Mirror Nuclei of 17O and 17F in Relativistic and Nonrelativistic Shell Model

Abstract

We have investigated energy levels mirror nuclei of the 17O and 17F in relativistic and nonrelativistic shell model. The nuclei 17O and 17F can be modeled as a doubly magic 17O = n + (N = Z = 8) and 17F = p + (N = Z = 8), with one additional nucleon (valence) in the ld5/2 level. Then we have selected the quadratic Hellmann potential for interaction between core and single nucleon. Using Parametric Nikiforov-Uvarov method, we have calculated the energy levels and wave function in Dirac and Schrodinger equations for relativistic and nonrelativistic, respectively. Finally, we have computed the binding and excited energy levels for mirror nuclei of 17O and 17F and compare with other works. Our results were in agreement with experimental values and hence this model could be applied for similar nuclei.

M. Mousavi 1 and M. R. Shojaei 1

Academic Editor:Ming Liu

Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran

Received 16 August 2016; Accepted 13 December 2016; 14 February 2017

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³ .

1. Introduction

Special interest resides in the study of masses and sizes for a given element along isotopic chains. Experimentally, their determination is increasingly difficult as one approaches the neutron drip-line; as of today, the heaviest element with available data on all existing bound isotopes is oxygen (Z = 8) [1]. Theoretically, the link between nuclear properties and internucleon forces can be explored for different values of N/Z and binding regimes, thus critically testing both our knowledge of nuclear forces and many-body theories [2]. The way shell closures and single-particle energies evolve as functions of the number of nucleons is presently one of the greatest challenges to our understanding of the basic features of nuclei. The properties of single particle energies and states with a strong quasi-particle content along an isotopic chain are moreover expected to be strongly influenced by the nuclear spin-orbit force [3].

The best evidence for single-particle behavior is found near magic (also called closed-shell) nuclei, where the number of protons or neutrons in a nucleus fills the last shell before a major or minor shell gap. For example, the nuclei ¹⁷ O and ¹⁷ F can be modeled as a doubly magic ¹⁷ O = n + (N = Z = 8) and ¹⁷ F = p + (N = Z = 8), with one additional (valence) nucleon in the ld_5/2 level. The ground state spin and parity of ¹⁷ O and ¹⁷ F are ^Jπ = 5/2⁺ , which corresponds to the spin and parity of the level where the valence nucleon resides [4]. The study of relativistic effects is always useful in some quantum mechanical systems. Therefore, The Dirac equation, which describes the motion of a spin-1/2 particle, has been used in solving many problems of nuclear and high-energy physics. The spin and the pseudo-spin symmetries of the Dirac Hamiltonian were discovered many years ago; however, these symmetries have recently been recognized empirically in nuclear and hadronic spectroscopes [5-7].

In this work we use relativistic and nonrelativistic shell model for calculation of the energy levels for ¹⁷ O and ¹⁷ F isotope. Since these isotopes have one nucleon out of the core, Dirac and Schrodinger equations are utilized to investigation them in relativistic and nonrelativistic shell model, respectively. These isotopes could be considered as a single particle. We apply the modified Hellmann potential [8, 9] between the core and a single particle because these potentials are important nuclear potentials for a description of the interaction between single nucleon and whole nuclei. Now that the N-N potential is selected; the next step is a solution of the Dirac and Schrodinger equation for the nuclei under investigation. We use the Parametric Nikiforov-Uvarov (PNU) method [10, 11] to solve them.

The scheme of paper is as follows: in Sections 2 and 3, energy spectrum in relativistic and nonrelativistic shell model is presented, respectively. Discussion and results are given in Section 4.

2. Energy Spectrum in Relativistic Shell Model

In the relativistic description, the Dirac equation of a single nucleon with the mass moving in an attractive scalar potential S(r) and a repulsive vector potential V(r) can be written as [17] [figure omitted; refer to PDF] where E is the relativistic energy, M is the mass of a single particle, and α and β are the 4×4 Dirac matrices.

The wave functions can be classified according to their angular momentum j and spin-orbit quantum number k as follows: [figure omitted; refer to PDF] where _{Fnr ,k} (r) and _{Gnr ,k} (r) are upper and lower components and ^Yjml (θ,[varphi]) and ^Yjml~ (θ,[varphi]) are the spherical harmonic functions. _nr is the radial quantum number and m is the projection of the angular momentum on the z-axis.

Under the condition of the spin symmetry, that is, Δ(r)=0 and Σ(r)=V(r)+S(r), the upper component Dirac equation could be written as [18] [figure omitted; refer to PDF] The quadratic Hellmann potential is defined as [19, 20] [figure omitted; refer to PDF] where the parameters a and b are real parameters, these are strength parameters, and the parameter α is related to the range of the potential.

Using the transformation _Fn,k (r)=r_Un,k (r), (3) brings into the form [figure omitted; refer to PDF] Equation (5) is exactly solvable only for the case of k=0,-1. In order to obtain the analytical solutions of (5), we employ the improved Pekeris approximation [21] that is valid for α<=1. The main characteristic of these solutions lies in the substitution of the centrifugal term by an approximation, so that one can obtain an equation, normally hypergeometric, which is solvable [18, 22]. [figure omitted; refer to PDF] Also this approximation in reverse order could be used.

We can write (5) by using improved Pekeris approximation as summarized below: [figure omitted; refer to PDF] where the parameters _χ2 , _χ1 , and _χ0 are considered as follows: [figure omitted; refer to PDF] Applying PNU method, we obtain the energy equation (with referring to [23, 24]) as [figure omitted; refer to PDF] With substituting (8) in (9) the energy equation is [figure omitted; refer to PDF] Let us find the corresponding wave functions. In referring to PNU method in [25, 26], we can obtain the upper wave function [figure omitted; refer to PDF] where N is the normalization constant; on the other hand, the lower component of the Dirac spinor can be calculated from (11) as [figure omitted; refer to PDF] And wave function for Dirac equation can be calculated from (2) as follows: [figure omitted; refer to PDF] where N is the normalization constant.

3. Energy Spectrum in Nonrelativistic Shell Model

The radial Schrodinger equation in spherical coordinates is given as [27, 28] [figure omitted; refer to PDF] By substituting quadratic Hellmann potential in (14) the radial Schrodinger equation is reduced as follows: [figure omitted; refer to PDF] Since the Schrodinger equation with the above potential has no analytical solution for l≠0 states, an approximation has to be made. Using improved Pekeris approximation [18, 22] that is presented in the previous section, (15) is as follows: [figure omitted; refer to PDF] where the parameters ^{χ2[variant prime]} , ^{χ1[variant prime]} , and ^{χ0[variant prime]} are considered as follows: [figure omitted; refer to PDF] Applying PNU method, we obtain the energy equation (with referring to [23, 24]) as follows: [figure omitted; refer to PDF] Substituting (17) in (18) the energy equation is [figure omitted; refer to PDF] And radial wave function for Schrodinger equation can be calculated with referring to PNU method in [23, 24] as follows: [figure omitted; refer to PDF] where ^{N[variant prime]} is the normalization constant.

4. Result and Discussion

We consider mirror nuclei ¹⁷ O and ¹⁷ F isotopes with a single nucleon on top of the ¹⁶ O and ¹⁶ F isotopes core. Since these isotopes have one nucleon out of the core, these isotopes could be considered as single particle model in relativistic and nonrelativistic shell model. Relativistic mean field (RMF) theory, as a covariant density functional theory, has been successfully applied to the study of nuclear structure properties [29]. In the relativistic mean field theory, repulsive and attractive effects at the same time have been combined, via vector and scalar potentials; also it involves the antiparticle solutions and spin-orbit interaction [30]. So we could use of Dirac equation for investigation them.

The ground state and first excited energies of mirror nuclei ¹⁷ O and ¹⁷ F isotopes are obtained in relativistic and nonrelativistic shell model by using (10) and (19), respectively. These results for relativistic and nonrelativistic shell model are compared with the experimental data and others work in Table 1.

Table 1: The ground state and the first excited energy of ¹⁷ F and ¹⁷ O isotopes in nonrelativistic and relativistic (with α=0.012 fm^-1 ).

Isotope	State	E Our (MeV)	E Other (MeV) [12]	E Exp. (MeV) [13]
Nonrelativistic	Relativistic
17 F	1d5/2	- 128.6460	- 128.5116	129.14	- 128.2196
2s1/2	- 128.2364	- 128.0045	--	- 127.7243
17 O	1d5/2	- 132.1423	- 131.9427	132.88	- 131.7624
2s1/2	- 131.3213	- 131.0455	--	- 130.8916

The difference between excited state energies and ground state energies of mirror nuclei ¹⁷ O and ¹⁷ F isotopes for relativistic and nonrelativistic shell model is compared with the experimental data and others work in Table 2.

Table 2: Comparison of the energies of excited states of ¹⁷ O and ¹⁷ F, relative to the ground state energies (the ^(5/2)1+ state of ¹⁷ O and ¹⁷ F): all entries are in MeV.

Isotope	Excited state	Others' work	Our work	Exp. [13]
N3 LO [14]	CD-Bonn [15]	V18 [16]	Nonrelativistic	Relativistic
17 F	( 1 / 2 ) 1 +	0.428	0.805	0.062	0.4096	0.5071	0.495
17 O	( 1 / 2 ) 1 +	- 0.025	0.311	- 0.390	0.8210	0.8972	0.870

The calculated energy levels have good agreement with experimental values. Therefore, the proposed model can well be used to investigate other similar isotopes and compare with experimental data.

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