Ab initio no-core shell model study of neutron-rich 181920C isotopes

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Ab initio no-core shell model study of neutron-rich

18,19,20C isotopes

Priyanka Choudhary1, Praveen C. Srivastava2

Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India

Abstract

We implement the ab initio no-core shell model approach to study neutron rich

18C, 19C and 20C isotopes. For this purpose, we employ charge-dependent Bonn

2000 (CDB2K), inside non-local outside Yukawa (INOY) and chiral next-to-

next-to-next-to-leading order (N3LO) nucleon-nucleon interactions. Low-lying

energy spectra, electromagnetic properties and point-proton radii for these nu-

clei up to basis space Nmax = 4 are calculated. Binding energies obtained with

INOY interaction are in better agreement with the experimental values as com-

pared to other ab initio interactions. We also show the behavior of ground

state energy and point-proton radii with the NCSM parameters, ~Ω and Nmax.

We report a strong sensitivity of the B(E2) values from the ﬁrst excited 2+to

ground state of 18C and 20C to the nuclear interaction. Shell model calcula-

tions with YSOX interaction are also performed, and corresponding results are

compared with ab initio one.

Keywords: No-Core shell model, Inside Non-Local Outside Yukawa, Basis

space

1. Introduction

The study of the nuclear structure of atomic nuclei far from the beta stabil-

ity line poses a challenge to theory as well as experiments due to their exotic

behavior, such as halo structure [1, 2] and change in the shell structure [3, 4, 5].

The neutron halo nuclei are characterized by weakly bound neutron(s), which

are spatially decoupled from the nuclear core leading to extended core-decoupled

wave functions (extended radial distribution). Ongoing development in the ex-

perimental facilities promotes the study of nuclei near to drip line.

Neutron-rich carbon isotopes are of interest due to their exotic features

across the isotopic chain. For instance, 19C is a one-neutron halo nucleus [6],

and 22C, which is a drip line nuclei, is also a candidate for a two-neutron halo

1pchoudhary@ph.iitr.ac.in

2Corresponding author: praveen.srivastava@ph.iitr.ac.in

Preprint submitted to Elsevier October 28, 2022

arXiv:2210.15227v1 [nucl-th] 27 Oct 2022

nucleus [7]. The radioactive nuclear beam facilities at GANIL, MSU, GSI, and

RIKEN have been used to study the structural properties of near drip line

carbon isotopes. The reduced electric quadrupole transition strength [B(E2)]

is an essential fundamental observable which provides information about the

weakly bound and decoupled neutrons from the core. The B(E2) from the ﬁrst

2+state to the ground state (g.s.) has been measured through several exper-

iments over the past years for even-even carbon isotopes and diﬀerent values

are reported for the same nucleus, obtained from diﬀerent measurements [8].

Imai et al. [9] have measured extremely quenched B(E2) value 0.63 e2fm4for

16C and claimed that this is due to increase of shell gap between proton p3/2

and p1/2orbitals from 14C (N= 8) to 16C (N= 10). However, based on

the measurement by Wiedeking et al [10], larger B(E2) value 4.15(73) e2fm4

is derived for 16C and after reanalyzing the data of Ref. [9], a larger but still

quenched B(E2)= 2.6 ±0.9 e2fm4was obtained [11]. Both values of B(E2)

do not favor the decoupling of valence neutron from the core. The B(E2)

(4.21+0.34

−0.26(stat)+0.28

−0.24(systBρ)+0.64

−0.00(systfeeding) e2fm4), obtained from a new life-

time measurement [12] agrees well with the previous measurement [10]. No-core

shell model (NCSM) [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] is an ab initio

microscopic approach which has successfully described properties of lighter nu-

clei. The NCSM calculations with several NN and NN +NNN interactions have

been carried out for 16C in Ref. [23]. The extrapolated B(E2) value is under-

estimated for 16C with the used CDB2K interaction. Energy levels of 16C are

also calculated, and the values obtained with the CDB2K and chiral N3LO NN

are in reasonable agreement with the experiment. Inclusion of NNN interaction

further improves the excitation energies of the states. Similarly, diﬀerent claims

have been made based on the diﬀerent experimental results for 18C and 20C.

Theoretical calculations with shell model [24, 25, 26], antisymmetrized molec-

ular dynamics (AMD) [27], the multi-Slater determinant AMD (AMD+MSD)

[28], deformed Skyrme Hartree-Fock [24] and NCSM [23] predicted diﬀerent val-

ues of B(E2) for these nuclei in the past. Motivated by these experimental data

and diﬀerent theoretical claims, we have performed ab initio NCSM calculations

for 18−20C isotopes using three realistic interactions i.e. CDB2K, INOY, and

N3LO. Apart from the ab initio calculations, we have carried out shell model

calculations in psd-model space for comparison. Earlier, Forss´en et al. [23]

have studied even-even carbon isotopes in the range A= 10 −20 within the

framework of the NCSM and presented the ground state (g.s) and the excited

1state energies, the B(E2; 2+

1→0+

1) transition rates and the 2+

1quadrupole

moments dependence on the NCSM parameters.

The elastic electron scattering and muonic atom x-ray spectroscopy are em-

ployed to measure the charge radius of the nucleus. Since these techniques are

applied only for stable isotopes, therefore, a diﬀerent method is required for

radii measurement of neutron-rich isotopes. The optical and Kαx-ray isotopes

shift (IS) method is the only way to determine the charge radii of short-lived

unstable nuclei, although it is also challenging to apply this method for the

nuclei lying in the range 4 < Z < 10 due to inadequate precision in the atomic

physics calculations and diﬃculty in low-energy isotopes production. In Ref.

[29], a new technique, viz. charge changing cross-section (σCC ) measurement,

has been used in probing the point-proton radii (rp) of 12−19C isotopes. The

measured rpfor 18C and 19C was 2.39(4) and 2.40(3) fm, respectively. The rpof

18C was in good agreement with those obtained from ab initio coupled-cluster

theory using chiral NN and NNN interactions (NNLOsat). Also, rpfor 12−16C

isotopes were extracted using σCC technique at the Research Center for Nuclear

Physics (RCNP) facility, Osaka, and the Glauber model within the optical-limit

approximation [30]. In Ref. [31], the smallest spin-orbit originated magic num-

ber at Z= 6 in 12−20C isotopes was evident from the rpdistribution and B(E2)

measurements. The rpfor 12−19C was observed to be almost constant, which

might be an indication of an inert proton core.

In the present paper, we discuss the evolution of nuclear structure for 18−20C

isotopes by using the ab initio NCSM method with realistic NN interactions.

We have calculated low-lying energy spectra of 18−20C isotopes and compared

them with the experimental data. In addition, electromagnetic observables of

these isotopes are predicted and compared with the experimental data wherever

available. We have reported proton and neutron occupancies for the ground

and ﬁrst excited states with each interaction. Also, point-proton radii of g.s. of

18−20C are evaluated and their variations with NCSM parameters are shown.

Thus, our work is a comprehensive study of 18−20C isotopes using the NCSM,

with 19C being investigated for the ﬁrst time.

This paper is divided into the following sections. In Section II, the NCSM

formalism is brieﬂy summarized. In Section III, we introduce the realistic NN

interactions. Further, we present the calculated results of energy spectra, spec-

troscopic properties, occupancies and point-proton radii of 18,19,20C isotopes

obtained from the NCSM approach in Section IV. Finally, we conclude the pa-

per in Section V.

2. Ab initio no-core shell model formalism

We consider a nuclear system that consists of Apoint like non-relativistic

nucleons interacting via realistic interaction. We have constrained our calcula-

tions up to two-body interactions. In NCSM approach [18, 20], these Anucleons

of the system are taken as the degree of freedom. The initial Hamiltonian for

this system is written as

HA=1

i<j

(~pi−~pj)2

2m+

i<j

VNN

ij ,(1)

where, the ﬁrst term indicates the relative kinetic energy operator, and the

second term is the NN interaction containing both nuclear and Coulomb parts.

Here, mis the mass of nucleon. In the NCSM method, harmonic oscillator

(HO) basis states are employed that are restricted by truncation parameter

Nmax. Nmax is deﬁned as the maximum number of allowed HO excitations

above the g.s. conﬁguration of Anucleons system. Because of the properties of

these realistic nuclear interactions, we require eﬀective NN interaction in order

to obtain convergent results. We add centre-of-mass (c.m.) HO Hamiltonian to

the starting Hamiltonian 1 to facilitate the derivation of eﬀective Hamiltonian.

Now, modiﬁed Hamiltonian will take the form

HΩ

A=HA+Hc.m. =

i=1 ~p2

2m+1

2mΩ2~r2

i

i<j VNN

ij −mΩ2

2A(~ri−~rj)2,

(2)

where, the ﬁrst term indicates a sum of single particle HO Hamiltonian. Here,

the c.m. HO Hamiltonian is Hc.m. =~

2mA +1

2AmΩ2~

R2,where, Ω represents

the HO frequency, ~

P=PA

i=1 ~piand ~

R=1

APA

i=1 ~ri.The Hamiltonian 2 can be

rewritten as

HΩ

I=1

hi+

i<j

VΩ,A

ij .

Hence, the modiﬁed Hamiltonian becomes frequency dependent. This modiﬁed

Hamiltonian has the same intrinsic eigenstates as obtained from the translation-

ally invariant Hamiltonian 1. To solve the Schr¨odinger equation of Anucleons

system, the full Hilbert space is split into two parts which are ﬁnite model space

(P) that contains all HO basis states up to Nmax and the remaining model space

Q(= 1 −P). Thus, ﬁnal NCSM calculations are performed in the Pmodel

space. To derive eﬀective Hamiltonian, two renormalization procedures are im-

plemented that are the Okubo-Lee-Suzuki (OLS) scheme [32, 33, 34, 35] and the

Similarity Renormalization Group (SRG) [36, 37]. For the interaction used in

our calculations, former unitary transformation is employed. Now, we replace

the second term in the Hamiltonian 2 by the eﬀective interaction and subtract

out the c.m. Hamiltonian, which is added prior. So, the eﬀective Hamiltonian

will have the form

HΩ

A,eﬀ =P





i<j "(~pi−~pj)2

2mA +mΩ2

2A(~ri−~rj)2#

i<j VNN

ij −mΩ2

2A(~ri−~rj)2eﬀ 





(3)

This eﬀective Hamiltonian has up to A-body terms and it is rather diﬃcult to

solve this A-body system, thus, a simple approximation which is a two-body

cluster OLS is used. Since Hamiltonian 2 is not translationally invariant, it

generates spurious eigenstates with excited c.m. motion. These spurious states

are projected upward in the energy spectrum by the inclusion of the Lawson

projection term βHc.m. −3

2~Ω[38] to the Hamiltonian 3, in the ﬁnal step.

In our calculation, βis taken 10 that is large enough to shift these spurious

states up to high energies. This term will not change the intrinsic eigenstates

of the system. In this way, our ﬁnal calculations will vary with HO frequency

and Nmax.

3. Details about the eﬀective NN interactions

In the present NCSM calculations, the CDB2K [39, 40, 41, 42], INOY [43,

44, 45] and N3LO [47, 46] realistic NN interactions have been adopted. These

interactions come from either meson-exchange theory or Quantum Chromody-

namics (QCD) via chiral eﬀective ﬁeld theory (χEFT). In the CDB2K interac-

tion, exchange particles between two nucleons are mesons with masses below

the nucleon mass. These mesons are pions (π±,0), rho (ρ±,0), η,ωetc. The

INOY interaction contains local character at long distance while it shows non-

local behavior at short distance (r < 3 fm) and the form of this potential can

be found in Refs. [43, 44]. Its non-local feature helps to reproduce correct

binding energies of 3H and 3He without the addition of three-body forces. The

Bonn-Bochum-J¨ulich group has developed the chiral NN potentials, which are

reported in Refs. [48, 49, 50, 51, 52]. The N3LO interaction, ﬁrstly presented by

the Idaho group, Entem and Machleidt [47, 46], is based on the expansion up

to fourth order of chiral perturbation theory. The CDB2K, INOY and N3LO

interactions present strong short range correlations, therefore, require the OLS

renormalization. NCSM calculations for these ab initio interactions are per-

formed using pAntoine [53, 54, 55].

Additionally, the shell model calculations with the phenomenological YSOX

interaction [56] have been carried out for comparison with the ab initio results.

In YSOX interaction, 4He is taken as the inert core and it contains psd-model

space. We have performed calculations in a psd-model space with 2~Ω excita-

tions using KSHELL code [57].

As reported in [58], the NNN force improves the excitation energy spectra

of light nuclei. However, it is also mentioned that the NNN force over-corrects

the deﬁciencies of the NN interaction in the case of the 12C and shifts the 7/2−

state of 13C upwards in the energy spectrum, which indicates that further im-

provement in the interaction is required. Also, the three-nucleon force leads to

an overbinding of the carbon isotopes, as reported in Ref. [59]; this issue is still

unresolved as to whether it is caused due to deﬁciencies of the N2LO approxi-

mation to the NN interaction, or, has to be ﬁxed by higher-order NNN force

contributions. Earlier, our group have successfully studied boron (10−14B) [60],

nitrogen (18−22N) [61], oxygen (18−23O) [62] and ﬂorin (18−24F) [62] isotopes

using NCSM method and we found that without inclusion of the three-body

forces, INOY interaction correctly reproduces g.s. spin-parity of 10B as 3+and

also predicts the correct location of drip-line for oxygen chain at 24O. Since

inclusion of nonlocality in the INOY NN interaction can account some of the

many-nucleon force eﬀects, thus, our aim in the present work is to check whether

or not this interaction improves the description of near drip line carbon isotopes.

Also, we want to test which interaction is more suitable for these nuclei. So, we

have used three diﬀerent realistic N N interactions.

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Abinitiono-coreshellmodelstudyofneutron-rich18;19;20CisotopesPriyankaChoudhary1,PraveenC.Srivastava2DepartmentofPhysics,IndianInstituteofTechnologyRoorkee,Roorkee247667,IndiaAbstractWeimplementtheabinitiono-coreshellmodelapproachtostudyneutronrich18C,19Cand20Cisotopes.Forthispurpose,weemploycharge-d...

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Ab initio no-core shell model study of neutron-rich 181920C isotopes

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