When temperature (T) exceeds 2,000 K inside the Earth, transfer of heat by diffusion of photons is important. However, the effective thermal conductivity (k_rad) associated with this process does not follow a T³ law. The dependence of scattering and emission spectra on grain size (d), and the non-linear dependence of absorption and emission spectra on frequency, result in a complex dependence of k_rad on T and d. Specifically, for large grains, k_rad rises to a local maximum near 1,300 K, followed by a local minimum near 2,000 K, followed by a gentle rise towards high temperature. For small grains, k_rad depends quadratically on T. Above ~ 2,000 K, k_rad is largest for d near 1 mm. Irrespective of possible grain sizes, k_rad has a minimum for the temperatures expected at 670 km: this critical point must impact convection above 670 km, as negative dk/dT is destabilizing. The effect of a critical point in the transition zone on mantle convection is being investigated through geodynamic models. Below 670 km, radiative transport dominates over phonon scattering and is stabilizing. To compare the relative importance of k_rad to that of viscosity, numerical simulations were made of constant k or of k strongly depending on T, each with a range of lateral viscosity contrasts due to T spanning from 10² to 10⁶. The lower bound is characteristic of what is expected in the lower mantle due to the high background temperature in the Arrhenius argument of the viscosity. We found that k_rad exerts greater control. In particular, for the low viscosity contrast and high k_rad expected for the lower mantle, convection is substantially weakened. Based on this result, we interpret the low heterogeneity inferred from the tomographic images of the middle of lower mantle as the signature of a stagnant layer.