Deformed quasiparticle random phase approximation formalism for single- and two-neutrino double $\beta$ decay

We use a deformed quasiparticle random phase approximation formalism to describe simultaneously the energy distributions of the single $\beta$ Gamow-Teller strength and the two-neutrino double $\beta$ decay matrix elements. Calculations are performed in a series of double $\beta$ decay partners with $A=48$, 76, 82, 96, 100, 116, 128, 130, 136, and 150, using deformed Woods-Saxon potentials and deformed Skyrme Hartree-Fock mean fields. The formalism includes a quasiparticle deformed basis and residual spin-isospin forces in the particle-hole and particle-particle channels. We discuss the sensitivity of the parent and daughter Gamow-Teller strength distributions in single $\beta$ decay, as well as the sensitivity of the double $\beta$ decay matrix elements to the deformed mean field and to the residual interactions. Nuclear deformation is found to be a mechanism of suppression of the two-neutrino double $\beta$ decay. The double $\beta$ decay matrix elements are found to have maximum values for about equal deformations of parent and daughter nuclei. They decrease rapidly when differences in deformations increase. We remark on the importance of a proper simultaneous description of both double $\beta$ decay and single Gamow-Teller strength distributions. Finally, we conclude that for further progress in the field, it would be useful to improve and complete the experimental information on the studied Gamow-Teller strengths and nuclear deformations.

Figure 1

Binding energy (MeV) as a function of the quadrupole deformation parameter $\beta$ obtained from deformed Hartree-Fock calculations with the Skyrme force Sk3. Experimental $\beta$ values from Refs. 26 and 27 are represented as the extreme values of the black boxes. The experimental difference of binding parent and daughter binding energies is given by the distance between the two horizontal lines (see text).

Figure 2

HF-Sk3 $B\left({\mathrm{GT}}^{-}\right)$ strength distributions $[g_A^2∕4\pi ]$ in ${}^{128}\mathrm{Te}$ and ${}^{136}\mathrm{Xe}$ calculated with the equilibrium deformation $({\beta }_{\mathrm{equil}})$ and with the deformation that fits the experimental $Q_{\beta \beta }$ values $({\beta }_{\mathrm{fit}})$.

Figure 3

HF-Sk3 Gamow-Teller strength distributions $[g_A^2∕4\pi ]$ in ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$ for various values of the coupling strength ${\chi }_{\mathrm{GT}}^{\mathit{ph}}\left(\mathrm{MeV}\right)$.

Figure 4

HF-Sk3 Gamow-Teller strength distributions $[g_A^2∕4\pi ]$ in ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$ for various values of the coupling strength ${\kappa }_{\mathrm{GT}}^{\mathit{pp}}\left(\mathrm{MeV}\right)$.

Figure 5

Gamow-Teller $B\left({\mathrm{GT}}^{-}\right)$ and $B({\mathrm{GT}}^{+})$ strength distributions $[g_A^2∕4\pi ]$ in ${}^{48}\mathrm{Ca}$ and ${}^{48}\mathrm{Ti}$ plotted as a function of the excitation energies of the corresponding daughter nuclei. Left panels show results from HF(Sk3) calculations without residual interaction (dashed lines) and with residual interactions with ${\chi }_{\mathrm{GT}}^{\mathit{ph}}=0.10\mathrm{MeV}$, ${\kappa }_{\mathrm{GT}}^{\mathit{pp}}=6∕A\mathrm{MeV}$ (solid lines). Right panels show results using WS potentials with ${\chi }_{\mathrm{GT}}^{\mathit{ph}}$ and ${\kappa }_{\mathrm{GT}}^{\mathit{pp}}$ from Ref. 8 and with two different values for the quadrupole deformation: ${\beta }_1$ from Ref. 26 (solid line) and ${\beta }_2$ from Ref. 27 (dashed line). Experimental data are from Ref. 30. Notice that no quenching factor has been included in the calculations.

Figure 6

Same as in Fig. 5 for ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$. Data in ${}^{76}\mathrm{Se}$ are from 31. Vertical lines in ${}^{76}\mathrm{Ge}$ are experimental data from 32.

Figure 7

Same as in Fig. 5 for ${}^{82}\mathrm{Se}$ and ${}^{82}\mathrm{Kr}$. Vertical lines in ${}^{82}\mathrm{Se}$ are experimental data from 32.

Figure 8

Same as in Fig. 5 for ${}^{96}\mathrm{Zr}$ and ${}^{96}\mathrm{Mo}$.

Figure 9

Same as in Fig. 5 for ${}^{100}\mathrm{Mo}$ and ${}^{100}\mathrm{Ru}$. Vertical lines in ${}^{100}\mathrm{Mo}$ are experimental data from 33.

Figure 10

Same as in Fig. 5 for ${}^{116}\mathrm{Cd}$ and ${}^{116}\mathrm{Sn}$. Vertical lines in ${}^{116}\mathrm{Cd}$ are experimental data from 33.

Figure 11

Same as in Fig. 5 for ${}^{128}\mathrm{Te}$ and ${}^{128}\mathrm{Xe}$. Vertical lines in ${}^{128}\mathrm{Te}$ are experimental data from 32.

Figure 12

Same as in Fig. 5 for ${}^{130}\mathrm{Te}$ and ${}^{130}\mathrm{Xe}$. Vertical lines in ${}^{130}\mathrm{Te}$ are experimental data from 32.

Figure 13

Same as in Fig. 5 for ${}^{136}\mathrm{Xe}$ and ${}^{136}\mathrm{Ba}$.

Figure 14

Same as in Fig. 5 for ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$.

Figure 15

$2\nu \beta \beta$-decay matrix elements of ${}^{96}\mathrm{Zr}$ as a function of both parent and daughter deformations. The two dashed horizontal lines correspond to experimental $M_{\mathrm{GT}}^{2\nu }$ extracted from Ref. 34 using $g_A=1.0$ and $g_A=1.25$. The thick segments in each curve correspond to the experimental values of $M_{\mathrm{GT}}^{2\nu }$.

Figure 16

Difference between parent and daughter quadrupole deformations in double-$\beta$ emitters. Dots are self-consistent results from Skyrme Sk3 calculations. Vertical lines indicate the maximum and minimum experimental differences (see Table ), which are used in WS calculations.

Figure 17

$2\nu \beta \beta$-decay matrix elements calculated with Woods-Saxon potentials as a function of the particle-particle interaction strength ${\kappa }_{\mathrm{GT}}^{\mathit{pp}}$. Dashed lines correspond to the results assuming spherical nuclei. Solid lines correspond to the results obtained by using the maximum and minimum differences between the experimental deformations of parent and daughter (see Table ). Horizontal lines are the experimental $M_{\mathrm{GT}}^{2\nu }$ extracted from Ref. 34 using $g_A=1.0$ and $g_A=1.25$.

Figure 18

$2\nu \beta \beta$-decay matrix elements obtained from Skyrme (Sk3) deformed Hartree-Fock calculations as a function of the particle-particle interaction strength ${\kappa }_{\mathrm{GT}}^{\mathit{pp}}$. Horizontal lines are the experimental $M_{\mathrm{GT}}^{2\nu }$ extracted from Ref. 34 using $g_A=1.0$ and $g_A=1.25$.

Figure 19

$2\nu \beta \beta$-decay matrix elements of ${}^{96}\mathrm{Zr}$ obtained from Skyrme (SG2) deformed Hartree-Fock calculations as a function of the particle-particle interaction strength ${\kappa }_{\mathrm{GT}}^{\mathit{pp}}$. The dotted curve corresponds to calculations using the prolate shapes for parent and daughter ${\beta }_{\mathrm{p}}=0.147$, ${\beta }_{\mathrm{d}}=0.119$ (see Table ). The dashed curve is for spherical shapes and the solid curve is for spherical parent and prolate daughter ${\beta }_{\mathrm{p}}=0.016$, ${\beta }_{\mathrm{d}}=0.119$ (see Table ). Horizontal lines are the experimental $M_{\mathrm{GT}}^{2\nu }$ extracted from Ref. 34 using $g_A=1.0$ and $g_A=1.25$.