Crystal Stracture

Solid materials may be classified according to the regularity with which atoms or
ions are arranged with respect to one another. A crystalline material is one in which
the atoms are situated in a repeating or periodic array over large atomic distances;
that is, long-range order exists, such that upon solidification, the atoms will position
themselves in a repetitive three-dimensional pattern, in which each atom is bonded
to its nearest-neighbor atoms. All metals, many ceramic materials, and certain polymers form crystalline structures under normal solidification conditions. For those
that do not crystallize, this long-range atomic order is absent; these noncrystalline
or amorphous materials are discussed briefly at the end of this chapter.
Some of the properties of crystalline solids depend on the crystal structure of
the material, the manner in which atoms, ions, or molecules are spatially arranged.
There is an extremely large number of different crystal structures all having longrange atomic order; these vary from relatively simple structures for metals to
exceedingly complex ones, as displayed by some of the ceramic and polymeric
materials. The present discussion deals with several common metallic crystal structures. Chapters 12 and 14 are devoted to crystal structures for ceramics and polymers, respectively.
When describing crystalline structures, atoms (or ions) are thought of as being
solid spheres having well-defined diameters. This is termed the atomic hard sphere
model in which spheres representing nearest-neighbor atoms touch one another.
An example of the hard sphere model for the atomic arrangement found in some
of the common elemental metals is displayed in Figure 3.1c. In this particular case
all the atoms are identical. Sometimes the term lattice is used in the context of crystal structures; in this sense “lattice” means a three-dimensional array of points
coinciding with atom positions (or sphere centers).
The atomic order in crystalline solids indicates that small groups of atoms form a
repetitive pattern.Thus, in describing crystal structures, it is often convenient to subdivide the structure into small repeat entities called unit cells. Unit cells for most
crystal structures are parallelepipeds or prisms having three sets of parallel faces;
one is drawn within the aggregate of spheres (Figure 3.1c), which in this case happens to be a cube. A unit cell is chosen to represent the symmetry of the crystal
structure, wherein all the atom positions in the crystal may be generated by translations of the unit cell integral distances along each of its edges. Thus, the unit cell
is the basic structural unit or building block of the crystal structure and defines
the crystal structure by virtue of its geometry and the atom positions within.
Convenience usually dictates that parallelepiped corners coincide with centers of
the hard sphere atoms. Furthermore, more than a single unit cell may be chosen for
a particular crystal structure; however, we generally use the unit cell having the
highest level of geometrical symmetry.
The atomic bonding in this group of materials is metallic and thus nondirectional
in nature. Consequently, there are minimal restrictions as to the number and position of nearest-neighbor atoms; this leads to relatively large numbers of nearest
neighbors and dense atomic packings for most metallic crystal structures. Also, for
metals, using the hard sphere model for the crystal structure, each sphere represents an ion core. Table 3.1 presents the atomic radii for a number of metals. Three
relatively simple crystal structures are found for most of the common metals: facecentered cubic, body-centered cubic, and hexagonal close-packed.
The crystal structure found for many metals has a unit cell of cubic geometry, with
atoms located at each of the corners and the centers of all the cube faces. It is aptly
called the face-centered cubic (FCC) crystal structure. Some of the familiar metals having this crystal structure are copper, aluminum, silver, and gold (see also Table 3.1).
Figure 3.1a shows a hard sphere model for the FCC unit cell, whereas in Figure 3.1b
the atom centers are represented by small circles to provide a better perspective of
atom positions. The aggregate of atoms in Figure 3.1c represents a section of crystal
consisting of many FCC unit cells.These spheres or ion cores touch one another across
a face diagonal; the cube edge length a and the atomic radius R are related through
This result is obtained in Example Problem 3.1.
For the FCC crystal structure, each corner atom is shared among eight unit cells,
whereas a face-centered atom belongs to only two. Therefore, one-eighth of each of
the eight corner atoms and one-half of each of the six face atoms, or a total of four
whole atoms, may be assigned to a given unit cell. This is depicted in Figure 3.1a,
where only sphere portions are represented within the confines of the cube. The
cell comprises the volume of the cube, which is generated from the centers of the
corner atoms as shown in the figure.
Corner and face positions are really equivalent; that is, translation of the cube
corner from an original corner atom to the center of a face atom will not alter the
cell structure.
Two other important characteristics of a crystal structure are the coordination
number and the atomic packing factor (APF). For metals, each atom has the same
number of nearest-neighbor or touching atoms, which is the coordination number.
For face-centered cubics, the coordination number is 12. This may be confirmed by
examination of Figure 3.1a; the front face atom has four corner nearest-neighbor
atoms surrounding it, four face atoms that are in contact from behind, and four other
equivalent face atoms residing in the next unit cell to the front, which is not shown.
The APF is the sum of the sphere volumes of all atoms within a unit cell (assuming the atomic hard sphere model) divided by the unit cell volume—that is
For the FCC structure, the atomic packing factor is 0.74, which is the maximum
packing possible for spheres all having the same diameter. Computation of this APF
is also included as an example problem. Metals typically have relatively large atomic
packing factors to maximize the shielding provided by the free electron cloud.
The Body-Centered Cubic Crystal Structure
Another common metallic crystal structure also has a cubic unit cell with atoms
located at all eight corners and a single atom at the cube center. This is called a
body-centered cubic (BCC) crystal structure. A collection of spheres depicting this
crystal structure is shown in Figure 3.2c, whereas Figures 3.2a and 3.2b are diagrams
of BCC unit cells with the atoms represented by hard sphere and reduced-sphere
models, respectively. Center and corner atoms touch one another along cube diagonals, and unit cell length a and atomic radius R are related through
Chromium, iron, tungsten, as well as several other metals listed in Table 3.1 exhibit
a BCC structure.
Two atoms are associated with each BCC unit cell: the equivalent of one atom
from the eight corners, each of which is shared among eight unit cells, and the single center atom, which is wholly contained within its cell. In addition, corner and
center atom positions are equivalent. The coordination number for the BCC crystal structure is 8; each center atom has as nearest neighbors its eight corner atoms.
Since the coordination number is less for BCC than FCC, so also is the atomic packing factor for BCC lower—0.68 versus 0.74.
The Hexagonal Close-Packed Crystal Structure
Not all metals have unit cells with cubic symmetry; the final common metallic
crystal structure to be discussed has a unit cell that is hexagonal. Figure 3.3a shows
a reduced-sphere unit cell for this structure, which is termed hexagonal closepacked (HCP); an assemblage of several HCP unit cells is presented in Figure
3.3b.1 The top and bottom faces of the unit cell consist of six atoms that form
regular hexagons and surround a single atom in the center. Another plane that
provides three additional atoms to the unit cell is situated between the top and
bottom planes. The atoms in this midplane have as nearest neighbors atoms in
both of the adjacent two planes. The equivalent of six atoms is contained in each
unit cell; one-sixth of each of the 12 top and bottom face corner atoms, one-half
of each of the 2 center face atoms, and all 3 midplane interior atoms. If a and c
represent, respectively, the short and long unit cell dimensions of Figure 3.3a, the
ca ratio should be 1.633; however, for some HCP metals this ratio deviates from
the ideal value.
The coordination number and the atomic packing factor for the HCP crystal
structure are the same as for FCC: 12 and 0.74, respectively. The HCP metals include
cadmium, magnesium, titanium, and zinc; some of these are listed in Table 3.1.
Some metals, as well as nonmetals, may have more than one crystal structure, a phenomenon known as polymorphism. When found in elemental solids, the condition
is often termed allotropy. The prevailing crystal structure depends on both the temperature and the external pressure. One familiar example is found in carbon:
graphite is the stable polymorph at ambient conditions, whereas diamond is formed
at extremely high pressures. Also, pure iron has a BCC crystal structure at room
temperature, which changes to FCC iron at C ( F). Most often a modification of the density and other physical properties accompanies a polymorphic
transformation.
Since there are many different possible crystal structures, it is sometimes convenient to divide them into groups according to unit cell configurations and/or atomic
arrangements. One such scheme is based on the unit cell geometry, that is, the shape
of the appropriate unit cell parallelepiped without regard to the atomic positions
in the cell. Within this framework, an x, y, z coordinate system is established with
its origin at one of the unit cell corners; each of the x, y, and z axes coincides with
one of the three parallelepiped edges that extend from this corner, as illustrated in
Figure 3.4. The unit cell geometry is completely defined in terms of six parameters:
the three edge lengths a, b, and c, and the three interaxial angles , , and . These
are indicated in Figure 3.4, and are sometimes termed the lattice parameters of a
crystal structure.
On this basis there are seven different possible combinations of a, b, and c, and
, , and each of which represents a distinct crystal system. These seven crystal
systems are cubic, tetragonal, hexagonal, orthorhombic, rhombohedral,2 monoclinic,
and triclinic. The lattice parameter relationships and unit cell sketches for each are
represented in Table 3.2.The cubic system, for which and
has the greatest degree of symmetry. Least symmetry is displayed by the triclinic
system, since and
From the discussion of metallic crystal structures, it should be apparent that
both FCC and BCC structures belong to the cubic crystal system, whereas HCP
falls within hexagonal. The conventional hexagonal unit cell really consists of three
parallelepipeds situated as shown in Table 3.2.
The rate at which this change takes place is extremely slow; however, the lower the temperature
(below C) the faster the rate. Accompanying
this white tin-to-gray tin transformation is an increase in volume (27 percent), and, accordingly, a
decrease in density (from 7.30 g/cm3 to 5.77 g/cm3).
Consequently, this volume expansion results in the
disintegration of the white tin metal into a coarse
powder of the gray allotrope. For normal subambient temperatures, there is no need to worry
about this disintegration process for tin products,
due to the very slow rate at which the transformation occurs.
This white-to-gray-tin transition produced
some rather dramatic results in 1850 in Russia.The
winter that year was particularly cold, and record
low temperatures persisted for extended periods
of time.The uniforms of some Russian soldiers had
tin buttons, many of which crumbled due to these
extreme cold conditions, as did also many of the
tin church organ pipes. This problem came to be
known as the “tin disease.”
When dealing with crystalline materials, it often becomes necessary to specify a particular point within a unit cell, a crystallographic direction, or some crystallographic
plane of atoms. Labeling conventions have been established in which three numbers or indices are used to designate point locations, directions, and planes.The basis
for determining index values is the unit cell, with a right-handed coordinate system
consisting of three (x, y, and z) axes situated at one of the corners and coinciding
with the unit cell edges, as shown in Figure 3.4. For some crystal systems—namely,
hexagonal, rhombohedral, monoclinic, and triclinic—the three axes are not mutually perpendicular, as in the familiar Cartesian coordinate scheme.
3.8 POINT COORDINATES
The position of any point located within a unit cell may be specified in terms of its
coordinates as fractional multiples of the unit cell edge lengths (i.e., in terms of a,
b, and c).To illustrate, consider the unit cell and the point P situated therein as shown
in Figure 3.5. We specify the position of P in terms of the generalized coordinates
q, r, and s where q is some fractional length of a along the x axis, r is some fractional
length of b along the y axis, and similarly for s. Thus, the position of P is designated
using coordinates q r s with values that are less than or equal to unity. Furthermore,
we have chosen not to separate these coordinates by commas or any other punctuation marks (which is the normal convention).
For some crystal structures, several nonparallel directions with different indices
are actually equivalent; this means that the spacing of atoms along each direction
is the same. For example, in cubic crystals, all the directions represented by the following indices are equivalent: [100], [ ], [010], [ ], [001], and [ ]. As a convenience, equivalent directions are grouped together into a family, which are enclosed
in angle brackets, thus: 81009. Furthermore, directions in cubic crystals having the
same indices without regard to order or sign, for example, [123] and [ ], are equivalent. This is, in general, not true for other crystal systems. For example, for crystals
of tetragonal symmetry, [100] and [010] directions are equivalent, whereas [100] and
[001] are not.