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 beryllium - II—The (2 s ) (2 p ) 3 P and 1 P excited states

1936 ◽  
Vol 154 (883) ◽  
pp. 588-607 ◽  
Keyword(s):  
Excited States ◽  
Normal State ◽  
Wave Functions ◽  
The Self ◽  
Energy Separation ◽  
Radial Wave ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field

In a recent paper we gave an account of the method and results of the solution of Fock’s equations of the self-consistent field, including exchange effects, for the normal state of neutral beryllium. The present paper is concerned with the extension of the calculations to the first two excited states, (1 s ) 2 (2 s ) (2 p ) 3 P and 1 P, of the same atom. This extension was undertaken for two reasons. Firstly, before going on to attempt the solution of Fock’s equations for a heavier atom, we wished to get some experience of the process of solution of Fock’s equations for a configuration involving wave functions which overlap to a greater extent than the wave functions (1 s ) and (2 s ) of the normal state, and for which exchange effects might be expected to be corre­spondingly greater; and secondly, for an atom with more than one electron outside closed ( nl ) groups, so that a given configuration gives rise to more than one term, the equations of the self-consistent field, when exchange effects are included, are no longer the same for the different terms, and it seemed likely to be of interest to examine the consequent difference between the radial wave functions for the different terms (here 3 P and 1 P), and the effect of this difference on the calculated energy separation between the terms.

scholarly journals Results of calculations of atomic wave functions. I.—Survey, and self-consistent fields for Cl - and Cu +

1933 ◽  
Vol 141 (844) ◽  
pp. 282-301 ◽  
Keyword(s):  
Radial Wave Function ◽  
Wave Functions ◽  
The Self ◽  
Radial Wave ◽  
Atomic Structures ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Electron Group ◽  
Working Out

An approximation to the structure of a many-electron atom can be obtained by considering each electron to be a stationary state in the field of the nucleus and the Schrodinger charge distribution of the other electrons, and rather more than five years ago I gave a method of working out atomic structures based on this idea, and called the field of the nucleus and distribution of charge so obtained the “self-consistent field.” The method of working out the self-consistent field for any particular atom involves essentially ( a ) the estimation of the contributions to the field from the various electron groups constituting the atom in question; ( b ) the solution of the radial wave equation for an electron in the field of the nucleus and other electrons, this solution being carried out for each of the wave functions sup­posed occupied by electrons in the atomic state considered; and ( c ) the calculation of the contribution to the field from the Schrodinger charge dis­tribution of an electron group with each radial wave function. The estimates of the contributions to the field have to be adjusted by trial until the agreement between the contributions finally calculated and those estimated is considered satisfactory.

scholarly journals Self-consistent field, including exchange and superposition of configurations, with some results for oxygen

1939 ◽  
Vol 238 (790) ◽  
pp. 229-247 ◽  
Keyword(s):  
Wave Functions ◽  
Radial Wave ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Approximate Wave ◽  
The One ◽  
The Individual ◽  
Single Configuration ◽  
Good Agreement

The calculation of approximate wave functions for the normal configurations of the ions O +++, O ++, O +, and neutral O, and the calculation of energy values from the wave functions, was carried out some years ago by Hartree and Black (1933)- In this work, the one-electron radial wave functions were calculated by the method of the selfconsistent field without exchange, but exchange terms were included in the calculation of the energy from these radial wave functions. In the energy calculations, the same radial wave functions were taken for each of the spectral terms arising from a single configuration; * consequently the ratios between the calculated intermultiplet separations were exactly those given by Slater’s (1929) theory of complex spectra, f The ratios between the observed intermultiplet separations, however, depart considerably from these theoretical values (for example, we have for 0 ++ ( 1 D - 1 S) / ( 3 P - 1 D), calc. 3 : 2, obs. 1.04 :1), although the energies of the individual terms, and particularly the intermultiplet separation between the lower terms, show quite a good agreement with the observed values.

scholarly journals Self-consistent field, with exchange, for Cu +

1936 ◽  
Vol 157 (892) ◽  
pp. 490-502 ◽  
Keyword(s):  
Complete Solution ◽  
Point Of View ◽  
Wave Functions ◽  
The Self ◽  
Radial Wave ◽  
Electron Atom ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field

Solutions of Fock’s equations for the self-consistent field of a many-electron atom, including exchange effects, have already been carried out for several atoms by Fock and Petrashen and the present authors. The heaviest atom for which results of such calculations have previously been published is Cl - ; Cu + was selected as the next atom for which to attempt the solution of Fock’s equations, for the following reasons. As already pointed out in Paper IV, the results of the solution of Fock’s equations are most interesting for atoms for which exchange effects are large; the self-consistent field without exchange, which is an almost necessary preliminary to the solution of Fock’s equation, had already been worked out for Cu + , and from this work it was known that the (3 d ) 10 group of Cu + is very sensitive to the atomic field, so that it is likely to be con­siderably affected by the inclusion of exchange terms in the equations. Further, in view of the interest of Cu from the point of view of metal theory, it is desirable to have as good wave functions for Cu + as possible, particularly for the outer groups, which are those likely to be most affected by the inclusion of exchange terms in the equation from which they are determined. The number of radial wave functions involved in the normal con­figuration of Cu + is perhaps almost the largest for which a complete solution of Fock’s equations is practicable, for the following reason.

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.

Spin-Wave Spectrum in the Sinusoidal Superlattice with Amplitude and Phase Inhomogeneities

2014 ◽  
Vol 215 ◽  
pp. 385-388
Author(s):  
Valter A. Ignatchenko ◽  
Denis S. Tsikalov
Keyword(s):  
Spin Wave ◽  
Density Of States ◽  
Wave Spectrum ◽  
The Self ◽  
One Dimensional ◽  
Consistent Approximation ◽  
Self Consistent ◽  
Greens Function ◽  
The One ◽  
Spin Wave Spectrum

Effects of both the phase and the amplitude inhomogeneities of different dimensionalities on the Greens function and on the one-dimensional density of states of spin waves in the sinusoidal superlattice have been studied. Processes of multiple scattering of waves from inhomogeneities have been taken into account in the self-consistent approximation.

scholarly journals Self-Consistent Calculation of Particle-Hole Diagrams on the Matsubara Frequency: Flex Approximation

1997 ◽  
Vol 08 (05) ◽  
pp. 1145-1158
Author(s):  
J. J. Rodríguez-Núñez ◽  
S. Schafroth
Keyword(s):  
Hubbard Model ◽  
Imaginary Part ◽  
Chemical Potential ◽  
Order Approximation ◽  
The Self ◽  
Green Functions ◽  
Matsubara Frequency ◽  
Self Energy ◽  
Peak Structure ◽  
Self Consistent

We implement the numerical method of summing Green function diagrams on the Matsubara frequency axis for the fluctuation exchange (FLEX) approximation. Our method has previously been applied to the attractive Hubbard model for low density. Here we apply our numerical algorithm to the Hubbard model close to half filling (ρ =0.40), and for T/t = 0.03, in order to study the dynamics of one- and two-particle Green functions. For the values of the chosen parameters we see the formation of three branches which we associate with the two-peak structure in the imaginary part of the self-energy. From the imaginary part of the self-energy we conclude that our system is a Fermi liquid (for the temperature investigated here), since Im Σ( k , ω) ≈ w2 around the chemical potential. We have compared our fully self-consistent FLEX solutions with a lower order approximation where the internal Green functions are approximated by free Green functions. These two approches, i.e., the fully self-consistent and the non-self-consistent ones give different results for the parameters considered here. However, they have similar global results for small densities.

scholarly journals NONCOMMUTATIVE U(1) SUPER-YANG–MILLS THEORY: PERTURBATIVE SELF-ENERGY CORRECTIONS

2004 ◽  
Vol 19 (25) ◽  
pp. 4231-4249 ◽  
Author(s):  
A. A. BICHL ◽  
M. ERTL ◽  
A. GERHOLD ◽  
J. M. GRIMSTRUP ◽  
L. POPP ◽  
...  
Keyword(s):  
Power Counting ◽  
The Self ◽  
Field Theories ◽  
Yang Mills ◽  
Self Energy ◽  
Supersymmetric Field Theories ◽  
Crucial Feature ◽  
The One ◽  
Infrared Divergences ◽  
Super Yang Mills Theory

The quantization of the noncommutative [Formula: see text], U(1) super-Yang–Mills action is performed in the superfield formalism. We calculate the one-loop corrections to the self-energy of the vector superfield. Although the power-counting theorem predicts quadratic ultraviolet and infrared divergences, there are actually only logarithmic UV and IR divergences, which is a crucial feature of noncommutative supersymmetric field theories.

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