COLLISIONAL DAMPING OF GIANT QUADRUPOLE RESONANCE*
A. Gokalp1, S. Yildirim1, O. Yilmaz1, S. Ayik2
Physics Department, Middle East Technical University, 06531 Ankara, Turkey 2Physics Department, Tennessee Technological University, Cookeville TN 38505, USA
agokalp@metu.edu.tr ovilmaz@metu.edu.tr
ABSTRACT
We calculate the collisional width of giant quadrupole excitation for 12"Sn nucleus at finite temperature in Thomas-Fermi approximation using a description for the distortion functions by parameterizing the velocity field in terms of the second order Bessel functions.
Recently, we have investigated the collisional damping of giant dipole resonance [1] and giant monopole resonance [2] in Thomas-Fermi approximation by employing the microscopic in medium cross-sections of Li and Machleidt and the phenomenological SkM* and Gogny forces. Here in this work we extend these calculations to the giant quadruple resonance.
Theoretical investigations of nuclear collective vibrations built on the ground and excited states of nuclei are mostly based on the random phase approximation (RPA) theory. However, since RPA approach is the small amplitude limit of the time dependent Hartree-Fock (TDHF), it is not suitable for describing the damping of the collective excitations [3], In order to describe nuclear collective response including damping, it is necessary to incorporate coupling between the collective states and the doorway configurations [4],
In Ref. [5] the collisional relaxation rates of giant quadrupole vibrations in nuclei are calculated using a semiclassical transport equation with a non-Markovian collision term. The isoscalar decay widths are given as Tx =\ dr Tx {r) [5] with
(1) Tx(r ) = -y^'uIP \dp2dpî dp4 [ ifpp + W + Wnn p n .
\2
X f \ f 2 J3f 4
* Presented at I. Eurasia Conference on Nuclear Science and its Aplication, Izmir-Turkey, 23-27 October 2000.
where Nx =\d?dp{xx f [ - 0 !de)f] is a normalization, A%x = x x (l) +%x ( 2 ) -x x (3)~%x (4),
Z = \S{ti(ox - Ae)-8{ticox + Ae)\lho)x , cox .is the mean-frequency of the RPA mode, and XxC) denotes the distortion factor of the phase-space density in the corresponding mode. In this expression, transition rates IVpp. I Vim. Wpn, associated with proton-proton, neutron-neutron, proton-neutron collisions are given in terms of the corresponding scattering cross-sections as
(2) W (12;34) 1 4 h da
(2n) m3 m2 dQ$(P\ + p 2 - p3 - P4) •
We apply formula (1) to calculate the collisional widths of giant quadrupole modes by determining the distortion functions in terms of the velocity field O(r) associated with the collective mode as % = (p.V)(p.V) ®(r) [6]. An accurate description of the quadrupole vibration is obtained by parameterizing the velocity field in terms of the second order Bessel function^(r)=j2(kr), with k as the radial wave number [6]. With such a velocity field the distortion function can be expressed as Xq J 2 j 2
kr (p.r)2 + -k-j2 P 2 kr 1
indicate the derivatives of j2(kr) with respect to its argument.
where primes
Figure 1: The collisional damping of giant quadrupole resonance in 120Sn as a function of temperature with bare nucleon mass. Dotted lines are calculations with the cross sections of Li and Machleidt, and solid and dashed lines are results with the SkM* and the Gogny cross-sections.
In order to assess how much of the total width of giant resonance excitations is exhausted by decay into the incoherent 2p-2h states, realistic in-medium cross-sections are needed and for this reason the microscopic in-medium cross-section calculated by Li and Machleidt is employed [1]. For comparison the results of calculations with phenomenological finite range Gogny force and zero range Skyrme force are also calculated. In calculations for the resonance energies
h®Q = 64A_1/3 M eV is used with the wave number k = 3.3421 /R [7].
In our work we carried out the calculation of the width of giant quadrupole resonance exactly by evaluating the momentum integrals without any approximation using a suitable change of variables. We evaluate the integrals by neglecting the angular anisotropy of the cross-sections
and making a replacement (da / dQ)pn ^ a pn / 4n [1]. In numerical calculations, we determine the nuclear density p(r) in Thomas-Fermi approximation using a Wood-Saxon potential with a depth Vo = -44 MeV, thickness a=0.67 fm and sharp radius Ro=1.27 A13 fm, calculate the position dependent chemical potential n(e, T)in the Fermi-Dirac function f (e, T)at each temperature. The results of these calculations are shown in Fig.1 and Fig.2 where the calculated GQR widths in 120Sn nuclei are plotted as a function of temperature for bare mass and effective mass respectively. For zero temperature our results compare well with the approximate results of Ref. [6].
Figure 2: The collisional damping of giant quadrupole resonance in 120Sn as a function of temperature with effective nucleon mass. Dotted lines are calculations with the cross-sections of Li and Machleidt, and dashed lines are results with the SkM* and the Gogny cross-sections.
REFERENCES
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2. A.Gokalp, S. Yildirim, O.Yilmaz, S. Ayik, Presented at Second International Balkan School on Nuclear Physics, 12-19 September 2000, Bodrum, Turkey
3. “Electric and Magnetic Resonances in Nuclei”, ed. By J Speth, World Scientific (1991). 4. A.Smerzi, A. Bonasera, M. DiToro, Phys. Rev. C 44, 1723 (1991).
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