$B\left(E2\right)$ value and configuration mixing in ${}^{32}$Mg

I demonstrate that the $B$($E$2) value in ${}^{32}$Mg can be understood with a model in which both the ground and 2${}^{+}$ first-excited states are predominantly of $sd$-shell character.

Figure 1

Solid curve is the value of $A^2$ that maximizes the ${}^{32}$Mg $B$($E$2) for a given value of $a^2$. Long dashes are the ratio of that $B$($E2$) to the one in ${}^{30}$Mg. Short dashes represent the ratio for equal 0${}^{+}$ and 2${}^{+}$ mixing $A^2$ $=$ $a^2$.

Figure 2

Calculated $B$($E$2) for ${}^{32}$Mg (solid curve) with the $\pm$ 1 $\sigma$ limits (dashed) vs $a^2$, the amount of $sd$-shell component in the gs. Open circles are at the experimental value extracted with a large correction for feeding from above. Solid squares are at the experimental value if the analysis included little or no such correction.

Figure 3

Squares are as in Fig. 2, but are now plotted vs $C^2$, using $a^2$ $=$ 0.75 and $b^2$ $=$ $B^2$ $=$ 0.25, and with the $C$ term (see text). The upper curve is for the $A$,$B$ combination that maximizes $B$($E$2) for given $C^2$, still with $a^2$ $=$ 0.75 and $b^2$ $=$ 0.25.

Figure 4

The value of $M$($sd$) (see text) necessary to fit the experimental $B$($E$2) $=$ 446(31) $e^2$fm${}^4$ for ${}^{32}$Mg, plotted vs $A^2$ for three different small values of $C^2$ $=$ 0.025 (long dashes), 0.05 (short dashes), and 0.10 (solid line). Open circles represent the experimental $M$(${}^{30}$Mg).

Figure 5

Mixed and unmixed 0${}^{+}$ and 2${}^{+}$ states in ${}^{32}$Mg. The left-hand column depicts the physical states, the middle column depicts the 0${}^{+}$ basis states from the mixing in Ref. [6], and the right-hand column depicts one possibility for the 2${}^{+}$ basis states.