Absorption and Emission of Radiation

Point a lamp at a jar of bromine vapor and the orange color visibly fades as light passes through. Somewhere inside that jar, molecules are absorbing photons and jumping to higher energy states. Some of those excited molecules will later throw the energy back out as light, in every direction, whenever they feel like it. This is the phenomenon behind every absorption spectrum, every laser, and every glow-in-the-dark sticker. Molecular spectroscopy is built on three simple radiative processes: stimulated absorption, spontaneous emission, and stimulated emission. Einstein classified them in 1917, decades before anyone had a quantum mechanical theory of the transition dipole moment. This article builds the phenomenological picture. You will meet the three processes. You will see how Einstein connected their rates using thermodynamic equilibrium. You will learn why spontaneous emission depends so strongly on frequency. This groundwork prepares you for the Beer-Lambert law, transition dipole moments, and fluorescence.

Key takeaways

Three radiative processes link any pair of quantized energy levels: stimulated absorption, spontaneous emission, and stimulated emission.
Stimulated absorption and stimulated emission both require an external radiation field and have rates proportional to its intensity.
Spontaneous emission happens with no external field; its rate is a fixed property of the excited state, the Einstein $A$ coefficient.
Einstein coefficients $B_{12}$ , $B_{21}$ , and $A_{21}$ are locked together by requiring blackbody equilibrium (Planck's law) at any temperature.
The ratio $A_{21}/B_{21}$ scales as $\nu^3$ , which is why spontaneous emission dominates in the visible/UV but is negligible in the microwave and radiofrequency.
Key idea: Absorption, spontaneous emission, and stimulated emission are three faces of the same coupling between a molecule and the electromagnetic field, related by fixed ratios fixed by thermal equilibrium.

The core insight

Any two quantized molecular energy levels are connected by exactly three radiative pathways — light-driven excitation up, light-driven de-excitation down, and spontaneous decay down with no light needed — and the relative importance of these three pathways is dictated not by the molecule alone but by the frequency of the transition and the intensity of the ambient radiation field.

The physical picture

Consider a molecule with two energy levels, a lower state $1$ with energy $E_1$ and an upper state $2$ with energy $E_2$ , separated by

\Delta E = E_2 - E_1 = h\nu

This follows directly from the Bohr frequency condition, the link between quantized energy levels and the electromagnetic spectrum introduced earlier. Three things can happen to a molecule sitting in one of these states, and each is a distinct physical event.

Stimulated absorption. A photon of frequency $\nu$ passes near the molecule in state $1$ . The oscillating electric field of the photon exerts a force on the molecule's charge distribution, and if the field oscillates at the right frequency and couples to the right molecular motion, the molecule absorbs the photon's energy and jumps to state $2$ . The photon disappears. This process cannot occur without a radiation field already present — no light, no absorption. The rate at which molecules undergo this transition is proportional to the number of molecules in state $1$ and to the intensity of radiation at the resonant frequency.

Stimulated emission. Now suppose a molecule sits in the upper state $2$ , and a photon of the resonant frequency passes by. Rather than being absorbed, this photon can trigger the molecule to drop to state $1$ , releasing a second photon that is coherent with the first — same frequency, phase, and direction. One photon comes in, two go out. This is the physical basis of the laser (Light Amplification by Stimulated Emission of Radiation). Like absorption, this process needs an ambient field to operate.

Spontaneous emission. An excited molecule in state $2$ can also drop to state $1$ entirely on its own, emitting a photon in a random direction with no external field required to trigger it. This is the same statistical inevitability behind radioactive decay: an individual excited molecule's lifetime is unpredictable, but a population of excited molecules decays with a well-defined, exponential time constant. Spontaneous emission is responsible for fluorescence afterglow, the glow of a neon sign after the discharge briefly cuts out, and the sodium D-line emission from streetlamps.

Figure

Three radiative processes linking two levels, state 1 (E₁) and state 2 (E₂ = E₁ + hν)

The second figure below shows why this matters practically: the balance between spontaneous and stimulated emission flips dramatically as you move across the electromagnetic spectrum, from radiofrequency NMR transitions to visible/UV electronic transitions.

Figure

Ratio A₂₁ divided by B₂₁ρ(ν) grows as ν³, log-log sketch across spectroscopic regions

The core relations

The Bohr frequency condition fixes the transition frequency between two states:

\Delta E = E_2-E_1 = h\nu

where: $h$ is Planck's constant, $\nu$ is the frequency of the photon absorbed or emitted. The three phenomenological rate laws proposed by Einstein for a two-level system are:

\left(\frac{dN_1}{dt}\right)_{\text{abs}} = -B_{12}\,\rho(\nu)\,N_1

where: $N_1$ is the population of the lower level, $\rho(\nu)$ is the spectral energy density of radiation at frequency $\nu$ , $B_{12}$ is the Einstein coefficient for stimulated absorption.

\left(\frac{dN_2}{dt}\right)_{\text{stim em}} = -B_{21}\,\rho(\nu)\,N_2

where: $N_2$ is the population of the upper level, $B_{21}$ is the Einstein coefficient for stimulated emission.

\left(\frac{dN_2}{dt}\right)_{\text{spont}} = -A_{21}\,N_2

where: $A_{21}$ is the Einstein coefficient for spontaneous emission, with units of $\text{s}^{-1}$ (a rate constant, giving an excited-state lifetime $\tau = 1/A_{21}$ ). The Einstein relations connecting the three coefficients, valid for a non-degenerate two-level system, are:

B_{12} = B_{21}, \qquad \frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}

where: $c$ is the speed of light in vacuum; the second relation shows spontaneous emission grows as the cube of the frequency relative to stimulated emission.

The core relations

Derivation ADeriving the Einstein relations from blackbody equilibrium

Goal. Find the fixed relationships between $B_{12}$ , $B_{21}$ , and $A_{21}$ by requiring that a collection of two-level molecules, bathed in blackbody radiation, reach a steady population ratio consistent with both the Boltzmann distribution and Planck's radiation law.

Step 1. Set up the steady-state condition. At thermal equilibrium the populations $N_1$ and $N_2$ are constant in time, so the total rate of upward transitions must equal the total rate of downward transitions:

B_{12}\,\rho(\nu)\,N_1 = B_{21}\,\rho(\nu)\,N_2 + A_{21}\,N_2

The left side is stimulated absorption; the right side is stimulated plus spontaneous emission.

Step 2. Solve for the radiation density. Rearranging for $\rho(\nu)$ isolates the field strength needed to sustain this balance:

\rho(\nu) = \frac{A_{21}}{ \dfrac{B_{12}N_1}{B_{21}N_2} - B_{21}/B_{21}}

More cleanly, divide both sides of the Step 1 equation by $N_2 B_{21}$ :

\rho(\nu) = \frac{A_{21}/B_{21}}{\left(\dfrac{B_{12}}{B_{21}}\right)\left(\dfrac{N_1}{N_2}\right) - 1}

Step 3. Insert the Boltzmann population ratio. For a system in thermal equilibrium at temperature $T$ , the ratio of populations in non-degenerate states follows the Boltzmann distribution:

\frac{N_1}{N_2} = e^{+h\nu/kT}

since state 2 lies higher in energy by $h\nu$ . Substituting:

\rho(\nu) = \frac{A_{21}/B_{21}}{\left(\dfrac{B_{12}}{B_{21}}\right)e^{h\nu/kT} - 1}

Step 4. Match to Planck's law. Planck's radiation law, an independently verified experimental and theoretical result, states:

\rho(\nu) = \frac{8\pi h\nu^3}{c^3}\cdot\frac{1}{e^{h\nu/kT}-1}

Step 5. Compare term by term. The two expressions for $\rho(\nu)$ must be identical at every temperature $T$ . This forces two conditions simultaneously: the prefactor multiplying $e^{h\nu/kT}$ inside the denominator must equal 1, and the overall constant out front must match. The first condition gives:

B_{12} = B_{21}

The stimulated absorption and stimulated emission coefficients between a given pair of non-degenerate states are exactly equal — a result with no classical justification, but forced purely by consistency with Planck's law. The second condition gives:

\boxed{\dfrac{A_{21}}{B_{21}} = \dfrac{8\pi h \nu^3}{c^3}}

This is the central result: spontaneous emission is not an independent, freely adjustable rate. It is locked to the stimulated-emission coefficient by a factor that depends only on the transition frequency, growing as $\nu^3$ .

Worked examples

Example 1 — Comparing spontaneous and stimulated emission for a rotational transition. A microwave rotational transition of a small polar molecule, such as CO, occurs at $\nu = 1.153 \times 10^{11}\ \text{Hz}$ (115.3 GHz, the $J=0\to1$ line). Estimate the ratio $A_{21}/B_{21}$ and compare the two contributions to emission at $T = 300\ \text{K}$ using $\rho(\nu)$ from Planck's law. Step 1: compute $A_{21}/B_{21}$ using

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

With $h = 6.626\times10^{-34}\ \text{J s}$ , $\nu^3 = (1.153\times10^{11})^3 = 1.533\times10^{33}\ \text{Hz}^3$ , and $c^3 = (2.998\times10^8)^3 = 2.694\times10^{25}\ \text{m}^3\text{s}^{-3}$ :

\frac{A_{21}}{B_{21}} = \frac{8\pi (6.626\times10^{-34})(1.533\times10^{33})}{2.694\times10^{25}}

Numerator: $8\pi = 25.13$ ; $25.13 \times 6.626\times10^{-34} = 1.665\times10^{-32}$ ; times $1.533\times10^{33}$ gives $2.553\times10^{1} = 25.53$ .

\frac{A_{21}}{B_{21}} = \frac{25.53}{2.694\times10^{25}} = 9.48\times10^{-25}\ \text{J s m}^{-3}

Step 2: compute $\rho(\nu)$ at $T=300\ \text{K}$ . First, $h\nu/kT$ :

h\nu = (6.626\times10^{-34})(1.153\times10^{11}) = 7.640\times10^{-23}\ \text{J}

kT = (1.381\times10^{-23})(300) = 4.143\times10^{-21}\ \text{J}

\frac{h\nu}{kT} = \frac{7.640\times10^{-23}}{4.143\times10^{-21}} = 0.01844

Since this is small, $e^{h\nu/kT} - 1 \approx 0.01844 + 0.000170 \approx 0.01861$ . Step 3: the ratio of the spontaneous rate to the stimulated rate for a molecule sitting in the upper state is

\frac{A_{21}}{B_{21}\rho(\nu)} = \frac{A_{21}}{B_{21}}\times\frac{1}{\rho(\nu)}

But it is cleaner to use directly:

\frac{A_{21}}{B_{21}\rho(\nu)} = e^{h\nu/kT} - 1 = 0.0186

(This follows immediately from Step 4 of the derivation, rearranged.) So spontaneous emission is only about 1.9% as fast as stimulated emission for this rotational transition at room temperature — stimulated processes dominate completely at microwave frequencies, which is why rotational (and NMR) spectroscopy relies on detecting absorption/stimulated processes rather than waiting for spontaneous glow.

Example 2 — Excited-state lifetime from spontaneous emission for an electronic transition. A molecule has an allowed electronic transition at $\lambda = 300\ \text{nm}$ with Einstein coefficient $A_{21} = 2.0\times10^{8}\ \text{s}^{-1}$ (typical of a strongly allowed UV transition). Find (a) the natural radiative lifetime of the excited state, and (b) the ratio $A_{21}/(B_{21}\rho(\nu))$ at $T = 300\ \text{K}$ , to confirm that spontaneous emission dominates in the UV. (a) The radiative lifetime is the reciprocal of the total spontaneous decay rate:

\tau = \frac{1}{A_{21}} = \frac{1}{2.0\times10^8\ \text{s}^{-1}} = 5.0\times10^{-9}\ \text{s} = 5.0\ \text{ns}

This is a realistic fluorescence lifetime for an allowed transition, consistent with typical aromatic chromophores (nanosecond scale) — a good sanity check. (b) First find $\nu$ :

\nu = \frac{c}{\lambda} = \frac{2.998\times10^8\ \text{m/s}}{3.00\times10^{-7}\ \text{m}} = 9.993\times10^{14}\ \text{Hz}

Then:

h\nu = (6.626\times10^{-34})(9.993\times10^{14}) = 6.622\times10^{-19}\ \text{J}

kT = 4.143\times10^{-21}\ \text{J (as before, at 300 K)}

\frac{h\nu}{kT} = \frac{6.622\times10^{-19}}{4.143\times10^{-21}} = 159.8

\frac{A_{21}}{B_{21}\rho(\nu)} = e^{h\nu/kT} - 1 \approx e^{159.8}

This number is astronomically large (roughly $10^{69}$ ). Physically: at room temperature, blackbody radiation at 300 nm is essentially absent ( $\rho(\nu) \approx 0$ ), so stimulated emission is completely negligible. An excited molecule at this energy relaxes almost exclusively by spontaneous emission (fluorescence) or by non-radiative pathways — never by being "stimulated down" by ambient thermal light. This is precisely why UV/visible spectroscopy is dominated by spontaneous emission (fluorescence, phosphorescence), while microwave and radiofrequency spectroscopy is dominated by stimulated processes.

Limiting cases & common traps

Regime	Dominant process	Why
Radiofrequency (NMR), ~MHz	Stimulated absorption/emission	$\nu^3$ factor tiny; $A_{21}$ negligible
Microwave (rotational), ~GHz	Mostly stimulated	$A_{21}/B_{21}$ small; some spontaneous
Infrared (vibrational)	Mixed, stimulated often still detected via absorption	Intermediate $\nu^3$
Visible/UV (electronic)	Spontaneous emission (fluorescence)	Huge $\nu^3$ ; ambient $\rho(\nu)$ at $h\nu\gg kT$ is negligible

Another trap: assuming $B_{12} = B_{21}$ always holds. This equality is derived here for non-degenerate levels. When the states have different degeneracies $g_1 \neq g_2$ , the correct relation becomes $g_1 B_{12} = g_2 B_{21}$ , a refinement covered when transition dipole moments and selection rules are introduced.

The chemist behind it

Albert Einstein's 1917 paper "Zur Quantentheorie der Strahlung" ("On the Quantum Theory of Radiation") introduced stimulated emission as a logical necessity, purely from thermodynamic consistency with Planck's law, decades before the physical mechanism was understood in quantum mechanical detail. He did not invent the concept out of laser physics — lasers did not exist until 1960 — but out of a demand for internal consistency: any theory of radiation-matter interaction that used only absorption and spontaneous emission could not reproduce Planck's blackbody spectrum. Stimulated emission was the missing ingredient that made the bookkeeping close. This single insight, developed with almost no experimental spectroscopic data, seeded the theoretical foundation for every maser and laser built forty years later.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · Introduced the A and B coefficients in 1917, predicting stimulated emission from pure thermodynamic consistency with Planck's law, decades before the laser existed.

Ask the tutor: How does the transition dipole moment determine the actual size of $B_{12}$ for a specific molecular transition?

Why does stimulated emission produce coherent light while spontaneous emission does not?

In stimulated emission, the emitted photon matches the triggering photon in frequency, phase, and direction of travel, because the process is driven by the existing field mode. Spontaneous emission has no external trigger; the photon is emitted in a random direction with a random phase, determined only by the molecule's internal dynamics. A laser works by amplifying one field mode via stimulated emission, so all emitted photons reinforce each other coherently, unlike the incoherent, omnidirectional glow of ordinary spontaneous fluorescence.

Does spontaneous emission violate energy conservation, since it seems to happen with no obvious trigger?

No. Energy conservation is fully respected: the molecule loses exactly $h\nu$ of internal energy and that energy appears as the emitted photon. What is unusual is that the timing of the event for any single molecule is fundamentally unpredictable, governed by quantum mechanical probability rather than a classical trigger. A population of excited molecules still decays with the definite exponential law $N_2(t) = N_2(0)e^{-A_{21}t}$ , so the statistics are perfectly predictable even though individual events are not.

How does temperature affect the balance between the three radiative processes?

Temperature enters only through the ambient radiation density $\rho(\nu)$ , which follows Planck's law and increases with $T$ at any fixed frequency. Higher $T$ means more ambient photons at frequency $\nu$ , boosting both stimulated absorption and stimulated emission rates equally, since $B_{12} = B_{21}$ . The spontaneous emission rate $A_{21}$ itself does not depend on temperature at all; it is a fixed molecular property, once the frequency is fixed.

How does this connect to the Beer-Lambert law used in real absorption spectroscopy?

The Beer-Lambert law describes how the intensity of a beam decreases as it passes through an absorbing sample, and its microscopic origin is exactly the net stimulated-absorption rate discussed here, summed over all the absorbing molecules along the light path. The absorption coefficient in Beer-Lambert is proportional to $B_{12}$ , the transition dipole moment, and the population difference $N_1 - N_2$ . Spontaneous emission plays essentially no role in that measurement, since it emits in random directions and does not contribute to the transmitted beam.

Why does fluorescence typically occur in the visible or UV, while microwave and radio spectroscopy never show a bright glow?

Fluorescence is spontaneous emission at electronic-transition frequencies, roughly $10^{14}$ to $10^{15}$ Hz, where the $\nu^3$ factor in the Einstein relation makes $A_{21}$ large, often $10^7$ to $10^9\ \text{s}^{-1}$ . Rotational and NMR transitions sit at $10^9$ to $10^{11}$ Hz, where the same $\nu^3$ factor makes $A_{21}$ vanishingly small. An excited nuclear spin state or rotational state essentially never decays by spontaneous photon emission on any laboratory timescale.

Absorption and Emission of Radiation

The physical picture

The core relations

The core relations

Worked examples

Limiting cases & common traps

The chemist behind it

Before this

Where this leads