Static Properties of Neutron-Rich and Proton-Rich Isotopes

Introduction

Investigating neutron-rich or proton-rich isotopes are one of the interesting subjects in nuclear science. These isotopes have some single nucleon out of the closed core in shell structure [1]. Determining the energy levels, charge radius and other static properties of nuclei, is one of the useful components to cognition the nuclear structure [2]. Studying the nucleon-nucleon interaction is given many important properties of multi-nucleon systems [3]. The multi- nucleon systems can be investigated in cluster structure [1]. Recently, the cluster model is applied to studying several different systems [3]. These theories have been modeled by mathematic equations like Schrodinger equation in nonrelativistic approach and Dirac equation for spin-1/2 particle and Klein-Gordon equation for spin-1 particles in relativistic approach [4]. Many nucleon-nucleon potentials 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.

There are two stable isotopes of carbon: 12C and 13C. 13C is used for instance in organic chemistry research, studies into molecular structures, metabolism, food labeling, air pollution and climate change. It is also used in breath tests to determine the presence of the helicobacter pylori bacteria which causes stomach ulcer. 13C can also be used for the production of the radioisotope 13N which is a PET isotope [12, 13].

Natural nitrogen consists of two stable isotopes, 14N, and 15N. Fourteen radioactive isotopes have also been found so far. The 13N decays with a half-life of ten minutes to 13C, emitting a positron [14]. Naturally occurring oxygen is composed of three stable isotopes, 16O, 17O, and 18O with 16O being the most abundant. 17O is a low-abundant, natural, stable isotope of oxygen [15]. Fluorine has 18 known isotopes, with atomic masses ranging from 14F to 31F. 19F is the only stable isotope of fluorine [16].

Sodium is the sixth most common element on Earth [17]. It is used as a heat exchanger in some nuclear reactors, and as a reagent in the chemicals industry. 23Na is the only stable isotopes of sodium [18]. The Earth’s crust contains 2.6% sodium, making it the fifth most abundant metal, behind aluminum, iron, calcium, and magnesium [19]. 23Na is created in the carbon-burning process in stars by fusing two carbon atoms together [20].

Neon is a chemical element and the second-lightest noble gas, after helium and very common element in the universe and solar system, but it is rare on earth [17, 21]. Stable isotopes of neon are produced in stars. Neon has three stable isotopes, 20Ne, 21Ne and 22Ne [21].

(a) (b) (c) (d) (e) (f) (g) (h) The 9Be, 9B, 13C, 13N, 17O, 17F, 21Ne, and 21Na isotopes can be modeled as a doubly magic (N=Z) with one additional nucleon out of core [22]. So, these isotopes can be investigated such as single particle in shell model. The experimental energy levels of these isotopes are shown in Figure 1 [23].

5/2- ------

1/2- ------

1/2- ------

1/2+ ------

7/2+ ------

7/2+ ------

1/2+ ------

5/2+ ------

5/2+ ------

3/2+ ------

5/2+ ------

3/2+ ------ 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 charge radius of 9Be, 9B, 13C, 13N, 17O, 17F, 21Ne, and 21Na isotopes by solving the Dirac equation using Eckert potential plus coulomb potential for interaction between single nucleon and core cluster.

The Ground State and Excited State Energy of Isotopes

The Dirac equation is one of the most significant equations in physics [24]. There is the exact solution of this equation just only for a few simple interactions. 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:

2 2 2 ( 1) 1 2 2 1 [ ( )( 8 )]F ( ) 0 0 , , , 2 2 2 2 2 2 (1 )

r V d k k e Mc E Mc E V r n k n k n k r r r r r dr r c e α $$ 2 - \frac {k (k + 1)}{r ^ {2}} - \frac {1}{\hbar^ {2} c ^ {2}} \left(M c ^ {2} + E _ {n _ {r}, k}\right) \left(M c ^ {2} - E _ {n _ {r}, k} + 8 V _ {0} \frac {e ^ {- 2 \alpha r}}{\left(1 - e ^ {- 2 \alpha r}\right) ^ {2}} + \frac {2 V _ {1}}{r}\right) ] F _ {n _ {r}, k} (r) = 0 $$ α (1) In equation (1), V0 and V1 are the actual parameter describing the potential well depth and the parameter α representing the potential range.

By suitable approximation and s = e-2𝛼x equation 1 can be written as equation 2 $$ \frac {d ^ {2} F}{d s ^ {2}} + \frac {1 - s}{s (1 - s)} \frac {d F}{d s} + \frac {1}{s ^ {2} (1 - s) ^ {2}} \left(- \xi_ {1} s ^ {2} + \xi_ {2} s - \xi_ {3}\right) F = 0 $$ (2) Where iξ are defined like bellow:

Click to enlarge

(3) By applying parametric Nikiforov-Uvarov method [22, 25, 26, 27, 28] the energy Eigen-value formula can be written as $$ (n + \frac {1}{2}) ^ {2} + \frac {1}{4} + (2 n + 1) \left(\sqrt {\xi_ {1} - \xi_ {2} + \xi_ {3} + \frac {1}{4}} + \sqrt {\xi_ {3}}\right) - \xi_ {2} + 2 \xi_ {3} + 2 \sqrt {\xi_ {1} - \xi_ {2} + \xi_ {3} + \frac {1}{4}} \sqrt {\xi_ {3}} = 0 $$ (4) The ground state and excited state energy of 9Be, 9B, 13C, 13N, 17O, 17F, 21Ne, and 21Na isotopes are compared with experimental results in Table 1.

In spin symmetry condition, the upper wave function is achieved in to the form $$ (e ^ {- 2 \alpha \mathrm {r}}) ^ {\sqrt {\xi_ {3}}} \left(1 - e ^ {- 2 \alpha \mathrm {r}}\right) ^ {\sqrt {\xi_ {1} - \xi_ {2} + \xi_ {3} + \frac {1}{4}} + \frac {1}{2}} P _ {n} ^ {\left(2 \sqrt {\xi_ {3}}, 2 \sqrt {\xi_ {1} - \xi_ {2} + \xi_ {3} + \frac {1}{4}}\right)} \left(1 - 2 e ^ {- 2 \alpha \mathrm {r}}\right) $$ $$ F _ {n, k} (r) = N \left(e ^ {- 2 \alpha \mathrm {r}}\right) ^ {\sqrt {\xi_ {3}}} \left(1 - e ^ {- 2 \alpha \mathrm {r}}\right) ^ {\sqrt {\xi_ {1} - \xi_ {2} + \xi_ {3} + \frac {1}{4}} + \frac {1}{2}} P _ {n} ^ {\left(2 \sqrt {\xi_ {3}}, 2 \sqrt {\xi_ {1} - \xi_ {2} + \xi_ {3} + \frac {1}{4}}\right)} \left(1 - 2 e ^ {- 2 \alpha \mathrm {r}}\right) $$ (5) Where N is the normalization constant [30], the lower component of the Dirac spinor can be calculated by equation (6) 2 2 $$ G _ {n, k} (r) = \frac {h ^ {2} c ^ {2}}{E + M c ^ {2}} \left(\frac {d}{d r} + \frac {k}{r}\right) F _ {n, k} (r) \tag {6} $$ And Wave function for Dirac equation can be calculated from equation (7) as $$ \frac {N}{r} \left[ \begin{array}{c} Y _ {\mathrm {j , m}} ^ {1} (\theta , \varphi) \\ \frac {i}{E + M c ^ {2}} \left(\frac {d}{d r} + \frac {k}{r}\right) Y _ {\mathrm {j , m}} ^ {- 1} (\theta , \varphi) \end{array} \right] \left(e ^ {- 2 \alpha \mathrm {r}}\right) ^ {\sqrt {\xi_ {3}}} \left(1 - e ^ {- 2 \alpha \mathrm {r}}\right) ^ {\sqrt {\xi_ {1} - \xi_ {2} + \xi_ {3} + \frac {1}{4}} + \frac {1}{2}} $$ l 1 Y ( , ) 1 2 3 j,m 2 r 2 r 3 2 ) l r Y ( , ) j,m 1 - + + N 4 i d k n,k ( + ) 2 dr r E+Mc $$ k (r, \theta , \varphi) = \frac {N}{\mathrm {r}} \left| \begin{array}{c c} Y _ {\mathrm {j}, \mathrm {m}} ^ {\mathrm {(0 , \varphi)}} & \\ \frac {i}{\left(\frac {d}{d} + k\right)} \tilde {Y} _ {\mathrm {(0 , \varphi)}} ^ {- 1} & \end{array} \right| \left(^ {- 2 \alpha \mathrm {r}}\right) ^ {\sqrt {\xi_ {3}}} \left(1 - e ^ {- 2 \alpha \mathrm {r}}\right) ^ {\sqrt {\theta}} $$ Ψ θ ϕ  1 2 2 - + +4 ( , ) P (1- 2e ) n

2 r ξ ξ ξ ξ −α

3 1 2 3 (7) The charge radius is determined from equation (8)

12 * 2 3 1 , , 2 2 * 3 , ,

    =     ∫ ∫

( ) ( ) r r r d r r r r d r ψ ψ (8)

n k n k r r

( ) ( ) ψ ψ n k n k r r

The charge radius of isotopes is compared by experimental data in Table 2.

Conclusion

In this paper, we have considered 9Be, 9B, 13C, 13N, 17O, 17F, 21Ne, and 21Na isotopes. Since these isotopes have one additional nucleon out of core, it can be investigated as single particle model in relativistic shell model. Therefore, we solved the spin symmetry Dirac equation with applying PNU method. By choosing suitable potential for N-cluster interaction, the ground state and excited state energy of isotopes are obtained. These results brought in table1 and compared with experimental data.

The charge radius obtained for ground state 9Be, 9B, 13C, 13N, 17O, 17F, 21Ne, and 21Na isotopes. As seen in table 2 our result has good agreement with experimental data.