Bravais point symmetry

Now that we have enumerated all of the 3D lattices, the 14 Bravais lattices, we can look in more detail at their symmetries. First of all, it must be recognized that every lattice point is a center of symmetry. The translation vectors tx, t2, and t3 are entirely equivalent to tj, -t2, and -t3, respectively. Therefore, in determining the point symmetry at each lattice point (which is what symmetry of the lattice means) we must include the inversion operation and all its products with the other operations. 

In Section 11.4 the fourteen 3D lattices (Bravais lattices) were derived and it was shown that they could be grouped into the six crystal systems. For each crystal system the point symmetry of the lattice was determined (there being one point symmetry for each, except the hexagonal system that can have either one of two). These seven point symmetries are the highest possible symmetries for crystals of each lattice type they are not the only ones. 

The operation of the allowed symmetry elements on the 14 Bravais lattices must leave each lattice point unchanged. The symmetry operators are thus representative of the point symmetry of the lattices. The most important lattice symmetry elements are given in Table 4.3. In all except the simplest case, two point group symbols are listed. The first is called the full Hermann-Mauguin symbol, and contains the most complete description. The second is called the short Hermann-Mauguin symbol, and is a condensed version of the full symbol. The order in which the operators within the symbol are written is given in Table 4.2. 

Trifluorides RF3 (R=Gd,..., Yb) belong to the Pnma (Dj ) space group, the Bravais lattice is orthorhombic, tiie point symmetry of rare-earfli positions being Cih = Q. An elementary cell contains four formula units, the coordinates of lattice points are as follows (three coordinates in units of a, b, c, correspondingly) ... 

Tablel.3-b Crystal families, crystal systems, crystallographic point groups, conventional coordinate systems, and Bravais lattices in three dimensions. Lattice point symmetries (holohedries) are given in bold...

Translation Symmetry of Crystals. Point Symmetry of Bravais Lattices. Crystal Class... 

The lattice types are labeled by P (simple or primitive), F (face-centered), I (body-centered) and A B,C) (base-centered). Cartesian coordinates of basic translation vectors written in units of Bravais lattice parameters are given in the third column of Table 2.1. It is seen that the lattice parameters (column 4 in Table 2.1) are defined only by syngony, i. e. are the same for all types of Bravais lattices with the point symmetry F and all the crystal classes F of a given syngony. 

Translation and Point Symmetry of Crystals 13 Table 2 1- Distribution of crystal classes F and Bravais lattices on singonies F°... 

BZ, where R are the elements of the holosymmetric point group F° (F° is the point-symmetry group of the Bravais lattice and defines the appropriate crystal system). 

The basis vectors Aj determine an LUC and a new Bravais superlattice. The LUC thus constructed has volume V), = LVa and consists of L primitive cells. The superlattice vectors A are linear combinations (with integral-valued coefficients) of the basis vectors Aj. The matrix 1 in (4.141) is chosen such that the point symmetry of the new superlattice is identical to that of the original lattice (the corresponding transformation (4.141) is called a symmetric transformation, see Sect. 4.2.1). The type of direct lattice can be changed if there are several types of lattice with the given point symmetry. The LUC is conveniently chosen in the form of a Wigner-Seitz (WS) cell, which possesses the point symmetry of the lattice. 

All crystal systems can be classified into one of the 14 Bravais lattices which can be subdivided into 32 crystal classes or point groups. If certain other translation operations that do not have point symmetry are considered, such as a translation combined with a mirror reflection (glide plane operation) or a translation combined with an n-fold rotation (screw axis), the 32 point groups can be subdivided into 230 possible space groups that completely describe the symmetry of all possible crystal systems. These are enumerated in the International Tables for Crystallography, vol A (Ed. Th. Hahn, 2006). 

It is the mentioned symmetry properties additional to the discrete translational symmetry that lead to a classification of the various possible point lattices by five Bravais lattices. Like the translations, these symmetry operations transform the lattice into itself They are rigid transforms, that is, the spacings between lattice points and the angles between lattice vectors are preserved. On the one hand, there are rotational axes normal to the lattice plane, whereby a twofold rotational axis is equivalent to inversion symmetry with respect to the lattice point through which the axis runs. On the other hand, there are the mirror lines (or reflection lines), which he within the lattice plane (for the three-dimensionaUy extended surface these hnes define mirror planes vertical to the surface). Both the rotational and mirror symmetry elements are point symmetry elements, as by their operation at... 

The Bravais lattices are classified according to the applying group of rotational and mirror symmetries. Figure 4.11 presents these classes with the translation vectors and the point symmetry elements indicated within one unit cell. 

Here, as in the discussion of the Smoluchowski effect, we have made expHdt reference to the discrete, periodic stracture of the lattice. The symmetry ofa periodic lattice is substantially lower than the continuous translational and rotational symmetry of the uniform jeUium model. Only translations involving a Bravais lattice vector and a discrete number of point symmetry operations remain and the constant potential has to be replaced by a crystal potential, which reflects the lowered symmetry. This has profound consequences for the electronic structure. 

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. 

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 ... 

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. 

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