Systematic shell-model study of Rn isotopes with $A=$ 207 to 216 and isomeric states

Fig. 1 shows the shell-model energy spectrum of ²⁰⁸Rn in comparison with the experimental data, where all the observed levels up to 5.2 MeV excitation energy are reported. In the calculated low-energy spectrum, we get a much higher level density compared to experimental data. We report only those yrast and non-yrast shell-model states corresponding to the experimental levels. The shell-model is giving close energy values for energy levels up to 2.5 MeV for all the yrast states. In our calculation, the experimental ${(4,5,6)^{+}}$ level at 1.580 MeV can be associated with either the $4_{3}^{+}$ or $6_{1}^{+}$ state. The calculated $5_{1}^{+}$ state is produced 300 keV higher than the experimental state at 1.580 MeV, and calculated $5_{2}^{+}$ is produced 500 keV higher than the experimental state at 1.658 MeV ( ${4}^{+}$, ${5}^{+}$). Therefore, we propose the assignment of calculated $5_{1}^{+}$ state to the closest experimental level at 1.658 MeV, excluding ${5}^{+}$ from the 1.580 MeV experimental state. For the negative parity state, the experimental ${(5^{-},6^{+})}$ level at 2.18 MeV can be associated with the calculated $5_{1}^{-}$ at 2.308 MeV.

The shell-model energy spectrum of ²¹⁰Rn in comparison with the experimental data is shown in Fig. 2, where all the observed levels up to 5.2 MeV excitation energy are reported. Experimentally, the relative energies between the ground state and many excited states are unknown in ²¹⁰Rn. The $(8_{1}^{+})$ state of these excited states is the lowest and observed at 1.665+x MeV. In this figure, these states are shown without any assumption of x ( i.e., x=0). The shell-model reproduces the energy levels well up to 4 MeV for both positive and negative parity states. Most of the levels in the case of ²¹⁰Rn are tentative. Therefore, a one-to-one correspondence has been estimated with the experimental data. The second excited state in the calculation is produced as ${2}^{+}$, 90 keV lower than the experimental energy value, which is an experimentally tentative $({4})^{+}$ state. The calculated $4_{1}^{+}$ state is produced at 1.593 MeV close to the tentative experimental level $({4})^{+}$ at 1.545 MeV. The calculated $6_{1}^{+}$, $8_{1}^{+}$ and $8_{2}^{+}$ are confirming the tentative experimental states with a difference of a few keV. The last two calculated negative parity states, $18_{1}^{-}$ and $19_{1}^{-}$ are lower in energies with respect to the experimental data by 436 keV and 572 keV, respectively. These levels might arise from core-excitations, and in some cases, significant admixtures of configurations without and with core-excitation beyond $Z=82$ and $N=126$ shell closure.

The shell-model energy spectrum of ²¹²Rn in comparison with the experimental data is shown in Fig. 3, where all the observed levels up to 6.2 MeV excitation energy are reported. In this work, we have reproduced all the new states identified in 212Rn2 , with a satisfactory quantitative agreement between our results and experimental data. A direct one-to-one correspondence between states can be established up to 2.300 MeV in positive parity states. Here we observe similar trend in ²⁰⁸Rn, ²¹⁰Rn and ²¹²Rn isotopes: a small energy gap between the ${6_{1}}^{+}$ and ${8_{1}}^{+}$ states, and a large gap between $8_{1}^{+}$ and $8_{2}^{+}$ states are featured. Our calculation confirms the tentative experimental $6_{3}^{+}$ and $13_{1}^{+}$ states with a difference of only a few keV. The calculated $6_{5}^{+}$ and $12_{1}^{+}$ states are 111 keV and 144 keV higher than the experimental data, respectively. This state could be ${6_{5}}^{+}$ state. For high-lying states, the compression becomes notable, except for the calculated $20_{2}^{+}$ state and confirms this tentative state. The $18_{1}^{-}$ state is 662 keV lower, and $19_{1}^{-}$ state is 1.065 MeV higher than the experimental data. The description of these high-lying states requires the core-excitation and maybe some mixing between the single-particle states and core-excitation above $Z=82$. and $N=126$ shell closure. In our calculation, the collective $3_{1}^{-}$ state at 2.856 MeV is 735 keV higher than the proposed experimental value, and arising from the configuration $\pi(h_{9/2})^{3}(i_{13/2})$. The single-particle orbitals with $\Delta l$ = 3, $\pi{f_{7/2}}-\pi{i_{13/2}}$ and $\nu{g_{9/2}}-\nu{j_{15/2}}$ in our model space and core-excitations across the two shell gaps at $Z=82$ and $N=126$ are responsible for the structure of this octupole vibration. The KHPE (results as reported in Ref. 212Rn2 ) and KHH7B interactions are giving almost similar results because we have only four valence protons beyond Z=82 and N=126. This is reflected from the similar wave functions we have obtained from these two interactions.

The shell-model energy spectrum of ²¹⁴Rn in comparison with the experimental data is shown in Fig. 4 with the two interactions KHPE and KHH7B, where all the observed levels up to 4.6 MeV excitation energy are reported. In ²¹⁴Rn, many of the energy levels are unidentified in terms of the spin-parity. These levels are not included in our figure. However, we have tried to interpret the first two unidentified levels. The shell-model results from the KHPE interaction are slightly better than the KHH7B interaction, as shown in Fig. 4. Similar to other even Rn isotopes, a small energy gap is observed between the $6_{1}^{+}$ and $8_{1}^{+}$ states, and a large gap between $8_{1}^{+}$ and $8_{2}^{+}$ states. The spin-parities of the experimental states at 1.332 MeV and 1.800 MeV are not assigned. In the experimental data, the first state decays to the $2_{1}^{+}$ state at 0.695 MeV, and the second state decays to the first unidentified state at 1.332 MeV. The calculated $2_{2}^{+}$ state is at 1.558 MeV and 1.509 MeV from the KHPE and KHH7B interaction, respectively. The calculated $6_{2}^{+}$ state is at 1.883 MeV and 1.842 MeV from the KHPE and KHH7B interaction, respectively. From the comparison with neighboring nuclei ²¹²Rn, ²¹⁶Rn from both our calculation and experimental data, and considering our results for the above two mentioned states, we suggest the spin-parities of the experimental state at 1.332 MeV to be $2_{2}^{+}$ and state at 1.800 MeV to be $6_{2}^{+}$. With both interactions the calculated $12_{1}^{-}$ state is very close to the tentative experimental state with only few keV differences. The calculated negative parity states above 3.0 MeV are highly compressed. The ²¹⁴Rn isotope is above the $Z=82$ and $N=126$ shell-closure with 4 valence protons and 2 valence neutrons. Therefore, it is important to consider sufficient orbitals around the shell-closure for core-excitation and required configuration mixing. Due to limited computational facilities, we are not able to include all orbitals without truncation.

The shell-model energy spectrum of ²¹⁶Rn in comparison with the experimental data is shown in Fig. 5 with two interactions KHPE and KHH7B up to 3.8 MeV excitation energy. Previously observed feature for a small energy gap between the $6_{1}^{+}$ and $8_{1}^{+}$ states, and a large gap between $8_{1}^{+}$ and $8_{2}^{+}$ states, are not observed here as prominently as in the other even Rn isotopes. However, as observed in the experiment, this pattern has vanished in ²¹⁶Rn due to the enhancement of quadrupole collectivity. As we move from ²¹⁴Rn to ²¹⁶Rn, the number of valence particles increases beyond $Z=82$ and $N=126$ shell closure, thus we need to include more orbitals in the model space apart from the core-excitation. Therefore, both the interactions are not being able to reproduce the levels in the whole energy range. Due to the model space requirement for any specific state, some levels are in good agreement with experimental data, while others could not be reproduced well. The spectrum of ²¹⁶Rn looks like vibrational one because low-lying states from $0^{+}$ - $8^{+}$ are equally spaced . In the case of vibrational spectra, the quadrupole moment should be smaller. In the simplest version of the vibrational model, the quadrupole moment of the $2^{+}$ state is predicted to be zero. Our shell model results give an equal spacing of low-lying states, although results for quadrupole-moments are large. However, there is no experimental data available for the quadrupole moment.

The seniority, configurations and half-life corresponding to different isomeric states for Rn isotopes as reported in Table 1 and Table 4 are discussed below. In spherical nuclei close to the magic number, the breaking of high-$j$ nucleon pairs produces isomeric states. The Rn isotopes considered in our calculations are spherical, thus it is possible to explain the isomers in terms of seniority quantum number. Seniority $\nu$ is the number of particles not in pairs coupled to angular momentum $J$ = 0. With shell-model, it is possible to extract information about seniority from the configurations. For the seniority isomer, the decay is hindered because of the same seniority of the initial and final states. Seniority isomers arise because $E2$ decay is hindered between the same seniority of the initial and final states. In the case of ^208,210,212Rn isotopes, $8^{+}$ state is the seniority isomer with a very small $B(E2$; $8^{+}$ $\rightarrow$ $6^{+}$) value. This is because of the same seniority $v=2$ for $8^{+}$ and $6^{+}$, which is coming from $\pi(h_{9/2})^{4}$ configuration.

In ²⁰⁸Rn, ²¹⁰Rn and ²¹²Rn the $8_{1}^{+}$ isomeric state is coming from $\pi(h_{9/2})^{4}$ configuration, in increasing probability with mass number. The $8_{1}^{+}$ isomeric state is formed by purely $h_{9/2}$ orbital with one pair breaking, thus $\nu$ = 2. The half-life of this isomeric state is also reproduced very close to the experimental data with $B(E2)$ transition for all three isotopes. The $10_{1}^{-}$ and $16_{1}^{-}$ states are coming from the same configuration $\pi(h_{9/2})^{3}(i_{13/2})$, and formed by one proton in $h_{9/2}$, $i_{13/2}$ each coupled with one pair in $h_{9/2}$ orbital, hence the seniority $\nu$ = 2. These two states are generated by $B(M1)$ transition and half-lives for these states are reproduced well with the experimental data. In ²¹⁰Rn, $11_{1}^{-}$[$\pi(h_{9/2})^{3}(i_{13/2})$] isomeric state is formed by one proton in $h_{9/2}$ and $i_{13/2}$ orbitals each, thus $\nu$ = 2. The ${14_{1}}^{+}$[$\pi(h_{9/2})^{3}(f_{7/2})$] isomeric state is formed by one pair breaking in $\pi(h_{9/2})$ orbital, and one proton in $h_{9/2}$ and $f_{7/2}$ orbitals each, thus $\nu$ = 4. The half-life for this state is reproduced in the same order as the experimental data 76(7) ns, with a higher magnitude, as the $B(E2)$ transition value in our calculation is smaller than the experimental data. The ${17_{1}}^{-}$[$\pi(h_{9/2})^{3}(i_{13/2})$] isomeric state is formed by one pair breaking in $\pi(h_{9/2})$ orbital, and one proton in $h_{9/2}$ and $i_{13/2}$ orbitals each, thus $\nu$ = 4. The $20_{1}^{+}$[$\pi(h_{9/2})^{2}(i_{13/2})^{2}$] isomeric state is formed by one pair breaking in $\pi(h_{9/2})$ and $\pi(i_{13/2})$ both the orbitals, thus $\nu$ = 4. In ²¹²Rn, the $6_{1}^{+}$ isomeric state is formed similar to the $8_{1}^{+}$ states in all even isotopes with the same seniority. The half-life of this state is almost the same as the experimental data, as our $B(E2)$ transition value and energy difference are reproduced very well. The $14_{1}^{+}$ and $17_{1}^{-}$ isomeric states are similar as in the ²¹⁰Rn, with almost double probability (98$\%$ and 99.59$\%$, respectively) as these states are pure states. The half-lives for $14_{1}^{+}$ and $17_{1}^{-}$ isomeric states are coming in the same order with higher magnitude because the calculated $B(E2)$ transition value is significantly less than the experimental data for the $14_{1}^{+}$ state, and our energy difference is very small for $17_{1}^{-}$. The $22_{1}^{+}$[$\pi(h_{9/2})^{3}(i_{13/2})\otimes\nu(p_{1/2})^{-1}(g_{9/2})$] isomeric state is formed by one pair breaking in $\pi(h_{9/2})$ orbital, and one proton in $h_{9/2}$ and $i_{13/2}$ orbitals each, and one neutron in $p_{1/2}$ and $g_{9/2}$ orbitals each, thus $\nu$ = 6. The $25_{1}^{-}$[$\pi(h_{9/2})^{2}(i_{13/2})^{2}\otimes\nu(p_{1/2})^{-1}(g_{9/2})$] isomeric state is formed by one pair breaking in $\pi(h_{9/2})$ and $\pi(i_{13/2})$ both the orbitals and one neutron in $p_{1/2}$ and $g_{9/2}$ orbitals each, thus $\nu$ = 6. In ²¹⁴Rn, the ${18_{1}}^{+}$[$\pi(h_{9/2})^{3}(i_{13/2})\otimes\nu(g_{9/2})^{2}$] isomeric state is formed by one pair breaking in $\pi(h_{9/2})$ or $\nu(g_{9/2})$ orbital, and one proton in $h_{9/2}$ and $i_{13/2}$ orbitals each, thus $\nu$ = 4. The $22_{1}^{+}$ isomeric state is arising from the configuration $\pi(h_{9/2})^{3}(i_{13/2})\otimes\nu(g_{9/2})(j_{15/2})$, and formed by one proton in $h_{9/2}$ and $i_{13/2}$ orbitals each, and one neutron in $g_{9/2}$ and $j_{15/2}$ orbitals each, thus $\nu$ = 4. For this isomeric state, seniority reduces as compared to the $22_{1}^{+}$ state in ²¹²Rn because of the role of $j_{15/2}$ orbital. It is important to mention here that to calculate $49/2_{1}^{+}$ and $63/2_{1}^{-}$ for ²¹¹Rn and $22_{1}^{+}$ and $25_{1}^{-}$ for ²¹²Rn, we have opened $2g_{9/2},1i_{11/2}$, and $1j_{15/2}$ neutron orbitals, allowing two-particle excitation in each orbital (with restriction on other orbitals also). Corresponding wavefunctions are shown in Table 1, and the calculations are performed using the NUSHELLX code.

Fig. 6 shows the shell-model energy spectrum of ²⁰⁷Rn in comparison with the experimental data, where all the observed levels up to 2.8 MeV excitation energy are reported. In ²⁰⁷Rn, most of the experimental levels above 1.5 MeV are not assigned with confirmed spin-parities. Therefore, we have calculated the same levels for both parities to see a resemblance between the experimental data and shell-model. For the known experimental states with both spin-parity, the shell-model reproduces the energy spectrum with good agreement. The calculated $3/2_{1}^{-}$ state is twice the energy of the experimental data. On the other hand, following the trend of our calculated spectrum, 100-200 keV range discrepancy in energy is negligible, and the experimental $3/2_{1}^{-}$ state is at a very small energy value of 0.123 MeV. The experimental state at 0.283 MeV can be associated with both $3/2_{2}^{-}$ and $1/2_{1}^{-}$ calculated states. Above 1.4 MeV, all the calculated states are compressed.

We can see from Fig. 6, our calculation can not predict precisely the unconfirmed experimental states, as they are equally likely be assigned in both the parities.

The shell-model energy spectrum of ²⁰⁹Rn in comparison with the experimental data is shown in Fig. 7, where all the observed levels up to 3.6 MeV excitation energy are reported. In ²⁰⁹Rn, the energy spectrum is dense for low-lying states, and the calculated spectrum has reproduced the experimental data with a quantitative agreement. We have calculated all of the low-lying experimental levels, but we have excluded few non-yrast calculated levels from Fig. 7, showing one-to-one feasibly neat correspondence with the experimental data. The calculated states are overpredicted up to 1.174 MeV experimental energy value and under predicted above this energy range for both the parities. In our calculation, the experimental level at 0.511 MeV is reproduced with large discrepancies 415 keV and 945 keV for $5/2_{2}^{-}$ and $3/2_{4}^{-}$ states, respectively, while $1/2_{2}^{-}$ is reproduced with 122 keV energy difference with respect to the experimental data. We suggest that this experimental level can be associated with either the $1/2_{2}^{-}$ or $5/2_{2}^{-}$ calculated state. For the experimental level at 0.867 MeV, the calculated state can be associated with $7/2_{2}^{-}$ or $9/2_{2}^{-}$. Above $\sim$ 1 MeV energy value, all the calculated states are compressed. The calculated $27/2_{1}^{+}$ state is lower in energy by 1 MeV with respect to the experimental data. The compression in energy in our calculated spectrum for high-spin states might be due to the need for core-excitation and significant configuration mixing with the higher orbitals beyond $Z=82$ and $N=126$ shell closure. As the neutron number increases in odd Rn isotopes, the calculated ${{1/2}_{1}}^{-}$, ${{3/2}_{1}}^{-}$ and ${{5/2}_{1}}^{-}$ states show strong single-particle nature with the dominant configuration $\nu(p_{1/2})^{-1}$ ($38\%$), $\nu(p_{3/2})^{-1}$ ($36\%$) and $\nu(f_{5/2})^{-1}$ ($39\%$), respectively.

The shell-model energy spectrum of ²¹¹Rn in comparison with the experimental data is shown in Fig. 8, where all the observed levels up to 3.9 MeV excitation energy are reported. Experimentally, the relative energies between the ground state and many excited states are unknown in ²¹¹Rn. The $(17/2_{1}^{-})$ state of these excited states is the lowest and observed at 1.578+x MeV. In this figure, these states are shown without any assumption of x ( i.e., x=0). Most of the levels in the case of ²¹¹Rn are tentative. For ²¹¹Rn, the shell-model spectrum reproduces very well the experimental data with an establishment of one-to-one correspondence in both positive and negative parity states, with good estimation for the tentative states. The calculated $5/2_{1}^{-}$ state is 147 keV higher than the experimental data, similar to other odd Rn isotopes for the first excited state. The tentative experimental state at 0.833 MeV is reproduced with only 31 keV energy difference in our calculation, suggesting this state as $3/2_{1}^{-}$. In the case of positive parity, the calculated $9/2_{1}^{+}$ is over-predicted, while $31/2_{1}^{+}$ and $35/2_{1}^{+}$ states are lower in energies. This may be due to our model space is not enough for these high-spin states. For the negative parity, the calculated ${{1/2}_{1}}^{-}$, ${{3/2}_{1}}^{-}$ and ${{5/2}_{1}}^{-}$ states show strong single-particle nature with the dominant configuration $\nu(p_{1/2})^{-1}$($48\%$), $\nu(p_{3/2})^{-1}$($42\%$) and $\nu(f_{5/2})^{-1}$($49\%$), respectively. Due to the restrictions of orbitals in the KSHELL code, as mentioned before, the KHH7B interaction could not reproduce high-lying states.

The shell-model energy spectrum of ²¹³Rn in comparison with the experimental data is shown in Fig. 9 with two interactions KHPE and KHH7B, where all the observed levels up to 3.5 MeV excitation energy are reported. Experimentally, the relative energies between the ground state and many excited states are unknown in ²¹³Rn. The $(25/2_{1}^{+})$ state of these excited states is the lowest and observed at 1.664+x MeV. In this figure, these states are shown without any assumption of x ( i.e., x=0). In ²¹³Rn, all levels are tentative, and many of the energy levels are unidentified in terms of the spin-parity. These unidentified levels are not included in the figure. Our calculation is supporting almost all of the tentative experimental states from both interactions. The shell-model results from the KHPE interaction are better than the KHH7B interaction for positive parity states, and the KHH7B interaction is slightly better for the negative parity states, as we can see in Fig. 9. This is due to the different model spaces in these interactions, which are responsible for generating the spectrum for positive or negative parity states. The ²¹³Rn isotope is just above the $Z=82$ and $N=126$ shell closure. Therefore the trend of a higher first excited state, as in other odd Rn isotopes, is not followed here as it is compressed because of collectivity. For the energy value up to 2.121 MeV, KHPE interaction gives overall good results, and KHH7B interaction gives compressed results. Above this energy range, KHPE interaction slightly overpredicts the states, but overall results agree with the with the experimental data, and KHH7B interaction gives highly compressed results. These compressed results from the KHH7B interaction indicate the importance of inclusion of the higher orbitals and core-excitation beyond $Z=82$ and $N=126$ shell closure. The KHPE interaction results for high-lying states are overpredicted because we have not taken into account core-excitation. The states above 3.0 MeV are compressed from the KHPE interaction. The KHH7B interaction has reproduced quite well the spectrum for the negative parity states. We can see from Fig. 9, our calculation can not predict precisely the unconfirmed ${27/2}$ experimental state, as this state is equally likely to be assigned in both the parities.

The shell-model energy spectrum of ²¹⁵Rn in comparison with the experimental data is shown in Fig. 10 with the two interactions KHPE and KHH7B, where all the observed levels up to 2.3 MeV excitation energy are reported. In ²¹⁵Rn, all levels are tentative except for the ground state. The shell-model result is supporting a few of the tentative experimental states from both the interactions. The first excited state is not confirmed experimentally. In our calculation, the ${{11/2}_{1}}^{+}$ state is coming as the first-excited state from both the interaction, which is an experimentally tentative second excited state. The calculated ${{7/2}_{1}}^{+}$ state has 119 keV and 246 keV difference and the ${{9/2}_{1}}^{+}$ state has 689 keV and 563 keV difference with the experimental value from KHPE and KHH7B interaction, respectively. Therefore, for the experimental state at 0.214 MeV, our calculation suggests the state as ${{7/2}_{1}}^{+}$. For the experimental negative parity state at 0.291 MeV, our calculation overpredicts all three tentative spins with large energy value differences from the experimental data from both the interactions.

In all the Rn isotopes, we have observed that most of the low-lying states are collective because they show large configuration mixing. Some of these states are highly admixture of different configurations that have almost the same probability. On the other hand, the high-lying states are less collective, and many states in ²¹²Rn are almost pure.