  Polycrystalline samples of underdoped (Ba1-xKx)Fe2As2 (x<=0.4) were
synthesized and studied by x-ray powder diffraction, magnetic susceptibility,
specific heat and 57Fe-Moessbauer-spectroscopy. The structural phase transition
from tetragonal to orthorhombic lattice symmetry shifts towards lower
temperatures, becomes less pronounced at x = 0.1-0.2 and is no longer present
at x = 0.3. Bulk superconductivity is observed in all samples except
(Ba0.9K0.1)Fe2As2 by resistivity and magnetic susceptibility measurements.
Specific heat data show a broad SDW phase transition in (Ba0.9K0.1)Fe2As2,
which is hardly discernible in (Ba0.8K0.2)Fe2As2. No SDW anomaly is found in
the specific heat of optimally doped (Ba0.6K0.4)Fe2As2, where C changes by 0.1
J/K at Tc = 37.3 K. 57Fe-Moessbauer-spectra show full magnetic hyperfine field
splitting, indicative of antiferromagnetic ordering at 4.2 K in samples with x
= 0-0.2, but zero magnetic hyperfine field in samples with x = 0.3. The spectra
of (Ba0.9K0.1)Fe2As2 and (Ba0.8K0.2)Fe2As2 in the phase transition regions are
temperature-dependent superpositions of magnetic and non-magnetic components,
caused by inhomogeneous potassium distribution. Our results suggest the
co-existence of AF magnetic ordering and superconductivity without mesoscopic
phase separation in the underdoped region and show unambiguously homogeneous
superconducting phases close to optimal doping. This is in contrast to recently
reported results about single crystal (Ba1-xKx)Fe2As2.
