# Einstein coefficients cannot depend on temperature?

I am looking at the Wikipedia article on the Einstein coefficients. There is this line about the detailed balance condition to hold at all temperatures.

Why is it that the Einstein coefficients cannot depend on the temperature?

They could depend on temperature, but allowing this in the model would complicate it and it wasn't necessary, the model works well without it. Einstein's model where they do not depend on temperature is simpler and useful. Like Schroedinger's model of an atom is simpler than relativistic quantum field theoretic attempts at such a model, but still very useful.

The coefficients are first and foremost functions of the transition frequency $$\nu_{12}$$. This is the frequency of radiation associated with transition between molecules' states 1 and 2.

This frequency is characteristic of a molecule, but isn't completely constant in all regimes. If the molecule gets into different environment (higher temperature means more intense EM noise), transition frequency and its line width can change a little. So will the effective coefficients $$A,B$$ in Einstein's kinetic equations. But these would be very weak and hard to measure effects in most cases, so it is not usually considered.

The Einstein coefficients are intrinsic properties of the atoms/ions and their energy eigenfunctions. They are proportional to the integration of the electric dipole operator between a final and an initial eigenstate.

These energy eigenfunctions do not change with temperature - except indirectly, for example one could consider perturbations associated with pressure-broadening.

But the argument in your question does not claim that the Einstein coefficients are independent of temperature, it merely says that the principle of detailed balnce holds at any temperature. This means that the coefficients of $$\exp(-h\nu/kT)$$ must balance in the equation and leads to the relationship between the Einstein coefficients shown.

• Technically, if the coefficients depend on temperature, then the argument in the question only gives the relationship between the coefficients in the $T \to \infty$ limit.
– Ian
Commented Mar 29, 2022 at 22:35
• If $A_{21}(T)g_2(1-\exp(-h\nu/kT))+B_{21}(T)g_2F(\nu)\exp(-h\nu/kT)=B_12(T)g_1F(\nu)$ applies at all $T, \nu$ then $A_{21}(T)=B_{21}(T)F(\nu)$ and $B_{21}(T)g_2 = B_{12}(T)g_1$ unless perhaps the coefficients depend exactly on $\exp(-h\nu/kT)$ ? @Ian Commented Mar 30, 2022 at 7:11