Silicon diffusion in molybdenum disilicide: correlation effects

Correlation factors for silicon diffusion by a vacancy mechanism in the silicon sublattice of the tetragonal MoSi₂ structure have been calculated by combining an analytical and a Monte Carlo approach. The ratio of the silicon diffusivity perpendicular to the tetragonal axis to that parallel to the tetragonal axis is also deduced. An effect of forward correlation of tracer atom jumps in the silicon sublattice with the corresponding partial correlation factor of 1.5 appears at small frequencies of silicon atom jumps along the tetragonal axis with respect to the jump frequencies in the silicon layer perpendicular to the tetragonal axis of the MoSi₂ structure. The anisotropy of silicon diffusion in MoSi₂ measured by Salamon et al. is explained in terms of correlation effects of silicon diffusion on its own sublattice.     

Analysis of interdiffusion via isolated vacancies in strongly ionic crystals

In this paper, we analyse chemical interdiffusion in strongly ionic crystals for diffusion couples AY _m –BY _m , where A and B have the same charge numbers. We employ the exact sum rule given by Moleko and Allnatt relating the phenomenological coefficients for diffusion in the multicomponent random alloy via the agency of monovacancies. It is shown that the ratio of the intrinsic diffusivities can be expressed very simply in terms of the atom–vacancy exchange frequencies without correlation terms. For the case of an immobile anion sublattice and making use of a highly accurate diffusion kinetics theory due to Moleko et al., it is shown that the interdiffusivity is principally proportional only to the off-diagonal phenomenological coefficient relating the two cations.     

Diffusion in ordered Fe-Si alloys

The measurement of the diffusional Mössbauer line broadening in single crystalline samples at high temperatures provides microscopic information about atomic jumps. We can separate jumps of iron atoms between the various sublattices of Fe-Si intermetallic alloys (D0₃ structure) and measure their frequencies. The diffusion of iron in Fe-Si samples with Fe concentrations between 75 and 82 at% shows a drastic composition dependence: the jump frequency and the proportion between jumps on Fe sublattices and into antistructure (Si) sublattice positions change greatly. Close to Fe₃Si stoichiometry iron diffusion is extremely fast and jumps are performed exclusively between the three Fe sublattices. The change in the diffusion process when changing the alloy composition from stoichiometric Fe₃Si to the iron-rich side is discussed.     

Application of random walk concept to the cyclic diffusion mechanisms for self-diffusion in intermetallic compounds

Huntington–Elcock–McCombie (HEM) mechanism involving six consecutive and correlated jumps, a triple-defect mechanism (TDM) involving three correlated jumps and an anti-structure bridge (ASB) mechanism invoking the migration of an anti-structure atom are the three mechanisms currently in vogue to explain the self- and solute diffusion in intermetallic compounds. Among them, HEM and TDM are cyclic in nature. The HEM and TDM constitute the theme of the present article. The concept of random walk is applied to them and appropriate expressions for the diffusion coefficient are derived. These equations are then employed to estimate activation energies for self-diffusion via HEM and TDM processes and compared with the available experimental data on activation energy for self-diffusion in intermetallic compounds. The resulting activation energies do not favour HEM and TDM for the self-diffusion in intermetallic compounds. A comparison of the sum of experimentally determined activation energies for vacancy formation and migration with the activation energies for self-diffusion determined from radioactive tracer method favours the conventional monovacancy-mediated process for self-diffusion in intermetallic compounds.     

Contribution a la theorie atomique de la diffusion chimique

Here we present an analysis and a development of the atomic theory of chemical diffusion as proposed by Manning for a binary system a/b.The general expression for the flux of a tracer in a concentration gradient is first established. This expression of the flux is identified with that deduced in the phenomenological theory. Thus a relationship between the partial correlation factors of vacancies with each of the a and b species is obtained.The effect of “vacancy flow” can be described in terms of these correlation factors. Thus the vacancy flow on species A leads to a correlation of the vacancy jumps with species B and vice versa.We shall see that the Nernst-Einstein equation can be extended to the case of chemical diffusion and that the ratio of the intrinsic diffusion coefficients is equal to the ratio of the mean jump frequencies W_A and W_B.Also, the activation energies of intrinsic diffusion coefficients are related very simply to the activation enthalpies of atomic jumps.In conclusion, we shall see that chemical diffusion in a binary system a/b can be completely described if either the thermodynamic factor and the coefficients of self diffusion, or the thermodynamic factor and the coefficients of intrinsic diffusion are known as functions of the concentration.     

Stochastic Model of PAC Nuclear Relaxation Caused by Defects Hopping on a Simple Cubic Lattice

A stochastic model is used to obtain an analytic approximation for the perturbation function caused by quadrupole interaction that applies when there is a fixed concentration of defects hopping on a simple cubic lattice, with the probe atom located at center of one of the cubes. A realization of this model is jumping of structural vacancies on one sublattice in the CsCl structure, with the probe atom on the other sublattice. PAC spectra obtained for Pd-poor PdIn using ¹¹¹In/Cd probes are refitted with the model perturbation function to obtain the vacancy jump frequency as a function of temperature. This revised version was published online in September 2006 with corrections to the Cover Date.     

A comment on Baker et al. ‘The time dependence of an atom-vacancy encounter due to the vacancy mechanism of diffusion’

Abstract

Baker et al. derived time-dependent expressions for calculating average number of jumps per encounter and displacement probabilities for vacancy diffusion in crystal lattice systems with infinitesimal vacancy concentrations. As shown in this work, their formulation is readily expanded to include finite vacancy concentration, which allows calculation of concentration-dependent, time-averaged quantities. This is useful because it provides a computationally efficient method to express lineshapes of nuclear spectroscopic techniques through the use of stochastic fluctuation models.     

Diffusion correlation effects of molybdenum and silicon in molybdenum disilicide

Diffusion data for both principal directions of silicon and molybdenum as well as germanium are briefly summarised. Analysis is performed of the defect formation energies (available from previous ab initio calculations and experimental measurements) for diffusion mechanisms via home and foreign sublattices. The home sublattice mechanism is shown to be the preferred one for both silicon and molybdenum. Tracer correlation factors for silicon and molybdenum diffusion via sublattice vacancies in the respective sublattices of the tetragonal C11_b structure of molybdenum disilicide are calculated by a direct Monte Carlo simulation technique. Correlation factors for Si diffusion on its sublattice are compared with literature values that were calculated using a more complicated Monte Carlo method based on the matrix approach. It is shown that there is no need for this complicated approach and that the direct Monte Carlo simulation technique gives highly accurate correlation factors. Correlation factors and anisotropy ratios of vacancy-mediated diffusion in both sublattices are deduced and compared with experimental data. Tracer correlation in the tetragonal direction is shown to contribute 0.40?eV (i.e. over 55%) of the migration energy of the corresponding Si diffusivity. Two possible jump rates for Si diffusion are separately estimated. Mo diffusion correlation factors are calculated using the direct Monte Carlo technique. A comparison with experiment is made and the ratio of two possible jump rates is also estimated.     

Extended random phase approximation for layered copper oxides antiferromagnets

The role of interlayer coupling constant on the Néel temperature of layered copper oxides and other magnetic properties have been studied. The theoretical framework is based on anisotropic Heisenberg model and the two sublattice approach. The higher-order Green functions are decoupled using the second random phase approximation. The Green’s function related to the localized spin correlation functions has been exploited. A self consistent expression for the sublattice magnetization and a corrected form for the Néel temperature are obtained. The free-spin-wave results are obtained under a specific approximation. The theoretical results are compared with that of the random phase approximation and existing experimental results.     

Solvent diffusion kinetics in the dilute random alloy

In this paper we address the problem of enhancement of the solvent diffusivity by the addition of very small amounts of solute. The random alloy model is employed at the dilute limit. No Monte Carlo calculations have been available for evaluating the available theoretical treatments. In the present study we report on very-high-precision results of the solvent correlation factors as obtained from Monte Carlo calculations with averaging over 6 210 7 atoms and very long runs (2000 jumps per atom). The results are expressed in terms of the usual solvent enhancement factors b 1 and b 2 . There is good agreement with the theory of Holdsworth and Elliott and excellent agreement with the theory of Moleko et al . but not with other available theories.     

Modeling band spectra of compound crystals in the basis of states of their sublattices

A technique for simulation of band spectra of compound crystals in the basis of states of their sublattices is developed. The Hamiltonian of the crystal is written as a sum of sublattice Hamiltonians and a perturbation operator resulting from hybridization of the sublattice states. The simplest models are proposed to account for the sublattice hybridization. The methods for calculating the average crystal potential are discussed to represent the band spectra of sublattices in a common energy scale when calculating band structures by the pseudopotential method. The band spectra of MgO and MgS crystals are calculated to show the application of the methods under consideration. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 81–84, July, 2006.     

Automatic numerical evaluation of vacancy-mediated transport for arbitrary crystals: Onsager coefficients in the dilute limit using a Green function approach

Abstract

A general solution for vacancy-mediated diffusion in the dilute-vacancy/dilute-solute limit for arbitrary crystal structures is derived from the master equation. A general numerical approach to the vacancy lattice Green function reduces to the sum of a few analytic functions and numerical integration of a smooth function over the Brillouin zone for arbitrary crystals. The Dyson equation solves for the Green function in the presence of a solute with arbitrary but finite interaction range to compute the transport coefficients accurately, efficiently and automatically, including cases with very large differences in solute-vacancy exchange rates. The methodology takes advantage of the space group symmetry of a crystal to reduce the complexity of the matrix inversion in the Dyson equation. An open-source implementation of the algorithm is available, and numerical results are presented for the convergence of the integration error of the bare vacancy Green function, and tracer correlation factors for a variety of crystals including wurtzite (hexagonal diamond) and garnet.     

Atomic structure of InSb(001) and GaAs(001) surfaces imaged with noncontact atomic force microscopy

Noncontact atomic force microscopy (NC-AFM) has been used to study the c(8x2) InSb(001) and the c(8x2) GaAs(001) surfaces prepared by sputter cleaning and annealing. Atomically resolved tip-surface interaction maps display different characteristic patterns depending on the tip front atom type. It is shown that representative AFM maps can be interpreted consistently with the most recent structural model of A(III)B(V)(001) surface, as corresponding to the A(III) sublattice, to the B(V) sublattice, or to the combination of both sublattices.     

Thermodynamics and point defects in the B2 intermetallic phase PdIn

The X-ray and bulk densities of PdIn alloys have been determined at ambient temperature on samples annealed at 1273 and 1373 K and quenched in water-ice mixtures. From these measurements the vacancy concentrations in this intermetallic phase have been obtained as a function of In concentration at these two temperatures. In addition, a generalized thermodynamic model is presented which considers the existence of antisite and vacancy defects on both sublattices without any dilute solution approximations. This model uses three energy parameters, whose values have been obtained for PdIn using the measured vacancy concentrations and the chemical potentials of In. This model is able to describe all the available vacancy data and is suitable for calculating these values at compositions and temperatures when such data are not available. In addition, the model permits the calculation of the concentrations of the minority point defects. It is shown that dilute-solution approximations, which have invariably been used in this kind of modelling, are unsatisfactory. The present results and their analysis confirm that PdIn is a near-triple-defect type of intermetallic and not a near-Schottky type, as has been claimed recently.     

Partial hypervirial theorems for atomic-molecular systems

A family of partial hypervirial theorems for physical properties of pairs of particles in three-and four-particle atomic-molecular systems is considered. The partial hypervirial theorems generalize the partial virial theorems proposed earlier. The sum rule is formulated, according to which the expectation values of the derivatives with respect to the interparticle distances are related to the products of the charges of pairs of the particles. This rule is used to check the accuracy of the calculation of variational wave functions for the positronium ion e ^? e ^? e ⁺. It is shown that the sum rule is two orders of magnitude more sensitive to the inaccuracy of the calculation of the wave function than the partial virial theorems and is five or six orders of magnitude more sensitive to it than the conventional virial theorem.     

The distribution of the valence electron density in predominantly ionic crystals with different Bravais sublattices

Based on the theory of the local-density functional, self-consistent valence electron densities are calculated for CaO with a rock-salt lattice, CaF₂ with a fluorite lattice, K₂O with an antifluorite lattice, and for their constituent sublattices. It is shown that in the crystals with different Bravais sublattices, the anionic sublattice is a framework with covalent bonds containing a metal sublattice inside of them. The coupling between the sublattices is characterized by the density difference, which is deflned as the difference between the total electron density and the densities of the individual sublattices. The density difference is found to be an order of magnitude smaller than the crystal and sublattice densities, which is evidence of weak hybridization of the sublattices and of the predominately ionic character of the bonding between them.     

The non-isotropic two-dimensional random walk

Abstract

A formula is obtained for the joint probability density function of the angle and length of the resultant of an N-step non-isotropic random walk in two dimensions for arbitrary step angle and radius probability density and for any fixed number of steps. The problem is attacked by applying the theory of generalized functions concentrated on smooth manifolds. The analysis is presented initially for the case where only the angles are random. The characteristic function is defined for the walk in terms of angular and radial frequencies and the inversion is obtained in terms of a sum of Hankel transforms. The Hankel transform sum is transformed by showing that it can be interpreted in terms of the motions of the two-dimensional Euclidean plane corresponding to the rotations and translations resulting from a sequence of fixed steps. This transformation results in an expression involving integrations over two manifolds defined by delta functions. The properties of the manifolds defined by the delta functions are then considered and this results in some simplification of the formulae. The analysis is then generalized to the case where both the phase and length of each step in the walk are random. Finally, seven examples are presented including the general two-step walk and three walks which lead to generalized K density functions for the resultant.     

Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

The theory of small-angle scattering is reviewed with special attention paid to the anomalous scattering and multiphase systems. A general equation is derived that describes the scattering of a multiphase system as a sum of scattering functions of each of the phases, as if it scattered alone in a two-phase system, and interphase interference scattering functions. These scattering functions depend only on the spatial distribution of the phase boundaries, but not on the scattering density. Contrast variation techniques are most rewarding when the scattering density of only one phase can be varied. For anomalous small-angle X-ray scattering (ASAXS), this means the most favourable is the case in which resonant atoms are contained in one phase only. The general equation involves n(p ? 1) unknown partial atomic number density differences, where p is the number of phases and n the number of the different atom types in the sample. These partial atomic number density differences can be found if a suitable structure model is applied to calculate the phase scattering functions. Then, the phase compositions and densities can be calculated by solving a system of linear equations incorporating the atom number conservation law. The partial structure factors formalism is also reviewed. Corresponding equations for a system of n types of atoms and p phases are derived. The number of independent partial structure factors is p(p ? 1)/2 and depends on the number of phases, but not on the number of the types of the atoms in the sample, as in the case of wide-angle scattering.     

Relaxation and movement of point defect clusters in copper by molecular dynamics simulation

Abstract

The structures of point defect clusters of both interstitial and vacancy type were examined by computer simulation using molecular dynamics and molecular statics with the DYNAMO code (Daw, Foiles and Baskes [6]). The code implements an isotropic potential of embedded atom method (EAM) developed by Daw and Baskes [5]. Interstitial clusters relax to either the immobile mixture of <100> dumbbell and bcc interstitials or a mobile platelet of parallel <110> interstitials. The latter cluster moves along <110> directions. A tri-vacancy relaxes to an un-collapsed stacking fault tetrahedron (sft) of Damask-Dienes type (3v-sft) containing a central atom that vibrates with a large amplitude. A hexa-vacancy relaxes to a stacking fault tetrahedron the structure of which fluctuates between a sft and void. Larger vacancy clusters are stable as a combination of sft and 3v-sft. In these vacancy clusters, atoms show significant vibration with large amplitude. Voids form only with the inclusion of gas-atoms into vacancy clusters.