Two types of data were obtained from the critical field
experiments on C_{4}KHg_{x}: H_{c2}
versus the angle *theta* at constant temperature, and
H_{c2} versus temperature at constant angle
*theta*. These two varieties of curves correspond to
different cross-sections of the three-dimensional phase
boundary, H_{c2}(*theta*, T), which separates
the superconducting and normal phases.

H_{c2}(*theta*) curves taken at about 0.44 K for
two C_{4}KHg GIC's are plotted in Figure .
Figure a) shows data taken by
Iye on a specimen with T_{c} = 0.73 K, along with
results from an MIT sample with a T_{c} of 0.95 K.
Figure b) shows results from two
MIT samples with T_{c}'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° (
*vec*H _|_ ^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 H_{c2} as a
function of the angle *theta* for 4 C_{4}KHg
GIC's at T 0.4 K. Fits (dotted lines) were calculated using
Equation . a) Data for an MIT
C_{4}KHg sample with T_{c} = 0.95 K
(*circ*) and also for a T_{c} = 0.73 K sample
(*bullet*) from Ref. [240]. For T_{c} = 0.95 K
sample, 1/*epsilon* = 10.0 and H_{c2}(0°) =
24 Oe with a residual *cal* R = 0.29. For data of Ref.
[240], 1/*epsilon* =
11.3, H_{c2}(0°) = 26 Oe, and *cal* R =
0.090. b) Data for two C_{4}KHg samples with
T_{c} 1.5 K. (*circ*), T_{c} = 1.53 K
with 1/*epsilon* = 10.2, H_{c2}(0°) = 46 Oe,
and *cal* R = 0.73; *bullet*, T_{c} = 1.54
K with 1/*epsilon* = 9.5, H_{c2}(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 H_{c2}(*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 *chi*^{2} 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- T_{c} samples in Figure
a) than for the higher- T_{c} 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
H_{c2}(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
C_{8}RbHg 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.

- Extrinsic Influences on Hc2(theta) in Anisotropic Superconductors
- Type I Superconductivity in C4KHg
- Type II/1 Superconductivity and C4KHg
- Tinkham's Formula Fits to Hc2(theta) in C4KHg
- Microscopic Models of Superconductivity and Hc2(theta) in C4KHg

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995