1,677 research outputs found
Non-exponential time-correlation function for random physical processes
The exponential correlation function is theoretically incorrect in the entire
frequency range of interest for processes described in terms of linear response
theory. The Lorentzian lineshape results from an inconsistent assumption of
exponential correlation at timescales smaller than the relaxation time. A new
correlation function is proposed that avoids the deficiencies of the
exponential function. Comparison on dielectric relaxation in gases shows that
the new correlation function can be used to fit data satisfactorily instead of
the exponential function. The new correlation function is theoretically
consistent for all processes described in terms of linear response theory. Its
additional mathematical superiority implies that it can be used instead of the
exponential function for all such processes.Comment: 14 pages, 1 Table, 4 figures, submitted for publication. Manuscript
rewritten to emphasise that exponential correlation is theoretically
incorrect for processes described in terms of linear response theor
Probabilistic method to determine electron correlation energy
A new method to determine electron correlation energy is described. This
method is based on a better representation of the potential due to interacting
electrons that is obtained by specifying both the average and standard
deviation. The standard deviation is determined from a probabilistic
interpretation of the Coulomb interaction between electrons. This leads to a
better representation of orbital energies as Ei (plus-minus) delta(Ei), where
Ei is the Hartree-Fock orbital energy and delta(Ei), the spread, is an
indicator of the magnitude of correlation energy. This new representation of
the potential when combined with an empirical constant leads naturally to a new
method to determine electron correlation energy. Correlation energy is
determined within the independent electron approximation without any
contribution from higher energy unoccupied states. A consistent physical
interpretation that an electron occupies a given position when other electrons
are farther than on average can be made. It is a general technique that can be
used to determine correlation energy in any system of particles with
inter-particle interaction V(r1, r2) and can be considered to be universal
first step beyond mean-field theory.Comment: 9 pages. Submitted for publicatio
Analysis of long range order
A first principles analysis of order-disorder transition in alloys shows that
ordering energy is a function of temperature due to thermal vibrations. The
inter-nuclear potential energy term converges if zero point vibrations are
incorporated and this method can replace the Ewald sum method. Core energy
contributions to the ordering energy are stored exclusively in superlattice
lines. The effect of electron-phonon interactions on ordering energy is of the
same order of magnitude as ordering energy near transition temperatures and
cannot be ignored. Ising model and variants are incorrect in explaining alloy
phase transitions as they ignore the role of electron-phonon interactions
without justification. A theoretical formalism that incorporates the
Debye-Waller Factor component of electron-phonon interactions in electronic
structure calculations already exists and must be adopted when modeling
temperature dependent phenomena. It is suggested that DWF correction will
account substantially for the discrepancy between experimental and theoretical
ordering energy in Ni3V. Thermal vibrations alter magnetic ordering energy at
finite temperatures. The role of electron-phonon interactions in alloy and
magnetic phase transitions cannot be ignored and must be incorporated in all
models. This will also ensure consistency with x-ray and electron diffraction
(alloy transitions) and neutron diffraction (magnetic transitions) results. An
isotope effect is predicted for (magnetic) phase transitions if the transition
temperature is below Debye temperature. Recent observations of an isotope
effect in magnetic phase transitions confirm our above conclusions and imply
that the role of electron-phonon interactions must be incorporated in all
theories and models of magnetism to avoid contradictions.Comment: revised manuscript, 31 pages, submitted for publicatio
Form of the exact partition function for the generalized Ising Model
The problem of N interacting spins on a lattice is equivalent to one of N
clusters linked in a specific manner. The energy of any configuration of spins
can be expressed in terms of the energy levels of this cluster. A new
expression is obtained for the probability of occurrence of any configuration.
A closed form expression is obtained for the partition function per spin in
terms of the energy levels of this cluster with the degeneracies being a
function of temperature. On physical grounds it is suggested to be the form of
the exact partition function per spin. The partition functions of Ising-like
systems all have a common form. For the 3D Ising model seven functions need to
be determined to describe the partition function completely. The key to
understanding second order phase transitions and critical phenomena lies in the
temperature dependence of various degeneracies. It is necessary to develop new
techniques to determine the partition function that account for this
temperature dependence, as it would represent the underlying physics correctly.Comment: 17 pages, 1 figure Submitted for publication. Improved figure.
Expanded discussion section adde
Role of thermal vibrations in magnetic phase transitions
We address and resolve the fundamental contradiction that has existed from
the earliest studies on magnetic phase transitions between theoretical models
that ignore the role of thermal vibrations and represent the exchange
interaction as a constant, Jij(0), and analysis of neutron diffraction data
that always incorporates thermal vibrations even though it is also possible to
analyze the same data by ignoring them. Of the two possibilities, ignoring
thermal vibrations in both theoretical models and analysis of diffraction data
leads to the latter giving different magnetic order parameters for different
reciprocal lattice lines. This appears to be the first report of a unique
consequence, viz. the assumption to neglect a physical phenomenon turns a
single-valued experimental observable into a multiple-valued one where all
values are equally valid. This assumption is clearly unacceptable and must be
rejected. The second possibility of incorporating thermal vibrations in both
leads to single-valued theoretical and experimental order parameters. Thus,
analysis of neutron diffraction data constrain the exchange interaction in all
theoretical models to be temperature dependent and represented as Jij(T).
Additional experimental and theoretical evidences in support of this conclusion
are presented.Comment: Submitted for publicatio
Correct form of the electron wavefunction in periodic solids
The Bloch wavefunction leads either to mathematically impossible consequences
or suggests that the ground state energy is a function of size and shape when
the geometry of large crystals is considered in detail. It is incompatible with
the assumption underlying the Born-von Karman periodic boundary condition. The
source of the difficulty is the incorrect dependence of the Bloch wavefunction
on the wavenumber index k. The mathematically impossible consequences can be
overcome if the periodic part of the electron wavefunction is represented as
un(r), which is dependent only on the band index, n, and is independent of the
wavenumber index k. This correct form of the wavefunction is consistent with
the Bloch theorem and with all other properties of Bloch wavefunctions. The
correct form is also consistent with the Born-von Karman periodic boundary
condition. The correct form of the electronic wavefunction in a periodic solid
has profound consequences. It simplifies the calculation of electronic
structure as only one wavefunction per band, un(r), needs to be evaluated. It
brings about a conceptual unification between the band picture favored by
physicists and the bond picture favored by chemists. The correct form of the
electron wavefunction will simplify the understanding of many phenomena
involving valence electrons.Comment: 15 pages, submitted for publicatio
Role of thermal vibrations in phase transitions
All theoretical models (Heisenberg, Ising etc.) assume a negligible role for
thermal vibrations in alloy and magnetic phase transitions. Analysis of
diffraction data conclusively proves that this assumption is incorrect. A
simple criterion emerges that theoretical models can ignore the role of thermal
vibrations only if the Debye-Waller Factor is ignored in the analysis of
diffraction data. Diffraction data constrain all theoretical models to
incorporate the role of thermal vibrations. This conclusion is also supported
by other experimental results, the effect of thermal vibrations on ordering
energy that is of the same order of magnitude as ordering energy and an isotope
effect on magnetic phase transitions. An electron-phonon interaction (EPI)
formalism that incorporates the Debye-Waller Factor in electronic structure
calculations already exists and must be adopted for a correct understanding of
phase transitions as it can account for all the different experimental results
mentioned above. The discrepancy between experimental and theoretical ordering
energy in Ni3V is direct evidence for the role of thermal vibrations in
altering ordering energy. The inter-nuclear potential energy term converges if
zero point vibrations are incorporated and this method can replace the Ewald
sum method. The three dimensional Ising model cannot represent order-disorder
transition in beta brass, CuZn. An isotope effect is predicted for magnetic
phase transitions if the transition temperature is below Debye temperature. The
long range order parameter obtained from diffraction data can only be compared
with predictions of models that incorporate the role of thermal vibrations and
not otherwise.Comment: 37 pages, submittd for publicatio
Relation between ab initio molecular dynamics and electron-phonon interaction formalisms
The relation between ab initio molecular dynamics formalism and the
electron-phonon interaction formalism [P.B. Allen and V. Heine, J. Phys. C 9,
2305 (1976)] is explored. The fundamental quantity obtained in the AIMD
formalism - total energy for any configuration - is also obtained from the
formalism (ES-DWF) that incorporates the role of Debye-Waller Factor in
electronic structure calculations. The two formalisms are exactly equivalent
and represent the direct and perturbation theory approaches to determine total
energy. This equivalence allows either formalism to be used depending on the
requirement - ES-DWF for a priori theoretical analysis and AIMD for ab initio
modeling of the effect of thermal vibrations. Combining the two formalisms
makes the ES-DWF formalism into an ab initio method and increases the range of
problems that can be modeled ab initio. It is also theoretically possible to
obtain self-consistent band structures from AIMD calculations. This study
clarifies the incorrect assumptions regarding the two formalisms that exist in
published literature. By combining the two formalisms and including self-energy
effects, more accurate results can be obtained, ab initio, within the adiabatic
approximation, than by using AIMD alone.Comment: 16 pages, manusript extensively rewritten, submitted for publicatio
Finite temperature electronic structure of Diamond and Silicon
The electron-phonon interaction contribution to the electronic energies is
included in density functional total energy calculations with ab initio
pseudopotentials via the Allen formalism [Phys. Rev. B 18, 5217 (1978)] to
obtain temperature dependent electronic structure of diamond and silicon. This
method allows us to obtain the thermally-averaged ab initio electronic
structure in a straightforward and computationally inexpensive way. Our
investigations on the finite temperature electronic structure of diamond and
silicon lead to a new criterion, temperature transferability, which is required
in the ab initio pseudopotentials for temperature dependent studies. The
temperature transferability of the Troullier-Martins pseudopotentials used in
this work is strongly dependent on the cut-off radius and the inclusion of the
unbound 3d state. The finite temperature indirect band gaps are highly
sensitive to the choice of cut-off radius used in the pseudopotentials. The
finite temperature band structures and density of states show that thermal
vibrations affect the electron energies throughout the valence and conduction
band. We compare our results on the band gap shifts with that due to the
Debye-Waller term in the Allen-Heine theory and discuss the observed
differences in the zero point and high temperature band gap shifts. Although,
the electron energy shifts in the highest occupied valence band and lowest
unoccupied conduction band enable to obtain the changes in the indirect and
direct band gaps at finite temperatures, the shifts in other electronic levels
with temperature enable investigations into the finite temperature valence
charge distribution in the bonding region. Thus, we demonstrate that the Allen
theory provides a simple and theoretically justified formalism to obtain finite
temperature valence electron charge densities that go beyond the rigid
pseudo-atom approximation.Comment: 15 pages, 9 figures, 8 table
Finite temperature external potential in crystalline solids
Thermal vibrations alter the external potential. Allen (Phys. Rev. B 18
(1978) 5217) proved that at finite temperatures the pseudopotential form
factors are corrected by a Debye-Waller Factor (DWF). We generalize this result
to the crystal potential. (The generalization to the all-electron case of the
nuclear potential fails due to the breakdown of the rigid-atom approximation.)
This finite temperature formalism only gives thermal-averaged properties and no
dynamical information can be obtained. Hence, it is labeled the Quasi Ab Initio
formalism. Analogous to the use of experimental lattice parameters in ab initio
studies, experimental DWF can also be used. The justification is identical; the
experimental parameters can be validated by separate ab initio studies. Our
work transforms, forty years later, Kasowski's empirical study (Phys. Rev. B. 8
(1973) 1378) into the first ab initio finite temperature band structure
calculation. This formalism opens the way to obtain ab initio finite
temperature thermal-averaged properties from a single calculation.Comment: 16 pages, submitted for publicatio
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