  The Casimir force is calculated analytically for configurations of two
parallel plates and a spherical lens (sphere) above a plate with account of
nonzero temperature, finite conductivity of the boundary metal and surface
roughness. The permittivity of the metal is described by the plasma model. It
is proved that in case of the plasma model the scattering formalism of quantum
field theory in Matsubara formulation underlying Lifshitz formula is well
defined and no modifications are needed concerning the zero-frequency
contribution. The temperature correction to the Casimir force is found
completely with respect to temperature and perturbatively (up to the second
order in the relative penetration depth of electromagnetic zero-point
oscillations into the metal) with respect to finite conductivity. The
asymptotics of low and high temperatures are presented and contributions of
longitudinal and perpendicular modes are determined separately. Serving as an
example, aluminium test bodies are considered showing good agreement between
the obtained analytical results and previously performed numerical
computations. The roughness correction is formally included and formulas are
given permitting to calculate the Casimir force under the influence of all
relevant factors.
