Editing BCS theory (section)

==Implications==
BCS derived several important theoretical predictions that are independent of the details of the interaction, since the quantitative predictions mentioned below hold for any sufficiently weak attraction between the electrons and this last condition is fulfilled for many low temperature superconductors - the so-called weak-coupling case. These have been confirmed in numerous experiments:
* The electrons are bound into Cooper pairs, and these pairs are correlated due to the [[Pauli exclusion principle]] for the electrons, from which they are constructed. Therefore, in order to break a pair, one has to change energies of all other pairs.<!-- Unclear how this is a consequence of the exclusion principle --> This means there is an energy gap for single-particle excitation, unlike in the normal metal (where the state of an electron can be changed by adding an arbitrarily small amount of energy). This energy gap is highest at low temperatures but vanishes at the transition temperature when superconductivity ceases to exist. The BCS theory gives an expression that shows how the gap grows with the strength of the attractive interaction and the (normal phase) single particle [[density of states]] at the [[Fermi level]]. Furthermore, it describes how the density of states is changed on entering the superconducting state, where there are no electronic states any more at the Fermi level. The energy gap is most directly observed in tunneling experiments<ref name="Giaever">Ivar Giaever - Nobel Lecture. Nobelprize.org. Retrieved 16 Dec 2010. http://nobelprize.org/nobel_prizes/physics/laureates/1973/giaever-lecture.html</ref> and in reflection of microwaves from superconductors.
* BCS theory predicts the dependence of the value of the energy gap Δ at temperature ''T'' on the critical temperature ''T''<sub>c</sub>. The ratio between the value of the energy gap at zero temperature and the value of the superconducting transition temperature (expressed in energy units) takes the universal value<ref name="Tinkham 1996 63">{{Cite book
| first=Michael| last=Tinkham| year=1996
| title=Introduction to Superconductivity | pages=63
| publisher=Dover Publications
| isbn=978-0-486-43503-9}}</ref> <math display="block">\Delta(T=0) = 1.764 \, k_{\rm B}T_{\rm c},</math> independent of material. Near the critical temperature the relation asymptotes to<ref name="Tinkham 1996 63"/> <math display="block">\Delta(T \to T_{\rm c})\approx 3.06 \, k_{\rm B}T_{\rm c}\sqrt{1-(T/T_{\rm c})}</math> which is of the form suggested the previous year by M. J. Buckingham<ref>
{{cite journal
 | last=Buckingham | first=M. J.
 | title=Very High Frequency Absorption in Superconductors
 |date=February 1956
 | journal=[[Physical Review]]
 | volume=101 | issue=4
 | pages=1431–1432
 | doi = 10.1103/PhysRev.101.1431
 | bibcode = 1956PhRv..101.1431B }}</ref> based on the fact that the superconducting phase transition is second order, that the superconducting phase has a mass gap and on Blevins, Gordy and Fairbank's experimental results the previous year on the absorption of millimeter waves by superconducting [[tin]].
* Due to the energy gap, the [[specific heat]] of the superconductor is suppressed strongly ([[exponential decay|exponentially]]) at low temperatures, there being no thermal excitations left. However, before reaching the transition temperature, the specific heat of the superconductor becomes even higher than that of the normal conductor (measured immediately above the transition) and the ratio of these two values is found to be universally given by 2.5.
* BCS theory correctly predicts the [[Meissner effect]], i.e. the expulsion of a magnetic field from the superconductor and the variation of the penetration depth (the extent of the screening currents flowing below the metal's surface) with temperature. 
* It also describes the variation of the [[upper critical field|critical magnetic field]] (above which the superconductor can no longer expel the field but becomes normal conducting) with temperature. BCS theory relates the value of the critical field at zero temperature to the value of the transition temperature and the density of states at the Fermi level.
* In its simplest form, BCS gives the superconducting transition temperature ''T''<sub>c</sub> in terms of the electron-phonon coupling potential ''V'' and the [[Debye frequency|Debye]] cutoff energy ''E''<sub>D</sub>:<ref name=BCS_theory/> <math display="block">k_{\rm B}\,T_{\rm c} = 1.134E_{\rm D}\,{e^{-1/N(0)\,V}},</math> where ''N''(0) is the electronic density of states at the Fermi level. For more details, see [[Cooper pairs]].
* The BCS theory reproduces the '''isotope effect''', which is the experimental observation that for a given superconducting material, the critical temperature is inversely proportional to the square-root of the mass of the [[isotope]] used in the material. The isotope effect was reported by two groups on 24 March 1950, who discovered it independently working with different [[mercury (element)|mercury]] isotopes, although a few days before publication they learned of each other's results at the ONR conference in [[Atlanta]]. The two groups are [[Emanuel Maxwell]],<ref>{{Cite journal|last=Maxwell|first=Emanuel|date=1950-05-15| 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}}</ref> and C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt.<ref>{{Cite journal|last1=Reynolds|first1=C. A.|last2=Serin|first2=B.|last3=Wright|first3=W. H.|last4=Nesbitt|first4=L. B.| date=1950-05-15|title=Superconductivity of Isotopes of Mercury|journal=Physical Review|volume=78|issue=4|pages=487|doi=10.1103/PhysRev.78.487|bibcode=1950PhRv...78..487R}}</ref> The choice of isotope ordinarily has little effect on the electrical properties of a material, but does affect the frequency of lattice vibrations. This effect suggests that superconductivity is related to vibrations of the lattice. This is incorporated into BCS theory, where lattice vibrations yield the binding energy of electrons in a Cooper pair.
* [[Little–Parks effect|Little–Parks experiment]]<ref name=Little>{{Cite journal | doi=10.1103/PhysRevLett.9.9| title=Observation of Quantum Periodicity in the Transition Temperature of a Superconducting Cylinder| year=1962| last1=Little| first1=W. A.| last2=Parks| first2=R. D.| journal=Physical Review Letters| volume=9| issue=1| pages=9–12| bibcode=1962PhRvL...9....9L}}</ref> - One of the first{{citation needed|date=January 2018}} indications to the importance of the Cooper-pairing principle.