Undergraduate
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  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Solid state physics → Crystal structure ↓


Bravais lattices


In the study of solid state physics, the structure of matter at the atomic level is of paramount importance. One of the central concepts in understanding how atoms are arranged in crystals is what is known as the Bravais lattice. Named after the French physicist Auguste Bravais, who first identified them in 1850, the Bravais lattice is a foundational element for the study of crystalline solids and lattice theory.

So, what exactly is a Bravais lattice? At a basic level, a lattice is an array of points (or nodes) in space. In crystal terms, this array is periodic, meaning it repeats itself in a regular pattern. A Bravais lattice is a group of these points arranged in such a way that the environment around each point is the same. In simple terms, if you sit at any of these points and look around, everything will look the same. This property makes the Bravais lattice a powerful tool for classifying crystal structures.

Understanding the concept of Bravais lattices

Bravais lattices help us classify crystal structures based on the symmetry and arrangement of their constituent particles. Simply put, these lattices show the different ways in which atoms, ions, or molecules can be arranged to form a solid. In three-dimensional space, there are exactly 14 unique Bravais lattices.

To understand the Bravais lattice, it is first necessary to understand some basic concepts about crystal structures. A crystal structure consists of two main components: the lattice and the basis. The lattice is the geometric arrangement of points in space, while the basis is the set of atoms associated with each lattice point. When a basis is attached to the lattice, the combination defines a crystalline solid.

Seven crystal systems

Each Bravais lattice belongs to one of seven crystal systems. These systems are classified based on the axial length and angles of the unit cell - the smallest repeating parts of the lattice that form the whole crystal when put together.

  1. Cube: All sides are equal, and all angles are 90 degrees. Example: NaCl (rock salt).
  2. Quadrilateral: Two sides are equal but the third is different; all angles are 90 degrees. Example: White tin.
  3. Orthorhombic: All sides are unequal, but all angles are 90 degrees. Example: Olivine.
  4. Hexagonal: Two sides are equal, the third is different; there is a 120 degree angle between the equal sides and a 90 degree angle between the third side. Example: Beryl.
  5. Triangular (rhombohedral): All sides are equal; angles are equal but not 90 degrees. Example: Quartz.
  6. Monoclinic: All sides are unequal; two angles are 90 degrees, and one is not. Example: Monoclinic sulfur.
  7. Triclinic: All sides are unequal, and all angles are unequal. Example: kyanite.

Fourteen Bravais lattices

In these seven crystal systems, the dots can be arranged in special patterns that form 14 different Bravais lattices. We'll explore each of these in detail.

1. Cube

  • Simple cubic (SC): The simplest form of cubic lattice. Points are located at each corner of the cube.
  • Body-centered cubic (BCC): In addition to the points at the corners of the cube, there is an additional point at the center of the cube.
  • Face-centered cube (FCC): The cube has points at each corner and at the center of each face.
        Cubic Lattice Example (Simple Cubic): Corner points at: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1), (1, 0, 1), (0, 1, 1), (1, 1, 1)

2. Quadrangular

  • Simple tetrahedron: Similar to a simple cube but extended along an axis. The points are at the corners.
  • Body-centered tetragonal: Similar to body-centered cubic but with a tetragonal cell shape.
        Tetragonal Lattice Example (Body-Centered Tetragonal): Corner points at: (0, 0, 0), (2, 0, 0), (0, 2, 0), (2, 2, 0), (0, 0, 2), (2, 0, 2), (0, 2, 2), (2, 2, 2) Center point at: (1, 1, 1)

3. Orthorhombic

  • Simple orthorhombic: points at the corners of an orthorhombic shaped cell.
  • Base-centered orthorhombic: additional points at the centers of each of the two bases.
  • Body-centered orthorhombic: an additional point at the center of the cell.
  • Face-centered orthorhombic: points located at the center of each face as well as at the corners.
        Orthorhombic Lattice Example (Face-Centered Orthorhombic): Corner points at: (0, 0, 0), (1, 0, 0), (0, 2, 0), (1, 2, 0), (0, 0, 3), (1, 0, 3), (0, 2, 3), (1, 2, 3) Face centers at: (0.5, 0, 1.5), (0.5, 2, 1.5), (0, 1, 1.5), (1, 1, 1.5), (0.5, 1, 0), (0.5, 1, 3)

4. Hexagonal

  • Simple hexagon: A six-sided prism with lattice points at the corners of the hexagon and in the top or bottom plane.
        Hexagonal Lattice Example (Simple Hexagonal): Corner points at: (0, 0, 0), (1, 0, 0), (0.5, √3/2, 0), (0, 0, c), (1, 0, c), (0.5, √3/2, c)

5. Triangular (rhombohedral)

  • Simple rhombus: A lattice in which each lattice vector is of equal length, and there are equal angles between them, but these angles are not right angles.
        Rhombohedral Lattice Example (Simple Rhombohedral): Corner points at: Lattice with each side of length 'a' Angles between vectors are less than 90 degrees

6. Monoclinic

  • Simple monoclinic: The cell has unequal sides and angles, with one angle not equal to 90 degrees, lying in between the other two 90 degree angles.
  • Base-centered monoclinic: one additional point at the center of the base in addition to the corners.
        Monoclinic Lattice Example (Base-Centered Monoclinic): Corner points at: (0, 0, 0), (a, 0, 0), (0, b, 0), (a, b, 0), (0, 0, c), (a, 0, c), (0, b, c), (a, b, c) Base center: (a/2, 0, 0), (a/2, b, 0)

7. Triclinic

  • Simple triclinic: the most generalized lattice form; sides and angles are all unequal and not necessarily perpendicular.
        Triclinic Lattice Example (Simple Triclinic): No symmetry requirements abound; all edges and angles random Closed structure from a single-particle view must repeat in all dimensions uniquely

Mathematical representation

The Bravais lattice can be mathematically described using three vectors, called lattice vectors. These vectors are represented by:

        a1, a2, a3 R = n1*a1 + n2*a2 + n3*a3

Here, n1, n2 and n3 are integers, while a1, a2 and a3 define the shape and size of the unit cell in the crystal lattice.

The angle and length of these vectors define the properties of the Bravais lattice and consequently the crystal system to which they belong.

Importance of Bravais lattices

The Bravais lattice forms the basic framework needed to study and understand more complex crystal structures. Here are some of the benefits of studying the Bravais lattice:

  • Classification: By understanding these lattices, we can systematically classify all possible crystal structures.
  • Mathematical foundation: Bravais lattices provide essential mathematical tools for further studies in solid state physics.
  • Understanding symmetry: Insight into the symmetry of materials which is important for understanding physical properties such as thermal, electrical, optical and mechanical behaviour.

Applications in the real world

Understanding Bravais lattices helps explain the properties of materials that are important in many technologies. For example:

  • Semiconductors: Understanding crystal structure helps design effective semiconductor materials that are the backbone of electronic devices.
  • Materials science: It helps predict how new materials, especially alloys, will behave.
  • Pharmaceuticals: Understanding crystal structure helps in the design and manufacture of drugs.

Conclusion

Understanding Bravais lattices is important in solid state physics and materials science. It provides a basis for understanding how different substances behave based on their atomic structures. The 14 types of Bravais lattices provide a framework for classifying and analyzing crystalline solids.


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Bravais lattices

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