Scattered wave functions and electronic states of dislocations

1982 ◽  
Vol 60 (5) ◽  
pp. 779-787
Author(s):  
J. Th. M. De Hosson
Keyword(s):  
Electronic States ◽  
Wave Model ◽  
Scattered Wave ◽  
Wave Functions ◽  
The Self ◽  
Atomic Configuration ◽  
Linear Elasticity Theory ◽  
Simulation Procedure ◽  
Self Consistent ◽  
Classical Linear Elasticity

This article outlines a model for calculating the localized electronic states of a [Formula: see text] edge dislocation in α-Fe and in Mo. A method is presented using a Wannier function approach for calculating the scattered wave function of a dislocated lattice. The model used for the calculations of the electronic structure is based on the multiple scattering model. In addition the computer simulation procedure for calculating the atomic configuration of dislocations is combined with classical linear elasticity theory and the self-consistent scattered wave model.

scholarly journals The relativistic self-consistent field

1935 ◽  
Vol 152 (877) ◽  
pp. 625-649 ◽  
Keyword(s):  
Wave Equation ◽  
Field Equation ◽  
Wave Functions ◽  
The Self ◽  
Magnetic Interactions ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Schrödinger's Equation

1—The method of the self-consistent field for determining the wave functions and energy levels of an atom with many electrons was developed by Hartree, and later derived from a variation principle and modified to take account of exchange and of Pauli’s exclusion principle by Slater* and Fock. No attempt was made to consider relativity effects, and the use of “ spin ” wave functions was purely formal. Since, in the solution of Dirac’s equation for a hydrogen-like atom of nuclear charge Z, the difference of the radial wave functions from the solutions of Schrodinger’s equation depends on the ratio Z/137, it appears that for heavy atoms the relativity correction will be of importance; in fact, it may in some cases be of more importance as a modification of Hartree’s original self-nsistent field equation than “ exchange ” effects. The relativistic self-consistent field equation neglecting “ exchange ” terms can be formed from Dirac’s equation by a method completely analogous to Hartree’s original derivation of the non-relativistic self-consistent field equation from Schrodinger’s equation. Here we are concerned with including both relativity and “ exchange ” effects and we show how Slater’s varia-tional method may be extended for this purpose. A difficulty arises in considering the relativistic theory of any problem concerning more than one electron since the correct wave equation for such a system is not known. Formulae have been given for the inter-action energy of two electrons, taking account of magnetic interactions and retardation, by Gaunt, Breit, and others. Since, however, none of these is to be regarded as exact, in the present paper the crude electrostatic expression for the potential energy will be used. The neglect of the magnetic interactions is not likely to lead to any great error for an atom consisting mainly of closed groups, since the magnetic field of a closed group vanishes. Also, since the self-consistent field type of approximation is concerned with the interaction of average distributions of electrons in one-electron wave functions, it seems probable that retardation does not play an important part. These effects are in any case likely to be of less importance than the improvement in the grouping of the wave functions which arises from using a wave equation which involves the spins implicitly.

scholarly journals Long-time discrete particle effects versus kinetic theory in the self-consistent single-wave model

2001 ◽  
Vol 64 (2) ◽  
Author(s):  
M-C. Firpo ◽  
F. Doveil ◽  
Y. Elskens ◽  
P. Bertrand ◽  
M. Poleni ◽  
...  
Keyword(s):  
Kinetic Theory ◽  
Wave Model ◽  
The Self ◽  
Single Wave ◽  
Discrete Particle ◽  
Self Consistent ◽  
Long Time

Vacancies in close-packed polyvalent metals

1970 ◽  
Vol 316 (1525) ◽  
pp. 201-222 ◽  
Keyword(s):  
Bloch Wave ◽  
Wave Functions ◽  
The Self ◽  
Radial Wave ◽  
Conduction Electrons ◽  
Self Energy ◽  
Crystal Density ◽  
Self Consistent ◽  
The One ◽  
Formation Energies

The difference in total energy of a crystal with and without a vacancy involves essentially three terms: (i) The change in the one-electron eigenvalues due to scattering of conduction electrons off the vacant site. (ii) The self-energy of the displaced charge. (iii) The change in exchange and correlation energies of the electron gas. We have investigated the contributions (i) to (iii) for Cu, Mg, Al and Pb. The change in the one-electron eigenvalues is shown to be insensitive to the Bloch wave character of the wave functions and also to the choice of the repulsive potential V ( r ) representing the effect of the vacancy on the conduction electrons. There is thus no difficulty in evaluating contribution (i) for metals of different valencies. In contrast, the self-energy of the displaced charge is shown to depend very sensitively on the choice of V ( r ), and it is, therefore, essential to make the calculation self-consistent. This we have done by properly screening the negative of the point ion fields for Cu + to Pb 4+ . The radial wave functions and phase shifts for the low order partial waves have been evaluated, and self-consistent displaced charges obtained. The exchange energy has been estimated from a Dirac–Slater type of approximation and is again not sensitive to the detailed form of the displaced charge, while the change in correlation energy is found to be unimportant in determining the vacancy formation energy. The formation energies for the polyvalent metals are in satisfactory agreement with experiment. Some results for excess resistivities due to vacancies in metals are also presented. Here, in contrast to the calculation of the formation energies, it is essential to account for the Bloch wave character of the electron waves scattered by the vacancy. It is also proposed that the displaced charge round a vacancy may be a useful building block (or pseudoatom) for forming the crystal density.

scholarly journals Self-consistent field, with exchange, for nitrogen and sodium

1948 ◽  
Vol 193 (1034) ◽  
pp. 299-304 ◽  
Keyword(s):  
Ionization Potential ◽  
Normal State ◽  
Wave Functions ◽  
The Self ◽  
Electron Atom ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field

Wave functions for the normal configurations of neutral nitrogen and N - have been calculated by the method of the self-consistent field with exchange (Fock’s equations). To the accuracy of the approximation represented by these equations, the N - ion would be unstable and liable to auto-ionization, but it is estimated that a better approximation of the treatment of a many-electron atom would give a small positive ionization potential for N - . Revised wave functions for Na + and the normal state of neutral Na have also been calculated. Tables of results are given.

Sign in / Sign up

Export Citation Format

Share Document