Pairing matrix elements and pairing gaps with bare, effective, and induced interactions

The dependence on the single-particle states of the pairing matrix elements of the Gogny force and of the bare low-momentum nucleon-nucleon potential $v_{\mathrm{low}\text{−}k}$—designed so as to reproduce the low-energy observables avoiding the use of a repulsive core—is studied for a typical finite, superfluid nucleus (${}^{120}\mathrm{Sn}$). It is found that the matrix elements of $v_{\mathrm{low}\text{−}k}$ follow closely those of $v_{\mathrm{Gogny}}$ on a wide range of energy values around the Fermi energy $e_F$, those associated with $v_{\mathrm{low}\text{−}k}$ being less attractive. This result explains the fact that around $e_F$ the pairing gap ${\Delta }_{\mathrm{Gogny}}$ associated with the Gogny interaction (and with a density of single-particle levels corresponding to an effective k mass $m_k\approx 0.7\hspace{1em}m$) is a factor of about 2 larger than ${\Delta }_{\mathrm{low}\text{−}k}$, being in agreement with ${\Delta }_{\exp }=1.4$ MeV. The exchange of low-lying collective surface vibrations among pairs of nucleons moving in time-reversal states gives rise to an induced pairing interaction $v_{\mathrm{ind}}$ peaked at $e_F$. The interaction $(v_{\mathrm{low}\text{−}k}+v_{\mathrm{ind}})\hspace{1em}{Z}_{\omega }$ arising from the renormalization of the bare nucleon-nucleon potential and of the single-particle motion (ω-mass and quasiparticle strength $Z_{\omega }$) associated with the particle-vibration coupling mechanism, leads to a value of the pairing gap at the Fermi energy ${\Delta }_{\mathrm{ren}}$ that accounts for the experimental value. An important question that remains to be studied quantitatively is to what extent ${\Delta }_{\mathrm{Gogny}}$, which depends on average parameters, and ${\Delta }_{\mathrm{ren}}$, which explicitly depends on the parameters describing the (low-energy) nuclear structure, display or not a similar isotopic dependence and whether this dependence is borne out by the data.

Figure 1

The nucleus ${}^{120}\mathrm{Sn}$. Diagonal pairing matrix elements of the induced interaction (upper panel, solid diamonds) and of the Gogny force (lower panel, solid circles), displayed as a function of the single-particle energy, $e_{\nu }$, of the state ν calculated using the bare nucleon mass and the single-particle wave functions of a Woods-Saxon potential with standard parameters (depth $V_0=-49$ MeV, diffusivity $a=0.65$ fm, and radius $R_0=6.16$ fm), including the spin-orbit term, parametrized according to Ref. 21. Also shown by means of vertical lines is the position of the Fermi energy, $e_F=-9.1\text{MeV}$. Note the different scale in the two figures.

Figure 2

Schematic model for the nucleus ${}^{120}\mathrm{Sn}$. The semiclassical pairing matrix elements of the Gogny force (solid line) are compared to the quantal matrix elements for the case in which the single-particle wave functions have been calculated making use of a harmonic oscillator potential without the spin-orbit term (and $m_k=m$). The diagonal and nondiagonal matrix elements are denoted by filled and open circles respectively. The crosses correspond to the weighted averages of the matrix elements within a shell.

Figure 3

The nucleus ${}^{120}\mathrm{Sn}$. The same as Fig. 2 (i.e., also with $m_k=m$) but for the fact that the single-particle wave functions have been calculated making use of the Woods-Saxon potential including the spin-orbit term already used for Fig. 1, where the diagonal matrix elements have already been shown. The nondiagonal matrix elements are plotted here at an energy $e_{\nu }$, which is the average between the energies of the initial and final states.

Figure 4

The matrix elements of the Gogny (solid curve) and of the $v_{\mathrm{low}\text{−}k}$ (dashed curve) interactions are plotted as a function of momentum.

Figure 5

The nucleus ${}^{120}\mathrm{Sn}$. The semiclassical matrix elements of the induced interaction, calculated according to Eq. (11) (dash-dotted curve), are compared with the matrix elements of the Gogny force (solid curve, cf. Fig. 3) and with those of the $v_{\mathrm{low}\text{−}k}$ interaction (dashed curve). Calculations are performed with $m_k=m$ and with the same Woods-Saxon potential used in Figs. 1 and 3.

Figure 6

The nucleus ${}^{120}\mathrm{Sn}$. The semiclassical matrix elements of the induced interaction, as a function of the single-particle energy (calculated with $m_k=m$) associated with the multipolarities $L=$ 2, 3, and 4 of the form factor [see Eq. (1)], are shown by a dashed, dotted, and solid line respectively. The sum of the three contributions is displayed by means of a dash-dotted line (cf. Fig. 5).

Figure 7

Pairing gaps calculated in neutron matter as a function of the Fermi momentum, obtained with the Gogny interaction (solid line), the Argonne $v_{14}$ potential (dash-dotted line), and the $v_{\mathrm{low}\text{−}k}$ potential (dashed line). The bare effective mass has been used in the calculation.

Figure 8

State-dependent pairing gaps of ${}^{120}\mathrm{Sn}$ calculated with a Woods-Saxon potential (with depth $V_0=-64$ MeV, diffusivity $a=$ 0.65 fm, and radius $R_0=$ 6.17 fm) as a function of the single-particle energy. The k-mass $m_k$ was set equal to 0.7 $m$. The Fermi energy is $e_F=-8.6$ MeV. Solid triangles (open squares) display the results of a HFB calculation with the Gogny interaction, with quantal (semiclassical) matrix elements. The solid diamonds refer instead to a HFB calculation using the semiclassical matrix elements of the $v_{\mathrm{low}\text{−}k}$ potential.

Figure 9

State-dependent pairing gap of ${}^{120}\mathrm{Sn}$ obtained making use of the Woods-Saxon potential used in calculating the results displayed in Fig. 8 (i.e., with $m_k=0.7\hspace{1em}m$) and different interactions. Diamonds and triangles show again the gaps obtained with the $v_{\mathrm{low}\text{−}k}$ and $v_{\mathrm{Gogny}}$ interactions, respectively. Stars show the gap associated with the matrix elements obtained, summing the matrix elements of $v_{\mathrm{low}\text{−}k}$ and $v_{\mathrm{ind}}$. The latter have been evaluated using Eq. (1).

Figure 10

Nucleons interact in a scattering experiment through the bare NN-interaction (a), the analysis of the associated differential cross section being made in terms of phase shifts. The main effect of this bare two-body interaction acting among nucleons inside the nucleus is to produce a mean field, sum of a local (b) (Hartree) and a non-local (c) (Fock) potential. In this potential, nucleons move independently of each other feeling the pullings and pushings of the other nucleons only when they try to leave the nucleus. In other words, they move in a single-particle potential produced by all the other nucleons bouncing elastically off the surface. However, this surface can vibrate as a whole, as testified by the existence of low-lying collective surface modes and of giant resonances in the inelastic spectrum of nucleus–nucleus collisions. Thus, from time to time a nucleon can bounce inelastically off the nuclear surface, setting it into vibration, changing state of motion and, at a later time, reabsorb the vibration returning to the original state as shown in (d). A similar process, but this time associated with the virtual excitation of the vacuum (e′) and with the Pauli principle is at the basis of the process depicted in (e), obtained from a simple change in the time ordering from process (d). A vibration excited by a nucleon can be reabsorbed by a second nucleon (f). Such a process leads to an induced interaction among nucleons, associated with the polarization of the nuclear medium.