(a) Purely radiative case: Consider a collection of two-level atoms in which only a radiative decay is present with the decay rate Yrad. Initially this collection of atoms is at the temperature of absolute zero with the upper-level population N = 0. At the time moment t = 0 this collection of atoms is suddenly moved into an enclosure with the walls held at a finite temperature Trad (i.e., is suddenly immersed into the radiative blackbody field of this temperature). Find formulas for the lower and upper level populations N(t) and N2(t), and for the temperature Ta(t) of the collection of atoms for t> 0. (b) Case with both radiative and nonradiative contributions: The collection of two-level atoms in a crystal is coupled both to the electromagnetic surroundings with the radiative decay rate Yrad and to the crystal lattice surroundings with the decay rate ynr. Let's assume (hyphotetically - without worrying whether this can be done in practice) that the electromagnetic surroundings are somehow held at the fixed temperature Trad which is different from the fixed temperature Tr of the crystal lattice. Derive a formula for the steady- state equilibrium value of the Boltzmann temperature Ta for the level populations of the two-level atoms in this case, as a function of the two surrounding temperatures Trad and Tr, the normalized energy gap hv/k, and the ratio Yrad/Ynr.