Unit cell cubic, simple/primitive

Some of the crystal systems have more than one kind of lattice. Apn/nrtrv lattice or simple lattice (denoted by P) is one in which lattice points occur only at the corners of the unit cell. A unit cell of a primitive lattice contains one basis (one-eighth of the basis at each corner). A body-centered lattice (denoted by I, for German imenzentriert) is one in which there is a lattice point at the center of the unit cell as well as at the corners. A face-centered lattice (denoted by F) is one in which there is a lattice point at the center of each face of the unit cell as well as at the corners. The sodium chloride lattice is a face-centered cubic lattice. A base-centered lattice or end-centered lattice (denoted by C) is one in which there is a lattice point at the center of one pair of opposite faces as well as at the corners. Table 28.1 and Figure 28.2 show the 14 possible lattices, which are called Bravais lattices. 

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... 

There are two allotropic forms of polonium, both primitive, with one atom per unit cell. a-Po is simple cubic (3PO) and each Po atom has six nearest neighbors (3.366 A) at the vertices of a regular octahedron. The simple cubic structure is the same as that of NaCl (Figure 3.6) with all atoms identical. The structure of (3-Po is a primitive rhombohedral lattice. Each Po atom has six nearest neighbors forming a slightly flattened trigonal antiprism (or a flattened octahedron). 

There are very many unit cell structures if we consider all the atoms or ions in the crystal. However, if we focus on just one atom or ion, we can reduce the number to just 14 primitive cells. Three of these, the simple cubic, face-centered cubic (fee), and body-centered cubic (bcc) unit cells, are shown in Figure 9-2. The lattice points, represented by small spheres in the drawings, correspond to the centers of the atoms, ions, or molecules occupying the lattice. 

Cesium chloride crystallizes with a structure derived from the simple cubic primitive cell. Ch ions occupy the 8 comer sites with Cs+ in the center of the cell note that this is not a body-centered cubic unit cell since the ion at the center is not the same as those at the comers. Thus there is one CsCl unit per unit cell and the coordination numbers of Cs+ and Ch are both 8. Crystals of CsBr and Csl adopt the CsCl structure, but all other alkali halides crystallize in the NaCl structure. 

The assignation of axes of reference in relation to the rotational symmetry of the crystal systems defines six lattices that, by definition, are primitive or P-lattices. To determine if new lattices can be formed from these P-lattices, one must determine if more points can be added so that the lattice condition is still maintained, and whether this addition of points alters the crystal system. For example, if one starts with a simple cubic primitive lattice and adds other lattice points in such a way that a lattice still exists, it must happen that the resulting new lattice still possesses cubic symmetry. Since the lattice condition must be maintained when new points are added, the points must be added to hightly symmetric positions of the P-lattice. These types of positions are (a) a single point at the body center of each unit cell, (b) a point at the center of each independent face of the unit cell, (c) a point at the center of one face of the unit cell, and (d) the special centering positions in the trigonal system that give a rhombohedral lattice. 

Three-dimensional lattices of the same symmetry class characterize the two polymorphic forms, but the unit cells exist with different coordination numbers and different coordination polyhedra. The classic example of this type of behavior is given by cesium chloride, which undergoes a reversible transformation from a cubic body-centered lattice to a cubic face-centered lattice at 445°C [14], On the body-centered modification, two simple primitive cubic lattices (one of Cs cations and one of Cl anions) are placed inside one another so that the comers of one kind of cube are situated at the centers of cubes of the other kind. The face-centered modification is built up from face-centered ionic lattices situated inside one another. 

If another sphere is added in the center of the simple cubic structure, body-centered cubic (bcc) is the result. If the added sphere has the same radius as the others, the size of the unit cell expands (relative to primitive cubic) because the radius of the central atom is larger than 0.73r so that the diagonal distance through the cube is 4r, where r is the radius of the spheres. The comer atoms are no longer in contact with each other. The new unit cell is 2.31r on each side and contains two atoms, because the body-centered atom is completely within the unit cell. This cell has two lattice points, at the origin (0, 0, 0) and at the center of the cell (j, j, j). 

These crystal systems are characterized by the three edge lengths, or lattice constants, a, b, and c and by the angles between these edges, a, /3, and y. (For the special case of the cubic unit cell, the geometry is particularly simple because all sides are equal and all angles are 90°.) A unit cell that contains only one lattice point is called a primitive unit crystal is better represented by... 

Solid COj belongs to the space group T — Fa3. The CO2 molecules form a face-centered cubic lattice (see Table II for molecular positions). However, the molecular symmetry axes of the CO molecules have four distinct directions in space, corresponding to the four body diagonals of the primitive, simple cubic unit cell. The correspondences between positions and axis orien tad ons are also clarified i n Table II. The symmetry operations of Tft are listed in Table A.V. The notadon is that of Zak (1969). 

Section 11.7 In a crystalline soUd, particles are arranged in a regularly repeating pattern. An amorphous solid is one whose particles show no such order. The essential structural features of a crystalline solid can be represented by its unit cell, toe smallest part of toe crystal that can, by simple displacement, reproduce the three-dimensional structure. The three-dimensional structures of a crystal can also be represented by its crystal lattice. The points in a crystal lattice represent positions in toe structure where there are identical environments. The simplest unit cells are cubic. There are three kinds of cubic unit cells primitive cubic, body-centered cubic, and face-centered cubic. 

Fig. 3.1. Primitive unit cell and Brillouin zone for simple cubic lattice...

Most of the crystal systems have more than one possible crystal lattice. A simple (or primitive) lattice has a unit cell in which there are lattice points only at the comers of the unit cell. Other lattices in the same crystal system have additional lattice points either within the body of the unit cell or on faces of the unit cell. As an example, we will describe the cubic crystal system in some detail. 

The cubic crystal system has three possible cubic unit cells simple (or primitive) cubic, body-centered cubic, and face-centered cubic. These cells are illustrated in Figure 11.32. A simple cubic unit cell is a cubic unit cell in which lattice points are situated only at the corners. A body-centered cubic unit cell is a cubic unit cell in which there is a lattice point at the center of the cubic cell in addition to... 

The cubic crystal system, for example, is separated into three Bravais lattices depending on whether the unit cell has species only at the corners (simple or primitive cubic)-, at the corners and the center of the unit cell (body-centered cubic)-, or at the faces of the unit cell (face-centered cubic). Note that for the body-centered cubic, the species (atom, ion, or molecule) in the center contributes one full member to the stoichiometry of the cell, and the atoms, ions, or molecules in the faces of the unit cell contribute of a member each. (Recall that species at the corners contribute I of a member each.) For face-centered cubic unit cells, the facial species contribute, overall, X 6 = 3 members to the stoichiometry of the unit cell. 

Fig. 13 (Color online) (a) The rhombohedral primitive eell (large atoms) is shown with reference to the fee structure. If the angle a is 60°, it is an ideal fee strueture. This angle is 70° in Hg. If this angle were 90°, then it would be a simple cubic lattice. An angle of 109.28° corresponds to a bee lattice, (b) The rhombohedral primitive cell is shown with reference to the conventional hexagonal unit cell described by two lattice parameters (here labeled and Chex)- The primitive cell in the hexagonal description is also shown (large atoms). The conventional cell described by hexagonal lattice parameters contains three atoms.

We turn our attention next to specific examples of real crystal surfaces. An ideal crystal surface is characterized by two lattice vectors on the surface plane. Hi = aix + aiyS, and H2 = 02xX -I- a2yy. These vectors are multiples of lattice vectors of the three-dimensional crystal. The corresponding reciprocal space is also two dimensional, with vectors bi, b2 such that b aj = IrrStj. Surfaces are identified by the bulk plane to which they correspond. The standard notation for this is the Miller indices of the conventional lattice. For example, the (001) surface of a simple cubic crystal corresponds to a plane perpendicular to the z axis of the cube. Since FCC and BCC crystals are part of the cubic system, surfaces of these lattices are denoted with respect to the conventional cubic cell, rather than the primitive unit cell which has shorter vectors but not along cubic directions (see chapter 3). Surfaces of lattices with more complex structure (such as the diamond or zincblende lattices which are FCC lattices with a two-atom basis), are also described by the Miller indices of the cubic lattice. For example, the (001) surface of the diamond lattice corresponds to a plane perpendicular to the z axis of the cube, which is a multiple of the PUC. The cube actually contains four PUCs of the diamond lattice and eight atoms. Similarly, the (111) surface of the diamond lattice corresponds to a plane perpendicular to the x -I- y -I- z direction, that is, one of the main diagonals of the cube. 

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