1. Elements of Materials Science and Engineering
ChE 210
Chapter 3

2. Crystalline Phases
Phase: is that part of a material that is recognized from other parts in structure and composition, e.g. ice-water
The same composition but different in state
Phase boundary: between the two states locates the discontinuinity in structure
Salt and saltwater (brine), both contain NaCl but form different phases (solid/solution)
Alcohol-water (miscible) different composition but form one phase
Oil-water (immiscible): two phases

3. Crystals
Metals, ceramics and certain polymers crystallize by solidification
Crystalline phases:
Posses a periodicity of long range order
i.e. The local atomic arrangement is repeated at regular intervals millions of times in 3-D of space.

4. Crystalline structure (NaCl)
The atomic (or ionic) coordination is repeated to give a long range periodicity. The centers of all the cube faces duplicate the pattern at the cube corners. At the center of each cell there is a Na+

5. Crystal Systems
Three dimensional system: x, y, z axes and , , and  are the axial angles
A, b, and c are the unit cell dimensions
Crystal systems vary in the axial angles and the unit cell dimensions

6. Crystal Systems  =  = 90O ≠  Axial Angle Axes System
a = b = c
Cubic
a = b ≠ c
Tetragonal
a ≠ b ≠ c
Monoclinic
 =  = 90O ≠ 
Triclinic
 ≠  ≠  ≠ 90O
orthorhombic
 =  = 90O;  =120O
a = a ≠ c
Hexagonal
 =  =  ≠ 90O
rhombhedral

7. Noncubic unit cell a a b c a=b≠c, angles =90o Tetragonal system
a = a ≠ c, angles =90o, 120o
hexagonal system
a ≠ b≠c, angles =90o
orthorhombic system

8. Lattices
The above 7 systems can be divided into 14 patterns of points, called Bravais lattices.
e.g Cubic systems:
Body centered cubic (BCC)
Face centered cubic (FCC)
Simple cubic

10. FCC lattice, e.g. solid methane.
CH4 -183oC
between K molecules rotate in their lattice sites.
Below 20 K, the molecules have identical alignments, as shown here.

11. Example 3-1.1: the unit cell of Cr metal is cubic and contains 2 atoms, if the density of Cr = 7.19 Mg/m3; determine the dimension of Cr unit cell
Mass of a unit cell = 2 Cr(52.0 g)/(6.02 x 1023 Cr)
= x g
volume = a3 = ( x g)(7.19 x 106 g/m3)
= 24 x m3
a = x 10-9 m

12. Cubic System (Body-centered cubic metals)
BCC of a metal
Show the location of the atom centers.
Model made from hard balls

13. (b.d) bcc metal = 4R = abcc metal √3
Body centered (bcc) Metal structure: two metal atoms/unit cell and the atomic packing factor of Since atoms are in contact along the body diagonals (b.d.) of the unit cell:
(b.d) bcc metal = 4R = abcc metal √3
abcc metal = 4R/√3
a is the lattice constant, R = the radius of the ion

14. The atomic packing factor (PF) of a bcc metal assuming spherical atoms (hard ball model) and calculating the volume fraction of the unit cell that is occupied by the atoms
There are 2 atoms per unit cell in a bcc metal, and we assuming spherical atoms

15. Face-centered cubic metals
Fcc metal structure, a) Schematic view showing location of atom centers,
b) hard ball model. Each unit cell contains 4 atoms
(f.d.) fcc metal = 4R = afcc metal √2
afcc metal = 4R/√2 and the PF = 0.74
This is because each atom has 12 neighbours

16. FCC metal structure: 4 metal atoms/unit cell
atomic packing = 0.74
atoms are in contact along the face diagonals
i.e. afcc = 4R/√2,
This is not applied to all fcc systems

17. Other face-centered cubic structure
e.g. molecular structure of solid methane with 5 atoms occupying the fcc positions as a molecular unit.
NaCl structure, since the atoms are unlike.
afcc NaCl = 2(rNa+ + RCl-)
Note that there is a gap between the Cl- ions, we can not apply the eqn used for lattice constant.

18. Diamond (fcc)

19. Simple cubic structures
It can be more complex than fcc or bcc
Simple means a crystal structure with repetition at only full unit cell increments.
bcc structure is replicated at the centre of the unit cell as well as the corners.
fcc is replicated at the face centers as well as the corners.
A simple cubic (sc) cell lacks at both these centered locations.
In CsCl, (sc), Cl- at the cell corners; Cs+ at the cell centre. But this is not bcc , because the cell centre possesses an ion different from that at the cell corner.

20. Simple cubic structures

22. Properties of simple cubic structures
Binary (Two component system)
The unit cell has two radii.
aCsCl-type = 2(rCs+ + RCl-)/√3
CN = 8, r/R = 0.73,
not in case of MgO rMg2+/RO2- = 0.47
Calcium titanate, CaTiO3 is also SC
- Ca2+ are located at the corners of the unit cell
Ti4+ are located at the center
O2- at the center of each face
Therefore, it is neither bcc nor fcc

23. Example 3-21 Calculate (a) the atomic packing factor of an fcc metal
(b) the ionic packing factor of fcc NaCl
There are 4 atoms/unit cell in an fcc metal structure
- But there are 4 Na+ and 4 Cl- per unit cell.
- Note: The lattice constants is related differently from fcc

24. Example p71
Copper has an fcc metal structure with an atom radius of nm. Calculate its density and check this value with the density listed in Appendix B.
Procedure: Since  = m/V, we must obtain the mass per unit cell and the volume of each unit cell. The former is calculated from the mass of four atoms; the latter requires the calculation of the lattice constant, a, for an fcc metal structure.

25. Example p71

26. Hexagonal Close-Packed Metals
Noncubic structures
Hexagonal Close-Packed Metals

27. Noncubic structures
Hexagonal Close-Packed Metals
Examples are Mg, Ti and Zn
Called the hexagonal closed-packed structure (hcp)
Each atom is located directly above or below interstices among three atoms in the adjacent layer below its plane
Each atom touches 3 atoms below and 6 atoms in the same plane.
Average of 6 atoms/ unit cell.
Such as the fcc CN =12
Therefore, the packing factor = 0.74

28. Other non cubic structure
Iodine  orthorhombic, because I2 is not spherically sym.
Polyethylene  orthorhombic, complex, due to the large and linear molecules.
Graphite  hexagonal structure

31. Iron carbide: Called cementite., Fe3C
The crystal lattice contains 2 elements in a fixed 3:1 ratio
Orthorhombic, 12 Fe and 4 C atoms in each unit cell.
Carbon in the interstices among the larger Fe atoms.
it greatly affects the properties of steel due to its superior hardness.

32. Barium Titanate: BaTiO3, Tetragonal system
Tetragonal BaTiO3: The base of the unit cell is square; but unlike in CaTiO3, c ≠ a. Also, the Ti4+ and O2- ions are not symmetrically located with respect to the corner Ba2+ ions. Above 120oC, BaTiO3 changes from
this structure to that of CaTiO3.

34. Fig

38. Polymorphism Diamond Graphite c
The existence of two crystal structure of the same substance is called polymorphism. The two forms are called polymorphs.
Heat + pressure
Diamond Graphite c

39. bcc fcc CN of bcc Fe = 8 at room temp., PF = 0.68 and R = 0.1241 nm
CN of fcc = 12 at 912oC, PF = 0.74 and R = nm
For bcc iron R = due to thermal expansion
The reaction involves volume changes and 1.4 V% (-0.47 l%) when the temperature is raised above 912oC.
Volume changes is due to larger CN in fcc atoms

40. Zirconia toughen alumina significantly, because it has 3 polymorphs
At high temperature changes from cubic to tetragonal with CaF2 structure
The next step it changes to monoclinic, involves dimensional strain that consumes energy before fracture

45. Unit Cell Geometry X- axis points to us. Y-axis points to our right
Z- axis points up ward
Origin is the lower left corner
Every point is identified by coefficients,
Thus the origin is 0,0,0.
The centre of the unit cell is at a/2, b/2, c/2
The indices of that location are ½, ½, ½
The far corner of the unit cell is 1,1,1, regardless of the crystal system.

47. Crystal indices
Transition from a selected site within the unit cell by integer multiple of lattice constants.
Replication occurs by ±a/2, ±b/2, ± c/2 in body centered systems.
i.e. a point with indices 0.3, 0.4, 0.2 duplicates at 0.8, 0.9, 0.7 and 0.8, -0.1, 0.7.
In face centered systems, replication occurs by translation of ±a/2, ±b/2, 0 of ±a/2, 0, ±c/2 and of 0, ±b/2, ±c/2.

51. Crystal directions
Can be represented by a vector from the origin to a point with the lowest integer index, h,k,l.
Specified by indices [h,k,l]

52. All the parallel directions use the same label or index
The parallel line that passes through 0,0,0 and since there are millions of points on any line. We choose the point with lowest set of integers.
[111] direction passes from 0,0,0 through 1,1,1 ans ½, ½, ½. Likewise, the [112] passes through ½,1/2,1/2.
Direction indices are written in square brackets [uvw], no commas, it can be positive or negative.
Negative directions are indicated by over bar.

53. Crystal directions

54. Angles between directions
The angle between [110] and [112] i.e. [110]  [112]) is the
In cubic crystals, we can determine cos [uvw]  [u`v`w`] by dot product
Repetition Distances
The vector between identical positions within a crystal
It differs from direction to direction and from structure to structure
Example: in bcc metal, there is a repetition each 2R (which is a√3/2).
The repetition distance is a√2 in the [110] direction of bcc but a/√2 in fcc.

55. Families of Directions
In the cubic system:
Are the same

56. Crystal planes Is defined by miller indices (hkl)
It is the reciprocals of the intersections plane axes.
Take the smallest possible integers
Negative is indicated by over bar.
Visualized as those outline the unit cell
All parallel planes are identified by the same indices.

57. Crystal planes

60. X-ray diffraction
X-rays is electromagnetic radiation but with very short wavelength compared to visible light.
Monochromatic x-rays have one and the same wavelength.
White x-rays have different wavelengths.
When two x-rays beams lie in plane a constructive interference occurs. i.e. P.d. = n 

61. Bragg’s Law
When monochromatic x-ray diffracted from two crystallographic planes
n .  = 2 . D. sin