Einstein Coefficients and Thermal Equilibrium Relationships

Hello again. Last lesson established the three elementary radiative processes and their rate forms: absorption raises an emitter from level $1$ to level $2$, while spontaneous and stimulated emission lower it. We also introduced the coefficients $A_{21}$, $B_{12}$, and $B_{21}$, without yet specifying how they are related.

This lesson supplies that connection. We will place a two-level ensemble and its radiation field in thermal equilibrium, combine Boltzmann populations with Planck’s blackbody spectrum, and derive the Einstein relations. The degeneracy factors will matter: gain is governed by population per available substate, not merely by comparing the total populations $N_2$ and $N_1$.

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The equilibrium thought experiment

Let the two levels have energies $E_1$ and $E_2$, separated by

$$E_2-E_1=h{\nu }_0.$$

We use $N_1$ and $N_2$ for the total number densities in the lower and upper levels. “Total” is important: if a level comprises several quantum substates with essentially the same energy, $N_i$ includes population in all of them.

For now, take the transition to be narrow and use ${\rho }_{\nu }({\nu }_0)$, the spectral electromagnetic energy density at the transition frequency, expressed per unit ordinary frequency. The three event rates per unit volume are

$$R_{\mathrm{abs}}=N_1{B}_{12}{\rho }_{\nu }({\nu }_0),$$ $$R_{\mathrm{sp}}=N_2{A}_{21},$$ $$R_{\mathrm{st}}=N_2{B}_{21}{\rho }_{\nu }({\nu }_0).$$

The Einstein coefficients are transition properties under this idealized free-space description:

• $A_{21}$ is the spontaneous-emission probability per unit time for an emitter initially in level $2$.
• $B_{12}$ converts spectral energy density into an absorption probability per unit time for an emitter in level $1$.
• $B_{21}$ does the analogous job for stimulated emission from level $2$.

For the derivation, imagine atoms and radiation enclosed in a perfectly equilibrated cavity at temperature $T$. This is explicitly not a laser medium: it has thermal populations and a blackbody radiation field. Equilibrium does not mean that transitions cease. Absorption, spontaneous emission, and stimulated emission all continue, but their gross rates balance, leaving $N_1$ and $N_2$ constant.

Therefore,

$$\frac{d{N}_2}{dt}=0,$$

so the upward absorption rate equals the sum of the downward rates:

$$N_1{B}_{12}{\rho }_{\nu }=N_2{A}_{21}+N_2{B}_{21}{\rho }_{\nu }.$$

A key distinction from the previous lesson: ${\rho }_{\nu }$ here refers to the isotropic equilibrium radiation field, integrated over all directions and polarizations. It is not the intensity of one collimated laser-like beam.

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Read the standard thermal derivation

The following section gives the same derivation in a compact textbook form, using Planck’s law, the Boltzmann population ratio, and the rate-balance condition. Read it once before following the slower algebraic unpacking below.

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Deriving the Einstein relations step by step

First solve the equilibrium rate balance for the radiation spectral energy density:

$$N_1{B}_{12}{\rho }_{\nu }-N_2{B}_{21}{\rho }_{\nu }=N_2{A}_{21},$$ $${\rho }_{\nu }({\nu }_0)=\frac{A_{21}}{\left(N_1/N_2\right)B_{12}-B_{21}}.$$

So far, this says only that a radiation field of a particular spectral density can hold the populations stationary. Thermal equilibrium now supplies two independent physical inputs.

1. Boltzmann statistics for the atoms

If level $i$ has degeneracy $g_i$, its equilibrium population is proportional to its number of available substates times the Boltzmann factor:

$$N_i\propto g_i\exp \left(-\frac{E_i}{k_{\mathrm{B}}T}\right).$$

Hence,

$$\frac{N_2}{N_1}=\frac{g_2}{g_1}\exp \left(-\frac{h{\nu }_0}{k_{\mathrm{B}}T}\right),$$

or equivalently,

$$\frac{N_1}{N_2}=\frac{g_1}{g_2}\exp \left(\frac{h{\nu }_0}{k_{\mathrm{B}}T}\right).$$

Substituting this into the rate-balance result gives

$${\rho }_{\nu }({\nu }_0)=\frac{A_{21}}{\left(\frac{g_1}{g_2}\right)B_{12}\exp \left(\frac{h{\nu }_0}{k_{\mathrm{B}}T}\right)-B_{21}}.$$

Factor out $B_{21}$ from the denominator:

$${\rho }_{\nu }({\nu }_0)=\frac{A_{21}/B_{21}}{\left(\frac{g_1{B}_{12}}{g_2{B}_{21}}\right)\exp \left(\frac{h{\nu }_0}{k_{\mathrm{B}}T}\right)-1}.$$

This is the spectral energy density that the Einstein-rate model predicts at equilibrium.

2. Planck’s law for the photons

Thermal radiation in a cavity has the Planck spectral energy density

$${\rho }_{\nu }({\nu }_0)=\frac{8\pi h{\nu }_0^3}{c^3}\frac{1}{\exp \left(\frac{h{\nu }_0}{k_{\mathrm{B}}T}\right)-1}.$$

The factor multiplying the thermal occupation term,

$$\frac{8\pi h{\nu }_0^3}{c^3},$$

arises from the density of electromagnetic modes in free space together with the energy $h{\nu }_0$ per photon. The denominator, especially its $-1$, is the signature that creates the need for stimulated emission in Einstein’s argument.

The Einstein prediction and Planck’s law must agree for every temperature $T$. Define

$$x=\frac{h{\nu }_0}{k_{\mathrm{B}}T}.$$

Then the two expressions have the forms

$${\rho }_{\nu }=\frac{A_{21}/B_{21}}{\left(g_1{B}_{12}/g_2{B}_{21}\right)e^x-1}$$

and

$${\rho }_{\nu }=\frac{8\pi h{\nu }_0^3/c^3}{e^x-1}.$$

Since $x$ changes continuously as the temperature changes, the denominators can match for all $T$ only if the coefficient of $e^x$ is exactly unity:

$$\boxed{g_1{B}_{12}=g_2{B}_{21}}$$

The remaining numerators must then be equal:

$$\boxed{\frac{A_{21}}{B_{21}}=\frac{8\pi h{\nu }_0^3}{c^3}}$$

or

$$\boxed{A_{21}=\frac{8\pi h{\nu }_0^3}{c^3}{B}_{21}.}$$

These are the Einstein relations when the spectral density is defined per unit ordinary frequency, ${\rho }_{\nu }$.

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What degeneracy changes physically

The first relation is often remembered incorrectly as $B_{12}=B_{21}$. That equality is true only when the two levels have equal degeneracy:

$$g_1=g_2.$$

In general,

$$B_{12}=\frac{g_2}{g_1}{B}_{21}.$$

This is natural once one remembers what a level means. A “level” may summarize several magnetic, angular-momentum, or other substates. With the conventional level-averaged Einstein coefficients, a larger number of accessible substates changes the aggregate transition probability.

For example, suppose

$$g_1=2,g_2=4.$$

Then

$$B_{12}=2B_{21}.$$

Even if the total populations happened to be equal, $N_1=N_2$, absorption would exceed stimulated emission:

$$R_{\mathrm{abs}}=2R_{\mathrm{st}}.$$

The relevant comparison for induced processes is obtained directly from their difference:

$$R_{\mathrm{st}}-R_{\mathrm{abs}}={\rho }_{\nu }\left(N_2{B}_{21}-N_1{B}_{12}\right).$$

Using the degeneracy relation,

$$R_{\mathrm{st}}-R_{\mathrm{abs}}=B_{21}{\rho }_{\nu }\left[N_2-\left(\frac{g_2}{g_1}\right)N_1\right].$$

Therefore, transparency occurs when

$$\frac{N_2}{g_2}=\frac{N_1}{g_1},$$

and stimulated emission dominates absorption only when

$$\frac{N_2}{g_2}>\frac{N_1}{g_1}.$$

This is the degeneracy-corrected population-inversion criterion. It will become central when we calculate gain, but already tells us why comparing $N_2$ and $N_1$ alone can be misleading.

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Frequency conventions: why references can look different

Laser texts use either ordinary frequency $\nu$ in Hz or angular frequency $\omega$ in radians per second:

$$\omega =2\pi \nu .$$

If the spectral density is written per angular frequency, it obeys

$${\rho }_{\omega }(\omega )d\omega ={\rho }_{\nu }(\nu )d\nu ,$$

so that

$${\rho }_{\omega }=\frac{{\rho }_{\nu }}{2\pi }.$$

The numerical value assigned to the corresponding $B$ coefficient must change consistently, because the physical rate $B\rho$ cannot change. Under the angular-frequency convention, the second Einstein relation becomes

$$\boxed{A_{21}=\frac{\hslash {\omega }_0^3}{{\pi }^2{c}^3}{B}_{21}^{(\omega )}.}$$

This is the form commonly encountered in laser-physics notes that use $\rho (\omega )$. It is not a different physical law. It is the same relation expressed using a different spectral variable and hence a differently normalized $B$ coefficient.

A reliable habit is to identify, before using a tabulated $B$ coefficient:

1. whether the spectral density is per Hz or per radian per second;
2. whether it is an energy density, intensity, or mean spectral radiance;
3. whether degeneracies have been included explicitly.

Mixing conventions is one of the most common sources of incorrect factors of $2\pi$, $4\pi$, or a degeneracy ratio.

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A useful physical consequence

The two Einstein relations also quantify the competition between the two downward processes in a thermal radiation field:

$$\frac{R_{\mathrm{sp}}}{R_{\mathrm{st}}}=\frac{A_{21}}{B_{21}{\rho }_{\nu }}.$$

Insert Planck’s law and the relation between $A_{21}$ and $B_{21}$:

$$\frac{R_{\mathrm{sp}}}{R_{\mathrm{st}}}=\exp \left(\frac{h{\nu }_0}{k_{\mathrm{B}}T}\right)-1.$$

At visible or near-infrared frequencies at ordinary temperatures, $h{\nu }_0$ is much larger than $k_{\mathrm{B}}T$, so spontaneous emission overwhelmingly dominates stimulated emission due to the ambient thermal field. A laser does not rely on the room-temperature blackbody field; it builds a highly occupied, selected optical mode and maintains a strongly nonequilibrium population distribution.

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Takeaways and next step

Thermal equilibrium links the three Einstein coefficients through two requirements: atomic populations follow Boltzmann statistics, and equilibrium radiation follows Planck’s spectrum. The resulting relations are

$$\boxed{g_1{B}_{12}=g_2{B}_{21}}$$

and, for spectral energy density per unit ordinary frequency,

$$\boxed{A_{21}=\frac{8\pi h{\nu }_0^3}{c^3}{B}_{21}.}$$

The degeneracy relation means that absorption and stimulated-emission coefficients are equal only for equally degenerate levels. More generally, optical gain requires the upper-level population per substate to exceed the lower-level population per substate.

Next, we will use these coefficients operationally: given a radiation spectral energy density and level populations, you will calculate the absorption and stimulated-emission rates and determine the net induced effect on a field.