Two types of data were obtained from the critical field experiments on C4KHgx: Hc2 versus the angle theta at constant temperature, and Hc2 versus temperature at constant angle theta. These two varieties of curves correspond to different cross-sections of the three-dimensional phase boundary, Hc2(theta, T), which separates the superconducting and normal phases.
Hc2(theta) curves taken at about 0.44 K for two C4KHg GIC's are plotted in Figure .
Figure
a) shows data taken by Iye on a specimen with Tc = 0.73 K, along with results from an MIT sample with a Tc of
0.95 K. Figure
b) shows results from two MIT samples with
Tc's of 1.53 and 1.54 K. One prominent feature of the MIT data is
the rather large scatter in the critical field values near theta =
90° ( vecH _|_ ^c). This scatter is attributed to
error in reading the angular orientation during the experiments. (See
Section
for an estimate of the measurement errors.) The
scatter in Iye's data is much smaller, probably due to a more reliable
alignment method, perhaps including a servo-controlled gearing system for
rotating the sample. The papers of Iye et al. do not mention what
method of sample rotation was employed.
Figure: Critical field Hc2 as a function of
the angle theta for 4 C4KHg GIC's at T 0.4 K. Fits
(dotted lines) were calculated using Equation . a) Data for
an MIT C4KHg sample with Tc = 0.95 K (circ) and also
for a Tc = 0.73 K sample (bullet) from Ref. [240]. For
Tc = 0.95 K sample, 1/epsilon = 10.0 and Hc2(0°)
= 24 Oe with a residual cal R = 0.29. For data of Ref. [240],
1/epsilon = 11.3, Hc2(0°) = 26 Oe, and cal R = 0.090.
b) Data for two C4KHg samples with Tc 1.5 K.
(circ), Tc = 1.53 K with 1/epsilon = 10.2, Hc2(0°) = 46 Oe, and cal R = 0.73; bullet, Tc = 1.54
K with 1/epsilon = 9.5, Hc2(0°) = 47 Oe, and cal R
= 1.18.
The dotted lines in Figure are fits to the
equation:
where theta is the angle defined in Figure a) and
epsilon is the critical field anisotropy parameter of Morris, Coleman
and Bhandari,[175], defined by:
The origin of the Hc2(theta) formula[175,127]
and the physical interpretation of the epsilon parameter are discussed
in more detail in Sections and
. The residual cal R parameter referred to in
Figure
is here defined by:
where the errors are estimated as:
and nu is the number of free parameters == (the number
of data points) - (the number of parameters used in the fit plus one).
This definition of the residual is similar to that of the reduced chi2
in standard books on statistical analysis of data.[21]
However, in the absence of any knowledge of the actual magnitude of the
errors in the critical field measurements, the residual parameter should be
thought of merely as a convenient figure-of-merit for intercomparison of
various fits, rather than as an absolute measure of the appropriateness of
Eqn. .
Qualitatively, Equation describes the experimental
points well. Quick comparison of the fits with the points shows that the
fit is better for the data of the lower- Tc samples in
Figure
a) than for the higher- Tc ones in
Figure
b). While the data points in a) deviate almost
randomly from the fit, the data in b) are systematically higher than the
fits in the angular region around theta = 0°. One can
attempt to improve the situation by raising the parameter Hc2(0°), but since there are only two parameters to vary, the
net result is inevitably to worsen agreement in the wings or near the peak,
resulting in about the same value of cal R. The situation appears to be
similar for some of the data taken by Iye and Tanuma on other
superconducting GIC's. For example, Iye and Tanuma's C8RbHg data in
Ref. [120] showed systematic deviations from the fits to
Eqn.
. Several reasons come to mind as explanations for the
systematic deviations seen here; some of these factors are due to extrinsic
experimental influences, but others are intrinsic to the physics of layered
superconductors, as discussed below.