Ab Initio No Core Shell Model Study of Neutron Rich Nitrogen Isotopes

In the present paper, we have calculated the energy spectra for neutron rich $^{18-22}$N isotopes using no core shell model (NCSM). To calculate the energy spectrum we have used three different $NN$ potentials, inside non-local outside Yukawa (INOY), next-to-next-to-next-leading order (N3LO) from chiral effective field theory and charge-dependent Bonn 2000 (CDB2K). The INOY potential, which is a two body interaction but also have the effect of three body forces by short range and non local character present in it. The calculations have been done at $\hbar\Omega$=20 MeV, 14 MeV and 12 MeV using INOY, N3LO and CDB2K potentials, respectively. Apart from this, we have also performed naive shell model calculations with the YSOX interaction. The results with INOY interaction show good agreement with the experimental data in comparison to other three interactions. We have also shown the occupancy of different orbitals involved corresponding to the largest model space ($N_{max}$= 4) in the present calculations.

In the present paper, we have calculated the energy spectra for neutron rich 18−22 N isotopes using no core shell model (NCSM). To calculate the energy spectrum we have used three different N N potentials, inside non-local outside Yukawa (INOY), next-to-next-to-next-leading order (N3LO) from chiral effective field theory and charge-dependent Bonn 2000 (CDB2K). The INOY potential, which is a two body interaction but also have the effect of three body forces by short range and non local character present in it. The calculations have been done at Ω=20 MeV, 14 MeV and 12 MeV using INOY, N3LO and CDB2K potentials, respectively. Apart from this, we have also performed naive shell model calculations with the YSOX interaction. The results with INOY interaction show good agreement with the experimental data in comparison to other three interactions. We have also shown the occupancy of different orbitals involved corresponding to the largest model space (Nmax= 4) in the present calculations.

I. INTRODUCTION
In nuclear physics, solving many body problem from first principle is computationally hard. But now a days, an advancement in computational facility made it possible. There are many ab initio methods are available to study nuclear properties. The no core shell model [1][2][3][4][5] is one of them. At present NCSM is well established technique used in nuclear physics to calculate nuclear properties. Here, we solve A-body Schrödinger equation for the particles treated as non relativistically and interacted by realistic two plus three body forces. With the NCSM, a detailed study has been done for even carbon isotopes where ground state energy, quadruple moment of 2 + 1 state, some B(E2) transitions and occupancies of 0 + 1 and 2 + 1 are calculated [6] using INOY [7,8] and CDB2K [9] interactions.
In the present work we will study the nitrogen isotopes and mainly focused on neutron rich side. The structure of neutron rich nuclei 18−22 N have been studied by in-beam γ-ray spectroscopy and spectra and other properties are compared with shell model calculations using WBT and WBTM interactions, where N = 14 closed sub shell is discussed [10]. The 22 N have halo structure in its ground state [11,12]. The properties of 23 N nucleus have been investigated in a three body model considering 21 N as a core and two valence neutrons and a small halo structure was suggested in the g.s. [13]. Although, for the drip line nucleus 23 N, the energy spectrum and the configuration for the structure have not been observed or predicted. So, considering as a three body system the two-neutron separation energy and other ground state properties have been calculated in Ref. [13]. Recently, the point proton radii of neutron rich 17−22 N isotopes have been measured * asaxena@ph.iitr.ac.in † Corresponding author: pcsrifph@iitr.ac.in from charge changing cross section in Ref. [14].
The study of neutron rich nuclei is very important to know the behaviour of nuclear forces because many properties changed as we go away from the line of stability, like disappearance of traditional magic numbers and appearance of new magic numbers. Previously, the shell model calculations for nitrogen isotopes using full psd shell with WBT interaction of Warburton and Brown are reported in Ref. [10]. More recently, Yuan and Suzuki etal , have done systematic study of B to O isotopes with a interaction YSOX which include (0-3) Ω excitations [15] in full psd model space. To the best of our knowledge for the first time we have done systematic NCSM calculations for nitrogen isotopes.
The present paper is organized as follows: In Sec. II, the theory and formalism of NCSM is given, In Secs. III and IV, we have discussed about effective interactions which are used in calculations and details of the calculations, respectively. The results and discussions part is in Sec. V and in the end we conclude the paper in Sec. VI.

II. NO CORE SHELL MODEL FORMALISM
The A-body Hamiltonian up to three body terms is given by: Where m is the nucleon mass. In the present work we have dealt with the two body part only. The V N N,ij is the NN interaction having nuclear and Coulomb part both.
We 1 2 AmΩ 2 R 2 , R = A i<j r/A.) to Eq. 1. As we use slater determinant basis, the Lawson Projection term is added to shift the spurious states to the Eq. 1. The Hamiltonian used in final calculations is given by: Where β is a parameter which is equal to 10.0 in the present calculations. The Eq. 2 is a Hamiltonian which we get after applying unitary transformation because we are not using soft interactions. So, we need a renormalization scheme to soften the interactions. Here, we use Okubo-Lee-Suzuki (OLS) scheme [16][17][18]. Now, we get an effective Hamiltonian which is in A-body space.
In the present paper, for the NCSM calculations, we have used the pAntoine [19,20] shell model code which is adapted to NCSM [21]. In the case of 22 N, for the largest model space N max = 4, the corresponding dimension is ∼ 6.4 × 10 7 . We have compared the NCSM results with the naive shell model calculations using YSOX interaction. For shell model calculations we have used KSHELL code [22].

III. EFFECTIVE N N INTERACTION
In the present work we have studied the neutron rich nitrogen isotopes with the three different N N interactions: INOY, CDB2K and N3LO [23][24][25]. When we use local N N interactions in many body systems, we also need higher body forces additionally for a better explanation. The magnitude of higher body forces decreases as we go from two body to higher body but still they are important to study some properties of nuclei for e.g. the drip-line in oxygen isotopes can be explained only with the inclusion of three body forces [26]. INOY potential, a non-local potential in coordinate space, is a mixture of local and non-local parts. The behaviour of INOY is local Yukawa tail at longer ranges (> 3 fm) and nonlocal at short range. The form of INOY N N interaction is given in Refs. [7,8]. This interaction reproduces the 3 H and 3 He binding energy accurately and results are in agreement with the experimental data without adding 3N force. The CDB2K interaction is also nonlocal interaction and charge dependent. The charge dependency is introduced due to pion mass splitting. This potential fits the p-p data below 350 MeV which was available in the year 2000. The N3LO interaction is from chiral effective field theory. Here, we use only N N part.

IV. DETAILS OF THE CALCULATIONS
In the present work we perform calculations for nitrogen isotopes. As we know NCSM calculations are variational, depend on HO frequency Ω and size of the model space N max . For seeing this dependence, we have calculated the g.s. energy with different N max and Ω, see Fig. 1. We are interested to see that region in which the dependence of g.s. energy on frequency is minimum (for largest model space). We select that frequency for our NCSM calculations. This procedure is called optimization of frequency. When we use this frequency, we get faster convergence rather than other values of frequencies. This is the benefit for doing optimization of frequency. So, we have done our calculations with frequency Ω= 20 MeV. For the other interactions we have chosen the frequency from the literature which is suitable in this mass region. We have chosen the frequency Ω=20 MeV for INOY and Ω=14 MeV for N3LO interaction [2]. In the case of CDB2K, for carbon isotopes we have taken Ω=12 MeV [6].       For 20 N, the results with the INOY ( Ω=22) interaction are better than other interactions. Although the g.s. is correctly reproduced by all the three interactions but the higher states are not in agreement with the N3LO and CDB2K interactions. The first 3 − state is close to the experimental data with INOY ( Ω=20) and 1 − is close to experimental data with INOY ( Ω=22).
In the case of 22 N, only INOY interaction can reproduce the correct g.s. 0 − and level ordering with both the frequencies. All the other interactions are not able to produce correct g.s. and level ordering of the energy states.
In the case of 19  For 21 N, the g.s. is correctly reproduced. Higher states are not yet been confirmed experimentally. All the interactions give first excited state as 3/2 − . Similarly, the second excited state seems to be 5/2 − . For higher states, we are not sure for spin prediction. So, from our NCSM calculations it is clear that INOY interaction which has the effect of three body forces is suitable to study the neutron rich nitrogen isotopes. The inclusion of 3N forces is important to reproduce correct spectra with CDB2K and N3LO interactions. In Fig. 4, we have shown the occupancy of first two states of nitrogen isotopes with the INOY interaction at Ω=20 MeV corresponding to N max = 4 model space size. Corresponding to N max = 4, we have taken 28 orbitals.
Here, we have shown the occupancy up to f p space because the occupancy of higher orbitals are very small to visualize. Although, the magnitude of occupancies of higher orbitals is very less but still they are important in the calculation. In Fig. 5, the calculated g.s. energy for 18−22 N isotopes follow the same trend as the experimental data. If we go to higher N max , the results will approach towards the experimental g.s. energies.

VI. CONCLUSIONS
In the present work, we have performed NCSM calculations with different interactions (INOY, N3LO and CDB2K) for neutron rich nitrogen isotopes. We have also compared our NCSM results with recently developed YSOX interaction for psd space from the Tokyo group. We have drawn following broad conclusions: • In 18 N, the INOY and YSOX interaction predict second excited state as 2 − .
• For 20 N, the results of INOY ( Ω=22) interaction are much better than YSOX interaction.
• For 22 N, the INOY results for ground and first excited states are better than YSOX interaction. The N3LO and CDB2K interactions are unable to predict correct ground state.
• For 19 N, the NCSM results with N3LO are much better.
• In the case of 21 N, the INOY results ( Ω=22) are near to the experimental data.
• These ab initio results are very important to confirm several tentative experimental levels for 18−22 N isotopes.