  We present a combined experimental and theoretical study of the evolution of
the low-temperature Fermi surface of lead telluride, PbTe, when holes are
introduced through sodium substitution on the lead site. Our Shubnikov-de-Haas
measurements for samples with carrier concentrations up to
$9.4\times10^{19}$cm$^{-3}$ (0.62 Na atomic $\%$) show the qualitative features
of the Fermi surface evolution (topology and effective mass) predicted by our
density functional (DFT) calculations within the generalized gradient
approximation (GGA): we obtain perfect ellipsoidal L-pockets at low and
intermediate carrier concentrations, evolution away from ideal ellipsoidicity
for the highest doping studied, and cyclotron effective masses increasing
monotonically with doping level, implying deviations from perfect parabolicity
throughout the whole band. Our measurements show, however, that standard DFT
calculations underestimate the energy difference between the L-point and
$\Sigma$-line valence band maxima, since our data are consistent with
occupation of a single Fermi surface pocket over the entire doping range
studied, whereas the calculations predict an occupation of the $\Sigma$-pockets
at higher doping. Our results for low and intermediate compositions are
consistent with a non-parabolic Kane-model dispersion, in which the L-pockets
are ellipsoids of fixed anisotropy throughout the band, but the effective
masses depend strongly on Fermi energy.
