Editing BCS theory (section)

==Overview==
At sufficiently low temperatures, electrons near the [[Fermi surface]] become unstable against the formation of [[Cooper pair]]s. Cooper showed such binding will occur in the presence of an attractive potential, no matter how weak. In conventional superconductors, an attraction is generally attributed to an electron-lattice interaction. The BCS theory, however, requires only that the potential be attractive, regardless of its origin. In the BCS framework, superconductivity is a macroscopic effect which results from the condensation of Cooper pairs. These have some bosonic properties, and bosons, at sufficiently low temperature, can form a large [[Bose–Einstein condensate]]. Superconductivity was simultaneously explained by [[Nikolay Bogolyubov]], by means of the [[Bogoliubov transformation]]s.

In many superconductors, the attractive interaction between electrons (necessary for pairing) is brought about indirectly by the interaction between the electrons and the vibrating crystal lattice (the [[phonon]]s). Roughly speaking the picture is the following:

<blockquote>An electron moving through a conductor will attract nearby positive charges in the lattice. This deformation of the lattice causes another electron, with opposite spin, to move into the region of higher positive charge density. The two electrons then become correlated. Because there are a lot of such electron pairs in a superconductor, these pairs overlap very strongly and form a highly collective condensate. In this "condensed" state, the breaking of one pair will change the energy of the entire condensate - not just a single electron, or a single pair. Thus, the energy required to break any single pair is related to the energy required to break ''all'' of the pairs (or more than just two electrons). Because the pairing increases this energy barrier, kicks from oscillating atoms in the conductor (which are small at sufficiently low temperatures) are not enough to affect the condensate as a whole, or any individual "member pair" within the condensate. Thus the electrons stay paired together and resist all kicks, and the electron flow as a whole (the current through the superconductor) will not experience resistance. Thus, the collective behavior of the condensate is a crucial ingredient necessary for superconductivity.</blockquote>

===Details===
BCS theory starts from the assumption that there is some attraction between electrons, which can overcome the [[Coulomb repulsion]]. In most materials (in low temperature superconductors), this attraction is brought about indirectly by the coupling of electrons to the [[crystal lattice]] (as explained above). However, the results of BCS theory do ''not'' depend on the origin of the attractive interaction. For instance, Cooper pairs have been observed in [[Ultracold atom|ultracold gases]] of [[fermion]]s where a homogeneous [[magnetic field]] has been tuned to their [[Feshbach resonance]]. The original results of BCS (discussed below) described an [[Atomic orbital|s-wave]] superconducting state, which is the rule among low-temperature superconductors but is not realized in many unconventional superconductors such as the [[Atomic orbital|d-wave]] high-temperature superconductors.

Extensions of BCS theory exist to describe these other cases, although they are insufficient to completely describe the observed features of high-temperature superconductivity.

BCS is able to give an approximation for the quantum-mechanical many-body state of the system of (attractively interacting) electrons inside the metal. This state is now known as the BCS state. In the normal state of a metal, electrons move independently, whereas in the BCS state, they are bound into Cooper pairs by the attractive interaction. The BCS formalism is based on the reduced potential for the electrons' attraction. Within this potential, a variational [[ansatz]] for the wave function is proposed. This ansatz was later shown to be exact in the dense limit of pairs. Note that the continuous crossover between the dilute and dense regimes of attracting pairs of fermions is still an open problem, which now attracts a lot of attention within the field of ultracold gases.

===Underlying evidence===
The hyperphysics website pages at [[Georgia State University]] summarize some key background to BCS theory as follows:<ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/solids/bcs.html|title=BCS Theory of Superconductivity|website=hyperphysics.phy-astr.gsu.edu|access-date=16 April 2018}}</ref>

* '''Evidence of a [[band gap]] at the Fermi level''' (described as "a key piece in the puzzle")
: the existence of a critical temperature and critical magnetic field implied a band gap, and suggested a [[phase transition]], but single [[electron]]s are forbidden from condensing to the same energy level by the [[Pauli exclusion principle]]. The site comments that "a drastic change in conductivity demanded a drastic change in electron behavior". Conceivably, pairs of electrons might perhaps act like [[boson]]s instead, which are bound by [[Bose–Einstein statistics|different condensate rules]] and do not have the same limitation. 
*'''Isotope effect on the critical temperature, suggesting lattice interactions'''
: The [[Debye frequency]] of phonons in a lattice is proportional to the inverse of the square root of the mass of lattice ions. It was shown that the superconducting transition temperature of mercury indeed showed the same dependence, by substituting the most abundant natural [[isotopes of mercury|mercury isotope]], <sup>202</sup>Hg, with a different isotope, <sup>198</sup>Hg.<ref name=maxwell1950>{{cite journal|last1=Maxwell|first1=Emanuel|title=Isotope Effect in the Superconductivity of Mercury|journal=Physical Review|volume=78|issue=4|pages=477|doi=10.1103/PhysRev.78.477|bibcode = 1950PhRv...78..477M |year=1950}}</ref>
* '''An [[Exponential growth|exponential rise]] in [[heat capacity]] near the critical temperature for some superconductors'''
: An exponential increase in heat capacity near the critical temperature also suggests an energy bandgap for the superconducting material. As superconducting [[vanadium]] is warmed toward its critical temperature, its heat capacity increases greatly in a very few degrees; this suggests an energy gap being bridged by thermal energy.
* '''The lessening of the measured energy gap towards the critical temperature'''
: This suggests a type of situation where some kind of [[binding energy]] exists but it is gradually weakened as the temperature increases toward the critical temperature. A binding energy suggests two or more particles or other entities that are bound together in the superconducting state. This helped to support the idea of bound particles – specifically electron pairs – and together with the above helped to paint a general picture of paired electrons and their lattice interactions.