## Sunday, July 24, 2011

### Einstein, thermodynamics, and elegance

Recently, in the course of other writing I've been doing, I again came to the topic of what are called Einstein A and B coefficients, and it struck me again that this has to be one of the most elegant, clever physics arguments ever made.  It's also conceptually simple enough that I think it can be explained to nonexperts, so I'm going to give it a shot.

Ninety-four years ago, one of the most shocking ideas in physics was the concept of the spontaneous, apparently random, breakdown of an atomic system.  Radioactive decay is one example, but even light emission from an atom in an excited state will serve.  Take ten hydrogen atoms, all in their first electronically excited state (electron kicked up into a 2p orbital from the 1s orbital).  These will decay back into the 1s ground state (spitting out a photon) at some average rate, but each one will decay independently of the others, and most likely at a different moment in time.  To people brought up in the Newtonian clockwork universe, this was shocking.  How could truly identical atoms have individually differing emission times?  Where does the randomness come from, and can we ever hope to calculate the rate of spontaneous emission?

Around this time (1917), Einstein made a typically brilliant argument:  While we do not yet know [in 1917] how to calculate the rate at which the atoms transition from the ground state "a" to the excited state "b" when we shine light on them (the absorption rate), we can reason that the rate of atoms going from a to b should be proportional to the number of atoms in the ground state (Na) and the amount of energy density in the light available at the right frequency (u(f)).  That is, the rate of transitions "up" = Bab Na u(f), where B is some number that can at least be measured in experiments.  [It turns out that people figured out how to calculate B using perturbation theory in quantum mechanics about ten years later.].  Einstein also figured that there should be an inverse process (stimulated emission), that causes transitions downward from b to a, with a rate = Bba Nb u(f).  However, there is also the spontaneous emission rate = AbaNb, where he introduced the A coefficient.

Here is the brilliance.  Einstein considered the case of thermal equilibrium between atoms and radiation in some cavity.  In steady state, the rate of transitions from a to b must equal the rate of transitions from b to a - in steady state, no atoms are piling up in the ground or excited states.  Moreover, from thermodynamics, in thermal equilibrium, the ratio of Nb to Na should just be a Boltzmann factor, exp(-Eab/kBT), where Eab is the energy difference between the two states, kB is Boltzmann's constant, and T is the temperature.  From this, Einstein shows that the two Bs were equal, was able to solve for the unknown A in terms of B (which can be measured and nowdays calculated), and to show that the energy density of the radiation (u(f,T)) is Planck's blackbody formula.

My feeble writing here doesn't do this justice.  The point is, from basic thermodynamic reasoning, Einstein made it possible to derive an expression for the spontaneous emission rate of atoms, many years in advance of the theory (quantum electrodynamics) that allows one to calculate it directly.  This is what people mean by the elegance of physics - in a few pages, from proper reasoning on fundamental grounds, Einstein was able to deduce relationships that had to exist between different physical parameters; and these parameters could be measured and tested experimentally.  For more on this, here is a page at MIT that links to a great Physics Today article about the topic, and an English translation of Einstein's 1917 paper.