The Chemistry
of Engineering
Materials
Part I: Crystals
RHONA C. ADAJAR
BASIC CONCEPTS OF
CRYSTAL STRUCTURES
Solids
Solids are characterized by an extended three-
dimensional arrangement of atoms, ions, or
molecules in which the components are
generally locked into their positions.
The components can be arranged in a regular
repeating three-dimensional array (a crystal
lattice), which results in a crystalline solid, or
more or less randomly to produce an
amorphous solid.
CRYSTAL STRUCTURES
Crystallography is the study of the formation, structure, and properties of crystals.
A crystal structure is defined as the particular repeating arrangement of atoms (molecules or
ions) throughout a crystal.
Structure refers to the internal arrangement of particles and not the external appearance of the
crystal. However, these are not entirely independent since the external appearance of a crystal is
often related to the internal arrangement.
For example, crystals of cubic rock salt (NaCl) are physically cubic in appearance. Only a few of
the possible crystal structures are of concern with respect to simple inorganic salts and these will
be discussed in detail, however, it is important to understand the nomenclature of crystallography.
CRYSTALLINE SOLIDS
Crystalline solids have regular ordered arrays of
components held together by uniform intermolecular
forces.
Crystalline solids, or crystals, have distinctive
internal structures that in turn lead to distinctive flat
surfaces, or faces. The faces intersect at angles that
are characteristic of the substance.
CRYSTALLINE SOLIDS
The characteristic angles do not depend on the size of the crystal; they reflect the regular
repeating arrangement of the component atoms, molecules, or ions in space.
When an ionic crystal is cleaved repulsive interactions cause it to break along fixed planes to
produce new faces that intersect at the same angles as those in the original crystal.
Crystals have sharp, well-defined melting points.
AMORPHOUS SOLIDS
The components of amorphous solids are not
arranged in regular arrays.
Amorphous solids have two characteristic properties.
When cleaved or broken, they produce fragments with
irregular, often curved surfaces; and they have poorly
defined patterns when exposed to x-rays because
their components are not arranged in a regular array.
AMORPHOUS SOLIDS
In an amorphous solid, the local environment, including both the distances to neighboring units
and the numbers of neighbors, varies throughout the material.
Different amounts of thermal energy are needed to overcome these different interactions.
Consequently, amorphous solids tend to soften slowly over a wide temperature range rather than
having a well-defined melting point like a crystalline solid.
If an amorphous solid is maintained at a temperature just below its melting point for long periods
of time, the component molecules, atoms, or ions can gradually rearrange into a more highly
ordered crystalline form.
Amorphous solids do not have sharp, well-defined melting points.
COMPARISON BETWEEN CRYSTALLINE AND
AMORPHOUS SOLIDS
CRYSTAL LATTICE
Lattice means a three-dimensional
array of points coinciding with
atom positions or sphere centers.
The constituents of a solid can be arranged in two
general ways: they can form a regular repeating three-
dimensional structure called a crystal lattice, thus
producing a crystalline solid, or they can aggregate with
no particular order, in which case they form an
amorphous solid (from the Greek ámorphos, meaning
“shapeless”).
UNIT CELLS
AND LATTICE
TYPES
The unit cell is repeated infinitely in all directions in the theoretical bulk
crystal solid. A unit cell is described by lattice points, which together
make up various lattice types. These are the points in space about
which the particles are free to vibrate in a crystal.
A unit cell can also be described using dimensions of length, width,
and height. The different lattice types can be differentiated by their
dimensions as well as the placement of atoms within the cell.
There are 14 different lattice types.
● The unit cell is the simplest repeating unit
in the crystal.
● Opposite faces of a unit cell are parallel.
● The edge of the unit cell connects
equivalent points
LATTICE TYPES
LATTICE TYPES
LATTICE TYPES
CUBIC UNIT CELL
❏ Simple Cubic
❏ Face-centered Cubic (FCC)
❏ Base-centered Cubic (BCC)
These unit cells are important for two reasons.
First, a number of metals, ionic solids, and
intermetallic compounds crystallize in cubic unit
cells.
Second, it is relatively easy to do calculations
with these unit cells because the cell-edge
lengths are all the same and the cell angles are
all 90.
CHARACTERISTICS OF CRYSTAL
STRUCTURES
There are two important characteristics of a crystal structure:
(a) the coordination number (for metals) wherein each atom has the same number of
nearest-neighbor or touching atoms;
(b) atomic packing factor (APF) which 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.
SIMPLE CUBIC
The simple cubic unit cell is the simplest repeating
unit in a simple cubic structure. Each corner of the unit
cell is defined by a lattice point at which an atom, ion,
or molecule can be found in the crystal.
By convention, the edge of a unit cell always connects
equivalent points. Each of the eight corners of the unit
cell therefore must contain an identical particle.
Other particles can be present on the edges or faces of
the unit cell, or within the body of the unit cell. But the
minimum that must be present for the unit cell to be Simple cubic structure:
classified as simple cubic is eight equivalent particles
on the eight corners. 8 corners x 1/8 = 1 atom
BODY-CENTERED CUBIC (BCC)
The body-centered cubic unit cell is the simplest
repeating unit in a body-centered cubic structure.
Once again, there are eight identical particles on the
eight corners of the unit cell. However, this time there
is a ninth identical particle in the center of the body of
the unit cell.
The unit cell length a and atomic radius (R) are related
by the way of:
Body-centered cubic structure:
(8 corners x 1/8) + 1 body = 2 atoms
FACE-CENTERED CUBIC (FCC)
The face-centered cubic unit cell also starts with
identical particles on the eight corners of the cube. But
this structure also contains the same particles in the
centers of the six faces of the unit cell, for a total of 14
identical lattice points.
The face-centered cubic unit cell is the simplest repeating
unit in a cubic closest-packed structure. In fact, the
presence of face-centered cubic unit cells in this structure
explains why the structure is known as cubic closest-
packed.
The cube edge length a and the atomic Face-centered cubic structure:
radius (R) are related through:
(8 corners x 1/8) + (6 faces x 1/2) = 4
atoms
HEXAGONAL CLOSE-PACKED CRYSTAL
STRUCTURE
The term "closest packed structures" refers to the most
tightly packed or space-efficient composition of crystal
structures (lattices).
Imagine an atom in a crystal lattice as a sphere. While
cubes may easily be stacked to fill up all empty space,
unfilled space will always exist in the packing of spheres.
To maximize the efficiency of packing and minimize the
volume of unfilled space, the spheres must be arranged as
close as possible to each other. These arrangements are
called closest packed structures.
HEXAGONAL CLOSE-PACKED CRYSTAL
STRUCTURE
In a hexagonal closest packed structure, the third layer
has the same arrangement of spheres as the first layer
and covers all the tetrahedral holes.
Since the structure repeats itself after every two layers,
the stacking for hcp may be described as "a-b-a-b-a-b."
The atoms in a hexagonal closest packed structure
efficiently occupy 74% of space while 26% is empty
space.
COORDINATION NUMBER AND NUMBER OF
ATOMS PER UNIT CELL
A unit cell is the smallest representation of an entire crystal. All crystal lattices are built of
repeating unit cells. In a unit cell, an atom's coordination number is the number of atoms it is
touching.
● The hexagonal closest packed (hcp) has a coordination number of 12 and contains 6 atoms
per unit cell.
● The face-centered cubic (fcc) has a coordination number of 12 and contains 4 atoms per unit
cell.
● The body-centered cubic (bcc) has a coordination number of 8 and contains 2 atoms per unit
cell.
● The simple cubic has a coordination number of 6 and contains 1 atom per unit cell.
DENSITY COMPUTATIONS
Number of atoms (n):
SCC = 1
BCC = 2
FCC = 4
SAMPLE PROBLEMS:
1.If the atomic radius of Pb=0.175nm (FCC), find the volume of the unit cell in cubic meter.
SAMPLE PROBLEMS:
2. Magnesium is HCP with c/a = 1.624, density =1.74 g/cm^3. Find the atomic radius of
magnesium (in pm).
SAMPLE PROBLEMS:
3.The edge length of the unit cell of Ta, is 330.6 pm; the unit cell is body-centered cubic. Tantalum
has a density of 16.69 g/cm3.
(a) Calculate the radius length of tantalum.
(b) Calculate the volume of tantalum.
(c) calculate the mass of a tantalum atom.
(d)Calculate the atomic weight of tantalum in g/mol.
SAMPLE PROBLEMS:
4.Palladium crystallizes in a face-centered cubic unit cell. Its density is 12.023 g/cm 3. Atomic
weight of 106.42 g/mol.
(a) Calculate the mass of an atom of palladium.
(b) Calculate the mass of 4 atoms of palladium in FCC.
(c) calculate the volume of palladium.
(d)Calculate the edge length a.
(e)Calculate the radius length of palladium.
TYPES OF CRYSTALS
1. IONIC CRYSTALS
2. COVALENT CRYSTALS
3. MOLECULAR CRYSTALS
4. METALLIC CRYSTALS
Each type has a different type of connection, or
bond, between its atoms. The type of atoms and
the arrangement of bonds dictate what type of
crystal is formed.
IONIC CRYSTALS
COVALENT CRYSTALS
MOLECULAR CRYSTALS
METALLIC CRYSTALS
The Chemistry
of Engineering
Materials
Part I: Crystals
RHONA C. ADAJAR
