The laser refrigeration of solid-state materials with nanoscale dimensions has been demonstrated for both semi- conducting (cadmium sulfide, CdS) and insulating dielectrics (Yb:YLiF4, YLF) in recent years. During laser refrigeration it is possible to observe morphology dependent resonances (MDRs), analogous to what is well- known in classical (Mie) light scattering theory, when the characteristic dimensions of the nanostructure are comparable to the wavelength of light used to initiate the laser cooling process. Mie resonances can create substantial increases for internal optical fields within a given nanostructure with the potential to enhance the absorption efficiency at the beginning of the cooling cycle. Recent breakthroughs in the laser refrigeration of semiconductor nanostructures have relied on materials that exhibit rectangular symmetry (nanoribbons). Here, we will present recent analytical, closed-form solutions to the energy partial differential equation that can be used to calculate the internal spatial temperature profile with a given semiconductor nanoribbon during irradiation by a continuous-wave laser. First, the energy equation is made dimensionless through the substitution of variables before being solved using the classical separation-of-variables approach. In particular, calculations will be presented for chalcogenide (CdS) nanoribbons using a pump wavelength of 1064 nm. For nanostructures with lower symmetry (such as YLF truncated tetragonal bipyramids) it is also possible to observe MDRs through numerical simulations using either the discrete dipole approximation or finite-difference time-domain simulations, and the resulting temperature profile can be calculated using the finite element method. Theoretical predictions are presented using parameters that will allow comparison with experimental data in the near future.