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I can't get a definitive explanation of why superconductivity happens and I am getting mixed explanations from my textbooks. I will tell you what I know and hopefully you can correct any misunderstandings I have:

  • A metal consists of an ionic lattice because the electrons in the valence band have 'jumped' to the conduction band

  • When a potential difference is applied across a conductor, electrons would move towards the positive potential but the lattice will become distorted as the ions are attracted to the electrons

  • This creates a region that has dense positive charge. Electrons would be attracted towards this region. However, electrons repel each other and don't want to be close to each other, additionally, Wolfgang Pauli's exclusion principle states that no two fermions can occupy the same quantum state so every two electrons have to combine and form a Bose Einstein condensate as bosons can occupy the same quantum state.

  • Cooper pairs arise because of the exchange of phonons. Phonons are a collection of excitations of atoms or molecules. The atoms or molecules have to be vibrating in some collective mode.

  • The bond between two electrons in a boson is very weak, 10^-3 eV. Therefore, the temperature of the metal must be 10 Kelvin for the cooper pairs to exist (using E=kT where k is 10^-4 eV).

  • Bosons do not interact with matter, hence, they aren't impeded by the lattice.

There are three things I do not understand about this explanation:

  1. Pauli's principle.

  2. Why bosons can be in the same energy state and fermions can't.

  3. Phonons.

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1) The Pauli exclusion principle says that either 0 or 1 identical fermion, but not more, may occupy the same quantum state. It's been originally induced from the properties of atoms (periodic table). Today, we may derive it from the first principle out of the antisymmetric wave functions for many fermions, those obeying $$\psi(\vec x_1,\vec x_2) = -\psi(\vec x_2,\vec x_1) $$ or, even more fundamentally, from the anticommuting ($ab=-ba$ obeying) nature of the corresponding quantum fields

2) Fermions are particles that obey this principle, bosons are those that don't. At the level of wave functions, bosons differ because their wave function is symmetric, not antisymmetric (the same identity as above but without the minus sign) or they're created by commuting ($ab=ba$) and not anticommuting fields. Pauli has linked the "statistics" – whether something is a boson or a fermion – to its having integer-valued or half-integer-valued spin, respectively.

3) Phonons are quanta of sound (the waves that are coming out of your phone) in the same sense as photons are quanta of light. In solids, crystals etc., the atoms may vibrate and the vibrations of some local group of atoms (including the nuclei) is described by a harmonic oscillator. However, everything in the world obeys postulates of quantum mechanics. For harmonic oscillators, it means that the energy stored in these vibrations is a multiple of $\Delta E = hf$ where $h$ is Planck's constant and $f$ is the frequency. While the packages are indivisible, it's still true that a wave composed of a large number of photons is nothing else than a sound wave in the solid. (Phonons require the environment of a solid; photons exist even in the vacuum.)

Much like photons may be interpreted as the "messengers" of the electromagnetic force, phonons are "messengers" of yet another force. The electric force repels like-sign charges, due to the odd spin of the photon. Phonons have an even spin (zero) so they attract like-sign charges. The mild attraction between two electrons is what produces the Cooper pairs, sort of analogous objects to atoms (bound states) in superconductivity, among other things.

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