Crystal lattices

CRYSTAL LATTICES

• Lattice: Regular 3-D arrangements of equivalent LATTICE POINTS in space.
• UNIT CELLS

• Smallest repeating internal unit that has the symmetry characteristic of the solid. 

1. Actual unit cell
2. Unit cell as drawn
3. Whole crystal

CRYSTAL LATTICE GEOMETRY

• Based on shape of unit cell ignoring actual atomic locations
• Unit cell = 3-dimensional unit that repeats in space
• Unit cell geometry completely specified by a, b, c (distance in x, y & z axis)& a, b, g (angles between a,b & c (lattice parameters or lattice constants)
• Seven possible combinations of a, b, c & a, b, g, resulting in seven basic crystal systems

14 Bravais Lattices: seven basic lattices and fourteen overall

14 Bravais Lattices from seven basic lattices

Cubic Unit Cells

There are 7 basic crystal systems, but we are only concerned with CUBIC.

DIFFERENT CUBIC UNIT CELLS

ATOM SHARING AT CUBE FACES AND CORNERS

       Atom shared in corner

à 1/8  inside each unit cell

Atom shared in face

à ½  inside each unit cell

Atoms shared on edges

à ¼ inside each unit cell

Atoms inside the unit cell

à 1 inside each unit cell

NUMBER OF ATOMS PER UNIT CELL

PACKING FRACTION

 

 

 

a = edge length of unit cell

R = radius of atoms( assuming they are hard spheres)

Z= no of atoms in one unit cell

PACKING FRACTION: SC

 

PACKING RACTION: BCC

PACKING FRACTION: FCC

Voids:- gaps between atoms in a crystal are called voids.

• There are two types of voids
  • Tetrahedral voids and octahedral voids

HOW TO LOCATE VOIDS

• No. of tetrahedral voids = 2 x effective atoms/unit cell

• No. of octahedral voids = 1 x effective atoms/unit cell

 COMPARISON OF CRYSTAL STRUCTURES