Generic features of the neutron-proton interaction

We show that fully aligned neutron-proton pairs play a crucial role in the low-energy spectroscopy of nuclei with valence nucleons in a high-$j$ orbital. Their dominance is valid in nuclei with valence neutrons and protons in different high-$j$ orbitals as well as in $N=Z$ nuclei, where all nucleons occupy the same orbital. We demonstrate analytically this generic feature of the neutron-proton interaction for a variety of systems with four valence nucleons interacting through realistic, effective forces. The dominance of fully aligned neutron-proton pairs results from the combined effect of (i) angular momentum coupling and (ii) basic properties of the neutron-proton interaction.

Figure 1

TBMEs for three different $\nu 0h_{11/2}\pi 0g_{9/2}$ interactions: (i) realistic (jj45pna, black circles), (ii) surface delta (SDI, blue triangles), and (iii) quadrupole ($Q_{\nu }·Q_{\pi }$, red squares).

Figure 2

Panels (a) and (b): The coefficient $C_{J_{\nu \pi }}$ in Eq. (1) for $j_{\nu }=11/2$ and $j_{\pi }=9/2$ as a function of $J_{\nu \pi }$ for $J=0$ and $J=2$. Panels (c) and (d): The energy contribution in the $\nu \nu \text{−}\pi \pi$ basis due to the individual TBMEs of the jj45pna interaction as a function of $J_{\nu \pi }$ for $J=0$ and $J=2$. Different symbols correspond to various couplings $(J_{\nu },J_{\pi },J_{\nu }^{\prime },J_{\pi }^{\prime })$ as indicated. Full (open) symbols indicate diagonal (off-diagonal) matrix elements in the $\nu \nu \text{−}\pi \pi$ basis.

Figure 3

Energies of the yrast eigenstates of the Hamiltonian (1) for two neutrons and two protons (${}^{128}\mathrm{Cd}$) as a function of $J_{\nu \pi }$, the angular momentum of the only nonzero component in the $\nu \pi$ interaction ($V_{\nu \pi }^{J_{\nu \pi }}\ne 0$). The results are for three different $\nu \pi$ interactions: (i) realistic (jj45pna, black), (ii) surface delta (SDI, blue), and (iii) quadrupole ($Q_{\nu }·Q_{\pi }$, red). Also the eigenenergies of the full Hamiltonian and the experimental spectrum of ${}^{128}\mathrm{Cd}$ are shown.

Figure 4

Graphical illustration of the (a) $\nu \nu \text{−}\pi \pi$ and (b) $\nu \pi \text{−}\nu \pi$ basis for a $0^{+}$ state. The dashed lines indicate the angular momentum vector adding scheme for different pairs. In (b) both couplings are identical but one pair is spin-flipped around its total angular momentum axis.

Figure 5

Squares of the components of the yrast eigenstates of ${}^{128}\mathrm{Cd}$ expressed in the $\nu \pi \text{−}\nu \pi$ basis $|J_{\nu \pi }^1{J}_{\nu \pi }^2;J\rangle$ for (a) $J=0$ and (b) $J=2$. The main figure shows ${\langle J_1|J_{\nu \pi }^1{J}_{\nu \pi }^2;J\rangle }^2$ as a function of $J_{\nu \pi }^1=J_{\nu \pi }^2$ for three different $\nu \pi$ interactions: (i) realistic (jj45pna, black), (ii) surface delta (SDI, blue), and (iii) quadrupole ($Q_{\nu }·Q_{\pi }$, red). In the inset of (b) this quantity is shown (for jj45pna) as proportional to the size of the squares as a function of $J_{\nu \pi }^1$ and $J_{\nu \pi }^2$.

Figure 6

Overlaps ${\langle J_{\mathrm{yrast}}|{\left(J_{\nu \pi }^{\mathrm{FAP}}\right)}^2;J\rangle }^2$ as a function of the angular momentum $J$ for (a) ${}^{128}\mathrm{Cd}$, (b) ${}^{212}\mathrm{Po}$, and (c) ${}^{96}\mathrm{Cd}$ calculated with a realistic interaction (black), and compared to the surface delta (SDI, blue) and quadrupole ($Q_{\nu }·Q_{\pi }$, red) interactions.

Figure 7

Same as Fig. 3 but for four neutrons and four protons (${}^{124}\mathrm{Pd}$).