Semiclassical unified description of wobbling motion in even-even and even-odd nuclei

A unitary description for wobbling motion in even-even and even-odd nuclei is presented. In both cases compact formulas for wobbling frequencies are derived. The accuracy of the harmonic approximation is studied for the yrast as well as for the excited bands in the even-even case. Important results for the structure of the wave function and its behavior inside the two wells of the potential energy function corresponding to the Bargmann representation are pointed out. Applications to ${}^{158}\mathrm{Er}$ and ${}^{163}\mathrm{Lu}$ reveal a very good agreement with available data. Indeed, the yrast energy levels in the even-even case and the first four triaxial superdeformed bands, TSD1, TSD2, TSD3, and TSD4, are realistically described. Also, the results agree with the data for the $E2$ and $M1$intra- as well as interband transitions. Perspectives for the formalism development and an extensive application to several nuclei from various regions of the nuclides chart are presented.

Figure 1

The $K$ amplitudes supplied by the diagonalization procedure and the coherent state, respectively.

Figure 2

The variable $t$ defined by Eq. (2.36) as a function of $x$ defined by Eq. (2.33).

Figure 3

The variable $x$, defined by Eq. (2.33), as function of $x$ given by Eq. (2.36).

Figure 4

The dependence of the potential on the variable $t$, defined by Eq. (2.38).

Figure 5

Eigenfunction $F_n$ as function of $t$, for $n=0$,1.

Figure 6

The $F_n$ as function of $t$ for $n=2$,3.

Figure 7

The $F_n$ as function of $t$ for $n=4$,5.

Figure 8

Comparison of the lowest exact eigenvalues obtained by diagonalization of $H_R$, the yrast solutions of the Schrodinger equation, and the yrast wobbling energies.

Figure 9

The maximal $K$ amplitude for the $i\mathrm{th}$ exact eigenfunction for a given $I$.

Figure 10

The lowest sixth solutions for a given angular momentum $I$, given by diagonalizing $H_R$ and by solving the Schrodinger equation, respectively.

Figure 11

The 7th to the 11th solutions for a given angular momentum $I$, given by diagonalizing $H_R$ and by solving the Schrodinger equation, respectively.

Figure 12

The first four solutions for the Schrodinger equations and the first four wobbling energies, respectively.

Figure 13

The fourth to the eighth solutions for the Schrodinger equations and the fourth to the seventh wobbling energies, respectively.

Figure 14

Results (Th.) obtained with Eq. (2.42) are compared with experimental data (Ex.) taken from Ref. [55].

Figure 15

The wobbling frequency for ${}^{158}\mathrm{Er}$ as function of the angular momentum.

Figure 16

The moments of inertia are reprezented as function of $\gamma$. The rigid value of $\gamma$, i.e., ${\gamma }_m={17}^0$, is also specified by a vertical line.

Figure 17

The canonicity parameter $k_1$ given by Eq. (3.28) as function of the total angular momentum.

Figure 18

The canonicity parameter $k_2$ given by Eq. (3.28) as function of the total angular momentum.

Figure 19

The phonon energy satisfying Eq. (3.35) as function of the total angular momentum $I$ for $j=\pi i13/2$.

Figure 20

The two solutions of Eq. (3.35), as function of the total angular momentum, for $j=\pi h_{9/2}$.

Figure 21

The excitation energies of the band TSD1. They correspond to the parameters $1/{\mathcal{J}}_0$ and $s$ obtained by fitting the experimental corresponding data taken from Refs. [31, 32].

Figure 22

The excitation energies of the band TSD2. They correspond to the parameters $1/{\mathcal{J}}_0$ and $s$ obtained by fitting the experimental corresponding data taken from Refs. [31, 32].

Figure 23

The excitation energies of the band TSD3. They correspond to the parameters $1/{\mathcal{I}}_0$ and $s$ obtained by fitting the experimental corresponding data taken from Refs. [31, 32].

Figure 24

The excitation energies of the band TSD4. They correspond to the parameters $1/{\mathcal{J}}_0$ and $s$ obtained by fitting the corresponding experimental data taken from Refs. [31, 32].

Figure 25

The function $f\left(\mathrm{\Omega }\right)$, defined in Eq. (3.34), is plotted together with the straight line, parallel with the abscissa axis having the ordinate equal to 1. The intersection of the two curves gives ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$. The frequencies for the uncoupled oscillators are just the poles of $f\left(\mathrm{\Omega }\right)$.