343·Mechanisms FIGURE 3.9.Hexagonal close-packed (HCP)crystal structure.The sixfold symmetry of the lattice is evident. One unit cell is shaded for clarity. There are three crystallographically equivalent possibilities for this unit cell.The stacking sequence (ABA), which will be explained later,is also depicted. -0 Coordination The coordination number is the number of nearest neighbors to Number a given atom.For example,the center atom in a BCC structure [Figures 3.8 and 3.10(a)]has eight nearest atoms.Its coordination number is therefore 8.The coordination numbers for some crys- tal structures are listed in Table 3.2. Atoms per The number of atoms per unit cell is counted by taking into Unit Cell consideration that corner atoms in cubic crystals are shared by eight unit cells and face atoms are shared by two unit cells.They count therefore only 1/8 and 1/2,respectively.As an example,the number of atoms associated with a BCC structure (assuming only one atom per lattice point)is 8 x 1/8 =1 corner atom and one (not shared)center atom,yielding a total of two atoms per unit cell.In contrast to this,an FCC unit cell has four atoms (8X 1/8+6 X 1/2).The FCC unit cell is therefore more densely packed with atoms than the BCC unit cell;see also Table 3.2. Packing The packing factor,P,is that portion of space within a unit cell Factor which is filled with spherical atoms that touch each other,i.e.: P=Auc*VA Vuc (3.1) FIGURE 3.10.Geometric arrangement of atoms considered to be hard spheres for a BCC structure (a)and an FCC structure (b). (a) (b)
The coordination number is the number of nearest neighbors to a given atom. For example, the center atom in a BCC structure [Figures 3.8 and 3.10(a)] has eight nearest atoms. Its coordination number is therefore 8. The coordination numbers for some crystal structures are listed in Table 3.2. The number of atoms per unit cell is counted by taking into consideration that corner atoms in cubic crystals are shared by eight unit cells and face atoms are shared by two unit cells. They count therefore only 1/8 and 1/2, respectively. As an example, the number of atoms associated with a BCC structure (assuming only one atom per lattice point) is 8 1/8 1 corner atom and one (not shared) center atom, yielding a total of two atoms per unit cell. In contrast to this, an FCC unit cell has four atoms (8 1/8 6 1/2). The FCC unit cell is therefore more densely packed with atoms than the BCC unit cell; see also Table 3.2. The packing factor, P, is that portion of space within a unit cell which is filled with spherical atoms that touch each other, i.e.: P , (3.1) Auc*VA Vuc FIGURE 3.9. Hexagonal close-packed (HCP) crystal structure. The sixfold symmetry of the lattice is evident. One unit cell is shaded for clarity. There are three crystallographically equivalent possibilities for this unit cell. The stacking sequence (ABA), which will be explained later, is also depicted. 34 3 • Mechanisms A B A c0 a0 FIGURE 3.10. Geometric arrangement of atoms considered to be hard spheres for a BCC structure (a) and an FCC structure (b). Coordination Number Atoms per Unit Cell Packing Factor (a) (b)
3.3.Arrangement of Atoms (Crystallography) 35 TABLE 3.2.Some parameters and properties of different crystal structures Crystal Coordination Atoms per Packing Mechanical structure number unit cell factor properties HCP 12* 2 0.74* Brittle FCC 12 4 0.74 Ductile BCC 2 0.68 Hard Simple cubic 6 1 0.52 No representative (SC) materials *Assuming a c/a ratio of 1.633. where Aue is the number of atoms per unit cell (see above), VA=3mr3 is the volume of an atom (assuming hard spheres of radius r),and Vie is the volume of the unit cell.The packing fac- tor in FCC and HCP structures is 0.74,whereas a BCC crystal is less densely packed,having a packing factor of 0.68.(Note that P=0.74 for HCP structures is only true if the c/a ratio equals the ideal value of 1.633.) Linear and The linear packing fraction is the portion of a line through the Planar centers of atoms in a specific direction that is filled by atoms. For example,the linear packing fraction in the direction of the Packing face diagonal of an FCC unit cell in which the corner atoms and Fractions the face atom touch each other [Figure 3.10(b)]is 100%.Simi- larly,a planar packing fraction can be defined as that portion of a given plane that is filled by atoms. Density The density,6,of a material can be calculated by using: 5=AuetMa=NatMa (3.2) Vuc*NoNo where Ma is the atomic mass of the atom (see Appendix IID),Na is the number of atoms per volume,and No is a constant called the Avogadro number (see Appendix II).Experimental densities of materials are given in Appendix III. Stacking Sequence A stacking sequence describes the pattern in which close-packed atomic planes are piled up in three dimensions.To explain this, let us assume for simplicity that the atoms are represented by hard spheres and are arranged on a plane in such a way to take up the least amount of space,as shown in Figure 3.11(a).The atoms are said to be situated in a close-packed plane which we designate here as plane A.We now attempt to construct a regu- lar and repetitive array of atoms in three dimensions.First we
where Auc is the number of atoms per unit cell (see above), VA 4 3 r3 is the volume of an atom (assuming hard spheres of radius r), and Vuc is the volume of the unit cell. The packing factor in FCC and HCP structures is 0.74, whereas a BCC crystal is less densely packed, having a packing factor of 0.68. (Note that P 0.74 for HCP structures is only true if the c/a ratio equals the ideal value of 1.633.) The linear packing fraction is the portion of a line through the centers of atoms in a specific direction that is filled by atoms. For example, the linear packing fraction in the direction of the face diagonal of an FCC unit cell in which the corner atoms and the face atom touch each other [Figure 3.10(b)] is 100%. Similarly, a planar packing fraction can be defined as that portion of a given plane that is filled by atoms. The density, , of a material can be calculated by using: , (3.2) where Ma is the atomic mass of the atom (see Appendix III), Na is the number of atoms per volume, and N0 is a constant called the Avogadro number (see Appendix II). Experimental densities of materials are given in Appendix III. A stacking sequence describes the pattern in which close-packed atomic planes are piled up in three dimensions. To explain this, let us assume for simplicity that the atoms are represented by hard spheres and are arranged on a plane in such a way to take up the least amount of space, as shown in Figure 3.11(a). The atoms are said to be situated in a close-packed plane which we designate here as plane A. We now attempt to construct a regular and repetitive array of atoms in three dimensions. First we Na*Ma No Auc*Ma Vuc*N0 3.3 • Arrangement of Atoms (Crystallography) 35 TABLE 3.2. Some parameters and properties of different crystal structures Crystal Coordination Atoms per Packing Mechanical structure number unit cell factor properties HCP 12* 2 0.74* Brittle FCC 12 4 0.74 Ductile BCC 8 2 0.68 Hard Simple cubic 6 1 0.52 No representative (SC) materials *Assuming a c/a ratio of 1.633. Linear and Planar Packing Fractions Density Stacking Sequence
363·Mechanisms a FIGURE 3.11.(a)A two- dimensional represen- tation of a three-di- mensional stacking sequence for close- packed lattice planes. (b)Close-packed planes for an HCP crystal structure.(c) Close-packed planes in an FCC structure. (b) (c) fill the depressions marked B,which are formed by each set of three atoms in plane A.This constitutes the second plane of atoms.When filling the third level with atoms,one has two choices.One may arrange the third level identical to the first one. The stacking sequence is then said to be ABAB,which represents the HCP crystal structure depicted in Figures 3.9 and 3.11(b).In other words,atoms in the first and third levels are exactly on top of each other.As an alternative,the hitherto unoccupied spaces, C,may be occupied on the third level.Only the fourth close- packed atomic plane is finally identical to the first one,which leads to a stacking sequence called ABCABC.The diagonal planes in an FCC crystal structure have this stacking sequence [Figure 3.11(c)]
fill the depressions marked B, which are formed by each set of three atoms in plane A. This constitutes the second plane of atoms. When filling the third level with atoms, one has two choices. One may arrange the third level identical to the first one. The stacking sequence is then said to be ABAB, which represents the HCP crystal structure depicted in Figures 3.9 and 3.11(b). In other words, atoms in the first and third levels are exactly on top of each other. As an alternative, the hitherto unoccupied spaces, C, may be occupied on the third level. Only the fourth closepacked atomic plane is finally identical to the first one, which leads to a stacking sequence called ABC ABC. The diagonal planes in an FCC crystal structure have this stacking sequence [Figure 3.11(c)]. FIGURE 3.11. (a) A twodimensional representation of a three-dimensional stacking sequence for closepacked lattice planes. (b) Close-packed planes for an HCP crystal structure. (c) Close-packed planes in an FCC structure. 36 3 • Mechanisms (a) (b) (c)
3.3.Arrangement of Atoms (Crystallography) 37 In conclusion,HCP and FCC structures both have the highest possible packing efficiency(Table 3.2),but differ in their stack- ing sequence.Both crystal structures are quite common among metals.The preference of one structure over the other is rooted in the tendency of atomic systems to assume the lowest possible energy level. Stacking In some instances,the regular stacking sequence may be inter- Faults and rupted.For example,a stacking sequence ABC AB ABC may oc- cur instead of ABC ABC ABC.This configuration,where one layer Twinning is missing,is called an intrinsic stacking fault.On the other hand, the sequence ABC B ABC,where one B layer has been added,is termed an extrinsic stacking fault.Another defect may involve the stacking sequence ABC ABC ABC BACBAC ABC ABC (Figure 3.12).Here,the order is inverted about the layers marked C due to a twin which is incorporated in the FCC crystal.Such config- urations may be formed during the heating of a previously de- formed FCC material (annealing twins)or by plastic deformation (deformation twins).We shall discuss the implications of stack- ing faults and twinning at a later point. Polymorphic Some materials have different crystal structures at lower tem- and Allotropic peratures than at higher temperatures.(The inherent energy of a crystal structure is temperature-dependent.)Materials that un- Materials dergo a transformation from one crystal structure to another are called polymorphic or,when referring to elements,allotropic.A well-known example is the allotropic transformation of room C B A Twinning B Matrix C Plane B C B Twin A Twinning FIGURE 3.12.Stacking faults in an Plane FCC metal to yield twinning.The B matrix and the twin match at the C Matrix interface,called the twinning plane. A The figure shows a (110)plane nor- B mal to the (111)twin plane,as we C shall explain below
In conclusion, HCP and FCC structures both have the highest possible packing efficiency (Table 3.2), but differ in their stacking sequence. Both crystal structures are quite common among metals. The preference of one structure over the other is rooted in the tendency of atomic systems to assume the lowest possible energy level. In some instances, the regular stacking sequence may be interrupted. For example, a stacking sequence ABC AB ABC may occur instead of ABC ABC ABC. This configuration, where one layer is missing, is called an intrinsic stacking fault. On the other hand, the sequence ABC B ABC, where one B layer has been added, is termed an extrinsic stacking fault. Another defect may involve the stacking sequence ABC ABC ABC BAC BAC ABC ABC (Figure 3.12). Here, the order is inverted about the layers marked C due to a twin which is incorporated in the FCC crystal. Such configurations may be formed during the heating of a previously deformed FCC material (annealing twins) or by plastic deformation (deformation twins). We shall discuss the implications of stacking faults and twinning at a later point. Some materials have different crystal structures at lower temperatures than at higher temperatures. (The inherent energy of a crystal structure is temperature-dependent.) Materials that undergo a transformation from one crystal structure to another are called polymorphic or, when referring to elements, allotropic. A well-known example is the allotropic transformation of room 3.3 • Arrangement of Atoms (Crystallography) 37 C A B C A B C B A C B A C A B C A B C Matrix Twin Twinning Plane Twinning Plane Matrix FIGURE 3.12. Stacking faults in an FCC metal to yield twinning. The matrix and the twin match at the interface, called the twinning plane. The figure shows a (110) plane normal to the (111) twin plane, as we shall explain below. Stacking Faults and Twinning Polymorphic and Allotropic Materials