is collapsing. A realistic description of nuclear matter properties can be obtained if an appropriate density dependence is introduced into the original M3Y interaction [11]. An energy and density-dependent effective NN interaction, named DDM3Y, has the following form [29]
.DDM3Y
(s,p,E) = f (p,E )vM3Y(s)-
(6)
The functional form of this density and energy-dependent factor, f (p, E) is chosen as [29]
f (p, E) = C(E)(l + a(E)) exp(-/3(E)p). (7)
flfl
It is usual to assume that local density is simply the sum of the two individual densities at that point: p(r) = pi(r\) + p2(r2). Fora nucleon at r\ in the projectile interaction with a nucleon at r2 in the target. This is called the frozen density approximation. This assumption ignores any readjustments due to their mutual interaction or to the Pauli Principle, and is really justified only for peripheral collisions where the density overlap is not large.
The energy-dependent parameters C(E), a(E), and ¡3(E) over the energy range from E = 0 to 150 MeV are taken from [29].
On the other hand, a very popular microscopic optical model description for obtaining the real and imaginary parts of the nucleon—nucleus optical potential is the JLM model [9].
The JLM effective NN interaction was parameterized to consist of the isoscalar, isovector, and Coulomb components of the complex nucleon— nucleus optical model potential for energies from 0 up to 160 MeV. The complex JLM effective NN interaction can, then, be factorized as
Uk (p2 ,E) =
(8)
E агзp2-1 Ej-1,
=
( 3
I
i,j=l
k = 1, for the real part
-1 4
C — 1 4
k = 2, for the imaginary part.
A projectile density-dependent factor fi(pi,E) is introduced [25—27]. This projectile density factors has the following functional form
(l - RE)p2/3, i = 1 for
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