Lecture 2: Crystal Structure

PHYS 430/603 material

Laszlo Takacs

UMBC Department of Physics

Unless we specify otherwise, “solid” means “crystalline,” at least on the microscopic scale

• Short range structure reflects the nature of bonds, but the crystal structure also has to conform to translational symmetry:

• If we shift the crystal by certain vectors of translation, T, every atom moves into the position of an identical atom.

• The possible vectors of translation are linear combinations with integer coefficients of three “primitive translational vectors”:

T = na + mb + pc• The entire structure can be described by a “unit cell” defined as

a parallelepiped defined by a, b, c and its repeated translations by a, b, c. There can be symmetries beyond translation.

• A smallest possible unit cell is the “primitive cell.”• The points in a lattice are mathematical points, we get the crystal

structure by putting identical groups of atoms - the basis - on each lattice point. In simple cases, the basis is a single atom.

The elementary vectors of translation, i.e. the unit vectors of our coordinate system

Find the unit cell

Maurits Cornelis Escher

Unit cell and symmetries

Crystal - glass

The fourteen Bravais lattices

Unit cells of the fcc structure

Interstitial sites in fcc structure

The hcp structure and its unit cell

Interstitial sites in hcp structure

Interstitial sites in bcc structure

CsCl (B2) structure

Packing based on hexagonal structure: AlB2 and WC