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Superconductors

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Certain metals and alloys, when cooled to extremely low temperatures,
suddenly lose all their electrical resistance and become superconductors.
The temperature at which the transition to the superconducting state
occurs is referred to as the critical temperature, T_{c}. Superconductors
also exhibit remarkable magnetic behavior and they can have their
superconductivity destroyed by the application of a sufficiently large
magmetic field.
For many purpsoses a superconductor can be viewed as consisting of
two interpenetrating fluids. One of the fluids is the normal fluid
and is composed of normal conduction electrons obeying Fermi-Dirac
statistics and the other is the superfluid. According to the BCS theory,
the superfluid consists of pairs of electrons (Cooper pairs) of equal
but opposite momentum and spin. The pairing is between electrons
which lie near the Fermi energy of the normal metal and arises because
of a subtle interaction between these electrons and the lattice. The
resulting pairs are in an energy state lower then the Fermi energy by
an amount corresponding to the binding energy of the pair. It can be
shown from quantum theory that it is energetically favourable for all
the Cooper pairs to be in the same quantum state and have the same
momentum. This is possible because a Cooper pair having integral
(zero) spin can be thought of as a Bose particle. This means that the
only scattering process which can reduce the current flow in a
superconductor is one in which the pairs are broken up, a process
which requires at least an energy equal to the binding energy of the
pair. For low current desnsities there is not a way in which this
energy can be imparted to the pairs and consequently the specimen
has no resistance. As with ordinary particles it is possible to
represent the motion of a Cooper pair by a wavefunction whose
gradient determines the magnitude of the current flowing in the
superconductor. This wavefunction has the form of a travelling wave
whose phase coherence extends over indefinitely large distances.
When superconductivity is destroyed, for example by a large current
or magnetic field, this phase coherence is lost.