6.1 Radiative decay processes

Radiative Decay Processes

Radiative decay is what happens when an excited molecule releases its extra energy by emitting a photon. The two main types, fluorescence and phosphorescence, differ in which excited state the photon comes from and how long the process takes. These processes underpin technologies from biological imaging to LED lighting, so understanding the mechanisms and the factors that control emission rates is essential.

Process of Fluorescence

Fluorescence is the emission of light from an excited singlet state ($S_1$). Because the transition from $S_1$ back to the ground state $S_0$ is spin-allowed, it happens fast, typically on the order of nanoseconds ($10^{-9}$ s).

A Jablonski diagram is the standard way to visualize what's going on. The sequence looks like this:

1. A photon is absorbed, promoting the molecule from $S_0$ to a higher vibrational level of $S_1$ (or $S_2$, etc.).
2. The molecule undergoes vibrational relaxation, quickly losing energy to surrounding molecules and settling into the lowest vibrational level of $S_1$.
3. From there, the molecule emits a photon and drops back to some vibrational level of $S_0$.

Because energy is lost during vibrational relaxation, the emitted photon always has less energy (longer wavelength) than the absorbed photon. The difference in wavelength between the absorption maximum and the emission maximum is called the Stokes shift.

Two quantities you need to know for characterizing fluorescence:

• Quantum yield ($\Phi_f$): the ratio of photons emitted to photons absorbed. A quantum yield of 1.0 means every absorbed photon leads to an emitted photon; real fluorophores are always less than that because non-radiative pathways compete.
• Fluorescence lifetime ($\tau_f$): the average time a molecule spends in $S_1$ before emitting. It's defined as the inverse of the total decay rate constant.

One more pattern worth remembering: the mirror image rule. The emission spectrum of many molecules is roughly a mirror image of the absorption spectrum. This happens because the spacing of vibrational levels in $S_0$ and $S_1$ is often similar, so the same set of vibronic transitions appears in both spectra, just reversed in energy order.

Mechanism of Phosphorescence vs. Fluorescence

Phosphorescence is emission from an excited triplet state ($T_1$). The transition $T_1 \rightarrow S_0$ is spin-forbidden, which is why phosphorescence is much slower than fluorescence, with lifetimes ranging from microseconds to seconds (or even longer).

To reach $T_1$, the molecule first has to undergo intersystem crossing (ISC), a non-radiative transition from $S_1$ to $T_1$. ISC requires a spin flip, which is normally unlikely. However, spin-orbit coupling can mix singlet and triplet character, making ISC much more efficient. Heavy atoms (like bromine, iodine, or transition metals) dramatically enhance spin-orbit coupling, which is why heavy-atom-containing molecules are often strong phosphors.

Because $T_1$ sits lower in energy than $S_1$, phosphorescence photons are lower in energy (longer wavelength) than fluorescence photons from the same molecule.

Here's a quick comparison:

Phosphorescence is often quenched at room temperature because the long-lived triplet state gives non-radiative processes (collisions, oxygen quenching) plenty of time to drain the energy. Rigid environments like frozen glasses or polymer matrices suppress molecular motion and protect the triplet state, which is why phosphorescence experiments are frequently run at 77 K.

Factors Influencing Radiative Decay Rates

Several interconnected factors determine how fast (and how efficiently) a molecule emits.

Einstein coefficients provide the fundamental framework. The spontaneous emission coefficient $A_{21}$ gives the rate at which an excited molecule emits a photon on its own, while $B_{12}$ and $B_{21}$ describe stimulated absorption and emission rates, respectively. A larger $A_{21}$ means a shorter radiative lifetime. For a two-level system, the radiative lifetime is simply $\tau_{rad} = 1/A_{21}$.

Oscillator strength ($f$) quantifies how strongly a transition couples to the electromagnetic field. Transitions with large oscillator strength have large $A_{21}$ values and therefore short radiative lifetimes. Fully allowed transitions typically have $f$ values near 1; forbidden transitions have $f$ values orders of magnitude smaller.

The Franck-Condon principle adds another layer. Transition probability depends on the overlap integral between the vibrational wavefunctions of the initial and final electronic states. Good overlap means a more probable (and therefore faster) transition. This is why the shape of emission spectra, and even the overall emission rate, depends on the geometry change between ground and excited states.

Environmental and structural factors also matter:

• Solvent polarity can shift energy levels and change the emission wavelength and rate. Polar solvents stabilize charge-transfer excited states, often causing red-shifted emission.
• Viscosity and rigidity reduce molecular motion, suppressing non-radiative decay and increasing the observed quantum yield.
• Temperature affects the competition between radiative and non-radiative pathways. Lower temperatures generally favor emission.
• Conjugation length influences the energy gap and oscillator strength. Extended conjugation typically red-shifts emission and can increase the radiative rate.
• Heavy atoms enhance spin-orbit coupling, boosting ISC and phosphorescence at the expense of fluorescence.

Keep in mind that the observed decay rate is the sum of radiative and non-radiative rate constants. Even if the radiative rate is large, a molecule won't fluoresce efficiently if non-radiative pathways (internal conversion, vibrational relaxation, energy transfer) are faster.

Applications of Radiative Decay Principles

These emission processes show up across a wide range of technologies:

• Fluorescence microscopy uses fluorescent labels to image specific structures inside cells with high spatial resolution. Techniques like confocal and super-resolution microscopy rely on carefully chosen fluorophores with known quantum yields and lifetimes.
• Fluorescent probes and sensors detect local conditions such as pH, metal ion concentrations ($Ca^{2+}$, $Zn^{2+}$), or oxygen levels. The probe's emission intensity or wavelength shifts in response to the analyte.
• Photodynamic therapy (PDT) uses photosensitizer molecules that undergo ISC to populate long-lived triplet states. These triplet states transfer energy to molecular oxygen, generating reactive singlet oxygen ($^1O_2$) that destroys cancer cells.
• LEDs and OLEDs convert electrical energy into light through electroluminescence. Phosphorescent emitters (often iridium or platinum complexes) are used in OLEDs because they can harvest both singlet and triplet excitons, pushing internal quantum efficiency toward 100%.
• Phosphorescent materials in glow-in-the-dark products (safety signs, watch dials) store energy in triplet states and release it slowly after the excitation source is removed.
• Laser technology relies on stimulated emission from a population-inverted medium. The Einstein $B$ coefficients govern the stimulated emission rate, and the gain medium's radiative properties determine the laser wavelength and efficiency.
• Analytical fluorescence techniques such as fluorescence spectroscopy, fluorescence lifetime imaging (FLIM), and Förster resonance energy transfer (FRET) measurements are standard tools for studying molecular structure, dynamics, and interactions.