Making use of surface superconductivity measurements, it is possible to obtain relatively precise values for the Ginzburg-Landau parameter $\kappa$ as a function of temperature in certain type-I materials. We have measured the ac susceptibility to obtain $H_{c3\mathrm{}}$, and the dc magnetization to obtain the bulk critical field, for Ta and three dilute alloys of Bi in Pb. The data are sufficiently accurate to distinguish between the various existing theories for $\kappa$; the Ginzburg extension of the Ginzburg-Landau theory to low temperature predicts $\kappa \mathrm{}\left(t\right)=\kappa {\mathrm{}\left(0\right)\mathrm{}}_{\mathrm{G}-\mathrm{L}}{(1+t^2)}^{-1\mathrm{}}$, while Gorkov predicts $\kappa \mathrm{}\left(t\right)=\kappa {\mathrm{}\left(0\right)\mathrm{}}_{\mathrm{G}}(1-0.24\mathrm{}{t}^2+0.04\mathrm{}{t}^4)$, where $t$ is the reduced temperature $\frac{T}{T_c}$. We find that the data fall on smooth curves which lie between the two theoretical predictions. Bardeen has used the two-fluid model to generate a temperature-dependent set of equations for the free energy, which yields the Ginzburg-Landau equations in the limit of $t\to 1$. Using the Bardeen two-fluid formulation, we find $\kappa \mathrm{}\left(t\right)=\kappa {\mathrm{}\left(0\right)\mathrm{}}_{\mathrm{B}}{(1+t^2)}^{-\frac{1}{2}}$. We find that the data fit the two-fluid temperature dependence more closely than they fit either the Gorkov or the Ginzburg relationship.

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