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Chapter3

# Chapter3 - Chapter 3 Crystal Structures Introduction...

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Introduction Bravais Lattice Crystals with One Atom per Lattice Point Miller Indices Densities and Packing Factors of Crystalline Structures Interstitial Positions and Sizes Crystals with Multiple Atoms per Lattice Site Liquid Crystals Single Crystals and Polycrystalline Materials Polymorphism X-Ray Diffraction Chapter 3 Crystal Structures

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Many materials crystallize in a regular array with basic building blocks being at repeated regular intervals. The structure is determined by the bond character and energy minimization. Metals, ceramics, polymers can all form crystal structures. Properties are dependent upon the type of bond and the structure. Chapter 3 Crystal Structures
Lattice points Unit cell Square Rectangular Hexagonal a = b, φ = 90° a b, φ = 90° a = b, φ = 120° Bravais Lattice and Unit Cells

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Tiling a General Parallelogram The tiles are space filling satisfying the need for translational periodicity of a lattice
Lack of Translational Periodicity Using a Pentagon

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Basis- Single character Basis- Double character Basis- Two characters with different orientations Lattice + Basis = Crystal structure Bravais Lattice and Unit Cells
a = b = c, α = β = γ = 90° a = b c, α = β = γ = 90° a b c, α = β = γ = 90° a = b c, α = β = 90°, γ = 120° Cubic Tetragonal Orthorhombic Rhombohedral a b c, α = γ = 90°≠ β a = b = c, α = β = γ ≠ 90° a b c, α ≠ β ≠ γ ≠ 90° Monoclinic Triclinic Hexagonal Bravais Lattice and Unit Cells

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x y z Bravais Lattice and Unit Cells- Face Centered Cubic (FCC) a 1 a 3 a 2 Each lattice point is surrounded by 12 equivalent lattice points. Three alternative vectors define a unit cell that is primitive, one lattice point per unit cell.
Equivalence of lattice points Position A is in the center of the black cube , and a corner of the blue cube Position B is in the center of the blue cube , and a corner of the black cube Bravais Lattice and Unit Cells

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Body Center Cubic (BCC) Lattice Each lattice point is surrounded by 8 equivalent lattice points. Three alternative vectors define a unit cell that is primitive, one lattice point per unit cell.
Basis Lattice Alternate Descriptions

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What is meant by the terms cross product and triple scalar product ? ˆ ˆ ˆ o a b c V a a ui vj wk × = = + + r r r r [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 3 3 3 100 010 001 010 001 0 1 0 100 0 0 1 100 100 o o o a b c a i j k a a × = × × = = = r r r [ ] [ ] [ ] ˆ ˆ ˆ 1 0 0 100 ˆ ˆ ˆ 0 1 0 010 ˆ ˆ ˆ 0 0 1 001 o o o o o o a a i j k a b a i j k a c a i j k a = + + = = + + = = + + = r r r o a [ ] 100 [ ] 010 [ ] 001 Non-primitive unit cell
[ ] [ ] [ ] [ ] [ ] [ ] 3 3 3 101 110 011 8 110 011 1 1 0 111 0 1 1 101 111 8 4 o p p p o o p p p a a b c i j k a a a b c × = × × = = × = = r r r r r r o a [ ] 101 2 o a [ ] 110 2 o a [ ] 011 2 o a Primitive unit cell

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Face Centered Cubic (FCC) Phase Lattice Hard sphere packing Fractions of atoms in the unit cell Distance between the centers of adjacent atoms Copper is an FCC material
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