ISB corrections in $78\leq A \leq 98$ nuclei

Our projected DFT approach can be used to predict isospin mixing in heavy nuclei. The calculated ISB corrections and $Q$-values in $78 \le A \le 98$ nuclei are listed in Table 4. The values of $\delta_{\rm C}$ are also shown in Fig. 12. Note that the predicted ISB corrections are here considerably smaller than those in $A$=70 and $A$=74 nuclei, see Tables 2 and 3. For the sake of comparison, Fig. 12 also shows predictions of Ref. [46] for the $^{82}$Nb $\rightarrow ^{82}$Zr transition using the VAMPIR approach with either charge-independent Bonn A potential or charge-dependent Bonn CD potential. Note that our prediction is only slightly below the Bonn A result and significantly lower than the Bonn CD value. For the sake of completeness, it should be mentioned that our $Q_\beta$-value of 10.379MeV for this transition agrees well with $Q_\beta=10.496$MeV (Bonn A) and 10.291MeV (Bonn CD) calculated within the VAMPIR approach.

Figure 12: The ISB corrections to the superallowed $0^+ \rightarrow 0^+$ transitions in heavy nuclei calculated in the present work (full dots). Vertical bars mark the ISB corrections to the $^{82}$Nb $\rightarrow$$^{82}$Zr transition calculated in Ref. [46] by using the VAMPIR formalism with the charge-independent Bonn A and charge-dependent Bonn CD interactions.

Table 4: Results of calculations for the superallowed $\beta$-decays in $78\leq A \leq 98$ nuclei: the isospin impurities in the parent and daughter nuclei; $\delta_{\rm C}$ for different shape-current orientations; averaged (recommended) $\delta_{\rm C}$; calculated equilibrium deformations $\beta_2$ and $\gamma$; and $Q_\beta$-values calculated here and estimated from the extrapolated masses of Ref. [47].

			$\quad$	$\alpha_{\rm C}^{{\rm (P)}}$	$\alpha_{\rm C}^{{\rm (D)}}$	$\quad$	$\delta_{\rm C}^{{\rm (X)}}$	$\delta_{\rm C}^{{\rm (Y)}}$	$\delta_{\rm C}^{{\rm (Z)}}$	$\delta_{\rm C}^{{\rm (SV)}}$	$\quad$	$\beta_2^{{\rm (SV)}}$	$\gamma^{{\rm (SV)}}$	$\quad$	$Q_\beta^{{\rm (th)}}$	$Q_\beta^{{\rm (exp)}}$
			$\quad$	(%)	(%)	$\quad$	(%)	(%)	(%)	(%)	$\quad$		(deg)	$\quad$	(MeV)	(MeV)
$T_z=0$	$\rightarrow$	$T_z = 1$
$^{78}$Y	$\rightarrow$	$^{78}$Sr		2.765	0.976		1.20	1.19	1.20	1.20(12)		0.004	60.0		10.471	10.650$^{\char93 }$
$^{82}$Nb	$\rightarrow$	$^{82}$Zr		3.099	1.408		0.70	0.91	0.70	0.77(13)		0.036	60.0		10.379	11.220$^{\char93 }$
$^{86}$Tc	$\rightarrow$	$^{86}$Mo		3.337	1.518		0.89	0.89	1.08	0.95(13)		0.122	0.0		10.965	11.350$^{\char93 }$
$^{90}$Rh	$\rightarrow$	$^{90}$Ru		3.525	1.608		0.99	0.99	1.09	1.02(11)		0.161	0.0		11.465	12.090$^{\char93 }$
$^{94}$Ag	$\rightarrow$	$^{94}$Pd		3.674	1.689		0.86	0.86	1.17	0.96(18)		0.136	0.0		11.896	13.050$^{\char93 }$
$^{98}$In	$\rightarrow$	$^{98}$Cd		3.805	1.771		0.89	0.89	1.36	1.05(25)		0.057	0.0		12.343	13.730$^{\char93 }$

Our calculated values of $\delta_{\rm C}$ are in heavy nuclei considerably smaller than those obtained from a perturbative expression [11,48,5]:

$$\delta_{\rm C} = 0.002645 \frac{Z^2}{A^{2/3}} (n+1)(n+ \ell +3/2)\, {\rm (\%)} ,$$ (24)

where $n$ and $\ell$ denote the number of radial nodes and angular momentum of the valence s.p. spherical wave function, respectively. Indeed, assuming the valence $1p_{1/2}$ state in $A=78$, Eq. (24) yields $\delta_{\rm C}$=1.54%. In heavier nuclei, where the spherical valence state is $0g_{9/2}$, Eq. (24) gives $\delta_{\rm C}$ values that increase smoothly from 1.30% in $A=82$ to 1.64% in $A=98$.