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To explain numerous experimental results which
deviated from the predictions of the London theory, Pippard [28]
proposed the following general nonlocal relation for the supercurrent
response

| |
(29) |

where , is the coherence length and is an effective
coherence length related to the electron mean-free path *l*
through the equation
| |
(30) |

where is a constant on the order of unity. For a pure
superconductor, .The response of the superconductor to the applied magnetic field is
nonlocal, in the sense that the value of measured at a point depends on the value of throughout a volume of radius surrounding the point .The Pippard kernel relating the kth Fourier component of to
the vector potential can be determined
from Eq. (2.30), and is given by
| |
(31) |

*Q*_{P} (*k*) is always smaller than the London kernel *Q*_{L} (*k*). As a
result, substituting the expression for *Q*_{P} (*k*) into Eq. (2.5)
will always yield a value for which is larger than .In particular, for
| |
(32) |

and for
| |
(33) |

Note that in the first limiting case, agrees with
the London prediction in a pure superconductor. On the other hand, the
second limiting case is completely independent of the electron mean-free
path. A superconductor described by the first equation, is called a
``type-II superconductor'', whereas one that is described
by the second equation is a ``type-I superconductor''.
Using an argument based on the uncertainty principle, Pippard estimated that
the coherence length in a pure metallic superconductor is

| |
(34) |

where *v*_{f} is the Fermi velocity and .In BCS theory, the response kernel is similar to
that in Pippard theory. The BCS coherence length
, is the range of the Fourier transform of and is defined as
| |
(35) |

where is the uniform energy gap at . Since
, the Pippard estimate is close to the
BCS coherence length.
In a conventional superconductor, *v*_{f} and hence is large.
The bound electron pairs which make up the superfluid have a
spatial extent on the order of .
The high-*T*_{c} materials differ markedly from conventional superconductors
in that they have much smaller coherence lengths. Consequently, these
materials are in the extreme type-II limit.
The small value of the coherence length also
means that the high-*T*_{c} compounds are generally in the clean limit, where
. Furthermore, both
fluctuation and boundary effects are much stronger in these short
superconductors.
In a *d*_{x2-y2}-wave superconductor where the energy gap is
anisotropic, one must define an angle dependent coherence length

| |
(36) |

where and is the maximum value of the energy gap.
The divergence
of Eq. (2.37) along the node directions
means that the extreme nonlocal
limit is obtained near the nodes (*i.e.* ).
Recently, Kosztin and Legget
[29] determined that nonlocal electrodynamics leads to
a crossover from a *T* to a *T*^{2} dependence for the penetration depth
in the Meissner state at extremely low temperatures. To observe this
experimentally, one must distinguish this effect from the *T*^{2}
dependence expected from sample impurities. Both the
temperature and magnetic field dependence of will be discussed later in
the thesis.

** Next:** 2.3 The Magnetic Field
** Up:** 2 The Characteristic Length
** Previous:** 2.1.3 Penetration Depth for d_{x-y}-Wave