Lattice elastic properties

U. Piesbergen, Heat Capacity and Debye Temperatures G.Giesecke, Lattice Constants J.R. Drabble, Elastic Properties... 

The obtained Ao gi = 5.7 x 10 is even larger than the value of Acr (Cu) X (= 4.7 X 10 A ), and of the hypothetical Co—Cu crystal with intermediate elastic properties than bulk cobalt and copper (4.1 x lO" A ). The derived effect of the effect of the lower coordination of the surface atoms on the mean-square relative displacement (perpendicular vs. parallel motions) is 1.4 times larger amplitude of the perpendicular vs. parallel motions, in agreement with lattice dynamics calculations. This SEXAFS study has produced a measure of the surface effect on the atomic vibrations. This has been possible due to the absence of surface or adsorbate reconstruction (i.e. no changes in bond orientations with respect to the bulk) and of intermixing. 

Despite a possible wide significance of such a topic, there is only one reported study of adatom-substitutional impurity atom interaction, where the interaction of a W adatom with substitutional Re atoms in a W lattice is studied by using a W-3% Re alloy as the substrate.182 The planes used in FIM studies of adatom behavior are usually quite small containing only a few hundred atoms. Thus a plane of a W-3% Re alloy is likely to contain a few Re substitutional atoms. The perturbation to the overall electronic and elastic properties of the substrate lattice should still be relatively small. Therefore the interaction of a single substitutional impurity atom with a diffusing adatom can be investigated. 

Other flexible framework calculations of methane diffusion in silicalite have been performed by Catlow et al. (64, 66). A more rigorous potential was used to simulate the motion of the zeolite lattice, developed by Vessal et al. (78), whose parameters were derived by fitting to reproduce the static structural and elastic properties of a-quartz. The guest molecule interactions were taken from the work of Kiselev et al. (79), with methane treated as a flexible polyatomic molecule. Concentrations of 1 and 2 methane molecules per 2 unit cells were considered. Simulations were done with a time step of 1 fs and ran for 120 ps. 

The human erythrocyte possesses a characteristic biconcave shape and remarkable viscoelastic properties. Electron microscopy studies performed on red blood cells (RBC), ghosts, and skeletons revealed a two-dimensional lattice of cytoskeletal proteins. This meshwork of proteins was thought to determine the elastic properties of the RBC. This... 

The simplest defect in a semiconductor is a substitutional impurity, such as was discussed in Section 6-E. There are also structural defects even in pure materials, such as vacant lattice sites, interstitial atoms, stacking faults (which were introduced at the end of Section 3-A) and dislocations (see, for example, Kittel, 1971, p. 669). They are always in small concentration but can be important in modifying conduction properties (doping is an example of this) or elastic properties (dislocations arc an example of this). 

Having developed and parameterized potential models, the final stage before their use in a simulation study should be their evaluation. Nonempirically derived potentials should be evaluated by reference to their ability to predict empirical crystal properties. For empirical potentials, it is clearly necessary to use data outside the range employed in the parameterization. We have already referred to the use of lattice dynamical data. Comparison with the results of high-pressure studies, in particular the variation of structural and elastic properties with pressure, is also of great value and... 

K. Barla, R. Herino, G. Bomchil, and J. C. Pfister, Determination of lattice parameter and elastic properties of porous silicon by X-ray diffraction, J. Cryst. Growth 68, 727, 1984. 

Conversion from a purely geometrical description, in terms of lattice parameters, to a thermodynamic description, in terms of strain, leads through to the elastic properties. Whereas there are up to six independent strains, e, (i = 1-6), for a crystal, there are up to 21 independent elastic constants, C (/, k = 1-6). Add to this the fact that strain states represent minima in a free energy surface, given by dGIdct = 0, while elastic constants... 

Figure 3. Spontaneous strains and elastic properties at the 422 < i> 222 transition in Te02. (a) Spontaneous strain data extracted from the lattice-parameter data of Worlton and Beyerlein (1975). The linear pressure dependence of (e - (filled circles) is consistent with second-order character for the transition. Other data are for non-symmetiy-breaking strains (e + 62) (open circles), 63 (crosses), (b) Variation of the symmetry-adapted elastic constant (Cn - Cu) at room temperature (after Peercy et al. 1975). The ratio of slopes above and below Po is 3 1 and deviates from 2 1 due to the contribution of the non-symmetry-breaking strains. (After Carpenter and Salje 1998).

In this work, we describe the results of theoretical studies of Poisson s ratio in disordered structures composed of two phases of disparate elastic properties applying a renormalization group approach to a model of percolation on a hierarchical cubic lattice. Although this approach has been described in detail elsewhere [160], we present it briefly here, for completeness. At the percolation... 

The results of calculations of the effective Poisson s ratio vp dependence on the bulk concentration of a rigid phase p at various values of a = log i/C/Au) are shown in Fig. 53. The calculations were made for Poisson s ratios of the phases ranging from 0.1 to 0.4. It can be seen that at percolation threshold Poisson s ratio of the isotropic fractal composite is vp = 0.2, when K jK > 0 it is also independent of the Poisson s ratios of the individual components of the composite. The Poisson s ratio obtained by us near the percolation threshold is in agreement with computer simulation results and the conjecture of Arbabi and Sahimi [161]. It has been shown that an approximate theoretical treatment of percolation on a cubic lattice exactly reproduces the Poisson s ratio obtained in computer simulation at the percolation threshold. This result may encourage one to use this approximation to describe various elastic properties of composites. It is worth noting that some critical indices have been calculated recently with a high degree of accuracy in the context of the present model. 

The considerations presented in this work suggest that at the percolation threshold the ratio of the shear modulus to the bulk modulus is a universal quantity, which does not depend on the elastic properties of the percolation phase. It is well known, however, that this ratio depends on the coordination number of the lattice on which the percolation takes place. Taking this into account, we conjecture that when the coordination number, Z, of the underlying lattice is more than four times larger than the dimension of the lattice, Z > Ad, Poisson s ratio near the percolation threshold should be negative, irrespective of the value of Poisson s ratio of the percolation phase. 

Timgsten has been of keen theoretical interest for electron band-structure calculations [1.14-1.25], not only because of its important technical use but also because it exhibits many interesting properties. Density functional theory [1.11], based on the at initio (nonempirical) principle, was used to determine the electronic part of the total energy of the metal and its cohesive energy on a strict quantitative level. It provides information on structural and elastic properties of the metal, such as the lattice parameter, the equilibrium volume, the bulk modulus, and the elastic constants. Investigations have been performed for both the stable (bcc) as well as hypothetical lattice configurations (fee, hep, tetragonal distortion). 

Ab Initio Calculations of Structural and Elastic Properties. Equilibrium lattice constants, equilibrium volumes, as well as bulk and shear moduli can be assessed based on ab initio electron-structure calculations. They are obtained from the calculated total energies as a fimetion of volume in the bcc or fee crystal structure and from respective volume-conserving distortions of the lattice. In most cases, they agree well with experiments (Table 1.6). 

Our starting point for the analysis of the thermal and elastic properties of crystals is an approximation. We begin with the assertion that the motions of the i atom, will be captured kinematically through a displacement field Uj which is a measure of the deviations from the atomic positions of the perfect lattice. It is presumed that this displacement is small in comparison with the lattice parameter, uj < C qq. Though within the context of the true total energy surface of our crystal (i.e. R2,. .., Rw)) this approximation is unnecessary, we will see... 

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