Microscopic calculation and local approximation of the spatial dependence of the pairing field with bare and induced interactions

The bare nucleon-nucleon interaction is essential for the production of pair correlations in nuclei, but an important contribution also arises from the induced interaction resulting from the exchange of collective vibrations between nucleons moving in time reversal states close to the Fermi energy. The pairing field resulting from the summed interaction is strongly peaked at the nuclear surface. It is possible to reproduce the detailed spatial dependence of this field by using either a local approximation, which fully takes into account finite size effects, or a contact interaction, with parameters that are quite different from those commonly used in more phenomenological approaches.

Figure 1

Diagrams showing the exchange of a vibration between two pairs of levels coupled to $J=0$.

Figure 2

Landau-Migdal parameters associated with the SLy4 force, calculated as a function of the distance from the center of the nucleus in ${}^{120}\mathrm{Sn}$.

Figure 3

Diagonal matrix elements ${\Delta }_{\mathit{nnlj}}$ as a function of the single-particle energy $e_{\mathit{nlj}}$. The squares, diamonds, and triangles refer to the gaps obtained, respectively, with the Argonne plus induced interaction $v_{\mathrm{Arg}+\mathrm{ind}}$, with the Argonne interaction $v_{\mathrm{Arg}}$, and with the induced interaction $v_{\mathrm{ind}}$. The vertical dashed line indicates the position of the Fermi energy, which turns out to be almost the same for the three calculations.

Figure 4

Abnormal density $\Phi (R_{\mathrm{c}.\mathrm{m}.},r_{12})$ for fixed values of $R_{\mathrm{c}.\mathrm{m}.}$. In (a) we show the results calculated with the induced pairing interaction, in (b) those obtained with the Argonne pairing interaction.

Figure 5

Root-mean-square radius of the Cooper pair as a function of the position of the center of mass, obtained with the Argonne+induced interaction $v_{\mathrm{Arg}+\mathrm{ind}}$ (solid line), the Argonne interaction $v_{\mathrm{Arg}}$ (dashed line), and the induced interaction $v_{\mathrm{ind}}$ (dash-dotted line).

Figure 6

Pairing field $\Delta (R_{\mathrm{c}.\mathrm{m}.},r_{12})$ for fixed values of $R_{\mathrm{c}.\mathrm{m}.}$. (a) Results calculated with the induced pairing interaction; (b) results obtained with the Argonne pairing interaction.

Figure 7

Pairing field [Eq. (16)] as a function of the position of the center of mass for different values of the relative momentum $k$, for (a) the Argonne plus induced interaction $v_{\mathrm{Arg}+\mathrm{ind}}$ and (b) the Argonne interaction $v_{\mathrm{Arg}}$.

Figure 8

Pairing field obtained with the semiclassical approximation [cf. Eq. (19)] for the three different pairing interactions: Argonne plus induced $v_{\mathrm{Arg}+\mathrm{ind}}$ (solid line), Argonne $v_{\mathrm{Arg}}$ (dashed line), and induced $v_{\mathrm{ind}}$ (dash-dotted line).

Figure 9

The local pairing gap calculated from Eq. (19) for the Argonne interaction, already shown in Fig. 8 (dashed line), compared to the gap obtained by using the simple LDA approximation (solid line).

Figure 10

(a) The diagonal matrix elements of the pairing gap associated with the Argonne interaction $v_{\mathrm{Arg}}$ (diamonds, already shown in Fig. 3) compared with those associated with the DDDI, zero-range interaction with the parameters $\alpha =0.66,\eta =0.84$ (circles). The semiclassical pairing gaps associated with the Argonne interaction (solid line) and with the zero-range interaction (dashed line) are shown in the insert. (b) The diagonal matrix elements of the pairing gap associated with the Argonne+induced interaction $v_{\mathrm{Arg}+\mathrm{ind}}$ (squares, already shown in Fig. 3) compared with the DDDI interaction with the parameters $\alpha =2.0,\eta =1.32$ (circles, cf. Table ). The semiclassical pairing gaps associated with the induced interaction (solid line) and with the zero-range interaction (dashed line) are shown in the insert.

Figure 11

(a) Spatial dependence of the different local pairing interactions introduced in this work to simulate the local pairing gaps [cf. Eq. (19)] obtained with the corresponding microscopic, nonlocal interactions: bare+induced interaction $v_{\mathrm{Arg}+\mathrm{ind}}$ (corresponding to the parameters $\alpha =2.0,\eta =1.32$; cf. Table ); bare+induced interaction neglecting the spin-dependent part ($v_{\mathrm{Arg}+\mathrm{ind}},G_0=0,G_0^{\text{'}}=0),(\alpha =1.79,\eta =1.0$; cf. Table in Appendix ); bare $v_{14}$ interaction $v_{\mathrm{Arg}}(\alpha =0.66,\eta =0.84$; cf. Table ); Gogny interaction $v_{\mathrm{Gogny}}$ ($\alpha =0.51,\eta =0.63$; cf. Table in Appendix ). (b) The spatial dependence of the bare+induced interaction with and without the spin-dependent part of the induced interaction, already shown in (a), compared with the volume, surface, and mixed interaction of Ref. [26] (see text).

Figure 12

(a) The values of the parameter $b_{\mathrm{attr}}$, obtained by fitting the Gaussian interaction [Eq. (23)], shown as a function of the center of mass $R_{\mathrm{c}.\mathrm{m}.}$ (filled dots) and compared with the function $0.14R_{\mathrm{nucl}}\frac{\mathit{dU}(r)}{{\mathit{dR}}_{\mathrm{c}.\mathrm{m}.}}$ (solid line). (b) The values of the parameter $b_{\mathrm{rep}}$, obtained by fitting the Gaussian interaction [Eq. (23)], also shown as a function of the center of mass $R_{\mathrm{c}.\mathrm{m}.}$ (filled squares).

Figure 13

(a) The diagonal matrix elements of the pairing gap associated with the induced interaction $v_{\mathrm{ind}}$ (triangles, cf. Fig. 3) compared with those associated with the Gaussian parametrization [circles, cf. Eq. (23)]. The spatial dependence of the semiclassical pairing gap associated with the induced interaction (solid line) and with the Gaussian interaction (dashed line) are shown in the insert. (b) The same as (a), but for the Argonne+induced interaction $v_{\mathrm{Arg}+\mathrm{ind}}$, shown by squares.

Figure 14

Different pairing gaps and Cooper pair wave functions obtained with the Gogny interaction. (a) Diagonal matrix elements ${\Delta }_{\mathit{nnlj}}$ as a function of the single-particle energy $e_{\mathit{nlj}}$. The vertical dashed line indicates the position of the Fermi energy. (b) Pairing gap $\Delta (R_{\mathrm{c}.\mathrm{m}.},r_{12})$ in coordinate space for fixed values of $R_{\mathrm{c}.\mathrm{m}.}$. (c) Abnormal density $\Phi (R_{\mathrm{c}.\mathrm{m}.},r_{12})$ in coordinate space for fixed values of $R_{\mathrm{c}.\mathrm{m}.}$. (d) Root-mean-square radius of the Cooper pair as a function of the position of the center of mass, for the Gogny interaction (dash-dotted curve), the Argonne interaction (dashed curve), and the Argonne+induced interaction (solid curve). (e) Pairing field [cf. Eq. (16)] as a function of the position of the center of mass for different values of the relative momentum $k$. (f) Pairing fields obtained with the semiclassical approximation [cf. Eq. (19)] for the Gogny interaction (dash-dotted curve) and for the Gogny interaction with rescaled matrix elements (dotted curve). They are compared with the pairing field associated with the Argonne+induced interaction (solid curve), already shown in Fig. 8.

Figure 15

Different pairing gaps and Cooper pair wave functions obtained by including only the spin-independent part of the induced interaction. (a) Diagonal matrix elements ${\Delta }_{\mathit{nnlj}}$ as a function of the single-particle energy $e_{\mathit{nlj}}$. The vertical dashed line indicates the position of the Fermi energy. (b) Pairing gap $\Delta (R_{\mathrm{c}.\mathrm{m}.},r_{12})$ in coordinate space for fixed values of $R_{\mathrm{c}.\mathrm{m}.}$. (c) Abnormal density $\Phi (R_{\mathrm{c}.\mathrm{m}.},r_{12})$ in coordinate space for fixed values of $R_{\mathrm{c}.\mathrm{m}.}$. (d) Root-mean-square radius of the Cooper pair as a function of the position of the center of mass, for the induced interaction (dash-dotted curve) and the Argonne+induced interaction (solid curve). (e) Pairing field [cf. Eq. (16)] as a function of the position of the center of mass for different values of the relative momentum $k$. (f) Pairing fields obtained with the semiclassical approximation [cf. Eq. (19)] for the induced interaction (dashed curve) and for the Argonne+induced interaction (solid curve).