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 ↓


Bravais lattices


The concept of a Bravais lattice is foundational in the study of crystal structures and condensed matter physics. Named after French physicist Auguste Bravais, who first described them in 1848, these lattices provide a systematic way to classify and understand the arrangement of atoms in crystalline solids. This arrangement plays a key role in determining the properties of materials, affecting everything from electrical conductivity to mechanical strength.

Understanding crystal structures begins with understanding the three-dimensional space lattice, which is a repetitive arrangement of points in space. These points represent equivalent positions in the crystal structure, and they provide a framework on which the actual motifs (groups of atoms) are placed. The concept of the Bravais lattice specifically refers to the 14 unique three-dimensional lattice types that can be created by translating a point through space, while maintaining symmetry and periodicity.

What is lattice?

A lattice in the context of crystal structures is a regular, repeating arrangement of points in space. Imagine a three-dimensional grid, such as a collection of orange crates stacked in a warehouse. Each corner of an orange crate corresponds to a lattice point, and the entire collection can extend infinitely in all directions. These points are connected by vectors that define the size and shape of the repeating unit.

The parameters that define the lattice include the lattice constants, which are the lengths of the edges of the unit cell and the angles between these edges. These constants help describe the periodicity and orientation of the crystal.

Understanding Bravais lattices

The 14 Bravais lattices are classified based on their symmetry properties and the lengths and angles of the unit cells. They can be grouped into seven crystal systems:

  • Cube
  • Square
  • Orthorhombic
  • Hexagonal
  • Mainly rava
  • Monoclinic
  • Triclinic

Each system has different Bravais lattices depending on the allowable symmetry operations such as rotation and reflection.

Seven crystal systems and their Bravais lattices

Cube system

In the cubic system, all edges of the unit cell are of equal length, and all angles are 90 degrees. The cubic system has three Bravais lattices:

  • Simple Cube (SC)
  • Body-Centered Cubic (BCC)
  • Face-centered cubic (FCC)
        
Simple Cubic Lattice:
- Position of points: corners of the cube
- Number of atoms per unit cell: 1

Body-Centered Cubic Lattice:
- Position of points: corners and one center point
- Number of atoms per unit cell: 2

Face-Centered Cubic Lattice:
- Position of points: corners and centers of faces
- Number of atoms per unit cell: 4

Quaternary system

The tetragonal system also has 90 degree angles, but two edges are equal, and the third is different. These include:

  • Simple quadrilateral
  • Body-centered quadrangular
        
Simple Tetragonal Lattice:
- Position of points: corners of the prism
- Number of atoms per unit cell: 1

Body-Centered Tetragonal Lattice:
- Position of points: corners and center
- Number of atoms per unit cell: 2

Orthorhombic system

In an orthorhombic lattice, all angles remain at 90 degrees, but all edges are unequal. These include:

  • Simple orthorhombic
  • Base-centered orthorhombic
  • Body-centered orthorhombic
  • Face-centered orthorhombic
        
Simple Orthorhombic Lattice:
- Position of points: corners
- Number of atoms per unit cell: 1

Base-Centered Orthorhombic Lattice:
- Position of points: corners, plus two opposite faces
- Number of atoms per unit cell: 2

Body-Centered Orthorhombic Lattice:
- Position of points: corners and one center
- Number of atoms per unit cell: 2

Face-Centered Orthorhombic Lattice:
- Position of points: corners and centers of all faces
- Number of atoms per unit cell: 4

Hexagonal system

In the hexagonal system, there are two equal edges at 120 degrees, and the third is different and perpendicular to the others. This system has only one Bravais lattice:

  • Simple hexagonal
        
Simple Hexagonal Lattice:
- Position of points: corners and center of hexagonal prism base
- Number of atoms per unit cell: 2

Rhomboid system

Also known as the triangular system, all sides are equal but the angles are not 90 degrees. It includes:

  • Simple rhomboid
        
Simple Rhombohedral Lattice:
- Position of points: corners of a rhombohedron
- Number of atoms per unit cell: 1

Monoclinic system

The monoclinic system has a unique angle that is not 90 degrees, while the others are at right angles. These include:

  • Simple monoclinic
  • Base-centered monoclinic
        
Simple Monoclinic Lattice:
- Position of points: corners
- Number of atoms per unit cell: 1

Base-Centered Monoclinic Lattice:
- Position of points: corners and two opposite faces
- Number of atoms per unit cell: 2

Triclinic system

The triclinic system has no right angles, and all edges are unequal. It includes:

  • Simple triclinic
        
Simple Triclinic Lattice:
- Position of points: corners
- Number of atoms per unit cell: 1

Visual depictions of lattice systems

The above illustration presents a simplified view of a cubic lattice, where each vertex of the cube represents a lattice point.

Applications and importance of Bravais lattices

Understanding Bravais lattices is important to many branches of science and technology. In crystallography, they provide the basis for identifying and classifying crystal structures. In materials science, knowing the structure helps predict the properties of materials. For example:

  • The arrangement of atoms in a metal can determine its electrical and thermal conductivity.
  • Polymorphism, or the ability of a substance to adopt more than one crystal structure, can affect the solubility and efficacy of a drug in medicine.

Conclusion

Studying the Bravais lattice gives a detailed understanding of how atoms stack and repeat in a crystalline structure. This knowledge is the cornerstone of condensed matter physics and provides essential insights that lead to the development of new materials, pharmaceuticals, and various technologies. By recognizing fundamental lattice structures, scientists and engineers can manipulate and optimize materials for desired properties, leading to innovations that impact daily life.


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Bravais lattices

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