First calculation of the $\gamma \gamma$-decay width of a nuclear $2_1^{+}$ state: The case of ${}^{48}\mathrm{Ca}$

The competitive double-$\gamma$ decay of the $2_1{}^{+}$ state of an even-even spherical nucleus is studied for the first time. The coupling of one-, two-, and three-phonon terms in the wave functions of excited states is taken into account within the microscopic model based on the Skyrme energy density functional. The approach enables one to perform the calculations in very large configurational spaces. We estimate the generalized electric dipole polarizabilities involved in the $\gamma \gamma /\gamma$ decay process and make a prediction for the branching ratio of the competitive $\gamma \gamma$-decay relative to its single $\gamma$-decay calculated to be $3\times {10}^{-8}$ for the case of ${}^{48}\mathrm{Ca}$.

Figure 1

The phonon-phonon coupling effect on the $B\left(E1\right)$ strength distribution. Experimental data are taken from Ref. [30].

Figure 2

(a) The $B\left(E1\right)$ strength distribution. The smoothing parameter 1 MeV is used for the strength distribution described by the Lorentzian function. Experimental data are taken from Refs. [31, 32]. The dashed, dotted, and solid lines correspond to the PPC2-$2_1{}^{+}$, PPC2, and PPC3 calculations, respectively. (b) Running sum of the polarizability ${\alpha }_D$. The shaded area indicates experimental upper and lower limits [32]. (c) Running sum of the generalized polarizability ${\alpha }_{E1E1}$ and the value $M_{E1E1}/E_{sc}$ (the dash-dotted line).

Figure 3

Reduced matrix elements of $E1$ transitions in the following energy interval: from 16 until 20 MeV (a) and from 22 until 25 MeV (b).

Figure 4

The ratio between polarizabilities ${\alpha }_D$ and ${\alpha }_{E1E1}$ as a function of the mean value $V$ of the matrix elements coupling one- and two-phonon configurations. The solid line corresponds to the PPC2-$2_1{}^{+}$ calculation taking into account scaling of the quasiparticle-phonon interaction. The results of the two-state model with $\mathrm{\Delta }{E}_u$ = 0 and 3.2 MeV are denoted by the dashed line and the dotted line, respectively. The PPC3 calculation of SLy5 is denoted by the circle.

Figure 5

The transition probability, the $\gamma$-decay width, and $\gamma \gamma /\gamma$ branching ratio of the $2_1{}^{+}$ state are given in panels (a), (b), and (c), respectively. PPC3 calculations are performed with different cutoffs of single-particle energies ($E_{\text{sp}}$). The solid lines are drawn as guide to the eye.

Figure 6

(a) The $B\left(E1\right)$ strength distribution. (b) Running sum of the polarizability ${\alpha }_D$. (c) Running sum of the generalized polarizability ${\alpha }_{E1E1}$. The dashed line corresponds to the calculation taking into account the orthogonalization of the spurious center-of-mass state to all physical states, the dotted line is the calculation without the orthogonalization, and the solid line is the calculation with the effective charges, $e_{\text{eff}}^n=-\frac{Z}{A}e$ and $e_{\text{eff}}^p=\frac{N}{A}e$.