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  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Condensed matter physics → Superconductivity ↓


Meissner effect and flux quantization


Superconductivity is a remarkable quantum phenomenon exhibited by certain materials when they are cooled below a critical temperature. In this state, the material exhibits zero electrical resistance and expels magnetic fields, allowing magnetic field quantization to occur. In this detailed explanation, we will take a deep look at two fundamental concepts associated with superconductivity: the Meissner effect and flux quantization.

Meissner effect

The Meissner effect, discovered by Walther Meissner and Robert Ochsenfeld in 1933, is the phenomenon by which a superconducting material expels all magnetic fields from within it. It occurs when a material changes to a superconducting state. Unlike perfect conductors, which simply freeze magnetic flux lines when cooled, superconductors actively repel magnetic fields.

Understanding the Meissner effect

The Meissner effect can be understood by considering the behavior of electrons in a superconductor. In the superconducting state, the electrons form so-called Cooper pairs due to attractive interactions mediated by lattice vibrations. This pairing allows the electrons to condense into a ground state characterized by a coherent quantum wave function.

When a superconductor is cooled below its critical temperature T c, it expels magnetic fields from within it. This phenomenon can be described by the London equations, which encompass the electromagnetic properties of superconductors. The first London equation is given as:

∂J/∂t = (n_s e²/m)E

Where:

  • J is the current density
  • E is the electric field
  • n_s is the density of superconducting electron pairs (Cooper pairs)
  • e is the charge of the electron
  • m is the mass of an electron

The second London equation is:

∇ × J = - (n_s e²/m) B

Where:

  • B is the magnetic field

These equations imply that the magnetic field in a superconductor decays exponentially from the surface over a characteristic length known as the London penetration depth λ_L. This results in the complete expulsion of the magnetic field from the bulk of the material, which is the essence of the Meissner effect.

Visual example of the Meissner effect

Imagine a spherical superconductor that has been cooled below its critical temperature while a uniform magnetic field is applied. In the Meissner state, the magnetic field lines are expelled from the interior, leaving only those that pass around the superconductor. This can be visualized as a field-free region inside the sphere, showing the expulsion effect in action.

For simplicity, consider the following illustrative diagram:

Magnetic field lines

Experimental observations

The Meissner effect is a hallmark of superconductivity and can be tested experimentally. Typically, a small magnet is placed above a superconducting sample, and as the sample cools below its critical temperature, the magnet rises due to the expulsion of its magnetic field from the superconductor.

This behavior is different from that of perfect conductors, which can trap magnetic fields. Active expulsion distinguishes superconductors from perfect conductors and is important in defining superconductivity as a distinct phase of matter.

Flux quantization in superconductors

An equally fascinating aspect of superconductivity is the quantization of the magnetic flux within a superconducting loop. When taken together with the Meissner effect, flux quantization provides insight into the macroscopic quantum nature of superconductors.

Concept of flux quantization

Flux quantization is due to the coherent quantum state occupied by Cooper pairs in the superconductor. The wave function describing this state must be single-valued, which means that the magnetic flux through the superconducting loop, Φ, is quantized in units of the flux quantum Φ 0:

Φ = n Φ 0

Where:

  • n is an integer
  • Φ 0 = h/2e is the flux quantum, where h is Planck's constant and e is the electron charge

This quantization arises from the condition that the wave function of the Cooper pairs, Ψ, must be single-valued because of the motion around a closed loop:

Ψ(r + L) = Ψ(r)

Visual example of flux quantization

Consider a thin superconducting ring that is exposed to an external magnetic field. Inside the superconductor, the magnetic field creates discrete flux tubes, each of which corresponds to one quantum of flux. This quantized nature is inherent in the complex quantum mechanical structure of superconductors.

For visualization, think of the flux tube as a bundle of quantized lines passing through a loop, as shown below:

Quantized flux lines

Applications and consequences

Flux quantization is crucial to understanding and designing superconducting devices. It is the basis for technologies such as superconducting quantum interference devices (SQUIDs), which use this quantization to detect extremely small magnetic fields.

Due to flux quantization, superconductors exhibit fascinating phenomena such as the Josephson effect, which forms the basis of the superconducting qubits used in quantum computers. The requirement for the wave function to be single-valued imposes restrictions on the dynamics of superconducting circuits, providing a rich playground for technological application and fundamental physics exploration.

Closing thoughts

The Meissner effect and flux quantization are central concepts in the study of superconductivity, each of which reveals the rich interplay between electromagnetism and quantum mechanics. The expulsion of magnetic fields and the quantized nature of the superconducting loop highlight the fundamental quantum nature of the superconducting state.

These phenomena highlight not only the unique properties of superconductors, but also their tremendous potential for technological applications ranging from magnetic levitation to quantum computing. By understanding these concepts, we gain a deeper insight into the power and beauty of quantum physics manifested in condensed matter systems.


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Meissner effect and flux quantization

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