Interweaving of elementary modes of excitation in superfluid nuclei through particle-vibration coupling: Quantitative account of the variety of nuclear structure observables

A complete characterization of the structure of nuclei can be obtained by combining information arising from inelastic scattering, Coulomb excitation, and $\gamma$-decay, together with one- and two-particle transfer reactions. In this way it is possible to probe both the single-particle and collective components of the nuclear many-body wave function resulting from the coupling of these modes and, as a result, diagonalizing the low-energy Hamiltonian. We address the question of how accurately such a description can account for experimental observations in the case of superfluid nuclei. Our treatment goes beyond the traditional approach, in which these properties are calculated separately, and most often for systems near closed shells, based on perturbative approximations (weak coupling). It is concluded that renormalizing empirically and on equal footing bare single-particle and collective motion of open-shell nuclei in terms of self-energy (mass) and vertex corrections (screening), as well as particle-hole and pairing interactions through particle-vibration coupling (PVC), leads to a detailed, quantitative account of the data, constraining the possible values of the $k$ mass, of the ${}^1{S}_0$ bare $NN$ interaction, and of the PVC strengths within a rather narrow window.

Figure 1

(a) State-dependent pairing gaps for the five valence orbitals of ${}^{120}\mathrm{Sn}$. The value of $\mathrm{\Delta }$ associated with the HFB solution of $v_{14}{(}^1{S}_0)(\equiv v_p^{\mathrm{bare}})$ is indicated by an arrow labeled ${\mathrm{\Delta }}^{\mathrm{HFB}}$. The pairing gaps calculated making use of the empirically renormalized density modes are shown in terms of open circles joined by a dashed line, while the corresponding results obtained including also spin modes, and thus corresponding to ${\tilde{\mathrm{\Delta }}}_{\nu }$ are shown by the solid dots joined by a continuous curve. The contributions ${\tilde{\mathrm{\Delta }}}_{\nu }^{\mathrm{bare}}$ and ${\tilde{\mathrm{\Delta }}}_{\nu }^{\mathrm{ind}}$ are displayed in terms of solid triangles and solid squares joined by dotted and by dashed lines respectively. (b),(c) Calculated two-particle transfer absolute differential cross sections associated with the reactions ${}^{120}\mathrm{Sn}{(p,t)}^{118}\mathrm{Sn}$(g.s.) and ${}^{122}\mathrm{Sn}{(p,t)}^{120}\mathrm{Sn}(\mathrm{g}.\mathrm{s}.)$ (continuous curves) in comparison with experimental data (solid dots) [38, 39]. (d) The absolute value of the deviation of the integrated theoretical absolute cross section from the experimental value in the case of the second reaction is given as a function of the strength of the bare pairing interaction (see footnote 3 below).

Figure 2

Absolute value of the difference between the experimental pairing gap and the theoretical value of ${\mathrm{\Delta }}_{h_{11/2}}$ calculated as a function of (a) the effective mass $m_k$ associated with different Skyrme forces (see footnote 4), (b) the ratio $G/G_0$ (see footnote 3), and (c) the ratio ${\beta }_2/{\left({\beta }_2\right)}_0$ (see footnote 2). (d) The lowest quasiparticle energy values as a function of ${\beta }_2/{\left({\beta }_2\right)}_0$ obtained from the full calculation as explained in the text referred to the energy of the $3/2^{+}$ state, in comparison with the experimental data. (e) Root mean-square deviation between the experimental and theoretical levels shown in (d) as a function of ${\beta }_2/{\left({\beta }_2\right)}_0$. (f) Root mean-square deviation between the experimental and theoretical energies, shown in (d) as a function of the ratio $G/G_0$. (g) The experimental energies of the members of the $h_{11/2}\otimes 2^{+}$ multiplet compared with the theoretical values, calculated as a function of the ratio ${\beta }_2/{\left({\beta }_2\right)}_0$. (h) Root mean-square deviation between the experimental and theoretical energies of the members of the $h_{11/2}\otimes 2^{+}$ multiplet shown in (g) as a function of ${\beta }_2/{\left({\beta }_2\right)}_0$.

Figure 3

(a) Experimental $B\left(E2\right)$ values [37] in units of $B_{\mathrm{sp}}$, of the quadrupole $\gamma$-decay following ${}^{119}\mathrm{Sn}{(\alpha ,{\alpha }^{\prime })}^{119}{\mathrm{Sn}}^{*}$ Coulomb excitation, connecting the low-lying states of ${}^{119}\mathrm{Sn}$. (b) Corresponding theoretical values calculated making use of the results of the full renormalized calculation as explained in the text. The energies are in MeV. (c) Root mean-square deviation between the experimental transition strengths associated with $E2$ decay from the $5/2^{+}$ levels, and the theoretical values calculated as a function of the ${\beta }_2$ parameter.

Figure 4

(a) Absolute finite range, full recoil DWBA theoretical differential cross sections associated with the low-lying fragments of the $h_{11/2},d_{3/2},s_{1/2}$, and $d_{5/2}$ valence states most strongly populated in the reaction ${}^{120}\mathrm{Sn}(d,p){}^{121}\mathrm{Sn}$, calculated with the help of state-of-the-art optical potentials and $v_{np}$ interaction [40], using the NG structure input, in comparison with the experimental data [46]. In the NG calculations the $d_{5/2}$ single-particle orbit in the SLy4 mean-field potential has been shifted toward ${\epsilon }_F$ by 0.6 MeV (see text). (b) ${}^{120}\mathrm{Sn}(p,d){}^{119}\mathrm{Sn}$ $(5/2^{+})$ absolute experimental differential cross sections [47], together with the DWBA fit used in the analysis of the data (right panel) in comparison with the DWBA calculations (left panel) carried out as mentioned in (a). (c) Comparison of the calculated strength function $S_{5/2}\{d[\sigma {(}^{120}\mathrm{Sn}{(p,d)}^{119}\mathrm{Sn}(5/2^{+}))+\sigma {(}^{120}\mathrm{Sn}{(d,p)}^{121}\mathrm{Sn}(5/2^{+}))]/dE\}$ with experimental data [46, 47]. The peaks have been folded with a Gaussian function of variance 0.25 MeV. (d) The difference between the centroid (width) of the experimental and of the calculated $d_{5/2}$ strength $S_{5/2}$ is shown as a function of the ratio ${\beta }_2/{\left({\beta }_2\right)}_0$ in terms of the solid (dashed) curve.