Effective quadrupole moments $q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ and $q_{22,\alpha}^{\mbox{\rm\scriptsize {eff}}}$

Figure 4: Experimental (closed symbols with error bars) and calculated (CRMF+NL1, open symbols) differential transition quadrupole moments for highly deformed bands in Ce, Pr, Nd, Pm, and Sm isotopes. The experimental data were taken from Refs. [21,20] and references quoted therein. The values of $\delta Q_t$ for SD band in $^{142}$Sm are shown in the inset. Dashed lines are drawn to guide the eye.

Table 2 displays the calculated effective s.p. transition quadrupole moments $q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$, cf. definitions (15-16). Based on the additivity principle, these values can be used to predict the total charge transition moments $Q_{t}(k)$ in highly deformed and SD bands of $A\sim130$ nuclei:

$$Q_{t}(k)=Q_{t}^{\mbox{\rm\scriptsize {core}}} + \sum_{\alpha}c_{\alpha}(k) q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}} ,$$ (30)

where the calculated CHF+SLy4 value for the core configuration in $^{131}$Cs is

$$Q_{t}^{\mbox{\rm\scriptsize {core}}} = 7.64\,\mbox{eb} .$$ (31)

Since the total calculated values are less precise than the relative ones which define the effective s.p. transition quadrupole moments $q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$, one may alternatively use in Eq. (30) the measured value [39],

$$Q_{t}^{\mbox{\rm\scriptsize {core,exp}}} = 7.4(3)\,\mbox{eb}.$$ (32)

Theoretical estimates of the total charge transition moments $Q_{t}(k)$ allow for predictions of $B(E2)$ values

$$B(E2)(I\rightarrow{I-2},k)= \frac{5}{16\pi} e^2 \langle I0\,20\vert I-2\,0\rangle Q^2_{t}(k),$$ (33)

and lifetimes [40].

In CHF+SLy4, the uncertainties of $q_{22,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ appear to be larger than those for $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$. In CRMF+NL1, however, those uncertainties are similar. This can be traced back to the different $\gamma$-softness of potential energy surfaces in CHF+SLy4 and CRMF+NL1 (see Ref. [18] and references quoted therein for the results obtained in different approaches); current analysis revealing large uncertainties for $q_{22,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ suggests that the potential energy surfaces are softer (and, thus less localized) in the CHF+SLy4 approach.

Although the values of $q_{22,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ are generally much smaller than $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$, large uncertainties in the determination of certain moments $q_{22,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ (especially, for $\nu [411]3/2^{\pm i}$, $\nu [532]5/2^{\pm i}$, $\nu 6_3^{-i}$, $\pi [301]1/2^{+i}$, and $\pi [550]1/2^{-i}$ orbitals, for which the errors exceed 0.1eb in the CHF+SLy4 approach) can lead to the deterioration of predicted $q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$. On the other hand, in many cases the uncertainties in $q_{22,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ are smaller than the experimental error bars; hence, they are less relevant when comparison with experiment is carried out. Currently available experimental data on relative transition quadrupole moments agree reasonably well with the CHF+SLy4 results [21,20].

Table 2 compares $q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ values obtained in CHF+SLy4 and CRMF+NL1 models. The results for proton orbitals are similar in both approaches: the differences between respective $q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ values do not exceed 0.1eb. Larger differences are seen for the neutrons: for about 50 percent of calculated orbitals ( $\nu [402]5/2^{\pm i}$, $\nu [411]1/2^{\pm i}$, $\nu [411]3/2^{+i}$, $\nu [413]5/2^{-i}$, $\nu [530]1/2^{+i}$, and $\nu [532]5/2^{+i}$), the difference between $q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$ values in CHF+SLy4 and CRMF+NL1 exceeds 0.1eb. Interaction (mixing) between those close-lying states (see Fig. 2), predicted differently in the two approaches, is the most likely reason for the deviations seen.

The results of CHF+SLy4 were compared with experimental transition moments in Refs. [21,20]. Here, we show in Fig. 4 a comparison between CRMF+NL1 and experiment for the relative transition quadrupole moments $\delta Q_t(k)$ in different highly deformed and SD bands in nuclei with $Z$=57-62 involving $i_{13/2}$ neutrons and/or $g_{9/2}$ proton holes. The agreement between experiment and theory is quite remarkable with all the experimental trends discussed in Refs. [21,20] well reproduced by calculations. One should note that the CRMF and CHF results are close to each other. The general pattern of decreasing $Q_t$ with increasing $Z$ and $N$ is consistent with the general expectation that as one adds particles above a deformed shell gap, the deformation-stabilizing effect of the gap is diminished. This trend continues until a new ``magic'' deformed number is reached. Such a situation occurs when going from $^{132}$Ce towards $Z$=62 and $N$=80 ($^{142}$Sm), where a large jump in transition quadrupole moment takes place marking the point at which it becomes energetically favorable to fill the high-$j$ $\pi i_{13/2}$ and $\nu j_{15/2}$ orbitals responsible for the existence of the $A\sim 142$ SD island.

It is gratifying to see that CRMF+NL1 reproduces the value of $Q_t$ in $^{142}$Sm based on the $^{131}$Ce core (see inset in Fig. 4). Earlier on, it was demonstrated in Refs. [21,20,9] that this $Q_t$ value can be also reproduced within CHF using either a $^{131}$Ce or a $^{152}$Dy core.

Table: Effective s.p. charge quadrupole moments $q_{20,\alpha}^{\mbox{\rm\scriptsize{eff}}}$ and $q_{22,\alpha}^{\mbox{\rm\scriptsize{eff}}}$ as well as the transition quadrupole moments $q_{t,\alpha}^{\mbox{\rm\scriptsize{eff}}}$ (all in eb) calculated in CHF+SLy4 and CRMF+NL1.

			$±$				$±$				$±$			$±$				$±$				$±$
			$±$				$±$				$±$			$±$				$±$				$±$
State		CSHF + SLy4						CRMF + NL1						$±$				$±$				$±$
[ ${\cal N}n_z\Lambda$]$\Omega^r$		$q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$		$q_{22,\alpha}^{\mbox{\rm\scriptsize {eff}}}$		$q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$		$q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$		$q_{22,\alpha}^{\mbox{\rm\scriptsize {eff}}}$		$q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$		$±$				$±$				$±$
$\nu$[402] $\frac{5}{2}^{+i}$		-0.35	$±$	0.01		0.14	$±$	0.06		-0.04	$±$	0.04	-0.26	$±$	0.02		-0.02	$±$	0.01		-0.25	$±$	0.02
$\nu$[402] $\frac{5}{2}^{-i}$		-0.34	$±$	0.02		0.08	$±$	0.08		-0.38	$±$	0.05	-0.26	$±$	0.02		-0.07	$±$	0.02		-0.22	$±$	0.03
$\nu$[411] $\frac{1}{2}^{+i}$		-0.15	$±$	0.02		-0.24	$±$	0.10		-0.01	$±$	0.06	-0.11	$±$	0.02		0.09	$±$	0.02		-0.16	$±$	0.02
$\nu$[411] $\frac{1}{2}^{-i}$		-0.12	$±$	0.01		0.06	$±$	0.06		-0.16	$±$	0.04	-0.06	$±$	0.02		-0.17	$±$	0.02		0.04	$±$	0.02
$\nu$[411] $\frac{3}{2}^{+i}$		-0.15	$±$	0.04		0.20	$±$	0.20		-0.26	$±$	0.12	-0.13	$±$	0.03		-0.02	$±$	0.03		-0.11	$±$	0.03
$\nu$[411] $\frac{3}{2}^{-i}$		-0.11	$±$	0.05		-0.05	$±$	0.24		-0.08	$±$	0.15	-0.12	$±$	0.03		0.02	$±$	0.03		-0.12	$±$	0.03
$\nu$[413] $\frac{5}{2}^{+i}$		-0.13	$±$	0.02		-0.05	$±$	0.10		-0.10	$±$	0.06	-0.13	$±$	0.03		-0.04	$±$	0.03		-0.10	$±$	0.03
$\nu$[413] $\frac{5}{2}^{-i}$		-0.12	$±$	0.03		-0.12	$±$	0.13		-0.05	$±$	0.08	-0.11	$±$	0.02		0.15	$±$	0.03		-0.20	$±$	0.03
$\nu$[523] $\frac{7}{2} ^{+i}$		0.03	$±$	0.01		-0.00	$±$	0.05		0.03	$±$	0.03	0.05	$±$	0.01		0.00	$±$	0.01		0.04	$±$	0.01
$\nu$[523] $\frac{7}{2} ^{-i}$		0.04	$±$	0.01		-0.01	$±$	0.05		0.05	$±$	0.03	0.01	$±$	0.02		-0.00	$±$	0.02		0.01	$±$	0.02
$\nu$[530] $\frac{1}{2}^{+i}$		0.22	$±$	0.01		-0.21	$±$	0.05		0.34	$±$	0.03	0.17	$±$	0.01		-0.09	$±$	0.01		0.22	$±$	0.01
$\nu$[530] $\frac{1}{2}^{-i}$		0.17	$±$	0.01		-0.01	$±$	0.05		0.18	$±$	0.03	0.19	$±$	0.01		0.10	$±$	0.01		0.13	$±$	0.01
$\nu$[532] $\frac{3}{2}^{+i}$		0.21	$±$	0.03		0.21	$±$	0.13		0.09	$±$	0.08	--		--		--					$±$
$\nu$[532] $\frac{3}{2}^{-i}$		0.17	$±$	0.03		0.03	$±$	0.13		0.15	$±$	0.08	--		--		--					$±$
$\nu$[532] $\frac{5}{2}^{+i}$		0.19	$±$	0.03		-0.08	$±$	0.20		0.24	$±$	0.12	0.17	$±$	0.03		-0.02	$±$	0.03		0.18	$±$	0.03
$\nu$[532] $\frac{5}{2}^{-i}$		0.24	$±$	0.03		-0.01	$±$	0.20		0.25	$±$	0.12	0.38	$±$	0.03		0.00	$±$	0.03		0.38	$±$	0.03
$\nu$[541] $\frac{1}{2}^{+i}$		0.35	$±$	0.03		-0.04	$±$	0.13		0.38	$±$	0.08	0.35	$±$	0.02		-0.00	$±$	0.02		0.35	$±$	0.03
$\nu$[541] $\frac{1}{2}^{-i}$		0.37	$±$	0.03		0.01	$±$	0.14		0.36	$±$	0.08	0.33	$±$	0.03		0.04	$±$	0.03		0.30	$±$	0.03
$\nu$ 6$_{1} ^{-i}$		0.38	$±$	0.01		0.21	$±$	0.03		0.26	$±$	0.02	0.40	$±$	0.01		0.12	$±$	0.01		0.33	$±$	0.01
$\nu$ 6$_{2} ^{+i}$		0.36	$±$	0.01		-0.01	$±$	0.04		0.37	$±$	0.03	0.36	$±$	0.01		-0.01	$±$	0.01		0.37	$±$	0.01
$\nu$ 6$_{3} ^{-i}$		0.35	$±$	0.05		-0.06	$±$	0.22		0.38	$±$	0.13	--		--		--					$±$
$\pi$[301] $\frac{1}{2}^{+i}$		0.51	$±$	0.05		-0.10	$±$	0.24		0.57	$±$	0.14	--		--		--					$±$
$\pi$[404] $\frac{9}{2}^{+i}$		-0.32	$±$	0.01		0.10	$±$	0.04		-0.38	$±$	0.02	-0.37	$±$	0.01		0.02	$±$	0.01		-0.38	$±$	0.01
$\pi$[404] $\frac{9}{2}^{-i}$		-0.32	$±$	0.01		0.09	$±$	0.04		-0.37	$±$	0.02	-0.37	$±$	0.01		0.02	$±$	0.01		-0.38	$±$	0.01
$\pi$[411] $\frac{3}{2}^{+i}$		-0.05	$±$	0.02		0.10	$±$	0.07		-0.10	$±$	0.05	--		--		--					$±$
$\pi$[411] $\frac{3}{2}^{-i}$		0.00	$±$	0.01		-0.22	$±$	0.07		0.12	$±$	0.04	--		--		--					$±$
$\pi$[422] $\frac{3}{2}^{+i}$		0.33	$±$	0.02		-0.27	$±$	0.10		0.48	$±$	0.06	0.33	$±$	0.03		-0.13	$±$	0.02		0.40	$±$	0.03
$\pi$[422] $\frac{3}{2}^{-i}$		0.34	$±$	0.02		0.14	$±$	0.10		0.25	$±$	0.06	0.28	$±$	0.02		0.16	$±$	0.02		0.19	$±$	0.02
$\pi$[532] $\frac{5}{2}^{+i}$		0.43	$±$	0.01		-0.05	$±$	0.05		-0.46	$±$	0.03	0.41	$±$	0.02		-0.04	$±$	0.01		0.43	$±$	0.02
$\pi$[532] $\frac{5}{2}^{-i}$		0.56	$±$	0.03		-0.07	$±$	0.09		0.60	$±$	0.05	0.54	$±$	0.03		0.05	$±$	0.03		0.51	$±$	0.04
$\pi$[541] $\frac{1}{2}^{-i}$		0.58	$±$	0.02		-0.01	$±$	0.10		0.59	$±$	0.06	--		--		--					$±$
$\pi$[541] $\frac{3}{2}^{+i}$		0.50	$±$	0.01		-0.05	$±$	0.06		0.52	$±$	0.04	0.48	$±$	0.01		-0.10	$±$	0.01		0.54	$±$	0.01
$\pi$[541] $\frac{3}{2}^{-i}$		0.57	$±$	0.01		-0.12	$±$	0.04		0.63	$±$	0.03	0.50	$±$	0.01		-0.10	$±$	0.01		0.56	$±$	0.01
$\pi$[550] $\frac{1}{2}^{-i}$		0.49	$±$	0.05		-0.06	$±$	0.22		0.52	$±$	0.14	0.47	$±$	0.04		-0.02	$±$	0.04		0.48	$±$	0.04
			$±$				$±$				$±$			$±$				$±$				$±$

Table: Effective s.p. angular momentum alignments $j^{\mbox{\rm\scriptsize{eff}}}_{\alpha}$ (in $\hbar$) of the active orbitals calculated in CHF+SLy4 and CRMF+NL1. In the second column, the bare s.p. angular momenta $j^{\mbox{\rm\scriptsize{bare}}}_{\alpha}$, calculated with CHF+SLy4 are also shown.

			$±$			$±$
			$±$			$±$
State	CHF+SLy4			CRMF+NL1		$±$
[ ${\cal N}n_z\Lambda$]$\Omega^r$	$j^{\mbox{\rm\scriptsize {bare}}}_{\alpha}$	$j^{\mbox{\rm\scriptsize {eff}}}_{\alpha}$		$j^{\mbox{\rm\scriptsize {eff}}}_{\alpha}$		$±$
$\nu$ [402] $\frac{5}{2}^{+i}$	$-$0.528	0.58	$±$	0.14	0.47	$±$	0.15
$\nu$ [402] $\frac{5}{2}^{-i}$	$-$0.493	0.51	$±$	0.20	0.38	$±$	0.26
$\nu$ [411] $\frac{1}{2}^{+i}$	0.411	0.67	$±$	0.24	0.64	$±$	0.17
$\nu$ [411] $\frac{1}{2}^{-i}$	0.380	0.40	$±$	0.15	0.09	$±$	0.16
$\nu$ [411] $\frac{3}{2}^{+i}$	$-$0.092	1.72	$±$	0.46	1.35	$±$	0.29
$\nu$ [411] $\frac{3}{2}^{-i}$	0.077	0.56	$±$	0.57	1.08	$±$	0.29
$\nu$ [413] $\frac{5}{2}^{+i}$	$-$0.316	$-$0.10	$±$	0.23	0.44	$±$	0.27
$\nu$ [413] $\frac{5}{2}^{-i}$	$-$0.428	0.12	$±$	0.30	0.14	$±$	0.26
$\nu$ [523] $\frac{7}{2} ^{+i}$	$-$0.908	$-$1.10	$±$	0.10	$-$1.24	$±$	0.12
$\nu$ [523] $\frac{7}{2} ^{-i}$	$-$0.974	$-$1.19	$±$	0.12	$-$0.92	$±$	0.18
$\nu$ [530] $\frac{1}{2}^{+i}$	1.548	1.19	$±$	0.11	1.86	$±$	0.09
$\nu$ [530] $\frac{1}{2}^{-i}$	0.564	0.88	$±$	0.11	0.93	$±$	0.10
$\nu$ [532] $\frac{3}{2}^{+i}$	0.171	$-$0.34	$±$	0.30	--
$\nu$ [532] $\frac{3}{2}^{-i}$	0.835	0.44	$±$	0.31	--
$\nu$ [532] $\frac{5}{2}^{+i}$	$-$0.331	$-$0.89	$±$	0.46	$-$0.95	$±$	0.29
$\nu$ [532] $\frac{5}{2}^{-i}$	0.417	$-$1.06	$±$	0.46	$-$1.29	$±$	0.30
$\nu$ [541] $\frac{1}{2}^{+i}$	1.793	0.92	$±$	0.31	0.95	$±$	0.25
$\nu$ [541] $\frac{1}{2}^{-i}$	0.466	0.89	$±$	0.32	$-$0.34	$±$	0.28
$\nu$ 6$_1^{-i}$	4.840	4.78	$±$	0.08	4.59	$±$	0.08
$\nu$ 6$_2^{+i}$	4.031	3.42	$±$	0.11	3.15	$±$	0.10
$\nu$ 6$_3^{-i}$	2.662	0.77	$±$	0.50	--
$\pi$ [301] $\frac{1}{2}^{+i}$	$-$0.432	1.23	$±$	0.55	--
$\pi$ [404] $\frac{9}{2}^{+i}$	$-$0.719	$-$0.00	$±$	0.09	0.09	$±$	0.09
$\pi$ [404] $\frac{9}{2}^{-i}$	$-$0.719	$-$0.00	$±$	0.09	0.11	$±$	0.09
$\pi$ [411] $\frac{3}{2}^{+i}$	$-$0.249	0.81	$±$	0.18	--
$\pi$ [411] $\frac{3}{2}^{-i}$	$-$0.062	0.65	$±$	0.16	--
$\pi$ [413] $\frac{5}{2}^{-i}$	$-$0.539	$-$1.52	$±$	0.53	--
$\pi$ [422] $\frac{3}{2}^{+i}$	$-$0.315	$-$0.19	$±$	0.25	$-$0.21	$±$	0.27
$\pi$ [422] $\frac{3}{2}^{-i}$	0.510	$-$0.84	$±$	0.23	$-$0.38	$±$	0.24
$\pi$ [532] $\frac{5}{2}^{+i}$	$-$0.253	$-$0.90	$±$	0.13	$-$1.11	$±$	0.16
$\pi$ [532] $\frac{5}{2}^{-i}$	$-$0.022	$-$0.67	$±$	0.20	--
$\pi$ [541] $\frac{1}{2}^{-i}$	0.944	1.75	$±$	0.23	--
$\pi$ [541] $\frac{3}{2}^{+i}$	1.743	1.57	$±$	0.13	1.18	$±$	0.11
$\pi$ [541] $\frac{3}{2}^{-i}$	$-$0.057	$-$0.54	$±$	0.10	$-$0.48	$±$	0.11
$\pi$ [550] $\frac{1}{2}^{-i}$	2.819	2.99	$±$	0.52	2.86	$±$	0.40