Crystal Structure of Solids

What is “Crystal” to the man on the street?

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.

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.

Under what categories could this class be grouped?

• 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

•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

Polycrystalline Solids

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

• 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

E (diagonal) = 273 GPa

E (edge) = 125 GPa

Single- Vs Poly- Crystal

-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

a

b

c

Lattice Parameters

Atoms in a Crystal

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.

• 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.

Unit cell

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.

1. Cubic (Isometric) System

Symmetry elements: Four 3-fold rotation axes along cube diagonalsa = b = c = = = 90o

3 Bravais lattices

ab

c

• 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)

Coordination Number = Number of nearest neighbors

One atom per unit cell

1/8 x 8 = 1

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

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

FCCCoordination number = 12

3 mutually perpendicular planes.4 nearest neighbors on each of the three planes.

• 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

Which one has most packing ?

Which one has most packing ?

For that reason, FCC is also referred to as cubic closed packed (CCP)

Symmetry element: One 6-fold rotation axisa = b ca= 120o

= = 90o

2. Hexagonal System

Only one Bravais lattice

• 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)

Symmetry element: One 4-fold rotation axisa = b ca= = = 90o

3. Tetragonal System

Two Bravais lattices

Symmetry element: One 3-fold rotation axisa = b ca= 120o

= = 90o

4. Trigonal (Rhombohedral) System

One Bravais lattice

5. Orthorhombic System

Symmetry element: Three mutually perpendicular 2-fold rotation axesa b ca = = = 90o

Four Bravais lattices

6. Monoclinic System

Symmetry element: One 2-fold rotation axisa b ca = = 90o, 90o

Two Bravais lattices

7. Triclinic System

Symmetry element: Nonea b ca 90o

One Bravais lattice