Crystal Lattice: MDCAT Chemistry notes
Crystal Lattice MDCAT notes: lattice points, unit cell, cell parameters, seven crystal systems, cubic cells, coordination number and NaCl structure.
Crystal lattice and unit cell
- Crystal lattice: a regular three-dimensional array of points showing how the atoms, ions or molecules are arranged in a crystal. Its shape depends on the arrangement of the particles.
- Lattice points (also called lattice sites): the positions occupied by the particles.
- Unit cell: the smallest repeating block of the lattice that carries all the information about the crystal structure. Repeating it in three dimensions builds the whole crystal.
Unit cell parameters
A unit cell is described by three edge lengths $a$, $b$, $c$ and three angles:
- $\alpha$ is the angle between edges $b$ and $c$
- $\beta$ is the angle between edges $c$ and $a$
- $\gamma$ is the angle between edges $a$ and $b$
The seven crystal systems
| System | Edges | Angles | Example |
|---|---|---|---|
| Cubic | $a=b=c$ | $\alpha=\beta=\gamma=90^\circ$ | NaCl, diamond |
| Tetragonal | $a=b\neq c$ | all $90^\circ$ | $\mathrm{SnO_2}$ |
| Orthorhombic (rhombic) | $a\neq b\neq c$ | all $90^\circ$ | Rhombic sulphur |
| Monoclinic | $a\neq b\neq c$ | $\alpha=\gamma=90^\circ$, $\beta\neq 90^\circ$ | Monoclinic sulphur |
| Hexagonal | $a=b\neq c$ | $\alpha=\beta=90^\circ$, $\gamma=120^\circ$ | Graphite, ice |
| Rhombohedral (trigonal) | $a=b=c$ | all equal, not $90^\circ$ | Calcite, $\mathrm{NaNO_3}$ |
| Triclinic | $a\neq b\neq c$ | all different, none $90^\circ$ | $\mathrm{CuSO_4\cdot 5H_2O}$ |
Both NaCl and diamond have cubic unit cells. Graphite layers are made of hexagonal rings.
Cubic unit cells and sharing of particles
| Position | Shared by | Contribution to one cell |
|---|---|---|
| Corner | 8 unit cells | $1/8$ |
| Edge | 4 unit cells | $1/4$ |
| Face centre | 2 unit cells | $1/2$ |
| Body centre | 1 unit cell | $1$ |
- Simple cubic: $8\times\frac18 = 1$ particle per cell, coordination number 6.
- Body-centred cubic (bcc): $1+1 = 2$ particles, coordination number 8. Examples: Na, K, Fe.
- Face-centred cubic (fcc): $1+3 = 4$ particles, coordination number 12. Examples: Cu, Al, Ag.
The coordination number is the number of nearest neighbours around a particle.
Structure of sodium chloride
- Face-centred cubic arrangement of $\mathrm{Cl^-}$ ions with $\mathrm{Na^+}$ ions in the gaps (two interpenetrating fcc lattices).
- Each $\mathrm{Na^+}$ is surrounded by six $\mathrm{Cl^-}$ ions and each $\mathrm{Cl^-}$ by six $\mathrm{Na^+}$ ions, arranged octahedrally. Coordination number 6:6.
- Each unit cell contains 4 $\mathrm{Na^+}$ and 4 $\mathrm{Cl^-}$, i.e. 4 formula units.
- A $\mathrm{Cl^-}$ at a face centre is shared between 2 unit cells.
Compare CsCl, where the coordination number is 8:8 because $\mathrm{Cs^+}$ is large.
Determining crystal structure
The positions of atoms in a crystal are found by X-ray diffraction, because X-ray wavelengths are comparable to the spacing between lattice planes.
Common MDCAT traps
- The unit cell, not the "crystal lattice", is the smallest repeating block.
- $\beta$ lies between $a$ and $c$, not between $a$ and $b$.
- Sodium metal is bcc (coordination number 8); the ion $\mathrm{Na^+}$ in NaCl has coordination number 6.
- Face-centre particles are shared by 2 cells, corners by 8.
Quick revision
- Lattice points are also called lattice sites.
- There are seven crystal systems; cubic is the most symmetrical.
- NaCl: fcc, 6:6 coordination, 4 formula units per cell.
- bcc has coordination number 8; fcc has 12.
- X-ray diffraction reveals crystal structures.