## Monday, May 14, 2012

### The unreasonable clarity of E. M. Purcell

Edward Purcell was one of the great physicists of the 20th century.  He won the Nobel Prize in physics for his (independent) discovery of nuclear magnetic resonance, and was justifiably known for the extraordinarily clarity of his writing.  He went on to author the incredibly good second volume of the Berkeley Physics Course (soon to be re-issued in updated form by Cambridge University Press), and late in life became interested in biophysics, writing the evocative "Life at Low Reynolds Number" (pdf).

Purcell is also known for the Purcell Factor, a really neat bit of physics.  As I mentioned previously, Einstein showed through a brilliant thermodynamic argument that it's possible to infer the spontaneous transition rate for an emitter in an excited state dropping down to the ground state and spitting out a photon.  The spontaneous emission rate is related to the stimulated rate and the absorption rate.  Both of the latter two may be calculated using "Fermi's Golden Rule", which explains (with some specific caveats that I won't list here) that the rate of a quantum mechanical radiative transition for electrons (for example) is proportional to (among other things) the density of states (number of states per unit energy per unit volume) of the electrons and the density of states of the photons.  The density of states for photons in 3d can be calculated readily, and is quadratic in frequency.

Purcell had the insight that in a cavity, the number of states available for photons is not quadratic in frequency anymore.  Instead, a cavity on resonance has a photon density of states that is proportional to the "quality factor", Q,  of the cavity, and inversely proportional to the size of the cavity.  The better the cavity and the smaller the cavity, the higher the density of states at the cavity resonance frequency, and off-resonance the photon density of states approaches zero.  This means that the spontaneous emission rate of atoms, a property that seems like it should be fundamental, can actually be tuned by the local environment of the radiating system.  The Purcell factor is the ratio of the spontaneous emission rate with the cavity to that in free space.

While I was doing some writing today, I decided to look up the original citation for this idea.  Remarkably, the "paper" turned out to be just an abstract!  See here, page 681, abstract B10.  That one paragraph explains the essential result better than most textbooks, and it's been cited a couple of thousand times.  This takes over as my new favorite piece of clear, brief physics writing by a famous scientist, displacing my long-time favorite, Nyquist's derivation of thermal noise.  Anyone who can be both an outstanding scientist and a clear writer gets bonus points in my view. Anonymous said...

Too bad said abstract is behind the paywall.

Douglas Natelson said...

Here is a transcript:
Spontaneous Emission Probabilities at Radio Frequencies. E. M. PURCELL, Harvard University. — For nuclear magnetic moment transitions at radio frequencies the probability of spontaneous emission, computed from A_{\nu}=(8 \pi \nu^{2}/c^{3})h \nu (8\pi^{3}\mu^{2}/3h^{2}) sec^{-1}, is so small that this process is not effective in bringing a spin system into thermal equilibrium with its surroundings.
At 300 K, for \nu =10^{7} sec.^{-1}, \mu=1 nuclear magneton, the corresponding relaxation time would be 5 \times 10^{21} seconds! However, for a system coupled to a resonant electrical circuit, the factor 8\pi \nu^{2}/c^{3} no longer gives correctly the number of radiation oscillators per unit volume, in unit frequency range, there being now one oscillator in the frequency range \nu/Q associated with the circuit. The
spontaneous emission probability is thereby increased, and the relaxation time reduced, by a factor f=3Q \lambda^{3}/(4 \pi^{2} V), where V is the vo1ume of the resonator. If a is a dimension
characteristic of the circuit so that V~a^{3}, and if \delta is the
skin-depth at frequency \nu, f~\lambda^{3}/a^{2}\delta. For a non-resonant circuit f~ \lambda^{3}/a^{3}, and for a \lessthan \delta it can be shown that f ~ \lambda^{3}/a \delta^{2}. If small metallic particles, of diameter 10^{-3} cm are mixed
with a nuclear-magnetic medium at room temperature, spontaneous emission should establish thermal equilibrium in a time of the order of minutes, for \nu=10^{7} sec.^{-1}.