ATOMIC ARRANGEMENTS

In document Fundamentals of Materials Science and Engineering 5th Edition pdf (Page 77-82)

The atomic arrangement for a crystallographic plane, which is often of interest, depends on the crystal structure. The (110) atomic planes for FCC and BCC crystal structures are represented in Figures 3.24 and 3.25; reduced-sphere unit cells are also included. Note that the atomic packing is different for each case. The circles represent atoms lying in the crystallographic planes as would be obtained from a slice taken through the centers of the full-sized hard spheres.

A ‘‘family’’ of planes contains all those planes that are crystallographically equivalent—that is, having the same atomic packing; and a family is designated by indices that are enclosed in braces—e.g.,兵100其. For example, in cubic crystals the (111), (111), (111), (111), (111), (111), (111), and (111) planes all belong to the

兵111其family. On the other hand, for tetragonal crystal structures, the兵100其family would contain only the (100), (100), (010), and (010) since the (001) and (001) planes are not crystallographically equivalent. Also, in the cubic system only, planes having the same indices, irrespective of order and sign, are equivalent. For example, both (123) and (312) belong to the兵123其family.

HEXAGONAL CRYSTALS

For crystals having hexagonal symmetry, it is desirable that equivalent planes have the same indices; as with directions, this is accomplished by the Miller–Bravais

z y x a b y b O c y f e (a) (b) x Point of intersection along y axis z g h (011) Plane A A B C D E F B C F D E (a) (b)

FIGURE3.24 (a) Reduced-

sphere FCC unit cell with (110) plane. (b) Atomic packing of an FCC (110) plane. Corresponding atom positions from (a) are indicated.

58Chapter 3 / Structures of Metals and Ceramics

system shown in Figure 3.21. This convention leads to the four-index (hkil) scheme, which is favored in most instances, since it more clearly identifies the orientation of a plane in a hexagonal crystal. There is some redundancy in that i is determined by the sum of h and k through

i⫽ ⫺(hk) (3.8)

Otherwise the three h, k, and l indices are identical for both indexing systems. Figure 3.22b presents several of the common planes that are found for crystals having hexagonal symmetry.

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It may be remembered from the discussion on metallic crystal structures (Section 3.4) that both face-centered cubic and hexagonal close-packed crystal structures have atomic packing factors of 0.74, which is the most efficient packing of equal- sized spheres or atoms. In addition to unit cell representations, these two crystal structures may be described in terms of close-packed planes of atoms (i.e., planes having a maximum atom or sphere-packing density); a portion of one such plane is illustrated in Figure 3.27a. Both crystal structures may be generated by the stacking of these close-packed planes on top of one another; the difference between the two structures lies in the stacking sequence.

Let the centers of all the atoms in one close-packed plane be labeled A. Associ- ated with this plane are two sets of equivalent triangular depressions formed by three adjacent atoms, into which the next close-packed plane of atoms may rest. Those having the triangle vertex pointing up are arbitrarily designated as B positions, while the remaining depressions are those with the down vertices, which are marked

C in Figure 3.27a.

A second close-packed plane may be positioned with the centers of its atoms over either B or C sites; at this point both are equivalent. Suppose that the B positions are arbitrarily chosen; the stacking sequence is termed AB, which is illustrated in Figure 3.27b. The real distinction between FCC and HCP lies in where

A B C E D (a) (b) A B D E C ⬘ ⬘ ⬘ ⬘ ⬘ ⬘ ⬘ ⬘ ⬘ ⬘ FIGURE3.25 (a) Reduced-sphere BCC unit cell with (110) plane. (b) Atomic packing of a BCC (110) plane. Corresponding atom positions from (a) are indicated.

3.15 Close-Packed Crystal Structures59

FIGURE3.27 (a) A portion

of a close-packed plane of atoms; A, B, and C positions are indicated. (b) The AB stacking sequence for close-packed atomic planes. (Adapted from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The

Structure and Properties of Materials, Vol. I, Structure,

p. 50. Copyright1964 by John Wiley & Sons, New York. Reprinted by permission of John Wiley & Sons, Inc.)

the third close-packed layer is positioned. For HCP, the centers of this layer are aligned directly above the original A positions. This stacking sequence, ABABAB . . . , is repeated over and over. Of course, the ACACAC . . . arrangement would be equivalent. These close-packed planes for HCP are (0001)-type planes, and the correspondence between this and the unit cell representation is shown in Figure 3.28. For the face-centered crystal structure, the centers of the third plane are situated over the C sites of the first plane (Figure 3.29a). This yields an ABCABCABC . . . stacking sequence; that is, the atomic alignment repeats every third plane. It is more difficult to correlate the stacking of close-packed planes to the FCC unit cell. However, this relationship is demonstrated in Figure 3.29b; these planes are of the (111) type. The significance of these FCC and HCP close-packed planes will become apparent in Chapter 8.

FIGURE3.28 Close-packed plane stacking sequence for hexagonal close-packed.

(Adapted from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties of

Materials, Vol. I, Structure, p. 51. Copyright

1964 by John Wiley & Sons, New York. Reprinted by permission of John Wiley & Sons, Inc.)

60Chapter 3 / Structures of Metals and Ceramics

CERAMICS

A number of ceramic crystal structures may also be considered in terms of close- packed planes of ions (as opposed to atoms for metals), as well as unit cells. Ordinarily, the close-packed planes are composed of the large anions. As these planes are stacked atop each other, small interstitial sites are created between them in which the cations may reside.

These interstitial positions exist in two different types, as illustrated in Figure 3.30. Four atoms (three in one plane, and a single one in the adjacent plane) surround one type, labeled T in the figure; this is termed a tetrahedral position, since straight lines drawn from the centers of the surrounding spheres form a four- sided tetrahedron. The other site type, denoted as O in Figure 3.30, involves six ion spheres, three in each of the two planes. Because an octahedron is produced by joining these six sphere centers, this site is called an octahedral position. Thus, the coordination numbers for cations filling tetrahedral and octahedral positions are 4 and 6, respectively. Furthermore, for each of these anion spheres, one octahe- dral and two tetrahedral positions will exist.

Ceramic crystal structures of this type depend on two factors: (1) the stacking of the close-packed anion layers (both FCC and HCP arrangements are possible, which correspond to ABCABC . . . and ABABAB . . . sequences, respectively), and (2) the manner in which the interstitial sites are filled with cations. For example, consider the rock salt crystal structure discussed above. The unit cell has cubic symmetry, and each cation (Na⫹ion) has six Cl⫺ion nearest neighbors, as may be verified from Figure 3.5. That is, the Na⫹ion at the center has as nearest neighbors FIGURE3.29 (a) Close-packed stacking sequence for face-centered cubic. (b) A

corner has been removed to show the relation between the stacking of close- packed planes of atoms and the FCC crystal structure; the heavy triangle outlines a (111) plane. (Figure (b) from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The

Structure and Properties of Materials, Vol. I, Structure, p. 51. Copyright1964 by John Wiley & Sons, New York. Reprinted by permission of John Wiley & Sons, Inc.)

3.15 Close-Packed Crystal Structures61

the six Cl⫺ions that reside at the centers of each of the cube faces. The crystal structure, having cubic symmetry, may be considered in terms of an FCC array of close-packed planes of anions, and all planes are of the {111} type. The cations reside in octahedral positions because they have as nearest neighbors six anions. Furthermore, all octahedral positions are filled, since there is a single octahedral site per anion, and the ratio of anions to cations is 1 : 1. For this crystal structure, the relationship between the unit cell and close-packed anion plane stacking schemes is illustrated in Figure 3.31.

Other, but not all, ceramic crystal structures may be treated in a similar manner; included are the zinc blende and perovskite structures. The spinel structure is one of the AmBnXptypes, which is found for magnesium aluminate or spinel (MgAl2O4).

With this structure, the O2⫺ions form an FCC lattice, whereas Mg2⫹ions fill tetrahe- dral sites and Al3⫹reside in octahedral positions. Magnetic ceramics, or ferrites, have a crystal structure that is a slight variant of this spinel structure; and the magnetic characteristics are affected by the occupancy of tetrahedral and octahedral positions兵(see Section 18.5).其

FIGURE3.30 The stacking of one plane of close-packed spheres (anions) on top

of another; tetrahedral and octahedral positions between the planes are designated by T and O, respectively. (From W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. 1, Structure. Copyright

1964 by John Wiley & Sons, New York. Reprinted by permission of John Wiley & Sons, Inc.)

FIGURE3.31 A section of the rock salt

crystal structure from which a corner has been removed. The exposed plane of anions (light spheres inside the triangle) is a {111}- type plane; the cations (dark spheres) occupy the interstitial octahedral positions.

62Chapter 3 / Structures of Metals and Ceramics

C R Y S T A L L I N E A N D N O N C R Y S T A L L I N E

In document Fundamentals of Materials Science and Engineering 5th Edition pdf (Page 77-82)