  We present supplementary information on the recent indirect measurement of
the Casimir pressure between two parallel plates using a micromachined
oscillator. The equivalent pressure between the plates is obtained by means of
the proximity force approximation after measuring the force gradient between a
gold coated sphere and a gold coated plate. The data are compared with a new
theoretical approach to the thermal Casimir force based on the use of the
Lifshitz formula, combined with a generalized plasma-like dielectric
permittivity which takes into account interband transitions of core electrons.
The theoretical Casimir pressures calculated using the new approach are
compared with those computed in the framework of the previously used impedance
approach and also with the Drude model approach. The latter is shown to be
excluded by the data at a 99.9% confidence level within a wide separation range
from 210 to 620 nm. The level of agreement between the data and theoretical
approaches based on the generalized plasma model, or the Leontovich surface
impedance, is used to set stronger constraints on the Yukawa forces predicted
from the exchange of light elementary particles and/or extra-dimensional
physics. The resulting constraints are the strongest in the interaction region
from 20 to 86 nm with a largest improvement by a factor of 4.4 at 26 nm.
