Eigenvalues Of Nuclear Shell Model Hamiltonian And Random Matrices

As the availability of large-scale facilities (such as RIA in USA and RIKEN-RIBF in Japan)in the last decade, nuclear physics has become one of the focuses in science. Although the nuclearshell model provide us with a firm foundation of nuclear structure theory, its configuration spaceis too gigantic to handle even for the best computers. Therefore, various truncation schemes arenecessary. Many efforts have been devote to truncate the huge shell model space to nucleon-pairsubspaces. In another context, the slow neutron scattering experiments showed numerous andnarrow resonances, due to which Wigner introduce random matrix theory. These two pictures giveus a complementary description of nuclear structure.In chapter one, we present a short introduction of the nuclear shell model and the randommatrix theory. We introduce the single particle model and the one-body plus two-body shell modelHamiltonian. We also introduce the Gaussian orthogonal ensemble and the two-body randomensemble as well as their applications in nuclear structure studies.In chapter two, we study the lowest eigenvalues of random Hamiltonian including two-bodyrandom ensemble and Gaussian orthogonal ensemble in both fermion and boson systems. We showthat an empirical formula, which can evaluate the lowest eigenvalues of random Hamiltonian, interms of energy centroid and widths of eigenvalues. It is applicable to many different systems. Weimprove the accuracy of the formula by considering the third central moment. We show that theseformulas are applicable not only to the evaluation of the lowest energy but also to the evaluationof excited energies of systems under random two-body interactions. Moreover, for the two-bodyrandom ensemble, we find a strong linear correlation between eigenvalues and diagonal matrixelements if both of them are sorted from the smaller values to larger ones. By using this linearcorrelation, we are able to reasonably predict all eigenvalues of a given Hamiltonian matrix with-out complicated iterations. For Gaussian orthogonal ensemble matrices, the hyperbolic tangentfunction improves the accuracy of predicted eigenvalues near the minimum and maximum.In chapter three, we investigate regular patterns of matrix elements of the nuclear shell modelHamiltonian, by sorting the diagonal matrix elements from the smaller to the larger values. Byusing simple plots of nonzero matrix elements and lowest eigenvalues of artificially constructed sub-matrices of Hamiltonian, we propose a new and simple truncation method, which predicts thelowest eigenvalue with remarkable precision. Based on this truncation method, we study extrap-olation approaches to evaluate energies of low-lying states for nuclei in the sd and pf shells, bysorting the diagonal matrix elements of the nuclear shell-model Hamiltonian. We introduce anextrapolation method with perturbation and apply our new method to predict both energies of low-lying states and E2transition rates between these states. Our predicted results arrive at an accuracyof the root-mean-squared deviations40–60keV for low-lying states. We also develop a newperturbation method of obtaining the lowest eigenvalues of the nuclear shell model Hamiltonian.Moreover, we obtain the effective Hamiltonian in the truncated space. We exemplify it by using afew realistic nuclei. Overlaps between the wave functions of our approach and those of the exactshell model calculation are presented. Some of the electromagnetic quantities are also discussed.In the appendix, we explain a few works in progress: Simulation of realistic shell modelHamiltonian by using random matrices (A); truncation approaches of random matrices (B); paritruncation schemes of some nuclei around208Pb (C).