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 ↓


X-ray diffraction


X-ray diffraction (XRD) is a powerful experimental technique used to investigate the crystal structure of solids. It enables scientists to understand the nature of crystalline materials by examining how X-rays are scattered by atoms in the material. In this explanation, we will learn in detail about the fundamentals of X-ray diffraction, the nature of crystals, and how the diffraction pattern provides insight into crystal structures.

Introduction to X-rays

X-rays are a form of electromagnetic radiation with wavelengths in the range of about 0.01 to 10 nanometers, which is roughly the size of an atom. Because of their short wavelength, X-rays are able to penetrate solid objects and are ideal for studying the atomic structure of substances. When X-rays collide with atoms within a crystal, they can scatter, creating a diffraction pattern that provides valuable information about the crystal's internal structure.

Nature of crystals

Crystals are solids in which atoms are arranged in a highly ordered, repetitive pattern. The smallest repeating unit in a crystal is called a unit cell. The way atoms are arranged in a crystal affects its physical properties, such as hardness, optical properties, and its interaction with X-rays.

To understand crystal structures, imagine a three-dimensional grid called a lattice, where each point on the grid represents an atom in the crystal. The unit cell is defined by the edges that span these points, and by repeating the unit cell in all three dimensions, the entire crystal structure can be described.

Lattice vectors: a, b, c
Angles: α, β, γ

The perfect crystal can be described as an infinitely repeating array of these unit cells. Each type of crystal structure is classified based on its symmetry and the shape of its unit cell, such as cubic, tetragonal, orthorhombic, and others.

Principles of X-ray diffraction

X-ray diffraction is based on Bragg's law, named after physicist William Henry Bragg and his son William Lawrence Bragg. This law relates the wavelength of the incident X-rays, the angle at which they are scattered, and the distance between atomic planes in the crystal.

Bragg's law: nλ = 2d sinθ
  • n is an integer (the order of diffraction).
  • λ is the wavelength of the X-ray.
  • d is the distance between the crystal planes.
  • θ is the angle of incidence which is equal to the angle of reflection.

For constructive interference to occur, the difference in the path lengths of the X-rays scattered by planes in the crystal must be an integer number of wavelengths. This results in a pattern of diffraction peaks that can be recorded and analyzed to infer the crystal structure.

X-ray of the incident Scattered X-rays Scattered X-rays

Crystal plane and Miller index

The planes of atoms in a crystal that scatter X-rays can be described by their Miller indices. These indices are a set of three integers (h, k, l) that indicate the orientation of the crystal plane. The Miller indices are determined by reversing the plane constraints along the unit cell axes.

For example, consider a cubic crystal structure, one of the simplest crystal structures. The Miller index (1, 0, 0) refers to a plane that intercepts the x-axis at the edge of the unit cell and is parallel to the y and z axes.

(1, 0, 0)

The importance of understanding and identifying Miller indices is that each set corresponds to a different atomic plane interval, which plays an important role in the analysis of X-ray diffraction patterns.

Construction of X-ray diffraction patterns

When a crystal is exposed to a beam of X-rays, the planes of the atoms scatter the radiation, resulting in a pattern of light and dark spots, known as a diffraction pattern. Each spot corresponds to a specific angle and reveals important information about the crystal structure.

To capture these diffraction patterns, an X-ray diffractometer is used. This instrument positions the crystal and directs a beam of X-rays at it, while also measuring the angle and intensity of the diffracted rays. By rotating or tilting the crystal, the diffractometer can obtain different patterns, revealing the three-dimensional arrangement of atoms within the crystal.

This diffraction pattern contains important information about the crystal lattice, since the position and intensity of the spots help to identify the parameters of the unit cell and the arrangement of atoms within it.

Interpretation of diffraction patterns

Interpreting an X-ray diffraction pattern is a process that involves determining the structure of a crystal - the arrangement and type of atoms - by analyzing the position and intensity of the spots recorded. This analysis involves the main steps:

  • Measuring the angle and intensity of the diffraction pattern.
  • Using Bragg's law to calculate the spacing between atomic planes based on the angles of the diffraction pattern.
  • This information is combined with knowledge of the Miller indices and crystal symmetry to predict the crystal structure.

In most cases, this interpretation involves calculations and the use of special software that can model different possible atomic arrangements and compare them to experimental data to determine the most probable structure.

Applications of X-ray diffraction

X-ray diffraction is widely used in many fields of science and technology because of its ability to provide detailed structural information about substances. Some of the major applications include:

  • Materials science: Identifying unknown materials, studying crystal defects, and analyzing stresses within materials.
  • Chemistry: Determining the molecular structure of compounds and analyzing chemical structures.
  • Biology: Studying the structure of biological molecules, such as proteins and nucleic acids, to understand their function within living organisms.
  • Physics: Investigating the properties of solids, phase transitions, and electron distribution.

These applications demonstrate the widespread influence of X-ray diffraction in fundamental scientific research and applied technological development. By revealing insights about structures at the atomic level, XRD provides valuable information that can lead to breakthroughs in material design and understanding biological processes.

Conclusion

In summary, X-ray diffraction is an essential tool in the study of crystalline solids, allowing scientists to explore the atomic-level structure of substances. By harnessing the principles of diffraction and using sophisticated equipment, researchers can obtain precise information about the arrangement and identity of atoms within crystals. This technique not only enhances our fundamental understanding of the physical world, but also leads to practical applications in many scientific fields.


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X-ray diffraction

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