New band mechanism of doubly-odd nuclei around mass 130

The nuclear structure of the $\Delta I=1$ doublet bands in doubly-odd nuclei ${}^{130}\mathrm{Cs}$, ${}^{132}\mathrm{Cs}$, ${}^{132}\mathrm{La}$, and ${}^{134}\mathrm{La}$ is studied in terms of a pair-truncated shell model, in which the collective nucleon pairs with angular momenta of zero and two are its basic ingredients. The effective interactions consist of single-particle energies and monopole and quadrupole pairing plus quadrupole-quadrupole interactions, whose strengths are determined so as to describe the level schemes of even-even nuclei in the mass $A~130$ region. The calculation reproduces well the energy levels of the doublet bands and the electromagnetic transitions, especially the staggering of the ratios $B(M1;I\to I-1)/B(E2;I\to I-2)$ for the yrast states. Through the analysis of the wave functions, the doublet bands turn out to be made of different angular momentum configurations of an unpaired neutron and an unpaired proton, weakly coupled with the quadrupole collective excitations of the even-even part of the nucleus.

Figure 1

Comparison of the experimental energy levels (expt.) with those of the PTSM for ${}^{130}\mathrm{Xe}$, ${}^{132}\mathrm{Xe}$, ${}^{132}\mathrm{Ba}$, and ${}^{134}\mathrm{Ba}$. The level sequences on the right represent the quasi-γ band, and the level sequences on the left the yrast band. Experimental data are taken from Refs. 6, 57, 55, 56, 54.

Figure 2

Comparison of the experimental energy levels (expt.) with those of the PTSM for ${}^{132}\mathrm{Cs}$. The two level sequences of $\Delta I=1$ on the right represent the yrast band, and the two level sequences of $\Delta I=1$ on the middle, the yrare band. The two levels on the left denote the third lowest energy states for each total spin I. Experimental data are taken from Ref. 35.

Figure 3

Same figure as in Fig. 2, but for ${}^{130}\mathrm{Cs}$. Experimental data are taken from Ref. 35.

Figure 4

Same figure as in Fig. 2, but for ${}^{134}\mathrm{La}$. Experimental data are taken from Ref. 33.

Figure 5

Same figure as in Fig. 2, but for ${}^{132}\mathrm{La}$. Experimental data are taken from Ref. 34.

Figure 6

Comparison of the experimental energy levels (expt.) with those of the PTSM for ${}^{130}\mathrm{Cs}$, ${}^{132}\mathrm{Cs}$, ${}^{132}\mathrm{La}$, and ${}^{134}\mathrm{La}$. Experimental data are taken from Refs. 58, 25, 26, 59, 27.

Figure 7

(Color online). Comparison of the calculated $B(M1)/B(E2)$ ratios for the yrast states of ${}^{130}\mathrm{Cs}$, ${}^{132}\mathrm{Cs}$, ${}^{132}\mathrm{La}$, and ${}^{134}\mathrm{La}$ with experiment. Experimental data are taken from Refs. 33, 35, 36. No experimental data are available for ${}^{132}\mathrm{La}$.

Figure 8

(Color online). Comparison of the calculated $B(M1){}_{\mathrm{In}}/B(M1){}_{\mathrm{Out}}$ ratios for the yrare states of ${}^{130}\mathrm{Cs}$, ${}^{132}\mathrm{Cs}$, ${}^{132}\mathrm{La}$, and ${}^{134}\mathrm{La}$ with the experimental data. Experimental data are taken from Refs. 33, 35. No experimental data are available for ${}^{132}\mathrm{La}$.

Figure 9

(Color online). The theoretical $B(E2)$ values for ${}^{132}\mathrm{Cs}$, ${}^{130}\mathrm{Cs}$, ${}^{134}\mathrm{La}$, and ${}^{132}\mathrm{La}$. The filled circles, open squares, open circles, and filled squares represent the transitions between yrast states, between yrare states, from yrast to yrare states, and from yrare to yrast states, respectively. Solid curves indicate the transitions between the odd-spin states, and dotted curves, the transitions between even-spin states.

Figure 10

(Color online). The theoretical $B(M1)$ values for ${}^{132}\mathrm{Cs}$, ${}^{130}\mathrm{Cs}$, ${}^{134}\mathrm{La}$, and ${}^{132}\mathrm{La}$.

Figure 11

(Color). Partial level scheme of ${}^{134}\mathrm{La}$ suggested by the PTSM calculation. The width of the green arrows indicates the absolute $E2$ ($\Delta I=2$) transition strengths [$B(E2)\hspace{1em}\ge 0.02$ $e^2{b}^2$], and that of the orange arrows indicates the absolute $M1$ ($\Delta I=1$) transition strengths [$B(M1)\hspace{1em}\ge \hspace{1em}0.40\hspace{1em}{\mu }_N^2$]. The numerals on the right-hand side of the $E2$ transitions denote the $B(E2)$ values (in ${10}^{-2}$ $e^2{b}^2$), and those beneath the $M1$ transitions denote the $B(M1)$ values (in ${\mu }_N^2$). Schematic illustrations of the chopsticks configurations are presented below each $\Delta I=2$ $E2$ band. The red and the blue arrows indicate the angular momenta of the neutron and the proton for the bandhead state, respectively. Their vector sum is indicated by black arrows. Four $\Delta I=2$ bands and the band with the bandhead state of $7_1^{+}$ have the configurations shown by the schematic illustrations below.

Figure 12

Absolute values of reduced matrix elements of $M1$ operators: the total reduced matrix elements (M) and the contributions from only the $0h_{11/2}$ orbitals ($M_j$); (a) $9_1^{+}\to 8_1^{+}$; (b) ${10}_1^{+}\to 9_1^{+}$; (c) ${11}_1^{+}\to {10}_1^{+}$; (d) ${13}_1^{+}\to {12}_1^{+}$; (e) ${15}_1^{+}\to {14}_1^{+}$; (f) ${17}_1^{+}\to {16}_1^{+}$; (g) ${12}_1^{+}\to {11}_2^{+}$; (h) ${14}_1^{+}\to {13}_2^{+}$; (i) ${16}_1^{+}\to {15}_{2(3)}^{+}$ for ${}^{130}\mathrm{Cs}$, ${}^{134}\mathrm{La}$, and ${}^{132}\mathrm{La}$ (for ${}^{132}\mathrm{Cs}$); (j) ${11}_2^{+}\to {10}_2^{+}$.

Figure 13

(Color online). The calculated g factors in the PTSM. Solid curves indicate the g factors for the yrast states, and dashed curves indicate those for the yrare states; dotted curves indicate the g factors contributed from only the $0h_{11/2}$ orbitals for the yrast states, and dot-dashed curves indicate those for the yrare states.

Figure 14

(Color online). The effective angles of two angular momenta calculated in the PTSM. Solid curves indicate the effective angles for the yrast states, and dotted curves indicate those for the yrare states.

Figure 15

(Color online). The squares of the core angular momentum calculated in the PTSM.

Figure 16

(Color online). Effective angles of two angular momenta calculated in the PTSM for the yrast states. Solid and dotted curves indicate the effective angles between the core and the neutrons in the $0h_{11/2}$ orbitals and those between the core and protons, respectively.

Figure 17

Band scheme predicted by the PTSM calculation in the weak coupling limit. The narrow arrows indicate $E2$ transitions, and the dashed narrow arrows denote $M1$ transitions. Schematic illustrations of two angular momenta of the unpaired neutron and the unpaired proton are presented below the band scheme. The gray wide arrows indicate the angular momenta of the neutron and the proton. Their vector sum is indicated by black wide arrows. The $\Delta I=2$ bands have the corresponding configurations shown by the schematic illustrations below (chopsticks configurations).