Rigid Lattice Approximation Research Articles

Decomposition kinetics of Fe-Cr solid solutions during thermal aging

The decomposition of Fe-Cr solid solutions during thermal aging is modeled by atomistic kinetic Monte Carlo simulations, using a rigid lattice approximation with pair interactions that depend on the local composition and temperature. The pair interactions are fitted on ab initio calculations of mixing energies and vacancy migration barriers at 0 K. The entropic contributions to the mixing of Fe-Cr alloys and to the vacancy formation and migration free energies are taken into account. The model reproduces the change in sign of the mixing energy with the alloy composition and gives realistic thermodynamic and kinetic properties, including an asymmetrical miscibility gap at low temperature and diffusion coefficients in good agreement with available experimental data. Simulations of short-range ordering and $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$${}^{\ensuremath{'}}$ decomposition are performed at 773 and 813 K for Cr concentrations between 10$%$ and 50$%$. They are compared with experimental kinetics based on three-dimensional atom probe and neutron scattering measurements. The possible effect of magnetic properties on diffusion in the $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{\ensuremath{'}}$ phases, and therefore on the decomposition kinetics, is emphasized.

Simulations of Decomposition Kinetics of Fe-Cr Solid Solutions during Thermal Aging

The decomposition of Fe-Cr solid solutions during thermal aging is modeled by Atomistic Kinetic Monte Carlo (AKMC) simulations, using a rigid lattice approximation with composition dependant pair interactions that can reproduce the change of sign of the mixing energy with the alloy composition. The interactions are fitted on ab initio mixing energies and on the experimental phase diagram, as well as on the migration barriers in iron and chromium rich phases. Simulated kinetics is compared with 3D atom probe and neutron scattering experiments.

Formation of oxide nanoclusters in nanostructured ferritic alloys during anisothermal heat treatment: A kinetic Monte Carlo study

Kinetic Monte Carlo simulations, based on rigid lattice approximation, have been performed to study the homogeneous precipitation in Fe–Y–O and Fe–Ti–O alloys during anisothermal heat treatments. Simulations predict different kinetic behaviors, including transient precipitation of metastable iron oxides followed by precipitation of Y 2O 3 or Ti 2O 3 nanoclusters. Then, precipitation kinetics in ternary alloys are compared. Significant differences in the precipitation behaviour of Y 2O 3 and Ti 2O 3 were observed as result of dramatically different diffusion rates of Y and Ti atoms.

Atomistic Kinetic Monte Carlo studies of microchemical evolutions driven by diffusion processes under irradiation

Atomistic Kinetic Monte Carlo (AKMC) simulations are a powerful tool to study the microstructural and microchemical evolution of alloys controlled by diffusion processes, under irradiation and during thermal ageing. In the framework of the FP6 Perfect program, two main approaches have been applied to binary and multicomponent iron based alloys. The first one is based on a diffusion model which takes into account vacancy and self-interstitial jumps, using simple rigid lattice approximation and broken-bond models to compute the point-defect jump frequencies. The corresponding parameters are fitted on ab initio calculations of a few typical configurations and migration barriers. The second method uses empirical potentials to compute a much larger number of migration barriers, including atomic relaxations, and Artificial Intelligence regression methods to predict the other ones. It is somewhat less rapid than the first one, but significantly more than simulations using “on-the-fly” calculations of all the barriers. We review here the recent advances and perspectives concerning these techniques.

Importance of Correlated Motions on the Low Barrier Rotational Potentials of Crystalline Molecular Gyroscopes

The energetic and structural changes taking place upon rotation of the central phenylene of 1,4-bis(3,3,3-triphenylpropynyl)benzene in the solid state were computed using molecular mechanics calculations. Pseudopolymorphic crystals of a benzene clathrate (1A) and a desolvated form (1B) were analyzed with models that account for varying degrees of freedom within the corresponding lattices. The calculated rotational barriers in a rigid lattice approximation, 78 kcal/mol for 1A and 72 kcal/mol for 1B, are about 5 times greater than those previously measured by variable-temperature 13C CPMAS NMR and quadrupolar echo 2H NMR line-shape analysis: 12.8 kcal/mol for 1A and 14.6 kcal/mol for 1B. The potential energy barriers calculated with a model that restricts whole body rotation and translational motions but allows for internal rotations give results that are near the experimental free-energy barriers. The calculated barriers for 1A and 1B are 15.5 and 16.2 kcal/mol, respectively. The differences between the rigid and partially relaxed models are attributed to the effect of correlated motions of the lattice and the rotating group, which are evident from the structural analysis of the atomic position data as a function of the dihedral angle of the rotator. The displacements of neighboring molecules near the rotary transition states for 1A and 1B can be as large as 2.7 and 1.1 A, respectively. The displacement and oscillation (C2) of interpenetrating phenyl rings from neighboring rotors proximal to the event are significant for both 1A and 1B. In addition, 6-fold (C6) benzene rotations in clathrate 1A were found to be directly correlated to the rotation of the phenylene rotator.

Cluster variation method in the computational materials science

Cluster Variation Method (CVM) has been very successful in the computations of alloy phase diagrams as well as in many problems of the materials science related to the phase transitions. Originally, CVM was developed in the framework of the so-called rigid lattice approximation, but it has recently been extended to include continuous atomic displacements due to thermal lattice vibration and local atomic distortion due to size mismatch of the constituent atoms. In the present study, we focus our attention on the latter continuous displacement treatment of CVM. The continuous displacement (CD) formulation of the CVM is applied to study the phase stability of the binary alloys. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r . The effects of continuous atomic displacement on the thermodynamic quantities and phase transitions of binary alloys are investigated in detail. We also discuss the extension of the CD treatment of CVM to the calculations of solid-liquid and gas liquid phases transitions.

The Mössbauer and magnetic resonances investigations of structural phase transitions in K ZnCl crystals

The experimental data carried out by Mossbauer and magnetic resonances investigations of the structural phase transitions in K2ZnCl4 crystals are discussed by a simple electrostatic model, calculating, the lattice contributions to the local electric potential V(r), electric field intensity E(r) and electric field gradient tensor, \(\)(r) and taking into account both the fractional electric point charges and rigid lattice approximations. The validity of the model is proved by a good fit of the computing results and experimental data of quadrupole splitting parameters at K sites obtained by 39K-NMR methods in high temperature incommensurate phase (\(\) Pnam symmetry). The experimental results obtained by Mossbauer and EPR methods in commensurate phase (Pna21symmetry) of iron and copper doped K2ZnCl4 crystals are explained by relaxing the rigid lattice approximation. The insertion of probe ions appear to be done on not-exactly-Zn2+ site.

Microscopic theory of the pseudogap and Peierls transition in quasi-one-dimensional materials.

The problem of deriving from microscopic theory a Ginzburg-Landau free-energy functional to describe the Peierls or charge-density-wave transition in quasi-one-dimensional materials is considered. Particular attention is given to how the thermal lattice motion affects the electronic states. Near the transition temperature the thermal lattice motion produces a pseudogap in the density of states at the Fermi level. Perturbation theory diverges and the traditional quasiparticle or Fermi-liquid picture breaks down. The pseudogap causes a significant modification of the coefficients in the Ginzbug-Landau functional from their values in the rigid-lattice approximation, which neglects the effect of the thermal lattice motion.

Interfacial energy calculation at interconnect-metal/barrier-metal interfaces for grain orientation control

The interfacial energy of interconnect-metal/barrier-metal interfaces in ultralarge-scale integrations have been investigated numerically by assuming a simple additive interaction potential within a rigid-lattice approximation. The calculation gave some noteworthy results: Al prefers the (111) orientation on a TiN(111) plane, but Cu did not have such a strong preference. Al(111) can be expected to show strong epitaxial growth on VNx(111), as did Cu on Nb(110). On the basis of the calculations, a Cu/Nb bilayered structure was sputter deposited sequentially by dc magnetron sputtering and examined for crystalline orientation by x-ray-diffraction analysis. Cu exhibited strong (111) orientation preference, and a narrow full width at half-maximum in (111) rocking angle around the substrate normal.

Spectroscopy of third harmonic generation in polyacetylene

Trans -polyacetylene exhibits the largest optical susceptability associated with third-harmonic generation (THG) of any conjugated polymer. However, the origin of the nonlinear response remains controversial; proposed mechanisms range from the nonlinear polarizability calculated by direct application of third order perturbation theory in a rigid lattice approximation to the nonlinear polarizability arising from virtual soliton-antisoliton pairs enabled by nonlinear zero point motion. We have addressed the question of the origin of the third harmonic response by measuring the THG over a wide range of fundamental (pump) frequencies in both the trans and cis isomers. The THG spectroscopy probes the two-photon and three-photon resonances; the comparison of the data for the trans and cis isomers separates the effect of the confinement forces which are important in cis -(CH) x but absent in trans -(CH) x.

Theoretical cluster-model study of line-broadening effects in core-level spectra

The core-level linewidths obtained in recent experiments for oxygen chemisorbed on Ni(100) have been analyzed with use of cluster-model calculations. Accurate quantum-mechanical calculations including electron correlation but within the rigid-lattice approximation are only able to reproduce part of the temperature dependence of the linewidths. The reason for the remaining discrepancy is discussed in detail. The origin for the non-temperature-dependent line broadening is suggested to be due to an unresolved, strong shake-up satellite. The main peak is due to a screening with an electron from the nickel sp band whereas the origin of the satellite line is a screening from the nickel 3d band. These two peaks are close to each other even though the chemisorption bonds are almost without 3d contributions.

New method of calculation of the surface enthalpy of solids

A new method of calculation of the surface enthalpy of solids E c in the so-called rigid lattice approximation is proposed. The formerly found [J. Solid State Chem. 57 (1985) 269, 291] empirical relationship between bond length and bond energy in oxide crystals is used for this purpose. It is sufficient to add the energies of all bonds cut in the process of surface formation over the surface unit cell and to divide the sum by double the unit cell area. The advantage of this widely applicable method consists in the fact that it is sufficient to know the structure of the considered crystal; for oxides all coefficients of the proposed equation have already been determined and listed in the quoted paper. Generalization of the method on the non-oxide (X) crystals is dependent on the determination of the respective coefficients for the element-X bonds. But at present the method is also applicable to non-oxide crystals of high symmetry (e.g. NaCl-type, CsCl-type, ZnS-type, fluorite-type). It is argued that E c may be thought of as active surface energy responsible for various aspects of the surface reactivity and cleavage, while the passive surface energy of the relaxed surface governs static phenomena. As the passive surface energy (or free energy) is not easy to be obtained either experimentally or theoretically, an attempt is made to predict the equilibrium form of crystals with Curie-Wulff plots using E c instead of the free energy. In most cases the predictions agree well with accessible literature data. Data involving cleavage planes and direct experimental determinations of the surface energy are also used in the evaluation of the proposed method. The paper is illustrated with numerical data of E hkl c calculated for a number of oxides and halides of the NaCl-type structure as well as for oxides of the rutile-type, TiO 2-anatase, V 2O 5 and MoO 3 including the reduced faces of the two latter oxides.

Direct calculation of interfacial energetics: Roles of axial commensuration and strain in epitaxial growth.

Static energetics of monolayer overgrowth on a crystalline substrate is considered in detail by going beyond the rigid-lattice approximation. Conditions necessary to obtain a coherent interface between two dissimilar structures are carefully examined in terms of axial commensuration. Obtaining the strain and the strain energy for lattice-mismatched interfaces, we investigate: (i) the nucleation size for stable microcrystallite formation, and (ii) the difference in epitaxial growth between positive misfit and negative misfit. Using model potentials obtained by superposing suitable two-body interaction terms, we make a connection with semiconductor superlattice growth for zinc-blende-type systems.

Adsorption states of light atoms (H, D, He) on quantum crystals (H2, D2, He, Ne)

A study is made of the adsorption states of light atoms (H,D,He) on perfect quantum crystals (H2,D2,He,Ne). The quantum nature of the crystal is taken into account at the outset of the theory and the adsorption states are calculated as the bound states of a single atom in the potential created by the semi-infinite solid (rigid lattice approximation). The three-dimensional Schrödinger equation is solved to obtain the binding energies of the various hydrogen and helium isotopes on the (111) and (100) faces of the following quantum crystals: H2, D2, He, Ne. The major sources of uncertainty in the results are discussed in the paper: by increasing importance they are the neglect of many body interactions, the neglect of the coupling of the adatom with the surface, and finally the uncertainties in most of the pair potentials. A good agreement is obtained with available experimental results.

Cluster model studies of oxygen intercalated graphite

Electronic structure calculations were carried out on clusters of atoms modeling oxygen intercalation of graphite using the modified intermediate neglect of differential overlap (MINDO) molecular orbital method. The graphite models consisted of both a full three dimensional representation as well as an independent layer model of the lattice. Four different positions for the oxygen atom were considered and consisted of 1. (1) an oxygen directly above a carbon of one layer and directly below that of the second (site A); 2. (2) an oxygen above a carbon of one layer and below the center of a hexagonal ring of the second layer (site B); 3. (3) an oxygen above a carbon-carbon bond and below a position midway between second nearest neighbors of the hexagonal ring (site C); 4. (4) an oxygen above and below a point midway between second nearest neighbors of the hexagonal rings in both layers (site D). The oxygen-graphite layer distances in the cluster models were optimized within a rigid lattice approximation resulting in the bond site being the most energetically favored position for the oxygen atom at a distance of 1.25 Å above the carbon-carbon bond. Additionally, oxygen interaction with the carbons of only one graphite plane predominated not only for the most favored position but also for all four sites considered, with the three dimensional results differing little from those obtained using the independent layer model for the various sites.

A Born Model Calculation of the Energies of Schottky Pair Formation and Cation Migration in Potassium Azide

AbstractCalculations are presented for the energy of Schottky pair formation and for the activation energy of cation vacancy migration in potassium azide (KN3). Parameters for the repulsive overlap energy of the azide molecule ion have been derived from a Born model treatment of the lattice interactions with the cohesive energy and its first and second derivatives used in the fitting procedure. Three different representations of the repulsive potential are considered. Point defect calculations are performed first in a rigid lattice approximation and then with the effects of polarization and relaxation included. The polarization energy is calculated by a generalized Mott and Littleton method, which accounts for the anisotropy of the lattice, in both zeroth and first order approximations. Relaxations include displacements and rotations of nearest neighbors of the defect configurations. The sensitivity of the results t o the choice of a repulsive potential are discussed.

On the energy of {100} coincidence twist boundaries in MgO

The energy of coincidence twist boundaries formed by rotation about a [100] axis in MgO crystals has been calculated using the Born-Mayer potential and rigid lattice approximation. It is found that the four prominent coincidence twist boundaries at 16.26, 22.62, 28.07, and 36.87° have local energy minima. The coincidence boundary formed by a rotation of 28.07° has a positive contribution to energy from the electrostatic term and therefore has the shallowest minima. This is in agreement with experimental observations on crystals of MgO smoke.

Twist grain boundaries in alkali halides

Abstract The surfaces of single crystals of sodium chloride and potassium bromide have been studied, by a decoration technique, in electron microscopy. Features on the decoration figures observed are interpreted as twist grain boundaries at (110) planes. Under most reasonable assumptions concerning the Burgers vectors of the dislocations forming the boundary the angles of twist are calculated and they always have the same values: 17′, 41′ or 50′. These rotation angles of the boundary are explained ä crystal positions of minimum energy. This has been done by using a rigid lattice approximation taking into account only the Coulomb interaction between some ions belonging to a cell of the superlattice created by the dislocation network associated with the grain boundary.

The theory of defects in irradiated semiconductors

Developments in the theory of defects in semiconductors rire reviewed, with emphasis on changes since the Santa Fe meeting and on the points which were controversial then. It has become clear that most of the desirable simplifications are invalid for vacancies in diamond and silicon, and probably also for similar systems. In particular one cannot use the one-electron approximation, the rigid lattice approximation, the Jahn-Teller instability with coupling within isolated levels alone, nor some of the conceptually acceptable treatments of the lattice distortion. The successes and weaknesses of the various approaches are reviewed, and some of the potential methods are described.

Grain Boundary Energies in NaCl in the Rigid Lattice Approximation

Grain boundary energies are computed for twist boundaries in NaCl with the assumption that there is no relaxation of the lattice at the boundary. The energy is 573 ergs/cm2 and is independent of angle over the range studied. Comparison is made with recent experimental results for twist boundaries in KCl.