The London equations whose derivation appears elsewhere
(see [13] for instance), are local equations (i.e.
they relate the current density at a point
to the vector
potential
at the same point), and hence define the superconducting
properties as such. However, early discrepancies between experimental
estimations of
(0) (the penetration depth at zero temperature)
for certain conventional superconductors, and those predicted by
Eq. (2.1) led Pippard [13,15] in 1950 to introduce
non-local effects into the London equations. Spatial changes of
quantities such as *n*_{s} in a superconductor may only occur on a finite
length scale, the *coherence length*
,
and not
over arbitrarily small distances. That is to say, whenever *n*_{s} is
varying in space its value may change significantly over distances of
order
.
Thus the coherence length defines the intrinsic
nonlocality of the superconducting state.
Using the uncertainty principle,
Pippard estimated the coherence length for a pure metal to be
[17,18]:

where

In the London
model, the local value of the magnetic flux density
and the supercurrent density
are assumed to vary slowly
in space on a length scale
,
and
have negligible variation over distances
.
In Fig. 2.2, the superconducting carrier density *n*_{s}jumps discontinuously from zero to a maximum value at the sample surface.
As already mentioned, *n*_{s} is expected to make a significant change like
this only over a distance on the order of
.
Thus in Fig. 2.2,
,
so that one would anticipate an exponential screening of the
magnetic field in a real superconductor only
when
.
Indeed,
the London model is valid only for
.

In a type-I superconductor, the magnetic penetration depth
is
much less than the coherence length
,
so that *n*_{s} does not
reach its maximum value near the surface.
That is to say, not all of the
electrons within a thickness
from the surface of the
superconductor contribute to the screening currents.
Consequently, the penetration
of the magnetic field does not follow the exponential form of
Eq. (2.2) until one ventures an appreciable distance
into the superconductor. Thus the usual London model is
inadequate in describing type-I superconductors
where,
.
A modification of the London equations which gives the magnetic flux and
supercurrent densities a more rapid variation in space, predicts
the actual penetration depth
in a type-I
superconductor to be [14]:

(5) |

where is larger than , but much smaller than .

2001-09-28