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#

Results and Discussion

The nonlocal model outlined in Section 5.2 fitted the SR
data on LuNi_{2}B_{2}C much better than the local London model used in combination with
a triangular vortex lattice [51]. This reflects the presence of a
highly ordered square flux lattice in LuNi_{2}B_{2}C at temperatures *T*
between
and
,
achieved through
field cooling under an
field applied parallel to the
crystal axis. The behaviour of the
penetration depth ,
the nonlocality parameter *C* and the core
radius
in LuNi_{2}B_{2}C under these field and temperature conditions
emerges readily from the fitted parameters.

Figure 6.1(a) displays the behaviour of the fitted penetration
depth
with temperature *T* under an external
field
.
The fitted penetration depth
increases
from
to
over the temperature *T* range from
to
.
The solid curve depicts the
temperature dependence anticipated from relation (5.4) and plotted in
Figure 5.1. Choosing a zero temperature penetration
depth
to optimally fit this BCS temperature variation to the
measured penetration depth
indicates a zero temperature penetration
depth
.
The observed penetration depth appears
fairly consistent with the exponentially-activated form expected for BCS
electron-phonon
coupling and *s*-wave order parameter symmetry in LuNi_{2}B_{2}C. However, the error
bar size also permits the interpretation of the penetration depth temperature
dependence
as a power law, as would be expected [7] if
nodes existed in the energy gap. The
penetration depth grows much more weakly with temperature in LuNi_{2}B_{2}C than in
YBa_{2}Cu_{3}O_{6.95} [52], known to possess an energy
gap with line nodes. While the penetration depth
measurements seem as
anticipated, the fitted nonlocality parameter C appears a little more
surprising.

Figure 6.1(b) depicts the variation of the fitted nonlocality
parameter *C* with
temperature *T* under an
applied field. As described in
Section 5.5, reasonable fits to the data require a nonzero *C*,
demonstrating the importance of nonlocal effects in LuNi_{2}B_{2}C. The solid curve in
Figure 6.1(b) is a fit to the predicted temperature
dependence (5.3) in the clean limit. As expected, the best fit
values of the nonlocality parameter *C* are fairly uniform at low temperatures.
At higher temperatures the fitted nonlocality parameter *C* changes
little, rather than dropping
monotonically as anticipated. Possible explanations for this slight discrepancy
include the presence of impurities and the approximate nature of the field
distribution model (5.2). Figure 5.1 illustrates the
behaviour of parameter *C* with temperature, as influenced by the
nonlocality radius ,
when the LuNi_{2}B_{2}C sample is completely free of
impurities. As the impurity level rises the nonlocality radius
becomes temperature independent [12], leading to a weaker temperature
dependence in parameter *C* that better agrees with that observed.
In fact, the constancy of the fitted nonlocality parameter *C*over the studied temperature range agrees with recent small angle neutron
scattering
(SANS) data [53] that shows the onset field *H*_{2} of the square to
hexagonal vortex lattice symmetry transition essentially holds constant below
temperature
.
Closer agreement between the anticipated and fitted temperature dependences of
parameter *C* might also result from the inclusion of higher
order nonlocal terms in expression (5.2) for the spatial field
profile
**B**(*r*). Refitting the data with the nonlocality parameter *C*
fixed to its expected clean limit temperature variation (5.3) induces
negligible change in the remaining fitted parameters and the inferred core
radius .

**Table:**
Fitted penetration depth ,
effective coherence length
and nonlocality parameter *C* for the field
distributions *n*(*B*) plotted in Figure 6.2.
London model |
Temperature *T* (
**K**) |
(Å) |
(Å) |
*C* |

Nonlocal |
**10** |
**1100(80)** |
57(1) |
0.25(9) |

Nonlocal |
**2.6** |
**940(30)** |
28(2) |
0.17(4) |

Local |
**2.6** |
**1494(9)** |
41(1) |
0 |

Figure 6.2 compares the internal magnetic field
distributions *n*(*B*) calculated for the fitted spatial field
profile (5.2) at temperatures
and
under an
external field.
The best fit values of the penetration depth ,
the effective coherence
length
and the nonlocality parameter *C* for the distributions *n*(*B*)
shown in this figure appear in Table 6.1. The
shape of the nonlocal field distributions *n*(*B*) differs qualitatively from
that associated with the traditional London model, as illustrated at
temperature
in the inset to this figure.
The distinct difference between these shapes explains the vast improvement in
fit quality the nonlocal London model (5.2) achieves over the
conventional one, as evident in Figure 5.7.
The basic London model emerges from the nonlocal model when the nonlocality
parameter *C* = 0,
and, in the case of a square vortex lattice, generates a characteristic low
field shoulder in the field distribution *n*(*B*). The nonlocal corrections
greatly reduce the spectral weight of this shoulder, to the point where at
temperature
the shoulder almost vanishes. They also
give rise to the small peak appearing at the lowest field *B* in the
distribution *n*(*B*). Such a peak is absent in the local London model, and its
presence reflects a flatter spatial field profile at the centre of the
square unit cell with a vortex in each corner. The clear rise in the maximum
field *B* of the distribution *n*(*B*) as the temperature *T* falls
from
to
reflects the shortening of
the vortex core radius .

Figure 6.3 compares the temperature dependence of the core
radius
measured in LuNi_{2}B_{2}C at external field
to that reported for NbSe_{2} [6] (
), YBaCu_{3}O_{6.95} [7]
(
)
and YBaCu_{3}O_{6.60} (
)
at
.
As the reduced temperature *T* / *T*_{c} rises from *T*/*T*_{c} =
0 to
,
the core radius
in YBaCu_{3}O_{6.95}and YBaCu_{3}O_{6.60} remains almost constant, consistent with the
attainment of the quantum limit as discussed in Section 2.4. In
NbSe_{2} the quantum limit temperature *T*_{0} occurs around
,
above which the core radius
expands linearly with temperature *T*, at a
rate much slower than anticipated from the proposed Kramer-Pesch effect. The
Kramer-Pesch effect in LuNi_{2}B_{2}C appears equally weak, with no low
temperature saturation in its core size evident over the temperature interval
studied. Overlapping the LuNi_{2}B_{2}C data with that for NbSe_{2}determines the zero temperature core radius
for LuNi_{2}B_{2}C to be
,
greatly exceeding the predicted value
.
The relation
,
displayed as a solid line in Figure 6.3 with
the zero temperature core radius
set to
,
best fits the LuNi_{2}B_{2}C data for slope
and quantum limit
temperature
.
This agrees with the expected quantum
limit temperature
calculated in
Section 3.2, and is similar to that recorded for NbSe_{2}.
The almost identical reduced temperature *T* / *T*_{c} dependence of the core
radius
in LuNi_{2}B_{2}C and NbSe_{2} suggests that
quasiparticles in these materials behave in the same way as a function of
reduced temperature *T* / *T*_{c}, despite the different dimensionalities of these
superconductors. It also implies that longitudinal disorder of vortices has
negligible effect on SR determinations of the core radius .
As with SR studies of the Kramer-Pesch effect in
NbSe_{2} and YBaCu_{3}O
,
the weak core shrinkage upon cooling and
order-of-magnitude larger than expected zero temperature core radius
observed in LuNi_{2}B_{2}C highlight the need to include interactions between
vortices and their zero point motion in theories concerning the Kramer-Pesch
effect.

** Next:** Conclusions
** Up:** Astria Price's M.Sc. Thesis, Oct. 2001
** Previous:** Fit Quality
*Jess H. Brewer *

2001-10-31