Primitive cubic lattice

The linking pattern of two zeolites is shown in Fig. 16.24. They have the /I-cage as one of their building blocks, that is, a truncated octahedron, a polyhedron with 24 vertices and 14 faces. In the synthetic zeolite A (Linde A) the /3-cages form a cubic primitive lattice, and are joined by cubes. j3-Cages distributed in the same manner as the atoms in diamond and linked by hexagonal prisms make up the structure of faujasite (zeolite X). 

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. 

The parallelepiped in Figure 2 is the unit cell of the ammonia crystal phase I. Thus, the ammonia crystal can be regarded as the combination of a pattern of four ammonia molecules (16 atoms) in the unit cell with all possible translations in a cubic primitive lattice. Considerations about crystalline symmetry lead to the conclusion that ammonia in phase I crystallizes according to space group P2i3. Letter P in the symbol stands for primitive lattice, and the other symbols denote the main symmetry operations. The last element in the symbol, 3, indicates the presence of a three-fold axis not aligned with the principal rotation axis (if it was, it would follow letter P), which further indicates that the lattice is cubic. A cubic unit cell is completely specified by just one... 

Primitive cubic crystal lattice. One unit cell is marked... 

The region within which k is considered (—n/a first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

First Brillouin zone for a cubic-primitive crystal lattice. The points X are located at k = it/a in each case... 

Two modifications are known for polonium. At room temperature a-polonium is stable it has a cubic-primitive structure, every atom having an exact octahedral coordination (Fig. 2.4, p. 7). This is a rather unusual structure, but it also occurs for phosphorus and antimony at high pressures. At 54 °C a-Po is converted to /3-Po. The phase transition involves a compression in the direction of one of the body diagonals of the cubic-primitive unit cell, and the result is a rhombohedral lattice. The bond angles are 98.2°. 

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... 

Similarly, for the body-centered cubic (bcc) lattice one might choose an orthogonal set for the fundamental translations giving a non-primitive unit cell with two lattice points per cell (Figure 16.2(b)) or one could choose a primitive unit cell with the fundamental translations... 

Figure 16.12. Brillouin zones, with symmetry points marked, of (a) the primitive cubic Bravais lattice and (b) the cubic close-packed or fee Bravais lattice.

The reciprocal lattice will then be a simple cubic lattice with primitive lattice... 

As a first example, consider the growth of a cubic lattice about the origin. For the simple primitive lattice , most familiar as the structure of the model crystal cubium , the cubic array... 

The type of the correlations is mostly the Al, A2, AO = cubic primitive AH = hexagonal primitive or Cl 1 type (cf. abbreviations at the end of the paper), but it may be deformed by other influences. These types are all translation lattices, and the proposal of translation lattices with uniform distances is the simplest conceivable possibility for a spatial correlation. 

Using this procedure for a set of reflections in an unknown cubic material gives the columns of data tabulated in Table 3.1. For a primitive lattice the first reflection is the 100 therefore the ratio is just ... 

The pattern we have just indexed is for a primitive lattice where all reflections are observed. In the other types of cubic lattice, certain types of reflection are absent and these are called systematic absences. 

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