  The 4.8 GHz formaldehyde masers are one of a number of rare types of
molecular masers in the Galaxy. The aim of the present calculations is to
explore a larger region of parameter space to improve on our previous
calculations, thereby to better understand the range of physical conditions
under which an inversion of the 4.8 GHz transition occurs. We solve the rate
equations of the first 40 rotational levels of o-formaldehyde using a
fourth-order Runge-Kutta method. We consider gas kinetic temperatures between
10 K and 300 K, H_2 densities between 10^4 and 10^6 per cc, and a number of
different dust temperatures and grey-body spectral energy density
distributions. Using a grey-body dust radiation field appropriate for Arp 220
we find that none of 4.8 GHz, 14 GHz, and 28 GHz transitions are inverted for
kinetic temperatures less than 100 K. Our calculations also show that in theory
the 4.8 GHz transition can be inverted over a large region of explored
parameter space in the presence of an external far-infrared radiation field.
Limiting the abundance of o-formaldehyde to less than 10^{-5}, however, reduces
the region where an inversion occurs to H_2 densities > 10^5 per cc and kinetic
temperatures >100 K. We propose a pumping scheme for the formaldehyde masers
which can explain why collisions play a central role in inverting the 4.8 GHz
transition, and therefore why an external radiation field alone does not lead
to an inversion. Collisions are an essential mechanism for the inversion of the
4.8 GHz transition. Our results suggest that 4.8 GHz formaldehyde megamasers
are associated with hot and dense gas typical of high mass star forming regions
rather than with cold material.
