Shell model results for $^{47-58}$Ca isotopes in the $fp$, $fpg_{9/2}$   and $fpg_{9/2}d_{5/2}$ model spaces

Bharti Bhoy; Praveen C. Srivastava; Kazunari Kaneko

arXiv:1906.00706·nucl-th·May 6, 2020

Shell model results for $^{47-58}$Ca isotopes in the $fp$, $fpg_{9/2}$ and $fpg_{9/2}d_{5/2}$ model spaces

Bharti Bhoy, Praveen C. Srivastava, Kazunari Kaneko

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TL;DR

This study uses shell-model calculations with realistic interactions to analyze calcium isotopes, highlighting the importance of specific orbitals in neutron-rich isotopes and comparing results with experimental data.

Contribution

It introduces systematic shell-model results for $^{47-58}$Ca isotopes using various model spaces and interactions, including IM-SRG derived forces, providing new insights into orbital roles.

Findings

01

Good agreement with experimental data in the $fp$ space

02

$g_{9/2}$ and $d_{5/2}$ orbitals are crucial for heavier isotopes

03

Spectroscopic factors align with recent experiments

Abstract

We have reported shell-model results for $^{47 - 58}$ Ca isotopes in the $f p$ , $f p g_{9/2}$ and $f p g_{9/2} d_{5/2}$ model spaces using realistic $N N$ interaction. We have also performed a systematic shell-model study using interactions derived from in-medium similarity-renormalization group (IM-SRG) targeted for a particular nucleus with chiral $N N$ and $3 N$ forces. The results obtained are in a reasonable agreement with the available experimental data in $f p$ model space with $N N$ interaction. It is shown that the $g_{9/2}$ and $d_{5/2}$ orbitals play an important role for heavier neutron-rich $^{54 - 58}$ Ca isotopes, while it is marginal for $^{47 - 52}$ Ca. We have also examined spectroscopic factor strengths using $N N$ and $N N + 3 N$ interactions for recently available experimental data.

Tables4

Table 1. Table 1: B ( E 2 ) 𝐵 𝐸 2 B(E2) value in calcium isotopes compared with experiment montanari ; Raman . The B ( E 2 ) 𝐵 𝐸 2 B(E2) values are calculated with IM-SRG and GXPF1Br+V MU ( f p 𝑓 𝑝 fp , f p g 9 / 2 𝑓 𝑝 subscript 𝑔 9 2 fpg_{9/2} , and f p g 9 / 2 d 5 / 2 𝑓 𝑝 subscript 𝑔 9 2 subscript 𝑑 5 2 fpg_{9/2}d_{5/2} model spaces) interactions. The units are in e 2 superscript 𝑒 2 e^{2} fm 4 .

Nuclei	Transition	Expt.	$f p$	$f p g_{9 / 2}$	$f p g_{9 / 2} d_{5 / 2}$	IM-SRG
⁴⁷Ca	$3 / 2^{-} \to 7 / 2^{-}$	4.0 $\pm$ 0.2	3.22	3.08	3.02	1.58
⁴⁸Ca	$2^{+} \to 0^{+}$	19 $\pm$ 6.4	10.35	10.50	10.60	11.82
⁴⁹Ca	$7 / 2^{-} \to$ 3/2^-	0.53 $\pm$ 0.21	3.53	3.27	3.28	0.001
⁵⁰Ca	$2^{+} \to 0^{+}$	7.4 $\pm$ 0.2	7.82	7.82	8.01	8.0
⁵¹Ca	$7 / 2^{-} \to$ 3/2^-	N/A	6.72	6.44	6.43	7.64
⁵²Ca	$2^{+} \to 0^{+}$	N/A	6.16	6.67	7.06	6.46
⁵³Ca	$7 / 2^{-} \to$ 3/2^-	N/A	4.48	5.28	6.10	2.09
⁵⁴Ca	$2^{+} \to 0^{+}$	N/A	6.34	7.95	8.55	6.13
⁵⁵Ca	$7 / 2^{-} \to$ 3/2^-	N/A	5.39	2.69	2.02	4.55
⁵⁶Ca	$2^{+} \to 0^{+}$	N/A	8.95	13.15	13.92	6.90
⁵⁸Ca	$2^{+} \to 0^{+}$	N/A	8.25	10.48	11.12	6.85

Table 2. Table 2: Comparison of experimental ruiz and theoretical quadrupole and magnetic moments of ground states. Shell model results obtained from IM-SRG and GXPF1Br+V MU ( f p 𝑓 𝑝 fp , f p g 9 / 2 𝑓 𝑝 subscript 𝑔 9 2 fpg_{9/2} , and f p g 9 / 2 d 5 / 2 𝑓 𝑝 subscript 𝑔 9 2 subscript 𝑑 5 2 fpg_{9/2}d_{5/2} model spaces) interactions. We have taken e n subscript 𝑒 𝑛 {e_{n}} = 0.5 e 𝑒 e and g s e f f superscript subscript 𝑔 𝑠 𝑒 𝑓 𝑓 g_{s}^{eff} = g s f r e e superscript subscript 𝑔 𝑠 𝑓 𝑟 𝑒 𝑒 g_{s}^{free} .

	$J^{π}$	Expt.	$f p$	$f p g_{9 / 2}$	$f p g_{9 / 2} d_{5 / 2}$	IM-SRG	Expt.	$f p$	$f p g_{9 / 2}$	$f p g_{9 / 2} d_{5 / 2}$	IM-SRG
A		$μ (μ_{N})$					Q(eb)
47	$7 / 2^{-}$	-1.4064(11)	-1.4618	-1.4661	-1.4679	-1.3640	+0.084(6)	+0.0675	+0.0673	+0.0675	+0.0790
49	$3 / 2^{-}$	-1.3799(8)	-1.3921	-1.3919	-1.3857	-1.3290	-0.0360(3)	-0.0386	-0.0386	-0.0386	-0.0452
51	$3 / 2^{-}$	-1.0496(11)	-1.0077	-1.0503	-1.0523	-1.0610	+0.036(12)	+0.0390	+0.0371	+0.0369	+0.0421
53	$1 / 2^{-}$	N/A	+0.5063	+0.5120	+0.5020	+0.4930
55	$5 / 2^{-}$	N/A	+1.0687	+0.3617⁴⁴4Although $1 / 2^{-}$ predicted as a g.s. from SM	+1.0316	+0.9870	N/A	-0.0550	-0.0476⁵⁵5Although $1 / 2^{-}$ predicted as a g.s. from SM	-0.0482	-0.0567
57	$5 / 2^{-}$	N/A	+0.486	+0.472	+1.1668	+1.176⁶⁶6Although $3 / 2^{-}$ predicted as a g.s. from SM	N/A	+0.003	+0.001	+0.001	-0.0012 ⁷⁷7Although $3 / 2^{-}$ predicted as a g.s. from SM

Table 3. Table 3: Comparison of experimental Crawford and theoretical spectroscopic factor strenghts obtained from IM-SRG and GXPF1Br+V MU ( f p 𝑓 𝑝 fp , f p g 9 / 2 𝑓 𝑝 subscript 𝑔 9 2 fpg_{9/2} , and f p g 9 / 2 d 5 / 2 𝑓 𝑝 subscript 𝑔 9 2 subscript 𝑑 5 2 fpg_{9/2}d_{5/2} model spaces) interactions for different transitions.

Level energy (keV)	$J^{π}$	$C^{2} S_{e x p}$	IM-SRG	$f p$	$f p g_{9 / 2}$	$f p g_{9 / 2} d_{5 / 2}$	N2LO_sat PRL_nav	NN + 3N (lnl) PRL_nav
⁴⁸Ca $\to$ ⁴⁷Ca
0	7/2^-	6.4 ${}^{+ 0.8}_{- 0.9}$	7.5	7.7	7.6	7.6
2014	3/2^-	$\leq$ 1.4	0.002	0.05	0.05	0.06
⁵⁰Ca $\to$ ⁴⁹Ca
0	3/2^-	2.1(3)	1.8	1.8	1.7	1.6
2023	1/2^-	0.28 ${}_{- 0.03}^{+ 0.05}$	0.04	0.09	0.1	0.1
3357Crawford	7/2^-	3.4 ${}_{- 0.3}^{+ 0.4}$	4.7	7.7	7.6	7.6
⁵²Ca $\to$ ⁵¹Ca
0	3/2^-	N/A	3.7	3.5	3.4	3.3
1718	1/2^-	N/A	0.04	0.1	0.1	0.2
2378	5/2^-	N/A	0.003	0.003	0.01	0.01
⁵⁴Ca $\to$ ⁵³Ca
0	1/2^-	N/A	1.9	1.8	1.4	1.3	1.56	1.58
1738PRL_nav	5/2^-	N/A	0.01	0.1	0.5	0.5	0.01	0.02
2220PRL_nav	3/2^-	N/A	3.8	3.5	3.4	3.4	3.12	3.17

Table 4. Table 4: The calculated shell model wave functions corresponding to 3 − superscript 3 3^{-} state in 48,50,52,54 Ca isotopes.

Nuclei	State	$f p g_{9 / 2}$	${(f p g_{9 / 2})}_{n}$	$f p g_{9 / 2} d_{5 / 2}$
⁴⁸Ca	$3^{-}$	91% $f_{7 / 2}^{7} g_{9 / 2}^{1}$	91% $f_{7 / 2}^{7} g_{9 / 2}^{1}$	90% $f_{7 / 2}^{7} g_{9 / 2}^{1}$
		3.4% $f_{7 / 2}^{6} p_{3 / 2}^{1} g_{9 / 2}^{1}$	3.5% $f_{7 / 2}^{6} p_{3 / 2}^{1} g_{9 / 2}^{1}$	3.7% $f_{7 / 2}^{6} p_{3 / 2}^{1} g_{9 / 2}^{1}$
		1.5% $f_{7 / 2}^{5} p_{3 / 2}^{2} g_{9 / 2}^{1}$	1.5% $f_{7 / 2}^{5} p_{3 / 2}^{2} g_{9 / 2}^{1}$	1.5% $f_{7 / 2}^{5} p_{3 / 2}^{2} g_{9 / 2}^{1}$
⁵⁰Ca	$3^{-}$	93.8% $f_{7 / 2}^{8} p_{3 / 2}^{1} g_{9 / 2}^{1}$	93.3% $f_{7 / 2}^{8} p_{3 / 2}^{1} g_{9 / 2}^{1}$	87% $f_{7 / 2}^{8} p_{3 / 2}^{1} g_{9 / 2}^{1}$
		1.4% $f_{7 / 2}^{6} p_{3 / 2}^{3} g_{9 / 2}^{1}$	1.4% $f_{7 / 2}^{6} p_{3 / 2}^{3} g_{9 / 2}^{1}$	2.6% $f_{7 / 2}^{8} p_{3 / 2}^{1} d_{5 / 2}^{1}$
		1.1% $f_{7 / 2}^{6} p_{3 / 2}^{1} f_{5 / 2}^{2} g_{9 / 2}^{1}$	1.1% $f_{7 / 2}^{6} p_{3 / 2}^{1} f_{5 / 2}^{2} g_{9 / 2}^{1}$	1.6% $f_{7 / 2}^{7} p_{3 / 2}^{2} g_{9 / 2}^{1}$
⁵²Ca	$3^{-}$	87% $f_{7 / 2}^{8} p_{3 / 2}^{3} g_{9 / 2}^{1}$	88% $f_{7 / 2}^{8} p_{3 / 2}^{3} g_{9 / 2}^{1}$	83.5% $f_{7 / 2}^{8} p_{3 / 2}^{3} g_{9 / 2}^{1}$
		3.3% $f_{7 / 2}^{8} p_{3 / 2}^{1} p_{1 / 2}^{2} g_{9 / 2}^{1}$	3.2% $f_{7 / 2}^{8} p_{3 / 2}^{1} p_{1 / 2}^{2} g_{9 / 2}^{1}$	3.7% $f_{7 / 2}^{8} p_{3 / 2}^{1} p_{1 / 2}^{2} g_{9 / 2}^{1}$
		2.5% $f_{7 / 2}^{8} p_{3 / 2}^{1} f_{5 / 2}^{2} g_{9 / 2}^{1}$	2.2% $f_{7 / 2}^{8} p_{3 / 2}^{1} f_{5 / 2}^{2} g_{9 / 2}^{1}$	2.6% $f_{7 / 2}^{8} p_{3 / 2}^{1} f_{5 / 2}^{2} g_{9 / 2}^{1}$
⁵⁴Ca	$3^{-}$	77.3% $f_{7 / 2}^{8} p_{3 / 2}^{3} p_{1 / 2}^{2} g_{9 / 2}^{1}$	77.5% $f_{7 / 2}^{8} p_{3 / 2}^{3} p_{1 / 2}^{2} g_{9 / 2}^{1}$	86.2% $f_{7 / 2}^{8} p_{3 / 2}^{4} p_{1 / 2}^{1} d_{5 / 2}^{1}$
		8.7% $f_{7 / 2}^{8} p_{3 / 2}^{3} f_{5 / 2}^{2} g_{9 / 2}^{1}$	9.1% $f_{7 / 2}^{8} p_{3 / 2}^{3} f_{5 / 2}^{2} g_{9 / 2}^{1}$	3.2% $f_{7 / 2}^{8} p_{3 / 2}^{2} f_{5 / 2}^{2} p_{1 / 2}^{1} d_{5 / 2}^{1}$
		3.5% $f_{7 / 2}^{8} p_{3 / 2}^{3} g_{9 / 2}^{3}$	2.9% $f_{7 / 2}^{8} p_{3 / 2}^{3} f_{5 / 2}^{1} p_{1 / 2}^{1} g_{9 / 2}^{1}$	1.2% $f_{7 / 2}^{6} p_{3 / 2}^{4} f_{5 / 2}^{2} p_{1 / 2}^{1} d_{5 / 2}^{1}$

Equations4

H

H

H = E_{0} + ij \sum f_{ij} {a_{i}^{†} a_{j}} + \frac{1}{4} ij k l \sum Γ_{ij k l} {a_{i}^{†} a_{j}^{†} a_{l} a_{k}} + \frac{1}{36} ij k l mn \sum W_{ij k l mn} {a_{i}^{†} a_{j}^{†} a_{k}^{†} a_{n} a_{m} a_{l}},

H = E_{0} + ij \sum f_{ij} {a_{i}^{†} a_{j}} + \frac{1}{4} ij k l \sum Γ_{ij k l} {a_{i}^{†} a_{j}^{†} a_{l} a_{k}} + \frac{1}{36} ij k l mn \sum W_{ij k l mn} {a_{i}^{†} a_{j}^{†} a_{k}^{†} a_{n} a_{m} a_{l}},

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Full text

Shell model results for 47-58Ca isotopes in the $fp$ , $fpg_{9/2}$ and $fpg_{9/2}d_{5/2}$ model

spaces

Bharti Bhoy1111E-mail address: [email protected], Praveen C. Srivastava1222Corresponding author: [email protected] and Kazunari Kaneko2333E-mail address: [email protected]

1Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247 667, India

2Department of Physics, Kyushu Sangyo University, Fukuoka 813-8503, Japan

Abstract

We have reported shell-model results for 47-58Ca isotopes in the $fp$ , $fpg_{9/2}$ and $fpg_{9/2}d_{5/2}$ model spaces using realistic $NN$ interaction. We have also performed a systematic shell-model study using interactions derived from in-medium similarity-renormalization group (IM-SRG) targeted for a particular nucleus with chiral $NN$ and $3N$ forces. The results obtained are in a reasonable agreement with the available experimental data in $fp$ model space with $NN$ interaction. It is shown that the $g_{9/2}$ and $d_{5/2}$ orbitals play an important role for heavier neutron-rich 54-58Ca isotopes, while it is marginal for 47-52Ca. We have also examined spectroscopic factor strengths using $NN$ and $NN+3N$ interactions for recently available experimental data.

pacs:

21.60.Cs, 21.30.Fe, 21.10.Dr, 27.20.+n, 27.30.+t

I INTRODUCTION

The study of neutron-rich calcium isotopes is a topic of ongoing interest to understand the shell evolution and the location of drip line nature_ruiz ; holtdrip ; nadya . The discovery of 60Ca and implication for the stability of 70Ca has been recently reported by RIKEN experimental group in the Ref. 022501 . In contrast to $ab~{}initio$ calculations including three-body forces and continuum effects predict that 59Ca 032502 ; forssen is unbound and 60Ca marginally bound and unbound 132501 . The mass measurement of 55-57Ca 022506 confirmed the $N$ =34 subshell closure in 54Ca. In the recent experiment, the robust characteristic of $N=34$ subshell closure has been reported in 52Ar 072502 .

The neutron-rich Ca isotopes have been previously investigated by the shell model with $NN$ and $NN+3N$ interactions in $fp$ and $fpg_{9/2}$ model spaces Holt2 . Shell-model calculations show that $fpg_{9/2}$ model space can reproduce reasonable spectra up to N $\leq$ 35 but fails to explain strong collectivity in nuclei around $N=40$ . To reproduce the enhanced collectivity, $d_{5/2}$ orbital should be included in $fpg_{9/2}$ model space, because collective behavior can be understood in terms of quasi-SU(3) Lenzi . The importance of the $d_{5/2}$ orbital is also reported in Ref. keneko . It has been proposed Caurier that for neutron-rich $fp$ -shell nuclei, the neutrons are excited to the $sdg$ orbitals coupled to the unfilled $f_{7/2}$ proton orbital is responsible for a new region of deformation. Recently, the shell-model interpretation of the first spectroscopy of 61Ti using LNPS interaction for $fpg_{9/2}d_{5/2}$ model space has been reported by Wimmer et al. in Ref. wimmer . It has been shown that the ground state configuration is dominated by particle-hole excitations to the $g_{9/2}$ and $d_{5/2}$ orbitals. Thus, in the neutron-rich $fp$ shell nuclei, the inclusion of $g_{9/2}$ and $d_{5/2}$ orbitals in the model space becomes crucial as we approach towards $N=40$ .

Earlier, it has been shown that the many-body perturbation theory (MBPT) with three-nucleon forces ( $3N$ ) is very important to explain the spectroscopy of neutron-rich Ca isotopes Holt1 . In addition, the $ab~{}initio$ calculations with other modern approaches: in-medium similarity renormalization group (IM-SRG) and coupled-cluster effective interaction (CCEI) with chiral $NN$ and $3N$ forces among valence nucleons are found to describe well the location of drip line Stroberg .

The neutron-rich calcium isotopes are particular attraction for investigating the shell formation. The importance of $3N$ forces are crucial for explaining spectroscopy of Ca chain as reported in Ref. Holt2 .

Motivated with recent experimental data for spectroscopic factor strengths for Ca isotopes, we perform shell-model calculations with $NN$ and $NN+3N$ interactions. The aim of the present manuscript is to investigate recently available experimental data for spectroscopy and nuclear observables for the Ca isotopes using shell-model calculations with $NN$ interaction for $fp$ , $fpg_{9/2}$ , and $fpg_{9/2}d_{5/2}$ model spaces. We have also reported shell model results with $NN+3N$ interaction for $fp$ space. The present study will add more information to earlier theoretical work reported in Refs. Holt1 ; Holt2 .

This paper is organized as follows. In Sec. II, we present details of theoretical formalism. Comprehensive discussions are reported in Sec. III. Finally, a summary and conclusions are drawn in Sec. IV.

II THEORETICAL FRAMEWORK

We can express the present shell-model effective Hamiltonian in terms of single-particle energies and two-body matrix elements numerically,

[TABLE]

where $\alpha=\{nljt\}$ denote the single-particle orbitals and $\varepsilon_{\alpha}$ stand for the corresponding single-particle energies. $\hat{N}_{\alpha}=\sum_{j_{z},t_{z}}a_{\alpha,j_{z},t_{z}}^{{\dagger}}a_{\alpha,j_{z},t_{z}}$ is the particle number operator. $\langle j_{\alpha}j_{\beta}|V|j_{\gamma}j_{\delta}\rangle_{JT}$ are the two-body matrix elements coupled to spin $J$ and isospin $T$ . $A_{JT}$ ( $A_{JT}^{{\dagger}}$ ) is the fermion pair annihilation (creation) operator.

In the present work, we perform shell-model calculations in ${fp}$ , ${fpg_{9/2}}$ , and ${fpg_{9/2}d_{5/2}}$ model spaces without any truncation. To diagonalize the matrices, the shell model code KSHELL Kshell has been used. The maximum dimension we have diagonalized in the case of 58Ca for the ground state is 2.7 x 108.

We have taken GXPF1Br+VMU interaction Tomoaki for all the three model spaces. Since the GXPF1Br+VMU interaction is made for $fpg_{9/2}d_{5/2}$ model space, thus while doing calculation for $fp$ and ${fpg_{9/2}}$ model spaces, we allow valence neutrons to occupy in the $f_{7/2}$ , $p_{3/2}$ , $f_{5/2}$ , $p_{1/2}$ orbitals, and further including $g_{9/2}$ orbital, respectively. To see the impact of modified single-particle energies on the higher mass side of Ca isotopes, we have also reported shell model results for $fpg_{9/2}$ model space with modified single-particle energy of $g_{9/2}$ orbital by increasing it with 2 MeV, corresponding results are shown in figures at the last column. This part of the calculation is denoted by $({fpg_{9/2}})_{n}$ . The $fp$ -shell matrix elements are taken from GXPF1Br steppenbeck . The GXPF1Br steppenbeck interaction is modified version of GXPF1B Honma2 with correction in monopole interaction for $<0f_{5/2}1p_{3/2}|V|0f_{5/2}1p_{3/2}>_{T=1}$ . The GXPF1B interaction Honma2 is upgraded version with the modification of five $T$ = 1 two-body matrix elements and the bare single-particle energy which involve the 1 $p_{1/2}$ orbital from GXPF1A Honma1 . The cross-shell two-body interaction between $fp$ and $gds$ -shell orbitals are taken from $V_{MU}$ vmu . In the Hamiltonian ( Eq. 1) we have added $\beta_{c.m.}$$H_{c.m.}$ term as proposed by Gloeckner and Lawson GL to remove the spurious center-of-mass motion due to the excitation beyond one major shell in the case of $fpg_{9/2}$ and $fpg_{9/2}d_{5/2}$ calculations. We have taken $\beta_{c.m.}$ = 10. There is no further effect on results of energy levels and occupancy of orbitals by increasing value of $\beta_{c.m.}$ . Further, all the two-body matrix elements are scaled by $(A/42)^{-0.3}$ as the mass dependence. We have used the harmonic-oscillator parameter $\hbar\Omega=41A^{-1/3}$ for all the calculations. Earlier we have reported the importance of $g_{9/2}$ orbital in the model space for Mn isotopes in the Ref. pcs_mn .

Stroberg $\it{et~{}al.}$ Stroberg presented a nucleus-dependent valence-space approach using the IM-SRG, which is normal ordered with respect to a finite-density reference state $|\Phi\rangle$ . This approach adopts a decoupled valence space Hamiltonian in which occupied orbitals are fractionalized. The effective Hamiltonian can be expressed in terms of single-particle energies and two and three-body matrix elements, as:

[TABLE]

where $E_{0},f_{ij},\Gamma_{ijkl}$ and $W_{ijklmn}$ are the normal ordered zero-, one-, two-, and three-body terms, respectively. The normal ordered strings of creation and annihilation operators obey $\langle\Phi|\{a_{i}^{\dagger}\ldots a_{j}\}|\Phi\rangle=0$ . Here chiral $NN$ interaction is taken from N3LO machleidt11_n3lo ; machleidt12_n2lo , and a chiral 3 $N$ interaction is taken from N2LO navratil . To make the calculation easier, the residual 3 $N$ interaction $W_{ijklmn}$ is neglected among valence nucleons leading to the normal-ordered two-body approximation.

III RESULTS AND DISCUSSION

The comparisons of energy levels with shell model calculations and experimental data are shown for 47-52Ca and 53-58Ca in Figs. 1 and 2, respectively.

In 47Ca, the $fp$ , ${fpg_{9/2}}$ , and ${fpg_{9/2}d_{5/2}}$ model spaces result for negative parity states are in a reasonable agreement with the experimental data.

In 48Ca, the first excited 2*+* state is higher than those of the neighboring Ca nuclei. All the three set of model space results are in a good agreement with the experimental data for positive parity energy states, while the 3*-* and 5*-* states in $fpg_{9/2}$ and $fpg_{9/2}d_{5/2}$ calculation are much higher than the experimental data. With increasing model space the energy levels are slightly compressing.

For 49Ca, all the calculations with $NN$ force reproduce well first excited $1/2^{-}$ state. The ground state in 49Ca is dominated by the single-particle $p_{3/2}$ state. The calculated first $5/2^{+}$ level is at 6.966 MeV in the calculations from $fpg_{9/2}d_{5/2}$ model space and 9.575 MeV with $fpg_{9/2}$ model space. For the other excited states, the calculation from all the three valence spaces reproduce reasonably energy levels. In IM-SRG, the first excited $1/2^{-}$ state lies at very high energy ( $\sim$ 1.7 MeV higher).

For 50Ca, the location of the first excited state $2{{}^{+}}$ in all the calculations have been predicted very well with experimental data except for IM-SRG result. IM-SRG calculations with $NN+3N$ forces underestimate the first excited state by $~{}500$ keV. Most of the experimental levels are tentative in case of 50Ca. The large energy difference between the ${2_{1}}^{+}$ and ${2_{2}}^{+}$ states is reproduced by all $NN$ interactions. The spin and parity of the third excited state have not been experimentally identified, but our calculations predict it ${1}^{+}$ state.

In 51Ca, there is no definite experimental information on the spins and parities of the excited states. The first excited ${1/2}^{-}$ state is indicative of the effective $p_{3/2}-p_{1/2}$ gap and is consistent with the experimental tentative spin assignment.

The experimental evidence of $N$ = 32 subshell closure for 52Ca was first time reported in Ref Huck . The calculation from all three valence space reproduces well higher $2_{1}^{+}$ level, consistent with the $N=32$ subshell closure. All the calculations predict the second excited state as ${1}^{+}$ , while this state is not yet observed experimentally. The observed (3-) state above 2 ${}_{1}^{+}$ is tentative.

In 53Ca, the ground state is dominated by $p_{1/2}$ hole. Hence the difference between the first excited ${5/2}^{-}$ and ${3/2}^{-}$ levels will be mostly due to effective $p_{1/2}-f_{5/2}$ and $p_{1/2}-p_{3/2}$ gaps. This suggests both the $N$ =32 and $N$ = 34 subshell closures. The $fp$ , $(fpg_{9/2})_{n}$ model space and IM-SRG calculations in $fp$ shell predict ${5/2}^{-}$ for the first excited state. In the valence spaces $fpg_{9/2}$ and $fpg_{9/2}d_{5/2}$ , the calculations predict ${3/2}^{-}$ . The calculation from $(fpg_{9/2})_{n}$ predicting ${5/2}^{-}$ (1.863 MeV) and ${3/2}^{-}$ (1.865 MeV) levels as almost degenerate.

For 54Ca, the first experimental spectroscopic study on low-lying states was performed with proton-knockout reactions at RIKEN steppenbeck . They observed ${2_{1}}^{+}$ state at 2.043 MeV. The calculations predict this state at 2.689 MeV in $fpg_{9/2}d_{5/2}$ space and at 2.569 MeV in $fpg_{9/2}$ space and at 2.408 MeV in $(fpg_{9/2})_{n}$ space. In the IM-SRG calculation, the first excited state lies much higher (at 3.352 MeV) than the experimental energy.

In 55-58Ca, only spin and parity of ground states are known except for 57Ca. We have calculated a few low-lying states using the shell-model. Our calculated results will be important for upcoming future experiments.

In 55Ca $fp$ , $fpg_{9/2}d_{5/2}$ , and IM-SRG predict ${5/2}^{-}$ as ground state, consistent with experimental data.

For 56,58Ca the first excited ${2}^{+}$ states are at high energy. As 54Ca is a doubly closed shell nucleus, a ${5/2}^{-}$ state is expected as a ground state for 55Ca, with excited states at higher energies. This agrees with the results from the $fp$ , $fpg_{9/2}d_{5/2}$ and IM-SRG, but contrasts with the $fpg_{9/2}$ model space results. The $fpg_{9/2}d_{5/2}$ calculation predicts a very high ${2}^{+}$ state in 56Ca at about 2.5 MeV, higher energy than doubly magic isotopes 52Ca and 54Ca. In 58Ca, with a ${2}^{+}$ state around 3 MeV excitation energy. To overcome this problem we have tried to modify the single-particle energy of $g_{9/2}$ orbital from 0.881 MeV to 2.881 MeV. We chose this particular energy from a series of different sets of calculation with modifying single-particle energy taking reference of ${2}^{+}$ state in 52Ca. The calculation from $(fpg_{9/2})_{n}$ shows that the $g_{9/2}$ orbital is crucial for higher mass region of Ca isotopes. With the modification of the single-particle energy value of $g_{9/2}$ orbital, the high ${2}^{+}$ state starts decreasing from 54Ca onwards, however, it show negligible effect below 54Ca. The energy levels with IM-SRG is stretched because $NN$ and $3N$ forces for this interaction were not consistently SRG evolved.

Thus we may conclude that results of $fp$ model space is sufficient to reproduce energy levels of 47-52Ca isotopes and the role of $d_{5/2}$ orbital is very small. Although, with increasing the neutron number, occupancies of the $g_{9/2}$ and $d_{5/2}$ orbitals increase as shown in the Fig. 3. The occupancies for $g_{9/2}$ and $d_{5/2}$ decrease for the excited states. This means the effects of $g_{9/2}$ and $d_{5/2}$ could become important for $B(E2)$ between the ground state and the first excited state beyond 54Ca. In fact, as seen in Table 1, the $B(E2)$ increases for $fpg_{9/2}$ and $fpg_{9/2}d_{5/2}$ model space beyond 52Ca, while they are almost similar to all model spaces below 52Ca.

In Table 1, we have shown $B(E2)$ values for transitions in Ca isotopes. Our calculated results are in a reasonable agreement with the available experimental data. In the calculations, the neutron effective charge is taken as ${e_{n}}$ = 0.5 $e$ . It is clearly seen in Table 1 that the $g_{9/2}$ and $d_{5/2}$ orbitals affect significantly the $B(E2)$ values for heavier 54-58Ca. Thus the $g_{9/2}$ and $d_{5/2}$ orbitals play an important role for 54-58Ca.

In Table 2, we present the calculated spectroscopic quadrupole moments and magnetic moments for odd-mass of calcium isotopes using GXPF1Br+VMU ( $fp,fpg_{9/2},fpg_{9/2}d_{5/2}$ model spaces) and IM-SRG ( $fp$ ) interactions. The overall calculated results are in good agreement with the experimental data for magnetic moments. For 53,55,57Ca, the experimental data are not available. For 47,49,51Ca isotopes, the single particle magnetic moment value is -1.913 corresponding to the last filled neutron in $f_{7/2}$ (47Ca) and $p_{3/2}$ (49,51Ca) orbitals. The calculated magnetic moments of 47Ca and 49Ca are somewhat close to the effective single particle moment, indicating less contribution of the higher orbitals. For 51Ca the difference in the single particle magnetic moment and theoretical calculation is large, showing collective effect of orbitals. For the 53,55,57Ca isotopes, the single particle magnetic moment corresponding to the last occupied orbital $f_{5/2}$ is +1.366. For 53Ca, the magnetic moment value is very less than the single particle magnetic moment, this shows that the ground state for this isotope has a mixed configuration. In the case of 55Ca for $fpg_{9/2}$ interaction there is a drop in magnetic moment value, which maybe related to the ground state prediction, here we have shown value for the ${5/2}^{-}$ state, for the obtained ground state ${1/2}^{-}$ (from SM) also magnetic moment value is very small 0.2249. The magnetic moment values obtained from all the interactions are almost same for 47,49,51Ca isotopes. For 57Ca calculated magnetic moment value becomes close to single particle magnetic moment for the $fpg_{9/2}d_{5/2}$ interaction. In the case of 57Ca the calculated magnetic moment for 5/2- state is larger with $fpg{{}_{9}/2}d_{5/2}$ model space in comparison to $fp$ and $fpg_{9/2}$ model spaces. This reflects the effect of inclusion of $d_{5/2}$ orbital in the model space.

The calculated spectroscopic quadrupole moments are also compared to the known experimental values. Here we can see, a good description of the data has been obtained from all the interactions. The value of single particle quadrupole moment (in $eb$ ) for 47,49,51,53,55,57Ca isotopes are 0.075, -0.046, 0.047, -0.049, -0.071, 0.073, respectively. Since single particle quadrupole moments are not very close to the experimental quadrupole moment for 47,49,51Ca, thus we may conclude that the single particle contribution is not very strong for these isotopes, although there is a small configuration mixing. The IM-SRG interaction values are much closer to the single particle quadrupole moment. For 55Ca and 57Ca isotopes calculated quadrupole moments are very less than the single particle quadrupole moments.

Next, we study spectroscopic factor strengths $C^{2}S$ associated with neutron-hole states in 47-53Ca from IM-SRG and GXPF1Br+VMU ( $fp$ , $fpg_{9/2}$ , and $fpg_{9/2}d_{5/2}$ model spaces) interactions. Experimental data are available for 48Ca $\rightarrow$ 47Ca and 50Ca $\rightarrow$ 49Ca transitions. The calculated results are compared with the experimental data Crawford in Table 3. For the 48Ca $\rightarrow$ 47Ca transition, IM-SRG result is $C^{2}S_{th}$ = 7.55 corresponding to observed $C^{2}S_{exp}$ = 6.4 ${}_{-0.9}^{+0.4}$ for the lowest ${7/2}^{-}$ state, while GXPF1Br+VMU is giving $C^{2}S_{th}$ $\sim$ 7.7. The calculated spectroscopic factor to the first excited ${3/2}^{-}$ state is too small in comparison with the experimental value. For 50Ca $\rightarrow$ 49Ca, the spectroscopic factor from the IM-SRG for the 3/2*-* and 7/2*-* states are reasonable as in the experimental data, while it is quite small for the 1/2*-* state. From GXPF1Br+VMU interaction the spectroscopic factor for the 3/2*-* and 1/2*-* states are reasonable as in the experimental data and giving a large value for 7/2*-* state. For 52Ca $\rightarrow$ 51Ca and 54Ca $\rightarrow$ 53Ca transitions, we have reported theoretical results of the spectroscopic factor for future experiment. Recently, the ab initio calculation for $C^{2}S$ corresponding to 54Ca $\rightarrow$ 53Ca transition using N2LOsat and NN + 3N (lnl) interactions are reported in the Ref. PRL_nav . In the present work, we have compared our calculated results with these ab initio results in the Table 1. The ab initio $C^{2}S$ PRL_nav are lower than the GXPF1 ones because due to collective excitations. Our results for $C^{2}S$ also become smaller once we increase model space from $fp$ to $fpg_{9/2}$ to $fpg_{9/2}d_{5/2}$ .

The poor spectroscopy produced by the IM-SRG might be due to (i) if we look Stroberg et al. Stroberg , in the Fig. 2, the calcium isotopes around 48Ca are overbound by something like 50 MeV. This is due to at least in part to the fact that the $NN$ and $3N$ forces for this interaction were not consistently SRG evolved. The inconsistent evolution led to the oxygen chain agreeing with experiment, but this is almost certainly due to a lucky cancellation for that mass region. As we can see, the C isotopes are underbound and the Ca and Ni are overbound. (ii) also inadequacies of the initial chiral EFT interaction, in the Ref. gazit it is reported that the way that the $3N$ force was fit for that interaction was a mistake. Basically, it was fit to the beta decay of the triton, but an incorrect conversion between the axial current and the $3N$ coupling was used.

Previously, shell-model calculation for Ca isotopes has been done in Ref. Holt2 with many-body perturbation theory (MBPT) using $NN+3N$ forces in the extended model space $pfg_{9/2}$ . In our present work, we have tried to get better energy spectra for neutron-rich calcium isotopes along with nuclear observables. As the result reported in Ref. Holt2 , energy spectra is very much compressed from 47Ca to 49Ca, although after inclusion of $3N$ forces, somehow it improves but still gives approximately 1 $\sim$ 2 MeV compressed spectra. In our calculations up to 54Ca where experimental data are known, we can obtain very reasonable energy spectra for natural parity states. The calculation from MBPT contrary to experiment giving $0^{+}$ as the first excited state in 48Ca and 54Ca. In the present work, we have $2^{+}$ as the first excited state in 48Ca and 54Ca as in the experimental data at similar energy difference from the g.s. In comparison with the Ref. Holt2 , our calculation predicts both $N$ =32 and 34 subshell closure very well. Also, the calculated first and second excited states of 51Ca and 53Ca suggest the subshell closure at $N$ =32 and 34.

IV Excitation Energies of $3_{1}^{-}$ state in even-even Ca isotopes

The wave function corresponding to $3_{1}^{-}$ state in the 48,50,52,54Ca isotopes with $fpg_{9/2}$ and $(fpg_{9/2})_{n}$ interactions are shown in the Table 4. The calculated $3_{1}^{-}$ state for 48,50,52,54Ca isotopes with $(fpg_{9/2})_{n}$ interaction is higher than that of the $fpg_{9/2}$ interaction due to increase in the single particle energy of $g_{9/2}$ orbital by 2 MeV. The occupancy of $p_{3/2}$ orbital is changing from 0.073 (48Ca) to 1.037 (50Ca) to 2.835 (52Ca) to 2.943 (54Ca) with original $fpg_{9/2}$ interaction. The $3_{1}^{-}$ state is dominated by $\nu(p_{3/2}^{1}g_{9/2}^{1})$ configuration.

Since the $(fpg_{9/2})_{n}$ interaction predicts $3_{1}^{-}$ state at 12.244 MeV for 48Ca, the calculated $J^{\pi}$ = $5/2^{+}$ , $7/2^{+}$ and $9/2^{+}$ states around 12 MeV in 49Ca are the multiplets coming from coupling of ( $3_{1}^{-}$$\otimes$$p_{3/2}^{1}$ )J. Similarly $(fpg_{9/2})_{n}$ interaction predicts $3_{1}^{-}$ state at 9.085 MeV for 50Ca, the multiplets of ( $3_{1}^{-}$$\otimes$$p_{3/2}^{1}$ )J is responsible for $J^{\pi}$ = $5/2^{+}$ , $7/2^{+}$ and $9/2^{+}$ states in 51Ca.

V CONCLUSIONS

In the present work, we have performed shell-model calculations with realistic $NN$ interactions for 47-58Ca isotopes. To see the importance of $g_{9/2}$ and $d_{5/2}$ orbitals, we have performed calculations for $fp$ , $fpg_{9/2}$ , and $fpg_{9/2}d_{5/2}$ model spaces. In our calculations, after 54Ca, we are getting a very high ${2}^{+}$ state in the case of 56,58Ca. Thus, to reduce the energy of ${2}^{+}$ state and to see the importance of $g_{9/2}$ orbital, we have increased the single-particle energy of this orbital by 2 MeV. With this modification, however, the calculated $3^{-}$ states become higher than those of the $fpg_{9/2}$ calculations. On the other way, it might be also possible to adjust the cross-shell pairing two-body matrix elements (TBMEs) $<(fp)^{2}|V|(g_{9/2}d_{5/2})^{2}>$ ( $J=0$ ) to reduce the binding energy of $0_{1}^{+}$ . The $3^{-}$ problem remains open question. Results corresponding to the modified single-particle energy shows that $g_{9/2}$ orbital is very crucial for heavier Ca isotopes. The significant increase of occupancy for the $g_{9/2}$ orbital is obtained above $N=34$ once we move towards heavier 54-58Ca isotopes. The calculations predict that the $d_{5/2}$ orbital also plays an important role for heavier 54-58Ca, while it is marginal for 47-52Ca. Our calculations support $N=32$ and $N=34$ subshell closures in the Ca isotopes for 52Ca and 54Ca, and also confirmed by the wavefunctions of 53Ca for ground state, first and second excited states. The results for the IM-SRG interaction targeted for a particular nucleus with chiral $NN$ and $3N$ forces are also reported.

ACKNOWLEDGEMENTS

B. Bhoy acknowledges financial support from MHRD, Government of India. PCS acknowledges a research grant from SERB (India), CRG/2019/000556. We have performed theoretical calculations at Prayag 5 node computational facility at IIT-Roorkee. PCS would like to thank T. Togashi and R. Stroberg for useful help.

Bibliography35

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Shell model results for 47-58Ca isotopes in the fpfpfp, fpg9/2fpg_{9/2}fpg9/2​ and fpg9/2d5/2fpg_{9/2}d_{5/2}fpg9/2​d5/2​ model

Abstract

pacs:

I INTRODUCTION

II THEORETICAL FRAMEWORK

III RESULTS AND DISCUSSION

IV Excitation Energies of 31−3_{1}^{-}31−​ state in even-even Ca isotopes

V CONCLUSIONS

ACKNOWLEDGEMENTS

Shell model results for 47-58Ca isotopes in the $fp$ , $fpg_{9/2}$ and $fpg_{9/2}d_{5/2}$ model

IV Excitation Energies of $3_{1}^{-}$ state in even-even Ca isotopes