ELEMENTARY CRYSTALLOGRAPHY

The two previous sections discussed the equivalency of nonparallel crystallographic
directions and planes. Directional equivalency is related to linear density in the sense
that, for a particular material, equivalent directions have identical linear densities.
The corresponding parameter for crystallographic planes is planar density, and
planes having the same planar density values are also equivalent.
Linear density (LD) is defined as the number of atoms per unit length whose
centers lie on the direction vector for a specific crystallographic direction; that is,
(3.8)
Of course, the units of linear density are reciprocal length (e.g., ).
For example, let us determine the linear density of the [110] direction for the
FCC crystal structure. An FCC unit cell (reduced sphere) and the [110] direction
therein are shown in Figure 3.12a. Represented in Figure 3.12b are those five atoms
that lie on the bottom face of this unit cell; here the [110] direction vector passes
from the center of atom X, through atom Y, and finally to the center of atom Z.
With regard to the numbers of atoms, it is necessary to take into account the sharing
of atoms with adjacent unit cells (as discussed in Section 3.4 relative to atomic packing factor computations). Each of the X and Z corner atoms are also shared with
one other adjacent unit cell along this [110] direction (i.e., one-half of each of these
atoms belongs to the unit cell being considered), while atom Y lies entirely within
the unit cell. Thus, there is an equivalence of two atoms along the [110] direction
vector in the unit cell. Now, the direction vector length is equal to 4R (Figure 3.12b);
thus, from Equation 3.8, the [110] linear density for FCC is
LD110  2 atoms (3.9)
4R

1
2R
nm1, m1
LD 
number of atoms centered on direction vector
length of direction vector
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In an analogous manner, planar density (PD) is taken as the number of atoms
per unit area that are centered on a particular crystallographic plane, or
(3.10)
The units for planar density are reciprocal area (e.g., ).
For example, consider the section of a (110) plane within an FCC unit cell as
represented in Figures 3.10a and 3.10b. Although six atoms have centers that lie on
this plane (Figure 3.10b), only one-quarter of each of atoms A, C, D, and F, and onehalf of atoms B and E, for a total equivalence of just 2 atoms are on that plane.
Furthermore, the area of this rectangular section is equal to the product of its length
and width. From Figure 3.10b, the length (horizontal dimension) is equal to 4R,
whereas the width (vertical dimension) is equal to , since it corresponds to
the FCC unit cell edge length (Equation 3.1). Thus, the area of this planar region
is and the planar density is determined as follows:
(3.11)
Linear and planar densities are important considerations relative to the process of
slip—that is, the mechanism by which metals plastically deform (Section 7.4). Slip
occurs on the most densely packed crystallographic planes and, in those planes,
along directions having the greatest atomic packing.
3.12 CLOSE-PACKED CRYSTAL STRUCTURES
You may remember from the discussion on metallic crystal structures 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 3.13a. 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. Associated 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 3.13a.
PD
110 
2 atoms
8R212 
1
4R212
14R212R122  8R212,
2R12
nm2, m2
PD 
number of atoms centered on a plane
area of plane
3.12 Close-Packed Crystal Structures • 61
Figure 3.12 (a) Reducedsphere FCC unit cell with
the [110] direction
indicated. (b) The bottom
face-plane of the FCC unit
cell in (a) on which is
shown the atomic spacing
in the [110] direction,
through atoms labeled X,

Close-Packed
Structures (Metals)
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 3.13b. The real distinction between FCC and HCP lies in
where 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 3.14.
For the face-centered crystal structure, the centers of the third plane are situated
over the C sites of the first plane (Figure 3.15a). 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 3.15b. These planes are of
the (111) type; an FCC unit cell is outlined on the upper left-hand front face of
Figure 3.15b, in order to provide a perspective. The significance of these FCC and
HCP close-packed planes will become apparent in Chapter 7.
The concepts detailed in the previous four sections also relate to crystalline ceramic and polymeric materials, which are discussed in Chapters 12 and 14. We may
specify crystallographic planes and directions in terms of directional and Miller indices; furthermore, on occasion it is important to ascertain the atomic and ionic
arrangements of particular crystallographic planes. Also, the crystal structures of a
number of ceramic materials may be generated by the stacking of close-packed
planes of ions
Single crystals exist in nature, but they may also be produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled.
If the extremities of a single crystal are permitted to grow without any external constraint, the crystal will assume a regular geometric shape having flat faces,
as with some of the gem stones; the shape is indicative of the crystal structure. A
photograph of a garnet single crystal is shown in Figure 3.16. Within the past few
years, single crystals have become extremely important in many of our modern technologies, in particular electronic microcircuits, which employ single crystals of silicon and other semiconductors.
Most crystalline solids are composed of a collection of many small crystals or grains;
such materials are termed polycrystalline. Various stages in the solidification of a
polycrystalline specimen are represented schematically in Figure 3.17. Initially, small
crystals or nuclei form at various positions. These have random crystallographic orientations, as indicated by the square grids. The small grains grow by the successive
addition from the surrounding liquid of atoms to the structure of each. The extremities of adjacent grains impinge on one another as the solidification process approaches completion. As indicated in Figure 3.17, the crystallographic orientation
varies from grain to grain. Also, there exists some atomic mismatch within the region where two grains meet; this area, called a grain boundary, is discussed in more
detail in Section 4.6.
The physical properties of single crystals of some substances depend on the crystallographic direction in which measurements are taken. For example, the elastic
modulus, the electrical conductivity, and the index of refraction may have different
values in the [100] and [111] directions. This directionality of properties is termed
anisotropy, and it is associated with the variance of atomic or ionic spacing with
crystallographic direction. Substances in which measured properties are independent of the direction of measurement are isotropic. The extent and magnitude of anisotropic effects in crystalline materials are functions of the symmetry of the
crystal structure; the degree of anisotropy increases with decreasing structural
symmetry—triclinic structures normally are highly anisotropic.The modulus of elasticity values at [100], [110], and [111] orientations for several materials are presented
in Table 3.3.
For many polycrystalline materials, the crystallographic orientations of the individual grains are totally random. Under these circumstances, even though each grain
may be anisotropic, a specimen composed of the grain aggregate behaves isotropically.
Also, the magnitude of a measured property represents some average of the directional
values. Sometimes the grains in polycrystalline materials have a preferential crystallographic orientation, in which case the material is said to have a “texture.”
The magnetic properties of some iron alloys used in transformer cores are
anisotropic—that is, grains (or single crystals) magnetize in a 81009-type direction
easier than any other crystallographic direction. Energy losses in transformer cores
are minimized by utilizing polycrystalline sheets of these alloys into which have
been introduced a “magnetic texture”: most of the grains in each sheet have a 81009-type crystallographic direction that is aligned (or almost aligned) in the same
direction, which direction is oriented parallel to the direction of the applied magnetic field. Magnetic textures for iron alloys are discussed in detail in the Materials of Importance piece in Chapter 20 following Section 20.9.
Historically, much of our understanding regarding the atomic and molecular
arrangements in solids has resulted from x-ray diffraction investigations; furthermore, x-rays are still very important in developing new materials. We will now give
a brief overview of the diffraction phenomenon and how, using x-rays, atomic
interplanar distances and crystal structures are deduced.
The Diffraction Phenomenon
Diffraction occurs when a wave encounters a series of regularly spaced obstacles
that (1) are capable of scattering the wave, and (2) have spacings that are comparable in magnitude to the wavelength. Furthermore, diffraction is a consequence of
specific phase relationships established between two or more waves that have been
scattered by the obstacles.
Consider waves 1 and 2 in Figure 3.18a which have the same wavelength
and are in phase at point . Now let us suppose that both waves are scattered
in such a way that they traverse different paths. The phase relationship between
the scattered waves, which will depend upon the difference in path length, is important. One possibility results when this path length difference is an integral number of wavelengths. As noted in Figure 3.18a, these scattered waves (now labeled
and ) are still in phase. They are said to mutually reinforce (or constructively
interfere with) one another; and, when amplitudes are added, the wave shown on
the right side of the figure results. This is a manifestation of diffraction, and we refer to a diffracted beam as one composed of a large number of scattered waves that
mutually reinforce one another.
Other phase relationships are possible between scattered waves that will not
lead to this mutual reinforcement. The other extreme is that demonstrated in
Figure 3.18b, wherein the path length difference after scattering is some integral
number of half wavelengths. The scattered waves are out of phase—that is, corresponding amplitudes cancel or annul one another, or destructively interfere (i.e., the resultant wave has zero amplitude), as indicated on the extreme right side of the
figure. Of course, phase relationships intermediate between these two extremes
exist, resulting in only partial reinforcement.
X-Ray Diffraction and Bragg’s Law
X-rays are a form of electromagnetic radiation that have high energies and short
wavelengths—wavelengths on the order of the atomic spacings for solids. When a
beam of x-rays impinges on a solid material, a portion of this beam will be scattered
in all directions by the electrons associated with each atom or ion that lies within
the beam’s path. Let us now examine the necessary conditions for diffraction of
x-rays by a periodic arrangement of atoms.
Consider the two parallel planes of atoms and in Figure 3.19, which
have the same h, k, and l Miller indices and are separated by the interplanar spacing . Now assume that a parallel, monochromatic, and coherent (in-phase) beam
of x-rays of wavelength is incident on these two planes at an angle . Two rays in
this beam, labeled 1 and 2, are scattered by atoms P and Q. Constructive interference of the scattered rays and occurs also at an angle to the planes, if the
path length difference between and ( ) is equal to a
whole number, n, of wavelengths. That is, the condition for diffraction is Equation 3.13 is known as Bragg’s law; also, n is the order of reflection, which
may be any integer (1, 2, 3, . . . ) consistent with not exceeding unity. Thus, we
have a simple expression relating the x-ray wavelength and interatomic spacing to
the angle of the diffracted beam. If Bragg’s law is not satisfied, then the interference will be nonconstructive in nature so as to yield a very low-intensity diffracted
beam.
The magnitude of the distance between two adjacent and parallel planes of
atoms (i.e., the interplanar spacing ) is a function of the Miller indices (h, k,
and l) as well as the lattice parameter(s). For example, for crystal structures that have
cubic symmetry,
(3.14)
in which a is the lattice parameter (unit cell edge length). Relationships similar to
Equation 3.14, but more complex, exist for the other six crystal systems noted in
Table 3.2.
Bragg’s law, Equation 3.13, is a necessary but not sufficient condition for diffraction by real crystals. It specifies when diffraction will occur for unit cells having atoms positioned only at cell corners. However, atoms situated at other sites
(e.g., face and interior unit cell positions as with FCC and BCC) act as extra scattering centers, which can produce out-of-phase scattering at certain Bragg angles.
The net result is the absence of some diffracted beams that, according to Equation
3.13, should be present. For example, for the BCC crystal structure, must
be even if diffraction is to occur, whereas for FCC, h, k, and l must all be either odd
or even.