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Crystal Structure of Solids
What is “Crystal” to the man on the street?
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Fundamental Properties of Matter
States of Matter
1. Solids – Definite volume, definite shape.
2. Liquids – Definite volume, no fixed shape. Flows.
3. Gases – No definite volume, no definite shape. Takes the volume and shape of its container.
Matter: - Has mass, occupies spaceMass – measure of inertia - from Newton’s first law of motion. It is one of the fundamental physical properties.
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STRUCTURE OF SOLIDS
•Can be classified under several criteria based on atomic arrangements, electrical properties, thermal properties, chemical bonds etc.
•Using electrical criterion: Conductors, Insulators, Semiconductors
•Using atomic arrangements: Amorphous, Polycrystalline, Crystalline.
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Under what categories could this class be grouped?
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• No regular long range order of arrangement in the atoms.
• Eg. Polymers, cotton candy, common window glass, ceramic.
• Can be prepared by rapidly cooling molten material. • Rapid – minimizes time for atoms to pack into a
more thermodynamically favorable crystalline state. • Two sub-states of amorphous solids: Rubbery and
Glassy states. Glass transition temperature Tg = temperature above which the solid transforms from glassy to rubbery state, becoming more viscous.
Amorphous Solids
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•Atomic order present in sections (grains) of the solid.
•Different order of arrangement from grain to grain. Grain sizes = hundreds of m.
•An aggregate of a large number of small crystals or grains in which the structure is regular, but the crystals or grains are arranged in a random fashion.
Polycrystalline Solids
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Polycrystalline Solids
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Atoms arranged in a 3-D long range order. “Single crystals” emphasizes one type of crystal order that exists as opposed to polycrystals.
Crystalline Solids
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• Properties of single crystalline materials vary with direction, ie anisotropic. • Properties of polycrystalline materials may or may not vary with direction.
If the polycrystal grains are randomly oriented, properties will not vary with direction ie isotropic.• If the polycrystal grains are textured, properties will vary with direction ie anisotropic
Single- Vs Poly- Crystal
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E (diagonal) = 273 GPa
E (edge) = 125 GPa
Single- Vs Poly- Crystal
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-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.
200 m
Single- Vs Poly- Crystal
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a
b
c
Lattice Parameters
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Atoms in a Crystal
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The Unit Cell Concept• The simplest repeating unit in a crystal is called
a unit cell. • Opposite faces of a unit cell are parallel.• The edge of the unit cell connects equivalent
points.• Not unique. There can be several unit cells of a
crystal.• The smallest possible unit cell is called
primitive unit cell of a particular crystal structure.
• A primitive unit cell whose symmetry matches the lattice symmetry is called Wigner-Seitz cell.
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• Each unit cell is defined in terms of lattice points.
• Lattice point not necessarily at an atomic site.• For each crystal structure, a conventional unit
cell, is chosen to make the lattice as symmetric as possible. However, the conventional unit cell is not always the primitive unit cell.
•A crystal's structure and symmetry play a role in determining many of its properties, such as cleavage (tendency to split along certain planes with smooth surfaces), electronic band structure and optical properties.
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Unit cell
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Bravais Lattice and Crystal System
Crystal structure: contains atoms at every lattice point.• The symmetry of the crystal can be more
complicated than the symmetry of the lattice.• Bravais lattice points do not necessarily
correspond to real atomic sites in a crystal. A Bravais lattice point may be used to represent a group of many atoms of a real crystal. This means more ways of arranging atoms in a crystal lattice.
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1. Cubic (Isometric) System
Symmetry elements: Four 3-fold rotation axes along cube diagonalsa = b = c = = = 90o
3 Bravais lattices
ab
c
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• Rare due to poor packing (only Po has this structure)• Close-packed directions are cube edges.
Coordination # = 6 (# nearest neighbors)
1 atom/unit cell
(1-a): Simple Cubic Structure (SC)
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Coordination Number = Number of nearest neighbors
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One atom per unit cell
1/8 x 8 = 1
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APF = Volume of atoms in unit cell*Volume of unit cell
*assume hard spheres• APF for a simple cubic structure = 0.52
APF = a3
43
(0.5a)31atoms
unit cellatom
volume
unit cellvolume
aR=0.5a
Adapted from Fig. 3.19, Callister 6e.
Atomic Packing Factor
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Adapted from Fig. 3.1(a), Callister 6e.
• Exhibited by Al, Cu, Au, Ag, Ni, Pt• Close packed directions are face diagonals.• Coordination number = 12• 4 atoms/unit cell
6 x (1/2 face) + 8 x 1/8 (corner) = 4 atoms/unit cell
(1-b): Face Centered Cubic Structure (FCC)
All atoms are identical
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FCCCoordination number = 12
3 mutually perpendicular planes.4 nearest neighbors on each of the three planes.
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• Exhibited by Cr, Fe, Mo, Ta, W• Close packed directions are cube diagonals.• Coordination number = 8
2 atoms/unit cell
(1-c): Body Centered Cubic Structure (BCC)
All atoms are identical
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Which one has most packing ?
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Which one has most packing ?
For that reason, FCC is also referred to as cubic closed packed (CCP)
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Symmetry element: One 6-fold rotation axisa = b ca= 120o
= = 90o
2. Hexagonal System
Only one Bravais lattice
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• Exhibited by …. • ABAB... Stacking Sequence• Coordination # = 12• APF = 0.74
3D Projection
2D Projection
A sites
B sitesA sites
Bottom layer
Middle layer
Top layer
Adapted from Fig. 3.3, Callister 6e.
Hexagonal Closed Packed Structure (HCP)
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Symmetry element: One 4-fold rotation axisa = b ca= = = 90o
3. Tetragonal System
Two Bravais lattices
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Symmetry element: One 3-fold rotation axisa = b ca= 120o
= = 90o
4. Trigonal (Rhombohedral) System
One Bravais lattice
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5. Orthorhombic System
Symmetry element: Three mutually perpendicular 2-fold rotation axesa b ca = = = 90o
Four Bravais lattices
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6. Monoclinic System
Symmetry element: One 2-fold rotation axisa b ca = = 90o, 90o
Two Bravais lattices
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7. Triclinic System
Symmetry element: Nonea b ca 90o
One Bravais lattice