  Most static 1D atmosphere models in the quiet Sun predict a rise of the gas
temperature at chromospheric layers, but numerical simulations only yield an
increase in the brightness temperature. We investigate the thermal structure in
the solar chromosphere as derived from an LTE inversion of Ca II H spectra in
QS and active regions. We investigate the temperature stratifications on
differences between magnetic and field-free regions in the QS, and between QS
and ARs. We determine the energy content of individual calcium bright grains
(BGs). The rms temperature fluctuations are below 100 K in the photosphere and
200-300 K in the chromosphere. The average temperature stratification in the QS
does not exhibit a clear chromospheric temperature rise, opposite to the AR
case. We find an energy content of about 7*10E18 J for BGs that repeat with a
cadence of about 160 secs. The precursors of BGs have a vertical extent of
about 200 km and a horizontal extent of about 1 Mm. The comparison of observed
with synthetic NLTE profiles confirms that the solar chromosphere in the QS
oscillates between an atmosphere in radiative equilibrium and one with a
moderate chromospheric temperature rise. Two-dimensional x-z temperature maps
exhibit nearly horizontal canopy-like structures with a few Mm extent around
photospheric magnetic field concentrations at a height of about 600 km. The
large difference between QS regions and ARs, and the better match of AR and
non-LTE reference spectra suggest that magnetic heating processes are more
important than commonly assumed. The temperature fluctuations in QS derived by
the LTE inversion do not suffice on average to maintain a stationary
chromospheric temperature rise. The spatially and vertically resolved
information on the temperature structure allows one to investigate in detail
the topology and evolution of the thermal structure in the lower solar
atmosphere.
