Contribution from the tensor terms to the binding energy

Fig. 3 shows the contribution to the total nuclear binding energy due to the tensor term, calculated by using the spherical Hartree-Fock-Bogoliubov (HFB) code HFBRAD [30] with the SLy4$_T$ functional. Contributions due to the isovector and isoscalar parts are depicted separately in the upper and middle panels, respectively. The total contribution, $B_T (N,Z)$, is shown in the lowest panel of the figure.

Figure 3: The isovector (upper), the isoscalar (middle) and the total (lower) tensor contribution to the nuclear binding energy. The calculations were done using the SLy4$_T$ interaction in the particle-hole channel and the volume-$\delta$ interaction in the particle-particle channel with spherical symmetry assumed. Vertical and horizontal lines indicate the single-particle tensorial magic numbers at spherical shape. See text for further details.

From these results one can see that the isovector component is rather weak. Hence, the topology of the total contribution to the energy is mostly determined by the isoscalar term that shows a strong shell dependence. Following the argumentation presented in Sec. 2, the strongest tensor effects are expected to appear for $N(Z)$=14, 32, 56, and 90. They correspond to nucleons filling the $1d_{5/2}$, $1f_{7/2}\oplus 2p_{3/2}$, $1g_{9/2}\oplus 2d_{5/2}$ and $1h_{11/2}\oplus 2f_{7/2}$ shells, respectively, which creates a maximum SUS filling. Since the level (shell) ordering, at least in well bound near-spherical nuclei, is rather well established experimentally and since it serves as a natural constrain for effective forces, the maximum SUS configurations are, to a large extent, parameterization independent. Hence, the $N(Z)$ numbers at which they appear are robust and can be therefore regarded as tensorial magic numbers.

However, as it is seen from Fig. 3, the $B_T (N,Z)$ does not follow the expected pattern exactly. This is due to (i) pairing-induced configuration mixing and (ii) changes in the s.p. ordering of levels caused by the combination of strong attractive tensor fields and strongly reduced SO field. Two such situations are visible in Fig. 3. For $N < 30$ the tensor contribution is, as expected, largest for $_{32}$Ge. For $40 < N < 50$, however, the minimum on the plot is shifted toward the $_{28}$Ni isotopes, which suggests a change in the order of the $1p_{3/2}$ and $1f_{5/2}$ proton sub-shells with increasing neutron excess. Of course, the sub-shell filling pattern in these nuclei is, to a large extent, determined by pairing. Nevertheless, the HF calculations confirm that for the $Z\sim 30$ chain of isotopes inversion of the $1\pi p_{3/2}$ and $1\pi f_{5/2}$ orbitals indeed takes place around $N\sim 46$ as predicted from Fig. 3. The figure also indicates that on the proton side, $Z = 50$ rather than $Z = 56$ is the tensorial magic number. Again, this suggests that the $1g_{7/2}$ proton sub-shell is filled before $1d_{5/2}$. Consequently, the tensorial magic numbers may slightly differ for neutrons ($N$ = 14, 32, and 56) and for protons ($Z$ = 14, 28, and 50). This effect, however, may strongly depend upon a rather delicate balance between the tensor and SO strengths and needs to be studied in detail.

The second major source of configuration mixing in atomic nuclei - nuclear deformation - is beyond the scope of this work. Our recent calculations [24] show, however, that although it further smears out the tensorial effects major topological features of Fig. 3 including the tensorial magic structure are, to a large extent, retained.