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Unit 2 - CrystallographyUnit 2 - Crystallography

In most solids, atoms fit into a In most solids, atoms fit into a regular 3-dimensional pattern called regular 3-dimensional pattern called a a crystalcrystal

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-Crystals are not small and simple like molecules are (e.g. H20, C02)-Theoretically a crystal can go on forever-Real crystals never do-However even the smallest crystal extends billions of atoms in all directions-Since crystals are so huge, how can we wrap our minds around the way crystals are structured?

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-The conceptual tool we use for this is the unit cell-The unit cell is the smallest possible repeating pattern of atoms in the crystal

Na+

Cl-

Unit cell of NaCl

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The unit cell is repeated to form the The unit cell is repeated to form the crystalcrystal

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The unit cell is repeated to form the The unit cell is repeated to form the crystalcrystal

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The unit cell is repeated to form the The unit cell is repeated to form the crystalcrystal

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Lattice constantsLattice constants are numbers that are numbers that characterize the size of the unit cellcharacterize the size of the unit cell

With a cubic geometry, only one lattice constant is needed. It is usually designed a

With a hexagonal geometry, two lattice constants are needed, usually called a and c

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There are seven common crystal geometriesThere are seven common crystal geometries

Cubic Hexagonal Tetragonal Rhombohedral

OrthorhombicMonoclinic Triclinic

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Most metals have one of these 3 crystal Most metals have one of these 3 crystal geometriesgeometries

Face-centered cubic (FCC)

Body-centered cubic (BCC)

Hexagonal close-packed (HCP)

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Face-Centered Cubic (FCC) Unit CellFace-Centered Cubic (FCC) Unit Cell

Reduced Sphere Representation Solid Sphere Representation

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Face-Centered Cubic (FCC) Lattice Face-Centered Cubic (FCC) Lattice StructureStructure

Examples:LeadCopperGoldSilverNickel

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If you know the atomic radius, you If you know the atomic radius, you know the size of an FCC unit cellknow the size of an FCC unit cell

a

4R

a2 + a2 = (4R)2

a = 81/2R

Example:Rgold = .144 nmagold = .407 nm

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Body-Centered Cubic (BCC) Unit CellBody-Centered Cubic (BCC) Unit Cell

Reduced Sphere Representation Solid Sphere Representation

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Body-Centered Cubic (BCC) Lattice Body-Centered Cubic (BCC) Lattice StructureStructure

The most familiar example of BCC is room temperature ironAlso tungsten and chromium

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The The coordination numbercoordination number is the is the number of other atoms touched by number of other atoms touched by

each atom in a latticeeach atom in a lattice

This atom touches …

1

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8 (atom opposite 5)

7(opposite 6)

6

5

2

Plus 4 more atoms in the next unit cell over

The coordination The coordination number for FCC atoms number for FCC atoms is 12is 12

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Atoms per unit cell in FCCAtoms per unit cell in FCC

6 x ½ = 38 x 1/8 = 1Total = 4

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Atomic Packing Factor (APF) measures Atomic Packing Factor (APF) measures the fraction of the unit cell volume the fraction of the unit cell volume actually occupied by atomsactually occupied by atoms

Notice all the empty space

Example for FCCAPF = Vatoms / Vunit cell

Vatoms = 4 x 4/3 R3

Vunit cell = a3 = 83/2 R3

Do the arithmetic:APF = 0.74

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Your turn: How many atoms are in Your turn: How many atoms are in a unit cell of BCC iron?a unit cell of BCC iron?

Also, how would Also, how would you go about you go about determining the determining the APF (this is a APF (this is a homework homework problem).problem).

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DensityDensity

The concept of atomic packing factor The concept of atomic packing factor allows us to relate atomic weight to allows us to relate atomic weight to macroscopically observed densitymacroscopically observed density

= nA

VcellNA

n = atoms/unit cellA = atomic weightVcell = volume of the unit cellNA = Avogadro’s number

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Hexagonal Close-packed (HCP) Unit CellHexagonal Close-packed (HCP) Unit Cell

Reduced Sphere Representation

HCP lattice structure