  The frequency-dependent linear response of a plasma is studied in the
finite-temperature Thomas-Fermi approximation, with electron dynamics described
using Bloch hydrodynamics. The variational framework of average-atoms in a
plasma is used. Extinction cross-sections are calculated for several plasma
conditions. Comparisons with a previously studied Thomas-Fermi Impurity in
Jellium model are presented. An Ehrenfest-type sum rule, originally proposed in
a full quantum approach is derived in the present formalism and checked
numerically. This sum rule is used to define Bremsstrahlung and collective
contributions to the extinction cross-section. It is shown that none of these
is negligible. Each can constitute the main contribution to the cross-section,
depending on the frequency region and plasma conditions. This result obtained
in the Thomas-Fermi-Bloch case stresses the importance of the self consistent
approach to the linear response in general. Some of the methods used in this
study can be extended to the linear response in the quantum case.
