Luận văn Nghiên cứu cấu trúc vỏ của các đồng vị giàu neutron ⁴⁹Cl và ⁴⁹Ar thông qua phản ứng loại một nucleon ⁵⁰Ar(p,2p) và ⁵⁰Ar(p,pn)

ections are obtained, using calculated C2S and single-particle cross sections from the TC and DWIA meth- ods described in the text. The excitation energy range is limited to keV, as no state above this energy with a sizeable spectroscopic factor has been obtained in the theoretical calculations. The last two rows compare the measured inclusive cross section σinc to the sum of exclusive cross sections ∑ σexi for the three employed calculations. Experiment σljsp(E∗, Einc) SDPF-MUs Eexp σexp ∆ℓ nlj σTC σDW State Energy C2S σth,TC σth,DWIA (keV) (mb) (mb) (mb) (keV) (mb) (mb) gs <17.2 (15) 1p3/2 7.4 8.0 3/2−1 gs 1.844 13.6 14.7 198 (3) 2.97 (64) 1 1/2−1 94 0.616 4.5 4.9 1050 (29) 1.52 (35) 5/2−1 923 0.006 1340 (14) 8.26 (66) 3 0f7/2 5.6 5.4 7/2−1 983 2.564 14.3 13.8 1466 (21) 6.34 (57) 1p3/2 6.4 7.0 3/2−2 1197 0.678 4.3 4.7 5/2−2 1460 0.233∑ σexi 35.5 37.5 Inclusive 36.3 (9) Cross-section analysis of 49Ar isotope is similar to 49Cl, experimental values are com- pared in Tab. 4.7 to the results of cross-section calculations σexi (E ∗) for excitation energies E∗ (see Eq. 4.1.4). Here, they were calculated with the TC [61] and DWIA [75] methods at 217 MeV/nucleon, which corresponds to the mid-target energy for 50Ar projectiles. Single-particle σljsp(E∗, Einc) values were determined for the removal of one neutron in the different orbitals and energies (1p3/2, gs), (0f7/2, 1340 keV) and (1p3/2, 1466 keV). Values are given in Tab. 4.7 and used in the calculation of σth for the 3/2−1 , 7/2 − 1 and 3/2 − 2 states. Due to the weak dependence with E∗, the same values were used for the 3/2−1 and 1/2 − 1 states. An overall agreement is observed with experimental values, except for the large value obtained for the population of the 7/2−1 state, which possibly suggests a smaller spectroscopic factor. Overall consistency may be tested through the reduction factor Rs = σinclusive / ∑ σexi , using the spectroscopic factor predictions of a shell model routinely used in this region like SDPF-MUs calculation in Tab. 4.7. The values Rs = 1.09(11) and 1.03(11) are found with the 87 Results and Discussion TC and DWIA reaction models, respectively, which places 50Ar(p,pn) (∆S = -17.8 MeV) in the general trend observed for the one nucleon knockout reactions (see [77] and Fig.2 in Ref. [164]). For the study of a new magic number at N = 32, 50Ar and the one neutron knockout is a good tool to test the persistence of this shell effect with two protons less. In the shell-model framework, one would expect, for a closed-shell nucleus at N = 32 in a spherical configuration, the ν0f7/2 and ν1p3/2 orbitals to be filled, while ν1p1/2 and ν0f5/2 orbitals to be empty. In the one-neutron removal, the occupancy number should be found 2j +1 for occupied orbitals and 0 for empty ones. Theoretical calculations have been performed for the one nucleon removal from 52Ca and 50Ar. Details of the calculations and the used interactions are given in Sec. 4.1 with joined reference, in addition with the predictions of the HFB-D1S calculation in Ref. [165]. 52Ca is predicted to be a spherical nucleus [165]. The 3/2−1 ground state and 7/2 − 1 state in 51Ca have large spectroscopic factors in all theoretical calculations (Fig. 4.16), very close to the 2j+1 limit, which is a signature of their single-particle character with a strong overlap with a neutron-hole in the ν1p3/2 and ν0f7/2 orbitals of 52Ca, respectively. This is consistent with filled orbitals in 52Ca. At variance, the 1/2−1 state has a large excitation energy and a small spectroscopic factor, if not null. This can be understood as the removal of a 1p1/2 neutron from a (ν1p3/2)2(ν1p1/2)2 configuration with a small weight in the 52Ca wave function and a large ν1p3/2−ν1p1/2 spherical gap. This is what we expect for a closed shell nucleus at N = 32 in the shell model framework. 0 10000 1 2 3 4 49Ar-SM 49Ar-IMSRG 49Ar-GGF1 49Ar-GGF2 51Ca-SM 51Ca-IMSRG 51Ca-GGF1 51Ca-GGF2 C2 S Energy (keV) 3/2- 0 1000 20000.0 0.2 0.4 0.6 0.8 1.0 Energy (keV) 1/2- 0 2000 40000 2 4 6 8 Energy (keV) 7/2- Figure 4.16: Spectroscopic factor distributions for 3/2− (left), 1/2− (middle) and 7/2− (right) states in 49Ar (full bars) and 51Ca (open bars) obtained with shell-model, IMSRG and GGF calculations using the original SDPF-MU interaction (red), 1.8/2.0 (EM) interaction (blue) and the NN + 3N(lnl) (green) and NNLOsat (black) interactions, respectively. A rather different picture arises for 50Ar. The potential energy surface in Ref. [165] is much softer than for 52Ca and a trend towards oblate deformation may be observed. This trend in the argon isotopes below the Z = 20 magic number, is favored by the gain in energy for the 88 Results and Discussion 3/2+ [202] proton orbital, in the Nilsson model framework, as observed in Fig. 4.17. A secondary shallow minimum is predicted at low energy and β ≃ −0.25. Figure 4.17: The different plots are taken from Ref. [165] for 50Ar with HFB calculation and the D1S interaction : (left) potential energy surface of 50Ar versus quadrupolar deformation; (right) energy of the proton and neutron orbitals versus the quadrupolar deformation β parameter. The orbitals discussed in text are labelled with the usual Nilsson K[N,nz,Λ]. The black dots stand for the Fermi level. The red arrows correspond to β = 0.25 which is approximately the position of the oblate minimum Considering axially deformed nuclei, such as for the quadrupolar degree of freedom, J is no longer a good quantum number, but the third component K of the total angular momentum on the symmetry axis. At a given deformation, the wave function of each Kπ orbital is a mixing of all the spherical orbitals with same Kπ, with phases related to the spectroscopic factors. For a given K, an intrinsic j orbital may be projected over several physical states Jπ, following −→ j = −→ J + −→ R . The band-head J = j state only, corresponding to R = 0, can be accessed in a prompt direct reaction, such as a one-step nucleon transfer. Depending on the one-step character of the (p,pn) reaction at 217 MeV/u or not, the states accessible will be also limited to R = 0 or beyond. For 50Ar, the spherical valence orbital ν1p3/2 is now splitted into two deformed orbitals, labeled 1/2−[321] and 3/2−[312] in Fig. 4.17. At low deformation on the oblate side (β < 0), the Fermi level coincides with the valence 1/2−[321] orbital. It is mainly of ν1p3/2 origin close to β = 0. When β increases, the 1/2−[321] orbital is rapidly repulsed downwards by 1/2−[301] from the spherical ν1p1/2, and further on by the others K = 1/2 orbitals , which results in a strong 89 Results and Discussion mixing of the wave function. This could explain, at low β values, the population of two states, 3/2−1 and 1/2 − 1 . It is not possible to predict qualitatively which one would be the ground state, but a low excitation energy for the excited state and high C2S values for both of them, while a 1/2− state in a spherical configuration should be found at much higher excitation energy, such as for 51Ca. The slightly more bound 3/2−[312] is the next valence orbital, just below 1/2−[321]. Even at low oblate deformation, 3/2−[312] is repulsed upwards by the 3/2−[321] orbital of ν0f7/2 origin. The mixing is favored by the quadrupolar correlations between the ν1p3/2 and ν0f7/2 orbitals, for which ∆ℓ = 2. This could explain the 7/2−1 and 3/2 − 2 states at moderate excitation energy. There is also a possible mixing due to the repulsion from the K = 3/2 orbital of ν0f5/2 origin, responsible for a 5/2− state with small C2S values at low deformation. The neutron removal from the deeper orbitals originating from the spherical ν0f7/2, such as 1/2−[312] or 3/2−[321], will populate states at much higher excitation energy, considering the large N = 28 spherical energy gap. 1000 2000 3000 4000 5000 60000.0 0.5 1.0 1.5 2.0 2.5 3.0 C2 S Energy (keV) 49Ar : 7/2- 49Ar : 3/2- 3500 4000 4500 50000 2 4 6 8 C2 S Energy (keV) 51Ca : 7/2- 0 1000 2000 3000 4000 50000.0 0.4 0.8 1.2 1.6 2.0 C2 S Energy (keV) 0 1000 2000 30000 2 4 C2 S Energy (keV) 51Ca : 3/2- Figure 4.18: Spectroscopic factor ||⟨A−1X + n ∥AX⟩||2 distributions for 7/2− and 3/2− states in 49Ar (red) and 51Ca (blue) obtained for the shell-model calculation with the SDPF-MUs interaction. We will first consider the results of the shell-model calculation with SDPF-MUs interaction. 7/2−states 90 Results and Discussion The C2S distribution is shown in Fig.4.18-(a) for the first 7/2− states in 49Ar. In spite of the high excitation energy range, the sum ∑ i(C 2S)i = 6.95 is lower than the value for 7/2−1 in 51Ca, with evidence for a fragmentation of the distribution. Two states, 7/2−1 and 7/2 − 4 , have a large C2S value. The 7/2−1 state has a low excitation energy E ∗ = 983 keV and a large value B(E2; 7/2−1 → 3/2−1 ) = 72.1 e2fm4 to the ground state. It is consistent with the neutron removal from a deformed valence orbital. From what precedes, the origin could be from the 3/2−[312], but also possibly from 1/2−[321], as a member of the ground state band with R = 2, if such a state is populated in the (p,pn) reaction. Moreover the excitation energy of the first 2+ state in 50Ar, E∗(2+1 ) = 1178 keV [9] is close to the 7/2 − 1 energy. The excitation energy E ∗ = 3245 keV of the 7/2−4 state, well above the S1n energy, is similar to the value obtained for 7/2 − 1 in 51Ca. It may be associated to the neutron removal from the spherical ν0f7/2 or the slightly deformed orbitals, such as 1/2−[330]. 3/2−states The C2S distribution for 3/2− states is shown in Fig.4.18-(b). Here again the distribution is more fragmented and the sum ∑ i(C 2S)i = 3.19 is lower than the value for 3/2−1 in 51Ca. Besides the 3/2−1 ground state, a 3/2 − 2 state is found at about 1 MeV, which could be associated to the removal from a deformed valence orbital. This is consistent with the measured cross sections. 1/2−states The energy difference between the ground and first excited stats is found to be quite small. Comparably small values are obtained with the original SDPF-MU or SDPF-MUr interactions. Whereas the C2S value for 3/2−1 , although large, is well reduced compared to the 51Ca case, the C2S value for 1/2−1 significantly increases from 51Ca to 49Ar which is confirmed by the experimental cross section measured for the first excited state, assumed to be 1/2−1 , in Tab. 4.7. Here again we observe a striking deviation from a closed shell nucleus which is better illustrated by 51Ca, for which the low C2S value of the 1/2−1 state at about 1 MeV is consistent with a weak occupancy of the spherical ν1p1/2 orbital. Numerical values for the neutron content Nν of the ν1p3/2 and ν1p1/2 orbitals are displayed in Tab. 4.8 for the shell model calculation with SDPF-MUs interaction of 51Ca and 49Ar. This change is not driven by the monopole gap between the ν1p3/2 and ν1p1/2 orbitals which does not significantly change from one nucleus to the other, as already mentioned in [9]. Considering the monopole matrix elements in the SDPF-MU hamiltonian = −0.66MeV and 91 Results and Discussion < π0d3/2−ν1p1/2 = −0.57MeV , the removal of two protons from 51Ca to 49Ar is not expected to impact much the energy gap ν1p3/2−ν1p1/2 and therefore Nν . The low excitation energy and sizeable C2S value of the 1/2−1 state is a clear indication of collective effects present in 49Ar. This change is not driven by the monopole gap between the ν1p3/2 and ν1p1/2 orbitals which does not significantly change from one nucleus to the other, as already mentioned in [9]. Considering the monopole matrix elements in the SDPF-MU hamiltonian = −0.66MeV and < π0d3/2 − ν1p1/2 = −0.57MeV , the removal of two protons from 51Ca to 49Ar is not expected to impact much the energy gap ν1p3/2 − ν1p1/2 and therefore Nν . The low excitation energy and sizeable C2S value of the 1/2−1 state is a clear indication of collective effects present in 49Ar Table 4.8: Neutron content Nν of the different fp orbitals in the gs wave function of 52Ca and 50Ar obtained in the shell-model calculation using the SDPF-MUs interaction. Nν(nlj) 52Ca 50Ar ν0f7/2 7.95 7.61 ν0f5/2 0.13 0.41 ν1p3/2 3.76 3.28 ν1p1/2 0.17 0.70 Quadrupolar correlations may be put forward. A comparison may be drawn with the less neutron-rich partners 48Ca and 46Ar nuclei at the N = 28 shell closure. It was suggested [20, 166] that 46Ar has a vibrational structure and an oblate minimum is visible in the PES calculated [165, 167] with the 5DCH interaction [168]. Coming back to 50Ar, the SDPF-MU calculation in [12] predicts a ratio E(4+1 )/E(2 + 1 ) = 2.05, consistent with a vibrational character. The PES shows a soft behaviour of the ground state vs deformation, so that the measurement of E(4+1 ) would provide information on the quadrupolar deformation of 50Ar. The deformed shallow minimum at low energy seen in Fig. 4.17 impacts the physical states 0+1 and 0 + 2 in also present in the shell model calculation, as shown in Tab.4.9. A stronger mixing 0p0h versus 2p2h is found in the wave functions of the 0+2 and 0 + 2 states in 50Ar compared to 52Ca. The resulting 0+2 state is predicted at much smaller excitation energy. This discussion is consistent with a fast reduction of the magic number N = 32 due to collective effects when protons are removed from 52Ca. It should be investigated in the further N = 32 isotone 48S, for which the proton valence orbital from π1s1/2 origin has a very different dependence versus quadrupolar deformation. 92 Results and Discussion Table 4.9: 0p0h and 2p2h components of the wave function for the 0+ states for 52Ca and 50Ar, obtained in the shell-model calculation using the SDPF-MU interaction. Energy 0p0h 2p2h (keV) % % 52Ca 0+1 gs 87.1 12.4 52Ca 0+2 4007 7.3 88.9 50Ar 0+1 gs 54.0 28.8 50Ar 0+2 2412 25.4 58.5 Ab initio calculations were performed in the context of the valence-space in-medium simi- larity renormalization group (VS-IMSRG) [67, 71] and the self-consistent Gorkov Green’s func- tion (GGF) [65, 89] approaches. Results are displayed respectively in Tab. 4.10 and 4.11. Table 4.10: Spin Jπ, excitation energies E∗ and spectroscopic factors ||⟨A+1X - n ∥AX⟩||2 for levels of 49Ar and 51Ca obtained in shell-model calculation with the SDPF-MUs interaction and IMSRG calculation with the 1.8/2.0 (EM) interaction. SM (SDPF-MUs) IMSRG (1.8/2.0 EM) 49Ar 51Ca 49Ar 51Ca State E∗ C2S E∗ C2S E∗ C2S E∗ C2S (keV) (keV) (keV) (keV) 3/2−1 gs 1.84 gs 3.57 155 1.37 gs 3.69 1/2−1 95 0.61 1552 0.14 gs 0.94 2025 0.09 5/2−1 924 <0.01 2298 <0.01 1150 <0.01 2068 0.00 7/2−1 983 2.56 3374 7.56 1852 1.89 4197 7.59 3/2−2 1197 0.68 2893 0.09 1439 0.91 3081 0.06 7/2−2 2317 0.05 4622 0.07 3271 0.09 The predictions of the VS-IMSRG calculation with the 1.8/2.0 EM interaction are similar to the SDPF-MU calculation for the low-lying states especially for 52Ca, except for i) a less compressed level scheme in 50Ar, as seen in Fig. 4.15; ii) a spin inversion for the ground state 1/2−1 instead of 3/2 − 1 , although the two states are also very close to each other in excitation energy. The spin inversion is also observed in the IMSRG calculations performed with other interactions, such as NNLO or N3LO interactions. 93 Results and Discussion Table 4.11: Spin Jπ, excitation energies E∗ and spectroscopic factors ||⟨A+1X - n ∥AX⟩||2 for levels of 49Ar and 51Ca obtained in GGF calculations with the NN + 3N(lnl) and NNLOsat. interactions. GGF (NN + 3N(lnl)) GGF (NNLOsat) 49Ar 51Ca 49Ar 51Ca State E∗ C2S E∗ C2S E∗ C2S E∗ C2S (keV) (keV) (keV) (keV) 3/2− gs 2.68 gs 3.36 gs 3.17 gs 3.34 1/2− 180 0.22 1610 <0.01 330 0.02 3060 0.03 5/2− 440 0.10 1628 <0.01 1890 <0.01 2980 <0.01 7/2− 3590 0.10 3020 0.02 2480 0.48 4152 0.02 3/2− 3620 0.01 1740 0.02 2690 0.02 3010 0.03 7/2− 4390 3.64 3720 6.75 3980 5.59 4660 6.05 GGF calculations were performed with two different interactions, namely NNLOsat [91] and NN + 3N(lnl) [89]. The calculated energies of the lowest-lying states in 49Ar (51Ca) and the associated spectroscopic factors for one-neutron removal from 50Ar (52Ca) are displayed3 in Tab. 4.11. As for the other theoretical approaches, the properties of the low-lying states in 51Ca suggest that 52Ca is a closed-shell nucleus. First, the 3/2− ground state has a large C2S value, similar to previous calculations. Second, the first 1/2− state appears at high excitation energy, especially with NNLOsat, and has very small C2S values. Furthermore, the main 7/2− fragment is also found at high excitation energy and has a C2S value that is close to the 2j+1 limit for both interactions. In contrast, corresponding results for 49Ar point to the emergence of a qualitatively different picture. The C2S values for the main 3/2− and 7/2− states decrease, although the change is quantitatively different with the two interactions. While the reduction is significant (around 30%) for NN + 3N(lnl), only a ∼ 5 − 10% decrease is observed for NNLOsat. This is presumably due to the more perturbative character of the former, which favours fragmentation of the spectral function at the current level of approximation in the GGF approach. Contrary to valence-space calculations, here the energy of the 7/2− state changes by only a few hundred keV and a component below 1 MeV is not observed for this angular momentum. Importantly, together with the reduction of spectroscopic factors, a low-energy 1/2− state appears with both interactions. In addition, the energy of the first 5/2− is lowered by more than 1 MeV. These features all signal an increase of collectivity in 49Ar. All together, the changes observed in GGF calculations when going from 51Ca to 49Ar are consistent with shell model results and point to the deterioration of the N = 32 gap in argon. The quantitative differences emerging in the two sets of results are very likely due to the missing collective degrees of freedom (i.e., lack of deformation and low truncation in the particle-hole expansion) in the GGF approach. 3Only states with C2S > 0.01 are reported here. 94 Conclusion and future perspectives In this work, the first in-beam gamma spectroscopy of 49Cl and 49Ar were investigated via one nucleon removal from a 50Ar beam at 217 MeV per nucleon at the mid of the target. During the experiment, the secondary beam of 50Ar were produced by in-flight fission of 70Zn-beam at the Radioactive Isotope Beam Factory, RIKEN Nishina Center, Japan. A liquid-hydrogen target coupled to a vertex tracker in the MINOS device was surrounded by the DALI2+ scintillator array. This array was used to measure γ-ray from the reaction. Due to the large acceptance of the SAMURAI spectrometer, the momentum distribution of one-nucleon removal reactions was also analyzed. In addition, inclusive cross-sections, as well as cross-sections for the excitation of a particular state, were performed. These results have been compared with the benchmark of other neutron-rich nuclei in the same experiment. The method and results of the PID of the SEASTAR experimental data are published in the article "Particle identification for Z = 25 – 28 exotic nuclei from SEASTAR experimental data" in Nuclear Science and Technology (ISSN 1810-5408), Vol. 7, No. 2, pp. 08-15 (2017). Analysis of experimental data revealed clear signatures of the restoration of the natural ordering of proton-hole states in 49Cl. In a simple single particle shell model framework, the proton shell in the 50Ar isotope occupies the sd shell valence with a mix configuration πs1/2 and πd1/2 orbitals. According to our analysis results, the ground state of 49Cl has the unpaired proton populated the πd3/2 orbital. Therefore, it is assigned to spin parity of 3/2+. The first excited state is found at 350 keV and transferred from 1/2+ to 3/2+gs. A weak transition at 970 keV includes direct and cascade transitions, which is an unknown spin parity. Spin-parity of 5/2+ assigned to the population to the excited state at 1515 keV. This standard ordering for 3/2+ versus 1/2+ states is similar to the recently observed 51K, while spin inversion is still under debate for less neutron-rich chlorine isotopes. Combining with other results of my analysis for 47Cl from multi-nucleon removal reactions, the investigation results are published in the article "Investigation of the ground-state spin inversion in the neutron-rich 47,49Cl isotopes" in Physics Review C 104, 044331 (2021). For neutron configuration in the 50Ar isotope, the valence orbitals are νp1/2 and νp3/2. In our experimental data combine theoretical calculation, the 49Ar ground state has spin parity 95 Results and Discussion of 3/2− that populated the νp3/2 orbital. The first excitation state at 198 keV is assigned to spin parity of 1/2−. The strength of 7/2− state at 1340 keV is very higher than the 5/2− state at 1050 keV or 1464 keV. The energy at 1466 keV was assigned at the second 3/2−. We found states at low excitation energy, possibly 1/2−1 and 7/2 − 1 , not compatible with a spherical 50Ar and closed orbitals at N = 32 as observed for 52Ca. The spectroscopic factors of the low energy spectroscopy of 49Ar are consistent with not only theoretical calculations but also collective effects when two protons are removed from 52Ca. This result has been summarized in a draft, which will be published in the Physics Review C journal in the future. To finish, it would also be interesting to study the structure of 47Cl, near N = 28. Al- though the structure of 47Cl was investigated in my article (Physics Review C 104, 044331 (2021)). 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