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Model for the Internal Field Distribution
The SR data is fitted to a London model [11] that incorporates
the effects of nonlocality though a kernel
Q(k), an extension
described in Section 2.1. As mentioned in that section,
the London model is inapplicable within a vortex core. Despite this the
theory can yield the vortex lattice structure belonging to high superconductors, such as LuNi_{2}B_{2}C, under weak external fields
,
because in these situations the intervortex spacing greatly surpasses
the diameter of
a core [10]. LuNi_{2}B_{2}C exhibits both square and triangular vortex
structures according to the field
H applied, as discussed in
Chapter 3. This nonlocal
theory is chosen to fit the SR data since it accounts for an observed
vortex lattice transformation between these two geometries in
LuNi_{2}B_{2}C, and because the data are collected under low field (
)
conditions.
Expanding [11] the BCS kernel
Q(k), for weak currents
J,
to first order in the small term
leads to the internal spatial
field profile
B(r) of a nonlocal London model:

(7.1) 
where the coordinates (x, y, z) coincide with the crystal frame
(a, b, c) and the external field
H is in the
direction. The average magnetic field inside the superconducting sample
is
.
The constant coefficients 0.0705 and 0.675 arise
from the evaluation of Fermi surface averages of products of Fermi
velocities according to LuNi_{2}B_{2}C band structure. The Gaussian cutoff
factor
in the sum over the reciprocal lattice
vectors
k of the vortex lattice compensates for the
failure of the London approach within the core region.
This model neglects vortexvortex interactions.
The parameter C reflects the strength
of the nonlocal effects, and contains several poorly known factors.
The nonlocality parameter C varies theoretically with temperature T
as [11]

(7.2) 
where
is the nonlocality radius, on the order of the BCS
coherence length [12]. Figure 5.1 shows
the numerically calculated behaviour of the penetration depth
,
the nonlocality radius
and the nonlocality parameter C(T)as functions of temperature.
The relation [12]

(7.3) 
where
are the Matsubara frequencies
and n is a nonnegative integer, gives the expected
temperature dependence of the penetration depth
in the clean
limit. Solving [13] the equation

(7.4) 
determines the BCS energy gap ,
where
is the Debye energy, N(0) is the density of states at the
Fermi level for electrons of one spin orientation, and V describes the
strength of the interaction potential for scattering a Cooper pair.
Evaluating the above integral at temperature T = T_{c} reveals the
constant N(0)V to be
where
is Euler's constant. The calculations plotted
in Figure 5.1 assume a critical temperature
and a Debye temperature
,
values appropriate for LuNi_{2}B_{2}C. The expression [12]

(7.6) 
generates the clean limit behaviour of the nonlocality radius
shown in this
figure. The above sums over
converge by
.
Combining the temperature dependence
of the penetration depth
and the nonlocality radius
according
to (5.3) produces the temperature variation of the internal field
model parameter C, displayed in Figure 5.1.
The next section describes the software utilised to fit this
internal field model to the experimental data.
Next: Fitting Software
Up: Analysis
Previous: Data Fitting in the Time Domain
Jess H. Brewer
20011031