Crystal lattice primitive translation vectors

A crystal is a physical object - it can be touched. However, an abstract construction in Euclidean space may be envisioned, known as a direct space lattice (also referred to as the real space lattice, space lattice, or just lattice for short), which is comprised of equidistant lattice points representing the geometric centers of the structural motifs. Any two of these lattice points are connected by a primitive translation vector, r, given by ... 

It has just been stated that a band stracture diagram is a plot of the energies of the various bands in a periodic solid versus the value of the reciprocal-space wave vector k. It is now necessary to discuss the concept of the reciprocal-space lattice and its relation to the real-space lattice. The crystal structure of a solid is ordinarily presented in terms of the real-space lattice comprised of lattice points, which have an associated atom or group of atoms whose positions can be referred to them. Two real-space lattice points are connected by a primitive translation vector, R ... 

The space groups are listed and described in the International Tables for X-Ray Crystallography (1962). For our purposes, we are interested only in the primitive unit cell, or the smallest unit cell that can be used to reproduce the crystal by means of translations only. The primitive unit cell often has lower symmetry than the conventional unit cell (Kittel, 1968). According to the international nomenclature for space groups, the letter P denotes a primitive lattice. If we deal with such a crystal, the conventional unit cell is the primitive cell. If, however, the group designation begins with another letter (e.g., I, f, C), then the primitive cell has to be determined and this is smaller and less symmetric than the conventional um t cell. The unit cell vectors of the primitive cell are called primitive translation vectors. 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

An infinite three-dimensional crystal lattice is described by a primitive unit cell which generates the lattice by simple translations. The primitive cell can be represented by three basic lattice vectors such as and h defined above. They may or may not be mutually perpendicular, depending on the crystal... 

The pattern points associated with a particular lattice are referred to as the basis so that the description of a crystal pattern requires the specification of the space lattice by ai a2 a3 and the specification of the basis by giving the location of the pattern points in one unit cell by K, i= 1,2,. .., (Figure 16.1(b), (c)). The choice of the fundamental translations is a matter of convenience. For example, in a face-centred cubic fee) lattice we could choose orthogonal fundamental translation vectors along OX, OY, OZ, in which case the unit cell contains (Vg)8 + (l/2)6 = 4 lattice points (Figure 16.2(a)). Alternatively, we might choose a primitive unit cell with the fundamental translations... 

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. 

Figure 2.6-2 Variation of the frequencies by the incorporation of a tetraatomic molecule with two degenerate vibrational states ( ) in a crystal lattice, a spectrum of the free molecule, R = rotations, T = translations b static influence of the crystal lattice. The degenerate states split, the free rotations change into librations L c dynamic coupling of the vibrations of molecules within a primitive unit cell with z = 2 molecules. Each vibrational level of a molecule splits into z components and 3 z - 3 translational vibrations TS and 3 z librations L appear d dependence of the vibrational frequencies on the wave vector k of the coupled vibrations of all unit cells in the lattice. The three acoustic branches arise from the three free translations with = 0 (for k 0) of the unit cell all vibrations of the unit cells with / 0 (for k 0) give optical branches .

Since every unit cell in the crystal lattice is identical to all others, it is said that the lattice can be primitive or centered. We already mentioned (Eq. 1.1) that a crystallographic lattice is based on three non-coplanar translations (vectors), thus the presence of lattice centering introduces additional translations that are different from the three basis translations. Properties of various lattices are summarized in Table 1.13 along with the international symbols adopted to differentiate between different lattice types. In a base-centered lattice, there are three different possibilities to select a pair of opposite faces, which is also reflected in Table 1.13. 

A crystal lattice is composed of atoms or molecules arranged in a periodic array and is defined by a primitive cell which is the cell of smallest volume which can reproduce the entire array by integral translations along a set of primitive vectors. To illustrate. Figure 27 shows a two-dimensional array of particles. Various cells will reproduce the lattice but the primitive cell (containing one particle) is that described by the vector... 

The primitive orthorhombic lattice can be thought of as arising from a primitive monoclinic lattice with the added restriction that the third angle is also 90°. In that case, all the unit cell translation vectors are 90° to one another but have different lengths. In the orthorhombic system, one can construct a C-centered cell, which can also be described as an A- or B-lattice by an interchange of the orthogonal axes. In addition, there can also be an all-face-centered F-lattice structure and a body-centered I-lattice. Thus for the orthorhombic crystal system there are four unique Bravais lattices, P, I, F, and C. 

Let us suppose that we are mainly interested iu electrons that inhabit the crystal. As we know, electrons glue particles of a solid. As far as electrons are concerned, it is convenient to describe the lattice by the primitive lattice translation vectors. A primitive unit cell, which can fiU up all space, is important in this case. Such a unit ceU in the real space is called the A gner-Seitz ceU. 

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. 

In (4.53), the index fi labels all AOs in the reference primitive unit cell (p = 1,2,..., M) and Rn is the translation vector of the direct lattice (for the reference primitive cell Rn = 0). The summation in (4.53) is snpposed to be made over the infinite direct lattice (in the model of the infinite crystal) or over the inner primitive translations ii of the cyclic cluster (in the cyclic model of a crystal). In the latter case, the sum of the two inner translation vectors fi -I- R = Ri may appear not to be the inner translation of the cychc cluster. However, the subtraction of the translation vector A of the cyclic cluster as a whole (in the cyclic model the vector A is... 

The plane waves exp(ik a ) seem to be the most convenient as interpolation functions for the integrand in the BZ integration, where a = rejUj are direct lattice translation vectors and a are primitive translations. It is easy with plane waves to take into account the translational and point symmetry of the crystal. 

In the periodic systems the basis sets are chosen in such a way that they satisfy the Bloch theorem. Let a finite number of contracted GTFs be attributed to the atom A with coordinate in the reference unit cell. The same GTFs are then formally associated with all translationally equivalent atoms in the crystal occupying positions rA + (In (In is the direct lattice translation vector). For the crystal main region of N primitive unit cells there are N tha Gaussian-type Bloch functions (GTBF)... 

A unit cell of a crystal consists of a volume which contains the array of atoms that are repeated. The smallest possible unit cell that can form the structure is a primitive unit cell although is it often convenient to describe to solid using nonprimitive unit cells that have higher symmetries. A unit cell is described by three lattice vectors that define the directions the cell can be propagated to form the crystal lattice or framework of the structure. The basis of the crystal describes the positions of the atoms within the unit cell. The basis plus the lattice defines the structure. TTae set of all possible translations along the lattice vectors, forms the translation group. 

A dislocation is defined as being perfect when b is a lattice translation vector and imperfect when it is not a lattice translation vector. A lattice translation vector is any vector which joins two points of the crystal lattice and when the line is extended these points are repeated at regular intervals along the line. Since plastic deformation by slip does not change the crystal structure, the Burgers vectors, that is, the slip vectors for the dislocations which accomplish slip, must be lattice translation vectors and the dislocations must be perfect dislocations. Furthermore, dislocations with the smallest possible lattice translation vectors, the primitive lattice translation vectors, are expected because these have the least energy. The Burgers vector for face-centered cubic metals and NaCl structured ionic crystals is thus a<110> and it is fl for body-centered cubic metals, where a is the lattice parameter of the unit cell. 

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