$Ab~initio$ description of collectivity for $sd$ shell nuclei

TL;DR

This paper applies ab initio shell model methods to sd-shell nuclei, demonstrating their effectiveness in describing nuclear deformation and comparing results with phenomenological interactions and experimental data.

Contribution

It introduces ab initio approaches (IM-SRG and CC) to sd-shell nuclei, showing their validity in capturing deformation phenomena in open-shell nuclei.

Findings

IM-SRG and CC results agree with experimental data for most isotopes.

Ab initio methods effectively describe deformation in sd-shell nuclei.

Comparison with SDPF-MU highlights the role of pf orbitals in nuclear structure.

Abstract

In the present work, we have reported shell model results for open shell nuclei Ne, Mg and Si isotopes with $10 \leq N \leq 20$ in $sd$-shell model space. We have performed calculations in $sd$ shell with two $ab~initio$ approaches: in-medium similarity renormalization group (IM-SRG) and coupled-cluster (CC) theory. We have also performed calculations with phenomenological USDB interaction and chiral effective field theory based CEFT interaction. The results for rotational spectra and $B(E2;2_1^+\rightarrow 0_1^+)$ transitions are reported for even-mass isotopes. The IM-SRG and CC results are in reasonable agreement with the experimental data except at $N$ =20. This demonstrates a validity of $ab~initio$ description of deformation for doubly open-shell nuclei for $sd$ shell. To see the importance of $pf$ orbitals, we have also compared our results with SDPF-MU interaction by taking account of $2p-2h$ and $4p-4h$ configurations in $sd$-$pf$-shell model space.

Equations

$$H(s)=U^{†}(s)H(0)U(s)=H^d(s)+H^{od}(s).$$

$$\frac{dH(s)}{ds}=[\eta (s),H(s)],$$

$$\eta (s)=\frac{dU(s)}{ds}{U}^{†}(s)=-{\eta }^{†}(s).$$

$$H_{CCEI}=H_0^{A_c}+H_1^{A_c+1}+H_2^{A_c+2}+...$$

$$B(E2)=\frac{1}{2J_i+1}\mid (J_f\mid \mid \sum _i{e}_i{r}_i^2{Y}_2({\theta }_i,{\varphi }_i)\mid \mid J_i){\mid }^2.$$

Full text

I Introduction

The structure and properties of nuclei significantly change once we move towards drip-lines Tsunoda ; 32Mg . The “island of inversion” of nuclei around A = 30 has been a subject of several experimental and theoretical studies. There are different portals of island of inversion, for example, they may occurs at $N=8$, $N=20$, $N=28$, $N=40$ and $N=50$. To decide structure of these nuclei, intruder orbitals also become important. Recently there are several experimental groups which involve to study structure of these nuclei, in particular the “island of inversion” region of light nuclei with $N\sim 20$. Thus, from theoretical side apart from naive shell model it is important to study these nuclei with $ab~{}initio$ approaches stroberg ; jansen . As it is now possible to study lower $sd$ shell nuclei using $ab~{}initio$ approaches due to advancement in the computational facility, it is challenging to test predictive power of $ab~{}initio$ calculations for doubly-open shell nuclei for the description of deformation in the medium-mass region.

In the present work our motivation is to test the $ab~{}initio$ Hamiltonians to calculate the spectra and $B(E2)$ transitions for the doubly-open $sd$ shell nuclei. Previously, we have reported electromagnetic properties, Gamow-Teller (GT) strengths and spectroscopic strengths of $sd$ shell nuclei in Refs. archana_prc ; archana_gt ; pcs_prc .

We organize present work as follows. In section 2, we present details about Hamiltonians for $ab~{}initio$ calculations. In section 3, we present theoretical results along with experimental data wherever it is available. Finally summary and conclusions are drawn in section 4.

II Formalism

In our studies of neutron-rich Ne, Mg and Si isotopes, we have performed calculations in $sd$ space. As it is well known that excitation to $pf$-shell is very important for nuclei which belong to “island of inversion” mg_brown , thus, we have also performed calculations in $sd$-$pf$ shell.

For the $sd$ shell, we use the $ab~{}initio$ Hamiltonian derived from two modern $ab~{}initio$ approaches: IM-SRG stroberg and CCEI jansen ; jan1 . We have also compared results with a phenomenological USDB effective interaction usdb . For $sd-pf$ shell we have performed calculations with SDPF-MU intercation utsuno12 . For the diagonalization of matrices we have used shell-model code KSHELL KShell .

Using IM-SRG approach bogner based on chiral two- and three-nucleon interactions, Stroberg $\it{et~{}al.,}$ derived mass-dependent Hamiltonians for $sd$ shell nuclei stroberg . The strategy is that we select a Hamiltonian in certain basis so that the energy states which strongly differ in energy range in off diagonal matrix elements should be eliminated. After applying unitary transformation we get a final Hamiltonian $H(s)$ from a initial Hamiltonian $H(0)$.

[TABLE]

Here, s is the flow parameter. $H^{d}(s)$ is the diagonal part of the Hamiltonian and $H^{od}(s)$ is the off diagonal part of the Hamiltonian. The flow of the Hamiltonian is obtained by differentiating Eq. (1) w.r.to ‘s’-

[TABLE]

where $\eta(s)$ is the anti-Hermitian generator of unitary transformation,

[TABLE]

Eq. (2) is known as the flow equation for the Hamiltonian. The off-diagonal matrix elements become zero as $s\rightarrow\infty$ for appropriate value of $\eta(s)$. Here $sd$ valence space decouple from the core and higher shells as $s\rightarrow\infty$. Now we use the resulting Hamiltonian in the shell model calculations with $\hbar\omega$=24 MeV. Further details about parameters are given in ref. stroberg .

The Hamiltonian developed from the Coupled Cluster Effective Interaction approach is $A$- dependent and can be extended as-

[TABLE]

Here, $H_{0}^{A_{c}}$, $H_{1}^{A_{c}+1}$, and $H_{2}^{A_{c}+2}$ are called core, one-body, and two-body cluster Hamiltonians, respectively. This expansion is known as valance cluster expansion. Any operator can be expanded in the valence space in the same way as the Hamiltonian for the shell model calculations. The Okubo-Lee-Suzuki (OLS) similarity transformation is used to calculate the two-body term. In this approach Hartree-Fock ground state in thirteen oscillator major shells with $\hbar\omega$=20 MeV is used. Recently, L. Huth et al. Huth derived a shell-model interaction from effective field theory, which will be referred as CEFT.

The $B(E2)$ values are calculated with the formula:

[TABLE]

Where, $J_{i}$ and $J_{f}$ are the initial and final state spins, respectively. The $B(E2)$ values are calculated with the effective charges $e_{p}$=1.35e and $e_{n}$=0.35e.

III Results and Discussions

The experimental measurement of $B(E2)$ values and nuclear moments are important to predict existence of the “island of inversion”. The nuclei lying in the “island of inversion” region show a drastic enhancement of quadrupole collectivity compared to neighboring nuclei from these experimental results.

20Ne provides a good example of rotational spectra doubly_open in the lower $sd$ shell. The comparison of rotational energy levels for 20Ne obtained by USDB, IM-SRG, CCEI, and CEFT interactions is shown in the Fig. 1. The rotational spectra 20Ne and 24Mg are reproduced in our $ab~{}initio$ calculations. The $E_{2_{1}^{+}}$ and $B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$ transitions using $ab~{}initio$ interactions and USDB interactions along with experimental data nndc ; be2 for even Ne isotopes with $N=10-20$ are shown in Fig. 2. The USDB results for $E({2_{1}^{+}})$ are close to experimental data up to $N=16$ but above this results are deviating. The IM-SRG results are the best and close to the experimental data from $N=10$ to $N=18$. At $N=20$, only CCEI shows the same pattern as the experimental data, while results of all the other interactions go upward deviating from the experiment. Experimentally, $N=18$ shows less collectivity in comparison to $N=16$ and an enhancement in collectivity is seen at $N=20$. The $B(E2)$ values obtained are not satisfactory especially at $N=20$ for Ne and Mg isotopes. For all the interactions the collectivity is decreasing from $N=18$ to $N=20$ in contrast to the experiment. From the literature the $N=20$ lies on the boundary of “island of inversion” 28Ne and $0\hbar\omega$ shell model calculations are not able to reproduce the enhancement in collectivity at $N=20$.

To see the importance of neutron excitations from $sd$ to $pf$ shell, we have shown results with $2p-2h$ and $4p-4h$ excitations in Fig. 3. The results with $4p-4h$ excitations show the same trends as in the experiment : there is an increase of $B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$ from $N=18$ to $N=20$ though not enough to reproduce the experimental data. The increase in the occupancy of $pf$ orbitals are also confirmed from Fig. 4.

For the Mg isotopes, the case of 32Mg is very interesting. Its g.s. is deformed and it shows rotational spectrum. Up to $4^{+}$, the CCEI results are very close to the experimental data but $6^{+}$ state is very high in energy. Also, for 24Mg, the rotational spectra is well reproduced by $ab~{}initio$ interactions up to $6^{+}$. In Fig. 2, the energy of $2_{1}^{+}$ state is near to the experimental data with IM-SRG effective interaction except at $N=14$ and $N=20$. The CCEI results are also good except at $N=14$. At $N=20$ the energy of $2_{1}^{+}$ state is correctly given by CCEI but $B(E2)$ value is far from the experimental data. This shows there is a problem with the wavefunction, that is, there is a large configuration mixing in this wavefunction. Here, with $ab~{}initio$ interactions it is not possible to show collectivity at $N=20$. The shell model results with $2p-2h$ and $4p-4h$ exciations for SDPF-MU interaction are shown in Fig. 3. They show smooth decrease in $B(E2)$ values from $N=12-16$ for both $2p-2h$ and $4p-4h$ excitations, however, the results of $4p-4h$ show the same trends as in the experiment at $N=20$ : there is an increase of $B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$ from $N=18$ to $N=20$.

In the case of 34Si isotope none of the interaction explain properly the spectra. For Si isotopes the trend of energy of $2_{1}^{+}$ state for $N=10$ to $N=18$ isotopes are well predicted by CEFT and USDB interactions. The $B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$ trend from $N=12$ to $N=18$ are showing reasonable agreement with the experiment for all the interactions. The $B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$ values for the case with $2p-2h$ and $4p-4h$ excitations show similar results.

In the present calculations the gap between $d_{3/2}$ orbital and $fp$ shell is large for the SDPF-MU interaction compared with the interaction in Ref. Tsunoda , where the neutron ESPE’s of $pf$-shell are very close to those $sd$-shell; the difference between $f_{7/2}$ and $d_{3/2}$ is as small as about 2 MeV for Z=12 (N=20). Therefore, even if we allow $4p-4h$ excitation from $sd$ to $pf$ shell, the occupancy of $fp$-shell are around 2.23 in 32Mg. As we can see in Ref. Tsunoda , where energies and $B(E2)$ are well reproduced up to $N=20$, the occupancy of $fp$-shell becomes as large as 3.5 for 32Mg.

Thus our results might be improved if we reduce the gap between $sd$ and $fp$ shells for the SPDF-MU interaction.

IV Summary and conclusions

In the present work, we have performed shell model calculations for open shell nuclei with $10\leq N\leq 20$ for Ne, Mg and Si isotopes in the $sd$ and $sd-pf$ spaces. For $sd$ shell we have taken two $ab~{}initio$ interactions, IM-SRG and CCEI. We have also performed calculations with phenomenological USDB and interaction based on chiral effective field theory. The degree of freedom of $sd$ and $pf$ shells are essential for the nuclei close to “island of inversion” so, we have also reported results of $sd-pf$ shell with SDPF-MU interaction. The results of $ab~{}initio$ interactions show reasonable agreement with the experimental data except at $N$ =20. For nuclei in the island of inversion such as 30Ne and 32Mg, the admixture of $pf$ shells is important to explain the lowering of the energies of 2${}_{1}^{+}$ states and the enhancement of the $B(E2)$ values. present study will add more information to earlier theoretical $B(E2)$ values of Ne, Mg and Si isotopes.

Acknowledgments

AS acknowledges financial support from MHRD (Govt. of India) for her Ph.D. thesis work. Vikas Kumar acknowledges partial support from CUK.