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

TL;DR

This study calculates energy spectra of neutron-rich nitrogen isotopes using the no core shell model with three different NN potentials, finding INOY potential aligns best with experimental data.

Contribution

It introduces a comprehensive NCSM analysis of nitrogen isotopes with multiple potentials, highlighting the effectiveness of INOY in matching experimental results.

Findings

INOY potential shows the best agreement with experimental data.

Shell model calculations reveal orbital occupancy patterns.

Different potentials yield varying energy spectra accuracy.

Abstract

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 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.

Tables

Table 1. Table 1: The g.s. energies (in MeV) for nitrogen isotopes using YSOX, INOY ( ℏ ​ Ω Planck-constant-over-2-pi Ω \hbar\Omega =20 MeV), N3LO ( ℏ ​ Ω Planck-constant-over-2-pi Ω \hbar\Omega =14 MeV), and CDB2K ( ℏ ​ Ω Planck-constant-over-2-pi Ω \hbar\Omega =12 MeV) interactions.

Nucleus	EXP	YSOX	INOY	N3LO	CDB2K
18N	-126.695	-127.344	-121.782	-112.036	-102.979
19N	-132.025	-133.083	-125.471	-117.084	-107.616
20N	-134.180	-134.556	-128.788	-119.857	-109.921
21N	-138.768	-139.637	-133.702	-124.769	-114.278
22N	-140.052	-140.657	-136.560	-127.114	-116.052

Equations

$$H_A=T_{rel}+V=\frac{1}{A}\sum _{i<j}^A\frac{({\vec{p}}_i-{\vec{p}}_j{)}^2}{2m}+\sum _{i<j}^A{V}_{NN,ij}$$

$$\displaystyle H_{A,eff}^{\Omega}=P\Bigg{\{}\sum_{i=1}^{A}\Bigg{[}\frac{({\vec{p}_{i}-\vec{p}_{j}})^{2}}{2mA}+\frac{m{\Omega}^{2}}{2A}({\vec{r}_{i}-\vec{r}_{j}})^{2}+V_{ij,eff}^{\Omega,A}\Bigg{]}$$

$$\displaystyle+\beta\left(H_{c.m.}-\frac{3}{2}\hbar\Omega\right)\Bigg{\}}P.$$

Full text

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 available to study nuclear properties. The no core shell model Navratil3 ; Barrett ; Navratil1 ; Navratil2 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 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 Carbon_isotopes using INOY Doleschall_1 ; Doleschall_2 and CDB2K CDB2K interactions.

In the present work we will study the nitrogen isotopes and mainly focused on neutron rich side. The structure of neutron rich nuclei 19-22N has been studied by in-beam $\gamma$-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 Nitrogen . The 22N has halo structure in its ground state 22N1 ; 22N2 . Recently, the point proton radii of neutron rich 17-22N isotopes have been measured from charge changing cross section in Ref. proton_radii . More recently, Yuan and Suzuki $\it{et~{}al}$, have done systematic study of B to O isotopes with a interaction YSOX which include (0-3) $\hbar\Omega$ excitations YSOX 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 starting A-body Hamiltonian is given by:

[TABLE]

$T_{rel}$ is the relative kinetic energy. The momenta of the individual nucleons are given by $p_{i}$ (i=1,…..A). The nucleon mass is given by $m$. In the present work we have dealt with the two body part only. The $V_{NN,ij}$ is the NN interaction having nuclear and Coulomb part both. Next, we divide the $A$-nucleon infinite HO basis space into finite active space ($P$) having all states of up to $N_{max}$ HO excitations above the unperturbed ground state and an excluded space ($Q=1-P$).

We add the center-of-mass (c.m.) harmonic oscillator (HO) Hamiltonian $H_{c.m.}$($=T_{c.m.}+U_{c.m.}$, $U_{c.m.}=\frac{1}{2}Am{\Omega}^{2}{\vec{R}}^{2}$, $\vec{R}=(\frac{1}{A})\sum_{i=1}^{A}{\vec{r}_{i}}$.) to Eq. 1. As we use slater determinant basis, the Lawson Projection term lawson is added to shift the spurious states (arises from the incorrect treatment of the centre-of-mass motion) to the Eq. 1. The Hamiltonian used in final calculations is given by:

[TABLE]

Where $\beta$ 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 (to soften the potential with the purpose of simplifying many-body calculations, these interactions are obtained by applying the unitary transformation to the two-nucleon system in momentum space with a regulator). So, we need a renormalization scheme to soften the interactions. Here, we use Okubo-Lee-Suzuki (OLS) scheme OLS1 ; Suzuki1 ; Suzuki2 . Now, we get an effective Hamiltonian which is in $A$- body space. In our calculations we have done NCSM calculations with the renormalized interactions keeping up to two-body cluster terms.

In the present paper, for the NCSM calculations, we have used the pAntoine Caurier1 ; CFors1 shell model code which is adapted to NCSM Caurier3 . In the case of 22N, for the largest model space $N_{max}$= 4, the corresponding dimension is $\sim 6.4\times 10^{7}$. We have compared the NCSM results with the shell model calculations using YSOX interaction. For shell model calculations we have used KSHELL code kshell .

III **Effective $NN$ Interaction **

In the present work we have studied the neutron rich nitrogen isotopes with the three different $NN$ interactions: INOY, CDB2K and N3LO Entem ; Machleidt2 ; Rup . 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 Odrip . In the 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 non-local at short range. The form of INOY $NN$ interaction is given in Refs. Doleschall_1 ; Doleschall_2 . This interaction reproduces the 3H and 3He 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 $NN$ 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 $\hbar\Omega$ and size of the model space $N_{max}$. To see this dependence, we have calculated the g.s. energy with different $N_{max}$ and $\hbar\Omega$, 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 ( computational time will be smaller) rather than other values of frequencies. This is the benefit for doing optimization of frequency. So, we have done our calculations with frequency $\hbar\Omega$= 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 $\hbar\Omega$=20 MeV for INOY and $\hbar\Omega$=14 MeV for N3LO interaction Barrett . In the case of CDB2K, we have taken $\hbar\Omega$=12 MeV Carbon_isotopes .

V **Results and discussions **

We have done calculations using INOY at $\hbar\Omega$= 20 MeV, CDB2K and N3LO interactions at 12 and 14 MeV, respectively. We have also compared our INOY results at $\hbar\Omega$= 22 MeV. The energy spectra are shown in Figs. 2 and 3. In the case of 18N, the g.s. is correctly reproduced by INOY $NN$ and YSOX interaction, while other two interactions give $2^{-}$ as a g.s. The order of energy states are correct with the INOY ($\hbar\Omega$=20 MeV) and YSOX only. The calculated $1_{2}^{-}$ state is at higher energy ($>$ 2.5 MeV) with INOY interaction (except for $N_{max}=0$). The NCSM results for $N_{max}=4$ with INOY ($\hbar\Omega$=22 MeV) are compressed in comparison to the CDB2K interaction.

For 20N, the results with the INOY ($\hbar\Omega$=22 MeV) 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 ($\hbar\Omega$=20 MeV) and $1^{-}$ is close to experimental data with INOY ($\hbar\Omega$=22 MeV).

In the case of 22N, 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 19N, INOY ($\hbar\Omega$=20 MeV) and the other interactions reproduce the correct g.s. $1/2^{-}$, though, all the states are not yet been confirmed experimentally. The g.s. and first two excited states are very compressed with the INOY at both the frequencies in comparison to the other interactions. The N3LO interaction gives the energy states better and level ordering is correct with the experimental one. Overall the INOY interaction gives compressed energy levels.

For 21N, 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 ($\hbar\Omega$=20 MeV), CDB2K, and N3LO interactions correspond to $N_{max}$= 4 model space size. For $N_{max}$= 4, we have taken 28 orbitals. Here, we have shown the occupancy up to $fp$ space because the occupancy of higher orbitals are very small to visualize. Although, the magnitudes of occupancies of higher orbitals are very small, still they are important in the calculation. The contribution of neutron occupancy from 0$d_{3/2}$ and 1$s_{1/2}$ orbitals for CDB2K and N3LO interaction is larger in comparison to INOY interaction. This larger occupancy is also reflected in the energy spectra. The CDB2K and N3LO results are similar for the g.s. spin and first excited state, however the occupancies for INOY interaction is different and for this interaction we are getting results which differ from other two interactions. In Fig. 5, the calculated g.s. energy for 18-22N isotopes using INOY and YSOX interactions follow the same trend as the experimental data. The g.s. energy for nitrogen isotopes with the other interactions are given in the Table 1 in which results with N3LO and CDB2K are very far from the experimental data. If we go to higher $N_{max}$, the results will come closer to 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. In 18N, the INOY and YSOX interaction predict second excited state as $2^{-}$. For 20N, the results of INOY ($\hbar\Omega$=22 MeV) interaction are better than YSOX interaction. For 22N, 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 19N, the NCSM results with N3LO are much better.

Acknowledgement:

AS acknowledges financial support from MHRD (Govt. of India) for her Ph.D. thesis work. We would like to thank Prof. Petr Navrátil for providing us his NN effective interaction code and Prof. Christian Forssén for pAntoine. We would also like to thank Prof. Ruprecht Machleidt for valuable comments on this article and Prof. Toshio Suzuki for the YSOX interaction. PCS acknowledges the hospitality extended to him during his stay at TRIUMF.