Shell model results for Ca isotopes and role of orbital
We present shell-model calculations in , and valence spaces with realistic interactions, and obtain a good agreement with the experimental data. It is shown that orbital plays an important role in spectroscopy for heavier neutron-rich Ca isotopes. 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 and forces to study spectroscopy of Ca isotopes. We have examined spectroscopic factor strengths using interactions for recently available experimental data. The results obtained are in a reasonable agreement with the available experimental data.
pacs:21.60.Cs, 21.30.Fe, 21.10.Dr, 27.20.+n, 27.30.+t
The study of neutron-rich calcium isotopes is a current topic for understanding the shell evolution and the location of drip line nature_ruiz (). The discovery of Ca and implication for the stability of Ca has been recently reported by RIKEN experimental group in the Ref. 022501 (). In contrast to calculations including three-body forces and continuum effects predict that Ca 032502 (); forssen () is unbound and Ca marginally bound and unbound 132501 (). The mass measurement of Ca 022506 () confirmed the =34 subshell closure in Ca. In the recent experiment, the robust characteristic of subshell closure has been reported in Ar 072502 ().
The neutron-rich Ca isotopes have been previously investigated by the shell model with and interactions in and model spaces Holt2 (). Shell-model calculations show that interaction can reproduce reasonable spectra up to N 35 but fails to explain strong collectivity in nuclei around . To reproduce the enhanced collectivity, orbital should be included in model space, because collective behavior can be understood in terms of quasi-SU(3) Lenzi (). The importance of the orbital is also reported in Ref. keneko (). It has been proposed Caurier () that for neutron-rich -shell nuclei, the neutrons are excited to the orbitals coupled to the unfilled proton orbital is responsible for a new region of deformation. Recently, the shell-model interpretation of the first spectroscopy of Ti using LNPS interaction for 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 and orbitals. Thus, in the neutron-rich shell nuclei, the inclusion of orbital in the model space becomes crucial as we approach towards .
Earlier, it has been shown that the many-body perturbation theory (MBPT) with three-nucleon forces () is very important to explain the spectroscopy of neutron-rich Ca isotopes Holt1 (). In addition, the calculations with other modern approaches: in-medium similarity renormalization group (IM-SRG) and coupled-cluster effective interaction (CCEI) with chiral and forces among valence nucleons are found to describe well location of drip line Stroberg ().
The neutron-rich calcium isotopes are particular attraction for investigating the shell formation. The importance of 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 and 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 interaction for model space, and also interaction for 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 effective Hamiltonian in terms of single-particle energies and two-body matrix elements numerically,
where denote the single-particle orbitals and stand for the corresponding single-particle energies. is the particle number operator. are the two-body matrix elements coupled to spin and isospin . () is the fermion pair annihilation (creation) operator.
In the present work, we perform shell-model calculations in , , and model spaces. To diagonalize the matrices, the shell model codes ANTOINE Antoine (), KSHELL Kshell () and NuShellX Nushellx () have been used. For the shell region, the KB3G interaction Poves () which is obtained on the basis of Kuo-Brown’s G-matrix interaction is used. The KB3G interaction is a modification of the KB3 interaction, which was derived from the bare nucleon-nucleon interaction. The modifications involve changing the single-particle energies and introducing a mass dependence in the interaction, to better reproduce the subshell closure at 28. The interaction is derived using Ca as a core and the single-particle energies are given by the experimental spectrum of Ca. Typically the KB3G is used for nuclei situated in the lower shell with A 50.
We use the interaction pcs () for space and the GXPF1Br interaction Tomoaki () for space. The -shell matrix elements are taken from GXPF1B Honma2 () for GXPF1Br. The GXPF1B Honma2 () interaction is the upgraded version of GXPF1A interaction Honma1 (), with the modification of five = 1 two-body matrix elements and the bare single-particle energy which involve the 1 orbit from GXPF1A. The cross-shell two-body interaction between and -shell orbits are taken from vmu (). Further, the two-body matrix elements are scaled by as the mass dependence.
Stroberg Stroberg () presented a nucleus-dependent valence-space approach using the IM-SRG, which is normal ordered with respect to a finite-density reference state . This approach adopts a decoupled valence space Hamiltonian in which occupied orbits are fractionalized. The effective Hamiltonian can be expressed in terms of single-particle energies and two and three-body matrix elements, as:
where and are the normal ordered zero-, one-, two-, and three-body terms, respectively. The normal ordered strings of creation and annihilation operators obey . Here chiral interaction is taken from NLO machleidt11_n3lo (); machleidt12_n2lo () , and a chiral 3 interaction is taken from NLO navratil (). To make the calculation easier, the residual 3 interaction is neglected among valence nucleons, leading to the normal-ordered two-body approximation due to overbinding that grows with an increase in valence particles.
Iii Results and Discussion
In Ca, the KB3G ( model space) results for negative parity states are in a reasonable agreement with the experimental data.
In Ca, the first excited 2 state is higher than those of the neighboring Ca nuclei. The KB3G results are in a good agreement with the experimental data, while the excited states in calculation are much higher than the data. Although, the KB3G calculation predicts the first excited state as 4. With space, the first excited 2 energy can be reproduced well. Thus, the other energy levels including higher excited states in space are improving.
For Ca, all the calculations with force reproduce well first excited state. The ground state in Ca is dominated by the single-particle state. Therefore, the first excited energy is in good agreement with the experimental data. The calculated level is slightly compressed in the calculations from model space compared with model space. For the other excited states, the calculation in valence space reproduces reasonably energy levels. In IM-SRG, the first excited state lies at very high energy ( 1.7 MeV higher).
For Ca, the location of the first excited state in all the calculations have been predicted very well with experimental data except for IM-SRG result. IM-SRG calculations with forces underestimate the first excited state by keV. Most of the experimental levels are tentative in case of Ca. The large energy difference between the and states is reproduced by all interactions. The spin and parity of the third excited state have not been experimentally identified, but our calculations predict it state.
In Ca, there is no definite experimental information on the spins and parities of the excited states. The first excited state is indicative of the effective gap and is consistent with the experimental tentative spin assignment.
The experimental evidence of = 32 subshell closure for Ca was first time reported in Ref Huck (). The calculation from valence space reproduces well higher level, consistent with the subshell closure. All the calculations predict second excited state , while this state is not yet observed experimentally. The observed (3) state above 2 is tentative.
In Ca, the ground state is dominated by hole. Hence difference between the first excited and levels will be mostly due to effective and gaps. This suggests both the =32 and = 34 subshell closures. The KB3G and IM-SRG calculations in shell predict for the first excited state. In the valence spaces and , the calculations predict . Since the results in valence space are better than those of other calculations, the state could be a candidate for first excited state.
For Ca, the first experimental spectroscopic study on low-lying states was performed with proton-knockout reactions at RIKEN steppenbeck (). They observed state at 2.043 MeV. The calculations predict this state at 2.689 MeV in space and at 2.583 MeV in space. In the IM-SRG calculation, the first excited state lies much higher than the experimental energy at 3.352 MeV. The difference between KB3G and GXPF1Br calculations can be seen clearly. The energies in KB3G and GXPF1Br are 700 keV below and 500 keV above the experimental data, respectively. This suggests an importance of the orbital in the model space.
In Ca, only spin and parity of ground states are known except for Ca. We have calculated a few low-lying states using the shell-model. Our calculated results will be important for upcoming future experiments.
In Ca KB3G, GXPF1Br, and IM-SRG predict as ground state, consistent with experimental data, while calculation shows as ground state.
In Table 1, we have shown values for selected transitions in Ca isotopes. Our calculated results are in a reasonable agreement with the experimental data. The occupancies of orbital in the Ca isotopes are shown in Fig. 3. Increasing the neutron number, the occupancy of the orbital increases.
In Table 2, we present the calculated spectroscopic quadrupole moments and magnetic moments for odd-mass of calcium isotopes using IM-SRG and GXPF1Br. In the calculations, the neutron effective charge is taken as = 0.5. Both the calculated results are in good agreement with the experimental data for quadrupole and magnetic moments. The agreement of GXPF1Br is better than IM-SRG for Ca, Ca and Ca isotopes. For Ca and Ca, the experimental data are not available, it might be useful for the future experiment in this region.
Next, we study spectroscopic factor strengths associated with neutron-hole states in Ca in the IM-SRG approach. Experimental data are available for Ca Ca and Ca Ca transitions. The calculated results are compared with the experimental data Crawford () in Table 3. For the Ca Ca transition, IM-SRG result is = 7.55 corresponding to observed = 6.4 for the lowest state. The calculated spectroscopic factor to the first excited state is too small in comparison with experimental value. For Ca Ca, the spectroscopic factor for the 3/2 and 7/2 states are in good agreement with experimental data, while it is quite small for the 1/2 state. For Ca Ca and Ca Ca transitions, we have calculated the spectroscopic factor for future experiment.
|Level energy (keV)||IM-SRG|
In the present work, we have performed shell-model calculations with realistic interactions for Ca isotopes. To see the importance of and orbitals, we have performed calculations for , , and model spaces. The significant increase of the occupancy for the orbital is obtained above once we move towards heavier Ca isotopes. The inclusion of orbital is crucial for heavier Ca isotopes above 34. Our calculations support and subshell closures in the Ca isotopes. This could be connected with the importance of orbital. The results for the IM-SRG interaction targeted for a particular nucleus with chiral and forces are also reported. With the IM-SRG, the results of spectroscopic factor strengths reproduce well available experimental data. We hope that the present shell-model study is helpful to confirm several experimentally tentative states.
B. Bhoy acknowledges financial support from MHRD (Govt. of India) for her Ph.D. thesis work.
- (1) R.F. Garcia Ruiz et al., Unexpectedly large charge radii of neutron-rich calcium isotopes, Nat. Phys. 12, 594 (2016).
- (2) O.B. Tarasov et al., Discovery of and Implications For the Stability of , Phys. Rev. Lett. 121, 022501 (2018).
- (3) G. Hagen, M. Hjorth-Jensen, G.R. Jensen and R. Machleidt, and T. Papenbrock, Evolution of Shell Structure in Neutron-Rich Calcium Isotopes, Phys. Rev. Lett. 109, 032502 (2012).
- (4) C. Forssen, G. Hagen, M. Hjorth-Jensen, W. Nazarewicz, and J. Rotureau, Living on the edge of stability, the limits of the nuclear landscape, Phys. Scr. T152, 014022 (2013).
- (5) G. Hagen, P. Hagen, H.-W. Hammer, and L.Platter, Efimov Physics Around the Neutron-Rich Ca Isotope, Phys. Rev. Lett. 111, 132501 (2013).
- (6) S. Michimasa et al., Magic Nature of Neutrons in : First Mass Measurements of , Phys. Rev. Lett. 121, 022506 (2018).
- (7) H.N. Liu et al., How robust is the subshell closure ? first spectroscopy of Ar, Phys. Rev. Lett. 122, 072502 (2019).
- (8) J.D. Holt et al., Three-nucleon forces and spectroscopy of neutron-rich calcium isotopes, Phys. Rev. C 90, 024312 (2014).
- (9) S. M. Lenzi et al., Island of inversion around Cr, Phys. Rev. C 82, 054301 (2010).
- (10) K Kaneko, Y. Sun, M. Hasegawa and T. Mizusaki, Shell model study of single-particle and collective structure in neutron-rich Cr isotopes, Phys. Rev. C 78, 064312 (2008).
- (11) E. Caurier, F. Nowacki, and A. Poves, Large-scale shell model calculations for exotic nuclei, Eur. Phys. J. A 15, 145-150 (2002).
- (12) K. Wimmer et al., First spectroscopy of Ti and the transition to the island of inversion at , Phys. Lett. B 792, 16 (2019).
- (13) J.D. Holt et al., Three-body forces and shell structure in calcium isotopes, J. Phys. G: Nucl. Part. Phys. 39, 085111 (2012).
- (14) S.R. Stroberg et al., Nucleus-Dependent Valence-Space Approach to Nuclear Structure, Phys. Rev. Lett. 118, 032502 (2017).
- (15) E. Caurier, G. Martínez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, The shell model as a unified view of nuclear structure, Rev. Mod. Phys. 77, 427 (2005).
- (16) Noritaka Shimizu, Nuclear shell-model code for massive parallel computation, KSHELL (private communication).
- (17) B.A. Brown, and W.D.M. Rae, The Shell-Model Code NuShellX@MSU, Nucl. Data Sheets 120, 115 (2014).
- (18) A. Poves, J. Sanchez-Solano, E. Caurier, and F. Nowacki, Shell model study of the isobaric chains A = 50, A = 51 and A = 52, Nucl. Phys. A 694, 157 (2001).
- (19) P.C. Srivastava, Nuclear structure study with core excitations in Ni region: for space, Mod. Phys. Lett. A 27, 1250061 (2012).
- (20) T. Togashi et al., Large-scale shell-model calculations for unnatural-parity high-spin states in neutron-rich Cr and Fe isotopes, Phys. Rev. C 91, 024320 (2015).
- (21) M. Honma et al., Shell-model description of neutron-rich Ca isotopes, RIKEN Accel. Prog. Rep. 41, 32 (2008).
- (22) M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki, Shell-model description of neutron-rich pf-shell nuclei with a new effective interaction GXPF1, Eur. Phys. J. A 25, 499 (2005).
- (23) T. Otsuka, T. Suzuki, M. Honma, Y. Utsuno, N. Tsunoda, Novel Features of Nuclear Forces and Shell Evolution in Exotic Nuclei, K. Tsukiyama, and M. Hjorth-Jensen, Phys. Rev. Lett. 104, 012501 (2010).
- (24) R. Machleidt and D.R. Entem, Chiral effective field theory and nuclear forces, Phys. Rep. 503, 1 (2011).
- (25) D.R. Entem and R. Machleidt, Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory, Phys. Rev. C 68, 041001(R) (2003).
- (26) P. Navratil, Local three-nucleon interaction from chiral effective field theory, Few-Body Syst. 41, 117 (2007).
- (27) A. Huck, G. Klotz, A. Knipper, C. Miehe, C. Richard-Serre, G. Walter, A. Poves, H. L. Ravn, and G. Marguier, Beta decay of the new isotopes K, Ca, and Sc; a test of the shell model far from stability, Phys. Rev. C 31, 2226 (1985).
- (28) http://www.nndc.bnl.gov/ensdf/.
- (29) D. Montanari, S. Leoni, D. Mengoni, J. J. Valiente-Dobon, G. Benzoni, N. Blasi, G. Bocchi, P. F. Bortignon, S. Bottoni, and A. Bracco, spectroscopy of calcium nuclei around doubly magic Ca using heavy-ion transfer reactions, Phys. Rev. C 85, 044301 (2012).
- (30) S. Raman, C. W. G. Nestor Jr., and P. Tikkanen, Tables of E2 transition probabilities from the first 2 states in even-even nuclei, At. Data Nucl. Data Tables 78, 1 (2001).
- (31) D. Steppenbeck et al., Evidence for a new nuclear magic number from the level structure of Ca, Nature 502, 207 (2013).
- (32) H. L. Crawford, A. O. Macchiavelli, P. Fallon, M. Albers, Unexpected distribution of 1f strength in Ca, Phys. Rev. C 95, 064317 (2017).
- (33) R.F. Garcia Ruiz et al., Ground-state electromagnetic moments of calcium isotopes, Phys. Rev. C 91, 041304(R) (2015).