1. INDUSTRIAL MATERIALS Instructed by: Dr. Sajid Zaidi
PhD in Advanced Mechanics, UTC, France
MS in Advanced Mechanics, UTC, France
B.Sc. in Mechanical Engineering, UET, Lahore
B.TECH Mechanical Technology
IQRA COLLEGE OF TECHNOLOGY (ICT)
INTERNATIONAL ISLAMIC UNIVERSITY, ISLAMABAD

2. Crystal Geometry
(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
INDUSTRIAL MATERIALS
(a,b,c) non-dense, random packing, (d) dense, regular packing

3. Concept of Crystal Geometry
Space Lattice
Atomic 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.
INDUSTRIAL MATERIALS
Crystal Geometry

4. Concept of Crystal Geometry
Unit Cell
The 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.
INDUSTRIAL MATERIALS
Crystal Geometry
By stacking identical unit cells, the entire lattice can be constructed.

5. Concept of Crystal Geometry
Lattice Parameters
The 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 = m = 10-8 cm
INDUSTRIAL MATERIALS
Crystal Geometry

6. Concept of Crystal Geometry
Crystal System
There are 7 unique arrangements of the unit cell that define the crystal system
INDUSTRIAL MATERIALS
Crystal Geometry
Axes
Angle b/w axes
Volume of unit cell
a = b = c
α = β = γ = 90º a3
Cubic
Tetragonal
a = b ≠ c
α = β = γ = 90º
a2c
Orthorhombic
a ≠ b ≠ c
α = β = γ = 90º
abc

7. Concept of Crystal Geometry
Crystal System
Axes
Angle b/w axes
Volume of unit cell
INDUSTRIAL MATERIALS
Crystal Geometry
Rhombohedral
a = b = c
α = β = γ ≠ 90º
a3(1-3cos2α+2cos3α)1/2
Monoclinic
a ≠ b ≠ c
α = β = 90º, γ ≠ 90º
abc sinγ
Triclinic
a ≠ b ≠ c
α ≠ β ≠ γ ≠ 90º
abc (1-cos2α-cos2β-cos2γ+2cosαcosβcosγ)1/2
Hexagonal
a = b ≠ c
α = β = 90º, γ = 120º
3*0.866a2c

8. Concept of Crystal Geometry
Bravais 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
INDUSTRIAL MATERIALS
Crystal Geometry

9. Concept of Crystal Geometry
Bravais Lattices
INDUSTRIAL MATERIALS
Crystal Geometry

10. Concept of Crystal Geometry
Atomic Radius vs Lattice Parameters
INDUSTRIAL MATERIALS
Crystal Geometry
a0 = 2r
a0√3 = 4r
a0√2 = 4r

11. Concept of Crystal Geometry
No. of Atoms per Unit Cell
The 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.
INDUSTRIAL MATERIALS
Crystal Geometry

12. Concept of Crystal Geometry
No. of Atoms per Unit Cell
A 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.
INDUSTRIAL MATERIALS
Crystal Geometry
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.

13. Concept of Crystal Geometry
No. of Atoms per Unit Cell
Number 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
INDUSTRIAL MATERIALS
Crystal Geometry

14. Concept of Crystal Geometry
Coordination Number
The coordination number is the number of atoms touching a particular atom, or the number of nearest neighbors for that particular atom.
INDUSTRIAL MATERIALS
Crystal Geometry
Simple Cubic

15. Concept of Crystal Geometry
Packing 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.
INDUSTRIAL MATERIALS
Crystal Geometry
Packing Factor of SC = 0.52

16. Principle Metallic Crystal Structures
Most elemental metals (about 90 percent) crystallize upon solidification into three densely packed crystal structures:
Body-Centered Cubic (BCC)
Face-Centered Cubic (FCC)
Hexagonal Close-Packed (HCP)
INDUSTRIAL MATERIALS
Crystal Geometry

17. Principle Metallic Crystal Structures
Most 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.
INDUSTRIAL MATERIALS
Crystal Geometry
Simple Cubic
Face-Centered Cubic
Body-Centered Cubic
Hexagonal Close-Packed

18. 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.
a0√3 = 4r
INDUSTRIAL MATERIALS
Crystal Geometry
Hard Sphere Unit Cell
Isolated Unit Cell

19. 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.
INDUSTRIAL MATERIALS
Crystal Geometry
Selected metals that have the BCC Crystal Structure at room temperature (20◦C) and their Lattice Constants and Atomic Radii

20. Face-Centered Cubic (FCC)
Number of atoms in a unit cell of FCC = 4
Coordination number = 12
Packing Factor = 0.74
That is, 74 percent of the volume of the BCC unit cell is occupied by atoms and the remaining 26 percent is empty space.
INDUSTRIAL MATERIALS
Crystal Geometry
Hard Sphere Unit Cell
Isolated Unit Cell
a0√2 = 4r

21. 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.
INDUSTRIAL MATERIALS
Crystal Geometry
Selected metals that have the FCC Crystal Structure at room temperature (20◦C) and their Lattice Constants and Atomic Radii

22. 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.
INDUSTRIAL MATERIALS
Crystal Geometry
Hard Sphere Unit Cell
Isolated Unit Cell

23. 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 = 6.
INDUSTRIAL MATERIALS
Crystal Geometry

24. 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
INDUSTRIAL MATERIALS
Crystal Geometry
Selected metals that have the HCP Crystal Structure at room temperature (20◦C) and their Lattice Constants, Atomic Radii and c/a ratios

25. 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.
INDUSTRIAL MATERIALS
Crystal Geometry

26. 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. z
(0,1,1)
INDUSTRIAL MATERIALS
Crystal Geometry
(0,0,1)
(1,0,1)
(1,1,1)
(0,1,0)
(0,0,0) y
(1,0,0)
(1,1,0) x
(1/2,1/2,1/2)
Body – Centered Cubic (BCC)

27. Atom Positions in Cubic Unit Cellsz
INDUSTRIAL MATERIALS
Crystal Geometry
(0,0,0) y x
Face – Centered Cubic (FCC)

28. Directions in Cubic Unit Cells
Directions in crystal lattices are especially important for metals and alloys with properties that vary with crystallographic orientation.
INDUSTRIAL MATERIALS
Crystal Geometry z
Direction of OS
Coordinates of S - Coordinates of O
(1,1,0) – (0,0,0) = (1,1,0) = [110]
Position coordinates of direction vector OS O
Direction indices of direction vector OS y S x

29. Directions in Cubic Unit Cells
Some directions in cubic unit cell
INDUSTRIAL MATERIALS
Crystal Geometry

30. 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.
INDUSTRIAL MATERIALS
Crystal Geometry

31. Planes in Cubic Unit Cells
Sometimes 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.
INDUSTRIAL MATERIALS
Crystal Geometry

32. Miller Indices
The 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:
Choose a plane that does not pass through the origin at (0, 0, 0).
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.
Form the reciprocals of these intercepts.
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
INDUSTRIAL MATERIALS
Crystal Geometry

33. Miller Indices
Some examples
INDUSTRIAL MATERIALS
Crystal Geometry

34. 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
INDUSTRIAL MATERIALS
Crystal Geometry

35. Crystallographic Planes in Hexagonal Unit Cells
Crystal 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◦ with
each other.
The fourth axis or c axis is the
vertical axis located at the center
of the unit cell.
INDUSTRIAL MATERIALS
Crystal Geometry

36. Crystallographic Planes in Hexagonal Unit Cells
The 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.
INDUSTRIAL MATERIALS
Crystal Geometry

37. Crystallographic Planes in Hexagonal Unit Cells
Basal Planes
As the basal plane on the top of
the 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.
INDUSTRIAL MATERIALS
Crystal Geometry

38. Crystallographic Planes in Hexagonal Unit Cells
Prism Planes
The intercepts of the front prism
Plane (ABCD) are a1 = +1, a2 = ∞,
a3 = −1, and c = ∞. Taking the
reciprocals of these intercepts gives
h = 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.
INDUSTRIAL MATERIALS
Crystal Geometry

39. Expected Questions
Calculate 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 nm. Calculate the lattice constant a for the cube edge of the iron unit cell.
INDUSTRIAL MATERIALS
Crystal Geometry

40. Expected Questions
Calculate 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 = nm and c = 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.
INDUSTRIAL MATERIALS
Crystal Geometry