  Aims. In this article we aim to show how the gradient of the thermal
millimetre continuum spectrum, as emitted from the quiet solar atmosphere, may
be used as a diagnostic for the optical thickness regime at the centre of the
observing frequency band.
  Methods. We show the theoretical derivation of the gradient of the millimetre
continuum for both logarithmic- and linear-scale spectra. We compare this
expression with the empirical relationship between the slope of the millimetre
continuum spectrum and the plasma optical thickness computed from both
isothermal and multi-thermal two-dimensional cylindrical radiative transfer
models.
  Results. It is found that the logarithmic-scale spectral gradient provides a
clear diagnostic for the optical thickness regime for both isothermal and
multi-thermal plasmas, provided that a suitable correction is made for a
non-constant gaunt factor over the frequency band. For the use of observers we
present values for this correction at all ALMA bands and at a wide range of
electron temperatures.
  Conclusions. We find that the spectral gradient can be used to find (a)
whether the source is fully optically thin, (b) the optical thickness of the
source if it lies within the transitional regime between optically thin and
thick plasma ({\tau} $\approx$ $10^{-1} - 10^{1}$), or (c) whether the source
is fully optically thick for an isothermal plasma. A multi-thermal plasma will
act the same as an isothermal plasma for case (a), however, the transitional
regime will only extend from {\tau} $\approx$ $10^{-1} - 10^{0}$ . Above {\tau}
= 1 the slope of the continuum will depend increasingly on the temperature
gradient, as well as the optical thickness, reducing the reliability of the
diagnostic.
