Cluster boundaries

The dynamic of the cluster boundaries explains also the strong dielectric response under small ac fields. The naturally diffuse behavior (temperature stability) with high dielectric... 

The cluster model approach assumes that a limited number of atoms can be used to represent the catalyst active site. Ideally, one would like to include a few thousands atoms in the model so that the cluster boundary is sufficiently far from the cluster active site thus ensuring that edge effects are of minor importance and can be neglected. Unfortunately, the computational effort of an ab initio calculation grows quite rapidly with the number of atoms treated quantum mechanically and cluster models used in practice contain 20 to 50 atoms only. It is well possible that with the advent of the N-scaling methods " this number can dramatically increase. Likewise, the use of hybrid methods able to decompose a very large system in two subsets that are then treated at different level of accuracy, or define a quantum mechanical and a classical part, are also very promising. However, its practical implementation for metallic systems remains still indeterminate. 

A clear definition of cluster boundaries, which works successfully at least in numerical simulations, was proposed from the Riemannian geometrization method of Hamiltonian dynamics. The idea was the following In the inside of the cluster the sectional curvature should be positive because the stability of... 

Figure 19. The image potential VjM(r) [Eq. (88)] for the motion of the electron bubble in an ( He)jy cluster with N = 1.88 x 10 and R = 127 A (upper panel). The lower panel shows the exponential dependence of the electron tunneling times [Eq. (87)] near the cluster boundary versus the distance r = R — of the centers of the bubble and of the cluster. The t data calculated from Eqs. (80) and (86) are presented in the range x = 10 -10 s.

For the motion of an electron bubble within the superfluid cluster the static, harmonic approximation (Section IV.E) breaks down. Consider an electron bubble initially located at the radial distance d from the cluster boundary. The dynamic spatial distribution p/(r) of the electron bubble in the image potential VW [Eq. (88)] in the absence of dissipation can be described by the probability of the bubble location at distance d cluster surface, where the bubble moves back and forth from d up to d in the image potential. 

The mechanisms of plastic deformation and mechanical states of polyclusters are described in Sect. 6.9. The comparatively high density of cluster boundaries (experimental data point to the fact that cluster sizes are about 102a, a being the average interatomic distance), peculiarities of structure and displacement of dislocations under the action of stress determine the dominant deformation mechanisms in some region of temperature T and stress a. 

Fig. 6.4. FIM-image of the alloy Fe4KNi4()B>(l. Dense chains of light points indicate the outlets on the surface of cluster boundaries decorated with boron atoms [6.18J...

Maimed and Klein [6.20] found two-dimensional defects (probably cluster boundaries) in the quasi-crystal Al Mn. The sizes of clusters (quasi-crystallites) comprise 102u, and the boundary thickness equals approximately a. 

The low-energy excitations described above contribute essentially to the reversible relaxation processes [6.32], to the internal friction [6.33], and to the specific heat. The expressions for the excitation contributions ii)—iii) to the specific heat are given in [6.29, 30], where it is also shown that cooperative rearrangements iii) contribute greatly to the melting of cluster boundaries which is treated as the glass-liquid transition in polyclusters (Sect. 6.8). 

Cluster boundary decoration by metalloid atoms may occur due to small-atom locations in noncoincident sites. It is similar to the decorating of the Fe-Ni-B alloy boundaries with boron atoms (Fig. 6.2). 

A criticism sometimes heard about this approach is that cluster boundaries are arbitrary. Naturally, someone must decide where on a dendrogram one cluster ends and another begins. These decisions are based on non-computational judgements, as are decisions about whether a particular clustering method provides a result that an observer would have produced by visual inspection alone. In that context, we have found that by including in our calculations the slopes between data points, the clustering results improve - the results are more similar to what one would expect based on visual examination of the expression patterns (unpublished observations). 

In the framework of stellar evolution models, Si burning has often been approximated in different ways relying on the QSE cluster concept. These approximations are not fully appropriate, particularly in view of the time dependence of the QSE cluster boundaries. This has been stressed by [15] who adopt instead for the QSE and NSE regimes a detailed Si-burning network coupled to the stellar evolution equations. 

Cluster models consisting of just few T-atoms are crude models for a description of the zeolite crystal. There are al least three potential sources of errors inherent to cluster models (i) wavefunction perturbation due to the cluster boundary, (ii) structural constraints, and (iii) neglect of the long range interactions. 

Adsorption energies on metals calculated in a cluster approach often show considerable oscillations with size and shape of the cluster models because such (finite) clusters describe the surface electronic structure insufficiently [257-260]. These models may yield rather different results for the Pauli repulsion between adsorbate and substrate, depending on whether pertinent cluster orbitals localized at the adsorption site are occupied or empty. The discrete density of states is an inherent feature of clusters that may prevent a correct description of the polarizability of a metal surface and thus hinders cluster size convergence of adsorption energies [257]. Even embedding of metal clusters does not offer an easy way out of this dilemma [260,261]. Anyway, the form of conventional moderately large cluster models may be particularly crucial. Such models are inherently two-dimensional with substrate atoms from two or three crystal layers usually taken into accormt thus, a large fraction of atoms at the cluster boundaries lacks proper coordination. 

Pai and Doren s cluster calculation can also he compared with the slab calculations. The BP functional is similar to those used in the slab calculations. The cluster calculation value for of 60 kcal/mol (including ZPE) may be compared to 58 kcal/mol from the slab calculations with a 2 x 1 unit cell or 52 kcal/mol with a 2 x 2 unit cell. Apparently, the small unit cell and cluster calculations give similar values, but effects of surface relaxation and adjacent dimers that are included in the larger unit cell are neglected in the cluster calculation. As noted above, different functionals and basis sets can affect the results as much as the difference between slab and cluster boundary conditions. 

Clearly, by artificially breaking bonds on the cluster boundary, one creates artificially large and positive AIM FF values of the boundary oxygens (Figs. 13, 14), and therefore one also affects the FF pattern of the interior atoms. In order... 

The development of a cavity cluster from a distribution of supercritical cavitation nuclei at their exposure to tensile stress is discussed. An approach to this problem was presented by Hansson et al. [1], and is the basis of further analysis and comparison of planar and spherical cavity cluster development. The stress penetration into the cluster depends primarily on the inter—cavity distance and on the cluster form. In interplay with the cavity dynamics it determines an acoustic impedance of the cluster boundary which approaches zero during cavity growth, and so the tensile stress at the boundary resulting from the incident and the reflected waves becomes small which indicates that not only this pressure but also the equilibrium pressures of the cavities are important for the cluster development. 

The velocity of the liquid at the cluster boundary x = 0 connected to a pressure disturbance (p)x=o is found from (4)... 

The boundary condition (p)x=o can be interpreted as the result of the reflection of an incident acoustic wave at the cluster boundary which exhibits a nonlinear acoustic impedance connected to the cavity dynamics [6]. The reflection coefficient is... 

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