Two-proton radioactivity and three-body decay. III. Integral formulas for decay widths in a simplified semianalytical approach

Three-body decays of resonant states are studied using integral formulas for decay widths. The theoretical approach with a simplified Hamiltonian allows semianalytical treatment of the problem. The model is applied to decays of the first excited $3/2^{-}$ state of ${}^{17}\mathrm{Ne}$ and the $3/2^{-}$ ground state of ${}^{45}\mathrm{Fe}$. The convergence of three-body hyperspherical model calculations to the exact result for widths and energy distributions are studied. The theoretical results for ${}^{17}\mathrm{Ne}$ and ${}^{45}\mathrm{Fe}$ decays are updated, and uncertainties of the derived values are discussed in detail. Correlations for the decay of the ${}^{17}\mathrm{Ne}$ $3/2^{-}$ state are also studied.

Figure 1

Energy conditions for different modes of the two-nucleon emission (three-body decay): (a) true three-body decay, (b) sequential decay.

Figure 2

Single-particle coordinate systems: (a) V system typical for a shell model. (b) T Jacobi system, where the diproton and core are explicitly in configurations with definite angular momenta $l_x$ and $l_y$. (c) Y Jacobi system, which for a heavy core, is close to the single-particle system (a).

Figure 3

Convergence of the ${}^{17}\mathrm{Ne}$ width in a simplified model in the Y Jacobi system. OFSI model with experimental position $E_{2r}=0.535$ KeV of the $s$-wave two-body resonance. Diamonds show the results of dynamic HH calculations. Solid curves correspond to calculations with effective FR potentials.

Figure 4

Convergence of widths in OFSI model for different positions $E_{2r}$ of the two-body resonance in the core-$p$ channel (Y Jacobi system). For $K_{max}>24,$ the value of $K_{max}$ denotes the size of the basis for the Feshbach reduction to $K_{max}=24$.

Figure 5

Width of the ${}^{17}\mathrm{Ne}$ $3/2^{-}$ state as a function of two-body resonance position $E_{2r}$. Dashed, dotted, and solid lines show cases of pure HH calculations with $K_{max}=24$, the same but with Feshbach reduction from $K_{max}=100$, and the corrected width ${\Gamma }_c$. Inset shows the same, but as a function of the potential depth parameter $V_{x0}$. Gray area shows the transition region from the three-body to the two-body decay regime. The gray curve shows the simple analytical dependence of Eq. (34).

Figure 6

Convergence of energy distribution for ${}^{17}\mathrm{Ne}$ in the Y Jacobi system.

Figure 7

Energy distributions for ${}^{17}\mathrm{Ne}$ in the Y Jacobi system for different two-body resonance positions $E_{2r}$. The three-body decay energy is $E_{3r}=0.344$ MeV. The distributions are normalized to have unity value on a maximum of three-body components. The values near the peaks show the fraction of the total intensity concentrated within the peak. Note the change of the scale on the vertical axis.

Figure 8

Convergence of the ${}^{17}\mathrm{Ne}$ width in a simplified model, Y Jacobi system. OFSI model with $s$-wave two-body resonance at $E_{2r}=0.535$ MeV; Gaussian potential and potential with repulsive core. TFSI model with $d$-wave two-body resonance at $E_{2r}=0.96$ MeV.

Figure 9

Convergence of the ${}^{17}\mathrm{Ne}$ width in a simplified model, T Jacobi system. Final state interaction describes $s$-wave $p$-$p$ scattering.

Figure 10

Energy distributions for ${}^{17}\mathrm{Ne}$ in the T Jacobi system (between two protons).

Figure 11

Comparison of the OFSI calculations for ${}^{45}\mathrm{Fe}$ in the T system with diproton model Eq. (36). Solid lines show effective equivalent channel radius $r_{\mathrm{ch}}(\mathit{dp})$ for “diproton emission” as a function of (a) radius ${\rho }_0$ of the three-body potential [Eq. (25)], where the value ${\rho }_0/\sqrt{2}$ should be comparable with typical nuclear sizes, and (b) position of the peak $Y_{\mathrm{peak}}$ in the three-body WF ${\Psi }^{(+)}$ in the $Y$ coordinate. The dashed lines are given to guide the eye.

Figure 12

Convergence of the ${}^{17}\mathrm{Ne}$ width for experimental positions $E_{2r}=0.535$ MeV of the $0^{+}$ two-body resonance in the $X$ subsystem and $E_{2r}=0.96$ MeV of the $2^{+}$ two-body resonance in the $Y$ subsystem (TFSI model).

Figure 13

Convergence of the ${}^{45}\mathrm{Fe}$ width for position of the $1^{-}$ two-body resonances in $X$ and $Y$ subsystems $E_{2r}=1.48$ MeV.

Figure 14

${}^{45}\mathrm{Fe}$ g.s. width as a function of $E_{2r}$. Dashed, dotted, and solid lines show, respectively, a pure HH calculation with $K_{max}=24$, the same but with Feshbach reduction from $K_{max}=100$, and the corrected width ${\Gamma }_c$. Gray area shows the transition region from three-body to two-body decay regime. Gray curve shows simple analytical dependence of Eq. (34).

Figure 15

Interpolation of ${}^{17}\mathrm{Ne}$ decay width obtained in full three-body calculations by means of TFSI convergence curves (see Fig. 8). Upper curves correspond to the TFSI case with Gaussian potential in the $s$-wave and compatible S1 case for the full three-body model. Lower curves correspond to the TFSI case with repulsive core potential in the $s$-wave and compatible GMZ case for the full three-body model.

Figure 16

Correlations for ${}^{17}\mathrm{Ne}$ decay in T and Y Jacobi systems. Three-body calculations with realistic (GMZ) potential.

Figure 17

Correlations for ${}^{17}\mathrm{Ne}$ decay in T and Y Jacobi systems. Three-body calculations with Coulomb FSIs only (all nuclear pairwise potentials are turned off).

Figure 18

Correlations for ${}^{17}\mathrm{Ne}$ decay calculated in simplified OFSI model in T (only $p$-$p$ FSI) and Y Jacobi systems (only $s$-wave core-$p$ FSI).

Figure 19

Lifetime of ${}^{45}\mathrm{Fe}$ as a function of the $2p$ decay energy $E_{3r}$. The plot is the analog of Fig. 6(a) from Ref. [4] with experimental data [9] (gray circle), [10] (gray rectangle), [12] (black rectangle), and improved theoretical results. Solid curves show the cases of practically pure $p^2$ and $f^2$ configurations, dashed curves stand for different mixed $p^2/f^2$ cases. The numerical labels on the curves show the weights of the $s^2$ and $p^2$ configurations in percents.

Figure 20

Compatibility of the measured width of the ${}^{45}\mathrm{Fe}$ with different assumptions about position $E_{2r}$ of the ground state in the ${}^{44}\mathrm{Mn}$ subsystem and structure of ${}^{45}\mathrm{Fe}$ [weights of the $p^2$ configuration $W(p^2)$ are shown on the vertical axis]. Central gray area corresponds to experimental width uncertainty $\Gamma =(1.62\pm 0.38)\times {10}^{-19}$ MeV [12]. The light gray area also takes into account the energy uncertainty $E_{3r}=1.154(16)$ MeV [12]. The vertical dashed line corresponds to the $E_{2r}$ used in Fig. 19.