We can write the rate equation for in steady state, in which case the time derivative goes to zero. Rearranging to solve for the ratio we obtain:
If we consider the particular case of equilibrium, we also know that
We can set these two equations equal to one another and solve for the electromagnetic energy spectral density . Doing so yields
Let’s consider a blackbody, which is an idealized object that is capable of absorbing or emitting any wavelength of light. One real-world approximation to a blackbody is a small hole in the side wall of a box made of light absorbing material. It can be shown that a blackbody always emits a characteristic spectrum that depends only on its temperature; fundamentally this is because the electronic energy states in the blackbody are filled according to thermodynamic principles. The electromagnetic spectral energy density produced by an ideal blackbody is:
Comparing this to the equation for we derived above, we see that the following relationships must hold amongst the and coefficients:
these are the Einstein relations, which hold for the A and B coefficients between any pair of states. One thing to note is that making an x-ray laser would be extremely difficult, as decreases with photon frequency cubed .