No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Consistency condition in the energy expression of thepseudopotential theory of metals
Abstract
The consistency condition for the energy expression of a metal obtained from the pseudopotential theory is found to be equivalent to the statement that the static and dynamic elastic constants must agree. If the band-structure energy and the coupling parameter are both confined to the second order of the perturbation theory then this consistency condition is violated. It is pointed out that the reason for this violation lies in the fact that the homogeneous deformation theory takes note of the change in the dielectric function due to strain, while the long wave theory partly ignores it. It is shown that by suitably coupling the local strain to the ionic coordinates one can get the missing terms in the long-wave theory and the consistency condition is satisfied. The effect of these terms on the phonon dispersion curves for Al is analyzed.
... In the solution of the compressibility problem the concept of local volume was introduced earlier [24, 25] with a specific form for the local volume strain, satisfying some general properties demanded by the lattice mechanical theory. The force constant elements redefined with some additional terms, ensure the equivalence of the compressibility and other elastic constants in "homogeneous deformation" and "long wave" methods [24]. Using the concept of local volume it may be, therefore, possible to describe ionic rearrangements under varying local volume keeping the average volume constant. ...
Computation of reliable effective interionic pair interaction for simple metals has been discussed on the basis of localised ab initio pseudopotential within linear screening approximation. The influence of different commonly used screening unctions on the shape of pair potential is examined and the temperature dependence of the interaction is also investigated. Various features of the potential are studied and compared with empirical and available first principle results. Importance of the present study and some related problems are also analysed.
The following notes, meant as a pedagogical introduction to the subject of perturbative quantum chromodynamics (pQCD), are based on the lectures by Ashoke Sen in the XVI SERC Main School on Theoretical High Energy Physics at the HRI, Allahabad. The notes were taken by Anindya Dutta, Debashis Ghoshal, Dileep Jatkar, Swapan Majhi, Partha Konar, Subhendu Rakshit and V. Ravindran; and compiled in their final form by Debashis Ghoshal and V. Ravindran.
Calculation of metallic pair interaction, extracted from higher order
pseudopotential energy in the case of Al shows excellent agreement with
recent ab initio calculation based on large set of experimental data.
The calculation performed with ab initio pseudopotential has provided
comprehensive unified study of various metallic properties of a number
of simple metals. Analysis of the perturbation calculation beyond second
order is shown to yield a reasonable estimate for the non-pairwise
interaction in these metals.
The second order pseudopotential theory suggests the possibility of a break-up of the total energy of simple metals into a purely volume-dependent part and an effective central pairwise interaction between ions. In the present paper finite contributions for these two parts of the energy have been extracted in a form convenient for calculation. Using the local Heine-Abarenkov model potential, a reliable effective ion-ion interaction is generated and the volume-dependent energy is calculated for Al. The relative contributions of the effective interaction and the volume-dependent energy term to various metallic properties are also calculated. The importance of volume dependence on the effective interaction is also discussed.
The interactomic force constants upto eighth neighbour are derived from the effective interaction and it is found that the force constants beyond the third neighbour are negligibly small. This result is also confirmed by the calculation of dispersion curves with force constants obtained from the effective interaction upto the third neighbour which is found to reproduce the results of the full pseudopotential calculations. The force constants obtained are also used to study some finite temperature properties of Al in the quasi-harmonic approximation and the limitations of the theory are pointed out.
We report the results of an explicit calculation of the leading contributions including O(alphas2) corrections to the asymptotic behavior of the quark form factor in QCD in the limit of large virtual mass of the photon. In contrast to previous investigations, we regularize infrared singularities by keeping either one (on/off), or both quarks (off) off shell, or both quarks on shell (on) and employing dimensional regularization in addition. Up to the investigated order, our final result is represented by the beginning of an exponential series, no matter which regularization prescription is chosen. We give a representation which comprises the three limits of the form factor in a uniform manner. Supported by ``Studienstiftung des deutschen Volkes''.
The consideration of two-body central forces cannot explain the correct ordering of dispersion branches in the I'KM direction in many h.c.p. metals. The importance of three-body forces in these metals is also indicated by the fact that for these metals the sum rule of Waren is not satisfied. Pseudopotential theory demonstrates that the total energy of a metal consists of an effective two-body central interaction and a purely volume dependent part. Starting from this basis, a model expression for the energy of a metal is developed. It is shown that the interpretation of the volume strain as the local strain, as demanded by the adiabatic hypothesis, leads to an effective three-body interaction between the ions. The model is applied to study the lattice dynamics of Mg and Be. It is found that the model predicts the correct ordering of the dispersion branches and fairly good agreement is obtained in the case of phonon frequencies and elastic data.
Die Betrachtung von Zwei-Körper-Zentralkräften können nicht die richtige ordnung der Dispersionszweige in der I'KM-Richtung in vielen h. d.p. Metallen erklären. Die Bedeutung von Dreikörperkräften in diesen Metallen wide auch durch die Tatsache deutlich, daß für diese Metalle die Summenregel von Waren nicht erfüllt ist. Pseudopotentialtheorie demonstriert, daß die Gesamtenergie eines Metalls aus einer effektiven Zwei-Körper-Zentralwechselwirkung und einem reinen volumenabhängigen Anteil besteht. Ausgehend von dieser Basis, wird ein Modellausdruck für die Energie eines Metalls entwickelt. Es wird gezeigt, daß die Interpretation der Volumen-Drei-Körper-Wechselwirkung zwischen den Ionen Führt. Das Modell wird auf die Untersuchung der Gitterdynamik von Mg und Be angewendet. Es wird gefunden, daß das Modell die richtige Ordnung des Dispersionszweigen vorhersagt und es wird eine befriedigend gute Übereinstimmung für die Phononenfrequenzen und elastischen Daten erhalten.
The localised version of the density functional based norm-conserving pseudopotential has already been applied to various metallic solids and aggregate state properties with some success. Application to the study of liquid phonons and to the surface dynamics are specially discussed in the present paper. An analytic fit to the pseudopotential is obtained and further application to molecular dynamics is indicated.
From a careful analysis of the energy expression of the second-order pseudopotential
theory, the reason for the breakdown of the consistency condition is pointed out. The
problem is resolved by introducing the concept of local volume strain which introduces
some modifications in the dynamical matrix and leads to the equality of static and
dynamic elastic constants. A unified study of the alkali metals with the Heine-Abarenkov
model pseudopotential in conjunction with Taylor's dielectric function, and taking
proper care of the consistency condition, yields satisfactory results for both the
static and dynamic properties of these metals. One interesting feature of the present
investigation is that the theory satisfying the consistency condition reproduces the
crossing of the longitudinal and transverse branches in the (100) direction for Li,
while reproducing the normal behaviour for the dispersion curve of other metals.
Our previous three-body force model [8] is modified and used to
calculate the elastic constants and phonon dispersion curves of
yttrium and scandium. It is demonstrated that the interpretation
of the volume strain as the local strain, as the adiabatic
hypothesis demands, leads to an effective three-body interaction
which is essential to guarantee the equality of the static and
dynamic elastic constants. This model satisfies the two
equilibrium conditions and the symmetry properties of the
dynamical matrix. The results obtained with this model are
in good agreement with the available experimental data. The
experimental phonon dispersion curves for these metals do not
provide a clear picture along the [1 1 2 0] direction. Out of
the six possible branches only two are measured; the other four
which would provide valuable informations about the importance
of three-body forces are not available. As for these branches,
the calculated results may provide useful information.
The present work is an attempt to identify the role of pseudopotential
third order energy in different lattice mechanical properties of HCP
metals. Although it plays an important role in explaining the typical
features of the phonon dispersion curves, the third order contribution
to the individual elastic constants is found to be unphysical. The root
of this unphysical contribution may be identified with the singularity
in the derivative of the g-function appearing in the expression for
the third order energy. A possible solution to this problem is
suggested. An alternative approach of a unified study in the framework
of a second order pseudopotential theory invoking the consistency
condition is also examined. This method simulates the effect of some
terms beyond the second order in lattice dynamics and explains typical
features of the phonon dispersion curves which are believed to be
manifestations of the many-ion effect.
Calculation of metallic pair interaction, extracted from higher order pseudopotential energy in the case of Al shows excellent agreement with recent ab initio calculation based on large set of experimental data. The calculation performed with ab initio pseudopotential has provided comprehensive unified study of various metallic properties of a number of simple metals. Analysis of the perturbation calculation beyond second order is shown to yield a reasonable estimate for the non-pairwise interaction in these metals.
The electrical and thermal transport properties of some simple metals are calculated using a unified local pseudopotential whose parameters have earlier been determined from various lattice mechanical properties. The transport properties include electrical and thermal resistivities both, in solid and liquid phases at different temperatures, resistivities due to vacancies, and thermoelectric power. The overall agreement with the experiment is good for Na, K, Al but for Li and to some extent Rb, the discrepancies in theory and experiments indicate the necessity of a comprehensive non-local theory and some refinements of transport equations.
Mit einem vereinheitlichten lokalen Pseudopotential, dessen Parameter bereits früher aus ver-schiedenen gittermechanischen Eigenschaften bestimmt wurden, werden die elektrischen und thermischen Transporteigenschaften einiger einfacher Metalle berechnet. Die Transporteigen-schaften schließen den elektrischen und thermischen Widerstand sowohl in der festen als auch flüssigen Phase bei verschiedenen Temperaturen, Widerstände aufgrund von Leerstellen und die Thermospannung ein. Die Gesamtübereinstimmung mit dem Experiment ist für Na, K und Al gut; jedoch für Li und im gewissen Maße für Rb zeigen die Diskrepanzen zwischen Theorie und Experiment die Notwendigkeit einer umfassenden nichtlokalen Theorie und einiger Verbesserungen der Transportgleichungen.
A Unified study of lattice-mechanical properties of lead using energy-dependent pseudopotential is carried out. Energy dependence
in pseudopotential is considered through the effective mass approximation; the pseudopotential model chosen is the local Heine-Abarenkov
model potential. Properties studied include cohesive energy, equilibrium lattice parameter, second-order elastic constants,
pressure derivative of second-order elastic constants, equation of state (atT=0 K), phonon-dispersion and effective two-body interaction. The results show fairly good agreement with experiment especially
with a modified Heine-Abarenkov potential.
A comprehensive unified study of fcc alkaline earth metal Ca, Sr and the bcc phase of Sr and Ba has been made. Properties
studied include equilibrium lattice parameter, total crystal energy, second order elastic constants, pressure derivative of
the second order elastic constants and phonon spectrum in the symmetry direction. The results obtained show an overall agreement
with experiment. The results partially reproduced the experimentally observed phonon-crossover in Ba, and this has been found
as the many-body effect in lattice dynamics.
A new method is described for setting up the effective potential for electrons in non-transition metals, including liquid metals and alloys. It is based on a model potential fitted to the spectroscopically measured energy levels of the free ions. The potential between the atomic cores is obtained from the dielectric screening calculation of Cohen and Phillips (1961) with some refinements. The method is tested on the band structures of the (solid) alkali metals. The Fermi surfaces of K, Rb and Cs are found to be considerably less distorted than calculated by Ham (1962), and thus in better agreement with experiment.
In this and a succeeding paper it is shown how a theory equivalent to the Bohm & Pines collective motion theory of the electron plasma can be derived directly from a perturbation series which gives in principle an exact solution of the many-body problem. This result is attained by making use of a diagrammatic method of analysis of the perturbation series. By a process analogous to the elimination of photon self-energy parts from the electrodynamic S matrix it is found possible to simplify the perturbation series, introducing a modified interaction between the particles. A useful integral equation for this modified interaction can be set up, and it is shown how the energy of the system can be expressed in terms of the modified interaction. The close connexion between this approach and the dielectric theory of plasma oscillations is indicated.
The complete set of six third-order elastic constants of single-crystal Al has been experimentally determined by measuring both hydrostatic-pressure and uniaxial-stress derivatives of the natural sound velocities using a two-specimen interferometric technique. The specimens were neutron-irradiated to eliminate dislocation effects from the uniaxial experiments. A self-consistent set of hydrostatic-pressure derivatives of the second-order elastic constants has been calculated from the measured third-order constants. The third-order elastic constants have also been used to calculate the thermal expansion in the anisotropic-continuum model at both high and low temperatures, and a comparison has been made with the directly measured expansion coefficients.
Using self-consistent-field techniques, we derive the dynamic dielectric constants appropriate to electrons and to test charges which differ with the inclusion of screened Hartree-Fock exchange. For large wave vector (in the static limit), the exchange contributions exceed the Coulomb contributions, and the electronic dielectric constant becomes less than unity. We also derive a screened exchange potential linearly dependent on the charge density, which seems preferable to the Slater rho13(r) approximation for energy-band calculations.
The dispersion relations for aluminum have been determined at 80 and 300°K by neutron spectrometry, using a three-axis crystal spectrometer. Particular attention was paid to precision, in order to investigate small effects-e.g., Kohn anomalies, phonon frequency widths, frequency shifts with temperature-and to establish an accurate experimental routine. A focusing method used throughout to optimize resolution is described, as well as a method for calculating energy resolution and extracting phonon widths from observed one-phonon resonances. Results are presented as dispersion curves for phonons in the three principal directions, accompanied by phonon widths, and as contour maps of phonon frequency on the surface of an elemental tetrahedron in q space. Measurements at points off the principal directions are utilized in the latter maps. Kohn anomalies have been observed, but are reported elsewhere. Phonon widths at 80°K are interpreted semiquantitatively in terms of the interaction between phonons and conduction electrons. Exceptionally large phonon widths in two regions may be due to singularities in the phonon-phonon interaction: the conservation rules for decay of a phonon into two phonons suggest a source for such singularities, but the suggestion has not been confirmed by computation.
The theory developed in a previous paper is applied to calculate the correlation energy of a free-electron gas. The theory involves on cut-off and gives a uniform description of collective motion effects in the long-range limit and of particle motion effects in the short-range limit. It is shown that in the lowest order the theory agrees with Bohm & Pines's plasma oscillation theory in the long-range limit, but is inadequate in the short-range limit. The theory is approximately evaluated to the next order, which is correct in the short-range limit, and is applied to calculate the correlation energy at several gas densities; the results are in good agreement with those of Bohm & Pines.
On the basis of the local-pseudopotential perturbation formulation of the energy of a metal, the elastic coefficients are calculated for a primitive cubic lattice by the method of homogeneous deformation and also by the method of long waves. Comparison of the two methods leads to two results which are valid for any lattice structure. Firstly, the Hubbard-Sham exchange and correlation screening parameter is determined by requiring agreement between the two calculations for the exchange and correlation contribution to the bulk modulus. Secondly, it is shown that certain screening corrections are contained to a higher order in the homogeneous-deformation calculation than in the long-waves calculation. This leads to a difference in the longitudinal elastic coefficients as calculated by the two methods; the difference does not represent a disagreement with the well-known equivalence of the two methods, but merely results from the use of a finite-order pseudopotential perturbation. These results are illustrated by a simple model calculation of the elastic coefficients and their pressure derivatives for sodium and potassium.
The phonon spectrum of sodium is calculated in the harmonic approximation. The effective interaction between the ions is separated into direct two-body forces between bare ions and effective attraction due to the presence of conduction electrons. Careful treatment of the two-phonon processes includes the calculation of the electron-phonon scattering based on the pseudo-potential method and the screening effects due to the interaction between electrons by the Hartree-Fock approximation with a screened exchange potential. Comparison of the results for sodium with neutron diffraction measurements and a previous calculation by Toya shows good agreement on the whole. It is hoped that this method may be applied to some other metals.
An electron fluid model is proposed for the lattice dynamics of metals which satisfies the requirement of translational invariance
and the lattice is in equilibrium without recourse to external forces. The model is applied to calculate the phonon dispersion
of sodium in the symmetry directions.
A new procedure for calculating the frequency- and wave-vector-dependent dielectric response function is described. It is based on decoupling and solving the equations of motion for the Green's functions of the charge-density operators by a moment-conserving method which is discussed. By use of this method an expression for the dielectric function in the static limit (ω→0) is obtained; it depends on a function G(k), for which numerical values are calculated and tabulated. Evidence that the procedure described here leads to reliable values of G(k) for small, intermediate, and large values of k is presented.
In this paper we derive expressions for a number of properties of simple metals, using a spatially local energy-independent pseudopotential to represent the electron-ion interaction. Quantities are expressed in terms of correlation functions for the homogeneous electron gas and matrix elements of the electron-ion pseudopotential; electron-electron interactions are included to all orders in perturbation theory, and electron-ion interactions to low order. First, we consider the long-wavelength finite-frequency dielectric function and generalize some results of Hopfield; the dielectric function is then used to derive expressions for optical properties, and the frequency and damping of long-wavelength plasma oscillations. Second, we consider the phonon spectrum; expressions for the phonon dynamical matrix are derived, taking into account the influence of the periodic ionic lattice on the motion of electrons. It is shown that to obtain results consistent with expressions for the elastic constants derived from expressions for the energy of the system calculated to the second order in the electron-ion interaction, one must include in the calculation of the phonon dynamical matrix some terms of third and fourth order in the electron-ion interaction. A detailed discussion of the long-wavelength behavior of the dynamical matrix is given.
In this paper we present a modification of an earlier theory of Singwi et al. of electron correlations at metallic densities. The modification consists in allowing for the change of the pair correlation function in an external weak field via the density derivative of the equilibrium pair correlation function. This results in a new expression for the local-field correction. The present theory has the merit of satisfying almost exactly the compressibility sum rule and of giving a satisfactory pair correlation function. Results of self-consistent numerical calculations for the static pair correlation function, correlation energy, compressibility, and plasmon dispersion relation for the electron liquid in the metallic-density range are presented. For those interested in the application of the results of the present paper, numerical values of the local-field correction as a function of wave number have been tabulated in the density range rs=1-6.
A model expression is developed for the energy in a metal with the help of which one may calculate a fairly large number of different properties of the metal with much greater facility than in a strictly pseudopotential calculation. The model is applied to calculate the phonon frequencies of the b. c. c. metal W and the f. c. c. metals Cu, Ag, and Ni. With a suitable expression for the two-body central interaction the model can be easily extended to calculate a fairly large number of both static and dynamic properties of metals.
Measurements of the elastic moduli of aluminum over the temperature range 300°–930°K are reported. These data together with those reported for T < 300°K, show that all elastic moduli, including the shear moduli, vary smoothly with the temperature, up to the melting point, in contrast to other measurements at lower ultrasonic frequencies. From the data, a high temperature Grüneisen constant of 2·17 is calculated. The temperature dependence of the shear moduli at room temperature is calculated from Leigh's model, and the results of this calculation are compared with the experimentally determined data, and with the pressure dependence of the elastic moduli.
The composite piezoelectric oscillator is employed to measure the adiabatic elastic moduli of crystalline aluminum over the temperature interval 63\ifmmode^\circ\else\textdegree\fi{}K to 773\ifmmode^\circ\else\textdegree\fi{}K. The data permit a valid extrapolation to 0\ifmmode^\circ\else\textdegree\fi{}K. The Debye characteristic temperature of aluminum at 0\ifmmode^\circ\else\textdegree\fi{}K, computed with these data, is 439\ifmmode^\circ\else\textdegree\fi{}K. Various novel procedures designed to facilitate the use of the method are described.
- K. S. Singwi