  We use Planck HFI data combined with ancillary radio data to study the
emissivity index of the interstellar dust emission in the frequency range 10 -
353 GHz, or 3 - 0.8 mm, in the Galactic plane. We analyse the region l=20 degr
- 44 degr and |b| \leq 4 degr where the free-free emission can be estimated
from radio recombination line data. We fit the spectra at each sky pixel with a
modified blackbody model and two spectral indices, beta_mm and beta_FIR, below
and above 353 GHz respectively. We find that beta_mm is smaller than beta_FIR
and we detect a correlation between this low frequency power-law index and the
dust optical depth at 353 GHz, tau_353. The opacity spectral index beta_mm
increases from about 1.54 in the more diffuse regions of the Galactic disk, |b|
= 3 degr - 4 degr and tau_353 ~ 5 x 10^{-5}, to about 1.66 in the densest
regions with an optical depth of more than one order of magnitude higher. We
associate this correlation with an evolution of the dust emissivity related to
the fraction of molecular gas along the line of sight. This translates into
beta_mm ~ 1.54 for a medium that is mostly atomic and beta_mm ~ 1.66 when the
medium is dominated by molecular gas. We find that both the Two-Level System
model and the emission by ferromagnetic particles can explain the results. The
results improve our understanding of the physics of interstellar dust and lead
towards a complete model of the dust spectrum of the Milky Way from
far-infrared to millimetre wavelengths.
