Experiment 01 · Reference Charts
Crystallography
A. The seven crystal systems
| No. | System | Lattice constants | Interaxial angles | Bravais lattices | Examples |
|---|---|---|---|---|---|
| 1 | Cubic | a = b = c | α = β = γ = 90° | P I F | Au, Cu, NaCl, CaF₂ |
| 2 | Tetragonal | a = b ≠ c | α = β = γ = 90° | P I | SnO₂, TiO₂, NiSO₄ |
| 3 | Orthorhombic | a ≠ b ≠ c | α = β = γ = 90° | P I F C | KNO₃, BaSO₄, MgSO₄ |
| 4 | Rhombohedral (Trigonal) | a = b = c | α = β = γ ≠ 90° < 120° | P | As, Sb, Bi, Calcite |
| 5 | Hexagonal | a = b ≠ c | α = β = 90°, γ = 120° | P | SiO₂, Zn, Mg, Cd, AgI |
| 6 | Monoclinic | a ≠ b ≠ c | α = γ = 90° ≠ β | P C | CaSO₄·2H₂O, FeSO₄, Na₂SO₄ |
| 7 | Triclinic | a ≠ b ≠ c | α ≠ β ≠ γ | P | K₂Cr₂O₇, CuSO₄·5H₂O |
| Total Bravais lattices | 14 | ||||
P primitive · I body-centred · F face-centred · C base-centred
B. Characteristics of Cubic Bravais Lattices
The unit cell characteristics for the cubic cell, with a the lattice constant, M the molecular weight and NA Avogadro's number.
| No. | Characteristic / parameter | SC | BCC | FCC |
|---|---|---|---|---|
| 1 | Unit cell volume (V) For a cubic cell the volume of the unit cell is the volume of a cube of side a. | V = a³ | V = a³ | V = a³ |
| 2 | Effective number of atoms per unit cell (Z) A corner atom is shared by 8 cells, a face atom by 2 cells, and a body-centred atom by 1 cell. | Z = 1 | Z = 2 | Z = 4 |
| 3 | Atomic radius (r) Half the distance between the centres of two nearest neighbouring atoms that are directly in contact. | r = a/2 | r = (√3/4) a | r = (√2/4) a |
| 4 | Nearest-neighbour distance (2r) The distance between the centres of two nearest neighbouring atoms that are directly in contact. | 2r = a | 2r = (√3/2) a | 2r = (√2/2) a |
| 5 | Coordination number (C.N.) The number of nearest neighbouring atoms simultaneously in contact with one atom. | C.N. = 6 | C.N. = 8 | C.N. = 12 |
| 6 | Atomic packing fraction (A.P.F.) The fraction of the cell filled by atoms: A.P.F. = Z · v / V. | A.P.F. = 0.52 = 52 % | A.P.F. = 0.68 = 68 % | A.P.F. = 0.74 = 74 % |
| 7 | Void space The fraction of the cell left empty: void space = 1 − A.P.F. | Void space = 0.48 = 48 % | Void space = 0.32 = 32 % | Void space = 0.26 = 26 % |
| 8 | Density (ρ) Mass over volume, with the mass of one atom taken as M/Nᴀ: ρ = Z · M / (V · Nᴀ). | ρ = M / (a³ Nᴀ) | ρ = 2M / (a³ Nᴀ) | ρ = 4M / (a³ Nᴀ) |
C. Symmetry Elements in a cubic crystal
Twenty-three elements in three families. Selecting any of them in the simulation draws it on the cube.
| Centre of Symmetry | Intersection point of body diagonals | ≡ 1 |
|---|---|---|
| Axes of Symmetry | 4-fold axes of symmetry (tetrad axis) | ≡ 3 |
| Axes of Symmetry | 3-fold axes of symmetry (triad axis) | ≡ 4 |
| Axes of Symmetry | 2-fold axes of symmetry (diad axis) | ≡ 6 |
| Planes of Symmetry | Planes of symmetry parallel to the faces | ≡ 3 |
| Planes of Symmetry | Planes of symmetry passing through the edges | ≡ 6 |
| Total number of symmetry elements | ≡ 23 | |