INDUSTRIAL MATERIALS

Instructed by: Dr. Sajid ZaidiPhD in Advanced Mechanics, UTC, France

MS in Advanced Mechanics, UTC, FranceB.Sc. in Mechanical Engineering, UET, Lahore

B.TECH Mechanical TechnologyIQRA COLLEGE OF TECHNOLOGY (ICT)INTERNATIONAL ISLAMIC UNIVERSITY, ISLAMABAD

Crystal GeometryIN

DU

STR

IAL

MAT

ERIA

LS

(a) Inert monoatomic gases have no regular ordering of atoms

(b,c) Some materials, including water vapor, nitrogen gas, amorphous silicon and silicate glass have short-range order

(d) Metals, alloys, many ceramics and some polymers have regular ordering of atoms/ions that extends through the material

(a,b,c) non-dense, random packing, (d) dense, regular packing

Concept of Crystal GeometrySpace LatticeAtomic arrangements in crystalline solids can be

described by referring the atoms to the points of intersection of a network of lines in three dimensions.

Such a network is called a space lattice. It can be described as an infinite three-dimensional array

of points.Each space lattice can be described by specifying the

atom positions in a repeating unit cell.The unit cell is the subdivision of a lattice that still

retains the overall characteristics of the entire lattice.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Concept of Crystal GeometryUnit CellThe basic structural unit of a crystal structure. Its geometry and atomic positions define the crystal

structure.A unit cell is the smallest component of the crystal that

reproduces the whole crystal when stacked together with purely translational repetition.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

By stacking identical unit cells, the entire lattice can be constructed.

Concept of Crystal GeometryLattice ParametersThe lattice parameters include

◦ Dimensions of the sides of the unit cell◦ Angles between the sides◦ The length is often given in nanometers

(nm) or Angstrom (Å) units, where:

1 angstrom (Å) = 0.1 nm = 10-10 m = 10-8 cm

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Concept of Crystal GeometryCrystal SystemThere are 7 unique arrangements of the unit cell that

define the crystal system

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Cubic

Tetragonal

Orthorhombic

Axes Angle b/w axes Volume of unit cell

a = b = c α = β = γ = 90º a3

a = b ≠ c α = β = γ = 90º a2c

a ≠ b ≠ c α = β = γ = 90º abc

Concept of Crystal GeometryCrystal System

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Rhombohedral

Monoclinic

Triclinic

Hexagonal

Axes Angle b/w axes Volume of unit cell

a = b = c α = β = γ ≠ 90º a3(1-3cos2α+2cos3α)1/2

a ≠ b ≠ c α = β = 90º, γ ≠ 90º abc sinγ

a ≠ b ≠ c α ≠ β ≠ γ ≠ 90º abc (1-cos2α-cos2β-cos2γ+2cosαcosβcosγ)1/2

a = b ≠ c α = β = 90º, γ = 120º 3*0.866a2c

Concept of Crystal GeometryBravais Lattices

Lattice points are located at the corners of the unit cells and, in some cases, at either faces or the center of the unit cell.

14 distinct arrangements of lattice points are possible. These unique arrangements of lattice points are known as the Bravais lattices.

Concept of a lattice is mathematical and does not mention atoms, ions or molecules.

One or more atoms can be associated with each lattice point

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Concept of Crystal GeometryB

ravais Lattices

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Concept of Crystal GeometryAtomic Radius vs Lattice Parameters

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

a0 = 2r

a0√3 = 4r

a0√2 = 4r

Concept of Crystal GeometryNo. of Atoms per Unit CellThe number of atoms per unit cell is the product of the

number of atoms per lattice point and the number of lattice points per unit cell.

In most metals, one atom is located at each lattice point.A unit cell has specific number of lattice points:

◦ at each corner of the cell

◦ at center of the cell (in case of body-centered)

◦ at each face of the cell (in case of face-centered)

It must be noticed that one lattice point may be shared by more than one unit cell.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Concept of Crystal GeometryNo. of Atoms per Unit CellA lattice point at a corner of one unit cell is shared by

seven adjacent unit cells (thus a total of eight cells); only one-eighth of each corner lattice point belongs to one particular cell.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Similarly, a lattice point at the face of one unit cell is shared by one adjacent unit cells (thus a total of two cells); only one-half of each face lattice point belongs to one particular cell.

Concept of Crystal GeometryNo. of Atoms per Unit CellNumber of total corner lattice points in a unit cell is

◦ (Total number of corners) x (lattice point per corner)= 8 x 1/8 = 1

Number of total body-centered lattice points in a unit cell is◦ (Total number of centers) x (lattice point per center)= 1 x 1 = 1

Number of total face-centered lattice points in a unit cell is◦ (Total number of faces) x (lattice point per face)= 6 x 1/2 = 3

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Concept of Crystal GeometryCoordination NumberThe coordination number is the number of atoms

touching a particular atom, or the number of nearest neighbors for that particular atom.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Simple Cubic

Concept of Crystal GeometryPacking Factor

The packing factor is the fraction of space occupied by atoms, assuming that atoms are hard spheres sized so that they touch their closest neighbor. The general expression for the packing factor is:

Packing factor = (number of atoms/cell) x (volume of each atom) / (volume of unit cell)

No commonly encountered engineering metals or alloys have the SC structure, although this structure is found in ceramic materials.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Packing Factor of SC = 0.52

Principle Metallic Crystal StructuresMost elemental metals (about 90 percent) crystallize

upon solidification into three densely packed crystal structures:a) Body-Centered Cubic (BCC)b) Face-Centered Cubic (FCC)c) Hexagonal Close-Packed (HCP)

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Principle Metallic Crystal StructuresMost metals crystallize in these dense-packed

structures because energy is released as the atoms come closer together and bond more tightly with each other. Thus, the densely packed structures are in lower and more stable energy arrangements.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Simple Cubic

Body-Centered Cubic

Face-Centered Cubic

Hexagonal Close-Packed

Body-Centered Cubic (BCC)

Number of atoms in a unit cell of BCC = 2 Coordination number = 8 Packing Factor = 0.68 That is, 68 percent of the volume of the BCC unit cell is

occupied by atoms and the remaining 32 percent is empty space.

The BCC crystal structure is not 100% close-packed structure since the atoms could be packed closer together.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

a0 √3 =

4rHard Sphere Unit Cell Isolated Unit Cell

Body-Centered Cubic (BCC)Many metals such as iron, chromium, tungsten,

molybdenum, and vanadium have the BCC crystal structure at room temperature.

Metals with mixed bonding, such as iron, may have unit cells with less than the maximum packing factor.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Selected metals that have the BCC Crystal Structure at room temperature (20◦C) and their Lattice Constants and Atomic Radii

Face-Centered Cubic (FCC)

Number of atoms in a unit cell of FCC = 4Coordination number = 12Packing Factor = 0.74That is, 74 percent of the volume of the BCC unit cell is

occupied by atoms and the remaining 26 percent is empty space.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Hard Sphere Unit Cell Isolated Unit Cella0√2 = 4r

Face-Centered Cubic (FCC)The PF of 0.74 is for the closest packing possible of

“spherical atoms.” Many metals such as aluminum, copper, lead, nickel,

and iron at elevated temperatures (912 to 1394◦C) crystallize with the FCC crystal structure.

Metals with only metallic bonding are packed as efficiently as possible.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Selected metals that have the FCC Crystal Structure at room temperature (20◦C) and their Lattice Constants and Atomic Radii

Hexagonal Close-Packed (HCP)Metals do not crystallize into the simple hexagonal

crystal structure because the PF is too low.The atoms can attain a lower energy and a more stable

condition by forming the HCP structure.The PF of the HCP crystal structure is 0.74, the same as

that for the FCC crystal structure since in both structures the atoms are packed as tightly as possible.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Hard Sphere Unit Cell Isolated Unit Cell

Hexagonal Close-Packed (HCP)The HCP has six atoms per unit cell.

◦ Three atoms form a triangle in the middle layer.◦ There are six 1/6 atom sections on both the top and bottom

layers, making an equivalent of two more atoms (2 × 6 × 1/6 = 2).

◦ Finally, there is one-half of an atom in the center of both the top and bottom layers making the equivalent of one more atom.

◦ The total number of atoms in the HCP crystal structure unit cell is thus 3 + 2 + 1 = 6.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Hexagonal Close-Packed (HCP) Coordination number is 12. The ratio of the height c of the hexagonal prism of the HCP

crystal structure to its basal side a is called the c/a ratio. The c/a ratio for an ideal HCP crystal structure consisting of

uniform spheres packed as tightly together as possible is 1.633.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Selected metals that have the HCP Crystal Structure at room temperature (20◦C) and their Lattice Constants, Atomic Radii and c/a ratios

Hexagonal Close-Packed (HCP)Cadmium and zinc have c/a ratios higher than ideality,

which indicates that the atoms in these structures are slightly elongated along the c axis of the HCP unit cell.

The metals magnesium, cobalt, zirconium, titanium, and beryllium have c/a ratios less than the ideal ratio. Therefore, in these metals the atoms are slightly compressed in the direction along the c axis.

Thus, for the HCP metals listed in the table, there is a certain amount of deviation from the ideal hard-sphere model.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Atom Positions in Cubic Unit Cells To locate atom positions in

cubic unit cells, rectangular x, y, and z axes are used.

In crystallography the positive x axis is usually the direction coming out of the paper, the positive y axis is the direction to the right of the paper, and the positive z axis is the direction to the top. Negative directions are opposite to those just described.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

x

y

z

(0,0,0)

(1,0,0)

(1,0,1)

(0,0,1)(0,1,1)

(1,1,0)

(0,1,0)

(1,1,1)

(1/2,1/2,1/2)

Body – Centered Cubic (BCC)

Atom Positions in Cubic Unit CellsIN

DU

STR

IAL

MAT

ERIA

LSC

ryst

al G

eom

etry

Face – Centered Cubic (FCC)

(0,0,0)

x

y

z

Directions in Cubic Unit CellsDirections in crystal lattices are especially important for

metals and alloys with properties that vary with crystallographic orientation.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

x

y

z

O

S

Direction of OSCoordinates of S - Coordinates of O(1,1,0) – (0,0,0) = (1,1,0) = [110]

Position coordinates of direction vector OS

Direction indices of direction vector OS

Directions in Cubic Unit CellsSome directions in cubic unit cell

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Significance of Crystallographic Directions

Metals deform more easily, for example, in directions along which atoms are in closest contact.

Another real-world example is the dependence of the magnetic properties of iron and other magnetic materials on the crystallographic directions.

It is much easier to magnetize iron in the [100] direction compared to [111] or [110] directions. This is why the grains in Fe-Si steels used in magnetic applications are oriented in the [100] or equivalent directions.

In the case of magnetic materials used for recording media, we have to make sure the grains are aligned in a particular crystallographic direction such that the stored information is not erased easily. Similarly, crystals used for making turbine blades are aligned along certain directions for better mechanical properties.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Planes in Cubic Unit CellsSometimes it is necessary to refer to specific lattice

planes of atoms within a crystal structure, or it may be of interest to know the crystallographic orientation of a plane or group of planes in a crystal lattice.

Metals deform easily along planes of atoms that are most tightly packed together.

The surface energy of different faces of a crystal depends upon the particular crystallographic planes. This becomes important in crystal growth. In thin film growth of certain electronic materials (e.g., Si or GaAs), we need to be sure the substrate is oriented in such a way that the thin film can grow on a particular crystallographic plane.

To identify crystal planes in cubic crystal structures, the Miller notation system is used.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Miller IndicesThe Miller indices of a crystal plane are defined as the

reciprocals of the fractional intercepts (with fractions cleared) that the plane makes with the crystallographic x, y, and z axes of the three nonparallel edges of the cubic unit cell.

The procedure for determining the Miller indices for a cubic crystal plane is as follows:1. Choose a plane that does not pass through the origin at (0, 0,

0). 2. Determine the intercepts of the plane in terms of the

crystallographic x, y, and z axes for a unit cube. These intercepts may be fractions.

3. Form the reciprocals of these intercepts.4. Clear fractions and determine the smallest set of whole

numbers that are in the same ratio as the intercepts. These whole numbers are the Miller indices of the crystallographic plane and are enclosed in parentheses

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Miller IndicesSome examples

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Miller Indices In cubic crystal structures the interplanar spacing

between two closest parallel planes with the same Miller indices is designated dhkl , where h, k, and l are the Miller indices of the planes. This spacing represents the distance from a selected origin containing one plane and another parallel plane with the same indices that is closest to it.

From simple geometry, it can be shown that for cubic crystal structures

dhkl = a S

√h2 + k2 + l2

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Crystallographic Planes in Hexagonal Unit CellsCrystal planes in HCP unit cells are commonly

identified by using four indices, Miller-Bravais indices, instead of three.

These are denoted by the letters h, k, i, and l and are enclosed in parentheses as (hkil).

These four-digit hexagonal indices are based on a coordinate system with four axes

There are three basal axes, a1,a2, and a3, which make 120◦ witheach other. The fourth axis or c axis is thevertical axis located at the centerof the unit cell.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Crystallographic Planes in Hexagonal Unit CellsThe a unit of measurement along the a1, a2, and a3 axes

is the distance between the atoms along these axes.The unit of measurement along the c axis is the height of

the unit cell.The reciprocals of the intercepts that a crystal plane

makes with the a1, a2, and a3 axes give the h, k, and i indices, while the reciprocal of the intercept with the c axis gives the l index.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Crystallographic Planes in Hexagonal Unit Cells

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Basal PlanesAs the basal plane on the top ofthe HCP unit cell is parallel to a1, a2, and a3, the intercepts of this plane with these axes will all be infinite. Thus, a1 = ∞, a2 = ∞, and a3 = ∞. The c axis is unity since the top basal plane intersects

the c axis at unit distance.Taking the reciprocals of these intercepts gives the

Miller-Bravais indices for the HCP basal plane. Thus h = 0, k = 0, = 0, and l = 1. The HCP basal plane is, therefore, a zero-zero-zero-one or (0001) plane.

Crystallographic Planes in Hexagonal Unit CellsPrism PlanesThe intercepts of the front prism Plane (ABCD) are a1 = +1, a2 = ∞, a3 = −1, and c = ∞. Taking the reciprocals of these intercepts givesh = 1, k = 0, i = −1, and l = 0, or the (1010) plane.The ABEF prism plane has the indices (1100) and the

DCGH plane the indices (0110).All HCP prism planes can be identified collectively as

the {1010} family of planes.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Expected QuestionsCalculate the packing factor for the SC, BCC and FCC.Calculate the percent volume change as zirconia (ZrO2)

transforms from a tetragonal to a monoclinic structure. The lattice constants for the monoclinic unit cells are: a =5.156Å, b=5.191Å, c=5.304Å , and the angle β is 98.9º. The lattice constants for the tetragonal unit cell are a=5.094Å and c=5.304Å. Does the zirconia expand or contract during this transformation?

Draw [121] direction and (210) plane in a cubic unit cell.

Iron at 20ºC is BCC with atoms of atomic radius 0.124 nm. Calculate the lattice constant a for the cube edge of the iron unit cell.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry

Expected QuestionsCalculate the volume of the zinc crystal structure unit

cell by using the following data: pure zinc has the HCP crystal structure with lattice constants a = 0.2665 nm and c = 0.4947 nm.

Draw the following direction vectors in cubic unit cells:◦ [100], [110], [112], [110], [321]

Draw the following crystallographic planes in cubic unit cells:◦ (101), (110), (221) ◦ Draw a (110) plane in a BCC atomic-site unit cell, and list the

position coordinates of the atoms whose centers are intersected by this plane.

IND

UST

RIA

L M

ATER

IALS

Cry

stal

Geo

met

ry