  Inner-shell K$\alpha$ x-ray lasers have been created by pumping gaseous,
solid, and liquid targets with the intense x-ray output of free-electron-lasers
(FELs). For gaseous targets lasing relies on the creation of K-shell core-holes
on a time-scale short compared with filling via Auger decay. In the case of
solid and liquid density systems, collisional effects will also be important,
affecting not only populations, but also line-widths, both of which impact the
degree of overall gain, and its duration. However, to date such collisional
effects have not been extensively studied. We present here initial simulations
using the CCFLY code of inner-shell lasing in solid density Mg, where we
self-consistently treat the effects of the incoming FEL radiation and the
atomic kinetics of the Mg system, including radiative, Auger, and collisional
effects. We find that the combination of collisional population of the lower
states of the lasing transitions and broadening of the lines precludes lasing
on all but the K$\alpha$ of the initially cold system. Even assuming
instantaneous turning on of the FEL pump, we find the duration of the gain in
the solid system to be sub-femtosecond.
