Solids are broadly classified into crystals, in which atoms, ions, or molecules are arranged regularly, and disordered amorphous solids. Understanding crystal structures is fundamental to materials science.
Basic Knowledge
Crystals are classified into four types according to their constituent particles and types of bonding.
① Ionic crystals: NaCl-type, CsCl-type, ZnS-type — hard, high-melting-point, and water-soluble
② Molecular crystals: I2, dry ice (CO2), naphthalene — soft, low-melting-point, and sublimable
③ Covalent network crystals: diamond, graphite, SiO2 — extremely hard and extremely high-melting-point
④ Metallic crystals: Fe, Cu, Al — electrical and thermal conductivity, malleability, and ductility
Representative structures of metallic crystals are the body-centered cubic lattice (bcc) (coordination number 8, packing fraction 68%), the face-centered cubic lattice (fcc) (coordination number 12, packing fraction 74%), and the hexagonal close-packed structure (hcp) (coordination number 12, packing fraction 74%).
Unit cell (smallest repeating unit composing a crystal)
Coordination number (number of particles nearest to one particle)
Packing fraction (ratio of atomic volume to unit-cell volume)
Body-centered cubic (bcc) (Fe(α), Na, K — coordination number 8, packing fraction 68%)
Face-centered cubic (fcc) (Cu, Al, Ag, Au — coordination number 12, packing fraction 74%)
Hexagonal close-packed (hcp) (Mg, Zn, Co — coordination number 12, packing fraction 74%)
Density calculation（ρ = (n·M)/(NA·V) ）
Further Study (Principles and Applications)
In the NaCl-type structure, all Na+ ions occupy the octahedral holes in the fcc arrangement of Cl- ions, so both the cation and anion have coordination number 6. In the CsCl-type structure, one ion occupies the center of a simple cubic arrangement, giving a coordination number of 8. In the ZnS-type structure, cations occupy tetrahedral holes, giving a coordination number of 4.
Diamond has a tetrahedral network of sp3 covalent bonds, while graphite consists of planar sp2 sheets stacked by weak π-stacking. They are allotropes. The semiconductor Si has a diamond-type structure, and its electrical properties are controlled by doping with impurities.
- Four crystal types = ionic, molecular, covalent network, and metallic
- bcc: 2 atoms per unit cell, diagonal 4r=√3a
- fcc: 4 atoms per unit cell, face diagonal 4r=√2a
- hcp and fcc both have a packing fraction of 74% (close-packed)
- NaCl-type structures have coordination number 6 for both ions
- Diamond and graphite are carbon allotropes
- Density ρ = nM/(NAV) can be used to determine crystal structure
Important Notes (Common Confusions and Frequent Mistakes)
① Number of atoms in a unit cell: body-centered = 2, face-centered = 4 (corners 1/8×8 + faces 1/2×6). ② Coordination number and packing fraction are different concepts. ③ Diamond is an insulator, whereas graphite conducts electricity because its π electrons can move. ④ Simple cubic structures have a packing fraction of 52% and are rare among real metals (only Po).
Practice
- State the number of atoms contained in a face-centered cubic unit cell.
- Find the density of copper (Cu, fcc, atomic radius 1.28×10-8 cm, atomic mass 63.5).
- Explain from their structures why diamond and graphite have different electrical conductivity.