Unit cell volumes

The unit cell volume is given by the following general equation which is calculated from the mixed product of the three lattice vectors  

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In this equation is the 6x6 matrix of second derivatives (elements Ide j), emd are the corresponding 3N x 6 and 6 x,3N mixed coordinate/strain matrices, is the 3iV X 3N second-derivative coordinate matrix and V is the unit cell volume. It is the second term in Equation (5.54) that accounts for internal atomic relaxations as the cell distorts. 

Concerning the VDW parameters, the ability to directly apply previously optimized values makes convergence criteria unnecessary. If VDW parameter optimization is performed based on pure solvent or crystal simulations, then the heats of vaporization or sublimation should be within 2% of experimental values, and the calculated molecular or unit cell volumes should be also. If rare gas-model compound data are used, the references cited above should be referred to for a discussion of the convergence criteria. 

The symmetry of the structure we are looking for is imposed on the field 0(r) by building up the field inside a unit cubic cell of a smaller polyhedron, replicating it by reflections, translations, and rotations. Such a procedure not only guarantees that the field has the required symmetry but also enables substantial reduction of independent variables 0/ the function F (f)ij k )- For example, structures having the symmetry of the simple cubic phase are built of quadrirectangular tetrahedron replicated by reflection. The faces of the tetrahedron lie in the planes of mirror symmetry. The volume of the tetrahedron is 1 /48 of the unit cell volume. 

Figure 1. Volume dependence of the total en gy of rutile (R), anatase (A), brookite (B) and columbite (C) phases. Experimental values of the unit cell volume at ambient conditions are shown with arrows in the following (Bder C-R-B-A (experimental scatter for anatase is illustrated with a box).

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... 

The muffin-tin potential around each atom in the unit cell has been calculated in the framework of the Local-Spin-Density-Approximation using the ASW method. The ASW method uses the atomic sphere approximation (ASA), i.e. for each atom a sphere radius is chosen such that the sum of the volumes of all the overlapping spheres equals the unit cell volume. The calculation yields the expected ferromagnetic coupling between Cr and Ni. From the self-consistent spin polarized DOS, partial and total magnetic moment per formula unit can be computed. The calculated total magnetic moment is 5.2 pg in agreement with the experimental value (5.3 0.1 e calculations presented here have been performed... 

Improvement of Mm(Ni - Co - A1 - Mn)x type alloys has been achieved in various ways. It was reported that alloys with a nonstoichiometric composition [Mm(Ni - Co - Mn - Al), 4.5 < jc < 4.8] had a larger discharge capacity than those with stoichiometric alloys [26-27J. Using X-ray diffraction analysis, it was found that the larger capacity is dependent on an increase in the unit cell volume of alloys with x=4.5-4.8. It was also reported that annealing treatment improved the durability of this type of alloy. 

For comparable metal volatilities, the tendency to dissociate decreases with shortening of B—B distances. The high thermal stability of YB4, GdB4, TbB4, DyB4, H0B4 and ErB4 results from both a relatively low metal volatility and a small unit cell volume. 

The distribution of the observed higher borides among the five structural types (MB2, MB4, MBg, MB]2 and Mg ) presented in Table 1, which shows correlations with the metallic radius r. values of which are in order of decreasing magnitude (r, corresponds to coordination number 12). In order to discuss the existence of the actinide borides, the table also shows the unit cell volume V of the borides MB4, MBg and MB,2. 

Inspection of Table 1 indicates that the unit cell volume of MB4, MB, MB,2 and MB borides varies as r except for the actinides. 

Table 3. Existence of the Actinides Borides Comparison of the Unit Cell Volume OF THE Actinide Tetraborides with Those of Comparable Rare-Earth Tetraborides...

Figure 1, Unit cell volume vs. Cu content in -rh boron.

The XRD data and Zn/Cu ratio are also given for a reference aurlchalclte specimen reported In the literature ( ). All d-spaclngs In the mineral and synthetic aurlchalclte matched the literature values within the lattice volume changes (<2%) reported In Table 1. Over 30 XRD peaks were used In the XRD comparisons. The XRD analysis established that the structure of the mineral and synthetic aurlchalclte was essentially Identical. The only distinguishing features were the higher Cu content and the 1.6Z smaller unit cell volume of the mineral sample compared to the synthetic sample. 

In this zeolitic material a very low percentage of Ti(IV), dispersed in a pure siliceous microporous matrix (with the MFI framework, the same as that of the ZSM-5 zeolite), is able to oxidize in mild conditions many substrate with extremely high activity and selectivity (see Sect. 2). However, after more than three decades, a complete picture of reaction mechanisms is still missing. Major problems related to characterization are due to the extremely high dilution of Ti(IV) in the zeolitic matrix and the presence of high amounts of water in the reaction media. The first point requires characterization techniques very sensitive and selective towards Ti(IV). For instance, XRD measurements have been able to recognize the presence of Ti(IV) in the framework only indirectly, via the measured unit cell volume increase [21,22], but attempts to... 

The symbols used are Vc, unit cell volume Bj, isotropic thermal factor of the yth atom. 

However, the calculation has taken account only of the change of the unit cell volume of the compound, AKhl = Fhs Fls which occurs due to the spin-state transition, the individual HS and LS complexes being treated as incompressible spheres. On the basis of detailed X-ray structure investigations [6], the crystal experiences not only a change in size but also a change in shape in the course of the HS LS transition. Every lattice vector x(T) may be expressed according to [70] ... 

Interesting is a comparison of the volumes occupied by individual complexes in solution and in the solid state. The partial molal volumes can be obtained from precise measurements of the solution densities of the complexes as a function of concentration [177]. These values may be subsequently compared with the unit cell volumes per complex molecule derived from the crystal structure. For Fe[HB(pz)3]2, the apparent molal volume in tetrahydrofuran solution was determined as 340.9 em mol Taking into account that the complex in solution forms an equilibrium between 86% LS and 14% HS isomers and employing the volume difference between the two spin states AF° = 23.6 cm mol S the volume of the LS isomer was calculated as 337.6 cm mol This value agrees closely with the volume of 337.3 cm mol for the completely LS complex in solid Fe[HB(pz)3]2 [105]. 

The volume expansion accompanying the spin-state transition has been demonstrated, at least in quantitative terms, on the basis of multi-temperature crystal structure studies [6]. An alternative method starts from the determination of the lattice constants as a function of temperature and therefrom, the volume of the unit cell may be calculated at each temperature. It has been shown [70] that the variation of the unit cell volume V(T) may be reproduced by ... 

Here, and Fl are the unit cell volumes of the pure HS and LS isomer at 0 K, respectively, and is the coefficient of thermal expansion which is assumed to be equal in both lattices. The expression of Eq. (128) may be rewritten as ... 

Fig. 38. Temperature dependence of the lattice constants a, b, c and the unit cell volume V of [Fe(2-pic)3]Cl2 CH3OH. The solid lines are calculated using the parameter values of Table 20. According to Ref. [39]...

The studies show that the observed crystal volume is in fact composed of the fractional contributions from the unit cell volumes of the HS and LS isomers of the compound and a linear volume change with temperature as expressed in Eq. (128). Similarly, the observed lattice constants are formed from a deformation contribution proportional to the HS fraction and a contribution from thermal expansion following Eq. (131). This is a convincing demonstration that it is the internal variation of the molecular units occurring in the course of the spin-state transition which determines, at least in principle, the observed crystal properties. 

Vh-amylose shows an increase in water molecules from 4 to 16 per unit cell)51). Eight of these guests occupy interstitial sites and eight are present within the helical canals (Fig. 13). All unit cell dimensions are increased over the Va-form, thereby increasing the unit cell volume from 2304 to 2604 A3. The sixfold helical arrangement has been confirmed for Vh-amylose. 

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