Lattice Bravais

In 1849 Bravais showed that there are only 14 ways to arrange identical points in space such that each point has the same number of neighbors at the same distances and in the same directions. They represent combinations of the seven crystal systems. 

Similar special symmetries arise in the 3D structures. For example, if a lattice point is placed in the center of a cube, the primitive lattice would be trigonal. But because of the 

Face-centered rectangular lattice in 2D. All the lattice points could be generated by translating along the oblique primitive lattice vectors a and b, or by the nonprimitive vectors a and b with a basis of (0,0) and (l/2,l/2). 

Frankheim in 1842 was the first to classify the possible crystal lattices including the special body-centered and face-centered nonprimitive lattices. However, he had mistakenly added a 15th structure that turned out to be redimdant. Auguste Bravais was the first in 1845 to correctly characterize the 14 unique lattices that now bear his name. 

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Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

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.

Fig. 3.8 Some basic Bravais lattices (a) simple cubic, (b) body-centred cubic, (c) face-centred cubic and (d) simple hexagonal close-packed. (Figure adapted in part from Ashcroft N V and Mermin N D 1976. Solid State Physics.

If atoms, molecules, or ions of a unit cell are treated as points, the lattice stmcture of the entire crystal can be shown to be a multiplication ia three dimensions of the unit cell. Only 14 possible lattices (called Bravais lattices) can be drawn in three dimensions. These can be classified into seven groups based on their elements of symmetry. Moreover, examination of the elements of symmetry (about a point, a line, or a plane) for a crystal shows that there are 32 different combinations (classes) that can be grouped into seven systems. The correspondence of these seven systems to the seven lattice groups is shown in Table 1. 

The crystal group or Bravais lattice of an unknown crystalline material can also be obtained using SAD. This is achieved easily with polycrystalline specimens, employing the same powder pattern indexing procedures as are used in X-ray diffraction. ... 

Fig. 2.47. Direct (left) and reciprocal (right) lattices for the five two-dimensional Bravais lattices (2.243).

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... 

All crystal structures are derived from the 14 Bravais lattices. The atoms in a unit cell are counted by determining what fraction of each atom resides within the cell. The type of unit cell adopted by a metal can be determined by measuring its density. 

Bravais lattices The 14 basic patterns of unit cells from which a crystal can be built. 

The task of predicting a reasonable structure for this alloy was carried out with no information about the powder X-ray diffraction pattern except that one group of investigators had said that it could not be indexed by any Bravais lattice. The prediction of the structure was made entirely on the basis of knowledge of the effective radii of metal atoms and the principles determining the structure of metals and intermetallic compounds. 

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. 

In two dimensions, five different lattices exist, see Fig. 5.6. One recognizes the hexagonal Bravais lattice as the unit cell of the cubic (111) and hep (001) surfaces, the centered rectangular cell as the unit cell of the bcc and fee (110) surfaces, and... 

If the binary constituents AC and BC both have the structure a as their stable form in the temperature range (7, then the alloy is of Type I and one observes a single Bravais lattice of the type a at all alloy compositions for which solid... 

These 14 Bravais Lattices are unique in themselves. If we arrange the crystal systems in terms of symmetry, the cube has the highest symmetry and the triclinic lattice, the lowest symmetry, as we showed above. The same hierarchy is maintained in 2.2.4. as in Table 2-1. The symbols used by convention in 2.2.4. to denote the type of lattice present are... 

If we now apply rotadonal nnmetxy (Factor II given in 2.2.1) to the 14 Bravais lattices, we obtain the 32 Point-Groups which have the factor of symmetry imposed upon the 14 Bravais lattices. The symmetry elements that have been used are ... 

Translatioiial symmetry operations generate the 14 Bravais lattices. 

The rotational operations generate a total of 32 Point Groups derived from these s)mimetry operations on the 14 Bravais lattices. 

In a three-dimensional lattice, we have observed planes of atoms (or ions) composing the lattice. Up to now, we have assumed that these planes maintain a certain relation to one another. That is. we have shown that there are a set of planes as defined by the hkl values, which in turn depends upon the type of Bravais lattice that is present. However, we find that it is possible for these rows of atoms to "slip" from their equilibrium positions. Hiis gives rise to another type of lattice defect called "line defects". In the following diagram, we present a hexagonal lattice in which a line defect is present ... 

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. 

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... 

Greek indices a, p = x,y,z of the Cartesian coordinate axes is meant). Minimization of expression (2.1.1) for an arbitrary two-dimensional Bravais lattice becomes possible since the Fourier representation in q implies the reduction of the double sum over j and/ to the single sum over q. Then the ground state energy is given by... 

Two-dimensional Bravais lattices with no higher than second-order axes of symmetry are characterized by a non-degenerate dipole ground state. On a rectangular lattice, the dipoles are oriented along the chains with the least intersite distances ax and antiferroelectric ordering in neighboring chains. As an example, for... 

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