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1
MSE 310Elec. Props of Matls
1Knowlton
Fundamentals of Crystallography
7 Crystal Systems:14 Bravais Lattices:
3
1
2
1(trigonal)
4
2
114 total
Additional Information:See: Chapter 3Materials Science and Engineering – An Introduction, William D. Callister, Jr. 6th Ed or 7th Ed (Wiley, 2003)
Materials Science and Engineering – An Introduction, William D. Callister, Jr. 6th Ed (Wiley, 2003)
MSE 310Elec. Props of Matls
2Knowlton
Fundamentals of Crystallography
Unit Cells of the Bravais Lattices:
Ashcroft & Mermin, Solid State Physics
One of these has a mistake. Which one?
2
MSE 310Elec. Props of Matls
3Knowlton
Fundamentals of CrystallographyCrystal Structure is described by: Bravais lattice + basis (atoms decorating lattice point)
Also called: Space lattice + MotifExamples of Crystal Structures: Diamond Cubic Lattice & Zinc Blende
Examples of: a) Si & Ge; b) ZnS, ZnSe, GaAsShackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
MSE 310Elec. Props of Matls
4Knowlton
Fundamentals of Crystallography
Unit Cells of the Bravais Lattices:Unit Cells are an array of lattice points in a specific Bravais lattice that, when periodically repeated, forms the entire lattice.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
3
MSE 310Elec. Props of Matls
5Knowlton
Fundamentals of Crystallography
Examples of Primitive Unit Cells:FCC & BCC
Blakemore, Solid State Physics 2nd Ed (Cambridge, 1985)
MSE 310Elec. Props of Matls
6Knowlton
Fundamentals of CrystallographyPEROVSKITES - Example of:
Primitive Unit CellCrystal Structure = Bravais Lattices/space lattice + Basis
Allen & Thomas, The Structure off Materials, (Wiley, 1999)
InformationFormula: M'M''X3Bravais Lattice: simple cubicBasis: 5 ions (1Ba2+, 1Ti4+, 3O2-)Atoms per Unit Cell: 5
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
4
MSE 310Elec. Props of Matls
7Knowlton
Fundamentals of CrystallographyCrystallographic Directions: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:
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.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.These 3 numbers are multiplied or divided by a common factor to reduce them to the smallest integer values.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)
MSE 310Elec. Props of Matls
8Knowlton
Fundamentals of CrystallographyCrystallographic Directions: an Example
Materials Science and Engineering – An Introduction, William D. Callister, Jr. 6th Ed (Wiley, 2003)
5
MSE 310Elec. Props of Matls
9Knowlton
Fundamentals of CrystallographyCrystallographic Directions: an Example
What directions are the bonds in Si & Zinc Blende materials?
Examples of: a) Si & Ge; b) ZnS, ZnSe, GaAsShackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
MSE 310Elec. Props of Matls
10Knowlton
Fundamentals of CrystallographyCrystallographic Planes: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:
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.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.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.If necessary, these 3 numbers are changed to the set of smallest integers by multiplication or division by a common factor.Finally, the integer indices (not separated by commas) are enclosed in parentheses: (hkl).
Foundations of Materials Science and Engineering, Smith 3rd Ed (McGraw Hill, 2003)
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MSE 310Elec. Props of Matls
11Knowlton
Fundamentals of CrystallographyCrystallographic Planes: Several Examples
Materials Science and Engineering – An Introduction, William D. Callister, Jr. 6th Ed (Wiley, 2003)
MSE 310Elec. Props of Matls
12Knowlton
Fundamentals of Crystallography
Crystallographic Planes: Equivalent planes Family of planesThose planes which are equivalent in the crystal by symmetryDesignated by { }{100} = (100) + (010) + (001) + (-100) + (0-10) + (00-1)
Materials Science and Engineering – An Introduction, William D. Callister, Jr. 6th Ed (Wiley, 2003)
7
MSE 310Elec. Props of Matls
13Knowlton
Fundamentals of Crystallography
Comparison of Packing of Atoms in the FCC & HCP Structures:
Plane AOr
A sites
Plane BOr
B sites
Plane COr
C sites
Stacking sequence in FCC Crystals for (111) planes:
ABCABC
FCCHCP
Blakemore, Solid State Physics 2nd Ed (Cambridge, 1985)
14
MSE 310/510Elec Props of Matls
Knowlton
Extra for those of you that are interested
8
MSE 310Elec. Props of Matls
15Knowlton
Fundamentals of CrystallographyCrystallographic Directions:
Example of the Hexagonal system:a2
a1
a3
a1 =-1
a3 =2
a2 =-1
a3 direction:1 120⎡ ⎤⎣ ⎦
[ ]
1 2 3
where the directionis written:
and the followingcondition imposed:
0
r ua va ta wc
uvtw
u v t
= + + +
+ + =
v v v v v
MSE 310Elec. Props of Matls
16Knowlton
Fundamentals of CrystallographyCrystallographic Directions:
Several Examples of the Hexagonal system: Barrett & Massalski, Structure of Metals 3rd Ed (Pergamon, 1980)
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MSE 310Elec. Props of Matls
17Knowlton
Fundamentals of CrystallographyHexagonal Indices for Planes:Hexagonal Indices for Planes:
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.For this reason, it is preferable to use the four axes:
o a1, a2, a3, cThe Miller indices are: (hkil)
where i=-(h+k) or h+k+i=0
0-211Reduce:
Miller Indices:
Reciprocal:
Intersection:
Axis:
0-211
0-211
∞-1/211
ca3a2a1
Example:
(1120) (11.0)or
Barrett & Massalski, Structure of Metals 3rd Ed (Pergamon, 1980)
MSE 310Elec. Props of Matls
18Knowlton
Fundamentals of CrystallographyDetermining Hexagonal Indices for Planes:Determining Hexagonal Indices for Planes:
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.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.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.If necessary, these 3 numbers are changed to the set of smallest integers by multiplication or division by a common factor.
0-211Reduce:
Miller Indices:
Reciprocal:
Intersection:
Axis:
0-211
0-211
∞-1/211
ca3a2a1
Example:
(1120) (11.0)or
Barrett & Massalski, Structure of Metals 3rd Ed (Pergamon, 1980)
10
MSE 310Elec. Props of Matls
19Knowlton
Fundamentals of CrystallographyPacking of Atoms in the FCC Structure:
Foundations of Materials Science and Engineering, Smith 3rd Ed (McGraw Hill, 2003)Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
MSE 310Elec. Props of Matls
20Knowlton
Fundamentals of CrystallographyPacking of Atoms in the BCC Structure:
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)Foundations of Materials Science and Engineering, Smith 3rd Ed (McGraw Hill, 2003)
11
MSE 310Elec. Props of Matls
21Knowlton
Fundamentals of Crystallography
Comparison of Packing of Atoms in the FCC & HCP Structures:
FCCHCP
Blakemore, Solid State Physics 2nd Ed (Cambridge, 1985)
Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
MSE 310Elec. Props of Matls
22Knowlton
Fundamentals of CrystallographyPacking of Atoms in the HCP Structure:
Materials Science and Engineering – An Introduction, William D. Callister, Jr. 6th Ed (Wiley, 2003)Shackelford, Intro to Materials Science for Engineers, 5th Ed. (Prentice Hall, 2002)
12
MSE 310Elec. Props of Matls
23Knowlton
Fundamentals of CrystallographyCrystalline and Noncrystalline Materials:
Question: How would you calculate the density of a material given the:o Lattice constant or Radius of Atomo Crystal structureo Atomic weight
where density is given by: mass/volumemass/volume?
Use Following Information: Cu, FCC, rCu = 0.128 nm, Atomic Mass=63.5g/mole.
MSE 310Elec. Props of Matls
24Knowlton
Fundamentals of CrystallographyCrystalline and Noncrystalline Materials:
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.
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:
• Periodicity of atoms in a crystal structure.• Density of a material of crystalline solids.• Carrier velocity.• Phonon velocity.
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MSE 310Elec. Props of Matls
25Knowlton
Fundamentals of CrystallographyCrystalline and Noncrystalline Materials:
Allotropy:o Elements that can exhibit more than one crystal structure are allotropic.
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)
MSE 310Elec. Props of Matls
26Knowlton
Fundamentals of CrystallographyCrystalline and Noncrystalline Materials:
Single Crystal:o A continuous periodic crystal structure only interrupted by the boundaries of
the solid
Amorphous:o Although short range periodicity may be present, long range periodicity is
absent. Hence, amorphous material is not crystalline.
Polycrystalline:o A material composed of from two to many single crystal grains.
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MSE 310Elec. Props of Matls
27Knowlton
Fundamentals of Crystallography
1 120⎡ ⎤⎣ ⎦
a1
a3
a1 =2
a3 =-1
a2 =-1
a2
a1
a3
a1 =-1
a3 =-1
a2 =2
a2