MSE 310
Elec. Props of
Matls
Fundamentals of
Crystallography
3
‰ 7 Crystal Systems:
‰ 14 Bravais Lattices:
1
2
Additional Information:
1
See: Chapter 3
Materials Science and Engineering – An
Introduction, William D. Callister, Jr. 6th Ed
or 7th Ed (Wiley, 2003)
(trigonal)
4
2
1
14 total
Materials Science and Engineering – An Introduction,
William D. Callister, Jr. 6th Ed (Wiley, 2003)
Knowlton
MSE 310
Elec. Props of
Matls
1
Fundamentals of Crystallography
‰ Unit Cells of the Bravais Lattices:
Knowlton
One of these has a mistake. Which one?
Ashcroft & Mermin, Solid State Physics
2
1
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Crystal Structure is described by: Bravais lattice + basis (atoms decorating lattice point)
9 Also called: Space lattice + Motif
‰ Examples of Crystal Structures: Diamond Cubic Lattice & Zinc Blende
Examples of: a) Si & Ge; b) ZnS, ZnSe, GaAs
Knowlton
MSE 310
Elec. Props of
Matls
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
3
Fundamentals of Crystallography
‰ Unit Cells of the Bravais Lattices:
9 Unit Cells are an array of lattice points in a specific Bravais lattice that, when
periodically repeated, forms the entire lattice.
9 There are two types of unit cells:
o Conventional unit cell
• Most geometrically convenient
• 1 or more lattice points per unit cell
o Primitive unit cell
• Smallest unit cell possible
• Only one lattice point
Knowlton
4
2
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Examples of Primitive Unit Cells:
9 FCC & BCC
Blakemore, Solid State Physics 2nd Ed (Cambridge, 1985)
Knowlton
5
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ PEROVSKITES - Example of:
9 Primitive Unit Cell
9 Crystal Structure = Bravais Lattices/space lattice + Basis
Information
Formula: M'M''X3
Bravais Lattice: simple cubic
Basis: 5 ions (1Ba2+, 1Ti4+, 3O2-)
Atoms per Unit Cell: 5
Knowlton
Allen & Thomas, The Structure off Materials, (Wiley, 1999)
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
6
3
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Crystallographic Directions: a crystallographic direction is defined as a line
between 2 pts, or a vector. The following steps are utilized in the determination of
the 3 directional indices:
9 A vector of convenient length is positioned such that it passes through the origin of the
coordinate system. Any vector may be translated throughout the crystal lattice without
alteration, if parallelism in maintained.
9 The length of the vector projection on each of the 3 axes is determined; these are
measured in terms of the unit cell dimensions a, b, and c.
9 These 3 numbers are multiplied or divided by a common factor to reduce them to the
smallest integer values.
9 The 3 indices (not separated by commas) are enclosed in square brackets:
o [uvw] where u, v, and w integers correspond to the reduced projection along the x, y, and z
axes, respectively.
Materials Science and Engineering – An Introduction,
William D. Callister, Jr. 6th Ed (Wiley, 2003)
Knowlton
MSE 310
Elec. Props of
Matls
7
Fundamentals of Crystallography
‰ Crystallographic Directions: an Example
Knowlton
Materials Science and Engineering – An Introduction,
William D. Callister, Jr. 6th Ed (Wiley, 2003)
8
4
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Crystallographic Directions: an Example
9 What directions are the bonds in Si & Zinc Blende materials?
Examples of: a) Si & Ge; b) ZnS, ZnSe, GaAs
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
Knowlton
MSE 310
Elec. Props of
Matls
9
Fundamentals of Crystallography
‰ Crystallographic Planes: a crystallographic planes in all but the hexagonal crystal system
are specified by 3 Miller indices: (hkl). Any 2 planes parallel to each other are equivalent &
have identical indices. The procedure used in determination of the h, k, and l Miller index
numbers is as follows:
9 If the plane passes through the selected origin, either another parallel plane must be constructed
within the unit cell by an appropriate translation, or a new origin must be established at the corner of
another unit cell.
9 At this point, the crystallographic plane either intersects or parallels each of the 3 axes; the length of
the planar intercept for each axis is determined in terms of the lattice constants a, b, and c.
9 The reciprocals of these numbers are taken. A plane that parallels an axis may be considered to have
an infinite intercept, &, therefore, a zero index.
9 If necessary, these 3 numbers are changed to the set of smallest integers by multiplication or
division by a common factor.
9 Finally, the integer indices (not separated by commas) are enclosed in parentheses: (hkl).
Knowlton
Foundations of Materials Science and Engineering,
Smith 3rd Ed (McGraw Hill, 2003)
10
5
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Crystallographic Planes: Several Examples
Materials Science and Engineering – An Introduction,
William D. Callister, Jr. 6th Ed (Wiley, 2003)
Knowlton
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Crystallographic Planes: Equivalent planes
Knowlton
11
‰ Family of planes
9 Those planes which are
equivalent in the crystal by
symmetry
9 Designated by { }
9 {100} = (100) + (010) +
(001) + (-100) + (0-10) +
(00-1)
Materials Science and Engineering – An Introduction,
William D. Callister, Jr. 6th Ed (Wiley, 2003)
12
6
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Comparison of Packing of Atoms in the FCC & HCP Structures:
Stacking sequence in FCC
Crystals for (111) planes:
ABCABC
HCP
Plane A
Or
A sites
FCC
Plane B
Or
B sites
Blakemore, Solid State Physics 2nd Ed (Cambridge, 1985)
Plane C
Or
C sites
Knowlton
13
MSE 310/510
Elec Props of Matls
Extra for those of you that are
interested
14
Knowlton
7
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
a2
‰ Crystallographic Directions:
9 Example of the Hexagonal system:
v
v
v
v
v
r = ua1 + va2 + ta3 + wc
where the direction
is written:
[uvtw]
a1 =-1
a3 =2
a1
and the following
condition imposed:
u +v+t = 0
a2 =-1
a3 direction:
a3
⎡⎣ 1 120 ⎤⎦
Knowlton
MSE 310
Elec. Props of
Matls
15
Fundamentals of Crystallography
‰ Crystallographic Directions:
9 Several Examples of the Hexagonal system:
Knowlton
Barrett & Massalski, Structure of Metals 3rd Ed
(Pergamon, 1980)
16
8
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Hexagonal Indices for Planes:
9 To determine the indices for planes in the hexagonal crystal system, the 3 axes
system used in the other crystal systems are not usually used because equivalent
planes do not have similar indices.
9 For this reason, it is preferable to use the four axes:
o a1, a2, a3, c
9 The Miller indices are: (hkil)
where i=-(h+k) or h+k+i=0
Example:
Axis:
a1
a2
a3
c
Intersection:
1
1
-1/2
∞
Reciprocal:
1
1
-2
0
Reduce:
1
1
-2
0
Miller
Indices:
1
1
-2
0
(1120) or (11.0)
Knowlton
Barrett & Massalski, Structure of Metals 3rd Ed (Pergamon, 1980)
MSE 310
Elec. Props of
Matls
17
Fundamentals of Crystallography
‰ Determining Hexagonal Indices for Planes:
9 If the plane passes through the selected origin, either another parallel plane must be constructed
within the unit cell by an appropriate translation, or a new origin must be established at the corner of
another unit cell.
9 At this point, the crystallographic plane either intersects or parallels each of the 4 axes; the length of
the planar intercept for each axis is determined in terms of the lattice constants a1, a2, a3, c.
9 The reciprocals of these numbers are taken. A plane that parallels an axis may be considered to have
an infinite intercept, &, therefore, a zero index.
9 If necessary, these 3 numbers are changed to the set of smallest integers by multiplication or
division by a common factor.
Example:
Axis:
a1
a2
a3
c
Intersection:
1
1
-1/2
∞
Reciprocal:
1
1
-2
0
Reduce:
1
1
-2
0
Miller
Indices:
1
1
-2
0
(1120) or (11.0)
Knowlton
Barrett & Massalski, Structure of Metals 3rd Ed (Pergamon, 1980)
18
9
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Packing of Atoms in the FCC Structure:
Knowlton
MSE 310
Elec. Props of
Matls
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
Foundations of Materials Science and Engineering,
Smith 3rd Ed (McGraw Hill, 2003)
19
Fundamentals of Crystallography
‰ Packing of Atoms in the BCC Structure:
Knowlton
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
Foundations of Materials Science and Engineering,
Smith 3rd Ed (McGraw Hill, 2003)
20
10
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Comparison of Packing of Atoms in the FCC & HCP Structures:
HCP
FCC
Blakemore, Solid State Physics 2nd Ed (Cambridge, 1985)
Knowlton
MSE 310
Elec. Props of
Matls
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
21
Fundamentals of Crystallography
‰ Packing of Atoms in the HCP Structure:
Knowlton
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
Materials Science and Engineering – An Introduction,
William D. Callister, Jr. 6th Ed (Wiley, 2003)
22
11
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Crystalline and Noncrystalline Materials:
9 Question: How would you calculate the density of a material given the:
o Lattice constant or Radius of Atom
o Crystal structure
o Atomic weight
where density is given by: mass/volume?
mass/volume
9 Use Following Information: Cu, FCC, rCu = 0.128 nm, Atomic Mass=63.5g/mole.
Knowlton
23
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Crystalline and Noncrystalline Materials:
9 Isotropic Materials:
o If the properties of the material are independent of the direction in which
they are measured, the material is categorized as isotropic.
9 Anisotropy Materials:
o If the properties of the material are dependent of the direction in which they
are measured, the material is categorized as anisotropic.
o Examples of these properties include:
•
•
•
•
Knowlton
Periodicity of atoms in a crystal structure.
Density of a material of crystalline solids.
Carrier velocity.
Phonon velocity.
24
12
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
‰ Crystalline and Noncrystalline Materials:
9 Allotropy:
o Elements that can exhibit more than one crystal structure are allotropic.
9 Polymorphism:
o Compounds that behave in the same manner as allotropic materials are
referred to as polymorphic.
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
Knowlton
MSE 310
Elec. Props of
Matls
25
Fundamentals of Crystallography
‰ Crystalline and Noncrystalline Materials:
9 Single Crystal:
o A continuous periodic crystal structure only interrupted by the boundaries of
the solid
9 Amorphous:
o Although short range periodicity may be present, long range periodicity is
absent. Hence, amorphous material is not crystalline.
9 Polycrystalline:
o A material composed of from two to many single crystal grains.
Knowlton
26
13
MSE 310
Elec. Props of
Matls
Fundamentals of Crystallography
a2
a2
a1 =-1
a2 =2
a3 =-1
a2 =-1
a3 =-1
a1
a1
a1 =2
a3
a3
⎡⎣ 1 120 ⎤⎦
Knowlton
27
14