PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Condensed matter physics ↓


Crystal structure and lattice


Crystal structure and lattice theory are fundamental components in condensed matter physics, which is the study of the physical properties of solids and liquids. These concepts help us understand a vast range of substances and phenomena. Crystal structure describes the orderly arrangement of atoms in a crystalline substance. When atoms are arranged in a repetitive pattern, we observe the formation of a lattice, which serves as a mathematical description of how these particles are arranged in space.

Introduction to crystal structures

The atoms in solids are arranged in a periodic pattern called a crystal lattice, which makes them distinct from amorphous substances where the structure lacks long-range order. This periodicity gives rise to the distinctive physical properties of crystalline materials.

Each point in a crystal lattice pattern represents one or a group of atoms. The distance between these points and their orientation in space define the crystal structure. To better understand these concepts, physicists classify crystal structures based on the symmetry of these lattice points.

For example, consider the simplest 2D example of a crystal lattice shown below, where each circle can represent an atom:

    O---O---O---O
    ,
    O---O---O---O
    ,
    O---O---O---O

Lattice and unit cells

A lattice is an arrangement of points (or particles) in an infinite periodic pattern. Each point is identical and has the same environment as all other points.

The smallest repeating unit in a crystal lattice that represents the complete structure of the lattice is known as the unit cell. The unit cell is fundamental because repeating it through space generates the lattice.

Types of unit cell

Unit cells can be classified based on their geometric shape and atomic arrangement:

  • Simple cube (SC)
  • Body-centered cubic (BCC)
  • Face-centered cubic (FCC)
  • Hexagonal close-packed (HCP)

The simple cubic (SC) structure is the simplest lattice system, where there is an atom at each corner of the cube, and the cube is repeated throughout the crystal.

    Simple Cube:

      Oh Oh
     ,
    O---O |
    | O-|-
    |/O
    Oh Oh

The body-centered cubic (BCC) structure includes an additional atom at the center of the cube, which makes the overall structure stronger by providing more bonding possibilities.

    Body-centered cubic:

      Oh Oh
     ,
    O---O |
    | O-|- O
    |/O
    Oh Oh

Additionally, in a face-centered cubic (FCC) lattice the atoms are centered on each face of the cube, which provides greater packing density.

    Face-centered cubic:

      Oh Oh
     /|O/|
    OOO |
    | O-|-
    |/O
    Oh Oh

Finally, the hexagonal close-packed (HCP) structure is characterized by a hexagonal arrangement that provides optimal packing efficiency.

Lattice parameters and symmetry

Lattice parameters are key measurements that describe the size and shape of a unit cell. These include:

  • Lengths of edges (a, b, c)
  • Angles between edges (α, β, γ)
Example formulas for simple cubic lattice volume: 
    V = a^3

Symmetry plays an important role in determining the properties of crystals. Symmetry operations include translational symmetry (shift the lattice by a vector and see the same arrangement) and rotational symmetry (rotation around an axis without changing the appearance of the system).

Bravais lattices

In three-dimensional space, there are 14 different Bravais lattices that describe all possible lattice structures. These lattices account for all possible symmetric arrangements of points in three-dimensional space.

X-ray diffraction and crystal structures

One of the essential techniques for studying crystal structures is X-ray diffraction. When X-rays are passed through a crystal, they scatter and form a pattern. Analyzing these patterns helps scientists determine the atomic arrangement of the crystal.

In this context the Bragg's law equation is essential: 
    nλ = 2d sin θ

Where:

  • n is the order of reflection.
  • λ is the wavelength of the incident X-rays.
  • d is the distance between the crystal planes.
  • θ is the angle of incidence.

Real-world applications

Understanding crystal structures is very important in materials science, including the development of new materials with specific properties. For example, knowledge of different crystal structures guides the creation of metals that resist corrosion or the fabrication of semiconductors used in electronic devices.

Summary

The study of crystal structures and lattices provides information about the fundamental nature of materials. By understanding how atoms in solids arrange and interact, scientists can optimize the properties of matter for a variety of innovations and practical applications. This knowledge connects theoretical physics and real-world technological advancements.


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Crystal structure and lattice

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