Base-centered lattice

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. 

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. ... 

In the monoclinic crystal system, the body-centered lattice can be converted into a base-centered lattice (C), which is standard. The face-... 

In the tetragonal crystal system the base-centered lattice (C) is reduced to a primitive (P) one, whereas the face-centered lattice (F) is reduced to a body-centered (I) cell both reductions result in half the volume of the corresponding unit cell (rule number three). 

By repeating the same process in combination with the base-centered lattice, C, two new space groups symmetry, Cm and Cc can be obtained. Therefore, the following four monoclinic crystallographic space groups... 

The two solutions (hexagonal and orthorhombic) are actually similar, since any hexagonal lattice can also be described in the lower symmetry orthorhombic base-centered lattice, as shown schematically in Figure 5.14. Obviously, the orthorhombic solution, found by DICVOL91, represents the primitive orthorhombic unit cell with 1/4 the volume of the conforming base-centered orthorhombic unit cell with the following unit cell dimensions o = 1/2i3hV3 bo = Ch, and Cq = Mlbn-... 

The gravimetric density of the crystals was not measured hut the content of the unit cell may be established by using Eq. 6.5 and the expectation that the reasonable value of p should be between 4 and 5 g/cm. The estimated density assuming NiMnOs composition has a reasonable value of 4.86 g/cm when Z = 4. The two closest numbers of formula units (Z = 3 or 5) are impossible due to the restrictions imposed by symmetry in a base-centered lattice, sites with odd multiplicities are impossible. The next two closest numbers (Z = 2 or 6) result in unrealistically low and high densities, respectively. Thus, we assume that there are 4 Mn, 4 Ni and 12 O atoms in the unit cell. 

A relatively large volume of the monoclinic unit cell translates into a considerable complexity of the diffraction pattern even in the case of a base-centered lattice, as can be seen from Figure 6.29. There are 10 reflections per degree at 20 = 50° and about 20 reflections per degree at 20 = 100°. 

The number of atoms per unit cell in any crystal is partially dependent on its Bravais lattice. For example, the number of atoms per unit cell in a crystal based on a body-centered lattice must be a multiple of 2, since there must be, for any atom in the cell, a corresponding atom of the same kind at a translation of from the first. The number of atoms per cell in a base-centered lattice must also be a multiple of 2, as a result of the base-centering translations. Similarly, the number of atoms per cell in a face-centered lattice must be a multiple of 4. 

Consider reflections from the (001) planes which are shown in profile in Fig. 4-2. For the base-centered lattice shown in (a), suppose that the Bragg law is satisfied for the particular values of A and 0 employed. This means that the path difference ABC between rays 1 and 2 is one wavelength, so that rays 1 and 2 are in phase and diffraction occurs in the direction shown. Similarly, in the body-centered lattice shown in (b), rays 1 and 2 are in phase, since their path difference ABC is one wavelength. However, in this case, there is another plane of atoms midway between the (001) planes, and the path difference DEF between rays 1 and 3 is exactly half of ABC, or one-half wavelength. Thus rays 1 and 3 are completely out of phase and annul each other. Similarly, ray 4 from the next plane down (not shown) annuls ray 2, and so on throughout the crystal. There is no 001 reflection from the body-centered lattice. 

In the orthorhombic crystal system the base-centered lattice merits special attention because of different possible settings. Let the initial base-centered lattice have the setting C. The transition to base-centered lattices with settings C and A (or B) gives different results (see Appendix A). The change of setting for the transition to other types of lattice does not give new superceUs. 

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. 

Using data from rotation and Laue photographs, it is shown that the unit of structure of sodalite, containing < NaiAlzSiiOi2Gl, has a0 = 8.87 A. The lattice is the simple cubic one, Fc the structure closely approximates one based on a body centered lattice, however. The atomic arrangement has... 

Orthorhombic crystals are similar to both tetragonal and cubic crystals because their coordinate axes are still orthogonal, but now all the lattice parameters are unequal. There are four types of orthorhombic space lattices simple orthorhombic, face-centered orthorhombic, body-centered orthorhombic, and a type we have not yet encountered, base-centered orthorhombic. The first three types are similar to those we have seen for the cubic and tetragonal systems. The base-centered orthorhombic space lattice has a lattice point (atom) at each comer, as well as a lattice point only on the top and bottom faces (called basal faces). All four orthorhombic space lattices are shown in Figure 1.20. 

The crystal descriptions become increasingly more complex as we move to the monoclinic system. Here all lattice parameters are different, and only two of the interaxial angles are orthogonal. The third angle is not 90°. There are two types of monoclinic space lattices simple monoclinic and base-centered monoclinic. The triclinic crystal, of which there is only one type, has three different lattice parameters, and none of its interaxial angles are orthogonal, though they are all equal. 

The structure is based on a body-centered lattice. At each lattice point there is a small atom (Zn, Al). It is surrounded by an icosahedron of twelve atoms (Fig. 11-14). This group is then surrounded by 20 atoms, at the corners of a pentagonal dodecahedron, each atom lying directly out from the center of one of the 20 faces of the icosahedron. The next 12 atoms lie out from the centers of the pentagonal faces of the dodecahedron this gives a complex of 45 atoms, the outer 32 of which lie at the corners of a rhombic triacontahedron. The next shell consists of 60 atoms, each directly above the center of a... 

A crystal lattice is an array of points arranged according to the symmetry of the crystal system. Connecting the points produces the lattice that can be divided into identical parallelepipeds. This parallelepiped is the unit cell. The space lattice can be reproduced by repeating the unit cells in three dimensions. The seven basic primitive space lattices (P) correspond to the seven systems. There are variations of the primitive cells produced by lattice points in the center of cells (body-centered cells, I) or in the center of faces (face-centered cells, F). Base-centered orthorhombic and monoclinic lattices are designated by C. Primitive cells contain one lattice point (8 x 1/8). Body-centered cells... 

The Seven Systems of Crystals are shown in Figure 2.2. The relationship between the trigonal and rhombohedral systems is shown in Figure B.la. The possibilities of body-centered and base-centered cells give the 14 Bravais Lattices, also shown in Figure 2.2. A face-centered cubic (fee) cell can be represented as a 60° rhombohedron, as shown in Figure B.lb. The fee cell is used because it shows the high symmetry of the cube. 

Why is there no face-centered tetragonal space lattice Why is there no base-centered tetragonal ... 

Bravais showed in 1850 that all three-dimensional lattices can be classified into 14 distinct types, namely the fourteen Bravais lattices, the unit cells of which are displayed in Fig. 9.2.3. Primitive lattices are given the symbol P. The symbol C denotes a C face centered lattice which has additional lattice points at the centers of a pair of opposite faces defined by the a and b axes likewise the symbol A or B describes a lattice centered at the corresponding A or B face. When the lattice has all faces centered, the symbol F is used. The symbol I is applicable when an additional lattice point is located at the center of the unit cell. The symbol R is used for a rhombohedral lattice, which is based on a rhombohedral unit cell (with a = b = c and a = ft = y 90°) in the older literature. Nowadays the rhombohedral lattice is generally referred to as a hexagonal unit cell that has additional lattice points at (2/3,1 /3, /s) and (V3,2/3,2/3) in the conventional obverse setting, or ( /3,2/3, ) and (2/3, /3,2/3) in the alternative reverse setting. In Fig. 9.2.3 both the primitive rhombohedral (.R) and obverse triple hexagonal (HR) unit cells are shown for the rhombohedral lattice. 

Within a given crystal system, a supplementary subdivision is necessary to be made, in order to produce the 14 Bravais lattices. In this regard, it is necessary to make a distinction between the following types of Bravais lattices, that is, primitive (P) or simple (S), base-centered (BC), face-centered (FC), and body-centered (BoC) lattices [1-3]. 

Unit cells are further subclassified as simple cubic/ face-centered cubic, body-centered cubic, base-centered rhombic, etc. but in order to avoid duplication in classification (Exercise 3), certain of the possibilities are left out (for example, face-centered tetragonal, side-centered rhombic), Actually 14 distinct types of space lattice are recognized. A number of cubic unit ceils and one body-centered tetragonal cell... 

In addition to the simple (primitive) space lattices listed on page 310, the following are known monoclinic system—base centered... 

Alternative approaches exist to explaine the mechanism of chemical interaction. It is accepted that water is formed as a result of mechanochemical interaction, solid reagents are dissolved in water, and the reaction proceeds via the dissolved state (hydrothermal-like process). On the other hand, it is assumed that the initial stages of the process involve the interaction between acid and base centers present at the surfaces in contact, and the next stages are connected with the process of calcium cations insertion into aluminum hydroxide lattice. 

Figure 2.14. Models of the 14 Bravais lattices. The various types of Bravais centering are given the symbols P (primitive/simple), F (face-centered), I (body-centered), and C (base-centered). The primitive rhombohedral Bravais lattice is often given its own symbol, R, and corresponds to a primitive unit cell possessing trigonal symmetry.

Base-centered unit cell (Figure 1.23) contains additional lattice points in the middle of the two opposite faces (as indicated by the vector pointing towards the middle of the base and by the dotted diagonals on both faces). This unit cell contains two lattice points since each face is shared by two neighboring unit cells in three dimensions. 

Body-centered unit cell Figure 1.24) contains one additional lattice point in the middle of the body of the unit cell. Similar to a base-centered, the body-centered unit cell contains a total of two lattice points. 

Figure 5.16. Alternative axes selection in the monoclinic crystal system. Open and hatched points represent lattice points. The open points are located in the plane, while the hatched points are raised by 1/2 of the full translation in the direction perpendicular to the plane of the projection. Unit cells based on the vectors a and c or a and d correspond to a base-centered (C) lattice, while the unit cell based on the vectors c and d corresponds to a body-centered lattice.

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