Effective charge quadrupole moments $q^{\mbox{\rm\scriptsize {eff}}}_{20,\alpha}$

Table 1 contains the values of CHF and CRMF effective s.p. charge quadrupole moments $q^{\mbox{\rm\scriptsize {eff}}}_{20,\alpha}$ for a number of s.p. orbitals in the vicinity of the deformed shell gaps at $Z$=58 and $N$=73 (see Figs. 2 and 3). There is an overall excellent agreement between $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ values for the two mean-field approaches employed. In the majority of cases, the uncertainties are small enough to allow determination of effective moments to two significant digits.

Table: Effective s.p. charge quadrupole moments $q^{\mbox{\rm\scriptsize{eff}}}_{20,\alpha}$ (in eb) for the s.p. orbitals active in the $A$$\sim$130 mass region of high- and superdeformation. Calculations were carried out with CHF+SLy4 and CRMF+NL1 approaches. The bare quadrupole moments $q^{\mbox{\rm\scriptsize{bare}}}_{20,\alpha}$ are also shown for CHF+SLy4. Theoretical errors resulting from the multivariate least-square fit are indicated. The results of previous calculations [5] pertaining to the $A$$\sim$150 mass region are displayed for comparison.

State		CHF+SkP	CHF+SkM*	CHF+SLy4				CRMF+NL1		$±$
[ ${\cal N}n_z\Lambda$]$\Omega^r$	$p/h$	$q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$	$q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$	$p/h$	$q^{\mbox{\rm\scriptsize {bare}}}_{20,\alpha}$	$q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$		$q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$		$±$
$\nu$ [402] $\frac{5}{2}^{+i}$	$p$	-0.44	-0.38	$p$	0.0	-0.35	$±$	0.01	-0.26	$±$	0.01
$\nu$ [402] $\frac{5}{2}^{-i}$	$p$	-0.44	-0.38	$p$	0.0	-0.34	$±$	0.02	-0.26	$±$	0.02
$\nu$ [411] $\frac{1}{2}^{+i}$	$h$		-0.18	$h$	0.0	-0.15	$±$	0.02	-0.11	$±$	0.02
$\nu$ [411] $\frac{1}{2}^{-i}$	$h$		-0.15	$h$	0.0	-0.12	$±$	0.01	-0.06	$±$	0.02
$\nu$ [411] $\frac{3}{2}^{+i}$	$h$			$h$	0.0	-0.15	$±$	0.04	-0.13	$±$	0.03
$\nu$ [411] $\frac{3}{2}^{-i}$	$h$			$h$	0.0	-0.11	$±$	0.05	-0.12	$±$	0.03
$\nu$ [413] $\frac{5}{2}^{+i}$	$h$	-0.16		$h$	0.0	-0.13	$±$	0.02	-0.13	$±$	0.03
$\nu$ [413] $\frac{5}{2}^{-i}$	$h$	-0.13		$h$	0.0	-0.12	$±$	0.03	-0.11	$±$	0.02
$\nu$ [523] $\frac{7}{2} ^{+i}$				$h$	0.0	0.03	$±$	0.01	0.05	$±$	0.01
$\nu$ [523] $\frac{7}{2} ^{-i}$				$h$	0.0	0.04	$±$	0.01	0.01	$±$	0.02
$\nu$ [530] $\frac{1}{2}^{+i}$				$p$	0.0	0.22	$±$	0.01	0.17	$±$	0.01
$\nu$ [530] $\frac{1}{2}^{-i}$				$p$	0.0	0.17	$±$	0.01	0.19	$±$	0.01
$\nu$ [532] $\frac{3}{2}^{+i}$				$p$	0.0	0.21	$±$	0.03	--
$\nu$ [532] $\frac{3}{2}^{-i}$				$p$	0.0	0.17	$±$	0.03	--
$\nu$ [532] $\frac{5}{2}^{+i}$				$h$	0.0	0.19	$±$	0.03	0.17	$±$	0.03
$\nu$ [532] $\frac{5}{2}^{-i}$				$h$	0.0	0.24	$±$	0.03	0.38	$±$	0.03
$\nu$ [541] $\frac{1}{2}^{+i}$				$h$	0.0	0.35	$±$	0.03	0.35	$±$	0.02
$\nu$ [541] $\frac{1}{2}^{-i}$				$h$	0.0	0.37	$±$	0.03	0.33	$±$	0.03
$\nu$ 6$_1^{-i}$				$h$	0.0	0.38	$±$	0.01	0.40	$±$	0.01
$\nu$ 6$_2^{+i}$				$p$	0.0	0.36	$±$	0.01	0.36	$±$	0.01
$\nu$ 6$_3^{-i}$	$h$	0.43	0.30	$p$	0.0	0.35	$±$	0.05	--
$\pi$ [301] $\frac{1}{2}^{+i}$	$h$	-0.15	-0.13	$h$	-0.08	0.51	$±$	0.05	--
$\pi$ [404] $\frac{9}{2}^{+i}$	$p$	-0.30	-0.28	$p$	-0.13	-0.32	$±$	0.01	-0.37	$±$	0.01
$\pi$ [404] $\frac{9}{2}^{-i}$	$p$	-0.30	-0.28	$p$	-0.13	-0.32	$±$	0.01	-0.37	$±$	0.01
$\pi$ [411] $\frac{3}{2}^{+i}$	$p$	0.11	0.10	$p$	0.06	-0.05	$±$	0.02	--
$\pi$ [411] $\frac{3}{2}^{-i}$	$p$	0.11	0.10	$p$	0.06	0.00	$±$	0.01	--
$\pi$ [413] $\frac{5}{2}^{-i}$				$p$	0.06	0.28	$±$	0.05	--
$\pi$ [422] $\frac{3}{2}^{+i}$				$h$	0.20	0.33	$±$	0.02	0.33	$±$	0.03
$\pi$ [422] $\frac{3}{2}^{-i}$				$h$	0.22	0.34	$±$	0.02	0.28	$±$	0.02
$\pi$ [532] $\frac{5}{2}^{+i}$				$p$	0.28	0.43	$±$	0.01	0.41	$±$	0.02
$\pi$ [532] $\frac{5}{2}^{-i}$				$p$	0.36	0.56	$±$	0.03	0.54	$±$	0.03
$\pi$ [541] $\frac{1}{2}^{-i}$				$p$	0.40	0.58	$±$	0.02	--
$\pi$ [541] $\frac{3}{2}^{+i}$				$h$	0.34	0.50	$±$	0.01	0.48	$±$	0.01
$\pi$ [541] $\frac{3}{2}^{-i}$				$h$	0.39	0.57	$±$	0.01	0.50	$±$	0.01
$\pi$ [550] $\frac{1}{2}^{-i}$				$h$	0.30	0.49	$±$	0.05	0.47	$±$	0.04

The two lowest neutron intruder orbitals 6$_1^{-i}$ and 6$_2^{+i}$ show significant signature splitting, and their effective charge quadrupole moment values differ by more than 5%. The extracted values confirm the general expectations for the polarization effects exerted by the intruder and extruder states [37,38]. The lowest neutron ${\cal N}$=6 orbitals, 6$_1^{-i}$ and 6$_2^{+i}$, have $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}\simeq$ 0.37eb, which indicates that their occupation drives the nucleus towards larger prolate deformation. The third intruder orbital, 6$_3^{-i}$, although calculated with relatively poor statistics, confirms this trend.

The proton $\pi [404]9/2^{\pm i}$ extruder high-$\Omega$ orbitals are oblate-driving; they have large negative values of $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$. Emptying them polarizes the nucleus towards more prolate-deformed shapes. Interestingly, their $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ values of around $-0.31$eb are close in magnitude to those of the ${\cal N}$=6 neutron intruders, in line with the findings of Ref. [18] that the holes in the proton $g_{9/2}$ orbitals are as important as the particles in the neutron $i_{13/2}$ orbitals in stabilizing the shape at large deformation. Due to their high-$\Omega$ content, the signature splitting of $\pi [404]9/2^{\pm i}$ routhians is extremely small and their $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ values are practically indistinguishable within error bars.

Our study indicates that proton $h_{11/2}$ states, such as $\pi [541]3/2^{\pm i}$ and $\pi [532]5/2^{\pm i}$ active below and above the $Z$=58 shell gap, respectively, play a significant role in the existence of this island of high deformation. Indeed, Table 1 attributes them to effective charge quadrupole moments in excess of 0.45eb - very significant values compared with other states listed.

The downsloping orbital $\pi [541]1/2^{-i}$, originating from mixed $\pi f_{7/2}\oplus h_{9/2}$ subshells, carries a large effective charge quadrupole moment of more than 0.5eb. Although one could expect it to play a role in the formation of large prolate deformation, this state appears too high in energy (above the $Z$=58 shell gap) and would therefore always stay unoccupied in most of the configurations of interest [21,20,19]. On the contrary, the strongly prolate-driving $\pi [550]1/2^{-i}$ orbital carrying $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$$\approx$0.47eb, is always occupied in the bands of interest.

Table 1 compares the values of $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ obtained in the present study with those from the additivity analysis of the SD bands in the $A$$\sim$150 region [5] based on the Skyrme SkP and SkM* energy density functionals. Note that some of the states, which are of particle character in the $A$$\sim$130 region, appear as hole states in the heavier region. For these states, conforming to our definitions of coefficients $c_{\alpha}$ (Sec. 2.2), we inverted signs of values shown in Table 1 of Ref. [5]. With few exceptions, $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ values are similar in both studies: only for the $\pi [301]1/2^{+i}$ orbital does the difference between $A\sim130$ and $A\sim 150$ results exceed 0.1eb. This result strongly suggests that the polarization effects caused by occupying/emptying specific orbitals are mainly due to the general geometric properties of s.p. orbitals and weakly depend on the actual parametrization of the Skyrme energy density functional; minor differences are likely related to interactions between close-lying s.p. states. These observations give strong reasons for combining the two regions into one, and interpreting the entire area of highly and SD rotational states in the mass range $A\sim 128-160$ within the united theoretical framework.

The results for $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ obtained in CHF+SLy4 and CRMF+NL1 models are indeed very similar (see Table 1). Only for the $\nu [402]5/2^{\pm i}$ and $\nu [532]5/2^{-i}$ orbitals, do the differences between $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ values come close to 0.1eb.

Table 1 compares the bare and effective s.p. charge quadrupole moments obtained in CHF+SLy4. In the majority of cases, these quantities differ drastically, underlying the importance of shape polarization effects. Large differences between bare and effective s.p. quadrupole moments have also been found in the CHF+SkP and CHF+SkM* calculations in the $A$$\sim$150 region of superdeformation [5].