  X-ray free-electron lasers (XFELs) may allow to employ the single particle
imaging (SPI) method to determine the structure of macromolecules that do not
form stable crystals. Ultrashort pulses of 10 fs and less allow to outrun
complete disintegration by Coulomb explosion and minimize radiation damage due
to nuclear motion, but electronic damage is still present. The major
contribution to the electronic damage comes from the plasma generated in the
sample that is strongly dependent on the amount of Auger ionization. Since the
Auger process has a characteristic time scale on the order of femtoseconds, one
may expect that its contribution will be significantly reduced for attosecond
pulses. Here, we study the effect of electronic damage on the SPI at pulse
durations from 0.1 fs to 10 fs and in a large range of XFEL fluences to
determine optimal conditions for imaging of biological samples. We analyzed the
contribution of different electronic excitation processes and found that at
fluences higher than $10^{13}$-$10^{15}$ photons/$\mu$m$^2$ (depending on the
photon energy and pulse duration) the diffracted signal saturates and does not
increase further. A significant gain in the signal is obtained by reducing the
pulse duration from 10 fs to 1 fs. Pulses below 1 fs duration do not give a
significant gain in the scattering signal in comparison with 1 fs pulses. We
also study the limits imposed on SPI by Compton scattering.
