Space lattice Bravais

Crystal System Bravais space lattice 1 ASTM notation Hermann-Mauguin Pearson notation... 

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

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... 

One of the concepts in use to specify crystal structures the space lattice or Bravais lattice. There are in all fourteen possible space (or Bravais) lattices. 

Our description of atomic packing leads naturally into crystal structures. While some of the simpler structures are used by metals, these structures can be employed by heteronuclear structures, as well. We have already discussed FCC and HCP, but there are 12 other types of crystal structures, for a total of 14 space lattices or Bravais lattices. These 14 space lattices belong to more general classifications called crystal systems, of which there are seven. 

The Bravais or space lattice does not distinguish between different types of local atomic environments. For example, neighbouring aluminium and silicon both take the same face-centred cubic Bravais lattice, designated cF, even though one is a close-packed twelve-fold coordinated metal, the other... 

Only fourteen space lattices, called Bravais lattices, are possible for the seven crystal systems (Fig. 328). Designations are P (primitive), / (body-centered), F (face-centered),34 C pace-centered in one set of laces), and R (rhombohedral) Thus our monoclinic structure P2Jc belongs to the monoclinic crystal system and has a primitive Bravais lattice. 

Periodic repclitions of a space lattice cell in three dimensions from the original cell vvill completely partition space without overlapping or omissions. El is possible to develop a limited number of such three-dimensional patterns. Bravais. in 1848. demonsirated geometrically that there were but fourteen types of space lattice cells possible, and that these fourteen types could be subdivided into six groups called systems. Each system may be distinguished hy symmetry features, which can be related lo four symmetry elements ... 

Up to this point we did not make any specific assumptions about the real space lattice. It could contain more than one atom per lattice point and more than more than one type of atoms. In such a case the lattice would be described using a Bravais lattice plus a basis (see Section 8.2.2. To obtain the intensity of the diffracted wave for crystals with a basis, we simply have to sum up the contributions from all scattering points within the unit cell. The scattering probability for a crystal of N unit cells with an electron density ne(r) is proportional to ... 

Auguste Bravais (1811-1863) first proposed the Miller-Bravais system for indices. Also, as a result of his analyses of the external forms of crystals, he proposed the 14 possible space lattices in 1848. His Etudes Cristallographiques, published in 1866, after his death, treated the geometry of molecular polyhedra. 

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. 

Auguste Bravais (1811-1863) corrects Frankenheim s number of crystal systems, noting that two were equivalent, and that the remaining 14 coalesced by pairs, thus proving that there are really seven distinct crystal systems derives the five two-dimensional and 14 three-dimensional space lattices. 

If the contents of a unit cell have symmetry, containing a number of units of pattern (atoms, molecules), the number of distinct types of space lattice becomes fourteen (Fig. 82) (Bravais, 1818). And when other symmetry operations are recognised (e.g. rotation of the lattice) there are found to be 230 distinct varieties of crystal symmetr ... 

There are only fourteen different arrangements of points in space that satisfy the definition of a lattice. These are known as the Bravais lattices, listed above under lattice types. Diagrams of these space lattices are found in introductory mineralogy texts. 

TABLE 3. The Fourteen Possible Space Lattices (Bravais Lattices)... 

Bravais proved that if the contents of a unit cell have symmetry, the number of distinct types of space lattices becomes fourteen. These are the only lattices that can fill all space and are commonly termed the 14 Bravais lattices. Since there are seven crystal systems, it might be thought that by combining the seven crystal systems with the idea of a primitive lattice a total of seven distinct Bravais lattices would be obtained. However, it turns out that the trigonal and hexagonal lattices so constructed are equivalent, and therefore only six lattices can be formed in this way. These lattices, which are given the label P, define the primitive unit cells (or P-cells) in each case. 

The idea that crystals are ordered assemblies of particles goes back to Hooke (Vol. II, p. 564) and was gradually extended by L. A. Seeber and G. Delafosse to the view of a space lattice , an arrangement of centres of structural units arranged as a network in an orderly manner, maintained in stable equilibrium by attractions and repulsions. A. Bravais proposed 14 types of lattice, later extended mathematically by taking account of symmetry... 

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