Lattice constants parameters

Table 2 Lattice parameter ao (in A) and elastic constants B, Cn, C12, and C44 for CoSi2 (in GPa), calculated using all-electron (FLAPW) and pseudopotential (VASP) techniques in the LDA and using GGC corrections.

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

At high temperatures, the module of the pyroelectric constants of both compounds increases more significantly and reaches an extremum at about 480K. Fig. 113 shows the temperature dependence of the pyroelectric coefficients. This phenomenon could be related to the change in dilatation mechanism that was observed while investigating the temperature dependence of the lattice parameters (see Fig. 102). 

Fichtner-Schmittler, H., Lohse, U., Engelhardt, G. et al. (1984) Unit-cell constants of zeolites stabilized by dealumination - determination of al-content from lattice-parameters, Cryst. Res. Technol., 19, Kl. 

Figure 8.8 One-dimensional diatomic chain with lattice parameter 2a and force constant K.

It has been noted that the conductivity and activation energy can be correlated with the ionic radius of the dopant ions, with a minimum in activation energy occurring for those dopants whose radius most closely matches that of Ce4+. Kilner et al. [83] suggested that it would be more appropriate to evaluate the relative ion mismatch of dopant and host by comparing the cubic lattice parameter of the relevant rare-earth oxide. Kim [84] extended this approach by a systematic analysis of the effect of dopant ionic radius upon the relevant host lattice and gave the following empirical relation between the lattice constant of doped-ceria solid solutions and the ionic radius of the dopants. 

Part of the alkali metal trend may be due to changes in lattice parameter. The lattice parameter for PTA is related to the cationic species separating the Keggin ions. For anhydrous PTA, the lattice constant is 10.8 A. while for Csj-PTA it expands to 13.7 A.10 For the hexahydrate, the lattice parameter is intermediate at 12.5 A.2... 

Figure 6.5 The approximation of a continuously varying lattice parameter into a series of lamellae of constant lattice parameter...

Such an approximation of periodicity was made for the calculations discussed in the next section (section 4). The supercells for these calculations were composed of either 12 or 32 primitive Li M02 unit cells (M = 3d TM ion 0 < X < 1) that contained various M defects. The lattice parameters of the supercells were kept constant at the parameters for the undetected structure, while the ionic coordinates were allowed to relax. A 2 x 2 x 2 Tr-point mesh was used for the calculations on the 12-unit supercells and a 1x1x1 Tr-point mesh for the 32-unit supercells. The primitive LiJV[02 unit cells used to construct the super cells had previously been calculated with full relaxation of lattice parameters as well as ionic coordinates. 

Direct experimental determinations of these quantities do not exist. The nearest approach seems to be in some observations made by Nicolson (26) in his work on surface tension. He found that when he made magnesium oxide particles by burning magnesium in air, their lattice constants were the same as those of the bulk material. When the crystals were made by the decomposition of magnesium carbonate in vacuo, the expected change in lattice parameter took place due to the surface tension. These negative results obtained in the first method of preparation were attributed to the presence of gases adsorbed from the air. 

Identification of unknown crystal structures and determination of phase fields by X-rays can be problematical if the characteristic patterns of the various phases are quite similar, for example in some b.c.c. A2-based ordered phases in noble-metal-based alloys. However, in many cases the characteristic patterns of the phases can be quite different and, even if the exact structure is not known, phase fields can still be well established. Exact determination of phase boundaries is possible using lattice-parameter determination and this is a well-established method for identifying solvus lines for terminal solid solutions. The technique simply requires that the lattice parameter of the phase is measured as a function of composition across the phase boimdary. The lattice parameter varies across the single-phase field but in the two-phase field becomes constant. Figure 4.12 shows such a phase-boundary determination for the HfC(i i) phase where results at various temperatures were used to define the phase boundary as a fimction of temperature (Rudy 1969). As can be seen, the position of is defined exactly and the method can be used to identify phase fields across the whole composition range. 

The calculations above allowed the positions of atoms to change within a supercell while holding the size and shape of the supercell constant. But in the calculations we introduced in Chapter 2, we varied the size of the supercell to determine the lattice constant of several bulk solids. Hopefully you can see that the numerical optimization methods that allow us to optimize atomic positions can also be extended to optimize the size of a supercell. We will not delve into the details of these calculations—you should read the documentation of the DFT package you are using to find out how to use your package to do these types of calculations accurately. Instead, we will give an example. In Chapter 2 we attempted to find the lattice constant of Cu in the hep crystal structure by doing individual calculations for many different values of the lattice parameters a and c (you should look back at Fig. 2.4). A much easier way to tackle this task is to create an initial supercell of hep Cu with plausible values of a and c and to optimize the supercell volume and shape to minimize... 

These experimental facts indicate that this system can be an excellent model of band-filling control, and will be a good candidate for making a superlattice composed of Mott insulator/superconductor hetero-junctions. It should be emphasized that their lattice parameters are nearly kept constant through such an anion modification, which is the most essential feature for achieving the successful tuning of Tc in an organic superconductor. 

In a similar vein it was later shown that all the internal parameters [x for M(2) and x,y,z for 0] could be determined by maximising the volume subject to the constraint of all four M-0 distances being kept equal and constant. Finally, taking the M-O distance from tables , the lattice parameter could be calculated. Table 5 lists parameters for the real, the maximum volume, and the ideal [metal atoms as in CusAu, O centering the... 

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