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1、Microscopic Description of the Breathing Mode and Nuclear Compressibility,Presented By: David Carson Fuls Cyclotron REU Program 2005 Mentor: Dr. Shalom Shlomo,Introduction,We use the microscopic Hartree-Fock (HF) based Random-Phase-Approximation (RPA) theory to describe the breathing mode in the 90Z
2、r, 116Sn, 144Sm, and 208Pb nuclei, which are very sensitive to the nuclear matter incompressibility coefficient K. The value of K is directly related to the curvature of the equation of state, which is a very important quantity in the study of properties of nuclear matter, heavy ion collisions, neut
3、ron stars, and supernova. We present results of fully self-consistent HF+RPA calculations for the centroid energies of the breathing modes in the four nuclei using several Skyrme type nucleon-nucleon (NN) interactions and compare the results with available experimental data to deduce a value for K.,
4、 fm-3, = 0.16 fm-3,E/A MeV,Nuclear Matter Incompressibility,The value of K is directly related to the second derivative of the equation of state (EOS) of symmetric nuclear matter. Once we know that a two-body interaction is successful in determining the centroid energy of the monopole resonance, we
5、can use that interaction to find the EOS and from that we can find the value of K.,E/A = -16 MeV,Classical Picture of the Breathing Mode,In the classical description of the breathing mode, the nucleus is modeled after a drop of liquid that oscillates by expanding and contracting about its spherical
6、shape. We consider the isoscalar breathing mode in which the neutrons and protons move in phase (T=0, S=0).,In the scaling model, we have the matter density oscillates as We consider small oscillations, so is very close to zero ( 0.1). Performing a Taylor expansion of the density we obtain,We have,
7、Where is equal to This nicely agrees with the transition density obtained from RPA calculations.,Microscopic Description of the Breathing Mode,Ground State The ground state of the nucleus with A nucleons is given by an antisymmetric wave function which is, in the mean-field approximation, given by a
8、 Slater determinant. In the spherical case, the single-particle wave function is given in terms of the radial , the spherical spin harmonic , and the isospin functions:,The total Hamiltonian of the nucleus is written as a sum of the kinetic T and potential V energies Where The total energy E,Now we
9、apply the variation principle to derive the Hartree-Fock equations. We minimize by varying with the constraint of particle number conservation, and obtain the Hartree-Fock equations,For the two-body nuclear potential Vij, we take a Skyrme type effective NN interaction given by, The Skyrme interactio
10、n parameters (ti, xi, , and Wo) are obtained by fitting the HF results to the experimental data. This interaction is written in terms of delta functions which make the integrals in the HF equations easier to carry out.,For a spherical case the HF equations can be reduced to, where the effective mass
11、 , the central potential , and the spin-orbit potential are written in terms of the Skyrme parameters, matter density, charge density, and current density.,Method of Solving the HF Equations,With an initial guess of the single-particle wave functions (usually the harmonic oscillator wave functions b
12、ecause they are known analytically) we can find the matter density, kinetic density, current density, and charge density. Once we know these values, we can use them to find the effective mass, central potential, and the spin- orbit potential. We then use these functions in the HF equations to find t
13、he new radial wave functions. We repeat the whole procedure with these new wave functions until convergence is reached.,Single-Particle Energies (in MeV) for 40Ca,*TAMU Skyrme Interaction: B. K. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C 72, 014310 (2005).,Giant Resonance In HF based RPA theory,
14、 giant resonances are described as coherent superpositions of particle hole excitations of the ground state. In the Greens Function formulation of RPA, one starts with the RPA-Greens function which is given by where Vph is the particle-hole interaction and the free particle-hole Greens function is d
15、efined as, where is the single-particle wave function, i is the single-particle energy, and ho is the single-particle Hamiltonian.,We use the scattering operator F where for monopole excitation, to obtain the strength function and the transition density.,A Note on Self-Consistency,In numerical imple
16、mentation of HF based RPA theory, it is the job of the theorist to limit the numerical errors so that these are lower than the experimental errors. Some available HF+RPA calculations omit parts of the particle-hole interaction that are numerically difficult to implement, such as the spin-orbit or Coulomb parts. Omission of these terms leads to self-consistency violation, and the s
