1. THE STRUCTURE OF CRYSTALLINE SOLID
CONTENT:
FUNDAMENTAL CONCEPTS
UNITS CELLS
METALLIC CRYSTAL STRUCTURES
The Face-centered Cubic Crystal Structure
The Body-centered Cubic Crystal Structure
The Hexagonal Close-packed Crystal Structure
DENSITY
POLYMORHISM AND ALLOTROPY
CRYSTALOGRAPHIC DIRECTION AND PLANE
LINEAR AND PLANER ATOMIC DENSITIES

2. FUNDAMENTAL CONCEPTS
Solid materials may be classified according to the regularity with which atoms or irons are arranged with respect to one another
Crystalline:
The atoms are situated in a repeating or periodic array over large atomic distance; long-range order exists (upon solidification), atoms will position themselves in a repetitive three-dimensional pattern, each atom bonded to its nearest-neighbor atom
All metal, many ceramic and certain polymers form crystalline under normal solidification condition
For those do not crystalline (noncrystalline or amorphous) – long-range atomic is absent

3. FUNDAMENTAL CONCEPTS Crystal structure:
The manner in which atoms, ions or molecules are spatially arranged
Atoms / ions are thought of as being solid spheres having well- defined diameters – atomic hard sphere model (figure) in which spheres representing nearest neighbor atoms touch one another
‘lattice’ – three-dimensional array of points coinciding with atom positions

4. UNIT CELL
In describing crystal structures, it is often convenient to subdivide the structure into small repeat entities called unit cells.
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 distance along each of its edges.
Unit cell is the basic structural init or building block of the crystal structure & defines the crystal structure by virtue of its geometry & the atom positions within.

5. METALLIC CRYSTAL STRUCTURE
The atomic bonding in this group of materials is:
metallic & nondirectional in nature.
no restrictions as to the number & position of nearest-neighbor atoms – lead to relatively large numbers of nearest neighbors & dense atomic packing for most metallic crystal structures.
Table1 presents the atomic radii for a number of metals.
Three relatively simple structure are found for most of the common metals:
Face-Centered Cubic, (FCC)
Body-Centered Cubic (BCC), and
Hexagonal Close-Packed (HCP)

6. Metal Crystal Structure Atomic Radius (nm) Structures
Table 1: Atomic Radii and Crystal Structures for 16 Metals.
Metal
Crystal
Structure
Atomic
Radius
(nm)
Structures
Aluminum
FCC
0.1431
Molybdenum
BCC
0.1363
Cadmium
HCP
0.1490
Nickel
0.1246
Chromium
0.1249
Platinum
0.1387
Cobalt
0.1253
Silver
0.1445
Copper
0.1278
Tantalum
0.1430
Gold
0.1442
Titanium (α)
Iron (α)
0.1241
Tungsten
0.1371
Lead
0.1750
Zinc
0.1332

7. CRYSTAL STRUCTURE The Face-cubic Crystal Structure (FCC)
Had found for many metals
Unit cell of cubic geometry, with atoms located at each of the corners & the centers of all the cube faces.
(a)
(b)
Figure (b)- the atoms centers are represented by small circles to provide a better perspective of atom positions.
Figure (a) – hard sphere model for the FCC unit cell,

8. CRYSTAL STRUCTURE The Face-cubic Crystal Structure (FCC)
These sphere or ion cores touch one another across a face diagonal; the cube edge length a & the atomic radius R are related through:
a = 2R(2)
Refer to Figure (a):
each corner atom is shared among eight unit cells
a face-centered atom belongs to only two, and
One atom located at a center of unit cell
TOTAL: 4 atoms may assigned to a given unit cell
(a)

9. CRYSTAL STRUCTURE
Two other important characteristics of a crystal structure:
coordination number, and
the atomic packing factor (APF).
For metals, each atom has the same number of the nearest-neighbor or touching atoms, which is the coordination number.
The APF is the fraction of solid sphere volume in a unit cell, assuming the atomic hard sphere model, or
For FCC:
The coordination number is 12.
Atomic packing factor (APF) is 0.74, which the max packing possible for sphere all having the same diameter.
APF = volume of atom in a unit cell
total unit cell volume

10. CRYSTAL STRUCTURE The Face-cubic Crystal Structure (FCC)
Coordination no. = 12
Total unit cell volume
Length across face-diagonal =
4R = 2. a
Atom radius, R = (2.a)/4
Each unit cell consist of =
(6 x ½) + (8 x 1/8) atom = 4 atom/cell
APF for FCC = π/(32) = 0.74 or 74%
No. of atom/unit cell
Volume/atom

11. CRYSTAL STRUCTURE The Body-Centered Cubic Crystal Structure (BCC)
Center and corner atoms touch one another along cube diagonals, thus unit cell length a and atomic radius R are related through:
a = 4R/ 3
Coordination no. = 8
Length across face-diagonal= 4R = 3.a
Each unit cell consist of=
1 + (8 x 1/8) atom = 2 atom/sel
Total unit cell volume
No. of Atom/unit sel
Volume/atom
Atom radius, R = (3.a) / 4
APF for BCC = π(3/8) = 0.68 or 68%

12. CRYSTAL STRUCTURE The Hexagonal Closed-packed Crystal Structure (HCP)
Top and bottom faces of unit cell consist of six atoms, single atom in the center and three additional atoms between the top and bottom plane
Each unit cell consist of 6 atoms
1/6 of each of the 12 atoms at top and bottom
½ of 2 atoms at center of top and bottom
3 midplane interior atoms
Coordination no: 12
AFP : 0.74

13. EXAMPLE: Crystal Structure Computation
Calculate the volume of an FCC unit cell in terms of the atomic radius R.
Show that the atomic packing factor for the FCC crystal structure is 0.74.

14. DENSITY
A knowledge of the crystal structure of a metallic solid permits computation of its theoretical density ρ through the relationship:
Where:
n = number of atoms associated with each unit cell
A = atomic weight
VC= volume of the unit cell
NA= Avogadro’s number (6.023 x 1023 atoms/mol)
ρ = nA / VC NA

15. EXAMPLE: Density Computation
Question:
Copper has an atomic radius of nm (1.28Å), an FCC crystal structure, and an atomic weight of 63.5 g/mol. Compute its density and compare the answer with its measured density.
Solution:
Since the crystal structure is FCC, n, the no. of atoms per unit cell is 4. Furthermore, the atomic weight Acu is given as 63.5 g/mol. The unit cell volume Vc for FCC is 16R3 2, where R the atomic radius is nm.
ρ = nAcu = nAcu
VC NA (16R3 2) NA
= (4 atom/unit cell)(63.5 g/mol)
[162(1.28x 10-8 cm)3/unit cell](6.032 x 1023 atoms/mol)
= 8.89 g/cm3

16. POLYMOPRHISM AND ALLOTROPHY
Polymorphism – a phenomenon of some metal & nonmetals have more than one crystal structure.
In the elemental solids, the condition is often termed allotropy.
The prevailing crystal structure depends on both:
the temperature &
the external pressure.
Example:
carbon: graphite is a stable polymorph at ambient conditions, whereas diamond is formed at extremely high pressures.
Iron: Has a BCC crystal structure at room temperature, but changes to FCC iron at 912°C.
Most a modification of density and other physical properties will lead to polymorphic transformation.

17. CRYSTAL SYSTEM b a β y α γ c x
According to unit cell geometry, crystal structure are groups within the seven crystal system (Figure 1)
Unit cell geometry is defined in terms of six parameters:
Three edge lengths a, b and c
Three interaxial angles α, β and γ
The cubic system has the greatest
degree of symmetry, and least
symmetry is display by triclinic system.
FCC and BCC belong to cubic crystal
structure and HCP fall within hexagonal.
Lattice parameters b a β y α γ c x

18. Figure 1: Lattice parameter relationships and unit cell geometries for the seven crystal systems.

19. CRYSTALLOGRAPHIC DIRECTION AND PLANE
It is often necessary to be able to specify certain directions and planes in crystals.
Many material properties and processes vary with direction in the crystal.
Directions and planes are described using three integers - Miller Indices

20. Point Coordinates
Point position specified in terms of its coordinates as fractional multiples of the unit cell edge lengths

21. EXAMPLE: Point Coordinates
Find the Miller indices for the points in the cubic unit cell below:
Note: J is on the left face of the cube, H is on the right face,
K is on the front face and I is on the back face
A :
B :
C :
D :
E :
F :
G :
H :
I :
J :
K :

22. General Rules for Lattice Directions, Planes & Miller Indices
Miller indices used to express lattice planes and directions
x, y, z are the axes (on arbitrarily positioned origin)
a, b, c are lattice parameters (length of unit cell along a side)
h, k, l are the Miller indices for directions and planes- expressed as:
directions: [hkl] and
planes : (hkl)
Conventions for naming
There are NO COMMAS between numbers
Negative values are expressed with a bar over the number
Example: -2 is expressed 2
Crystallographic direction:
– [123]
– [100]
– … etc.
Crystallographic plane:
– (123)
– (100)
– … etc.

23. CRYSTALLOGRAPHIC DIRECTION: Miller Indices for Direction
Defined as a line between two points, or a vector.
Methods:
Draw vector that pass through the origin of the coordinate system
The length of the vector projection on each of the three axes is determined; these are measured in terms of the unit cell dimensions a, b and c.
Remove fractions by multiplying or divided by a common factor to reduce them to the smallest integer values
Enclose in square brackets [uvw]. The u, v and w integers correspond to the reduced projection along the x, y and z axes, respectively.
Negative direction, for example, the [111] direction would have a component in the –y direction.

24. EXAMPLE: Crystallographic Direction
[???] =
A: D: B: E: C: F:

25. CRYSTALLOGRAPHIC DIRECTION: Families of Directions
Equivalence of directions
<123> Family of directions
[123], [213], [312], [132], [231], [321]
only in a cubic crystal
In the cubic system directions having the same indices regardless of order or sign are equivalent.

26. CRYSTALLOGRAPHIC PLANES: Miller Indices for Planes
(hkl) Crystallographic plane
{hkl} Family of crystallographic planes
e.g. (hkl), (lhk), (hlk) … etc.
In the cubic system planes having the same indices regardless of order or sign are equivalent
Hexagonal crystals can be expressed in a four index system (uvtw)
Can be converted to a three index system using formulas

27. CRYSTALLOGRAPHIC PLANES: Miller Indices for Planes
Method
If the plane passes through the origin, select an equivalent plane or move the origin
Determine the intersection of the plane with the axes in terms of a, b, and c
Take the reciprocal (1/∞ = 0)
Convert to smallest integers (optional)
Enclose by parentheses

28. EXAMPLE: Crystallographic Planes

29. CRYSTALLOGRAPHIC PLANES
Planes and their negatives are equivalent
In the cubic system, a plane and a direction with the same indices are orthogonal

30. LINEAR AND PLANAR ATOMIC DENSITIES
Linear Density
Number of atoms per length whose centers lie on the direction vector for a specific crystallographic direction.
Planar Density
Number of atoms per unit area that are centered on a particular crystallographic plane.
LD = No. of atoms centered on a direction vector
length of direction vector
PD = No. of atoms centered on a plane
area of plane

31. EXAMPLE Calculate the linear density of the [100] direction for BCC
Calculate the planar density of the (100) plane for FCC

32. LINEAR AND PLANAR ATOMIC DENSITIES
Why do we care?
Properties, in general, depend on linear and planar density.
Examples:
Speed of sound along directions
Slip (deformation in metals) depends on linear and planar density
Slip occurs on planes that have the greatest density of atoms in direction with highest density (we would say along closest packed directions on the closest packed planes)