Bravais rule

Row 9 of Table 12-3 presents the maximum density of planes in the main symmetry directions. As already noted in Ref. [13], there are twofold planes denser than the densest fivefold or threefold planes even though experimental evidence indicates that the fivefold sputtered and annealed surfaces are the most stable. In the light of this condition we propose a modification to the Bravais rule to take into account the layers of planes orthogonal to the main symmetry directions. 

Z. Papadopolos thanks G. Kasner for his programming that was crucial for the realisation of ideas on surfaces since 1997, and P. Pleasants for his rigorous mind through which the formulation of the Bravais rule by quasicrystals became accepted,... 

The Bravais empirical rule, which states that there is a close correlation between the polyhedral forms of a crystal and the lattice type. 

The rules for determining Miller-Bravais planar indices are similar to those for Miller indices with three axes. 

As previously mentioned, the primitive unit cell is the smallest unit of a crystal that reproduces itself by translations. Figure 1-37 illustrates the difference between a primitive and a centered or nonprimitive cell. The primitive cell can be defined by the lines a and c. Alternatively, we could have defined it by the lines a and c. Choosing the cell defined by the lines a" and c" gives us a nonprimitive cell or centered cell, which has twice the volume and two repeat units. Table 1-11 illustrates the symbolism used for the various types of lattices and records the number of repeat units in the cell for a primitive and a nonprimitive lattice. The spectroscopist is concerned with the primitive (Bravais) unit cell in dealing with lattice vibrations. For factor group selection rules, it is necessary to convert the number of molecules per crystallographic unit cell Z to Z, discussed later, which is the number of molecules per primitive cell. For example,... 

Turning to the crystal structure of compounds of unlike atoms, we find that the structure is built up on the skeleton of a Bravais lattice but that certain other rules must be obeyed, precisely because there are unlike atoms present. Consider, for example, a crystal of A Bj, which might be an ordinary chemical compound, an intermediate phase of relatively fixed composition in some alloy system, or an ordered solid solution. Then the arrangement of atoms in A By must satisfy the... 

The reader may have noticed in the previous examples that some of the information given was not used in the calculations. In (a), for example, the cell was said to contain only one atom, but the shape of the cell was not specified in (b) and (c), the cells were described as orthorhombic and in (d) as cubic, but this information did not enter into the structure-factor calculations. This illustrates the important point that the structure factor is independent of the shape and size of the unit cell. For example, any body-centered cell will have missing reflections for those planes which have ft + k + 1) equal to an odd number, whether the cell is cubic, tetragonal, or orthorhombic. The rules we have derived in the above examples are therefore of wider applicability than would at first appear and demonstrate the close connection between the Bravais lattice of a substance and its diffraction pattern. They are summarized in Table 4-1. These rules are subject to... 

As one of our central missions is to uncover the relation between microscopic and continuum perspectives, it is of interest to further examine the correspondence between kinematic notions such as the deformation gradient and conventional ideas from crystallography. One useful point of contact between these two sets of ideas is provided by the Cauchy-Bom rule. The idea here is that the rearrangement of a crystalline material by virtue of some deformation mapping may be interpreted via its effect on the Bravais lattice vectors themselves. In particular, the Cauchy-Bom mle asserts that if the Bravais lattice vectors before deformation are denoted by Ej, then the deformed Bravais lattice vectors are determined by e = FEj. As will become evident below, this mle can be used as the basis for determining the stored energy function W (F) associated with nonlinear deformations F. 

There are only 14 possible three-dimensional lattices, called Bravais lattices (Figure 2.8). Bra-vais lattices are sometimes called direct lattices. Bravais lattices are defined in terms of conventional crystallographic bases and cells, (see Section 2.1). The rules for selecting the preferred lattice are determined by the symmetry of the lattice, (see Chapters 3, 4 for information on symmetry). In brief, the main conditions are ... 

The 14 Bravais lattices are shown in Figure 5.1. For reasons of symmetry (Rule 1 above) we do not always choose a primitive cell. The face-centered cubic cell may be referred to the rhombohedral cell (which is primitive), but the cubic cell reflects the higher symmetry of the lattice. 

Disregarding any detailed morphological models proposed for the lamellar crystals in semicrystalline polymers in the past half centiny, crystal imit cells in semicrys-talUne poisoners exactly obey the rules of classical crystallography, which consist of 7 crystal systems and 14 Bravais lattices (more precisely, there are only 6 crystal systems in polymer crystals, and the cubic system does not exist). Most of the unit cell determinations of the semicrystalline polymers rely on the wide-angle x-ray diffraction (waxd) experiments on oriented poisoner fibers and films. This is be-canse of small crystal sizes in polymers, which lead to a difficult experimental task to obtain single-crystal waxd results on semicrystalfine polymers. It is also possible to nse electron diffraction (ed) method in transmission electron microscopy (tern) to determine polymer lamellar crystal unit cell structures, dimensions, and symmetries. 

The Bravais-Friedel-Donnay-Harker Rule and Attachment Energy Models.167... 

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