  We have recorded dose-fractionated electron cryo-microscope images of thin
films of pure flash-frozen amorphous ice and pre-irradiated amorphous carbon on
a Falcon~II direct electron detector using 300 keV electrons. We observe Thon
rings \cite{Thon1966} in both the power spectrum of the summed frames and the
sum of power spectra from the individual frames. The Thon rings from amorphous
carbon images are always more visible in the power spectrum of the summed
frames whereas those of amorphous ice are more visible in the sum of power
spectra from the individual frames. This difference indicates that while
pre-irradiated carbon behaves like a solid during the exposure, amorphous ice
behaves like a fluid with the individual water molecules undergoing
beam-induced motion. Using the measured variation in the power spectra
amplitude with number of electrons per image we deduce that water molecules are
randomly displaced by mean squared distance of $\sim$ 1.1 \AA$^{2}$ for every
incident 300 keV e$^{-}$/\AA$^2$. The induced motion leads to an optimal
exposure with 300 keV electrons of 4.0 e$^{-}$/\AA$^2$ per image with which to
see Thon rings centred around the strong 3.7{\AA} scattering peak from
amorphous ice. The beam-induced movement of the water molecules generates
pseudo-Brownian motion of embedded macromolecules. The resulting blurring of
single particle images contributes an additional term, on top of that from
radiation damage, to the minimum achievable B-factor for macromolecular
structure determination.
