Crystal Lattices and Unit Cells I

Crystal System: There are

230-crystal forms

32-point groups

7- Crystal systems

14- Bravais Lattice

There are seven types of unit cell formed. These are Cubic, Tetragonal, Orthorhombic, Monoclinic, Hexagonal, Rhombohedral or Trigonal and Triclinic. The 7 crystal systems can be remembered as   C   T   O   M   T   R   H

Crystal System	Relation between a, b, c,	Relation between α, β, ɣ	No. of Planes and Axes
Cubic	a = b = c	α = β = ɣ = 90⁰	9 planes + 4 axes
Tetragonal	a = b ≠ c	α = β = ɣ = 90⁰	5 planes + 5 axes
Orthorhombic	a ≠ b ≠ c	α = β = ɣ = 90⁰	3 planes + 3 axes
Monoclinic	a ≠ b ≠ c	α = β = 90⁰ ≠ ɣ	1 plane + 1 axis
Triclinic	a ≠ b ≠ c	α ≠ β ≠ ɣ ≠ 90⁰	0
Rhombohedral	a = b = c	α = β = ɣ ≠ 90⁰	7 planes + 7 axes
Hexagonal	a = b ≠ c	α = β = 90⁰, ɣ = 120⁰	7 planes + 7 axes

Bravais lattices: The Bravais lattice are the distinct lattice types which when repeated can fill the whole space. In three dimensions there are 14 Bravais lattices

Primitive	Side Centered	Body Centered	Face Centered
Cubic
a ≠ c		a ≠ c
Tetragonal
a ≠ b ≠ c	a ≠ b ≠ c	a ≠ b ≠ c	a ≠ b ≠ c
Orthorhombic
α ≠ 90⁰, β, γ = 90⁰	α ≠ 90⁰, β, γ = 90⁰
Monoclinic
Hexagonal	α = β = γ ≠ 90⁰ Trigonal	α, β, γ ≠ 90⁰ Triclinic

Crystal	Description	Co. Number
Rock Salt (NaCl)	Cl⁻ ions are in CCP and Na⁺ ions occupy octahedral voids	Cl⁻ → 6 Na⁺ → 6
Zincbled (ZnS)	S²⁻ ions are in CCP Zn²⁺ ions occupied tetrahedral voids.	Zn²⁺  → 4 S²⁻ → 4
CsCl	Cl⁻ ions are in HCP Cs⁺ ions in the body of the cube.	Cl⁻ → 8 Cs⁺ → 8
Wurtzrite (ZnS)	S²⁻ ions are in HCP Zn²⁺ ions occupies tetrahedral void.	Zn²⁺ → 4 S²⁻ → 4
Florite (CaF₂) (AB₂)	Ca²⁺ ions are in CCP F⁻ ions occupies all the tetrahedral voids	Ca²⁺  → 8 F⁻ → 4
Anti Florit (A₂B)	A⁺ ions occupies all the tetrahedral voids, B²⁻ ions are in CCP	A⁺ → 4 B²⁻ → 8

Structure of CsCl:

1. Total no: of ions = 9 (1Cs⁻ + 8 Cl⁻ [Or] 1 Cl⁻ + 8 Cs⁺)

2. No: of Cs⁺ per unit cell = 0/8 + 0/4 + 0/2 + 1/1 = 1

3. No: of Cl⁻ per unit cell = 8/8 + 0/4 + 0/2 + 0/1 = 1

4. No: of CsCl formula units per unit cell = 1.

5. Co-ordination number

 Each ion is surrounded by eight ions of its counterpart.

 So, coordination number for

Cs⁺ → 8

Cl⁻ → 8

6. Body diagonal = √3a = 2 (r _Cs⁺ + r _Cl⁻)

Structure of NaCl:

a. Total no: of ions = 27 (13 Na⁺ + 14 Cl⁻ [Or] 14 Na⁺ + 13 Cl⁻)

b. No: of Na⁺ ions per unit cell = 8/8 + 12/4 = 4 {1 is at body centered 12 are at edges}

c. No: of Cl⁻ ions per unit cell = 6/2 + 8/8 = 3 {8 at corners and 6 at face center}

d. No: of formula units per unit cell = 4

e. Co-ordination no: of Na⁺ = 6

f. Cl⁻ = 6

g. Edge length = a = 2 (r_Na⁺ + r_Cl⁻)