Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  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


In solid state physics, the concept of crystal structure is important. It is the cornerstone of understanding how materials behave at the microscopic level. Understanding crystal structures involves understanding the arrangement of atoms within a solid. This arrangement determines many properties of the material, such as electrical conductivity, magnetism, and optical properties.

Basic concepts

A crystal is a solid material whose atoms are arranged in a highly ordered, repeating pattern. This pattern extends in all three spatial dimensions. The smallest unit of this structure, which can be repeated without any change, is known as the unit cell. Unit cells can be thought of as the building blocks of a crystal.

To visualize a crystal structure, imagine a three-dimensional grid (like graph paper in 3D), where each intersection point represents an atomic position. The unit cell is a small box that can completely tile this grid by translation.

Forged

The concept of a lattice is central to understanding crystal structures. A lattice is a regular, repeating arrangement of points in space. Each point represents one or more atoms. In solid state physics, we classify lattices based on how the atoms are arranged in three-dimensional space.

The simple diagram above shows a 2D lattice, where the dots are lattice points, and these can be connected to atoms. Each line is a connection from a point to its neighbour, showing the regular pattern.

Unit cell

In crystallography, the purpose of the unit cell is to describe the crystal structure. Each unit cell contains not only the arrangement of atoms, but also the symmetry and dimensions of the crystal. The angles between the edges of the unit cell and the lengths of these edges are the main aspects that define its geometry.

Atom

The image above shows a 2D representation of a unit cell, where a red atom is placed at the center. In a 3D representation, you would expect to see identical layers stacked on top of each other.

Crystal systems and Bravais lattices

In three dimensions, crystals are classified into seven crystal systems, which describe the shape of the unit cell:

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

Each of these systems is defined by specific restrictions on the angles and lengths of the unit cell edges. In addition to these seven crystal systems, Auguste Bravais identified 14 unique three-dimensional lattices, known as Bravais lattices.

Example: cubic system

The cubic crystal system is characterized by a unit cell where all sides are equal in length, and all angles are 90 degrees. Within this system, there are several types of lattice arrangements, which are usually referred to as:

  • Simple Cube (SC)
  • Body-centered cubic (BCC)
  • Face-centered cube (FCC)

Simple Cube

In a simple cubic lattice, the atoms are located at each corner of the cube:

This simple layout allows us to see how repeating unit cells fit together to form larger structures, although simple cubicity is not common in natural crystals.

Body-centered cubic (BCC)

The BCC arrangement has atoms at each cube corner and an additional atom at the center of the cube. This structure is denser than a simple cube.

Visual example of BCC

In the BCC lattice, you can see how the central atom (in red) maximizes the space utilization of the unit cell.

Face-centered cubic (FCC)

The FCC arrangement places atoms at the corner of each cube and at the center of each face of the cube. This is the densest arrangement.

Visual example of FCC

The face-centered cubic lattice uses intermediate states to fill more space, making this structure common in metals such as aluminum and copper.

Density calculation

The atomic packing factor (APF) is a measure of the density of atoms within the lattice. It is calculated as:

APF = (Volume of atoms in a unit cell) / (Total volume of unit cell)

For the FCC structures, the APF is about 0.74, which shows that it has a higher density compared to the BCC structure, which has an APF of about 0.68.

Each type of crystal lattice has unique properties, such as strength, ductility, and conductivity. These properties result from the way the atoms are packed and their corresponding interactions.

Anisotropic

Crystals often exhibit anisotropy, meaning that their properties differ depending on the direction of measurement. For example, atomic arrangements along different axes in a crystal can lead to differences in physical or mechanical properties.

Materials with a face-centered cubic structure are more isotropic, while materials with less regular symmetry, such as monoclinic or triclinic, can exhibit significant asymmetry.

Defects in the crystal structure

No crystal is perfect. Imperfections can significantly affect the properties of a material. Common defects include:

  • Point defects such as vacancies and interstitials
  • Line defects, such as dislocations
  • Planar defects, such as grain boundaries

These imperfections can affect the material’s strength, electrical conductivity and optical properties.

Application

The understanding of crystal structures has important applications in many areas:

  • Semiconductor manufacturing depends on precise crystal growth for materials such as silicon.
  • Understanding materials in metallurgy to enhance their strength and ductility.
  • Nanotechnology and the design of new materials at small scales.

Modern materials science relies heavily on crystallographic understanding, influencing technological advances from electronics to biotechnology.

Conclusion

The study of crystal structures in solid state physics provides profound insights into the physical world. From identifying basic unit cells to grappling with the intricacies of real-world crystalline defects, this subject provides a foundational perspective. It not only helps us unlock the potential of existing materials but also drives the development of new, advanced materials, impacting everything from industrial production to cutting-edge technology.


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Crystal structure

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