Goodness of Generalized Seniority in Semimagic Nuclei
Abstract
Symmetry plays an important role in understanding the nuclear structure properties from the rotation of a nucleus to the spin, parity and isospin of nuclear states. This simplifies the complexity of the nuclear problems in one way or the other. Seniority is also a well known quantum number which arises due to the symmetry in the pairing interaction of nuclei. We present empirical as well as theoretical evidences based on decay rates which support the goodness of seniority at higher spins as well as in nrich or, ndeficient nuclei. We find that the generalized seniority governs the identical trends of highspin isomers in different semimagic chains, where different set of nucleon orbitals from different valence spaces are involved.
I Introduction
Understanding the complex nuclear structure of the atomic nucleus is an outstanding problem of nuclear physics. The pairing of nucleons is well known for several decades, which enables one in describing the ground states of eveneven nuclei, and the extra stability of nuclei with an even number of nucleons than the nuclei with an odd number, etc. In atomic physics, pairing was understood in terms of seniority, first introduced by Racah in 1940s Racah1943 (). In 1950s, Racah and Talmi Racah1952 (), and Flowers Flowers1952 () independently introduced the seniority scheme in nuclear physics. In simple terms, seniority may be defined as the number of unpaired nucleons in a given state, generally denoted as . It is now known that remains a good quantum number for states emanating from a purej configuration with j7/2. However, its validity for higherj values has also been suggested.
Arima and Ichimura Arima1966 (), and Talmi Talmi1971 () further extended the seniority picture in singlej to the generalized seniority in multij. Generalized seniority takes care of the presence of multij orbitals in a given state. Semimagic nuclei provide a fertile ground to study the various properties on the basis of the seniority scheme. One of the most interesting results from this scheme is the formation of seniority isomers. Seniority isomers are one of the well known categories of the nuclear isomers, i.e. the longer lived excited states, where the hindrance to their decays have been explained in terms of seniority selection rules Jain2015 (). It has been a general belief that the seniority isomers arise only in decays between the same seniority states due to the vanishing decay probabilities at the midshell in seniority scheme. We have recently used the simple quasispin scheme to obtain the generalized seniority results in a mixed configuration coming from several degenerate orbitals, and applied it to the highspin isomers in Sn isotopes Maheshwari2016 (). Hence, we have found for the first time oddtensor decaying isomers in Sn isotopes, a new category of isomers. We have then used the same scheme to understand the first excited states in Sn isotopes and explained the asymmetric twin parabolas Maheshwari20161 ().
In this paper, we have applied the “generalized seniority formalism” to the highspin nuclear isomers in the semimagic nuclei, particularly Z=50 isotopes, N=82 isotones and Z=82 isotopes. We find that the generalized seniority remains a reasonably good quantum number for a set of states, particularly the highspin isomers in these semimagic nuclei. This further governs the identical behavior of these isomeric states in different semimagic chains, having different set of active orbitals. We start by presenting a few empirical evidences and understanding the reasons behind, in section 2, which makes a good ground for the seniority calculations. Thereafter, we follow these empirical findings with our generalized seniority calculations and results for both even and oddA semimagic nuclei. The overall conclusions of the present work have been presented in the last section.
Ii Experimental evidences
The level schemes of the Sn isotopes have been studied by using the reactions induced by light ions, deep inelastic reactions, or fission fragment studies by several researchers daly80 (); fogelberg81 (); daly86 (); lunardi87 (); broda92 (); mayer94 (); daly95 (); pinston00 (); zhang00 (); lozeva08 (). Many isomer systematics have been identified for Snisotopes, and the isomeric states and have been characterized as seniority =2 and =3 states in these studies, where denotes the seniority quantum number. Pietri et al. pietri11 () recently identified and confirmed the highspin and highseniority =4, isomeric state in Sn. More recently, Astier et al. astier132 (); astier125 () reported detailed highspin level schemes in the Sn isotopes by using the binary fission fragmentation induced by heavy ions. Iskra et al. iskra14 () have also focused on highseniority states in neutronrich, eveneven Snisotopes. It may be noted that there exists some deformed collective states giving rise to a full or a part of rotational band in the eveneven light mass Sn isotopes with , interpreted as proton configuration bron79 (); poelgeest80 (); harada88 (); savelius98 (); gableske01 (); wolinska05 (); wang10 (). But the yrast isomeric states discussed in the present paper are not part of any rotational structure fotiades11 (). More recently, the studies on Sn isotopes have been pushed much beyond the shell closure and isomers in the Snisotopes have been populated by Simpson et al. simpson14 () which shed a new light on the effective interaction in nrich nuclei maheshwari15 ().
The and isomers have also been identified, in the isotonic chain from , Dy to , Hf, as seniority =2 and =3 isomers coming from the proton orbital mcneill89 (). Recently, the highspin structure of five isotones with has also been reported by Astier et al. astier126 (), where the isomers have been described as broken pairs of protons from the and orbitals in the evenmass isotones.
We plot the excitation energies of the isomers relative to states and the isomers relative to states for the isotopes and the isotones in the top and bottom panels of Fig. 1, respectively. It may be noted that the same valence orbitals are involved in both the and chains. While the neutrons occupy these orbitals in the isomers, the protons take over the role in the isomers. We find that all the main features observed in the isotopic chain are also present in the isotonic chain and both appear to be nearly identical to each other. We can see that the energy gap is almost constant and particle number independent which is a well known signature of nearly good seniority talmi93 (); lawson80 (); talmi03 (). The and the isomers, belonging to the eveneven and evenodd nuclei respectively, are seen to follow each other very closely throughout the chains, if one puts the and states on equal footing. This suggests that the nuclear configurations and structure for the and the isomers should be very similar without any oddeven effect which also suggests the aligned nature of the involved nucleons.
Fig. 1 (color online) Variation of the experimental energy values of the and isomers in isotopes and isotones.
We have plotted the measured halflives (in s) of these isomers with increasing nucleon numbers in the top and bottom panels of Fig. 2, for the and chains, respectively. The halflives of the and isomers exhibit a rise near the middle of the active valence space (from neutron/proton numbers to ), attain a maximum value, and fall with increasing nucleon number. The and isomeric states, for the isotopes, exhibit a maximum at the neutron numbers and respectively, where the neutron orbital becomes halffilled mayer94 (); zhang00 (); lozeva08 (). On the other hand, for the isomers, the peaks are observed at and for the and isomeric states respectively, where the proton orbital becomes halffilled mcneill89 (). This happens because the electric quadrupole (E2) transition probabilities between a state and another state with same seniorities vanish at the middle talmi93 (); lawson80 (); Maheshwari2016 (). We can, therefore, foresee that the isomeric halflives at the middle of the active valence space in the and chains are most affected by the seniority selection rules. We also notice that the halflives of oddA Sn isotopes, i.e. the isomers are lower than the neighboring evenA Sn isotopes, i.e. the isomers as expected from pairing consideration.
Fig. 2 (color online) Variation of the experimental halflife values of the and isomers in isotopes and isotones. The vertical scale is logarithmic.
We have also plotted the excitation energies of the isomers relative to states and the isomers relative to states in the top panel of Fig. 3 for the , Pb isotopes. All the experimental data in this paper have been adopted from our atlas Jain2015 (), the ENSDF (Evaluated Nuclear Structure Data File) ENSDF (), and the XUNDL (Unevaluated Nuclear Data List) XUNDL () data sets. We can again see that both the Pbisomers, evenA and oddA, closely follow each other, and do not show any oddeven effect, as in the cases of Sn isotopes and N=82 isotones. This again supports the empirical evidences of good seniority states with a particle number independent nature of the states. We, therefore, expect them to have similar origins in terms of their wave functions and nuclear configurations. We have also plotted their halflives (in units of s) in the bottom panel of Fig. 3, where one can observe that the halflives show their increment towards mass number (), which may be the middle of the active valence space for these Pb isomeric states.
Table 1 Comparison of the experimentally measured and transitions in eveneven and oddA Snisotopes for 66 and their ratio . Also, compared are the and transitions involving states which decay to the , isomers and their ratio . All the energies are in MeV.
Isotope  Isotope  

Sn  Sn  
Sn  1.237  Sn  1.179  0.95  
Sn  1.190  Sn  1.083  0.91  
Sn  1.103  Sn  1.043  0.95  
Sn  1.047  Sn  0.924  0.88  

Table 2 Same as Table 1, but for the isotones with Z 66. All the energies are in MeV.
Isotope  Isotope  

Dy  1.932  Ho  
Er  1.446  Tm  1.332  0.92  
Yb  Lu 
Table 3 Same as Table 1, but for the isotopes with N 108. Comparison of the experimentally measured and transitions in eveneven and oddA Pbisotopes and their ratio . All the energies are in MeV.
Isotope  Isotope  

Pb  0.774  0.897  Pb  0.818  1.056 
Pb  0.853  0.874  Pb  0.881  1.032 
Pb  0.965  0.932  Pb  0.969  1.004 
Pb  1.049  0.959  Pb  1.006  0.959 
We have further listed the experimental gamma energies associated with the transitions and for the eveneven and oddA Snisotopes in Table 1 Jain2015 (); astier132 (); astier130 (); ENSDF (); XUNDL (). The ratio of these transitions denoted as / is observed to be 1 for the Sn isotopes. This suggests a complete alignment of the oddneutron in the orbital, producing the spin state. This supports the observation that the state in oddA Snisotopes and the state in the neighboring eveneven Snisotopes have great similarity in their wave functions. Similarly, the observed gamma transitions and in eveneven and oddA Snisotopes have also been listed in Table 1 for Snisotopes astier132 (); astier130 (). Fotiades et al. fotiades11 () have compared the almost identical energies and similar structure involved in the and transitions within the same isotope for Sn, and suggested that the isomeric state comes from the two aligned neutrons in the orbital. We have calculated the ratio of the transitions in oddA Snisotope and its eveneven core Snisotope, denoted as / , which is also observed to be 1. The known gamma transition energies for the isotones, have also been listed in Table 2 astier130 (). The ratios and again have the value 1, wherever these could be obtained.
Fig. 3 (color online) Variation of the experimental energy and halflife values of the and isomers in isotopes.
It is obvious from the observed values that the seniority remains as for the states, as coming from the unique parity orbital of the active valence space. We may also infer from these observations that the and isomeric states, in both the chains, are maximally aligned decoupled states having similar wave functions and nuclear configurations. That is why the and states closely follow each other in excitation energy without exhibiting any oddeven effect, as shown in Fig. 1. We, therefore, foresee the seniority as =0 for the states, =1 for the states, =2 for the states and =3 for the states. Similarly, we may assign the seniority =2 for the states, =3 for the states, =4 for the states, and =5 for the states for these nrich Sn isotopes. The same seniority difference =2 between the and states, and for the and states gives their corresponding ratio as 1. The difference =2 also holds for the and states, and for the and states, which makes the ratio 1.
We have also listed the experimental gamma energies associated with the transitions and for the eveneven and oddA Pbisotopes in Table 3 Jain2015 (); ENSDF (); XUNDL (). The ratio of these transitions denoted as / is observed to be 1 for the Pb isotopes. This suggests a complete alignment of the oddneutron in the orbital, producing the spin state. This supports the observation that the state in oddA Pbisotopes and the state in the neighboring eveneven Pbisotopes have great similarity in their wave functions, very similar to the isomers in other two semimagic chains.
Similarly, the observed gamma transitions have also been listed in Table 3 for the eveneven Pb isotopes; however, no measurements are available for in oddA Pb isotopes. We, therefore, could not calculated the ratio of the transitions in oddA Snisotope and its eveneven core Snisotope, denoted as / , which is also expected to be 1. Therefore, similar arguments work for all the highspin isomers in these and semimagic chains, while their respective valence spaces and intruder orbitals are different. This similarity hints towards the goodness of seniority. Keeping our previous results for evenA Sn isotopes in mind Maheshwari2016 (), we have further done the generalized seniority calculations for decay rates to confirm the identical situation of various isomers from different nuclear regions.
Iii Theoretical interpretation
We briefly present the formulas used in the calculations using the generalized seniority scheme, and successfully applied to the , and isomers in Sn isotopes in Maheshwari2016 (). In this paper, we extend our studies to the isomers in isotones and the isomers in isotopes. We further apply these results for the oddA semimagic nuclei, particularly for , , and isomers in Sn isotopes, isomers in isotones and isomers in isotopes.
iii.1 rates from generalized seniority
The values, between and states, in a mixed configuration along with the corresponding total pair degeneracy by using the generalized seniority scheme can be written as follows
(1) 
This implies that the values show a parabolic behavior in the multi case depending upon the seniority of the states involved in the transition. We rewrite the seniority reduction formula for the reduced matrix elements with seniority conserving transitions between the initial and final states for the completeness of the text in paper. The relations are as follows:
(2) 
The values, which depends on the particle number , the generalized seniority and the corresponding total pair degeneracy , can be calculated by using these formulas. These formulas take care of mixing of the active orbitals in the valence space. We present details of the calculations and results in the next section. Note that some information on radial integrals along with the matrix elements of spherical harmonics is hidden in the constant of proportionality.
iii.2 EvenA Semimagic Nuclei
We have successfully shown that the highspin decaying , and decaying isomers are similar in their decay trends for eveneven Sn isotopes, see paper Maheshwari2016 () for details. We now apply the same formalism Maheshwari2016 () to the isomers in other semimagic chains, particularly, the isomers in isotones, and the isomers in isotopes. Both the isomeric chains decay by transitions as shown in Fig. 5 and 6, and their measured trends are quite similar to the isomers in the Snisotopes (follow fig. 4). We choose the active valence space as , and in the isotones and i, f and p in the Pbisotopes. The resultant and values become and in the isotones, similar as in the case of isotopes, since both the and chains share same orbitals in their active valence space of particle number . We consider as core by assuming the and orbitals as completely filled.
Therefore, we fit the situation at , Dy for the seniority isomers in the chain, and get the resultant parabolic trend as shown in Fig. 5. The calculated trends reproduce the experimental data quite well; gives the best fit to the data, a different situation as compared to the , Z=50 isomers, where gives the best fit. This can be understood in terms of different valence particles, particularly the involvement of protons in the case of N=82 isotones. It appears to be closer to the pure seniority scheme, as quoted in the previous literature talmi93 ().
Fig. 4 (color online) Variation of the values of the and isomers in isotopes Maheshwari2016 (). All the values are shown in the units of .
On the other hand, the resultant and values become and for the Pbisotopes, where . The active valence space of for these isotopes consists of , , , , and orbitals. We consider as completely filled, so the next three active orbitals , and will be filled at Pb. We fix the proportionality constants by fitting the measured values for Pb for the seniority isomers in the chains, respectively. The calculated results reproduce the experimental trends for both these isomeric chains as shown in Fig. 6. The calculations have been done by considering the transitions as seniority conserving ones () along with generalized seniority of these states as . One can also expect the occurrence of these generalized seniority isomers in the gaps of experimental data for both the and chains extending towards protondeficient and neutrondeficient sides, respectively (follow the figs. 5 and 6 for the same).
Fig. 5 (color online) Variation of the values of the isomers in isotones.
Fig. 6 (color online) Variation of the values of the isomers in Pbisotopes.
One can, therefore, observe the identical behavior of the s in the highspin isomers for all the three semimagic chains, , , and . This is due to their identical generalized seniorities and the transitions. It is interesting to note that the valence particles in chain are neutrons while protons become active in the case of chain, and they share same orbitals in the active valence space of . On the other hand, the chain has a different neutron valence space along with different set of active orbitals. In spite of these differences with each other, highspin isomers in all the three chains appear to follow the same microscopic scheme of generalized seniority. This highlights the importance of configuration mixing required in the generation of these states. Hence, the generalized seniority behaves almost as a good quantum number for these states in all three semimagic chains.
We note that the seniority isomers are only known in the isotopes up to now. Due to the strong validity of generalized seniority in all the three semimagic chains, one can expect and predict the highspin and highseniority isomers in the remaining two chains as well. Measurements in this direction should be made to confirm this scenario.
iii.3 OddA Semimagic Nuclei
We now study the , and isomers of oddA Snisotopes in light of the generalized seniority scheme. We assume that the and orbitals are completely filled up to Sn; hence, the remaining active orbitals for the Snisomers are , and orbitals in the 5082 valence space. We have performed the generalized seniority calculations assuming , and transitions and by fitting the value of Sn isotope, as shown in Fig. 7. We calculate the values for the isomers using values of , and corresponding to , ; , ; , , respectively. We present the calculated and experimental values for the isomers in Fig. 7 for comparison. One can see that the calculated values for fit the experimental data reasonably well. This confirms that the isomers behave as generalized seniority isomers having transitions to the lower lying states, and support the mixing of all the three , and orbitals.
We also plot the values vs. A (mass number) for the and isomers, in the top and bottom panels of Fig. 8, respectively. Experimental data for these isomers have been taken from the recent measurements of Iskra et al. Iskra2016 () and the references therein. We find that the calculated results from value, and generalized seniority with transitions, are again able to explain the experimental trend reasonably well. Note that we fit the experimental value of Sn in calculations for both the and isomers. This confirms that the , isomers decay to the same seniority states, and are generalized seniority isomers (involving , and orbitals). Note that the seniority , , isomers, and seniority , isomers in the eveneven Sn isotopes have already been explained by using the same mixed configuration having value of Maheshwari2016 (). We can, therefore, conclude that the highspin isomers in Sn mass region arise from the mixing of all the available , and orbitals for both even and oddA isotopes.
On the other hand, we find that the inclusion of orbital along with , and orbitals is required for explaining the first states in these Snisotopes Maheshwari20161 (). However, the configuration changes for the first excited states showing the octupole character having orbitals unpublished (). It is quite obvious that the dominance of orbital increases, while going towards the highspin and highseniority states. This type of information by generalized seniority guides us to infer the nature of configuration mixing and wave functions involved in the generation of a given set of states. Also, the structure information for both even and oddA nuclei are similar, irrespective of the mass region, involved valence particles, orbitals, etc., except for an extra nucleon in oddA systems. To sum up, we find that the seniority and generalized seniority provides a fingerprint evidence for the similarity between various isomers in the semimagic chains.
Fig. 7 (color online) Variation of the values of the isomers in Snisotopes unpublished ().
On the basis of this interpretation, we have also analyzed the single measured value at Sn, for the higher seniority , isomer. We have fitted the value of Sn, and calculated the values for the other neighboring isotopes assuming transitions using and the corresponding mixed configuration. We have taken Sn as core; this implies that the first location to have seniority state corresponds to Sn. The and the related mixed configuration can not be fitted using the experimental value at Sn as it leads to a zero value for the coefficient in the middle(). We have plotted in Fig. 9, the trend for the seniority , isomer using value and the respective mixed configuration. These calculations, therefore, help us to predict some unknown values also. It is quite obvious that these highseniority , isomers in oddA nrich Snisotopes can be related to the seniority isomers in evenA nrich Snisotopes.
Fig. 8 (color online) Variation of the values of the , and isomers in Snisotopes unpublished ().
We further study the isomers in oddA isotones using the same generalized seniority scheme. We have already pointed out that the Sn isotopes and isotones have same set of orbitals in the active valence space of nucleon number , while neutrons are active in Sn isotopes and protons are active in the isotonic chain. We assume that the and orbitals are completely filled up to Gd (Z=64); hence, the remaining active orbitals for these isomers are , and orbitals. We have performed the generalized seniority calculations assuming , and transitions and by fitting the value of Ho isotone, as shown in Fig. 10. We calculate the values for the isomers using values of , and corresponding to , ; , ; , , respectively, as in the case of Sn isotopes. We present the calculated and experimental values for the isomers in Fig. 10 for comparison. One can see that the calculated values for best fit the experimental data. This confirms that the isomers behave as more like pure seniority isomers having transitions to the lower lying states, and support the dominance of orbital. The same argument has already been shown to work for evenA N=82 isotones too.
We next present a comparison of the isomers in Pbisotopes on the same footing, where we again calculate the values for these isomers using , and as active orbitals. We use the possible mixed configurations: (), (), (), and . We assume that the lowest lying orbital is full, and therefore, the other three orbitals become full at Pb. Since we do not have experimental data at lower mass neutrondeficient side, we fit the value of Pb by assuming Pb as the completely full configuration, using generalized seniority for transitions. The calculated values from having mixing of all the three active orbitals are able to explain the experimental data quite closely (See Fig. 11).
Fig. 9 (color online) Variation of the values of the isomers in Snisotopes unpublished ().
Fig. 10 (color online) Variation of the values of the isomers in isotones.
We, therefore, conclude that the isomers are seniority isomers in Pbisotopes, similar to the isomers in Snisotopes. Their similar trends can be explained in terms of the involvement of same seniority and generalized seniority, though coming from the different orbitals and different valence spaces. However, more measurements are required to obtain experimental data for the remaining nuclei and complete the picture. The same argument has been presented for the isomers in evenA nuclei. The seniority and generalized seniority hence play a unifying role in explaining the similar behavior in different sets of semimagic nuclei.
Fig. 11 (color online) Variation of the values of the isomers in Pbisotopes unpublished ().
Iv Conclusion
We have used the quasispin formalism for degenerate multi orbitals to calculate the reduced electric transition probabilities in the semimagic isomers. We find that the configuration mixing is essential to fully describe the , isomers in the evenA Snisotopes, and the , , , and , isomers in the oddA Snisotopes and to explain the values in all these Snisomers. This formalism reproduces the experimental trend quite well and is also capable to predict some numbers for the gaps in the measurements. On the other hand, the situation for the , and the , isomers in chain becomes different and highlights the dominance of orbital only (pureseniority scheme). These isomers have also been compared with the identical trends for the , isomers, and the , isomers in the chain. The identical behavior of all the highspin isomers in various semimagic chains strongly supports the goodness of seniority and generalized seniority up to very highspin. This simple scheme of calculating the values may also be used to estimate the halflives in unknown cases and, hence predict new isomers.
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