Lecture#9
Review to:
- Materials and packing - metals
- Theoretical density
- Miller Indices
Dr.Haydar Al-Ethari
- Polymorphism and Allotropy
References:
1- Michael F. Ashby, David R. H. Jones, 1996, Engineering Materials,
An Introduction to their Properties and Applications, Second Edition,
Butterworth &Heinemann.
2- Derek Hull, David Bacon, (2001), Introduction to Dislocations, fourth
edition, Butterworth-Heinemann.
3- Hosford W.F, (2005), Mechanical Behavior of Materials, Cambridge
4- Meyers M.A. and Chawla K.K., (2009). Mechanical Behavior of
Materials, Prentice- Hall.
5- Dieter G.E., (1986), Mechanical Metallurgy, McGraw-Hill
MATERIALS AND PACKING
Crystalline materials:- atoms pack in periodic, 3D arrays
• typical of : -metals
-many ceramics -some polymers
Noncrystalline materials:- atoms have no periodic packing
• occurs for: - complex structures - rapid cooling
"Amorphous" = Noncrystalline
METALLIC CRYSTALS tend to be densely packed. They have several
reasons for dense packing:
-Typically, only one element is present, so all atomic radii are the same.
-Metallic bonding is not directional.
-Nearest neighbor distances tend to be small in order to lower bond
energy.
• METALS have the simplest crystal structures.
We will look at three such structures :-
1- SIMPLE CUBIC STRUCTURE (SC)
• Rare due to poor packing (only Po has this structure)
• Close-packed directions are cube edges.
* close-packed directions a, R=0.5a, contains 8 x 1/8 = 1 atom/unit cell
2- BODY CENTERED CUBIC STRUCTURE (BCC)
(Alpha-iron, columbium, tantalum, chromium, Mo, W……)
3- Face centered cubic : FCC (Al, Cu, Ag, Pb, Ni, Au……….)
THEORETICAL DENSITY,
ρ=
nA
VC N A
n- atoms/unit cell; A- Atomic weight (g/mol); Vc- Volume/unit cell
(cm3/unit cell); NA- Avogadro's number (6.023 x 1023 atoms/mol)
Example: Copper
• crystal structure = FCC: 4 atoms/unit cell
• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)
• atomic radius R = 0.128 nm
Compare to actual: cu = 8.94 g/cm3
Result: theoretical cu = 8.89 g/cm3
The densest packing:
The shaded planes are close packed in both structures.
What is the difference in the stacking sequence of the close packed planes
that makes HCP different from FCC? The figure below provides the
mechanism for producing HCP and FCC crystals with packing of closed
packed planes.
Possible stacking of close packed planes.
Stacking sequences of the type ABABAB or ACACAC are the same (they
can be transformed into each other by rotating the crystal by 180 deg. and
produce the HCP structure. It is much easier to see this with 3D ball models.
In conclusion an
FCC structure: ABCABCABC...
HCP structure: ABABAB.. or ACACAC.
Miller Indices: To determine the Miller indices of a direction:
Find the coefficients l,m, n of the vector r such that: r = la1 + ma2 + na3
Reduce these 3 numbers to the smallest integers having the same ratio
Enclose these in square brackets.
[231]
[210]
[ 1 01]
[166]
The directions for the vectors shown are r1 = [011], r2 = [1 1 0], r3 = [ 1 01]
and r4 = [01 1 ].All directions shown in the Fig. are close packed directions.
The equivalent family of directions in the FCC structure are the < 110 >
directions. < 110 > represents the directions [110], [1 1 0], [101], [10 1 ],
[011], [01 1 ] and their negative [ 1 1 0], etc.
The close-packed directions in bcc are the directions < 111 > (body
diagonal of the unit cell). The Miller indices of all possible specific close
packed directions in the BCC structure are [111], [1 1 1], [ 1 11], [11 1 ],
[ 1 1 1 ], 1 1 1 ], [1 1 1 ], [ 1 1 1].
MILLER INDICES FOR PLANES IN CUBIC STRUCTURES,
Miller Indices for Planes: To find the Miller indices of a plane:
Find its intercepts with the a1, a2 and a3 axes.
Take the reciprocals of these intercepts.
Reduce all fractions to the lowest common denominators.
Enclose the 3 numbers which are numerators of these fractions in round
parentheses ( ) to represent the specific plane.
Family of planes
All equivalent planes in a structure are conventionally represented by one set
of Miller indices in curly brackets { }.
All close packed planes in the FCC crystal can be represented by writing
‘(111) type planes’. In FCC {111} represents: (111), (1 1 1), (11 1 ), ( 1 11),
( 1 1 1 ), (¯11 1 ), (¯1 1 1), (1 1 1 ). To find another set of indices for a set
of parallel planes, take the negative of the known indices. For example the
following planes are parallel to each other: (111) and ( 1 1 1 ), (1 1 1) and
( 1 1 1 ), (11 1 ) and ( 1 1 1), and ( 1 11) and (1 1 1 ).
Octahedral plane in an FCC or BCC structure
Note that for the cubic systems that the [hkl] direction is always normal to
the (hkl) plane. This is not true for other crystal systems.
Miller Indices for Hexagonal structures:
Consider the Miller indices of the planes A,B,and C in a hexagonal structure
based on the 3 axis system shown. It is easy to show that plane A=(100),
plane B=(010) and plane C=(1 1 0). These planes are actually equivalent in
the hexagonal system, yet using the three-axis system, they have different
Miller indices. Note that planes A, B, C are equivalent crystallographic
planes since there is a 6-fold symmetry around the a3 axis. A 3600/6 = 60 0
rotation brings plane A into plane B, etc. Similarly note that in the hexagonal
structure shown, the [100] direction is not normal to the plane (100)! By
convention a 4-axis –4-index system is used to describe the hexagonal
structure which removes these apparent discrepancies. The Miller indices
(l,m, n, 0) of the planes A, B, C are found as before from the intersections of
the plane with the 4-axes. They are plane A=(10 1 0), plane B=(01 1 0) and
plane C=(1 1 00). Planes A, B, C are {1 1 00} type planes in the 4-index
system.
Basal plane in HCP: The plane perpendicular to the a4 axis is called the
basal plane. Its Miller indices are (0001). Note that the basal plane is closepacked plane in HCP.
Miller Indices for directions in hexagonal structures
To determine indices for directions in hexagonal structures: [hkil] these
indices are translations parallel to the 4 axes that cause motion in the
required direction. Since the ‘unit vectors’ in the a1, a2, a3 and c directions
are not independent, we need to maintain the relation i = (h + k).
The close packed directions in the HCP structure are of the < 2 1 1 0 > type.
All of them lie in the basal plane.
The HCP structure. The basal plane (0001) is close packed (not ideally) and
the close packed direction is < 11 2 0 >.
Polymorphism and Allotropy:
Polymorphism is the property of a material to exist in more than one type of
space lattice in the solid state. If the change in structure is reversible, then
the polymorphism change is known as allotropy. At least fifteen metals
show this property and the best-known example is iron. When iron
crystallizes at 2800o it is BCC ( Fe), at 25540 F the structure changes to
FCC ( Fe), and at 16700 F it again becomes BCC ( Fe).