Equivalent Crystal Theory Research Articles

Surface energy calculation of hcp metals using the analytical equivalent crystal theory

We have used the newly formulated analytical equivalent crystal theory method which is a modification of the ECT method to calculate the surface energies for the (001) surface of 20 hcp metals. Our calculated AECT surface energies show excellent agreement with the ECT. The data may be used as a consistent starting point for models of surface science phenomena. The accuracy of the surface energy results is established from a comparison with other theoretical results and experiment.

Surface energy calculation of bcc metals using the analytical equivalent crystal theory method

We have used the newly formulated analytical equivalent crystal theory method which is a modification of the ECT method to calculate the surface energies for three low-index surfaces of bcc alkali metals Li, Na, K, Rb, Cs and the bcc transition metals V, Nb, Ta, Cr, Mo, W and Fe. The data may be used as a starting point for models of surface science phenomena. The accuracy of the surface energies is established from a comparison with other theoretical results and experiment. The surface energy of each (hkl) plane in alkali metals is much lower than that in transition metals. For all bcc metals, the order of the three low-index surface energies is σ110<σ111<σ100 in good agreement with the MEAM of Baskes [Phys. Rev. B 46 (1992) 2727] except for Cr and Mo where the ordering obtained in this work is different from that of Baskes. The calculated results show that the surface energy is anisotropic and that the closed-packed plane (110) is the lowest of the three low-index surfaces.

Interfacial Phase Stability in TiV Multilaminate Thin Films

The influence of the interfacial energy on the material phase stability is investigated for a series of TiV multilaminate thin films. Experiments reveal that at a higher layer thickness, the α (hcp) phase is the most stable. As the layer thickness is reduced, a transformation from the α (hcp) phase to the β (bcc) phase occurs. Atomic-scale characterization of the transformed specimen by atom probe tomography reveals V interfacial diffusion between the layers. Equivalent crystal theory based calculations confirm the V interfacial diffusion mechanism. The predicted segregation profiles match those obtained experimentally.

Surface energies of hcp metals using equivalent crystal theory

Abstract The equivalent crystal theory method of Smith et al. [Phys. Rev. B 44 (1991) 6444] originally formulated for fcc and bcc metals, and semiconductors, is here extended to hcp metals and applied to calculate surface energies. The (0 0 1) surface energies obtained for 22 hcp metals are in good agreement with the results of both experiment and ab initio calculations.

Energetics of free pure metallic nanoclusters with different motifs by equivalent crystal theory

The energy of metallic Ni, Cu, Pd, Ag, Pt, and Au nanoparticles up to 5000 atoms are studied by equivalent crystal theory ECT, a quantum approximate method QAM that describes the ground state structure and the surface properties of metals and semiconductors at zero temperature. ECT relies on the universal binding energy relation to predict with precision and speed the energy of a crystal in a specific configuration. For each pure metallic nanoparticle of each chosen motif icosahedron, octahedron, and decahedron, the energy variation with the number of atoms Nat is studied. Crossover and minimum energy values are calculated and/or estimated and compared with the results obtained by molecular dynamics MD. Our results confirm the qualitative behavior i.e., icosahedron shapes are less energetic for small sizes, decahedron for medium sizes, and octahedron for bigger sizes predicted by MD, but the calculated crossover and minimum energy values are, in general, larger for all metals and geometries examined. Also, we studied the trends in relaxation between layers and the behavior of the average radius Rav of each relaxed nanoparticle as Nat was increased. For each motif, the most stable structures i.e., with the best truncation follow a simple law of Rav in terms of Nat. This simple law is unchanged for the four different motifs and can be extended for all six metals after a simple parametrization is performed.

Analytical algorithm for Equivalent Crystal Theory

Equivalent crystal theory (ECT) is a semi-empirical technique used for the calculation of defect energetics in metals and semiconductors. The implementation of the method involves the solution of transcendental equations. Although this is not a problem for simple defects, for complex defects, Monte Carlo and molecular dynamics calculations in large systems, it could be the speed-determining limitation in a calculation. In this paper we propose a procedure for bypassing this step and obtaining the desired result directly. The form of the particular transcendental equations suggest a different approach, namely, the equation can be cast in the form of the Lambert function that can be readily evaluated from standard routines. We test this scheme by evaluating the surface energies for a variety of metallic elements and by standard numerical approaches, and demonstrate that they agree to within a few ppm.

Gradient equivalent crystal theory

This paper presents an extension of the formalism of equivalent crystal theory (ECT) byintroducing an electron density gradient term so that the total model density becomes amore accurate representation of the real local density. Specifically, we allow for the electrondensity around a lattice site to have directionality, in addition to an average value, asassumed in ECT. We propose that an atom senses its neighbouring density asa weighted sum—the weights given by the its own electronic probability. As abenchmark, the method is used to compute vacancy migration energy curves ofiron. These energies are in good agreement with previously published results.

Surface relaxation and surface energy of face–centered cubic metals

Surface relaxation and surface energy are calculated for Rh, Ir, Pb, Ca, Sr, and Th, for three low-index surfaces, using the equivalent crystal theory (ECT) method. The surfaces are the (100), (110) and (111) faces. In this study, we only considered perpendicular relaxation where the relaxation was determined by the minimization of the total energy. The results obtained are analyzed with good agreement to experiment and first – principles calculations where available. Journal of Applied Sciences and Environmental Management Vol. 10(1) 2006: 37-42

Generalization of equivalent crystal theory to include angular dependence

The original equivalent crystal theory is a semi-empirical method for calculating the configurational energy of atomic systems. Each atomic site in the real crystal with defects is assigned an equivalent lattice constant, in general different from the ground state value. This parameter corresponds to a local compression or expansion of the perfect lattice. The basic method considers these volumetric transformations and, in addition, introduces the possibility that the reference lattice is anisotropically distorted. These distortions however, were introduced ad-hoc. In this work, we generalize the original Equivalent Crystal Theory by introducing site-dependent directional distortions of the lattice which account for the dependence of the energy on anisotropic local density variations. This is done in the spirit of the original framework of ECT, but includes a gradient in the density. This approach is introduced to ECT and may apply to other semi-empirical methods by making use of readily available first-principles results to fix parameters. We develop here the basic framework, and apply it to the calculation of Fe(1 1 0) and Fe(1 1 1) surface energy formation, and to the vacancy energy curve. The results, compared with first-principles calculations, show an improvement over previous semi-empirical approaches.

Force-Strain Curves Of Microcapsules With Atomic Force Microscopy

ABSTRACTWe develop an algorithm to measure elastic properties of microcapsules with Atomic Force Microscopy (AFM). The AFM is used as an indenter and presses down on a spherical microcapsule. We study the system from an atomic point of view (considering interactions between the atoms in the system via Equivalent Crystal Theory) and calculate the force produced by the system to balance the external AFM force. We plot this force as a function of the indentation depth, and from that curve we extract the interatomic parameters of ECT that are related with elastic constants. Our calculations model measurements of force-strain curves including non-linear effects. This is relevant as classical elasticity theory breaks down in the AFM indentation regime, when atomic interactions must be considered explicitly.

DETERMINATION OF THE GROUND-STATE GEOMETRIES OF COPPER CLUSTERS BY SIMULATED ANNEALING WITHIN AN EQUIVALENT CRYSTAL APPROACH

We determine the lowest energy structures of small (11–20 atoms) copper clusters. The semi-empirical Equivalent Crystal Theory (ECT) is used in conjunction with the Metropolis Monte Carlo algorithm to determine the equilibrium geometry of each cluster via simulated annealing. The optimum structures of the clusters in this size range are found to be derived from icosahedral structures. The 13-atom cluster is an icosahedron and the 19-atom is a double-icosahedron. The other sizes show structures related to these. The 11-atom clusters, however, show somewhat different structures. We propose the ECT as an efficient tool for developing starting structures for more chemically accurate, first principles and therefore computationally very demanding, approaches.

Determination of the Ground State Geometries of Copper Clusters by Simulated Annealing

We determine the lowest energy structures of small (10–20 atoms) copper clusters. The semi-empirical Equivalent Crystal Theory (ECT) is used in conjunction with the Metropolis Monte Carlo algorithm to determine the equilibrium geometry of each cluster via simulated annealing. The optimum structures of the clusters in this size range are found to be derived from icosahedral structures. The 13-atom cluster is an icosahedron and the 19-atom one a double-icosahedron. The other sizes show structures related to these. The 10- and 11-atom clusters, however, show somewhat different structures. We report the variation of binding energy, as obtained from ECT, with cluster size.

A Molecular Full-Potential LMTO Calculation for Copper Clusters

We study the electronic properties of small (10–20 atoms) copper clusters using the newly-developed molecular full-potential linearized muffin-tin orbital two-centre-fit (TCF) method of Methfessel and van Schilfgaarde. The geometric structures of the clusters had earlier been determined by us through simulated annealing using the Equivalent Crystal Theory to compute total energies. We report the variation of the binding energy, as obtained from the TCF calculations, with cluster size and compare these to the binding energies determined, for the same structures, from the ECT. We also show the variation of the HOMO-LUMO gap with cluster size, and the pseudo-density of states for select cluster sizes.

Calculations on Ni Clusters: An Equivalent Crystal Theory and LCAO Approach

The structure and electronic properties of small nickel clusters are studied using Equivalent Crystal Theory (ECT) and a parametrised LCAO method. The minimum energy structure is obtained using ECT with Monte Carlo techniques. The LCAO method is then used to study the electronic properties of the optimised clusters.

Avalanche in Adhesion at Metal Interfaces

Simulations have shown that as two metal surfaces approach each other, the surface layers can avalanche together when the rigid interfacial spacing falls below a critical distance. This is accompanied by a discontinuous decrease in the adhesive energy. Here we present an examination of this phenomenon for the bcc metals Fe and W using the Equivalent Crystal Theory. In order to identify the circumstances under which avalanche might be inhibited, the effect of loss of registry between the two surfaces is investigated in detail. The avalanche is inhibited when the two surfaces are sufficiently far out of registry and when only a few layers near the surface are allowed to relax. As the relaxing slabs get thicker a sharp avalanche reappears. However, as the loss of registry increases the energy released in the avalanche decreases.

The relaxation of a reconstructed metallic surface and its influence on the EFG acting on a substitutional PAC probe

The structure of the 1×2 reconstructed Au(110) surface, calculated using equivalent crystal theory, is studied. We perform point-charge calculations to estimate the electric field gradient (EFG) acting on a substitutional111In probe deposited in that surface. The relaxation breaks the local symmetry producing an asymmetric EFG but the calculated asymmetry is lower than the experimental one. Comparing the computed orientations of the principal axes of the EFG to the experimental values we conclude that In probes prefer to substitute gold into the high coordinated sites at the bottom of the channels.

A SEMI-EMPIRICAL STUDY OF SMALL COPPER CLUSTERS

We study the structure and electronic properties of small copper clusters using Equivalent Crystal Theory (ECT) and a parametrised LCAO method. The cluster geometry is optimised using ECT with Monte Carlo techniques. The LCAO method, particularly the parametrisation scheme due to Harrison, is then used to study the electronic properties of the optimised clusters.

Simulation of tip - sample interaction in the atomic force microscope

Recent simulations of the interaction between planar surfaces and model atomic force microscope (AFM) tips have suggested that there are conditions under which the tip may become unstable and `avalanche' toward the sample surface. Here we investigate via computer simulation the stability of a variety of model AFM tip configurations with respect to the avalanche transition for a number of fcc metals. We perform Monte Carlo simulations at room temperature using the equivalent-crystal theory (ECT) of Smith and Banerjea. Results are compared with recent experimental results as well as with our earlier work on the avalanche of parallel planar surfaces. Our results on a model single-atom tip are in excellent agreement with recent experiments on tunnelling through mechanically controlled break junctions.

Modelling of surfaces. I. Monatomic metallic surfaces using equivalent crystal theory

We present a detailed description of equivalent crystal theory, focusing on its application to the study of surface structure. While the emphasis is on the structure of the algorithm and its computational aspects, we also present a comprehensive discussion on the calculation of surface energies of metallic systems with equivalent crystal theory and other approaches. Our results are compared to experiment and other semi-empirical as well as first-principles calculations for a variety of fcc and bec metals.

Multilayer relaxation and surface energies of metallic surfaces

The perpendicular and parallel multilayer relaxations of fee (210) surfaces are studied using equivalent crystal theory (ECT). A comparison with experimental and theoretical results is made for Al(210). The effect of uncertainties in the input parameters on the magnitudes and ordering of surface relaxations for this semiempirical method is estimated. A new measure of surface roughness is proposed. Predictions for the multilayer relaxations and surface energies of the (210) face of Cu and Ni are also included.