Giant-Dipole Resonance and the Deformation of Hot, Rotating Nuclei

The development of nuclear shapes under the extreme conditions of high angular momentum and/or temperature is examined. Scaling properties are used to demonstrate universal properties of both thermal expectation values of nuclear shapes as well as the minima of the free energy, which can be used to understand the Jacobi transition. A universal correlation between the width of the giant-dipole resonance and quadrupole deformation is found, providing a novel probe to measure the nuclear deformation in hot nuclei.

Figure 1

Dependence of normalized shape parameters as a function of $\xi =J/A^{5/6}$ for ${}^{\mathrm{44}}\mathrm{T}\mathrm{i}$ (×), ${}^{\mathrm{90}}\mathrm{Z}\mathrm{r}$ (+), ${}^{\mathrm{120}}\mathrm{S}\mathrm{n}$ (◇), ${}^{\mathrm{168}}\mathrm{E}\mathrm{r}$ (□), ${}^{\mathrm{208}}\mathrm{P}\mathrm{b}$ (○) at $T=1\mathrm{M}\mathrm{e}\mathrm{V}$.

Figure 2

$\xi$ dependence of (left) normalized expectation values of deformation from $T=1–4\mathrm{M}\mathrm{e}\mathrm{V}$ for ${}^{\mathrm{44}}\mathrm{T}\mathrm{i}$ (×), ${}^{\mathrm{90}}\mathrm{Z}\mathrm{r}$ (+), ${}^{\mathrm{120}}\mathrm{S}\mathrm{n}$ (◇), ${}^{\mathrm{168}}\mathrm{E}\mathrm{r}$ (□), ${}^{\mathrm{208}}\mathrm{P}\mathrm{b}$ (○) indicating $T$ independence. Solid lines are Eqs. (4, 5, 6, 7, 8, 9). (Right) Free energy minima $({\beta }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\gamma }_{\mathrm{m}\mathrm{i}\mathrm{n}})$ for the same nuclei as a function of $\xi$ for $T=1–4\mathrm{M}\mathrm{e}\mathrm{V}$. Dashed lines are for guidance. The Jacobi transition is evident at $\xi \sim 1.2$.

Figure 3

Temperature dependence of moments of deformation at $J=0$ (symbols) compared to predictions from Eqs. (5, 6). Symbols are ${}^{\mathrm{44}}\mathrm{T}\mathrm{i}$ (×), ${}^{\mathrm{90}}\mathrm{Z}\mathrm{r}$ (+), ${}^{\mathrm{120}}\mathrm{S}\mathrm{n}$ (◇), ${}^{\mathrm{168}}\mathrm{E}\mathrm{r}$ (□), ${}^{\mathrm{208}}\mathrm{P}\mathrm{b}$ (○). For a given $T$, the ordering of the symbols is not monotonic with $A$, and is well reproduced by Eq. (5).

Figure 4

Location of the minimum of the free energy surface for hot, rotating nuclei. The circles are shown in increments of $\xi =0.2$, starting with $\xi =0$ at the origin.

Figure 5

Linear relation of average nuclear deformation to the GDR width $\Gamma$, for $44\le A\le 208$, $T\le 4\mathrm{M}\mathrm{e}\mathrm{V}$, and $J\le 60\hslash$, illustrating that $\Gamma$ can be used as a direct experimental measure of nuclear deformation for any $J$, $T$, $A$ where shell corrections are unimportant. Symbols represent the same nuclei as in Figs. 1, 2, 3.

Figure 6

Average nuclear deformation $⟨\beta ⟩$ extracted from experiment [1, 2, 3, 4] using Eq. (9), as a function of $J$ and $T$, compared to the scaling prediction (7) (dashed line) and the full calculation (3) (solid line).