### Condensed Matter I

Types of cohesive force

1. Ionic bonding: transfer of electrons from one atom to another, leading to charged atoms with filled shells, e.g. Na+Cl.
2. Covalent bonding: unpaired valence electrons mix (hybridise) to give a lower energy state. Covalent bonding is very directional.

s, px, py, pz -> sp3 hybridisation, tetrahedral diamond structure

3. Metallic bonding: outer atomic orbitals overlap several other atoms, so the electrons become mobile and delocalised (reducing KE). Metallic bonding is not directional, and is weaker than covalent bonding.
4. Hydrogen bonding: covalent bonds between different atomic species lead to polar molecules with electrostatic attraction between δ+ and δ of different molecules. Hydrogen bonding is much weaker than ionic or covalent bonding.

5. Van der Waals forces: the random motion of electrons in an atom leads to a temporarily induced dipole. This inturn induces a dipole on neighbouring atoms.

E-field at p2: $E\sim\frac{p_1}{r^3}$ induces $p_2 \propto E$.

Energy of interaction $=p_2E=\frac{p_1p_2}{r^3}\propto \frac{1}{r^6}$

Lattice: an infinite array of points, each with an identical environment. Each point has position r=n1a1+n2a2+n3a3 where the ai are lattice vectors.

Basis: the structural unit (group of atoms) associated with one lattice point.

Crystal structure: the convolution of a lattice with a basis, e.g.

lattice x basis = crystal structure (each dot is a lattice point)

Unit cell: a polyhedral region of space, such that when many identical unit cells are tessellated, they form a crystal structure (i.e. the ‘building block’ of said structure). The unit cell in 3D can be specified by three vectors, a, b and c. Positions within the unit cell can be specified by fractional coordinates (i.e fractios of a, b and c).

1. Conventional unit cell: can contain more than one lattice point (is usually an easy to work with shape like a square or cube that makes the maths easier).
2. Primitive unit cell: contains only one lattice point.
3. Wigner-Seitz unit cell: contains one lattice point, and displays the same point symmetry as the lattice.