A unified view of the first-excited 2 + and 3 − states of Cd, Sn and Te isotopes
aa r X i v : . [ nu c l - t h ] J u l A unified view of the first-excited 2 + and 3 − states of Cd, Sn and Te isotopes Bhoomika Maheshwari ∗ University of Malaya, Kuala Lumpur, Malaysia.August 7, 2020
Abstract
Symmetries are known to play an important role in the low lying levelstructure of Sn isotopes, mostly in terms of the seniority and generalizedseniority schemes. In this paper, we revisit the multi-j generalized senior-ity approach for the first excited 2 + and 3 − states in the Cd ( Z = 48),Sn ( Z = 50) and Te ( Z = 52) isotopes, where the Cd and Te isotopesrepresent two-proton hole and two-proton particle nuclei, thus involvingboth kind of particles (protons and neutrons) in contrast to Sn isotopes.Interestingly, the approach based on neutron valence space alone is able toexplain the B(E2) and B(E3) trends respectively for the 2 + and 3 − statesin all the three Cd, Sn and Te isotopes. The new results on the invertedparabolic behavior of B(E3) values in Cd and Te isotopes are understoodin a manner identical to that of Sn isotopes by using the generalized se-niority scheme. No shell quenching is supported by these calculations;hence, the neutron magic numbers, N = 50 and N = 82, remain robust inthese isotopic chains. It is quite surprising that the generalized senioritycontinues to be reasonably successful away from the semi-magic region,thus providing a unifying view of the 2 + and 3 − states. Symmetries provide deep theoretical insights of the structure of matter andfundamental forces, and offer a unified view of many phenomena. The connec-tion between symmetries and conservation laws plays a key role in this respect.For example, conservation laws for energy and momentum can be understooddue to the symmetries with respect to translations in time and space. Thefirst studied symmetries in nuclear physics include charge independence andcharge symmetry, which led to the concept of isospin, first introduced by the ∗ Email: [email protected] (Bhoomika Maheshwari) + states in Sn isotopes were made [13] which led tothe similar understanding of first 2 + states in Cd and Te isotopes [18]. In thispaper, we review the generalized seniority results of first 2 + states for not onlysemi-magic nuclei but also away from semi-magic nuclei, particularly for the Cdisotopes (2-proton holes) and the Te isotopes (2-proton particles) in comparisonto the Sn isotopes (Z=50 closed shell). We also extend the study of first 3 − states in Sn isotopes [16], in the chains of Cd and Te isotopes, for the first time.Presence of both protons and neutrons in the valence space makes the studyvery interesting for the first excited 2 + and 3 − states in all the three Cd, Snand Te isotopic chains.We have divided the paper into five sections. Section 2 presents some es-sential theoretical details and key features of the generalized seniority scheme.Section 3 provides an empirical test to the generalized seniority. Section 4 dis-cusses the calculated results for B(EL; L=2 or 3) trends of the first excited 2 + and 3 − states in Cd, Sn and Te isotopes and compares it with the experimentaldata wherever available. Section 5 summarizes the work. Seniority scheme is usually credited to Racah [4] but Flowers [19] also introducedit almost simultaneously though independently. Seniority ( v ) may be definedas the number of unpaired nucleons for a given state [5, 6, 7]. The quasi-spinscheme of Kerman [20] and Helmers [21], for identical nucleons in single-j shellsatisfies the SU(2) algebra formed by the pair operators S + j , S − j and S j and iswell suited to describe the seniority scheme, where S + j = ( − ( j − m ) a + jm a + j, − m , S − j = ( − ( j − m ) a j, − m a jm and S j = ( n j − Ω). Here n j and Ω are thenumber operator and pair degeneracy of the single-j shell. The details of thisalgebra and corresponding selection rules can be found in the book of Talmi [8].2he beauty of seniority lies in the reduction of n nucleons (identical) problemto the v nucleons problem by transforming the reduced matrix elements in j n configuration to the reduced matrix elements in j v configuration.When n identical nucleons occupy a multi-j space then the correspondingreduced matrix elements can be calculated in the generalized seniority scheme,which was first introduced by Arima and Ichimura [22] for many degenerate or-bitals. The corresponding generalized pair creation operator can be defined as S + = P j S + j , where the summation over j takes care of the multi-j situation [8].Such generalized pair operators also satisfy the SU(2) algebra. Talmi further in-corporated the non-degeneracy of the multi-j orbitals by using S + = P j α j S + j ,where α j are the mixing coefficients [23, 24]. Our recent extension of this schemefor multi-j degenerate orbitals by defining S + = P j ( − l j S + j , as proposed byArvieu and Moszokowski [25], led to a new set of generalized seniority selec-tion rules [12, 13, 14, 15, 16, 17, 18]. Here l j denotes the orbital angularmomentum of the given-j orbital. An important non-trivial consequence wasthe discovery of a new category of seniority isomers, decaying via odd tensortransitions [12]. The seniority in single-j changes to the generalized seniority v in multi-j with an effective-j defined as ˜ j = j ⊗ j ′ .... having a pair degeneracyof Ω = P j j +12 = (2˜ j +1)2 . The shared occupancy in multi-j space is akin tothe quasi-particle picture. However, the number of nucleons n = P j n j and thegeneralized seniority v = P j v j remain an integer. In this paper, we show thegeneralized seniority scheme with quasi-spin operators as S + = P j ( − l j S + j [25], where S + j = P m ( − j − m a + jm a j, − m [8]. A simple pairing Hamiltonian inmulti-j space can hence be defined as H = 2 S + S − , with the energy eigen values[2 s ( s + 1) − (Ω − n )(Ω + 2 − n )] = [( n − v )(2Ω + 2 − n − v )]. Here, s = P j s j is the total quasi-spin of the state.Kota [26] has recently shown that for each set of α j values (= ( − l j ),there exists a corresponding symplectic algebra Sp(2Ω) arising from U(2Ω) ⊃ Sp(2Ω) with Ω = P j j +12 . This one-to-one correspondence between Sp(N) ↔ SU(2) leads to special selection rules for electro-magnetic transition operatorsconnecting n − nucleon states having good generalized seniority, which are inagreement with the selection rules obtained by us [12]. The most prominent signatures of good seniority states show up in the be-haviour of the excitation energy, electromagnetic transition rates like B(EL)and B(ML) values, Q-moments and magnetic moments, or g-factor values. Wecan summarize the features of good generalized seniority states as follows: • The excitation energies of good generalized seniority states are expectedto have a valence particle number independent behaviour, similar to thegood seniority states arising from single-j shell. It is rather easy to showthis by extending the proof for the single-j seniority scheme by defininga multi-j effective configuration as ˜ j , as shown in [18]. Consequently, the3nergy difference remains independent of the valence particle number for agiven multi-j configuration. For example, the first excited 2 + states in Snisotopes are observed at nearly constant energy throughout the isotopicchain from N=52 to 80. • The magnetic dipole moments i.e. g-factors for a given generalized senior-ity state show a constant trend with respect to particle number variation.In general, the magnetic transition probabilities support a particle numberindependent behaviour for both the even and odd multipole transitions. • The electric transition probabilities exhibit a parabolic behaviour for boththe odd and even multipole transitions. We recall these developments bythe following expressions for electric multipole L (even or odd) operatorsas:(a) For generalized seniority conserving (∆ v = 0) transitions h ˜ j n vlJ || X i r Li Y L ( θ i , φ i ) || ˜ j n vlJ i = " Ω − n Ω − v h ˜ j v vlJ || X i r Li Y L ( θ i , φ i ) || ˜ j v vlJ i (1)(b) For generalized seniority changing (∆ v = 2) transitions h ˜ j n vlJ || X i r Li Y L ( θ i , φ i ) || ˜ j n v ± lJ i = "s ( n − v + 2)(2Ω + 2 − n − v )4(Ω + 1 − v ) h ˜ j v vlJ || X i r Li Y L ( θ i , φ i ) || ˜ j v v ± lJ i (2) • For L = 2, Eq.(1) can directly be related to the electric quadrupole mo-ments Q = h ˜ j n J || ˆ Q || ˜ j n J i = h ˜ j n J || P i r i Y || ˜ j n J i with the following con-clusions: The Q-moment values depend on the pair degeneracy (Ω), parti-cle number ( n ) and the generalized seniority ( v ) as per the square bracketshown in Eq.(1). The Q − moment values follow a linear relationship with n . The Q-values change from negative to positive on filling up the givenmulti-j shell with a zero value in the middle of the shell due to Ω − n Ω − v term.This is in direct contrast to the Q-moment generated by collective defor-mation which is expected to be the largest in the middle of the shell. • The dependence of the p B ( E
2) with particle number n in Eq.(2) forthe generalized seniority changing transitions is different than the case of Q − moments. The p B ( E
2) values for ∆ v = 2 transitions exhibit a flattrend throughout the multi-j shell, decreasing to zero at both the shellboundaries. A nearly spherical structure is supported at both the ends forthe given multi-j shell. • The first excited 2 + states with generalized seniority v = 2 usually de-cays to the ground 0 + states, a fully pair-correlated state with generalized4eniority v = 0. Such E v = 2), where the corresponding B(E2) values can beobtained as follows: B ( E
2) = 12 J i + 1 |h ˜ j n vlJ f || X i r i Y ( θ i , φ i ) || ˜ j n v ± l ′ J i i| (3)The involved reduced matrix elements can similarly be obtained by usingEq.(2) between initial J i and final J f states with respective parities of l and l ′ . • Similarly, the first excited 3 − states with generalized seniority v = 2 usu-ally decays to the ground 0 + states. Such E v = 2), where the correspondingB(E3) values can be obtained as follows: B ( E
3) = 12 J i + 1 |h ˜ j n vlJ f || X i r i Y ( θ i , φ i ) || ˜ j n v ± l ′ J i i| (4)The involved reduced matrix elements can similarly be obtained by usingEq.(2) between initial J i and final J f states with respective parities of l and l ′ . Figure 1 exhibits the experimental [27] energy variation of the first excited 2 + states with the neutron number in Cd, Sn and Te isotopes. One may note anearly constant energy trend in all the three isotopic chains throughout N =52 −
80. An abrupt rise in the energies of Cd isotopes at N = 52 ,
54 can beseen, which is very similar to case of Sn isotope at N = 52. No experimentalvalue is available for Te isotope at N = 52; though, a similar case of energy risecan be seen in Te isotopes having N ≥
76. This may be due to approachingthe respective neutron closed shell configurations at N = 50 and N = 82.The nearly constant energies for these 2 + states on going from N = 52 to N = 80 strongly hint towards the goodness of the generalized seniority in allthe three isotopic chains. A small hump in the energy variation around middle( N = 64) can be noticed, which hints towards a sub-shell gap in the single-particle energies of the respective neutron orbitals. The active neutron orbitalsin the N = 50 −
82 valence space are g / , d / , d / , h / and s / , respectively.Out of which, g / and d / lie lower in energy than the remaining d / , h / and s / . As soon as the neutrons start to occupy this valence space, the higherprobability is to occupy g / and d / ; however, once these two orbitals freezeout around N = 64, the dominance of h / can be observed. So, this smallchange around N = 64 is related to the change in filling of the orbitals forthe first excited 2 + states in Cd, Sn and Te isotopes. This feature of changeat N = 64 −
66 can also be followed from the shell model calculations of Sn5 E ne r g y ( M e V ) Neutron Number Cd Sn Te
Figure 1: (Color online) Empirical energy systematics [27] of first excited 2 + states in Cd, Sn and Te isotopic chains, respectively.isotopes [28]. Also, the same argument was given by Morales et al. [29] andby us [12], while explaining the B(E2) trends of the first 2 + states; which willbe discussed in next section. These energy values in Cd and Te isotopes areconsistently lower (nearly half) in comparison to the energies in Sn isotopes(with Z = 50 closed shell), due to moving away from the closed shell. However,the generalized seniority remains constant as v = 2 leading to a nearly particlenumber independent energy variation of the first excited 2 + states in full chainof isotopes, as shown in Fig. 1.Fig.2 exhibits the empirical energy variation for the first excited 3 − states ineven-even Cd, Sn and Te isotopes, wherever data are available. Again, a nearlyconstant variation is visible in all the three Cd, Sn and Te isotopic chains with adip at neutron number 66, i.e. the middle of the neutron valence space 50 − − states may support octupole character, if they arisefrom the d − h orbitals of neutron valence space.6 E ne r g y ( M e V ) Neutron Number Cd Sn Te 3 - states Figure 2: (Color online) Empirical energy systematics [27] of first excited 3 − states in Cd, Sn and Te isotopic chains, respectively. In this section, we discuss the generalized seniority calculated results for the firstexcited 2 + states and 3 − states in even-A Cd, Sn and Te isotopes. The pairdegeneracies Ω = 9 , ,
11 and 12 correspond to the configurations { d / ⊗ h / } , { g / ⊗ d / ⊗ d / ⊗ s / } , { d / ⊗ d / ⊗ h / } , and { d / ⊗ h / ⊗ d / ⊗ s / } ,respectively, in the following discussion. First we review our results on two asymmetric B(E2) parabolas for the first ex-cited 2 + states in Cd, Sn and Te isotopes [13, 18]. Recent B(E2) evaluation [30]has been used to obtain the experimental systematic trends of the B(E2) valuesfor Cd and Te isotopes. The evaluated B(E2) values for Sn isotopes is used from[13]. We note a nearly identical pattern of two asymmetric B(E2) parabolas for7 Exp. Gen. Seniority , = 10 Gen. Seniority , = 12 Cd isotopes B ( E ) ( W e i ssk op f U n i t s ) Neutron Number
Figure 3: (Color online) Comparison of the experimental [30] and generalizedseniority calculated B(E2) trends for the first excited 2 + states in Cd isotopes.The asymmetry in the overall trend is explained by the filling of different orbitalsbefore and after the mid-shell, resulting in a dip around middle. The chosenset of multi-j configuration in the generalized seniority calculations are chosenas Ω = 10 (before the middle) and Ω = 12 (after the middle), corresponding to g / ⊗ d / ⊗ d / ⊗ s / , and d / ⊗ h / ⊗ d / ⊗ s / , respectively.all the three Cd, Sn and Te isotopic chains as shown in Figs. 3, 4 and 5,respectively. A dip around the middle in the experimental B(E2) values is alsovisible for all the three isotopic chains.The generalized seniority calculations use the multi-j configuration corre-sponding to Ω = 10 (before the middle) and Ω = 12 (after the middle), respec-tively. The active set of orbitals is mainly dominated by g / and d / orbitalsbefore the middle, while the h / orbital dominates after the middle. The cal-culated trends depend on the square of the coefficients in Eq.(2), since the 0 + to 2 + transitions are generalized seniority changing ∆ v = 2 transitions. To takecare of other structural effects, we fit one of the experimental data and restrict8 B ( E ) ( W e i ssk op f U n i t s ) Neutron Number
Exp. Gen. Seniority, =10 Gen. Seniority, =12
Sn isotopes
Figure 4: (Color online) Same as Fig. 3, but for Sn isotopes.the values of radial integrals and involved 3 j − and 6 j − coeffcients as a constant,which should be the case for an interaction conserving the generalized seniority.It is interesting to note that the generalized seniority calculated values ex-plain the overall trends of the experimental data in all the three Cd, Sn and Teisotopic chains, see Figs. 3, 4 and 5, respectively. Asymmetry in the invertedparabola before and after the middle has again been attributed to the differencein filling the two sets of orbitals. The dominance of g / orbital gets shifted tothe h / orbital near the middle of the shell resulting in a dip. However, thegeneralized seniority remains constant at v = 2 leading to the particle numberindependent energy variation for the 2 + states throughout the full chain. Thegeneralized seniority, hence, governs the electromagnetic properties not only inSn isotopes but also in the Cd and Te isotopes, which are not semi-magic nu-clei. One may note that the calculations only consider the active orbitals of N = 50 −
82 valence space. No signs of shell quenching have, therefore, beenwitnessed for these first excited 2 + states in all the three Cd, Sn and Te isotopes.However, we note that the influence of two proton holes/ particles cannot be9 B ( E ) ( W e i ssk op f U n i t s ) Exp. Gen. Seniority , = 10 Gen. Seniority , = 12 Neutron Number
Te isotopes
Figure 5: (Color online) Same as Fig. 3, but for Te isotopes.ignored; the overall trend is largely obtained by changing the neutron numberin total E2 transition matrix elements.
Figs. 6, 7 and 8 exhibit the experimental and generalized seniority calculatedB(E3) trends for Cd, Sn and Te isotopic chains. The experimental data aretaken from the compilation of Kibedi and Spear [31]. The calculations aredone by using the pair degeneracy of Ω = 9 and 11 corresponding to the d / ⊗ h / and d / ⊗ d / ⊗ h / configurations, respectively. An inverted parabolicB(E3) trend is visible in all the three Cd, Sn and Te isotopic chains. Thecalculated results explain the gross experimental trend in the best possible way,wherever available. However, one can notice that the Fermi surface for the givenconfigurations seems to be different for each chain. Therefore, the peaks belongto different neutron number in each chain, such as N = 58 −
60 for Cd isotopicchain, N = 66 for Sn isotopic chain and N = 70 for Te isotopic chain. The10 B ( E ) ( W e i ssk op f U n i t s ) Neutron Number Exp. = 9 = 11Cd isotopes
Figure 6: (Color online) B(E3) variation with neutron number for the firstexcited 3 − states in Cd isotopes by using generalized seniority scheme with Ω =9 and 11 corresponding to the d / ⊗ h / and d / ⊗ d / ⊗ h / configurations,respectively.deviation of many points from the generalized seniority are more pronouncedin Cd isotopes. More precise measurements are needed to improve/test thetheoretical arguments. We have revisited the B(E2) and B(E3) trends for the first excited 2 + and 3 − states in Cd, Sn and Te isotopes and examined them on the basis of the multi-jgeneralized seniority scheme. Two asymmetric B(E2) parabolas for the first 2 + states have been noticed in Cd (two-proton holes) and Te (two-proton particles)isotopes, and explained in terms of generalized seniority identical to the case ofSn isotopes. The consistency of configuration on going from two-holes to two-11 Exp. = 9 = 11 B ( E ) ( W e i ssk op f U n i t s ) Neutron NumberSn isotopes
Figure 7: (Color online) Same as Fig. 6, but for the Sn isotopes.particles is remarkable, which results in a nearly particle number independentenergy variation for the first 2 + states in all the three isotopic chains.The generalized seniority scheme further explains the inverted parabolicB(E3) trend for the first excited 3 − states, not only in Sn isotopes but alsoin Cd and Te isotopes, for the first time. New and more precise measurementsare needed to confirm these results, particularly in Cd and Te isotopes. No shellquenching is suggested for these low lying states. The role of the symmetriesrelated to pairing shows up in the goodness of generalized seniority quantumnumber, which offers a unified view of the 2 + and 3 − states in Cd, Sn and Teisotopes. The author would like to thank Prof. A. K. Jain for innumerable thought-provoking discussions. The work was done during the author’s post-doctorate12
Exp. = 9 = 11 B ( E ) ( W e i ssk op f U n i t s ) Neutron NumberTe isotopes
Figure 8: (Color online) Same as Fig. 6, but for Te isotopes.at University of Malaya.
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