  We present a study of hydrogen, helium, and carbon millimeter-wave
radio-recombination lines (RRLs) toward ten representative positions throughout
the Orion Nebula complex, using the Yebes 40m telescope in the Q band (31.3 GHz
to 50.6 GHz) at an angular resolution of about $45\arcsec$ ($\sim$0.09\,pc).
The observed positions include the Orion Nebula (M42) with the Orion Molecular
Core 1, M43, and the Orion Molecular Core 3 bordering on NGC 1973, 1975, and
1977. While hydrogen and helium RRLs arise in the ionized gas surrounding the
massive stars in the Orion Nebula complex, carbon RRLs stem from the neutral
gas of the adjacent photo-dissociation regions (PDRs). The high velocity
resolution ($0.3\,\mathrm{km\,s^{-1}}$) enables us to discern the detailed
dynamics of the RRL emitting neutral and ionized gas. We compare the carbon
RRLs with SOFIA/upGREAT observations of the [CII] $158\,\mu\mathrm{m}$ line and
IRAM 30m observations of the $^{13}$CO (J=2-1) line. Using the [CII] and
[$^{13}$CII] intensities with the carbon RRL intensities, we can infer physical
conditions (electron temperature and electron density) in the PDR gas using
non-LTE excitation models. Our observations are sensitive enough to detect
faint lines toward two positions in OMC1, that may be attributed to RRLs of
C$^+$ or O$^+$. In general, the RRL line widths of both the ionized and neutral
gas, as well as the [CII] and $^{13}$CO line widths, are broader than thermal,
indicating significant turbulence in the interstellar medium, which transitions
from super-Alfv\'enic and subsonic in the ionized gas to sub-Alfv\'enic and
supersonic in the molecular gas. At the scales probed by our observations, the
turbulent pressure dominates the pressure balance in the neutral and molecular
gas, while in the ionized gas the turbulent pressure is much smaller than the
thermal pressure.
