Several objections may be raised to the type I
superconductivity hypothesis; firstly, the value of
H_{c} obtained from the t = 0.57
H_{c2}(*theta*) fit in Fig. 4 (43 gauss) is
considerably lower than that predicted from the specific heat
(75 gauss). The ratio between the value of H_{c}
calculated from the specific heat measurements and the
H_{c} obtained from the
H_{c2}(*theta*) fits is about 1.8 for two
T_{c} = 1.5 K specimens and the three temperatures
where H_{c2}(*theta*) measurements were
performed. A second problem with the type I explanation is
that the H_{c2}(*theta*) data are not
perfectly flat near *theta* = 0°, although
H_{c} should be absolutely constant as a function of
angle. Both of these discrepancies may be caused by
angle-dependent demagnetization effects, which are hard to
account for quantitatively.[17]

In order to extract believable values for H_{c} from
the H_{c2}(*theta*) fits, demagnetization
corrections need to be applied. The angular dependence of the
demagnetization factor of an oblate spheroidal superconductor
has been worked out by Denhoff and Gygax.[17] However, the application of
this formula to the C_{4}KHg data is not justified
since too many parameters which are poorly known (dimensions
of the intercalated sample) or not independently known (
and *epsilon*) are involved. Therefore the
demagnetization for the C_{4}KHg samples has been
estimated only for the high-symmetry directions by
approximating the sample shape as ellipsoidal.

The samples used in the H_{c2}(*theta*)
measurements were flat plates with typical dimensions on the
order of (2 mm × 2 mm × 0.5mm). Consultation of
standard tables[43] shows
that a demagnetization correction of about 3 is anticipated
for an ellipsoid with radii in the ratios 1:1:0.25 when the
field is applied along the perpendicular to the smallest
dimension. If this correction is applied to the H_{c}
values determined from the H_{c2}(*theta*)
fits, one obtains H_{c}(0) 203 gauss, which is
now a factor of 1.8 *higher* than the H_{c}(0)
of 112 gauss calculated from the specific heat data. Since
and , a higher H_{c} for the
pink T_{c} = 1.5 K specimens implies that they have a
higher density of states N(0) than the gold specimens, which
is in keeping with their higher T_{c}.

Now that an estimate for the true value of H_{c} has
been obtained, it is interesting to re-examine Fig. 4. marks the corrected value of the thermodynamic
critical field at **t** = 0.57. The implication
of this H_{c} estimate is that C_{4}KHg is
type I in the approximate range *theta* < 80 °,
and type II for *theta* > 80 °. Magnetization
measurements on C_{4}KHg are desirable to positively
identify type I superconductivity. Specific heat measurements
on T_{c} = 1.5 K pink C_{4}KHg specimens
would also be useful.

The extended linearity of H_{c2}(T) reported here for
T_{c} = 1.5 K pink C_{4}KHg specimens is not
surprising in light of previous studies of layered
superconductors. Positive curvature or extended linearity of
H_{c2}(T) are phenomena common to almost all
anisotropic superconducting compounds.[59] Depending on the specific
type of superconductor in question, many different
explanations might be considered for the anomalous
temperature dependence of H_{c2}. For example,
mechanisms ranging from coupling-dimensionality crossover[35] to
proximity-effect-induced curvature[7] to magnetoresistive anomalies[16] have been cited as the
cause of positive curvature in various superconductors.
However, each of these esplanations for its own reasons seems
to be inappropriate for GIC's.[10] In order to choose an
appropriate model from the many that are available, the best
procedure is to consider what is already known about GIC's.

There are two types of models of anisotropic
superconductivity that seem to contain the right features for
the GIC superconductors: these are the anisotropic Fermi
surface models[8,61] and the multiband
superconductivity models.[22,2] Anisotropic Fermi surface models
are an obvious choice because the quasi-2D band structure of
graphite is responsible for many of the characteristic
properties of GIC's. The simplest anisotropy-based model
available is that developed by Butler[8] to fit H_{c2}(T) in Nb.
This model was adapted by Dalrymple and Prober to fit the
critical fields of NbSe_{2}, which show extended
linearity and positive curvature[13] similar to that seen in
GIC's. The NbSe_{2} Fermi surface has cylindrical
pieces at the hexagonal Brillouin zone boundary, and bears a
great deal of resemblance to the proposed Fermi surface of
many of the GIC's.[28,18] The band structure of
C_{4}KHg has been calculated,[27] but unfortunately not
enough quantitative information about the Fermi surface has
been reported to allow a detailed comparison between the
Butler model and the data for C_{4}KHg.

Another well-established feature of the GIC's Fermi surface
is the presence of multiple bands. Some of the bands at the
Fermi surface of graphitic origin are nearly two-dimensional
in character. Other bands, which may be either of intercalant
or graphitic origin, are more 3D in character. In
C_{4}KHg these 3D bands are derived from hybridized K
and Hg levels.[27]
Models for H_{c2}(T) which incorporate the
participation of two bands in the superconductivity therefore
appear to be a logical choice for GIC's. Entel and Peter have
fit H_{c2}(T) data for
Cs_{0.1}WO_{2.9}F_{0.1}, a tungsten
fluoroxide bronze, using a two-band Fermi surface model.[22] Al-Jishi[3] has proposed a similar model
specifically tailored to fit critical field data on
C_{8}K, although calculations using this model are
still preliminary. More theoretical work is needed to
interpret the large body of critical field data already
available.[30,9]

The idea that both graphitic and intercalant bands
participate in GIC superconductivity is sensible for two
reasons. One reason is that there are superconducting GIC's
(specifically the binary compounds C_{8}K (Ref. [37]) and C_{8}Rb (Ref.
[36])) which are
synthesized from non-superconducting starting materials.
Since the KHg-GIC's are synthesized from KHg amalgams which
are themselves superconducting, it might be thought that this
argument for multiband superconductivity is invalid for them.
However, the increase of T_{c} with stage in the
KHg-GIC's is contrary to expectations of the superconducting
proximity effect[57,14] if the carbon layers
are not also participating in the superconductivity. The
large critical field anisotropy seen in GIC's is also
difficult to explain if graphitic electrons are not
involved.[49,2] If the graphitic electrons are
implicated in GIC superconductivity, then it follows that
speculation about a coupling-dimensionality crossover in
KHg-GIC's[30] due to
decoupling of the intercalant layers is not sensible. If the
graphitic bounding layers are also superconducting, then it
may be their separation which will determine a
coupling-dimensionality change. Considerations of this type
may be important in discussion of dimensionality change in
the high-T_{c} ceramic superconductors.

It is reasonable to expect a common origin for the extended
linearity of H_{c2}(T) and the deviations from Eq. 1
in the H_{c2}(*theta*) fits. However, the
deviations in H_{c2}(T) from the HW theory occur at
the lowest obtainable reduced temperatures. The
H_{c2}(*theta*) fits to Eq. 1, on the other
hand, worsen as the temperature is increased toward
T_{c}.[10]
Worsening agreement between Eq. 1 and the data as temperature
is increased is expected if type I superconductivity is the
origin of the discrepancy. This behavior has already been
observed for TaN.[56] All
the available evidence therefore points to type I
superconductivity as the explanation of the poor quality of
the fits shown in Fig. .

It was not necessary to make allowance for type I character
to fit the H_{c2}(*theta*) curves of the gold,
mixed-phase samples, a finding which is consistent with the
observations of Iye and Tanuma.[30] Type II character for all
orientations of low-T_{c} C_{4}KHg is due to
the fact that the T_{c} = 0.8 K samples have critical
fields almost as high as the T_{c} = 1.5 K samples.
To be more specific, the typical value of for a
T_{c} = 1.5 K C_{4}KHg specimen is only about
750 gauss, while that for the T_{c} = 0.8 K specimens
is about 650 gauss.[30] The
critical field slope at T_{c} in the HW model is
dH_{c2}/ dT , = , 4 k_{B} / *pi* e D,
where k_{B} and e are the usual fundamental constants
and D = v_{F} *l*/3 is the diffusivity.[42,47] Therefore the higher
dH_{c2}/dT in the mixed-phase samples suggests that
they have either a lower mean-free path *l* or a
smaller Fermi velocity v_{F}. The greater in-plane
disorder of the mixed-phase samples would appear to favor the
lower mean-free path explanation.