Static Properties of Neutron-Rich and Proton-Rich Isotopes

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Zari Binesh* Research Article Department of Nuclear Physics, Faculty of Physics and Nuclear Engineering, Shahrood Volume 4 Issue 2 University of Technology, Shahrood, Iran Received Date: September 28, 2020 Published Date: October 28, 2020 *Corresponding author: Zari Binesh, Department of Nuclear Physics, Faculty of Physics and Nuclear Engineering, Shahrood University of Technology, Shahrood, Iran, Tel: 09102878009; Email: [email protected]

Abstract

In this paper, we have investigated some static properties of 9Be, 9B, 13C, 13N, 17O, 17F, 21Ne, and 21Na isotopes. The ground state and excited state energy of these isotopes have determined in the relativistic shell model and compared with experimental data. The calculated charge radius of them is in good agreement with experimental results.

Keywords: Charge radius; Energy levels

Introduction There are two stable isotopes of carbon: 12C and 13C. 13C is used for instance in organic chemistry research, studies Investigating neutron-rich or proton-rich isotopes are into molecular structures, metabolism, food labeling, air one of the interesting subjects in nuclear science. These pollution and climate change. It is also used in breath tests isotopes have some single nucleon out of the closed core in to determine the presence of the helicobacter pylori bacteria shell structure [1]. Determining the energy levels, charge which causes stomach ulcer. 13C can also be used for the radius and other static properties of nuclei, is one of the production of the radioisotope 13N which is a PET isotope useful components to cognition the nuclear structure [2]. [12,13]. Studying the nucleon-nucleon interaction is given many important properties of multi-nucleon systems [3]. The multi- Natural nitrogen consists of two stable isotopes, 14N, and nucleon systems can be investigated in cluster structure [1]. 15N. Fourteen radioactive isotopes have also been found so far. Recently, the cluster model is applied to studying several The 13N decays with a half-life of ten minutes to 13C, emitting different systems [3]. These theories have been modeled a positron [14]. Naturally occurring oxygen is composed of by mathematic equations like Schrodinger equation in three stable isotopes, 16O, 17O, and 18O with 16O being the most nonrelativistic approach and Dirac equation for spin-1/2 abundant. 17O is a low-abundant, natural, stable isotope of particle and Klein-Gordon equation for spin-1 particles in oxygen [15]. Fluorine has 18 known isotopes, with atomic relativistic approach [4]. Many nucleon-nucleon potentials masses ranging from 14F to 31F. 19F is the only stable isotope such as Woods-Saxon potential [5,6], Frost-Musulin potential [7], Pöschl–Teller potential [8] and Eckart potential [1,9] are being used by different researchers. of fluorineSodium [16]. is the sixth most common element on Earth [17]. It is used as a heat exchanger in some nuclear reactors, and 9Be is only stable isotopes of beryllium and strongly as a reagent in the chemicals industry. 23Na is the only stable deformed [10]. The cluster structure of 9Be is interesting for isotopes of sodium [18]. The Earth’s crust contains 2.6% studying the dynamics of reactions with weakly bound nuclei and also nuclear astrophysics [10]. 9B has a structure similar aluminum, iron, calcium, and magnesium [19]. 23Na is created to 9Be, with the odd-neutron replaced by the odd-proton [11]. sodium,in the carbon-burning making it the process fifth most in stars abundant by fusing metal, two behindcarbon atoms together [20].

Static Properties of Neutron-Rich and Proton-Rich Isotopes Phys Sci & Biophys J 2 Physical Science & Biophysics Journal

Neon is a chemical element and the second-lightest The 9Be, 9B, 13C, 13N, 17O, 17F, 21Ne, and 21Na isotopes can be noble gas, after helium and very common element in the modeled as a doubly magic (N=Z) with one additional nucleon universe and solar system, but it is rare on earth [17,21]. out of core [22]. So, these isotopes can be investigated such Stable isotopes of neon are produced in stars. Neon has three as single particle in shell model. The experimental energy stable isotopes, 20Ne, 21Ne and 22Ne [21]. levels of these isotopes are shown in Figure 1 [23].

4.5

4 + + - 5/2 ------5/2 ------5/2 ------5/2 ------3/2------3.5 3/2------

+ - - 3 1/2 ------1/2 ------1/2 ------1/2 ------5/2 + ------1/2 ------2.5 - + + 5/2 ------5/2 ------1/2 ------1/2 ------2 MeV 7/2+ ------+ 1/2+ ------7/2 ------1.5 ? ------

1 1/2+ ------+ 0.5 1/2 ------+ 5/2 ------5/2 + ------0 3/2------3/2 ------1/2 ------1/2 ------5/2 + ------5/2 + ------3/2 + ------3/2 + ------

-0.5 0 2 4 6 8 10 12 14 16 (a) (b) (c) (d) (e) (f) (g) (h) Figure 1: The experimental energy levels of a) 9Be, b) 9B, c) 13C, d) 13N, e) 17O, f) 17F, g) 21Ne, h) 21Na.

In this paper, we have calculated the energy levels and written as equation 2 charge radius of 9Be, 9B, 13C, 13N, 17O, 17F, 21Ne, and 21Na isotopes by solving the Dirac equation using Eckert potential plus (2) coulomb potential for interaction between single nucleon and core cluster.

Where The Ground State and Excited State Energy ξi of Isotopes are defined like bellow: equations in physics [24]. There is the exact solution of this Theequation Dirac just equation only for isa few one simple of the interactions. most significant So, the kinds of various methods have been used for the solution of this equation, exemplar, the super symmetric method [24], Nikiforov-Uvarov method [8,25] and so on. By submitting suitable potential in spin symmetry Dirac equation can be written as:

(3) 22− αr By applying parametric Nikiforov-Uvarov method [22,25-28] d kk(+ 1) 1 22 e 2V [ − −(Mc + E)( Mc −+ E8 V +1 )] F (r ) = 0 2 2 22 nk,, nk 0 −2αr 2 nk, the energy Eigen-value formula can be written as dr r c rr(1− e ) r r

(1)

In equation (1), V and V are the actual parameter 0 1 (4) representing the potential range. The ground state and excited state energy of 9Be, 9B, describing the potential well depth and the parameter α 13C, 13N, 17O, 17F, 21Ne, and 21Na isotopes are compared with -2 x By suitable approximation and s = e 훼 equation 1 can be experimental results in Table 1.

Isotopes fm-1) V V State E (Mev) E (Mev) [29] 훼( 0 1 our exp 1p -58.1659 -58.164 9Be 0.0289 0.069 -410.821 1f 3/2 -55.7804 -55.735

1p5/2 -56.3134 -56.3136 9B 0.0288 0.07 -414.427 1f 3/2 -53.9537 -53.9686

1p5/2 -97.1324 -97.1326 13C 0.0399 2.2256 -1595.6 1/2 2p -93.4231 -93.4486

1p3/2 -94.1469 -94.1467 13N 0.0398 2.269 -1610.18 1/2 2p -90.4895 -90.6447

1d3/2 -131.7623 -131.7624 17O 0.0435 2.8859 -1679.01 2p5/21/2 -128.6877 -128.707 1d -128.2352 -128.2353 17F 0.043 2.902 -1698.53 2p5/21/2 -125.155 -125.1313 1d -167.404 -167.4047 21Ne 0.0322 0.108 -482.271 1g3/2 -165.2153 -165.6597

1d7/2 -163.0454 -163.0463 21Na 0.032 0.128 -525.442 1g3/2 -161.1837 -161.3303

Table 1: The ground state and excited state energy of isotopes. 7/2

In spin symmetry condition, the upper wave function is The charge radius of isotopes is compared by achieved in to the form experimental data in Table 2.

1 1 1 ξ+−ξ++ ξ ξ ξ −ξ +ξ −α2rξ −α 2r 123 (,223 123 + ) −α2r ) 324 4 1 1 Fn,k (r) = N(e (1 - e ) Pn (1 - 2e ) Isotopes r2 2 () fm r2 2 () fm (5) 0ur exp

9 Where N is the normalization constant [30], the lower Be 2.5181 2.5190 [30] component of the Dirac spinor can be calculated by equation 9B 2.5378 2.81 [31] (6) 13 hc22 d k C 2.4443 2.4614 [30] G (r) = 2 ( + )Fn,k (r) (6) n,k E + Mc dr r 13N 2.4718 ------

17O 2.6588 2.6932 [30] And Wave function for Dirac equation can be calculated from equation (7) as 17F 2.7008 ------

l 1 1 21 Y (,)θϕ ξξ-++ ξ + N j,m −α2rξ −α 2r 12 3 Ne 2.9665 2.9695 Ψ θϕ ) 324 n,k (r, , ) = i dk  (e (1 - e ) r ( +)Yl (,)θϕ 2 j,m E+Mc dr r 21Na 3.0130 3.0136 [30] 1 (,2ξ 2 ξξ -++ ξ ) 3 12 34 −α2r Pn (1 - 2e ) Table 2: The charge radius for 9Be, 9B, 13C, 13N, 17O, 17F, 21Ne, (7) and 21Na isotopes for ground state energy. The charge radius is determined from equation (8) Conclusion 1 2 *2 3 (8) 9 9 13 13 17 1 ψψ()rr () rdr In this paper, we have considered Be, B, C, N, O, 2 2 ∫ nkrr,, nk r = 17F, 21Ne, and 21Na isotopes. Since these isotopes have one *3 ψψ()r () rdr additional nucleon out of core, it can be investigated as ∫ nkrr,, nk

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