Crystals primitive lattice vectors

For the bcc ciystal structure, the (110) crystal plane is the most densely packed plane. The bcc(llO) surface can be pictured as a distorted hexagonal stracture. It is often convenient to describe this surface using the centered rectangular unit cell highlighted in Fig. 5a rather than the primitive unit cell the primitive unit cell can be transformed into the centered rectangular cell by multiplying the primitive lattice vectors by the matrix ... 

A crystal is described in real space in terms of the primitive lattice vectors ai, 82, as and the positions of atoms inside a primitive unit cell (PUC). The lattice vectors R are formed by aU the possible combinations of primitive lattice vectors, multiplied by integers ... 

The foundation for describing the behavior of electrons in a crystal is the reciprocal lattice, which is the inverse space of the real lattice. The reciprocal primitive lattice vectors are defined by... 

The characteristic feature of crystal surfaces is that the atoms on the surface assume positions different from those on a bulk-terminated plane. The differences can be small, which is referred to as surface relaxation , or large, producing a structure that differs drastically from what is encountered in the bulk, which is referred to as surface reconstruction . The changes in atomic positions can be such that the periodicity of the surface differs from the periodicity of atoms on a bulk-terminated plane of the same orientation. The standard way to describe the new periodicity of the surface is by multiples of the lattice vectors of the corresponding bulk-terminated plane. For instance, a i x 2 reconstruction on the (klm) plane is one in which the lattice vectors on the plane are and 2 times the primitive lattice vectors of the ideal, uiueconstructed, bulk-terminated (klm) plane. Simple integer multiples of the primitive lattice vectors in the bulk-terminated plane often are not adequate to describe the reconstruction. It is possible, for example, to have reconstructions of the form x /n2, or c( i x 2), where c stands for centered . 

The rocksalt stmcture is illustrated in figure Al.3.5. This stmcture represents one of the simplest compound stmctures. Numerous ionic crystals fonn in the rocksalt stmcture, such as sodium chloride (NaCl). The conventional unit cell of the rocksalt stmcture is cubic. There are eight atoms in the conventional cell. For the primitive unit cell, the lattice vectors are the same as FCC. The basis consists of two atoms one at the origin and one displaced by one-half the body diagonal of the conventional cell. 

The empirical pseiidopotential method can be illustrated by considering a specific semiconductor such as silicon. The crystal structure of Si is diamond. The structure is shown in figure Al.3.4. The lattice vectors and basis for a primitive cell have been defined in the section on crystal structures (ATS.4.1). In Cartesian coordinates, one can write G for the diamond structure as... 

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... 

Thus, the reciprocal lattice of a simple cubic lattice is also simple cubic. It is shown in Fig. 5.7 in the xy plane, where it is clear that the bisectors of the first nearest-neighbour (100) reciprocal lattice vectors from a closed volume about the origin which is not cut by the second or any further near-neighbour bisectors. Hence, the Brillouin zone is a cube of volume (2n/a)2 that from eqn (2.38) contains as many allowed points as there are primitive unit cells in the crystal. The second, third, and fourth zones can... 

Just as a reminder The dots between the vectors denote the scalar (inner) product and the crosses denote the cross (outer) product of the vectors. These vectors 6 are in units of nr, which is proportional to the inverse of the lattice constants of the real space crystal lattice. This is why one calls the three-dimensional space spanned by these vectors the reciprocal space and the lattice defined by these primitive vectors is called the reciprocal lattice. These primitive reciprocal vectors have the following properties ... 

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 regular orbit displayed in Figure 2.7, is the geometry on the unit sphere such that the bond length , the Euclidean distance between adjacent vertices, is constant. This restriction is not necessary from a symmetry viewpoint it may be relaxed subject only to the requirement that the local four, three and two-fold symmetries are maintained. One important example of such a relaxation occurs for the regular orbit of the Oh Crystallographic point group. In the simplest model crystal of Oh point symmetry, the primitive cubic array, for example, as in cubium, lattice points are distributed as dictated by the lattice vector Rmnp such that... 

The introduction of lattice centering makes the treatment of crystallographic symmetry much more elegant when compared to that where only primitive lattices are allowed. Considering six crystal families Table 1.12) and five types of lattices Table 1.13), where three base-centered lattices, which are different only by the orientation of the centered faces with respect to a fixed set of basis vectors are taken as one, it is possible to show that only 14 different types of unit cells are required to describe all lattices using conventional crystallographic symmetry. These are listed in Table 1.14, and they are known as Bravais lattices. ... 

The 1st Brillouin zone follows from the reciprocal lattice by construction of the planes which are perpendicular to the lines connecting neighbouring points in the reciprocal lattice at their midpoints. The smallest closed volume which is bounded by these planes is the 1st BZ. For the naphthalene crystal, we find from the lattice parameters at T = 300 K (Table 2.3) the following magnitudes for the reciprocal lattice vectors a = lit 0.145 A ) = 2jt 0.167 A c = 2jr 0.138 A and for the volume V of the primitive... 

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 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. 

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