Simple explicit expressions have been derived for the linewidth enhancement factor α and the change in refractive index with injected carrier density in direct-gap semiconductors. Both strict and partial conservation of crystal momentum have been considered. Both models yield the same expressions for α. Excellent agreement is found both between these expressions and numerical Kramers-Kronig transforms and with experimental data for λ = 1.5 μm InGaAsP. As a consequence of this analysis the factors which influence the linewidth enhancement factor can be readily understood for the first time. In particular the band-to-band contribution to α is shown to depend, in a simple way, only on the temperature, the position of the conduction band quasi-Fermi level and the difference between the photon and band-gap energies.