Crystal Structure of Solids What is “Crystal” to the man on the street?

Fundamental Properties of Matter Matter: - Has mass, occupies space Mass – measure of inertia - from Newton’s first law of motion. It is one of the fundamental physical properties. 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.

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?

Amorphous Solids 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.

Polycrystalline 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

Crystalline Solids Atoms arranged in a 3-D long range order. “Single crystals” emphasizes one type of crystal order that exists as opposed to polycrystals.

Single- Vs Poly- Crystal • 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

Single- Vs Poly- Crystal 200 mm -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic.

Lattice Parameters a b c   

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 3 Bravais lattices Symmetry elements: Four 3-fold rotation axes along cube diagonals a = b = c  =  =  = 90o a b c   

(1-a): Simple Cubic Structure (SC) • Rare due to poor packing (only Po has this structure) • Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) 1 atom/unit cell

Coordination Number = Number of nearest neighbors

One atom per unit cell 1/8 x 8 = 1

Atomic Packing Factor • APF for a simple cubic structure = 0.52 a Adapted from Fig. 3.19, Callister 6e.

(1-b): Face Centered Cubic Structure (FCC) • Exhibited by Al, Cu, Au, Ag, Ni, Pt Close packed directions are face diagonals. Coordination number = 12 4 atoms/unit cell All atoms are identical Adapted from Fig. 3.1(a), Callister 6e. 6 x (1/2 face) + 8 x 1/8 (corner) = 4 atoms/unit cell

3 mutually perpendicular planes. FCC Coordination number = 12 3 mutually perpendicular planes. 4 nearest neighbors on each of the three planes.

(1-c): Body Centered Cubic Structure (BCC) • Exhibited by Cr, Fe, Mo, Ta, W Close packed directions are cube diagonals. Coordination number = 8 All atoms are identical 2 atoms/unit cell

Which one has most packing ?

Which one has most packing ? For that reason, FCC is also referred to as cubic closed packed (CCP)

Only one Bravais lattice 2. Hexagonal System Only one Bravais lattice Symmetry element: One 6-fold rotation axis a = b  c = 120o  =  = 90o

Hexagonal Closed Packed Structure (HCP) 2D Projection Exhibited by …. ABAB... Stacking Sequence Coordination # = 12 APF = 0.74 3D Projection Adapted from Fig. 3.3, Callister 6e.

3. Tetragonal System Symmetry element: One 4-fold rotation axis Two Bravais lattices Symmetry element: One 4-fold rotation axis a = b  c =  =  = 90o

4. Trigonal (Rhombohedral) System One Bravais lattice Symmetry element: One 3-fold rotation axis a = b  c = 120o  =  = 90o

5. Orthorhombic System Four Bravais lattices Symmetry element: Three mutually perpendicular 2-fold rotation axes a  b  c =  =  = 90o

6. Monoclinic System Symmetry element: One 2-fold rotation axis Two Bravais lattices Symmetry element: One 2-fold rotation axis a  b  c =  = 90o,   90o

7. Triclinic System Symmetry element: None a  b  c      90o One Bravais lattice Symmetry element: None a  b  c      90o