In an effort to understand unusual power-law spectral slopes observed in some hypercompact HII regions , we consider the radio continuum energy distribution from an ensemble of spherical clumps .
An analytic expression for the free-free emission from a single spherical clump is derived .
The radio continuum slope ( with F _ { \nu } \propto \nu ^ { \alpha } ) is governed by the population of clump optical depths N ( \tau ) , such that ( a ) at frequencies where all clumps are thick , a continuum slope of +2 
is found , ( b ) at frequencies where all clumps are optically thin , a flattened slope of -0.11 is found , and ( c ) at intermediate frequencies , a power-law segment of significant bandwidth with slopes between these two limiting values can result .
For the ensemble distribution , we adopt a power-law distribution N ( \tau ) \propto \tau ^ { - \gamma } , and find that significant power-law segments in the SED with slopes from +2 
to -0.11 result only for a relatively restricted range of \gamma 
values of 1 to 2 .
Further , a greater range of clump optical depths for this distribution leads to a wider bandwidth over which the intermediate power-law segment exists .
The model is applied to the source W49N-B2 with an observed slope of \alpha \approx + 0.9 , but that may be turning over to become optically thin around 2 mm .
An adequate fit is found in which most clumps are optically thin and there is little “ shadowing ” of rearward clumps by foreground clumps ( i.e. , the geometrical covering factor C \ll 1 ) .
The primary insight gained from our study is that in the Rayleigh-Jeans limit for the Planck function that applies for the radio band , it is the distribution in optical depth of the clump population that is solely responsible for setting the continuum shape , with variations in the size and temperature of clumps serving to modulate the level of free-free emission .