The Bravais Lattice

Similarly Kaplan (327) has shown that a b.c.c. magnetic lattice, with /1, J2y Jz (all negative) interaction constants corresponding to nearest, next-nearest, and next-next-nearest neighbors, consists of 

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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 common periodic structures displayed by surfaces are described by a two-dimensional lattice. Any point in this lattice is reached by a suitable combination of two basis vectors. Two unit vectors describe the smallest cell in which an identical arrangement of the atoms is found. The lattice is then constructed by moving this unit cell over any linear combination of the unit vectors. These vectors form the Bravais lattices, which is the set of vectors by which all points in the lattice can be reached. 

The unit cell and the Bravais lattice type for IM-5 were obtained from tilt series of SAED patterns such as that shown in Figure 2 (a = 14.3 A, b = 57.4 A, c - 20.1 A with... 

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

Microdiffraction is the pertinent method to identify the crystal system, the Bravais lattices and the presence of glide planes [4] (see the chapter on symmetry determination). For the point and space group identifications, CBED and LACBED are the best methods [5]. 

Electron Diffraction (CBED) and Large-Angle Convergent-Beam Electron Diffraction (LACBED) allow the identification of the crystal system, the Bravais lattice and the point and space groups. These crystallographic features are obtained at microscopic and nanoscopic scales from the observation of symmetry elements present on electron diffraction patterns. 

The Bravais lattice can be identified, on some specific Zone-Axis Patterns, from the observation of the shift between the reflection net located in the ZOLZ and the one located in the FOLZ. This shift is easily observed by considering the presence or the absence of reflections on the mirrors. Thus, in the example given on figure 1, some reflections from the ZOLZ are present on the four mi, m2, m3 and mirrors. This is not the case in the FOLZ where reflections are present on the m3 and m4 mirrors but not on the mi and m2 mirrors. Simulations given in reference [2] allow to infer the Bravais lattice from such a pattern. It is pointed out that Microdiffraction is very well adapted to this determination due to its good angular resolution (the disks look like spots). 

An example is the (110) plane of III-V semiconductors, such as GaAs(llO). The only nontrivial symmetry operation is a mirror reflection through a line connecting two Ga (or As) nuclei in the COOl] direction, which we labeled as the X axis. The Bravais lattice is orthorhombic primitive (op). In terms of real Fourier components, the possible corrugation functions are... 

The reflections are indexed, and from the systematic absences the Bravais lattice and... 

Nickel crystallizes in a cubic crystal system. The first reflection in the powder pattern of nickel is the 111. What is the Bravais lattice ... 

X-ray powder data for NaCl is listed in Table 2.5. Determine the Bravais lattice, assuming that it is cubic. 

Table 1.1 gives the structures of the elements at zero temperature and pressure. Each structure type is characterized by its common name (when assigned), its Pearson symbol (relating to the Bravais lattice and number of atoms in the cell), and its Jensen symbol (specifying the local coordination polyhedron about each non-equiyalent site). We will discuss the Pearson and Jensen symbols later in the following two sections. We should note,... 

Again, our first concern must be to see how many ways there are in which the translation vectors can be related to one another (relative lengths, angles between them) to give distinct, space-filling patterns of equivalent points. We have seen (Section 11.2) that in 2D there were only 5 distinct lattices. We shall now see that in 3D there are 14. These are often designated eponymously as the Bravais lattices and are shown in Figure 11.11, in the form of one unit cell of each. 

It is not always possible to choose a unit cell which makes every pattern point translationally equivalent, that is, accessible from O by a translation a . The maximum set of translationally equivalent points constitutes the Bravais lattice of the crystal. For example, the cubic unit cells shown in Figure 16.2 are the repeat units of Bravais lattices. Because nt, n2, and w3 are integers, the inversion operator simply exchanges lattice points, and the Bravais lattice appears the same after inversion as it did before. Hence every Bravais lattice has inversion symmetry. The metric M = [a, a ] is invariant under the congruent transformation... 

Example 16.1-1 Find the Bravais lattices, crystal systems, and crystallographic point groups that are consistent with a C3z axis normal to a planar hexagonal net. 

We now remove the inconvenience of the translation subgroup, and consequently the Bravais lattice, being infinite by supposing that the crystal is a parallelepiped of sides Aja,-where ay, j 1,2,3, are the fundamental translations. The number of lattice points, N1N2N3, is equal to the number of unit cells in the crystal, N. To eliminate surface effects we imagine the crystal to be one of an infinite number of replicas, which together constitute an infinite system. Then... 

Find the Bravais lattice and crystallographic point groups that are compatible with a C2 axis. [Hint Use eq. (16.1.17).]... 

Bravais then showed that in three dimensions, there are only 14 different lattice types, currently named the Bravais lattices, which are grouped in seven crystal systems [1-3] (see Table 1.1). 

Among the 14 cells that generate the Bravais lattices (see Figure 1.4), only the P-type cells are considered primitive unit cells. It is possible to generate the other Bravais lattices with primitive unit cells. However, in practice, only unit cells that possess the maximum symmetry are chosen (see Figure 1.4 and Table 1.2) [1-6]. 

The representatives of the known structure types are arranged according to a structure symbol which indicates the Bravais lattice and the number of metal and non-metal atoms within the unit cell. The Bravais lattice is designated by a capital letter following the suggestion of the ASTM Committee (435) ... 

First symbol refers to the Bravais lattice P = primitive lattice C = centered lattice F = face-centered lattice I = body-centered lattice... 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

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