Investigation On The Energy Levels And The Fine Structure Of Lithium-like Atoms

In this paper, the energy levels and the fine structure of lithium-like atoms have been investigated. Firstly, the method for calculating the non-relativistic energy levels of atoms with three valence electrons is presented; general expressions for the matrix elements of the non-relativistic atomic Hamiltonian are derived by using angular momentum coupling theory and irreducible tensor theory. The corresponding results are applied to analyzing the term energies of n₁p²n₂p configuration and calculating the non-relativistic energies of lithium-like atoms ground states and low-excited states. Secondly, based on the theory of non-relativistic energy structure, the relativistic corrections and the fine structure for the low-excited states of lithium-like atoms are studied with the help of irreducible tensor theory. Analytical method for calculating the relevant angular integration and spin summation is established. Thirdly, in order to improve the calculated results for the non-relativistic energy levels, MCHF (multi-configuration Hatree-Fock) method is applied to calculating the non-relativistic energies and the ionization potentials for the 1s²2s 2^S states of lithium-like atoms. There are four chapters in this paper. The main contents are listed as following. In chapter one, the basic form of Hamiltonian for multi-electron atoms is introduced, the solution structure of the corresponding eigenvalue equations and the construction of Racah functions for atoms with three valence electrons are discussed; and the method for rewriting the Hamiltonian in terms of spherical tensors is presented briefly. In chapter two, a method for calculating the non-relativistic energy levels of atoms with three valence electrons is presented; general expressions for the matrix elements of the non-relativistic atomic Hamiltonian are derived. The angular integration and spin summation are carried out explicitly and the matrix elements are presented in terms of 3j, 6j, 9j symbols and Slater-Condon integrals. The method is applied to analyzing the term energies of n₁ p²n₂p configuration and calculating the non-relativistic energies of lithium-like atoms ground states 1s² 2s²S and low-excited states 1s² 2p 2^P, 1s² 3p 2^P, 1s² 3d 2^D. The calculated results are compared with other theoretical works. In chapter three, the method for calculating the relativistic corrections and the fine structure for 1s²nl 2^L states of lithium-like atoms is presented, general expressions for relativistic corrections to non-relativistic energy levels for 1s 2np2P states and fine structure of 1s²np2^P,1s²nd2^D states in lithium-like atoms are established. The angular integration and spin summation involved in the expressions are carried out with the help of irreducible tensor theory and the results are expressed as a sum of some radial integrals. Furthermore, by using the radial functions obtained in chapter two, the relativistic corrections of 1s² 2p2^P states and the fine structure of 1s²2p2^P, 1s² 3d2^D states in lithium-like atoms are calculated. The results show that the relativistic corrections agree with experimental values well, especially for high-Z ions (for example, the difference between the calculated result and the experimental data is 0.8% for N4⁺ ion 1s² 2p2^P state). It is found, however, that the calculated results of the fine structure for light atoms are not accurate enough. An important reason is that the radial functions obtained in chapter two are not precise enough. In chapter four, the non-relativistic energies and the ionization potentials for the 1s²2s2^S states of lithium-like atoms (Z=9-16) are calculated using MCHF (multi-configuration Hatree-Fock) method. The calculated results are compared with experimental values, and the differences of non-relativistic energies are less than 0.0012%.