2. Crystal Lattice

Table of Contents

1. Solids
2. Crystals
   1. Lattice + Basis
   2. Unit Cell: the Building Block
   3. System Types
   4. 7 Crystal Systems
3. Bravais Lattices

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Solids

Definition

Crystal: Atoms having long range order in a periodic arrangement. (Example: Crystalline $Si{O}_2$ (quartz))

Definition

Amorphous Materials: Atoms having no long range order, but have short and medium range orders. (Example: Amorphous $Si{O}_2$ (glass))

Crystals

• Definition: Atoms or molecules are situated in a repeating or periodic array over large atomic distances, i.e., long-range order.
• Two major characteristics: Melting Point ($T_m$) & Anisotropy (meaning the properties of material is depended on direction)

Lattice + Basis

• A lattice ⇒ A collection of points, called lattice points, which are arranged in a periodic pattern so that the surroundings of each point in the lattice are identical.
• Basis/Motif ⇒ A group of one or more atoms, located in a particular way with respect to each other.

Unit Cell: the Building Block

• Unit Cell ⇒ The subdivision of a lattice that still retains the overall characteristics of the entire lattice, and it is the smallest identical unit in the lattice.
  • Usually parallelepipeds
  • Represent the symmetry of the crystal structure
  • Edge lengths $a$, $b$, $c$, and the interaxial angles, $\alpha$, $\beta$, $\gamma$, are called lattice parameters.
  • Combinations of the lattice parameters give rise to the 7 crystal systems.

System Types

1. Cubic, $a=b=c$, $\alpha =\beta =\gamma =90°$
2. Tetragonal, $a=b\ne c$, $\alpha =\beta =\gamma =90°$
3. Orthorhombic, $a\ne b\ne c$, $\alpha =\beta =\gamma =90°$
4. Rhombohedral, $a=b=c$, $\alpha =\beta =\gamma \ne 90°$
5. Hexagonal, $a=b\ne c$, $\alpha =\beta =90°$, $\gamma =120°$
6. Monoclinic, $a\ne b\ne c$, $\alpha =\gamma =90°\ne \beta$
7. Triclinic, $a\ne b\ne c$, $\alpha \ne \beta \ne \gamma \ne 90°$

7 Crystal Systems

1. Cubic (ex. Fluorite)
2. Tetragonal (ex. Rutile)
3. Orthorhombic (ex. Barite)
4. Triclinic (ex. Rhodonite)
5. Hexagonal (ex. Corundum)
6. Rhombohedral (Trigonal) (ex. Cerussite)
7. Monoclinic (ex. Borax)

Bravais Lattices

14 possible basic crystal structure unit cell types

• Represents the symmetry of the lattice
• Minimum volume
• Maximum number of equal edge lengths and interaxial angles
• If right angles are present, the unit cell should have the maximum number of right angles