Giant dipole resonance and Jacobi transition with exact treatment of fluctuations

In a macroscopic approach to giant dipole resonance (GDR) in hot and rotating nuclei, the observables are related to the nuclear free energy surface with consideration of thermal shape fluctuations. This formalism is revisited with more exact methods. The Nilsson-Strutinsky (NS) method extended to high spin and temperature is used for free energy calculations. Various approaches to calculate shell corrections at finite temperature and spin are compared. The GDR built on the states determined by the NS method is studied with a macroscopic model comprising anisotropic harmonic oscillator potential with separable dipole-dipole interaction. Methods to parametrize the free energy, such as the Landau theory, for easier evaluation of thermal fluctuations, are discussed along with a scheme to evaluate thermal fluctuations exactly. The Landau theory is found to work well even in the extreme limits of spin, however, in the absence of strong shell effects. GDR as a probe for Jacobi transition leading to hyperdeformation is analyzed in the case of zirconium isotopes.

Figure 1

Shape dependence of $\tilde{a}$ in the nucleus ${}^{208}\text{Pb}$. The shaded circle represents the minimum and the contour line spacing is $0.1\mathrm{MeV}$.

Figure 2

Comparison of shell corrections calculated using different methods for ${}^{208}\text{Pb}$.

Figure 3

GDR cross sections and free energy surfaces for the nuclei ${}^{90}\text{Zr}$ and ${}^{92}\text{Mo}$. Left: Experimental data represented by solid squares are taken from Ref. 39. The solid lines represent calculations with orientation and thermal fluctuations, dashed lines correspond to thermal fluctuations alone, and the dash-dotted lines correspond to most probable shapes. Right: The contour line separation is $0.5\mathrm{MeV}$ and the most probable shape is represented by a solid circle. The first two minima are shaded.

Figure 4

GDR width at different temperatures for the nucleus ${}^{120}\text{Sn}$ obtained by NS calculations (solid line), liquid-drop model calculations (dashed line) in comparison with the experimental data points [40] (solid squares).

Figure 5

GDR width at different angular momentum for the nucleus ${}^{120}\text{Sn}$. The experimental data points represented by solid squares are taken from Ref. 41. The theoretical results are represented by the solid line.

Figure 6

The GDR width at different spins for the nucleus ${}^{194}\text{Hg}$. The experimental GDR widths represented by solid squares are taken from Ref. 42. The solid line represents our theoretical results.

Figure 7

GDR width at different temperatures for the nucleus ${}^{208}\text{Pb}$ obtained in present calculations with Landau theory (dash-dotted lines) and exact method (solid lines) in comparison with other theoretical values from Ref. 14 (dashed lines) and Ref. 40 (dotted lines). Black and gray correspond to the NS and liquid-drop calculations, respectively. The experimental data (solid squares) are those from Ref. 40.

Figure 8

Comparison between the experimental GDR cross sections in hot ${}^{45}\text{Sc}$ at different temperatures and spins. The experimental data denoted by solid squares are taken from Ref. 43. The theoretical results from exact fluctuation calculations are denoted by solid lines and those obtained using Landau theory are denoted by dashed lines.

Figure 9

Hodographs depicting the shape transitions occurring in rapidly rotating ${}^{84}\text{Zr}$ at different temperatures. The solid squares represent the most probable shapes and the open circle and solid circles represent the thermal averaged shapes calculated with Landau theory and exact method, respectively.

Figure 10

Same as Fig. 9, but for the nucleus ${}^{90}\text{Zr}$.

Figure 11

Theoretical GDR cross sections for the nucleus ${}^{84}\text{Zr}$ at $T=0.5\mathrm{MeV}$ and $T=1.0\mathrm{MeV}$ using the Landau theory (dashed lines) and exact method (solid lines) for thermal fluctuations. The existence of the Jacobi transition is clearly seen from the magnitude of splitting.

Figure 12

Same as Fig. 11 but at $T=2.0\mathrm{MeV}$ and $T=3.0\mathrm{MeV}$.