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 ↓


Phonons and lattice vibrations


In solid state physics, it is important to understand the behavior of solids at the microscopic level. A solid is made up of atoms or molecules arranged in a structured, repeating pattern known as a crystal lattice. In this lattice, the atoms are not stationary; they vibrate around their equilibrium positions. These vibrations are known as lattice vibrations.

The concept of lattice vibrations

Let's start by considering a simple one-dimensional chain of atoms, equally spaced and bound to each neighbour by equal forces. Each atom can move about its position, oscillating slightly back and forth. These oscillations are lattice vibrations.

1D chain of atoms

Each circle represents an atom, and the distance between them represents the bonding forces. These atoms can vibrate in different ways, causing waves to flow through the material.

Lattice vibration modeling

To understand lattice vibrations mathematically, consider atoms connected by springs, which behave according to Hooke's law. The displacement of the atoms from their equilibrium position determines the force experienced by each atom.

The equation of motion for an atom in this spring system can be written as:

        m * d²x/dt² = -k * (x - x₀)

Here, m is the mass of the atom, x is the displacement, x₀ is the equilibrium position, and k is the spring constant.

What are phonons?

Phonons are quantized modes of vibration occurring in a rigid crystal lattice, which serve as a quantum mechanical description of lattice vibrations. Just as photons are quantized units of light waves, phonons are quantized units of vibration waves.

Phonons are important for understanding thermal and electrical conduction in materials. They play a role in specific heat, thermal expansion, and the propagation of sound through solids.

Why use phonons?

The concept of phonons is useful because it allows us to consider vibrational energy as being made up of discrete units. This quantization helps apply statistical mechanics to better calculate the properties of a material.

The problem of lattice vibrations using phonons can be simplified by considering them as particle-like entities, allowing us to use the toolbox of quantum mechanics.

Acoustic and optical phonons

There are mainly two types of phonons: acoustic phonons and optical phonons.

Acoustic phonon

Acoustic phonons are associated with vibrations where all the atoms in the lattice move in phase with each other. For acoustic phonons, consider a wave where the entire lattice oscillates with low frequencies. They are called acoustic because they can transmit sound through the lattice.

Stepped motion

Acoustic phonon modes

Optical phonon

Optical phonons are created when adjacent atoms in the lattice move out of phase with each other. These vibrations typically have a higher frequency than acoustic phonons and can interact with electromagnetic radiation.

out-of-phase motion

Optical phonon modes

The phase difference results in the center of mass of optical phonons becoming zero, which distinguishes them from acoustic phonons.

Frequency and dispersion relations

The frequency of the phonon is important for understanding how these particles behave in different materials. The relationship between the phonon frequency (ω) and the wave vector (q) is described by the dispersion relation.

In its simplest form, this relation is given as:

        ω = v * |q|

Here, ω is the angular frequency, v is the phase velocity, and q is the wave vector, which represents the phase shift over a distance for the wave.

Why ω

Dispersion curve

This dispersion relation helps us understand the wave nature of phonons and their propagation through different materials. Different materials have different dispersion relations, resulting in different thermal and acoustic properties.

Understanding phonon interactions

Phonons interact in complex ways with each other and with other particles such as electrons. These interactions control many thermal and electrical properties of materials.

Phonon-phonon interactions

Phonons can dissociate from one another. This process is important for understanding thermal conductivity in materials. At higher temperatures, the increased interaction of phonons leads to a drop in thermal conductivity.

Phonon–electron interactions

Phonons can also interact with electrons in the lattice. This interaction affects electrical resistance in conductors and plays an important role in superconductivity.

Applications and significance

Understanding phonons and lattice vibrations is important in materials science and technology. Some major applications include:

Thermoelectric materials

In thermoelectric materials, controlling phonon vibrations helps optimize heat transfer and conversion efficiency. Materials are engineered to have specific phonon dispersion properties that enhance performance.

Semiconductors

The behavior of phonons affects the thermal performance of semiconductor devices. In high power and microelectronics, understanding phononic behavior is essential to effectively manage heat.

Thus, understanding phonons and lattice vibrations is fundamental to advancing physical research and developing new technologies.

Conclusion

Phonons and lattice vibrations are fundamental concepts in solid state physics. By treating vibrations as quantized phonons, scientists have been able to advance the understanding of the thermal and electrical properties of materials. Through the study of interactions, dispersion relations, and acoustic and optical phonons, the understanding of phonons continues to evolve, leading to technological innovations in a variety of fields.


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Phonons and lattice vibrations

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