Elastic constants first-principles calculation

In table 2 and 3 we present our results for the elastic constants and bulk moduli of the above metals and compare with experiment and first-principles calculations. The elastic constants are calculated by imposing an external strain on the crystal, relaxing any internal parameters (case of hep crystals) to obtain the energy as a function of the strain[8]. These calculations are also an output of onr TB approach, and especially for the hep materials, they would be very costly to be performed from first-principles. For the cubic materials the elastic constants are consistent with the LAPW values and are to within 1.5% of experiment. This is the accepted standard of comparison between first-principles calculations and experiment. An exception is Sr which has a very soft lattice and the accurate determination of elastic constants is problematic. For the hep materials our results are less accurate and specifically in Zr the is seriously underestimated. ... 

If an accurate equation for the interatomic potential is known, the elastic constants can be calculated from first principles. This analysis is straightforward for cubic ionic crystals. The potential for a pair of positive and negative ions is often written in the form... 

The elastic constants of iron have been studied experimentally and theoretically at low temperature and high pressure (Mao et al. 1998 Soderlind et al. 1996 Steinle-Neumann et al. 1999 Stixrude and Cohen 1995), but there has not yet been a first principles calculation of the full elastic constant tensor at inner core conditions (see Nye 1985 for a review of elastic constants). Laio et al. (2000) developed a clever hybrid method that combines first principles total energy and force calculations for a limited number of time steps with a semi-empirical potential fit to the first principles results. These authors investigated a number of properties with their ab initio method including... 

Table 2 Elastic constants and bulk moduli for 4d cubic elements. Comparison is made between the results of our tight-binding parametrization (TB), first-principles full potential LAP., results (LAPW), where available, and experiment (Exp.). Calculations were performed at the experimental volume.

Computer simulations therefore have several inter-related objectives. In the long term one would hope that molecular level simulations of structure and bonding in liquid crystal systems would become sufficiently predictive so as to remove the need for costly and time-consuming synthesis of many compounds in order to optimise certain properties. In this way, predictive simulations would become a routine tool in the design of new materials. Predictive, in this sense, refers to calculations without reference to experimental results. Such calculations are said to be from first principles or ab initio. As a step toward this goal, simulations of properties at the molecular level can be used to parametrise interaction potentials for use in the study of phase behaviour and condensed phase properties such as elastic constants, viscosities, molecular diffusion and reorientational motion with maximum specificity to real systems. Another role of ab initio computer simulation lies in its interaction... 

The elastic constants are of importance in theory of solid state. The point is that their values can be calculated from first principles or from other models of solids and be compared with the experimental data. 

The calculated binding energies in oxides (rows from 11 to 13 of Table 9.1) also agree weU with experimental data. The first-principle simulation correctly predicts the greatest cohesive energy for CaO and similar values for MgO and SrO. Precise measurements of the bulk moduli for oxides is difficult. The reported values are in agreement with elastic constants measured by ultrasonic techniques. 

Table 10.2 The parameters of unit cells, bulk moduli, and elastic constants of NisAI-based solid solutions. The data were calculated from first principles. Values of B, Cn, Cu, and C44 in GPa.

Laminate anafysis is based ui>on the principle of strain compatibility, which means that the in-plane strains in neighbouring layers must be equal. The stresses required in each layer to produce a given strain are first calculated from the previously computed elastic constants of the layer for the appropriate angle to the fibre direction, and then integrated over all layers. The results define the elastic constants of the whole laminate. Simple computer programs take the drudgery out of these calculations. 

Elastic constants measured as a function of temperature are available for most of the lanthanides in polycrystalline form (Rosen, 1967, 1968) and for Tb, Dy, Ho and Er single crystals (Palmer, 1970 Palmer and Lee, 1973 and du Plessis, 1976). For a summary of the elastic properties of the lanthanides reference can be made to Taylor and Darby (1972, section 2.4) and to ch. 8, section 9. If a suitable lattice dynamical model were devised, we should be able to calculate Cl from first principles. This was done for Gd, Dy and Er metals (Sundstrom, 1968), but at the time of these calculations, elastic constants were available only for polycrystalline samples at a few fixed temperatures. Nevertheless the results obtained did indicate that Lounasmaa s (1964a) interpolation idea was reasonable. With the elastic constant data available today it should be possible to calculate Cl for the entire region of interest, although this appears not to have attracted much attention, presumably because the uncertainty involved in separating off the contributions in experimental heat capacity results makes comparison with theory unrewarding as far as Cl is concerned. 

Figure 10. Elastic constants of hep iron at a density of 13 Mg m, typical of that in the Earth s iimer core, from first principles particle in a cell method calculations. There are five independent elastic constants in a hexagonal material. Results for C6 l2 cii-ci2) are shown for comparison with the other shear modulus C44.

The recent study of CoSi2 by Stadler et al. illustrates the current capabilities to calculate the cohesive, elastic, and dynamical properties of a material by first-principles pseudopotential and all-electron techniques based on DFT. On the LDA level, the lattice constant is slightly smaller than experiment, namely 5.292 A (FLAPW), 5.283 A (pseudopotential plane wave) and 5.365 A (experiment). On the GGA level, the calculated lattice constant is closer to experiment, namely 5.350 A. The difference in the calculated bulk modulus between LDA (168.4 GPa) and GGA (169.0 GPa) is smaller than that between the LDA FLAPW (171.6 GPa) and LDA pseudopotential plane wave (168.4 GPa) calculations. All results are in excellent agreement with the experimental value of 171.5 3.4-GPa. 

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