  We present an implementation of the Maxwell-Bloch (MB) formalism for the
study of x-ray emission dynamics from periodic multilayer materials whether
they are artificial or natural. The treatment is based on a direct
Finite-Difference-Time-Domain (FDTD) solution of Maxwell equations combined
with Bloch equations incorporating a random spontaneous emission noise. Besides
periodicity of the material, the treatment distinguishes between two kinds of
layers, those being active (or resonant) and those being off-resonance. The
numerical model is applied to the problem of $K\alpha$ emission in multilayer
materials where the population inversion could be created by fast inner-shell
photoionization by an x-ray free-electron-laser (XFEL). Specificities of the
resulting amplified fluorescence in conditions of Bragg diffraction is
illustrated by numerical simulations. The corresponding pulses could be used
for specific investigations of non-linear interaction of x-rays with matter.
