Relativistic Corrections to the Fine Structure of Helium

1965 ◽  
Vol 140 (5A) ◽  
pp. A1498-A1504 ◽  
Author(s):  
Kee Yong Kim
Keyword(s):  
Fine Structure ◽  
Relativistic Corrections

Perturbation Methods

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Perturbation Theory ◽  
Quantum Mechanics ◽  
Fine Structure ◽  
Dirac Equation ◽  
Perturbation Methods ◽  
Relativistic Quantum ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Perturbation Expansions ◽  
Relativistic Corrections

Perturbation theory has been one of the most frequently used and most powerful tools of quantum mechanics. The very foundations of relativistic quantum theory—quantum electrodynamics—are perturbative in nature. Many-body perturbation theory has been used for electron correlation treatments since the early days of quantum chemistry, and in more recent times multireference perturbation theories have been developed to provide quantitative or semiquantitative information in very complex systems. In the beginnings of relativistic quantum mechanics, perturbation methods based on an expansion in powers of the fine structure constant, α = 1/c, were used extensively to obtain operators that would provide a connection with nonrelativistic quantum mechanics and permit some evaluation of relativistic corrections, in days well before the advent of the computer. This seems a reasonable approach, considering the small size of the fine structure constant—and for light elements it has been found to work remarkably well. Relativity is a small perturbation for a good portion of the periodic table. Perturbation expansions have their limitations, however, and as well as successes, there have been failures due to the highly singular or unbounded nature of the operators in the perturbation expansions. Therefore, in recent times other perturbation approaches have been developed to provide alternatives to the standard Breit–Pauli approach. This chapter is devoted to the development of perturbation expansions in powers of 1/c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy–Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit–Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties.

THE HEUN–SCHRÖDINGER RADIAL EQUATION WITH TWO AUXILIARY PARAMETERS FOR H-LIKE ATOMS

2002 ◽  
Vol 16 (25) ◽  
pp. 937-941 ◽  
Author(s):  
V. F. TARASOV
Keyword(s):  
Fine Structure ◽  
Coulomb Potential ◽  
Radial Function ◽  
Standard Form ◽  
Nuclear Motion ◽  
Heun Equation ◽  
Radial Equation ◽  
Schrödinger’S Equation ◽  
Relativistic Corrections ◽  
Schrödinger's Equation

This article deals with the connection between the confluent Heun equation (with two singularities 0 and ∞) and Schrödinger's equation for H-like atoms with two "auxiliary" parameters μ and ν which "influence" the spectrum of eigenvalues, the Coulomb potential and also the radial function. The case of μ = ν = 1 corresponds to the "standard" form of Schrödinger's equation. In terms of the parameter ν, for example, "relativistic corrections" of the type: Dirac's formula of the fine structure, the nuclear motion, etc., may be considered.

scholarly journals Energy, fine structure, hyperfine structure, and transitions for the high-lying multi-excited 4Pe,o states of B-like ions

2016 ◽  
Vol 94 (5) ◽  
pp. 448-457
Author(s):  
Chun Mei Zhang ◽  
Yan Sun ◽  
Chao Chen ◽  
Feng Wang ◽  
Bin Shao ◽  
...  
Keyword(s):  
Perturbation Theory ◽  
Fine Structure ◽  
Hyperfine Structure ◽  
Excited States ◽  
Quantum Electrodynamic ◽  
Higher Order ◽  
Variation Method ◽  
First Order ◽  
Relativistic Corrections ◽  
First Order Perturbation

The energies of the high-lying multi-excited states 1s22s2pnl and 1s22p2nl 4Pe,o (n ≥ 2) for B-like C+, N2+, F4+, and Mg7+ ions are calculated using Rayleigh–Ritz variation method with multiconfiguration interaction, and the inclusion of mass polarization and relativistic corrections. The fine structure and hyperfine structure for these systems are investigated using first-order perturbation theory. The configuration structure of the high-lying multi-excited series is identified not only by energy, but also by its contribution to normalization of the angular spin components, and it is further tested by the addition of relativistic corrections and fine structure splittings. Transition wavelengths including the quantum electrodynamic effects and higher-order relativistic corrections are determined.

On the use of the Breit—Pauli approximation in the study of relativistic effects in electron-atom scattering

1975 ◽  
Vol 277 (1273) ◽  
pp. 587-622 ◽  
Keyword(s):  
Fine Structure ◽  
Scattering Problem ◽  
Relativistic Effects ◽  
Present Theory ◽  
Collision Strength ◽  
Relativistic Corrections ◽  
Electron Atom ◽  
Atom Scattering ◽  
Relativistic Collision ◽  
Semi Empirical

The electron-atom scattering problem is formulated by using the Breit-Pauli hamiltonian, and the Kohn variational principle is derived for this hamiltonian. Two distinct types of relativistic corrections are considered separately: (1) relativistic corrections due to the motion of the colliding electron and its interaction with the target; (2) relativistic corrections due to breakdown of LS -coupling in the target. In both of these cases it is shown that within the Breit-Pauli approximation a collision strength may be written Q vel ( i,j ) = Q nr (i,j) + a 2 C (2) rel ( i,J ),where Q rel is the collision strength including relativistic corrections and Q nr is the non-relativistic collision strength. The quantities C (2) rel are contributions of orders a 2 and a 4 respectively, relative to Q nr . In the case of corrections of type (1), consistency problems render it difficult to calculate the term a 4 C (4) rel reliably. O n the other hand, strong semi-empirical evidence suggests that in the case of corrections of type (2), the a 4 correction can be reliably estimated within the framework of existing theory. By means of Racah algebra it is demonstrated that fine structure interactions between colliding electron and target give no contributions of order a 2 provided that that Q rel ( i,j ) is summed over the fine structure levels of the initial and final target terms.Breakdown of L-S -coupling in the target (due to fine structure interactions among the the target electrons) gives contribution of order a 2 to the total collision strength. However, these contributions do not vanish when the collision strengths are summed over the fine structure levels of the initial and final terms. Asymptotic expansions for the dependence of Q rel upon the nuclear charge Z of the target are derived for corrections of types (1) and (2). The present work is discussed in relation to recent work by Carse & Walker (1973) and Walker (1974), who have studied the studied the electron-hydrogen scattering problem in a formulation based upon the Dirac equation. Practical procedures for carrying out calculations in the framework of the present theory are discussed, and one such procedure is formulated in some detail.

Fine Structure of Platelet Interaction with a New Microcrystalline Collagen Preparation

1974 ◽  
Vol 32 ◽  
pp. 58-59
Author(s):  
W. H. Zucker ◽  
R. G. Mason
Keyword(s):  
Fine Structure ◽  
Platelet Aggregation ◽  
Hydrochloric Acid ◽  
Whole Blood ◽  
Platelet Adhesion ◽  
Hemostatic Agent ◽  
Native Collagen ◽  
Fibrillar Morphology ◽  
Clinical Interest ◽  
New Form

Platelet adhesion initiates platelet aggregation and is an important component of the hemostatic process. Since the development of a new form of collagen as a topical hemostatic agent is of both basic and clinical interest, an ultrastructural and hematologic study of the interaction of platelets with the microcrystalline collagen preparation was undertaken.In this study, whole blood anticoagulated with EDTA was used in order to inhibit aggregation and permit study of platelet adhesion to collagen as an isolated event. The microcrystalline collagen was prepared from bovine dermal corium; milling was with sharp blades. The preparation consists of partial hydrochloric acid amine collagen salts and retains much of the fibrillar morphology of native collagen.

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