Shell model description of $^{119-126}$Sn isotopes and different seniority states

In the present work recently available experimental data up to high-spin states of $^{119-126}$Sn isotopes with different seniority ($v$), including $v$ = 4, 5, 6, and 7 have been interpreted with shell model, by performing full-fledged shell model calculations in the 50-82 valence shell composed of $1g_{7/2}$, $2d_{5/2}$, $1h_{11/2}$, $3s_{1/2}$, and $2d_{3/2}$ orbitals. The results have been compared with the available experimental data. These states are described in terms of broken neutron pairs occupying the $h{11/2}$ orbital. Possible configurations of seniority isomers in these nuclei are discussed. The breaking of three neutron pairs have been responsible for generating high-spin states. The isomeric states $5^-$, $7^-$, $10^+$ and $15^-$ of even Sn isotopes, and isomeric states $19/2^+$, $23/2^+$, $27/2^-$ and $35/2^+$ of odd Sn isotopes, are described in terms of different seniority. For even-Sn isotopes, the isomeric states $5^-$, $7^-$, and $10^+$ are due to seniority $v$ = 2; the isomeric state $15^-$ is due to seniority $v$ = 4, and in the case of odd-Sn isotopes, the isomeric states $19/2^+$, $23/2^+$, and $27/2^-$ are due to seniority $v$ = 3, and the isomeric state $35/2^+$ in $^{123}$Sn is due to seniority $v$ = 5.


Introduction
The Sn region is one of the important regions, where many experimental and theoretical studies such as Gamow-Teller decay of the doubly magic nucleus 100 Sn [1], measurement of electromagnetic properties of different excited states [2], upcoming measurements for definite spin assignments [3], population of high-spin states [4] and ab initio study of lighter Sn isotopes [5] are going on. Recent studies report lowering of the νg 7/2 orbital in comparison to the νd 5/2 for the 101 Sn. It is possible with direct spin assignments, together with magnetic moment measurements, to probe the wave function of the ground states of the 101−107 Sn isotopes. This may help accurately determine the ordering of the νd 5/2 − νg 7/2 orbitals. Several new experimental findings focused on the measurement of B(E2) strengths pattern in the Sn isotopes. For the heavier 126,128,130 Sn isotopes smooth decreases of the B(E2 ↑) values towards the major shell closure are shown and these are well described by the shell model. For the lighter Sn isotopes it is very difficult to explain measured B(E2 ↑) values with standard effective charges. Thus, there are two possibilities: either increasing the effective charges for the lighter Sn isotopes by taking robust 100 Sn core, or taking 80 Zr/ 88 Sr core with the standart effective charges. Also experimentally measured B(E2 ↑) values for the nuclei beyond Sn in gdsh shell are highly desired. The parabolic pattern for the transition strengths of the Sn isotopes between 50 − 82 major shell have been reported in the generalized seniority scheme [6]. The aim of the present work is systematic study of the high-spin states with the ν = 4, 5, and 6 seniority in 119−126 Sn, using the full-fledged shell model in 50 − 82 model space for description of the recent available experimental data. At the end we will also discuss for the first time large scale shell model results of B(E2 ↑) values in full gdsh shell without any truncation. This will help to understand the previous limitations of theoretical calculations. The 119−126 Sn isotopes have recently been populated as the fragments of binary fission induced by heavy ions [4]. Previously, high-spin states of nuclei below 120 Sn populated by fusion-evaporation reactions induced by heavy-ions were reported in Ref. [7].
The aim of this work is to study several newly populated high-spin states by Astier et al [4] within the framework of shell model. For even 120,122,124,126 Sn isotopes the isomeric states are 10 + , 5 − , 7 − and 15 − , while for odd 119,121,123,125 Sn isotopes the they are 27/2 − , 19/2 + , and 23/2 + . The aim of this experiment was to built high-spin states above the long-lived isomeric states lying around 4.5 MeV. The data for odd neutron-rich 119,121,123,125 Sn are recently reported by Iskra et al. in Ref. [9]. This work is organized as follows: comprehensive comparison of shell-model results and experimental data is given in Section 2. In Section 3 comparison of the calculated transition probabilities and some predicted values of quadrupole moments for isomeric states are given. Finally, concluding remarks are drawn in Section 4.

Analysis of spectra of even isotopes of Sn
Since 100 Sn core is used in this work, neutron excitations are important among the 1g 7/2 , 2d 5/2 , 2d 3/2 , 3s 1/2 and 1h 11/2 orbitals for the 119−126 Sn isotopes. The valence neutrons contribute in the structure of these nuclei because of the Z = 50 shell closure. In this section we perform shell model calculations for the even-even isotopes in the 50-82 shell, in order to describe the positive and negative parity levels of these nuclei. The even-even isotopes of Sn are discussed first. The odd isotopes 119,121,123,125 Sn have been studied within shell model in Ref. [9]. We sketch the results for the odd isotopes for the completeness and comparison in subsection 2.2, including some more recently measured states.

120 Sn:
Comparison of the calculated spectrum of 120 Sn with the experimental data is shown in figure 1. The calculated 2 + and 4 + levels are 67 keV lower and only 10 keV higher, respectively than those in the experiment. Then, there are gaps both in the experiment and calculation gaps (490 keV and 400 keV, respectively) between the 4 + and 6 + levels, calculated one being less. In the calculation, 6 + , 8 + and 10 + triple of the levels is slightly lower and more compressed than the experiment one: the differences between the experimental 6 + and 8 + , and 8 + and 10 + are 152 keV and 66 keV, respectively, while the calculated values are 87 keV and 67 keV, respectively. All other calculated levels, except 18 + , which is higher than in the experiment, are lower than those in the experiment. The experimental differences between the pair of levels 10 + and 12 + , 12 +  1127 keV, 454 keV, 639 keV, while corresponding calculated differences are 993 keV, 1054 keV, 588 keV, 1068 keV, i.e. trend in the differences are similar to the experimental one, except the last one, which is large in the calculation. We will see in the next subsections the agreement of the calculated differences with those of the experimental ones gradually improve as we move towards the heavier isotopes.
For the negative parity levels, the 5 − and 7 − levels are 118 keV and 95 keV higher, respectively, as compared to those of the experimental ones. The calculated 9 − level is 537 keV lower than in the experiment.

122 Sn:
Comparison of the calculated values with the experimental data is shown in figure 2. Comparing figures 1 and 2 one can see that the positive parity spectrum of the 122 Sn is very similar to that of 120 Sn. As is visually seen, for all respective positive and negative parity levels the agreement between the calculated and experimental values are improved as compared to that of 120 Sn. This can be seen especially in the differences of the energy levels of two neighboring levels.
The 2 + level is predicted 46 keV lower and 4 + level is only 26 keV higher than the experimental values, i.e. the values of the both energy levels are decreased with respect to the ground state as compared to those of 120 Sn. The values of the respective experimental and calculated energy gaps between 4 + and 6 + are 412 keV and 345 keV and in better agreement than in case of 120 Sn. The 6 + , 8 + and 10 + triplet of the levels in the calculation is still slightly lower and more compressed than in the experiment: the differences in the values of the experimental 6 + and 8 + , and 8 + and 10 + levels are 136 and 76 keV, respectively, while the calculated values are 85 and 63 keV, respectively. All other calculated levels, including 18 + , which was higher than in the experiment for 120 Sn, are lower than those in the experiment. The experimental differences between the pair of levels 10 + and 12 of 120 Sn. For the negative parity levels, the 5 − and 7 − levels are 85 and 73 keV higher, respectively, as compared to those of the experimental ones. For these two levels the calculations are better than in 120 Sn case. The calculated 9 − level is 299 keV lower than in the experiment. As compared to 120 Sn, in the experiment 11 − , 13 − and 15 − levels are still almost equidistant, which are 264 keV, 242 keV far from each other. The values of the calculated 13 − and 15 − levels are very close to the experimental ones being 25 keV and 41 keV larger, respectively, while the level 11/2 − is 36 keV lower than in the experiment. The experimental equidistant picture of triple of these levels is much better described than for 120 Sn. The 17 − level is 431 keV lower than in the experiment. The 6651 keV experimental level, for which there is no spin assignment, is close to the 19/2 − calculated level, being 129 keV higher than the calculated value.
Overall agreement of the calculated levels is in much better agreement with experimental data as compared to 120 Sn discussed in subsection 2.1.1.

124 Sn:
Comparison of the calculated values with the experimental data for the 124 Sn is shown in figure 3.
Comparing figures 1, 2 and 3 shows that the energies of the positive and negative parity levels of all the experimental levels with respect to the ground state ones are decreased as compared to those of the even isotopes discussed in the previous subsections 2.1.1 and 2.1.2. As compared to 122 Sn, in the calculation only the energy of 2 + level is increased to 3 keV and all other energies of the levels are decreased with respect to ground state like in the experiment The 2 + and 4 + levels are only 35 keV and 15 keV lower, respectively, than the experimental ones which shows better agreement as compared to that of 120,122 Sn. The values of the respective experimental and calculated energy gaps between the 4 + and 6 + levels are 352 keV and 339 keV. They are also in better agreement with the experiment than for 120,122 Sn. The 6 + , 8 + and 10 + triplet of the levels in the calculation is still slightly lower and more compressed than in the experiment: differences in the values of the experimental 6 + and 8 + , and 8 + and 10 + levels are 124 and 79 keV, respectively, while the calculated values of these differences are 89 and 52 keV, respectively. The experimental difference between the 6 + and 8 + levels is decreased while, the difference between the 8 + and 10 + levels is increased as compared to that of 122 Sn. Reverse trend is seen in the differences of the calculated levels: the difference between the 6 + and 8 + level is increased, the difference between 8 + and 10 + is decreased as compared to that of 122 Sn.  experiment, are lower than those in the experiment. The 18 + level was higher than in the experiment for 120 Sn and slightly lower for 122 Sn. The experimental differences between the pair of levels 10 + and 12 + , 12 + and 14 + , 14 + and 16 + , 16 + and 18 + are 1046 keV, 996 keV, 490 keV, 763 keV, while corresponding calculated differences are 943 keV, 1014 keV, 521 keV, 930 keV, i.e. trend in the differences is similar to the experimental one. Very close to the calculated 20 + level there is experimental level with 6976 keV energy which is 61 keV lower than in the calculated one.
For the negative parity levels, the 5 − and 7 − levels are 56 and 62 keV higher, respectively, as compared to those of the experimental ones. For these two levels the calculations are clearly better than 120,122 Sn cases. As compared to 120 Sn, in the experiment, the 11 − , 13 − and 15 − levels are still almost equidistant, which are 252, 229 keV far from each other. Now the energy values of the calculated 13 − and 15 − levels are very close to the experimental ones being 35 keV and 25 keV larger, respectively, while the calculated energy value of the 11 − level is now 124 keV larger than the experiment one (for the previous two nuclei they were less). The experimental equidistant picture of triple of these levels is not better described than for 122 Sn, because of the 11 − shifted to larger value with respect to its experimental counterpart as compared to 122 Sn. The 17 − level is 285 keV lower than in the experiment.
Overall agreement of the calculated positive and negative parity levels are in better agreement with the experimental data as compared to 120,122 Sn discussed in subsections 2.1.1 and 2.1.2.

126 Sn:
Comparison of the calculated values with the experimental data for 126 Sn is shown in figure 4.
As is seen from figure 4 adding two more neutrons to 124 Sn leads to the increasing back both experimental and calculated energies of the 2 + state of 126 Sn with respect to the ground state energy as compared to those of 124 Sn. Energies of all other positive and negative parity levels are decreased with respect to the ground state energy as compared to those of 124 Sn.
The shell model calculation predicts energies of the 2 + and 4 + levels only 17 keV and 38 keV lower, respectively, than the experimental ones. This shows slightly better agreement as compared to that for 124 Sn. The values of the gaps between 4 + and 6 + are 324 and 333 keV in the experiment and calculation, respectively. They are also in better agreement with the experiment than in 120,122,124 Sn. The 6 + , 8 + and 10 + triple of the levels in the calculation is still slightly lower and more compressed than in the experiment: differences in the values of the experimental 6 + and 8 + , and 8 + and 10 + levels are 115  keV, respectively, while the calculated values are 132 keV and 30 keV, respectively. As is seen from these differences the experimental triplet of the levels is more compressed as compared to that of 122 Sn. The difference in the calculated values of 6 + and 8 + levels is increased instead, which leads to the better agreement of these levels. However, the difference in the calculated values of 8 + and 10 + is decreased too much as compared to the experimental one. All other calculated levels, are lower than those in the experiment, however the pattern looks like stable and very much like to the experimental one.
The experimental and calculated negative parity patterns are exactly the same. The calculated values of the negative parity levels are in excellent agreement with the experimental ones for this nucleus. For the 6257 keV level no spin assigned yet. The calculated level is only 21 keV higher. The spin predicted by shell model calculation for this level is 19 − .
In general description of the whole spectra of even Sn isotopes are gradually improving as we move from A=120 to A=126.

Analysis of spectra of odd isotopes of Sn
For the odd isotopes of Sn unpaired neutron interchanges the position of the positive and negative parity bands as compared to the even-even isotopes. Calculation gives 11/2 − as the ground state for the all odd isotopes of Sn considered here. For some isotopes, in the experiment, the 11/2 − level is slightly higher than the ground states of these nuclei.

119 Sn:
For the 119 Sn in Fig.5 we have presented the calculation up to 35/2 − , taking into account the experimental levels to which no spin and parity are assigned yet. The calculation gives 11/2 − as the ground state of 119 Sn, while in the experiment 11/2 − is the excited stated with 89 keV energy.
The calculated values of the 15/2 − , 19/2 − , 23/2 − , 27/2 − are 265 keV, 310 keV, 240 keV, 285 keV are lower than their experimental counterparts, though the calculated pattern of the 119 Sn spectrum is very similar to the experimental one. In the experiment there is the level with the 3978 keV for which there is no spin assignment yet. This level is 129 keV higher than the calculated one. Then there is another 31/2 − for which spin is assigned tentatively. This level is 431 keV higher than the calculated one. Finally, there is yet another spin not assigned level, which 216 keV higher than 35/2 − level.
There is only one measured positive parity level which is 81 keV lower than the calculated one.

121 Sn:
The spectrum of 121 Sn is given in Fig. 6. The calculation gives 11/2 − as the ground state of 121 Sn, while in the experiment the energy of this level is 6 keV.
The There are two calculated 39/2 + 1 and 39/2 + 2 energy levels near to the measured 5611 keV level, for which there is no spin assignment yet. The first of them is 259 keV lower and the second is 124 keV higher than this level. The experimental (39/2 + ) level at 6314 keV is 962 keV and 579 keV higher than 39/2 1 and 35/2 + 2 , respectively.

123 Sn:
The spectrum of 123 Sn is given in Fig. 7 Fig. 7. Comparison of experimental and calculated excitation spectra for 123 Sn using SN100PN interaction. experimental one, while the similarity was up to 27/2 − and 31/2 − for 119,121 Sn, respectively. The calculated values of 15/2 − , 19/2 − , 23/2 − , 27/2 − , 31/2 − are 111 keV, 252 keV, 44 keV, 98 keV, 206 keV lower than their experimental counterparts. From these differences it is also seen that agreement of the calculated values of the energy levels of 123 Sn are much better than those of 119,121 Sn. Unlike 119,121 Sn case, the second 31/2 − 2 appears in right order with the experiment which is 271 keV higher than measured (31/2 − ) level. It is seen that starting from these level three pairs of levels 31/2 − 1 and 31/2 − 2 , 35/2 − 1 and 35/2 − 2 and the pair with 5478 keV and 5520 keV energies come in the experiment, where the energies of the pairs are very close and they are separated by the large energy gaps. This pattern is seen also in the calculation, however the energies of two levels in the pair of levels are separated by larger amount. The possibility of the last two level spins, for which no spin have been assigned yet, being 39/2 − 1 and 39/2 − 2 are very high, according to the shell model calculation.
More experimental data are available for the positive parity levels of 123 Sn. The calculated pattern is similar to experimental one still up to 35/2 + . The first two positive parity levels are 117 keV, 101 keV higher as compared to the experimental ones. There is the calculated 39/2 1 energy level near to the measured 5644 keV level, for which there is no spin assignment yet. The calculated level is 405 keV lower than this level. Another experimental spin not assigned level is at 6231 keV. The calculated 41/2 + and 45/2 + levels are 23 keV lower and 136 keV higher than this level.
All negative and positive parity levels of 123 Sn are better described as compared to 119,121 Sn by the shell model calculation.

125 Sn:
As is seen from figures 3 both calculated and experimental ground states are 11/2 − as it was for 123 Sn. For the 119,121 Sn 11/2 − experimental levels energy values were 89 keV and 6 keV, respectively. Also, all respective positive and negative parity excited states energies are lower both in the experiment and calculation with respect to ground state as compared to 119,121 Sn isotopes. There are two positive parity levels 19/2 + and 23/2 + measured for this nucleus.
The calculate values are 79 keV and 74 keV higher, respectively, than the experimental ones. This is better agreement of the experimental and calculated values of these two levels as compared to that of 119,121,123 Sn.

Configuration of the isomeric states
Since yrast states of lightest Sn isotopes may be formed by collective states, identifying the ν(h 11/2 ) n component of the neutron configuration was impossible for them [17]. 119−126 Sn isotopes, which contain more than 68 neutrons, are good for studying high spin states since they do contain ν(h 11/2 ) n with ν = 4, 5, 6 and the high spin states cannot be formed only by νs 1/2 and νd 3/2 orbitals themselves.
In Ref. [17], the 119−126 Sn isotopes have been produced as fragments of binary fission induced by heavy ions. New results were reported for these nuclei. Among them, isomeric states have been established from the delayed coincidences between fission fragment detectors and the gamma array. All the observed states treated in terms of broken neutron pairs occupying the ν(h 11/2 ) orbital.
Configuration of the isomeric 10 + , 5 − , 7 − , 15 − and 19 − states of even isotopes of Sn and 27/2 − , 19/2 + and 23/2 + odd isotopes, emerging from the current shell model calculation, are given separately in Table 1. The seniorities given in this Table are proposed in [17]. As is seen from this Table all configurations, except that of 19 − of 122 Sn, for which it is g 7 7/2 h 7 11/2 , are in accord with [17]. The 10 + states of all even isotopes and 27/2 − states of all odd isotopes are formed by breaking pairs in pure ν(h 11/2 ) orbital with ν = 2 and ν = 3, respectively. The d 3/2 and s 1/2 orbitals also participate in the formation of other isomeric states of the Sn isotopes.

Transition probabilities and quadrupole moments
The comparison of the transition probabilities with the experiment for B(E2 ↑ ; 0 + → 2 + ) transitions are given in Fig. 9. In the present work we have used three different set of effective charges for B(E2 ↑; 0 + → 2 + ) calculations. The results with e n =1.2e are more close to experimental data for lighter Sn isotopes, while for the heavier isotopes the value of effective charge e n =1.0e is more suitable [2], this is because we have not considered excitation across    100 Sn core in our model space, thus larger value of neutron effective charge is needed. In the Table 3, we have also shown electromagnetic properties from isomeric states with seniority ν = 4, 5, and 6 for 119−126 Sn isotopes, which have larger values than expected from isomeric states. These calculated values of E2 transition probabilities are important for future experiments. For the odd isotopes of Sn there some experimental data. Comparison of the calculated B(E2) transion probabilities with these experimental date are given in Table 4.

Summary
In the present work we have performed full-fledged shell model calculations for 119−125 Sn isotopes using SN100PN effective interaction for recently populated high-spin states. Present shell model results show good agreement with the experimental data. The high-spin states of the 119−125 Sn isotopes are very well described by the shell model. The breaking of three neutron pairs have been responsible for generating high-spin states. As expected the structure of these isomers are due to ν = 4, 5, and 6. We have also reported B(E2) and quadrupole moments.