On anharmonicities of giant dipole excitations

The role of anharmonic effects on the excitation of the double giant dipole resonance is investigated in a simple macroscopic model. Perturbation theory is used to find energies and wavefunctions of the anharmonic oscillator. The cross sections for the electromagnetic excitation of the oneand two-phonon giant dipole resonances in energetic heavy ion collisions are then evaluated through a semiclassical coupled-channel calculation. It is argued that the variations of the strength of the anharmonic potential should be combined with appropriate changes in the oscillator frequency, in order to keep the giant dipole resonance energy consistent with the experimental value. When this is taken into account, the effects of anharmonicities on the double giant dipole resonance excitation probabilities are small and cannot account for the well known discrepancy between theory and experiment. Typeset using REVTEX 1 The double giant dipole resonance (DGDR) has attracted considerable interest in the last decade. Several experiments to measure the DGDR cross section using relativistic heavy ion beams have been performed [1–6]. Comparison with the predictions of the harmonic oscillator model has clearly demonstrated a systematic discrepancy. The experimental values for the DGDR cross sections exceed the theoretical predictions by a considerable amount. One of the attempts to explain these differences was made by Bortignon and Dasso [7], using a macroscopic anharmonic oscillator model. These authors found that with a small anharmonic perturbation of the r-type one can reproduce both the experimentally observed DGDR excitation energy (which only marginally differs from that obtained in the harmonic approximation) and the DGDR cross section for the Pb+ Pb collision at 640A·MeV. They reached a similar conclusion for the Xe + Pb collision at 700A·MeV, where a much greater discrepancy from the harmonic model appears [2]. The purpose of this paper is to point out that this model does not lead to the enhancement found in Ref. [7], if proper renormalization of the oscillator frequency is performed in order to guarantee that the theoretical giant dipole resonance (GDR) excitation energy is kept at the experimental value. The model of Refs. [7,?] is based on the following Hamiltonian H = H0 + F (x, y, z; t), (1) where H0 is the anharmonic oscillator describing the intrinsic motion of the projectile, H0 = 1 2D (p2x + p 2 y + p 2 z) + C 2 (x + y + z) + B 4 (x + y + z), (2) where D is the mass parameter, C is the oscillator strength and B is the strength of the anharmonicity. Here, we take the mass parameter to be the reduced mass for the motion of the protons against the neutrons, D = NZ A m0, where m0 is the average nucleon mass. The beam is assumed to be parallel to the x−axis 2 and the coupling interaction F is derived from the Lienard-Wiechert potential [10] in the projectile frame φ(x, y, z, t) = ZT eγ [γ2(x− vt) + (y − b) + z2]1/2 , (3) were ZT e is the charge of the target, b is the impact parameter, and γ is the Lorentz factor, γ = 1/ √ 1− (v/c). To be specific, we study the Pb + Pb collision at 640A·MeV. We first solve the Schrödinger equation for the intrinsic motion, described by H0. For this purpose it is convenient to recast the intrinsic Hamiltonian into the following equivalent form