Face-centered lattice

In the sodium chloride crystal, the Na+ ion is slightly too large to fit into holes in a face-centered lattice of Cl- ions (Figure 9.18). As a result, the Cl- ions are pushed slightly apart so that they are no longer touching, and only Na+ ions are in contact with Cl- ions. However, the relative positions of positive and negative ions remain the same as in LiCk Each anion is surrounded by six cations and each cation by six anions. 

The structure can be instructively compared with that of fluorite, CaF2. In fluorite the calcium ions are arranged at face-centered lattice points, and each is surrounded by eight fluorine ions at cube corners. 

H20 or Si3Al9013 0H)w F, Cl)3, and assigned the crystal the symmetry of space group T, although his data indicated a face-centered lattice. 

The special positions include the following points and those obtained from them by the translational operations of the face-centered lattice ... 

Fluorite is calcium fluoride, Cap2, with a cubic face-centered lattice, while each fluorine ion is at the center of one of the smaller cubes obtained by dividing the unit cube into eight parts. Each Ca is coordinated by eight F ions and each F is surrounded by four Ca ions arranged at the corners of a regular tetrahedron. 

In Figure 1, a is plotted vs. ax for a face-centered lattice (a close-packed lattice) and for a simple cubic lattice (a loose-packed lattice). We notice that (1) the dependence of a on ax can be regarded as being practically the same for both lattices and that, (2) tx undergoes a rapid change around x = xc, which is the point at which a = 0 (Fig. 1). However, p/a does not attain the value it would have for the case of the unrestricted random walk model at x = xc, since at this point, p/a > 1 (Fig. 2), while for unrestricted chain pja = 1. Moreover, the dependence of p/a on ax is not the same for the two lattices while a as a function of ax is practically independent of the lattice. 

Fig. 4. The dispersion of z per chain element, z2> — 0.2 indicates uncertainty in the data, (See Ref. 2.)...

Fig. 5. The excess entropy per chain element versus x for the face-centered lattice.

Finally, there is a rare type of glide called a diamond glide (plane or operation).-This can occur only for 1 or F lattices where a translation of, for example, a/4 + b/4 can move a corner lattice point to a face centering lattice point (or vice versa). The symbol for a diamond glide is d. 

A-centered lattice (A) B-centered lattice (B) C-centered lattice (C) Face-centered lattice (F)... 

Figure 9.10. The face-centered lattice of AuCu3 with Cu in the face centers is shown.

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. 

First symbol refers to the Bravais lattice P = primitive lattice C = centered lattice F = face-centered lattice I = body-centered lattice... 

Double-spaced, face-centered lattice of chemisorbed oxygen in (110) azimuth at 17 volts. Multiply current scale by 6... 

The K3C60 crystal can be described as having the BiF3 structure, with the Ceo molecules replacing Bi at the cubic face-center lattice points 0 0 0, etc. (two different and random orientations), four K atoms located at the centers of the octahedral interstices,... 

For a face-centered lattice F, some of the lattice points are at xyz = mnp some are at... 

Show that the zinc blende structure can be described as having zinc and sulfide ions each in face-centered lattices, merged so that each ion is in a tetrahedral hole of the other lattice. 

FIGURE 21.9 Centered lattices, like all lattices, have lattice points at the eight corners of the unit cell. A body-centered lattice has an additional lattice point at the center of the cell, a face-centered lattice has additional points at the centers of the six faces, and a side-centered lattice has points at the centers of two parallel sides of the unit cell. (Note The colored dots in the lattice diagrams represent lattice points, not atoms.)... 

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). 

The latter example is illustrated in Figure 1.27, where a tetragonal face-centered lattice is reduced to a tetragonal body-centered lattice, which has the same symmetry but half the volume of the unit cell. The reduction is carried out using the transformations of basis vectors as shown in Eqs. 1.6 through 1.8. 

For example, considering the crystal structure of copper, which has cubic face-centered lattice Figure 6.2) and a total of 4 atoms in the unit cell, its Pearson s symbol is cF4. On the other hand, if the material has Pearson s symbol oI32, this means that its crystal structure is orthorhombic, and one body-centered unit cell contains a total of 32 atoms. 

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. 

Here the terms corresponding to the face-centering translations appear in the first factor the second factor contains the terms that describe the basis of the unit cell, namely, the Na atom at 0 0 0 and the Cl atom at The terms in the first bracket, describing the face-centering translations, have already appeared in example (d), and they were found to have a total value of zero for mixed indices and 4 for unmixed indices. This shows at once that NaCl has a face-centered lattice and that... 

Fig. 3. Peak current versus logio (pressure x time) for five diffraction beams obtained after an intermediate anneal. Oxygen exposure occurred at 26°C. Curve 1 Typical beam from clean nickel in the (110) azimuth at about 28 volts. Curve 2 Typical beam from clean nickel in the (001) azimuth at about 58 volts. Curve 3 Typical beam from a double-spaced, face-centered lattice of chemisorbed oxygen in the (110) azimuth at about 17 volts. Curve 4 Typical beam from a sss lattice in the (001) azimuth at about 27 volts. Curve 5 Typical beam from a NiO lattice in the (110) azimuth at about 22 volts. [From Farnsworth and Tuul (28).]...

Jescribe the face-centered lattice. Note that this determination of lattice points is independent of the angles between axes. 

Mathematics (Hassel, 1830) has shown that there are only 32 combinations of symmetry operations (rotation, inversion, and reflection) that are consistent with a three-dimensional crystal lattice. These 32 point groups, or crystal classes, can be grouped into one of the seven crystal systems given in Table 2.1. There are four types of crystal lattices primitive (P), end-centered (C, B, and A), face-centered (/O, and body-centered (/). The primitive lattice contains a lattice point at each comer of the unit cell, the end-centered lattice has an additional lattice point on one of the lattice faces, the face-centered lattice has an extra lattice on each of the lattice faces, and the body-centered lattice has an extra lattice point at the center of the crystal lattice. By combining the seven crystal systems with the four lattice types (P, C, I, F), 14 unique crystal lattices, also known as Bravais lattices (Bravais, 1849), are produced. 

Many oxide minerals can be visualized as a face-centered oxide ion lattice, with cations distributed within the tetrahedral and octahedral holes. Calculate the lattice constant, a, for a face-centered lattice. If cations occupy all the octahedral holes in MgO and CaO, calculate a for these minerals. Use data in Table 10-1. 

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