Pairing in nuclear matter and finite nuclei

Effects of pairing with isospin $T=0$ and $T=1$ are systematically studied in a model, which is based on a realistic nucleon-nucleon interaction and allows us to describe the transition from infinite nuclear matter to finite nuclei. Special attention is paid to the development of the spin-orbit term in the mean field of nucleons in finite nuclei. The spin-orbit term yields a drastic suppression of $T=0$ proton-neutron pairing but does not lead to a complete disappearance in finite nuclei. Arguments are presented as to why no clear evidence of $T=0$ pairing can be observed in the binding energies of finite nuclei.

Figure 1

The density profile for the model of nuclear matter calculated in a spherical box of radius $R=15\text{fm}$, assuming various Fermi momenta $k_F$. The solid lines represent the density profiles calculated for plane waves with boundary conditions according to Eq. (3) while the dashed lines have been obtained using the alternating boundary conditions discussed in the text. The arrows at the right axis indicate the densities for homogeneous matter according to Eq. (20).

Figure 2

Single-particle energies of nuclear matter calculated in a spherical box of radius $R=10\text{fm}$. The left panel shows the kinetic energies $t_{\mathrm{kin}}$ and the single-particle energies $\varepsilon \left(k\right)$ calculated for a Fermi momentum $k_F=1{\text{fm}}^{-1}$. The right panel shows values of ${\varepsilon }_{il}$ as a function of the Fermi momentum $k_F$. Results for various orbital angular momenta $l$ are identified by the color of the line (energies increase with increasing $l$). Solid, dashed, and dotted lines represent the values for the lowest three momenta according to the boundary conditions of Eq. (3).

Figure 3

Pairing gaps $\mathrm{\Delta }$ calculated for nuclear matter with various Fermi momenta $k_F$. The triangles represent results for $T=0$ pairing obtained from Eq. (16) while the square boxes represent the corresponding imaginary parts of the $pphh$ RPA eigenvalues [see Eq. (15)]. Results for isospin $T=1$ derived from Eq. (16) are shown in terms of x signs. Note that the single-particle energies have been approximated by kinetic energies.

Figure 4

Expansion coefficients of the bound states for $T=0$ and $T=1$ pairing in nuclear matter with $k_F=1{\mathrm{fm}}^{-1}$. The states of the two-particle basis are represented by the average momentum $k_{12}$ defined in Eq. (21).

Figure 5

Value for the pairing gap $\mathrm{\Delta }$ for $T=0$ and $T=1$ pairing as a function of the cutoff momentum for the particle states are represented by solid lines while the number of basis states is indicated by the corresponding dashed lines. All values are normalized with respect to the result, both gap and number of basis states, at the maximal cutoff $k_{\mathrm{cut}}$ of 2.5 ${\mathrm{fm}}^{-1}$.

Figure 6

Pairing gaps $\mathrm{\Delta }$ calculated for nuclear matter with various Fermi momenta $k_F$. The symbols represent results for $T=0$ pairing obtained from Eq. (16). While the diamonds represent results obtained for the corresponding HF single-particle energies, the triangles were obtained approximating these energies by kinetic energies $t_k$.

Figure 7

Depth of Woods-Saxon potentials $U_0$, defined in Eq. (7) to simulate quasinuclear systems corresponding to ${}^{16}\mathrm{O}$ and ${}^{40}\mathrm{Ca}$ for various radii as discussed in the text.

Figure 8

Single-particle energies of low-lying states in quasinuclear systems corresponding to ${}^{16}\mathrm{O}$ of different size. The energies appear in the empirical order.

Figure 9

Spin-orbit term defined by the difference in energy between single-particle states with angular momentum $j=l-1/2$ and $j=l+1/2$ for orbital angular momenta $l=1$, 2, and 3 for quasinuclei ${}^{40}\mathrm{Ca}$ of various sizes.

Figure 10

Pairing gaps $\mathrm{\Delta }$ calculated for various quasinuclear systems are presented as a function of the radius of the Woods-Saxon potential. While for the two panels on the left side (${}^{16}\mathrm{O}$ and ${}^{12}\mathrm{C}$) the wave functions and energies resulting from the ${}^{16}\mathrm{O}$ set have been used, the results for the other panels have been obtained using the ${}^{40}\mathrm{Ca}$ sets. Results for $T=0$ and $T=1$ pairing are displayed in terms of squares and triangles, respectively.

Figure 11

Level scheme for low-lying single-particle energies for protons in ${}^{16}\mathrm{O}$ and ${}^{40}\mathrm{Ca}$ obtained in the Hartree-Fock approximation. The sequence of levels corresponds to the empirical ordering.