  We present results from a quantitative spectroscopic analysis conducted on
archival Keck/HIRES high-resolution spectra from the California-$Kepler$ Survey
(CKS) sample of transiting planetary host stars identified from the $Kepler$
mission. The spectroscopic analysis was based on a carefully selected set of Fe
I and Fe II lines, resulting in precise values for the stellar parameters of
effective temperature (T$_{\rm eff}$) and surface gravity (log $g$). Combining
the stellar parameters with $Gaia$ DR2 parallaxes and precise distances, we
derived both stellar and planetary radii for our sample, with a median internal
uncertainty of 2.8$\%$ in the stellar radii and 3.7$\%$ in the planetary radii.
An investigation into the distribution of planetary radii confirmed the bimodal
nature of this distribution for the small radius planets found in previous
studies, with peaks at: $\sim$1.47 $\pm$ 0.05 R$_{\oplus}$ and $\sim$2.72 $\pm$
0.10 R$_{\oplus}$, with a gap at $\sim$ 1.9R$_{\oplus}$. Previous studies that
modeled planetary formation that is dominated by photo-evaporation predicted
this bimodal radii distribution and the presence of a radius gap, or
photo-evaporation valley. Our results are in overall agreement with these
models. The high internal precision achieved here in the derived planetary
radii clearly reveal the presence of a slope in the photo-evaporation valley
for the CKS sample, indicating that the position of the radius gap decreases
with orbital period; this decrease was fit by a power law of the form R$_{pl}$
$\propto$ P$^{-0.11}$, which is consistent with photo-evaporation and
Earth-like core composition models of planet formation.
