Shining Light on the Math: A Guide to Quantum Yield and Lifetime

Shining Light on the Math: Quantum Yield & Fluorescence Lifetime Guide

In photophysics, Quantum Yield (Φ) and Fluorescence Lifetime (τ) are foundational parameters. They reveal both the efficiency of light emission and the dynamics of the excited state—crucial for fluorophores, quantum dots, sensors, and photovoltaics.

1. Quantum Yield (Φ)

Φ is the fraction of absorbed photons that are re-emitted as fluorescence:

$$\Phi = \frac{\text{photons emitted}}{\text{photons absorbed}}$$

Φ = 0.80 means 80% of absorbed photons produce fluorescence; the rest are lost to non-radiative processes (heat, vibrations, etc.).

Relative Method (Common Lab Practice)

$$\Phi_{\text{sample}} = \Phi_{\text{ref}} \left( \frac{I_{\text{sample}}/A_{\text{sample}}}{I_{\text{ref}}/A_{\text{ref}}} \right) \left( \frac{n_{\text{sample}}^2}{n_{\text{ref}}^2} \right)$$

(I = integrated fluorescence intensity, A = absorbance at excitation wavelength, n = refractive index. Use optical density correction 1–10^–A for accuracy if OD > ~0.05.)

2. Fluorescence Lifetime (τ)

τ is the average time a molecule remains in the excited state before decaying to the ground state — typically 0.5–10 ns for organic fluorophores.

3. Connecting the Dots: Decay Rates

Two main pathways compete:

Radiative decay (fluorescence): rate constant kr
Non-radiative decay: rate constant knr (internal conversion, intersystem crossing, quenching, etc.)

$$\tau = \frac{1}{k_r + k_{nr}}$$

$$\Phi = \frac{k_r}{k_r + k_{nr}}$$

Combining these yields the most powerful relationship:

$$\Phi = k_r \cdot \tau$$

Interactive Calculator: kr & knr from Φ and τ

Calculate Radiative & Non-Radiative Rates

Quantum Yield Φ (0 – 1) Typical range: 0.1 – 0.95 for good emitters

Lifetime τ (ns) Common range: 1–10 ns

Quick Presets Fills fields with realistic literature values

Enter Φ and τ above (or choose preset) → see kr, knr, total rate.

Summary Table

Metric	Symbol	Definition	Typical Units / Range
Quantum Yield	Φ	Photons emitted / absorbed	0 – 1
Lifetime	τ	Average excited-state duration	nanoseconds (ns)
Radiative Rate	kr	Rate of photon emission	10⁷ – 10⁹ s⁻¹ (often 10–1000 × 10⁶ s⁻¹)
Non-Radiative Rate	knr	Rate of non-emissive decay	s⁻¹ (varies widely)

Got your own Φ and τ data? Paste values or ask for interpretation — happy to help derive quenching mechanisms or compare to literature!