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 ↓


Superconductivity


Superconductivity is a fascinating phenomenon in condensed matter physics, characterized by the complete absence of electrical resistance in certain materials when cooled below a specific critical temperature. The phenomenon was first discovered by Dutch physicist Heike Kamerlingh Onnes in 1911. Since then, it has baffled scientists with its paradoxical properties and potential technological applications. In this lesson, we will explore the fundamentals, theoretical explanations, and practical applications of superconductivity in a simple and detailed way.

Fundamentals of superconductivity

To understand superconductivity, it is important to understand why electrical resistance exists. In normal conductor materials such as copper or aluminum, free electrons pass through a lattice of metal atoms. As they move, they collide with these atoms, which are arranged in a regular pattern. Each collision causes resistance, resulting in a loss of energy as heat.

Common Conductors Superconductors

In contrast, superconductors allow electric current to flow without any resistance. This means that no energy is lost during transmission, making superconductors highly efficient for certain applications. The primary condition for a material to become superconductive is that it must be cooled below its specific critical temperature ((T_c)).

Meissner effect

A special feature of superconductivity is the Meissner effect. Discovered in 1933 by Walther Meissner and Robert Ochsenfeld, this effect describes how a superconductor expels a magnetic field from within itself during the transition to the superconducting state. This happens regardless of whether a magnetic field is present before cooling.

expelled magnetic field

The Meissner effect distinguishes superconductors from perfect conductors. In a perfect conductor, the magnetic field present before cooling will remain trapped. In contrast, a superconductor actively expels all magnetic fields, demonstrating its diamagnetic nature.

Theoretical explanation: BCS theory

The first comprehensive theoretical description of superconductivity was provided by John Bardeen, Leon Cooper, and Robert Schrieffer in 1957. This description, known as the BCS theory, introduced the concept of the Cooper pair.

According to the BCS theory, electrons move through the lattice at low temperatures in the form of pairs, known as Cooper pairs. These pairs form due to an attractive interaction mediated by lattice vibrations called phonons. This pairing allows the electrons to overcome the repulsive forces between negative charges, forming a stable configuration that maintains zero resistance.

Electron Phonon Cooper pair formation

The BCS theory not only explains the origin of zero resistance but also describes the energy gap that forms at the Fermi surface when a material becomes a superconductor. This energy gap (( delta )) represents the energy required to break a Cooper pair into two separate electrons.

        E = Delta_0 (1 - (T / T_c)^2) ^ 0.5, for T < T_c

High temperature superconductor

The discovery of high-temperature superconductors in 1986 by Johannes Bednorz and Karl Müller, who were awarded the Nobel Prize in Physics in 1987, substantially broadened the scope of research and application. These materials can exhibit superconductive properties at relatively high temperatures compared to conventional superconductors, which typically require liquid helium cooling.

High-temperature superconductors are often copper-oxide ceramics, known as cuprates, and have complex crystal structures. These materials can be superconductive at temperatures above 77 K, which is an important threshold because it allows cooling with liquid nitrogen, a less expensive and more accessible coolant.

77's High Temperature Superconductor

Despite their promise, the mechanism of high-temperature superconductivity remains a subject of active research. Unlike the BCS theory, which applies well to conventional superconductors, high-temperature superconductivity often involves more complex interactions, possibly including electron correlation and antiferromagnetic ordering.

Applications of superconductivity

Superconductivity holds immense potential for a variety of technological applications due to its unique properties. Some notable applications include:

Magnetic resonance imaging (MRI)

MRI machines typically use superconducting magnets to generate the strong magnetic fields needed for imaging. The zero-resistance property reduces energy loss and cooling requirements, improving efficiency and reducing operating costs.

Magnetically levitated trains

Superconductors are used in magnetic levitation (maglev) trains, which can float above tracks using the repulsion force generated by the superconductor. This levitation reduces friction, allowing trains to reach high speeds with minimal energy loss.

to illustrate:

Track (superconductor) train

Electric power systems

Superconductors can revolutionize the efficiency of the power grid. Superconducting cables can transmit electricity over long distances without any resistance-related losses. In addition, superconducting magnetic energy storage systems provide rapid response capabilities, stabilize the power grid and integrate renewable energy sources.

Particle accelerators

Superconductors enable the construction of powerful electromagnets that are used in particle accelerators such as the Large Hadron Collider. These magnets maintain strong fields with high precision, which is important for guiding and focusing particle beams.

Challenges and future directions

Despite their potential, many challenges must be overcome to enable widespread adoption of superconductors. Key challenges include:

  • Materials costs: Many superconductors require rare and expensive elements, which increases production costs.
  • Temperature limitation: Even high-temperature superconductors require cooling, which can be expensive in large-scale applications.
  • Brittleness of the material: Many superconducting materials are brittle and are difficult to shape into wires or coils without losing superconductivity.

Ongoing research aims to develop new superconducting materials that operate at even higher temperatures and are easier to manufacture and implement. These advances could lead to breakthroughs in energy efficiency and the development of new technologies.

Superconductivity remains a vibrant area of research, continually pushing the boundaries of our understanding of quantum mechanics and physics. Its potential to revolutionise industries and solve critical challenges in energy and transport underlines its importance in science and technology.

In short, superconductivity represents a quantum phenomenon where electric current flows without resistance due to the formation of Cooper pairs. The Meissner effect, BCS theory and the discovery of high-temperature superconductors highlight its unique properties and widespread importance. While challenges remain, the future of superconductivity promises transformational technological advancements and scientific discoveries.


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Superconductivity

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