Rotational motion of triaxially deformed nuclei studied by the microscopic angular-momentum-projection method. II. Chiral doublet band

In the sequel of the present study, we have investigated the rotational motion of triaxially deformed nucleus by using the microscopic framework of angular-momentum projection. The Woods-Saxon potential and the schematic separable-type interaction are employed as a microscopic Hamiltonian. As the first example, nuclear wobbling motion was studied in detail in part I of the series. This second part reports on another interesting rotational mode, chiral doublet bands: two prototype examples, ${}^{128}\mathrm{Cs}$ and ${}^{104}\mathrm{Rh}$, are investigated. It is demonstrated that the doublet bands naturally appear as a result of the calculation in this fully microscopic framework without any kind of core, and they have the characteristic properties of the $B\left(E2\right)$ and $B\left(M1\right)$ transition probabilities, which are expected from the phenomenological triaxial particle-rotor coupling model.

Figure 1

Schematic figure representing the selection rules of the electromagnetic transitions for the ideal chiral geometry considered in Ref. [12], where the thick (thin) arrow denotes the large (small) transition rate.

Figure 2

Cranking moments of inertia of the three intrinsic axes, $x,y$, and $z$, which are the short, medium, and long axes, (denoted by dotted, solid, and dashed lines, respectively) as functions of the triaxiality parameter $\gamma$ for the even-even core nucleus ${}^{128}\mathrm{Xe}$ of ${}^{128}\mathrm{Cs}$. The deformation parameters are ${\beta }_2=0.30,{\beta }_4=0.0$ and the pairing gaps are ${\mathrm{\Delta }}_{\mathrm{n}}=0.85$ MeV and ${\mathrm{\Delta }}_{\mathrm{p}}=1.07$ MeV, which are are employed for the study of chirality in ${}^{128}\mathrm{Cs}$. The $\gamma$ parameter of the Woods-Saxon potential is utilized in (a) and that of the density distribution, Eq. (5), in (b).

Figure 3

The same as Fig. 2 but for the even-even core nucleus ${}^{104}\mathrm{Pd}$ of ${}^{104}\mathrm{Rh}$. The deformation parameters are ${\beta }_2=0.25,{\beta }_4=0.0$ and the pairing gaps are ${\mathrm{\Delta }}_{\mathrm{n}}=0.95$ MeV and ${\mathrm{\Delta }}_{\mathrm{p}}=0.76$ MeV, which are employed for the study of chirality in ${}^{104}\mathrm{Rh}$.

Figure 4

Energy spectrum for ${}^{128}\mathrm{Cs}$ calculated by the angular-momentum-projection method from the noncranked mean-field with the configuration (i) in Eq. (13). A rigid-rotor reference energy $0.013I(I+1)$ MeV is subtracted.

Figure 5

Left: Energy spectrum for ${}^{128}\mathrm{Cs}$ calculated by the angular-momentum-projected configuration-mixing method with the two mean-field configurations (i) and (ii) in Eq. (13). Right: Comparison of the calculated and experimental chiral doublet bands in ${}^{128}\mathrm{Cs}$. Experimental data are taken from Ref. [27].

Figure 6

The calculated expectation values of the angular-momentum vector in the intrinsic frame for the configuration-mixed calculation of ${}^{128}\mathrm{Cs}$ corresponding to the spectrum in Fig. 5. The left panel shows the expectation values of the total vector for the yrast (filled symbols) and yrare (open symbols) $\mathrm{\Delta }I=1$ bands, while the right panel shows the neutron (filled symbols) and proton (open symbols) contributions in Eq. (8) separately for the yrast band. Note that the $x,y$, and $z$ axes are the short, medium, and long axes, respectively.

Figure 7

Angular-momentum vectors in the intrinsic frame for the $I=11$, 16, and 21 yrast (top) and yrare (bottom) states in ${}^{128}\mathrm{Cs}$ according to the expectation values shown in Fig. 6.

Figure 8

The calculated $B(E2:I\to I-2)$ values for the yrast band (solid lines) and the yrare band (dashed lines) as functions of spin in ${}^{128}\mathrm{Cs}$. The left and right panels show the in-band and out-of-band values, respectively. Note that the ordinate scale in the right panel is different from that in the left panel. Shown are the results of the configuration-mixed projection calculation corresponding to Fig. 5.

Figure 9

The calculated $B(E2:I\to I-1)$ and $B(M1:I\to I-1)$ values for the yrast band (solid lines) and the yrare band (dashed lines) as functions of spin in ${}^{128}\mathrm{Cs}$. The top (bottom) panels show the in-band (out-of-band) transition rates. These are the results of the configuration-mixed projection calculation corresponding to Fig. 5.

Figure 10

Left: The calculated $B(M1:I\to I-1)/B(E2:I\to I-2)$ ratios inside the yrast band (solid lines) and inside the yrare band (dashed lines) as functions of spin for ${}^{128}\mathrm{Cs}$. Right: The ratio of the $I\to I-1$ in-band and out-of-band $M1$ transitions, $B{\left(M1\right)}_{\mathrm{in}}/B{\left(M1\right)}_{\mathrm{out}}$, where the in-band transitions are those inside the yrare band and the out-of-band transitions are those from the yrare to the yrast band. As for $B{\left(M1\right)}_{\mathrm{in}}/B{\left(M1\right)}_{\mathrm{out}}$ the experimental data are also included [27]. These are the results of projection with the configuration mixing corresponding to Fig. 5.

Figure 11

The same as Fig. 10 but the calculation with using $\gamma \left(\mathrm{WS}\right)={25}^{\circ }$.

Figure 12

Left: Energy spectrum for ${}^{104}\mathrm{Rh}$ calculated by the angular-momentum-projection method from the noncranked mean-field with the configuration (i) in Eq. (13). A rigid-rotor reference energy of $0.017I(I+1)$ MeV is subtracted. The lowest fourteen bands for both even-$I$ and odd-$I$ sequences are shown. Right: Comparison of the calculated and experimental chiral doublet bands in ${}^{104}\mathrm{Rh}$. Experimental data are taken from Ref. [37].

Figure 13

The calculated expectation values of the angular-momentum vector in the intrinsic frame for the noncranked mean-field of ${}^{104}\mathrm{Rh}$ corresponding to the spectrum in Fig. 12. The left panel shows the expectation values of the total vector for the yrast (filled symbols) and yrare (open symbols) $\mathrm{\Delta }I=1$ bands, while the right panel shows the neutron (filled symbols) and proton (open symbols) contributions in Eq. (8) separately for the yrast band. Note that the $x,y$, and $z$ axes are the short, medium, and long axes, respectively.

Figure 14

The calculated $B(E2:I\to I-2)$ values for the yrast band (solid lines) and the yrare band (dashed lines) as functions of spin in ${}^{104}\mathrm{Rh}$. The left and right panels show the in-band and out-of-band transition rates, respectively. Note that the ordinate scale in the right panel is different from that in the left panel. These are the results of projection from the non-cranked lowest configuration (i) in Eq. (13) corresponding to Fig. 12.

Figure 15

The calculated $B(E2:I\to I-1)$ and $B(M1:I\to I-1)$ values for the yrast band (solid lines) and the yrare band (dashed lines) as functions of spin in ${}^{104}\mathrm{Rh}$. The top (bottom) panels show the in-band (out-of-band) transition rates. These are the results of projection from the noncranked lowest configuration (i) in Eq. (13) corresponding to Fig. 12.

Figure 16

Left: The calculated $B(M1:I\to I-1)/B(E2:I\to I-2)$ ratios inside the yrast band (solid lines) and inside the yrare band (dashed lines) as functions of spin for ${}^{104}\mathrm{Rh}$. Right: The ratio of the $I\to I-1$ in-band and out-of-band $M1$ transitions, $B{\left(M1\right)}_{\mathrm{in}}/B{\left(M1\right)}_{\mathrm{out}}$, where the in-band transitions are those inside the yrare band and the out-of-band transitions are those from the yrare to the yrast band. These are the results of projection from the noncranked mean-field corresponding to Fig. 12.

Figure 17

The same as Fig. 16 but the calculation with using $\gamma \left(\mathrm{WS}\right)={25}^{\circ }$.