Pair counting, pion-exchange forces and the structure of light nuclei

A simple but useful guide for understanding the structure of light nuclei is presented. It is based on counting the number of interacting pairs in different spin-isospin ($S,T$) states for a given spatial symmetry and estimating the overall binding according to the sum of ${\sigma }_i·{\sigma }_j{\tau }_i·{\tau }_j$ expectation values, as suggested by one-pion exchange. Applied to s- and p-shell nuclei, this simple picture accounts for the relative stability of nuclei as A increases and as T changes across isobars, the saturation of nuclear binding in the p shell, and the tendency to form $d,t$, or α subclusters there. With allowance for pairwise tensor and spin-orbit forces, which are also generated or boosted by pion exchange, the model explains why mixing of different spatial symmetries in ground states increases as T increases across isobars and why, for states of the same spatial symmetry, the ones with greater S are lower in the spectrum. The ordering of some sd-shell intruder levels can also be understood. The success of this simple model supports the idea that one-pion exchange is the dominant force controlling the structure of light nuclei.

Figure 1

Experimental spectrum for $A=6$ nuclei; $T=0$ ($T=1$) states are shown by blue (red) solid lines and breakup thresholds by black dotted lines.

Figure 2

Experimental spectrum for $A=7$ nuclei; $T=1/2$ ($T=3/2$) states are shown by blue (red) solid lines.

Figure 3

Experimental spectrum for $A=8$ nuclei: $T=0$, 1, and 2 states are shown by blue, red, and green solid lines, respectively, and isospin mixed states by magenta lines.

Figure 4

Experimental spectrum for $A=9$ nuclei: $T=1/2$, 3/2, and 5/2 states are shown by blue, red, and green lines respectively; solid lines denote natural-parity states and dashed lines unnatural-parity states.

Figure 5

Simplified experimental spectrum for $A=10$ nuclei; only stable natural-parity states are shown for ${}^{10}\mathrm{Be}$ and ${}^{10}\mathrm{B}$.

Figure 6

Detailed experimental spectrum for ${}^{10}\mathrm{Be}$ and ${}^{10}\mathrm{B}$ nuclei.

Figure 7

Ground-state binding energies for $A\le 16$ s- and p-shell nuclei.