The Universal Cross-Section of Photonic Interaction

In 1917, Einstein published his paper [1] renowned for the discovery of stimulated emission. However, it also contained the principles which allowed the calculation of the universal photonic cross-section = λ/2π. Any heavy electronic system will have this cross-section for photonic excitation and stimulated emission in its rest frame. The modifications of this cross-section due to the recoil of the emitter have not yet been calculated, but are in general expected to be second order in recoil velocity.


Introduction
When Einstein first found the term for stimulated emission, he based that discovery on some very simple physics, including the principle that the radiant spectrum of blackbody equilibrium was universal.We still believe all these principles today.Using those same results and some algebra, we find that they also imply a universal cross-section for every type of photonic interaction, from nuclear to atomic to superconducting.

Short Overview of Einstein's Calculation
A very brief summary is presented here to highlight Einstein's logic, which we will then extend.He began with a two state electronic system in thermal equilibrium at temperature T, which produced a probability P 1 of being in the ground state and probability P 2 = 1 − P 1 of being in the excited state.For simplicity, he assumed that the excitable system had identical excitation and emission wavelengths, which meant that it had no recoil during emission or excitation (absorption) and was thus infinitely massive.This was a simplifying assumption and will also apply to the derivation which follows.
He also applied the known physics for the blackbody spectrum BB(ν), where ν = the frequency of light, and the known equilibrium thermal probability ratio P 2 /P 1 for the two states as summarized below, where the additional physics of E = hν was also known and used.(2) Then he assumed three processes that would change the electronic state back and forth: spontaneous emission, excitation and finally stimulated emission.Stimulated emission was unknown to physics at that time but was required in order to achieve the proper equilibrium given by the mean rate of excitation minus de-excitation being zero.Since he assumed everything would be in thermal equilibrium, the rate of change of the probability of being excited must be zero on average.This was his key physics principle shown in Equation ( 3).Here the three constants A 21 (spontaneous emission), B 21 (stimulated emission) and B 12 (excitation) can all be adjusted to make this equation work.For later reference we also point out that constant A 12 = 1/τ decay , where τ decay is the exponential decay time of state 2 into state 1.This is the known decay form for a state at rest without other disturbances as described by quantum mechanics, and was also well known in Einstein's time.This type of exponential decay produces a familiar line shape, called a Lorentz line shape, which we will use later.
which is identically zero if and only if the following two conditions apply, as summarized in Equations (5a) and (5b).

Computing the Universal Photonic Cross-Section
Back in 1917 the main attention focused on the B 21 term for stimulated emission, which was new to physics.
Here however, we wish to focus on the first two terms which will allow us to compute the universal cross-section of a photonic interaction.Our process will be to write the decay rate in two different, physically equivalent forms: 2 2 21 12 decay d .d The first form in Equation ( 6) represents the rate of decay of an infinitely massive excited state alone in a vacuum, and it is correctly described by a single time constant τ decay , which produces an exponentially decaying emission probability, called Lorentz decay.In fact, A 21 = 1/τ decay .This type of decay is well known and produces a probability distribution of frequencies called a Lorentz emission spectrum, with a frequency probability given in Equation (7a) [2], where ν o = the center frequency of the decay spectrum, and Δν L = the half-width of the emission line.This equation is normalized so that it integrates to unity and is thus a probability distribution of the emitted frequencies.

( ) ( )
where Equation (7a) is the emission spectrum of an excited state into a single lower energy state, while to compute the excitation rate we need the reverse-the absorption spectrum of the lower state into the higher state.Fortunately the two are identical, a principle often used in astronomy to measure absorption lines of atoms and molecules by using their emission caused by illumination from a nearby star [3].
To compute the effective frequency bandwidth of the emission line, we simply renormalize this distribution so that the center frequency ν o has value 1-indicating 100% sensitivity.To accomplish this new normalization, we simply multiply Equation (7a) by πΔν L /2.Since Equation (7a) integrates to one, the new function will integrate to πΔν L /2-the effective frequency bandwidth of the emission as given by Equation (8).We can then substitute the value of Δν L = 1/(2πτ decay ) from Equation 7(b) to get A 21 in terms of the effective frequency bandwidth in Equation (9).
Solving for B 21 , we get the following result with a little algebra, which must also equal B 12 since the two are equal from Equation (5b).
The excitation rate per second (ER) is then To identify the term we seek, the general form to compute an excitation rate is by taking the probability of being unexcited (P 1 ) times the number of photons/sec/cm 2 /Hz (P FR ) times the effective bandwidth of the excitation (Δf eff ) times the cross-section of interaction (C photon ) for the photon to give photons/sec excited, as shown in Equation ( 14).We see immediately that the cross-section of the photon for every excitation of a heavy electronic system in its rest frame must be where λ = the wavelength in the rest frame of our infinitely heavy atom.
very physical in Equation (11) when we compute the rate of excitation (ER) for an unexcited atom as defined in Einstein's paper above.
this equation in terms of the P FR (photon flux rate per unit area per frequency), which is given simply as Equation (12).