Relativistic corrections to He I transition ratesThis paper was presented at the International Conference on Precision Physics of Simple Atomic Systems, held at École de Physique, les Houches, France, 30 May – 4 June, 2010.

2011 ◽  
Vol 89 (1) ◽  
pp. 129-134 ◽  
Author(s):  
Donald C. Morton ◽  
Paul Moffatt ◽  
G. W.F. Drake
Keyword(s):  
Nuclear Charge ◽  
Wave Functions ◽  
Oscillator Strengths ◽  
Matrix Elements ◽  
Nuclear Mass ◽  
Atomic Systems ◽  
Relativistic Corrections ◽  
Doubly Excited ◽  
Accurate Wave ◽  
The Continuum

The relativistic corrections to the theoretical oscillator strengths of light elements such as helium are typically less than 0.1% and usually are ignored. However, they can be important for comparisons with the most accurate experiments, and they rapidly increase in magnitude with increasing nuclear charge. We have begun with the spin-forbidden electric-dipole transitions of neutral helium, using calculations consisting of (1) extremely accurate wave functions without relativistic corrections for both infinite and finite nuclear mass, (2) spin-changing matrix elements through the perturbations of the wave functions by the spin-orbit and spin-other-orbit Breit operators, (3) the use of pseudostates in the sums over all the intermediate states including the continuum, and (4) the inclusion as perturbers of the 1S0 and 3S1 states the pseudostates corresponding to the doubly excited npn′p 3P 0e and npn′p 1P 1e terms, respectively. As examples of these calculations, we present oscillator strengths for the transitions 1 1S0–2 3P1, 2 1S0–2 3P1, 2 3S1–2 1P1, 2 1P1–3 3D1,2, and 2 3P1,2–3 1D2.

MULTIPOLE MATRIX ELEMENTS 〈nl|rβ|El'〉v FOR H-LIKE ATOMS AND THEIR APPLICATIONS (AS β = 1, n ≤ 4, Enl < 0 ≤ E ≤ 1)

2008 ◽  
Vol 22 (08) ◽  
pp. 937-965 ◽  
Author(s):  
V. F. TARASOV
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Atomic Physics ◽  
Nuclear Charge ◽  
Dipole Moments ◽  
Oscillator Strengths ◽  
Matrix Elements ◽  
Auxiliary Parameter ◽  
Radial Functions ◽  
Line Intensities

This article deals with the connection between multipole matrix elements 〈nl|rβ|n'l'〉ν and 〈nl|rβ|El'〉ν for H-like atoms, where ν is the so-called "auxiliary" parameter of Heun's differential equation and [Formula: see text] is the "effective" nuclear charge, and new properties of Appell's function F2(x,y) to the vicinity of the singular point (1, 1) and in addition, here, first V. A. Fock's idea for the continuous spectrum is taken into consideration. Such an approach allows us to get the explicit expressions for squares of the dipole moments and the certain physical characteristics in atomic physics and also their exact numerical values, e.g., the average oscillator strengths [Formula: see text] and the line intensities J(nl, El'), etc., as n ≤ 4, l'= l ± 1 and 0 ≤ E ≤ 1 (see Tables 1–3). Besides, diagrams of certain radial functions for the discrete-continuous transitions are given here.

Towards accurate wave functions and oscillator strengths

2000 ◽  
Author(s):  
Alan Hibbert
Keyword(s):  
Wave Functions ◽  
Oscillator Strengths ◽  
Accurate Wave

Inelastic Scattering of High Energy Electrons by Helium

1973 ◽  
Vol 51 (3) ◽  
pp. 311-315 ◽  
Author(s):  
S. P. Ojha ◽  
P. Tiwari ◽  
D. K. Rai
Keyword(s):  
Ground State ◽  
High Energy ◽  
Wave Functions ◽  
Oscillator Strengths ◽  
Final States ◽  
Interaction Type ◽  
Hartree Fock ◽  
High Energy Electrons ◽  
Approximate Wave ◽  
Accurate Wave

Generalized oscillator strengths and the cross section for excitation of helium by electron impact have been calculated in the Born approximation. Transitions from the ground state to the n1P (n = 2 and 3) states have been considered. Highly accurate wave functions of the Hartree–Fock and "configuration–interaction" type have been used to represent the ground state. Approximate wave functions due to Messmer have been employed for the final states. The results are compared with other calculations and with experiment.

Mikroskopische Theorie der Kernreaktionen für einen Mehrteilchen-Aufbruch / Microscopic Theory of Nuclear Reactions for a Multi-particle-break-up

1974 ◽  
Vol 29 (6) ◽  
pp. 859-866 ◽  
Author(s):  
A. Grauel
Keyword(s):  
Nuclear Reactions ◽  
Microscopic Theory ◽  
Bound State ◽  
Wave Functions ◽  
Matrix Elements ◽  
S Matrix ◽  
The Matrix ◽  
Nucleon Emission ◽  
The Continuum ◽  
Continuum Wave

Introducing correlated continuum wave functions for the two- and re-particle-continuum a microscopic theory of nuclear reactions based on a method of Fano is developed. The S-matrix-elements are given by the matrix-elements between correlated continuum wave functions and bound state wave functions. The antisymmetrization of the continuum wave functions with more than one particle in the continuum is included. The theory can be straightforwardly applied on the n-nucleon-emission process following photo- and particle excitations.

The Two Centre Dirac Equation

1976 ◽  
Vol 31 (1) ◽  
pp. 1-30 ◽  
Author(s):  
Berndt Müller ◽  
Walter Greiner
Keyword(s):  
Dirac Equation ◽  
Heavy Ions ◽  
Nuclear Charge ◽  
Basis Functions ◽  
Wave Functions ◽  
Matrix Elements ◽  
Prolate Spheroidal ◽  
Prolate Spheroidal Coordinates ◽  
Spheroidal Coordinates ◽  
Dirac Hamiltonian

During collisions of heavy ions with heavy targets below the Coulomb barrier, adiabatic molecular orbitals are formed for the inner electrons. Deviations from adiabaticity lead to coupling between various states and can be treated by time-dependent perturbation theory. For high charges ( Z1+Z2 ≧ 60) the molecular electrons are highly relativistic. Therefore, the Dirac equation has to be used to obtain the energies and wave functions. The Dirac Hamiltonian is transformed into the intrinsic rotating coordinate system where prolate spheroidal coordinates are introduced. A set of basis functions is proposed which allows the evaluation of all matrix elements of the Dirac Hamiltonian analytically. The resulting matrix is diagonalized numerically. The finite nuclear charge distribution is also taken into account. Results are presented and discussed for various characteristic systems, e. g. Br-Br, Ni-Ni, I-I, Br-Zr, I-Au, U -U, etc.

An expansion method for calculating atomic properties X. 1 s 2 1 S -1 snp 1 P transitions of the helium sequence

1967 ◽  
Vol 301 (1466) ◽  
pp. 253-260 ◽  
Keyword(s):  
Cross Sections ◽  
Nuclear Charge ◽  
Expansion Method ◽  
Probable Error ◽  
Oscillator Strengths ◽  
Matrix Elements ◽  
Isoelectronic Sequence ◽  
First Order ◽  
Hartree Fock ◽  
Atomic Properties

The electric dipole matrix elements connecting the 1 s 2 1 S and 1 snp 1 P states of the helium isoelectronic sequence are calculated exactly to first order in inverse powers of the nuclear charge Z and the differences from the Hartree-Fock approximation are shown to correspond to virtual transitions of the 1 s electrons. Comparison of the oscillator strengths predicted by a screening approximation with more accurate values reveals a regular variation in the error contained in the screening approximation, the correction of which allows the prediction of oscillator strengths and probabilities of 1 s 2 1 S – 1 snp 1 P transitions for all values of n and all values of Z within a probable error of 2% (table 5). Values of the photoionization cross-sections at the spectral heads are also presented.

Systematic Trends in Atomic Oscillator Strengths: The Helium Isoelectronic Sequence

1972 ◽  
Vol 50 (12) ◽  
pp. 1363-1369 ◽  
Author(s):  
M. Cohen ◽  
R. P. McEachran
Keyword(s):  
Electric Dipole ◽  
Nuclear Charge ◽  
Principal Quantum Number ◽  
Wave Functions ◽  
Oscillator Strengths ◽  
Isoelectronic Sequence ◽  
Dipole Oscillator ◽  
Good Agreement ◽  
Comparison Data ◽  
Dipole Length

Electric dipole oscillator strengths (f values) have been calculated for a large number of singlet and triplet S–P, P–D, and D–F transitions in the helium isoelectronic sequence through O+6. The analytical orbital wave functions employed were of frozen-core type, and generally produce very good agreement between length and velocity values of the calculated oscillator strengths. A conspicuous exception occurs in many cases where the principal quantum number remains unchanged in the transition, and the more reliable dipole length values have been adopted for such transitions. The smooth variation of the calculated f values as functions of the inverse of the nuclear charge Z provided a sensitive check on the accuracy of the computations and indicated a considerable number of P–D transitions where the velocity values seemed the more reliable. Wherever comparison data are available, our calculated oscillator strengths are in excellent agreement with the most accurate values; in other cases, the absolute uncertainty in the f values should in no case exceed 5%.

scholarly journals Accurate Exponential Representations for the Ground State Wave Functions of the Collinear Two-Electron Atomic Systems

Atoms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 4
Author(s):  
Evgeny Z. Liverts ◽  
Nir Barnea
Keyword(s):  
Ground State ◽  
Wave Functions ◽  
Linear Combinations ◽  
Simple Method ◽  
Atomic Systems ◽  
State Wave ◽  
Wolfram Mathematica ◽  
Model Wave ◽  
Accurate Wave

In the framework of the study of helium-like atomic systems possessing the collinear configuration, we propose a simple method for computing compact but very accurate wave functions describing the relevant S-state. It is worth noting that the considered states include the well-known states of the electron–nucleus and electron–electron coalescences as a particular case. The simplicity and compactness imply that the considered wave functions represent linear combinations of a few single exponentials. We have calculated such model wave functions for the ground state of helium and the two-electron ions with nucleus charge 1≤Z≤5. The parameters and the accompanying characteristics of these functions are presented in tables for number of exponential from 3 to 6. The accuracy of the resulting wave functions are confirmed graphically. The specific properties of the relevant codes by Wolfram Mathematica are discussed. An example of application of the compact wave functions under consideration is reported.

MULTIPOLE MATRIX ELEMENTS ν FOR H-LIKE ATOMS, THEIR ASYMPTOTICS AND APPLICATIONS (AS β=1, n ≤ 4, n' ≤ 10)

2009 ◽  
Vol 23 (08) ◽  
pp. 2041-2067 ◽  
Author(s):  
V. F. TARASOV
Keyword(s):  
Singular Point ◽  
Transition Probabilities ◽  
Nuclear Charge ◽  
Oscillator Strengths ◽  
Matrix Elements ◽  
Auxiliary Parameter ◽  
Radial Equation ◽  
Effective Nuclear Charge ◽  
Line Intensities

This article deals with the connection between multipole matrix elements <nl|rβ|n'l' >ν for H-like atoms and new properties of Appell's function F2(x,y) to the vicinity of the singular point (1, 1), where ν is the so-called "auxiliary" parameter of Heun-Schrödinger's radial equation, |1 - ν| = o(1), [Formula: see text] is the "effective" nuclear charge. Exact numerical values for the dipole matrix elements, the average oscillator strengths, the transition probabilities and the line intensities, as n ≤ 4 and n' ≤ 10, in the form of regular rational fractions are given (in Tables 1–4), that make more precise the well-known Tables 13–16 by Hans A. Bethe.

Calcul des forces d'oscillateur des premières transitions permises des donneurs dans le silicium et comparaison avec l'expérience

1985 ◽  
Vol 63 (4) ◽  
pp. 437-444 ◽  
Author(s):  
M. Bara ◽  
M. Astier ◽  
B. Pajot
Keyword(s):  
Point Of View ◽  
Wave Functions ◽  
Oscillator Strengths ◽  
Matrix Elements ◽  
Linear Combinations ◽  
Electron Effective Mass ◽  
Angular Momenta ◽  
Hydrogenic Donor ◽  
Ritz Procedure ◽  
Donor States

The calculation of the oscillator strength of the optical transitions between two electronic states necessitates the knowledge of the wave functions of these states. In the case of hydrogenic donor impurities in semiconductors, these wave functions are eigenvectors of a matrix whose elements couple hydrogenic basis wave functions of appropriate parity, angular momenta, and projection of angular momentum through a one-electron effective mass Hamiltonian, provided the set of eigenvalues of interest has been minimized with respect to the variational parameters included in the basis wave functions. In his well-known paper, Faulkner has given the eigenvalues of the first donor states for Si and Ge, but neither the eigenvectors nor the variational parameters. We have reproduced the calculations of Faulkner for Si, finding quite similar results for the eigenvalues. The coefficients of the linear combinations of the basis functions show that except for the very first levels, the labeling of the eigenvectors does not correspond to the basis function with the greatest weight, and that the labeling is somewhat arbitrary from that point of view. The eigenvectors are used to calculate the electric-dipole matrix elements. The values so obtained for the oscillator strengths are in qualitative agreement with the relative intensities of the observed lines for substitutional donors in Si. This seems to demonstrate that the eigenvectors obtained through the Rayleigh–Ritz procedure are reliable enough for the evaluation of spectroscopical quantities where they are needed.

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