  We present measurement results for the gradient of the Casimir force between
an Au-coated sphere and an Au-coated plate obtained by means of an atomic force
microscope operated in a frequency shift technique. This experiment was
performed at a pressure of 3x10^{-8} Torr with hollow glass sphere of 41.3 mcm
radius. Special attention is paid to electrostatic calibrations including the
problem of electrostatic patches. All calibration parameters are shown to be
separation-independent after the corrections for mechanical drift are included.
The gradient of the Casimir force was measured in two ways with applied
compensating voltage to the plate and with different applied voltages and
subsequent subtraction of electric forces. The obtained mean gradients are
shown to be in mutual agreement and in agreement with previous experiments
performed using a micromachined oscillator. The obtained data are compared with
theoretical predictions of the Lifshitz theory including corrections beyond the
proximity force approximation. An independent comparison with no fitting
parameters demonstrated that the Drude model approach is excluded by the data
at a 67% confidence level over the separation region from 235 to 420 nm. The
theoretical approach using the generalized plasma-like model is shown to be
consistent with the data over the entire measurement range. Corrections due to
the nonlinearity of oscillator are calculated and the application region of the
linear regime is determined. A conclusion is made that the results of several
performed experiments call for a thorough analysis of the basics of the theory
of dispersion forces.
