In the following sections on the temperature dependence of
the upper critical field, the theories of choice for
explaining the data are those which include anisotropic gap
or multiband effects. Logical consistency demands that these
effects also be considered in attempts to explain the
anomalies in H_{c2}(*theta*).

Many theorists have calculated the effects of Fermi surface
anisotropy, gap anisotropy, or multiband contributions to
superconductivity on H_{c2}(T).[274,33,80,5] Unfortunately, most of these
theorists have confined their calculations to one or two
high-symmetry directions, so that a great deal of work
remains to extract H_{c2}(*theta*) from their
equations.

The only model to explicitly consider the case of a general
field orientation is that developed by K. Takanaka.[238] This model incorporates
both Fermi surface anisotropy and gap anisotropy, and is
discussed further in Section .
Takanaka gives a full expression for
H_{c2}(*theta*, T) which could be used to fit
the H_{c2}(*theta*) data. Takanaka's model was
withdrawn, though, because of the unphysical divergence of
H_{c2}(t) below about t = 0.7.[114] While Youngner and Klemm[274] calculated a corrected
H_{c2}(T) that utilized some of Takanaka's ideas,
they did not produce an improved H_{c2}(*theta*)
equation.

Despite the unphysical behavior of Takanaka's model at low
temperatures, one might consider fitting it to the
experimental data at high reduced temperatures. Takanaka does
show H_{c2}(*theta*) curves in Ref. [238] for t = 0.9 which
incorporate gap anisotropy effects. Toyota *et al.*[253] fit their
NbSe_{2} data to the Takanaka equation, but it is not
obvious that the Takanaka curve describes their data better
than the AGL theory. Furthermore, if anisotropy-induced
anomalies are responsible for the deviations of the
C_{4}KHg H_{c2}(*theta*) data from the
AGL model, one would expect the deviations to be greatest at
the lowest reduced temperatures.[274] Figure
shows that the deviations are in fact greater at higher
reduced temperatures, where the simple AGL model should be
most applicable. The reason is that the assumptions that go
into Ginzburg-Landau theories are justifiable only near
T_{c},[252] so if
a more microscopic model is needed to fit the data, it should
describe low-temperature anomalies. To put the matter into
common-sense terms, one would expect anisotropy-derived
deviations in H_{c2}(*theta*) in the same
temperature range where the anisotropy is causing anomalies
in H_{c2}(t), *i.e.*, at low reduced
temperatures. The larger deviations from the AGL theory at
higher reduced temperatures is therefore in conflict with
expectations from anisotropic-gap and multiband models, but
consistent with type I behavior for small *theta*, as
discussed in Section .

The general problem with trying to fit more microscopic
models to the H_{c2}(*theta*) curves is that
many parameters (tilt, mosaic spread, H_{c},
*epsilon*, H_{c2}(0°)) have already been
identified as being relevant. 5 parameters is already a large
number for slightly noisy curves of about 40 or 50 points.
Considering that the situation is somewhat simpler with the
H_{c2}(T) data, it is hardly surprising that not many
detailed theories of H_{2}(*theta*) have
appeared.

Since the H_{c2}(*theta*) curves deviate from
the AGL theory near T_{c} and the H_{c2}(T)
curves deviate from the AGL theory near T=0, it does not seem
too far-fetched to attribute different causes to these
anomalies. With this decoupling in mind, the most convincing
explanation for the shape of the H_{c2}(*theta*)
curves in C_{4}KHg would seem to be that based on
type I superconductivity near *vec*H || ^c. This
explanation is embodied by Eqn. ,
which allows for type I behavior. The anomalies in
H_{c2}(T) are discussed in the section that follows.

alchaiken@gmail.com (Alison Chaiken)

Wed Oct 11 22:59:57 PDT 1995