【作者】 胡健；

【导师】 黄时中；

【作者基本信息】 安徽师范大学 ， 原子与分子物理， 2007， 硕士

【摘要】 本论文借助角动量耦合理论、对角和法则和不可约张量理论,导出了类硼离子基态和第一激发态非相对论性哈密顿在拉卡基函数之间的矩阵元的表达式,给出了计算类硼离子基态和第一激发态非相对论性能量的方法,并应用此法计算了类硼离子基态和第一激发态非相对论性能量。在此基础上,利用多电子原子哈密顿算符的球张量形式和不可约张量理论,进一步研究了类硼离子基态和第一激发态非相对论能级的相对论修正和精细结构,给出了轨道-轨道、自旋-轨道、自旋-自旋、自旋-其它轨道等相互作用所涉及的所有角向积分和自旋求和的解析计算方法,清晰地展示了多电子原子结构计算的过程,得到了较为精确的理论计算结果。全文共分五章,其主要内容如下。在第一章中,给出了角动量算符的基本性质和角动量耦合的基本理论,为研究类硼离子的非相对性论能量、相对论性修正和精细结构奠定理论基础。在第二章中,借助角动量耦合理论,将多电子原子哈密顿算符中的自旋-自旋、自旋-其它轨道、自旋-轨道以及轨道-轨道相互作用改用球张量表示,这种球张量形式的优点在于将原子哈密顿算符中的径向、角向和自旋部分完全分开,从而便于计算矩阵元,其中的角向矩阵元可以利用不可约张量理论来计算。在第三章中,利用对角和法则,导出了类硼离子基态及第一激发态非相对论能量的解析表达式,完成涉及到的角向积分和自旋求和后,利用变分原理求出了非相对论性波函数中的待定参数,从而确定了非相对论性波函数,并获得类硼离子基态及第一激发态非相对论能量。在第四章中,利用不可约张量理论具体地导出了类硼离子基态以及第一激发态非相对论能级的相对论修正,包括相对论质量修正、达尔文修正、电子间接触相互作用以及轨道-轨道相互作用对非相对论能量进行修正的理论计算式,在这些推导过程中我们完成了所有角向积分和自旋求和计算,相对论修正最终表示为径向积分之和。在第五章中,进一步给出了类硼离子基态以及第一激发态精细结构的计算方法。具体地说,利用不可约张量理论导出了类硼离子基态以及第一激发态的精细结构的表达式,给出了自旋-轨道、自旋-自旋、自旋-其它轨道相互作用等所涉及的所有角向积分和自旋求和的解析计算方法,完成了角向积分、自旋求和的计算,使类硼离子基态以及第一激发态精细结构能级最终可以表示为径向积分之和;在此基础上利用以前所得到的非相对论性波函数,进一步完成了所有径向积分的计算,从而得到了对应能级的精细结构的理论计算值。理论计算结果与实验数据比较表明:总能量的计算误差均小于0.4%,但能级间劈裂的误差较大,这主要是因为第三章中所得到的非相对论性波函数还不够精确。更多还原

【Abstract】 With the aids of the angular momentum coupling theory, the diagonal sum rule and the irreducible tensor theory, a formulism for calculating the matrix elements of Boron-like atoms’non-relativistic Hamilton between the Racah wave functions have been derived. Based on this approach, the non-relativistic energies of Boron-like atoms in the ground and the first excited states have been obtained. Afterwards, using the spherical tensor form of many-electron atom Hamilton and the irreducible tensor theory, the relativistic and fine-structure energy correction to the non-relativistic energy of Boron-like atoms in the ground and the first excited states have been further investigated. A theoretically analytic way has been given as to the calculation of all the radial integrations and spin summations, which concern interactions such as orbit-orbit, spin-orbit, spin-spin and spin-other-orbit. Therefore, the whole process of calculating the many- electron atom energy structures has been explicitly revealed and the results are relatively precise and satisfactory. This paper is divided into five parts as follows.In Chapter one, the fundamental qualities of angular momentum and the angular momentum coupling theory has been introduced, which lays the theoretic foundations of studying the non-relativistic energies, the relativistic and fine-structure energy corrections of Boron-like atoms.In Chapter Two, aided by the angular momentum coupling theory, the items of orbit-orbit, spin-orbit, spin-spin and spin-other-orbit in the many-electron atom Hamilton have all been transformed into spherical tensor forms, which have the advantage that the radial, angular and spin parts of the Hamilton can be completely separated and so the matrix elements can be easily worked out. The angular and spin matrix elements can be obtained using the irreducible tensor theory.In Chapter Three, based on the diagonal sum rule, a formulism for calculating the matrix elements of Boron-like atoms’non-relativistic Hamilton between the Racah wave functions have been derived. After all the angular integrations and spin summations completed, the needed parameters in the non-relativistic wave functions have been acquired using the variational method and so the functions themselves have been established in this way. And the non-relativistic energies for boron-like atoms in the ground and first excited states have been obtained.In Chapter Four, using the irreducible tensor theory, an expression for the relativistic energy correction to the non-relativistic energies of boron-like atoms in the ground and first excited states has been established, which includes mass correction, Darwin corrections, spin-spin-contact and orbit-orbit corrections. We’ve completed all the angular integrations and spin summations involved and make the final expression for relativistic correction in a combination of radial integrations.In Chapter Five, an approach to calculate the fine-structure energies of boron-like atoms in the ground and first excited states has been provided. Concretely speaking, using the irreducible tensor theory, we deuced the formulism for fine-structure energies of boron-like atoms in the ground and first excited states, which involves spin-orbit, spin-spin and spin-other-orbit interactions; completing all the angular integrations and spin summations needed, we make the final expression for fine-structure energies of boron-like atoms in the ground and first excited states in a combination of radial integrations. Afterwards, employing the wave functions we got in Chapter Three, we accomplished all the radial integrations and so obtained the fine-structure energies. The results show that the errors for the total energies all less than 0.4%, yet the errors for splitting are a little striking, which is mainly caused by the inaccuracy of the wave functions.更多还原