H_{c2}(T) data for a T_{c} = 1.5 K
C_{4}KHg are shown in Figs. 5 and 6. The data were
taken with the applied field both parallel (Fig. 5) and
perpendicular (Fig. 6) to the c-axis of the graphene planes.
Two fits are plotted along with each set of data. With the
data are plotted a linear and a quadratic fit.
If the hypothesis discussed in the previous section about
type I superconductivity for is correct, then the
data should be approximately quadratic rather
than linear. The residuals were calculated using Eq.
2, only now with the error of the **i**th data
point, *sigma*_{i}, estimated to be 10% of the
experimentally measured critical field. The linear
temperature dependence gives a lower residual than the
quadratic fit, but the difference between the two residuals
is not statistically significant.

Another check on the fits is to look at the
magnitude of T_{c} obtained from the extrapolation of
H_{c2} to zero. As expected, the T_{c}
obtained from extrapolation of the linear fits
usually agrees well with the T_{c} value measured in
zero-field temperature sweeps. In contrast, extrapolation of
the linear fits to H_{c2} = 0 consistently
gives K for specimens whose T_{c} was 1.5 K
according to zero-field temperature sweeps.[10] On the other hand,
extrapolation of the quadratic fits to = 0
always results in T_{c} about 1.5 K, in agreement
with the zero-field sweeps. At this point the functional form
of remains unresolved.

In Fig. 7, one of the fits is a simple straight line, while
the other is calculated from the Helfand-Werthamer (HW)
equation, which describes the temperature dependence of the
upper critical field of a typical type II superconductor.[26,47] For the T_{c} =
1.5 K samples, the residuals for the linear fit are
consistently at least a factor of 2 lower than those for the
Helfand-Werthamer fit. Thus we conclude that
H_{c2}(T) for the T_{c} = 1.5 K pink
C_{4}KHg samples exhibits extended linearity. The
question of extended linearity cannot be decided for the
lower-T_{c} mixed-phase samples because their data
(not shown) extends over a smaller reduced temperature range
due to the limitations imposed by the cryogenic apparatus.

The most convincing demonstration of the deviation of the
H_{c2}(T) data above the typical type II
superconductor curve is displayed in Fig. 8. By plotting the
reduced field, versus reduced temperature t (== T/
T_{c}), all the H_{c2}(T) data for 5
different C_{4}KHg samples can be displayed together.
The residual index for the linear fit is 3/4 that of the HW
fit, indicative of about a 90% probability that the linear
fit describes the data better.[6] Figure 8 also shows that at
(the lowest accessible reduced temperature)
the data have already reached 0.7, the
zero-temperature value of h^{*} calculated using the
Helfand-Werthamer formalism.[26] Therefore, while it would be
interesting to measure H_{c2}(t) to lower reduced
temperatures and see larger deviations from the HW theory,
the available data convincingly demonstrate a deviation from
the HW functional form.

If the critical field is truly linear for both field
orientations, then the anisotropy parameter *epsilon*
should be a constant:

1/*epsilon* = dH_{c2, perp c}/dT/dH_{c2,
par c}/dT

If *epsilon* is temperature-independent, the only
temperature-dependent quantity left in Eq. 1 should be
H_{c2}(0°). Therefore plots of
H_{c2}(*theta*) / H_{c2}(0)°
versus *theta* should lie directly on top of one
another, since all temperature dependence has presumably been
removed. A plot of H_{c2}(*theta*) /
H_{c2}(0)° is shown in Fig. 9 for a T_{c}
= 1.5 K C_{4}KHg sample at three different reduced
temperatures. As the Figure shows, the curves for t = 0.76
and t = 0.55 do indeed lie on top of one another, implying a
nearly temperature-independent anisotropy in this temperature
range. However, the curve for t = 0.29 clearly lies above the
other two curves near *theta* = 90°, which
suggests a small increase in 1/*epsilon* at low
temperatures. A similar plot of the
H_{c2}(*theta*) data of Iye and Tanuma[30] on C_{8}KHg shows
that *epsilon* is also temperature-dependent in the
stage 2 KHg-GIC.[9,10] This increase in
*epsilon* could be caused either by a slight upturn
from linearity in (positive curvature) or a slight
downturn from linearity in . Since is
expected to be quadratic for type I behavior, the second
alternative seems more consistent with our experience with
the H_{c2}(*theta*) fits. It should be noted,
however, that strong positive curvature of has been
observed by Iye and Tanuma[30] in C_{8}RbHg.