Characteristics of collectivity along the yrast line in even-even tungsten isotopes

The collective nature of high-spin yrast states in even-even ${}^{160-190}\mathrm{W}$ isotopes was systematically investigated by means of pairing self-consistent Woods-Saxon-Strutinsky calculations using the total Routhian surface approach in $({\beta }_2,\gamma ,{\beta }_4)$ deformation space. The calculated ground-state deformations are consistent with previous calculations and available experimental data. The deformation energy curves are presented to show the shape and softness evolutions, in particular in the triaxial direction. The backbending or upbending behavior in moment of inertia is attributed to band crossing. It is found that the neutron rotation alignment is preferred for most of the W isotopes (e.g., in ${}^{164-180}\mathrm{W}$), while in other nuclei the competition between the neutron and proton alignments may occur, even the proton alignment is favored in the very neutron-deficient nucleus ${}^{160}\mathrm{W}$. In addition, the evolution and transition between vibrational and rotational collective modes along the yrast line are investigated on the basis of the new centipedelike E-GOS (E-Gamma Over Spin) curves introduced by us, which to some extent explains the existing differences (e.g., in the moment of inertia) between theory and experiment.

Figure 1

The available phenomenological quantities $R_{4/2}$ ratio (a), P factor (b), and $E_S/E\left(2_1^{+}\right), E_S=E\left(2_2^{+}\right)-E\left(4_1^{+}\right)$ (c) for even-even W isotopes as a function of the neutron number $N$. The dashed lines represent some corresponding critical points. See text for more details. The data are taken from Refs. [8, 41, 42].

Figure 2

Deformation Routhian curves against ${\beta }_2$ for even-even ${}^{160-190}\mathrm{W}$ nuclei at several selected rotational frequencies $\omega$ = 0.00, 0.15, 0.30, and 0.45 $\mathrm{MeV}/\hslash$. At each ${\beta }_2$ point, the energy was minimized with respect to $\gamma$ and ${\beta }_4$.

Figure 3

Similar to Fig. 2 but against $\gamma$ deformation.

Figure 4

The experimental (filled circle symbols) and calculated (open square symbols) kinematic moments of inertia $J^{\left(1\right)}$ for even-even nuclei ${}^{160-190}\mathrm{W}$ as a function of the rotational frequency $\hslash \omega$. The experimental data are taken from Ref. [41].

Figure 5

The calculated aligned angular momenta $I_x$ (squares), including the proton $I_{xp}$ (circles) and neutron $I_{xn}$ (triangles) components, for even-even nuclei ${}^{160-190}\mathrm{W}$ as a function of the rotational frequency $\hslash \omega$.

Figure 6

Similar to Fig. 4, but for three selected nuclei ${}^{164,172,180}\mathrm{W}$. The open symbols (squares and triangles) denote the TRS results (with and without QQ-pairing interaction, respectively). The left (a) and right (b) panels show the calculated results with the pairing strengths obtained by the average gap method ($G_0$) and those adjusted, on average, for mean-field and blocking effects ($G=F{G}_0$), respectively. For reasons of simplicity, we use the constant factors $F_{\nu }=1.08$ (neutrons) and $F_{\pi }=1.05$ (protons) for the local mass region of the studied nuclei [18].

Figure 7

(a) The typical E-GOS curves for a perfect harmonic vibrator, $\gamma$-soft, and axially symmetric rotors with first $2^{+}$ excitations of 500, 300, and 100 keV, respectively [68]. (b) Similar to (a), four sets of such E-GOS curves at the selected spins $I=4,10,16$, and $22\hslash$, together with the experimental data (filled circles) for ${}^{180}\mathrm{W}$. Note that at each selected spin point, the red, green, and blue lines denote the E-GOS curves with vibrational, $\gamma$-soft, and rotational modes, respectively.

Figure 8

Similar to the ${}^{180}\mathrm{W}$ case of Fig. 7, but the centipedelike E-GOS curves along the yrast sequences for even-even ${}^{160-190}\mathrm{W}$ isotopes.