Structures and Energetics of Metallic and Ionic Solids - Part I

Amorphous solids
- Atoms or molecules have order over only short distances (a few atomic or molecular spacings).
- Do not exhibit a crystal structure.
- Characterized by an irregular bonding pattern.
- Examples: Rubber, plastic, glass.
  - Rubber (cis-polyisoprene): May be soft and rubbery when formed by long, tangled molecules, held by intermolecular forces.
  - Vulcanization: Process involving sulfur linkages ( $S_x$ ) to form a more rigid rubber.
  - Glassy solids: Hard and brittle, formed by atoms irregularly joined by covalent bonds.
Semi-crystalline/-amorphous solids
- Crystallinity is usually specified as a percentage of the volume of the material that is crystalline.
- Examples: Ceramics and some polymers.
Crystalline solids
- Atoms or molecules are arranged in a regular, periodic manner (long-range order).

Formed by electrostatic attraction between anions and cations, creating a crystal lattice.
Each ion is surrounded by ions of the opposite charge.
Come in three basic forms based on interstitial holes:
- Trigonal holes
- Tetrahedral holes
- Octahedral holes
Smaller ions typically fill these holes, while larger ions form the primary structure.
Extremely stable: Significant energy is required to break the strong ionic bonds, leading to high melting points.
Example: $NaCl$ (Halite structure).

Formed by positively charged nuclei of metal atoms held together by valence electrons (metal bond).
Electrons are "delocalized", meaning they are not bound to specific atoms.
These delocalized electrons can move throughout the solid, described by the "electron sea model" (positive nuclei float in a sea of negative electrons).
Characteristics:
- High thermal and electrical conductivity.
- Typically hard, shiny, and ductile.
Example: $Cu$ (Face-centered cubic (fcc) structure).

Consist of atoms held together by strong covalent bonds.
Characteristics:
- Incredibly hard.
- High melting points.
- Poor conductors of heat and electricity.
Examples:
- $SiO_2$
- Diamond ( $C$ )
- Many gemstones.

Weakest intermolecular force.
A temporary attractive force resulting from temporary dipoles formed when electrons in two adjacent atoms briefly occupy positions that make the atoms polar.
This temporary dipole in one atom can induce a dipole in a nearby atom (induced dipole-induced dipole attraction).
These attractive forces cause nonpolar substances to condense into liquids and freeze into solids when the temperature is lowered sufficiently.

Comprised of discrete molecules held together by intermolecular forces, which are considerably weaker than intramolecular (covalent) forces.
Intermolecular forces involved:
- London Dispersion Forces
- Dipole-dipole interactions
- Hydrogen bonds
Characteristics:
- Fairly soft.
- Poor electrical and thermal conductors.
- Low to moderate melting points.
Examples:
- $I_2$
- $S_8$

Regular arrangement of spheres where every sphere is in contact with six other spheres, forming a hexagonal arrangement in a layer.
Building layers:
- The second layer is built by placing spheres in the hollows of the first layer.
- In the second layer, two types of hollows are formed:
  - Hollows that lie directly over hollows in layer $1$ .
  - Hollows that lie directly over spheres in layer $1$ .

Two different close-packed arrangements are possible:
- Hexagonal close-packing (hcp): Characterized by two repeating layers (e.g., $ABABAB…$ sequence).
- Cubic close-packing (ccp): Characterized by three repeating layers (e.g., $ABCABCABC…$ sequence).
Coordination number for both hcp and ccp is $12$ . ( $6$ in the same layer, $3$ above, $3$ below).

A fundamental concept in solid-state chemistry.
The smallest repeating unit of a crystal structure.
When unit cells are stacked in 3D space, they describe the bulk arrangement of atoms in the crystal.
Represented by its lattice parameters:
- Lengths of the cell edges: $a, b, c$
- Angles between them: $\alpha, \beta, \gamma$
Types of Cubic Unit Cells:
- Simple cubic (P)
- Body-centered cubic (I or bcc)
- Face-centered cubic (F or fcc)
Relationship between ccp and fcc: The ccp arrangement (ABCABC…) is clearly reflected by the face-centered cubic (fcc) unit cell. The hcp arrangement (ABAB…) is easily recognized in its own unit cell.

Common arrangements for metals include:
- Face-centered cubic (fcc), which is also cubic close-packing (ccp).
- Body-centered cubic (bcc).

These are spaces between the spheres in a packed structure.
Tertrahedral holes:
- Surrounded by $4$ spheres.
- Spheres lie at the corners of a tetrahedron.
Octahedral holes:
- Surrounded by $6$ spheres.
- Spheres lie at the corners of an octahedron.
Numerical relationship: In a close-packed structure, there is one octahedral hole per sphere and twice as many tetrahedral holes as octahedral holes.
Size comparison: Octahedral holes are larger than tetrahedral holes.

Hexagonal close-packing (hcp): $74\%$ space filling.
Cubic close-packing (ccp) / Face-centered cubic (fcc): $74\%$ space filling.
Body-centered cubic (bcc): $68\%$ space filling.

These arrangements have lower space-filling efficiencies and coordination numbers.
Simple cubic lattice:
- Coordination number = $6$
Body-centered cubic (bcc) lattice:
- Coordination number = $8$