Prerequisites

Crystal structure

The lattice is an ideal periodic reference; a real crystal also has a basis and defects. Keep those roles separate when interpreting a diffraction or transport problem.

A crystal is described by a lattice and a basis. Primitive translation vectors generate equivalent points:

\mathbf R=n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3.

The reciprocal lattice is useful for diffraction and band theory. Constructive diffraction follows Bragg's law:

2d\sin\theta=m\lambda.

Real crystals contain vacancies, interstitials, substitutional impurities, dislocations, and grain boundaries. Defects can dominate diffusion, strength, conductivity, and optical response.

Phonons and thermal properties

Lattice vibrations are quantized as phonons with energy

E=\hbar\omega\left(n+\frac12\right).

At long wavelength, acoustic phonons behave like sound waves. Optical phonons occur when a unit cell has multiple atoms moving relative to one another. Heat capacity, thermal conductivity, and expansion depend on the phonon spectrum and scattering from boundaries, defects, and other phonons.

Electrons in solids

Periodic potentials split atomic levels into bands. The Fermi–Dirac distribution is

f(E)=\frac{1}{e^{(E-\mu)/(k_BT)}+1}.

Materials are commonly classified by the relationship between the Fermi level and available states:

Material	Electronic picture	Typical behavior
Metal	Partially filled band or overlapping bands	High electrical conductivity
Semiconductor	Small band gap	Conductivity increases with temperature or doping
Insulator	Large band gap	Very low ordinary conductivity

For a simple carrier model,

\mathbf J=\sigma\mathbf E, \qquad \sigma=nq\mu.

Doping creates donors or acceptors and changes carrier concentration. In a magnetic field, moving charges experience the Lorentz force and can produce a Hall voltage.

Magnetism and phase transitions

Diamagnetism opposes an applied field; paramagnetism comes from unpaired moments; ferromagnetism supports spontaneous alignment. Antiferromagnets and ferrimagnets have ordered but opposing sublattices.

Superconductors exhibit zero DC resistance and magnetic flux expulsion below a critical temperature. Their behavior is limited by critical field, current, and temperature.

Near a continuous phase transition, fluctuations become important and macroscopic properties often follow power laws. Order parameters distinguish phases, while symmetry breaking explains why phases have different properties.

Problem-solving workflow

Identify the structure, carriers, or collective mode involved.
Decide whether a microscopic, band, lattice, or continuum model is appropriate.
State approximations such as low temperature, weak disorder, or parabolic bands.
Check dimensions and compare the result with the relevant energy or length scale.

Summary

Condensed matter physics is organized around structure, excitations, and collective behavior. Diffraction probes structure, transport probes motion, and thermodynamic or magnetic measurements reveal phases and transitions.

Sources

OpenStax University Physics
Physics LibreTexts
Kittel, Introduction to Solid State Physics
Ashcroft and Mermin, Solid State Physics