11.4: Einstein Coefficients and Stimulated Emission

An extremely useful way to view the processes of absorption is to consider the specific types of interactions between a photon and an atom. This view will become the preferred one when we consider problems where we cannot assume LTE. In quantum mechanics, it is customary to think of the probability of the occurrence of a specific event rather than the classical picture of the frequency with which the event occurs. These are really opposite sides of the same coin, but they yield somewhat philosophically different pictures of the processes in question. In considering the transition of an atom from one excited state to another, Einstein defined a set of coefficients that denote the probability of specific transitions taking place. These are known as the Einstein coefficients. All radiative processes and the equation of radiative transfer itself may be formulated in terms of these coefficients. The coefficients are determined by the wave functions of the atom alone and thus do not depend in any way on the environment of the atom. (As with most rules, there is an exception. If the density is sufficiently high that the presence of other atoms distorts the wave functions of the atom of interest, then the Einstein coefficients of that atom can be modified.)

Consider an atomic transition which takes an atom from a lower-energy state of excitation \(n'\) to a higher-energy state \(n\) (or vice versa). The upper state \(n\) may be a bound or continuum state, and the transition will involve the absorption or emission of a photon. The number of transitions that will occur in a given time interval for an ensemble of atoms will then depend on the probability that one transition will occur times the number of atoms available to make the transition. There types of transitions can occur:

1. Spontaneous emission, where the electron spontaneously makes a downward transition with the accompanying emission of a photon.
2. Stimulated absorption, where a passing photon is absorbed, producing the resulting transition.
3. Stimulated emission, where the electron makes a downward transition with the accompanying emission of a photon. If this occurs in the presence of a photon of the same type as that emitted by the transition, the probability of the event is greatly enhanced.

The symmetric process of spontaneous absorption simply cannot occur because one cannot absorb what is not there.

We can describe the number of atoms undergoing these processes in terms of the probabilities of the occurrence of a single event: \[\begin{aligned}
& N_{n \rightarrow n^{\prime}}=N_n A_{n n^{\prime}} d t \\
& N_{n^{\prime} \rightarrow n}=N_{n^{\prime}} B_{n^{\prime} n} I\left(\nu_{n n^{\prime}}\right) d t \\
& N_{n \rightarrow n^{\prime}}=N_n B_{n n^{\prime}} I\left(\nu_{n^{\prime} n}\right) d t
\end{aligned}\label{11.3.1}\]

The dependence of the stimulated processes on the presence of photons is made clear by the inclusion of the specific intensity, corresponding to the energy of the transition, in the last two equations. The coefficients that appear in these equations are known as the Einstein coefficient of spontaneous emission \(\mathrm{A}_{n n^{\prime}}\), the Einstein coefficient of absorption (stimulated) \(\mathrm{B}_{n^{\prime} n}\), and the Einstein coefficient of stimulated emission \(\mathrm{B}_{n n^{\prime}}\). Since there is only one kind of absorption processes, the adjective stimulated is usually dropped from the coefficient \(\mathrm{B}_{n^{\prime} n}\).

a. Relations among Einstein Coefficients

The three Einstein coefficients are not linearly independent. Indeed, the specification of any one of them allows the determination of the other two. To show this, we construct an environment where we know something about the rates at which processes should take place. Since the Einstein coefficients are atomic constants, they are independent of the environment, and thus we are free to create any environment that we choose as long as it is physically self-consistent. With this in mind, consider a gas that is in STE. Under these conditions, the atomic transition rates in and out of each level are equal (detailed balancing). If this were not the case, cyclical processes could exist that would provide for a flow of energy from one frequency to another. But the assumption of STE requires that the photon energy distribution be given by the Planck function and that situation would not be preserved by an energy flow in frequency space. Therefore, it can not happen in STE. In addition, in STE the Boltzmann excitation formula holds for the distribution of atoms among the various states of excitation. Thus, the second of equations \ref{11.3.1} must be equal to the sum of the other two. \[N_{n^{\prime}} B_{n^{\prime} n} B_\nu(T)=N_n\left[A_{n n^{\prime}}+B_{n n^{\prime}} B_\nu(T)\right]\label{11.3.2}\]

But since the Boltzmann formula must hold, \[\frac{N_n}{N_{n^{\prime}}}=\frac{g_n}{g_{n^{\prime}}} e^{-h\nu_{n n^{\prime}} /(k T)}\label{11.3.3}\]

Substituting in the correct form for the Planck function [see equation \ref{1.1.24}] and noting that the frequency \(\nu\) is the same as the frequency that appears in the excitation energy of the Boltzmann formula \(\nu_{nn^{\prime}}\) we get \[A_{n n^{\prime}} \frac{g_n}{g_{n^{\prime}}}=\frac{2 h\nu^3}{c^2} B_{n^{\prime} n} \frac{e^{h\nu /(k T)}-B_{n n^{\prime}} g_n /\left(B_{n^{\prime} n} g_{n^{\prime}}\right)}{e^{h\nu /(k T)}-1}\label{11.3.4}\]

Now the Einstein coefficients are atomic constants and therefore must be independent of the temperature. This can happen only if the numerator of the rightmost fraction in equation 11.3.4 is identically 1. Thus, \[\frac{B_{n n^{\prime}} g_n}{B_{n^{\prime} n} g_{n^{\prime}}}=1 \quad A_{n n^{\prime}}=\left(\frac{2 h\nu^3}{c^2} \frac{g_{n^{\prime}}}{g_n}\right) B_{n^{\prime} n}\label{11.3.5}\]

These relationships must be completely general since the Einstein coefficients must be independent of the environment.

b. Correction of the Mass Absorption Coefficient for Stimulated Emission

In deriving the equation of radiative transfer, we took no notice of the concept of stimulated emission. The mass emission coefficient \(j_\nu\) implicitly contains the notion since it represents the total energy emitted per gram of stellar material. However, the mass absorption coefficient \(\kappa_\nu\) was calculated as the effective cross section per gram of stellar material and thus counts only those photons absorbed. Should a passing photon stimulate the production of an additional photon, that processes should be counted as a "negative" absorption. Indeed, some authors call the coefficient of stimulated emission the coefficient of negative absorption. Now it is a property of the stimulated emission process that the photon produced by the passage of another photon has exactly the same direction, energy, and phase. Indeed, this is the mechanism by which lasers work and which we discuss in greater depth in the chapters dealing with line formation. To correct the absorption coefficient for the phenomenon of stimulated emission, we need only conserve energy.

Consider the total energy produced within a cubic centimeter of a star and flowing into a solid angle dΩ. \[j_\nu \rho d v d \Omega=h\nu N_n\left(A_{n n^{\prime}}+B_{n n^{\prime}} I_\nu\right)=N_n A_{n n^{\prime}} h\nu\left(1+\frac{c^2 I_\nu}{2 h\nu^3}\right)\label{11.3.6}\]

This must be equal to the total energy absorbed in that same cubic centimeter from the same solid angle. \[I_\nu \kappa_\nu \rho d v d \Omega=N_{n^{\prime}} B_{n^{\prime} n} I_\nu h\nu\label{11.3.7}\]

Now consider an environment that is in LTE and in which scattering is unimportant. The source function for such an atmosphere is then \[\begin{aligned}
S_\nu\left(\tau_\nu\right) & =\frac{j_\nu}{\kappa_\nu}=\frac{N_n}{N_{n^{\prime}}} \frac{2 h\nu^3}{c^2} \frac{g_{n^{\prime}}}{g_n}\left(\frac{1+c^2 I_\nu}{2 h\nu^3}\right) \\
S_\nu & =B_\nu\left(1-e^{-h\nu /(k T)}\right)+I_\nu e^{-h\nu /(k T)}
\end{aligned}\label{11.3.8}\]

Now, if we insert this form for the source function into the plane-parallel equation of radiative transport, we get \[\mu \frac{d I_\nu}{d \tau_\nu}=I_\nu-S_\nu=\left(I_\nu-B_\nu\right)\left(1-e^{-h\nu /(k T)}\right)\label{11.3.9}\]

However, our original equation for this problem had the form \[\mu \frac{d I_\nu}{d \tau_\nu^{\prime}}=I_\nu-B_\nu\label{11.3.10}\]

We can bring equation \ref{11.3.9} into the same form as equation \ref{11.3.10} by simply redefining the mass absorption coefficient \(\kappa_\nu\) as \[\kappa_\nu^{\prime}=\kappa_\nu\left(1-e^{-h\nu /(k T)}\right) \quad d \tau_\nu^{\prime}=-\kappa_\nu^{\prime} \rho d x\label{11.3.11}\]

Modifying the mass absorption coefficient by the factor \(1-e^{\mathrm{h}\nu/\mathrm{kT}}\) simply corrects it for the effects of stimulated emission. Once again we are correcting atomic parameters that are independent of their environment and so the result is independent of the details of the derivation. That is, the correction term is a general one and applies to all problems of radiative transfer. Thus, one should always be sure that the absorption coefficients used are corrected for stimulated emission. This is particularly true when one is using tabular opacities that are generated by someone else.