Giant dipole resonance in highly excited nuclei

The evolution of the giant dipole resonance's (GDR) width and shape at finite temperature $T$ and angular momentum $J$ is described within the framework of the phonon damping model (PDM). The PDM description is compared with the established experimental systematics obtained from heavy-ion fusion and inelastic scattering of light particles on heavy target nuclei, as well as with predictions by other theoretical approaches. Extended to include the effect of angular momentum $J$, its strength functions have been averaged over the probability distributions of $T$ and $J$ for the heavy-ion fusion-evaporation reaction, which forms the compound nucleus $^{88}$Mo at high $T$ and $J$. The results of theoretical predictions are found in excellent agreement with the experimental data. The predictions by PDM and the heavy-ion fusion data are also employed to predict the viscosity of hot medium and heavy nuclei.


I. INTRODUCTION
The GDR built on highly excited compound (CN) nuclei was first observed in 1981 [1], and at present rich experimental systematics has been established for the GDR widths at finite temperature T and angular momentum J in various medium and heavy nuclei formed in heavy ion fusions, deep inelastic scattering of light particles on heavy targets, and α induced fusions [2,3]. The common features of the hot GDR are: (1) Its energy is nearly independent of T and J, (2) Its full width at half maximum (FWHM) remains mostly unchanged in the region of T ≤ 1 MeV, but increases sharply with T within 1≤ T ≤ 2.5 -3 MeV, and seems to saturate at T ≥ 4 MeV. As a function of J, a significant increase in the GDR width is seen only at J ≥ 25 -27 . In Ref. [4], by adding the pre-equlibrium γ emission in reanalyzing some GDR data, it was claimed that the GDR width does not saturate. However, it was realized later that the pre-equilibrium emission is proportional to the asymmetry between projectiles and targets and lowers the CN excitation energy. This may alter the conclusion on the role of pre-equlibrium emission. The recent measurements in 88 Mo at T ≥ 3 MeV and J > 40 did not show any significant effect of pre-equilibrium emission on the GDR width [5]. The evaporation width due to the quantal mechanical uncertainty in the energies of the CN states was also proposed to be added into the total GDR width [6], whose effect may become noticeable only at much higher values of T (≫ 3.3 MeV) and J (≫ 30 ) [7].
From the classical representation of the GDR as a damped spring mass system, it is clear that the damping width of the oscillator should be smaller than its frequency otherwise the spring mass system cannot make any oscillation. This means that the GDR width in the classical picture is upper-bounded by its energy. This implies the saturation of the GDR width.
The present contribution summarizes the achievements of the Phonon Damping Model (PDM) [8] in the description of the the GDR width and shape at finite T and J. The GDR parameters predicted by the PDM and experimentally extracted are also used to calculate the shear viscosity of finite hot nuclei.

II. DAMPING OF GDR IN HIGHLY EXCITED NUCLEI
The PDM's Hamiltonian consists of the independent single-particle (quasiparticle) field, GDR phonon field, and the coupling between them. The Woods-Saxon potentials at T = 0 are used to obtain the single-particle energies ǫ k . The GDR width Γ(T ) is a sum: Γ(T ) = Γ Q + Γ T of the quantal width, Γ Q , and thermal width, Γ T . In the presence of superfluid pairing, the quantal and thermal widths are given as [8]  which, for medium and heavy nuclei, can be well approximated with the Fermi-Dirac distribution for independent quasiparticles, n k = [exp(E k /T ) + 1] −1 . The parameter F 1 is chosen so that Γ Q at T = 0 is equal to GDR's width at T = 0, whereas the parameter F 2 is chosen so that, with varying T , the GDR energy E GDR does not change significantly.
The latter is found as the solution of the equation the energy of the GDR phonon before the coupling between the phonon and single-particle mean fields is switched on, and P q (ω) is the polarization operator owing to this coupling, whose explicit expression in given in Refs. [8]. The GDR strength function is calculated as  Fig. 1 (c) [9].
To describe the non-collective rotation of a spherical nucleus, the z-projection M of the total angular momentum J is added into the PDM Hamiltonian as −γM , where γ is the rotation frequency [10]. The latter and the chemical potential are defined, in the absence of pairing, from the equation , where N is the particle number and f ± k are the single-particle occupation numbers, The GDR width obtained within the PDM for 88 Mo is plotted against E * in Fig. 1 (e) in comparison with the available GDR experimental widths for molybdenum isotopes. rather strong, but at E * > 80 MeV the width increase is weaker because of the saturation of J max [11].

III. SHEAR VISCOSITY OF HOT NUCLEI
In the verification of the condition for applying hydrodynamics to nuclear system, the quantum mechanical uncertainty principle requires a finite viscosity for any thermal fluid.
Kovtun, Son and Starinets (KSS) [12] conjectured that the ratio η/s of shear viscosity η to the entropy volume density s is bounded below for all fluids, namely the value η/s = /(4πk B ) is the universal lower bound (KSS bound or unit). From the viewpoint of collective theories, one of the fundamental explanations for the giant resonance damping is the friction term (or viscosity) of the neutron and proton fluids. By using the Green-Kubo's relation, it has been shown in Ref. [13] that the shear viscosity η(T ) at finite T is expressed in terms of the GDR's parameters at zero and finite T as The predictions for the shear viscosity η and the ratio η/s by the PDM, pTSFM, AM, and  [14], where the same lower value η(0) =0.6u was used.

IV. CONCLUSIONS
The PDM generates the damping of GDR through its couplings to ph configurations, causing the quantal width, as well as to pp and/or hh configurations, causing the thermal width. This leads to an overall increase in the GDR width at low and moderate T , and its saturation at high T . At very low T < 1 MeV the GDR width remains nearly constant because of thermal pairing. The PDM predictions agree well with the experimental systematics for the GDR width and shape in various medium and heavy nuclei. The PDM also predicts the shear viscosity to the entropy-density ratio η/s between (1.3 -4.0) KSS units for medium and heavy nuclei at T = 5 MeV, almost the same at that of the quark-gluon-plasma like matter at T > 170 MeV (1.5 -2.5 KSS) discovered at RHIC and LHC.