Systematic study of multi-quasiparticle $K$-isomeric bands in tungsten isotopes by the extended projected shell model

Background: The interplay between collective and single-particle degrees of freedom is an important structure aspect to study. The nuclei in the $A\approx 180$ mass region are often denoted as good examples to study such problems because these nuclei are known to exhibit many rotational bands based on multi-quasiparticle $K$ isomers.

Purpose: A large set of high-quality experimental data on high-$K$ isomeric states in the $A\approx 180$ mass region has accumulated. A systematic description of them is a theoretical challenge as it requires a method going beyond the usual mean field with multi-quasiparticle configurations built in the shell-model basis. The $K$-isomer data provide an ideal testing ground for theoretical models.

Method: The recently extended projected shell model (PSM) by the Pfaffian method is employed with a sufficiently large configuration space including up to 10 quasiparticles. The restoration of rotational symmetry which is broken in the deformed mean field is obtained by means of angular-momentum projection. With axial symmetry in the basis deformation, each multi-quasiparticle state, classified by a $K$ quantum number, represents the major component of a rotational $K$ band. Shell-model diagonalization in such a projected basis defines the $K$ mixing, which is the key ingredient of the present method.

Results: Quasiparticle structure and rotational properties of high-$K$ isomers in even-even neutron-rich ${}^{174-186}\mathrm{W}$ isotopes are described. The rotational evolution of the yrast and near-yrast bands is discussed with successive band crossings. Multi-quasiparticle $K$ isomers and associated rotational bands in each W isotope are studied with detailed quasiparticle configurations given. Electromagnetic transition properties are also studied and the calculated $B\left(E2\right),B\left(M1\right)$, and $g$-factors are compared with experiment if data exist.

Conclusions: Many nuclei of the $A\approx 180$ mass region exhibit properties of an axially symmetric shape and $K$ is approximately a good quantum number. For such nuclei, the extended PSM assuming an axially symmetric basis but including $K$ mixing through diagonalization of the two-body Hamiltonian is an appropriate method to study multi-quasiparticle $K$ isomers and $K$ violations in these states. For special examples where one finds highly $K$-forbidden transitions the present model needs to be further improved.

Figure 1

Experimental ratio of the first $4^{+}$ and $2^{+}$ energies for W isotopic chain. Dashed horizontal lines are plotted at 3.33 and 2.5 for guidance, representing an axially deformed rotor and triaxial rotor, respectively .

Figure 2

Band diagrams for (a) ${}^{174}\mathrm{W}$, (b) ${}^{176}\mathrm{W}$, (c) ${}^{178}\mathrm{W}$, (d) ${}^{180}\mathrm{W}$, (e)${}^{182}\mathrm{W}$, (f) ${}^{184}\mathrm{W}$, and (g) ${}^{186}\mathrm{W}$. Note that only even-spin states are plotted for clarity in these curves. See the text for details.

Figure 3

Calculated moments of inertia for the yrast band of (a) ${}^{174}\mathrm{W}$, (b) ${}^{176}\mathrm{W}$, (c) ${}^{178}\mathrm{W}$, (d) ${}^{180}\mathrm{W}$, (e) ${}^{182}\mathrm{W}$,(f) ${}^{184}\mathrm{W}$, and (g) ${}^{186}\mathrm{W}$.

Figure 4

Comparison of calculated energies with available data for ${}^{174}\mathrm{W}$. Experimental values are taken from Refs. [42, 44, 45].

Figure 5

Comparison of calculated energies with available data for ${}^{176}\mathrm{W}$. Experimental values are taken from Refs. [46, 47].

Figure 6

Comparison of calculated energies with available data for ${}^{178}\mathrm{W}$. Experimental values are taken from Refs. [49, 50].

Figure 7

Comparison of calculated energies with available data for ${}^{180}\mathrm{W}$. Experimental values are taken from Ref. [55].

Figure 8

Comparison of calculated energies with available data for ${}^{182}\mathrm{W}$. Experimental values are taken from Refs. [56, 57, 58].

Figure 9

Comparison of calculated energies with available data for ${}^{184}\mathrm{W}$. Experimental values are taken from Refs. [63, 64].

Figure 10

Comparison of calculated energies with available data for ${}^{186}\mathrm{W}$. Experimental values are taken from Refs. [66].

Figure 11

Comparison of calculated $B\left(E2\right)$ values (in W.u.) with available data for the yrast band of (a) ${}^{174}\mathrm{W}$ (data taken from Ref. [45]), (b) ${}^{176}\mathrm{W}$, (c) ${}^{178}\mathrm{W}$ (data taken from Ref. [50]), (d) ${}^{180}\mathrm{W}$ (data taken from Ref. [55]), (e)${}^{182}\mathrm{W}$ (data taken from Ref. [58]), (f) ${}^{184}\mathrm{W}$ (data taken from Ref. [64]), and (g) ${}^{186}\mathrm{W}$ (data taken from Ref. [66]).

Figure 12

Comparison of calculated $g$-factor values with available data for the yrast band of (a) ${}^{174}\mathrm{W}$, (b) ${}^{176}\mathrm{W}$, (c) ${}^{178}\mathrm{W}$, (d) ${}^{180}\mathrm{W}$, (e)${}^{182}\mathrm{W}$, (f) ${}^{184}\mathrm{W}$, and (g) ${}^{186}\mathrm{W}$. Experimental values are taken from Ref. [72].

Figure 13

Comparison of calculated $B\left(M1\right)/B\left(E2\right)$ ratios of sideband 1 in Fig. 5 with available data for ${}^{176}\mathrm{W}$. Experimental values are taken from Ref. [46].