Reticular density

In Fig. 4.1 we depict three lattice types of the cubic system and the crystal faces with the highest reticular density (the density of lattice points per unit area) in each t5q)e. 

Lattice type is the basis of calculation of reticular density in the Bravais empirical law. In lattice types, only the symmetry elements with no translation, i.e. the... 

Figure 4.1. Crystal faces with the highest rank in the order of morphological importance (with the highest reticular density) for P. F. and I lattice types of the cubic system (a) P lattice, 100) (b) F lattice. Ill) (c) I lattice. (110).

The above remarks refer only to the relative dimensions of crystals. A consideration of the indices of the principal bounding faces may lead to further conclusions, at any rate for molecular crystals. The bounding faces on crystals are apparently those planes having the greatest reticular density of atoms or molecules the indices of the bounding faces may therefore indicate the general arrangement of the molecules. For instance, when a crystal is found to be bounded entirely or mainly by faces of 110 type (110, Oil, 101, etc.) it is likely that there are molecules... 

When ionic or polar forces play an important part in binding atoms or molecules together in a crystal, matters are more complex, since the rates of growth of crystal faces appear to be influenced by the distribution of electric charges as well as the reticular density (Kossel, 1927). The subject has not so far received much attention, and it is unwise to attempt to formulate generalizations. 

Zero-charge Potential and Reticular Density of Planes... 

Being semimetals, Bi and Sb show anomalies in the correlation of Ea=o with the surface atomic density [15, 16, 22], explained in terms of face-specific space charge effects. However, the definite dependence of Ea=o on the reticular density of plane has been established in a good agreement with general tendency. 

Data for single-crystal planes of sp metals have not been plotted in Fig. 12 as there are no correct values in the literature. For that reason the so-called electrochemical work function values have been obtained according to Frumkin et al. and Trasatti s conception discussed in Refs. [1, 71,72]. According to the data in Fig. 13, there is a good linear correlation between and Ea=o values for singlecrystal planes of sp metals, with the same slope as for PC sp metals. The dependence of AX on the crystallographic orientation of plane is weak, but AX seems to increase with the decrease of the reticular density of planes, thus in the same direction as for fee sd metals. 

The surface energy and the rate of growth of a face, however, should be inversely proportional to the reticular or lattice density of the respective lattice plane, so that faces having low reticular densities would grow rapidly and eventually disappear. In other words, high index faces grow faster than low. 

Figure 11.3 Outlineofacrystal with the tetragonal symmetryat equilibrium. A and 6 represent the reciprocal reticular density of each surface. From Ref [28].

Table 11.2 Experimentally measured growth rate of various surfaces of a Mohr salt with cubic symmetry and the corresponding reticular density of each surface. Note that all the values are normalized by that of the (001) surface. After Ref. [29]..

The reticular density has the identical meaning with the Ns that was introduced in Eq. (1) in the previous section, and its reciprocal is proportional to the energy of the corresponding surface. 

The use of surface energies instead of reticular densities was based on the fact that the former was roughly inversely proportional to the latter Equation (12) is commonly called the Wulfftheorem, and the analytical description of the Wulff theorem was later given by Laue (30] as ... 

Thus, there was historically concluded how the crystals grow in a perpendicular (normal) direction to its sides at a speed as much lower as the reticular density on tiiat face is higher, naturally defining the crystal face as the reticular plane with maximum density (of its structural points, inhibiting the further advancement of the crystalline form in the normal direction at the plane-face in cause). 

For instance, the e q)erience knowledge said that the crystal faces correspond to some planes, usually of high reticular density, while the edges correspond to the dense reticular strings. As long as a certain plane or string has a fixed position in the lattice, the result is that also the angles between them are constant too. 

Mostly, the Miller indices are integers and small numbers, rarely larger than 3 and even rarely greater than 5 (as based on empirical observations). These characteristics have substance in the Bravais observation that only the planes (reticular layers) with large reticular density will appear as natural faces, so these faces being simple planes in the unit cell with small Miller indices. 

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