-
In the present study, the three-dimensional angular momentum projection technique was successfully adopted to study the
γ -vibrational band structures in triaxially deformed nuclei. The TPSM approach has also provided theoretical support for observation of theγγ -band in those nuclei [33]. Because the studied118−128Xe isotopes are of the even-even type with even protons and neutrons, the TPSM basis chosen for carrying out the present study is composed of the projected 0-qp state (or qp-vacuum), two-proton qp, two-neutron qp, and 4-qp configurations, namely,ˆPIMK|Φ>,ˆPIMKa†p1a†p2|Φ>,ˆPIMKa†n1a†n2|Φ>,ˆPIMKa†p1a†p2a†n1a†n2|Φ>,
(1) where
|Φ> represents the triaxial vacuum state. The three-dimensional angular-momentum projection operator [34] is given byˆPIMK=2I+18π2∫dΩDIMK(Ω)ˆR(Ω),
(2) with the rotational operator
ˆR(Ω)=e−ıαˆJze−ıβˆJye−ıγˆJz,
(3) Ω denotes a set of Euler angles (α,γ = [0,2π ],β = [0,π ]) and J denotes angular-momentum operators. In this present approach, triaxiality is included in the deformed basis, which helps perform an exact three-dimensional angular momentum projection. In this manner, the deformed vacuum state obtained is significantly improved, as it permits all possible components of K. The projected shell model version employed for the description of axially deformed nuclei uses the pairing plus quadrupole-quadrupole Hamiltonian [35] with a quadrupole-pairing term also included, as follows:ˆH=ˆH0−12χ∑μˆQ†μˆQμ−GMˆP†ˆP−GQ∑μˆP†μˆPμ,
(4) and the equivalent triaxial Nilsson mean-field Hamiltonian is given by
ˆHN=ˆH0−23ℏω{ϵˆQ0+ϵ′ˆQ+2+ˆQ−2√2},
(5) where the first term in Eq. (4) is the single-particle spherical Hamiltonian, which has a proper spin-orbit force represented by Nilsson parameters [36]. The force strength quadrupole-quadrupole (QQ)
χ is determined in such a way that it holds a self-consistent relation with the quadrupole deformation. In Eq. (5),ϵ andϵ′ specify the deformation parameters corresponding to the axial and triaxial deformation, respectively, and they are related to the triaxiality parameter byγ =tan−1 (ϵ′ϵ) . The calculations were performed with deformation parameters displayed in Table 1. The pairing strength, i.e, monopole pairing strength (GM ), is adjusted to obtain the known energy gaps:Xe ϵ ϵ′ γ 118 0.215 0.162 37° 120 0.228 0.158 35° 122 0.217 0.1250 30° 124 0.210 0.135 33° 126 0.158 0.113 35° 128 0.145 0.113 38° Table 1. Axial deformation parameter
ϵ and triaxial deformation parameterϵ′ , employed in calculation for 118-128Xe. Theγ deformation is related to the two parameters throughγ =tan−1 (ϵ′ϵ) .GM=(G1∓G2N−ZA)1A(MeV),
(6) with the '–' sign is for neutrons and the '+' sign for protons. The present study on
118−128Xe isotopes was performed with both neutron and proton major shells numbers N = 3, 4, and 5 and with pairing strengthsG1 = 21.24 andG2 = 13.86. The quadrupole pairing strengthGQ is assumed to be proportional toGM , with the proportionality constant set to 0.18 for all isotopes considered in this study. These calculations were conducted with the same set of input parameters for all isotopes under consideration. -
Calculations were carried out for
118−128Xe isotopes within the quasiparticle basis space, which is formed by the use of deformation parametersϵ andϵ′ (as shown in Table 1). The axial quadrupole deformation parameterϵ has been selected in the present calculations with the aim to reproduce the correct shell filling of the nucleus. The nonaxial deformation parameter,ϵ′ , is assumed in such a way that the behaviour of theγ -band is properly explained, and its value is in accordance with those obtained from the minimum of the potential energy surface (PES), as shown in Fig. 1. In this figure, the projected ground-state energy is plotted as a function of the triaxial parameterϵ′ with the axial deformation parameterϵ held fixed. These deformation parameters are required to solve the triaxial potential that is used to construct the deformed basis states in present calculations. The discussion on calculated TPSM results for118−128Xe isotopes and the comparison to their experimental counterparts on various nuclear structure properties is presented in the following sub-sections. -
The collection of projected energies for various intrinsic configurations leads to the formation of a diagram known as the band diagram. In the present TPSM calculations, these projected energies arise from the diagonal matrix elements used before mixing of the configurations. These band diagrams are known to be quite useful and result oriented, as they play a major role in explaining the underlying intrinsic structures of various bands. Zero-quasiparticle, two-quasiparticle, and four-quasiparticle configurations have been employed for calculating the angular-momentum projected energies with the use of deformation parameters (given in Table 1), and these projected energies are plotted in Figs. 2(a)–(f) for the Xenon isotopes of this study. The projection from the 0-qp configuration provides band structures corresponding to K = 0, 2, 4, denoted as (0, 0), (2, 0), and (4, 0) bands, which relate to the ground,
γ , andγγ band, respectively. These bands are considered to be the main components of the band diagram, as they are responsible for obtaining the lowest energy bands after configuration mixing. It is abundantly clear from the present set of calculations that the projection from the 0-qp state gives rise to the formation of band with only even values of K, i.e, K = 0, 2, 4,... . No odd values of K are obtained here because of the symmetry requirement for the vaccum configuration.Figure 2. (color online) Band diagrams for 118-128Xe isotopes. Labels (0, 0), (2, 0), (4, 0), (1, 2n), (3, 2n), (1, 2p), (3, 2p), (2, 4), and (4, 4) correspond to ground,
γ , 2γ , and 2n-alignedγ band on this 2n-aligned state, 2p-alignedγ band on this proton-aligned state, (2n+2p)-aligned band, andγ band built on this four-quasiparticle state.For
118Xe (Fig. 2(a)), the band head energy of theγ -band was obtained at an excitation energy of 0.93 MeV from the ground state band. Here, the two-quasineutron configuration states with K = 1 and 3 cross the ground-state band at I = 10. These bands represent theγ -band built on the two-quasineutron-aligned configurations. The two quasiproton states (1, 2p) and (3, 2p) are closer in energy compared to the two-neutron states, and therefore, cross the ground state band at I = 14. Finally, at I = 16 and above, the 4-qp structures (two-quasineutron plus two-quasiproton) with K = 2 and 4 cross the ground-state band, thereby giving rise to the yrast states that originate from these quasiparticle configurations for at least up to the last calculated spin values shown in the figure. In the case of120Xe in Fig. 2(b), the band head energy of theγ -band is at an energy of 0.85 MeV above the ground state from the present calculation. Notably, the ground state band (0, 0) is crossed by two quasiparticle bands with (1, 2n) and (3, 2n) configurations at I = 12, and is yrast up to this spin value. Further, the two-aligned proton bands with configurations (1, 2p) and (3, 2p) and a 4-qp state (2, 2n2p) band cross the ground state band at I = 18 and decrease in energy until the last calculated spin values.For Fig. 2(c) in
122Xe , the calculated band head energy of the (2, 0) band is about 0.86 MeV from the ground state band (0,0). The ground-state band is crossed by two quasiparticle bands with (1, 2n) and (3, 2n) configurations at I = 10. The figure shows that the two-proton aligned band with (1, 2p) configuration also crosses the ground-state band at I = 14, and from I = 16 the two-quasiparticle proton band (3, 2p) and four-quasiparticle state (2n2p) attain the lowest energy, leading to the formation of yrast. In the band diagram for124Xe , the obtained energy of the band head of theγ -band is at about 0.86 MeV above the the ground state band. The ground-state band with the (0, 0) configuration is crossed at I = 10 by two two-neutron aligned configurations, (1, 2n) and (3, 2n). The two-proton (1, 2p) and (3, 2p) bands decrease in energy and cross the ground band at I = 14. Beyond this spin, it is expected that four-quasiparticle configurations dominate the yrast band along with all the above-mentioned bands.Similarly for
126Xe , the theoretically calculated band head energy of theγ -band is 0.93 MeV above the ground state. The two two-neutron bands (1, 2n) and (3, 2n) cross the ground state band (0, 0) at spin I = 10, and both qp-neutron bands become degenerate in energy. At spin values I = 14 and I = 16, the two 2-qp proton bands and two four-quasiparticle bands, respectively, cross the ground band and decrease in energy, thereby, joining the two 2-qp neutron band and becoming yrast states. Finally, the band diagram for128Xe is also presented in Fig. 2(f),where the calculated band head of theγ -band is at an excitation energy of 1.2 MeV above the ground band. The crossing of the (0, 0) and (2, 0) bands by the two-neutron aligned bands (1, 2n) and (3, 2n), respectively, occurs at I = 8. After spin I = 12 and I = 16, the two-quasiparticle proton bands and four-quasiparticle states (2n2p) attain the lowest energy and become yrast at larger values of angular momentum. -
Our next aim is to obtain the energies of states obtained after configuration mixing. To this end, the shell model Hamiltonian of Eq. (3) is diagonalized with the obtained projected energies, giving rise to the energies of the states known as yrast states. The energies of the yrast states for
118−128Xe are presented in Fig. 3 in comparison with the experimental data [37–42]. For the purpose of discussion in terms of the relevant physics, only the lowest three bands from the 0-qp configuration and the lowest two bands for other configurations are shown in the band diagrams. However, during the diagonalization of the Hamiltonian, the basis states employed are more numerous, which includes, for example, those with K = 1, 3, 5, and 7 withκ = 1 and K = 0, 2, 4, 6, and 8 withκ = 0, where the indexκ indicates the basis states. The lowest three bands, obtained after diagonalization for each angular momentum, that only contribute to the formation of yrast energies, are shown in Fig. 3 for all studied Xe isotopes. From the figure, the agreement between the TPSM and experimental energies for the yrast andγ -bands is excellent for all the chosen118−128Xe isotopes. The calculated spectra for these isotopes are well fitted to experimental data for low as well as high-spin states. The predictedγ -bands are calculated for the spins higher than the available experimental data, and we hope that future high-spin experimental studies are able to populate these bands. -
From the experimental point of view, the evolution of nuclear structure of the Xe isotopes with A = 118 to 128 can be inferred from Fig. 4, where the variation of
[E(2+1)] is shown with respect to neutron numbers. Fig. 4 shows that the energy of the first-excited state[E(2+1)] is at its minimum for N = 66 (for120Xe ) and then rises smoothly from N = 68 to N = 74. Both the experimental and calculated results follow the same trend, indicating satisfactory agreement for[E(2+1)] . Moreover, a dip in energy[E(2+1)] is clearly observable for120Xe , and as per Grodzin's rule [22], this isotope120Xe can be considered to be the most deformed among all other studied isotopes because of the inverse variation of[E(2+1)] with the quadrupole moment (ϵ2 ). Further, the experimental and calculated energy ratiosR4/2 =[E(4+1)/E(2+1)] with neutron numbers are presented in Table 2. Casten et al. [43] showed that for even-even nuclei, the shape phase transition is associated with a sudden change in nuclear collective behaviour, as a result of which the ratioR4/2 =[E(4+1)/E(2+1)] suddenly increases from the spherical value of 2.0 to the deformedγ -soft nuclei value of 2.5, and the maximum value of 3.33 for an ideally symmetric rotor. Table 2 indicates that the comparison of observed energy ratios with experimental data is quite satisfactory, and the experimentalR4/2 ratios for the118−128Xe isotopes lie between 2.33 and 2.5. These theoretical values suggest that the values ofR4/2 increase gradually from 2.4 to 2.5 with the increase in mass number A = 118 to 122, and then decreases from 2.5 to 2.33 on moving towards128Xe , indicating the presence ofγ -soft shapes in these nuclei. This confirms the triaxial nature of the chosen Xenon isotopes in the present study. Further, the heavier Xe isotopes beyond122Xe are less deformed than the lighter ones.The ratio attains its maximum value in122Xe near120Xe , indicating nuclei with a large deformation and more rotational properties.Xe R4/2 =[E(4+1)/E(2+1)] Expt. TPSM 118 2.4 2.5 120 2.46 2.51 122 2.5 2.53 124 2.48 2.49 126 2.42 2.48 128 2.33 2.42 Table 2. Comparison of observed energy ratios of energies of first two excited states
R4/2 =[E(4+1)/E(2+1)] with experimental data. -
The nature of triaxial shapes in all chosen
118−128Xe isotopes can be interpreted in terms of the staggering parameter, defined asS(I)=[E(I)−E(I−2)]−[E(I−1)−E(I−2)]E(2+1),
(7) S(I) is plotted for the
γ -bands in Fig. 5. The attempt has been made in the present study to describe the odd-even staggering in theγ -bands obtained after configuration mixing of the studied Xe isotopes. Fig. 5 shows that the experimental staggering parameter for the known energy levels is reproduced quite accurately by the TPSM calculations, except for118,120Xe . Furthermore, the staggering S(I) is quite small at lower spins. This is attributed to the fixed deformation parameters used in the calculations, and shape fluctuations are not taken into account. Above I = 8, the staggering increases for almost all isotopes under consideration, which might be due to the crossing of the two-quasineutron-aligned band withγ -bands (2, 0) at this spin value (as noted from the earlier discussion presented in Section 3.1 for band diagrams). In particular, all studied Xe isotopes can be considered to possess the same staggering phase in S(I). However, the results obtained are particularly interesting after a full mixing of quasiparticle configurations at spin valuesI≈14−18 for118−128Xe , where an inversion in the staggering pattern is observed. A study by R. Bengtsson et al. [44] proposed that the signature inversion, i.e., exchange of energetically favored and unfavored spins, might be evidence for a triaxial shape in those nuclei. Thus, the presence of inversion in the staggering pattern of the118−128Xe isotopes also provides an important indication of the triaxiality in the chosen nuclei. -
The essential process of the backbending phenomena [45] is considered as a quasiparticle level crossing between an vacant high-j intruder orbital and the most high-lying populated orbital, resulting in a sudden increase of the moment of inertia along the yrast level and decrease in the rotational frequency. K. Higashiyama et al. [45] have likewise successfully demonstrated the shell model description of the backbending phenomena in Xe isotopes in the
A∼130 region. In Fig. 6, the results of moment of inertiaJ(1) and the square of the rotational frequencyω2 are plotted for nuclei118−128Xe , whereJ(1) andω2 are defined as follows:Figure 6. (color online) Plots of kinematic moment of inertia
(2J(1)/ℏ2) as a function of(ℏω)2 for118−128Xe .2J(1)/ℏ2=4I−2EI−EI−2,
(8) ℏ2ω2=(I2−I+1)(EI−EI−2)(2I−1)2.
(9) The sudden drop in the value of rotational frequency of the yrast band at the band crossing is well reproduced for the even-even
118−128Xe isotopes. Fig. 6 clearly shows that for the low spin region, the g-band is the dominant band in all118−128Xe isotopes, and the slope of calculated backbend atI≈8−12 is more prominent than that of the experimental backbend, where the first 2-qp band crossing takes place for all isotopes under study. For118Xe , the first backbend is obtained at spin I = 10, which is the same spin value of the first band crossing. This occurs because the rotational frequency of the yrast band decreases suddenly at a band crossing and thus leads to the S-shaped (backbending) diagram. However, the experimental backbend occurs at spin I = 16. Similarly for120Xe , the calculated backbend is obtained at spin I = 10; however, no experimental backbend is observed as the data at high spins is inadequate. Furthermore for122Xe , the calculated and experimental backbends take place at spin I = 10, where the first band crossing is also reported in the band diagram for122Xe . Further, for124Xe , the calculated backbend is obtained at spin I = 10. However, due to the lack of experimental data at high spins, its accuracy cannot be guaranteed. For126Xe , the successive band crossings take place in succession as the spin increases at spin values I = 8, 10. In contrast, the experimental backbend is obtained at I = 12. Finally, for128Xe , the first calculated backbend is obtained at I = 8, whereas the experimental observed backbend is at I = 10. The backbending phenomena in these isotopes are, thus, considered to be caused by the crossing of the two-quasi-particle neutron bands and the ground-state band. At higher spins, the two quasi-proton aligned state is favourable in energy and crosses the g-band in all Xenon isotopes in the present study. One of the most critical factors for the sharpness of the backbend is the crossing angle of the g-band and other multi-quasi particle bands at the crossing spin. Finally, we conclude that the two 2-qp neutron aligned bands, which crossed the g-band at the (first) band crossing, remain as yrast bands until they are crossed by other 2-qp and 4-qp bands (usually a2ν⊗2π 4-qp band). The agreement between theory and experiment for backbendng is well reproduced for some, but not all of the isotopes under study. These results may improve with the inclusion of higher quasiparticle states in the present applied model. The theoretical calculations and analyses show that the backbending phenomenon may be the result of the crossing of the ground-state band and the band with two-quasi-neutrons in all Xenon isotopes. -
Another part of the present TPSM calculations involves the study of the electric quadrupole transition probabilities B(E2). In this study, B(E2) values are calculated according to the following procedure. The projected states thus obtained are employed as new basis states for diagonalization of the shell model Hamiltonian in the TPSM basis. Then, with the help of the diagonalized Hamiltonian, eigenfunctions are obtained, which are used to calculate the electric quadrupole transition probabilities as follows:
B(E2:(Ii,Ki)→(If,Kf))=12Ii+1|⟨ΨKfIf||ˆQ2||ΨKiIi⟩|2,
joining initial state
(Ii,Ki) and final states(If,Kf) . In the present set of calculations for B(E2), the proton and neutron effective charges are denoted byeπ andeν , respectively, which are defined through the relations aseπ=1+eeff andeν=eeff . In the present set of calculations, the value ofeeff is taken as 0.5. In Fig. 7, the calculated B(E2) transition probabilities along the yrast band are compared with known experimental values [46, 47] and HFB results [46]. The transition probabilities are efficiently reproduced by present TPSM calculations. In particular, the increasing trend of B(E2) for high-spin states are well described by TPSM calculations, and the observed drop in the B(E2) transitions with spin is likewise correctly reproduced, except for118Xe . Furthermore, the B(E2) values for120Xe are excellent, and this can be attributed to the presence of mid shell neutrons, N = 66, where the the number of active neutrons and hence the neutron-proton correlations are at their maximum. The transition probabilities in the band-crossing region are clearly reduced, as these values of B(E2) are evaluated between the predominant ground state and two-quasiparticle aligned configurations. The decrease in B(E2) values is related to the crossing of the two-quasiparticle neutron configuration with the ground-state band. Thus, the known experimental values and the calculated B(E2) values are efficiently reproduced by the present study.Figure 7. (color online) Comparison of calculated (TPSM) B(E2) values with experimental (Expt.) and other HFB ([46]) data in
118−128Xe .
Xe | |||
118 | 0.215 | 0.162 | 37° |
120 | 0.228 | 0.158 | 35° |
122 | 0.217 | 0.1250 | 30° |
124 | 0.210 | 0.135 | 33° |
126 | 0.158 | 0.113 | 35° |
128 | 0.145 | 0.113 | 38° |