Spontaneous Emission Rate Calculator

Formula Used

For an electric-dipole transition in vacuum, the spontaneous emission rate is:

A = (ω³ |μ|²) / (3π ε₀ ħ c³),   with ω = 2πν and ν = c/λ

When using oscillator strength f with wavelength λ (vacuum reference):

A = (2π e² / (ε₀ mₑ c)) · (f / λ²)

Medium scaling here uses a simple A ≈ n·A(vacuum) option as an approximation.

How to Use This Calculator

1. Select a method that matches your available data.
2. Enter either wavelength or frequency to set the transition.
3. Provide dipole moment or oscillator strength when required.
4. Optionally enable medium scaling and set the refractive index.
5. Press Calculate to view results above the form.
6. Use export buttons to save CSV or PDF outputs.

Technical Article

1) Why spontaneous emission matters

The spontaneous emission rate A sets how quickly an excited state relaxes by radiating a photon. In lasers, LEDs, and single-photon sources, A controls radiative efficiency, modulation bandwidth, and brightness. A larger A means a shorter lifetime and a broader natural linewidth.

2) Connecting wavelength to time

Optical transitions typically span λ ≈ 200–2000 nm (ultraviolet to near‑infrared), corresponding to ν ≈ 150–1500 THz. For many allowed electric‑dipole transitions, lifetimes often fall in the 1–100 ns range, implying A ≈ 10⁷–10⁹ s⁻¹.

3) Dipole moment method overview

When you know the transition dipole moment μ, the calculator uses A = (ω³|μ|²)/(3π ε₀ ħ c³). Molecular dipoles are often reported in Debye, with 1 D = 3.33564×10⁻³⁰ C·m. Because A ∝ ω³, shorter wavelengths can increase emission rates dramatically.

4) Oscillator strength method overview

Spectroscopy often reports oscillator strength f, commonly ranging from 10⁻³ (weak) to 1 (strong). The calculator applies A = (2π e²/(ε₀ mₑ c))·(f/λ²). For example, at λ ≈ 600 nm, increasing f by ten increases A by ten.

5) Lifetime, linewidth, and resolution

Radiative lifetime is τ = 1/A. The natural (lifetime‑limited) linewidth is Δν = A/(2π). A lifetime of 10 ns corresponds to A = 1×10⁸ s⁻¹ and Δν ≈ 1.6×10⁷ Hz, which is relevant in precision spectroscopy and narrowband filtering.

6) Energy checks using photon energy

The calculator also reports E = hν in joules and electronvolts. A quick check: λ = 620 nm corresponds to about 2.0 eV. Matching energy and wavelength helps validate inputs and avoids unit mistakes.

7) Medium and refractive index considerations

Real emitters often sit in dielectrics, waveguides, or cavities. Emission can scale with refractive index and local fields, and can be enhanced by the Purcell effect. This tool offers a simple A ≈ n·A(vacuum) option for quick comparisons, but detailed design may require a full photonic‑environment model.

8) Practical workflow for reliable results

Start with wavelength or frequency, choose the method matching your data, and keep units consistent. Compare A against expected ranges (e.g., 10⁷–10⁹ s⁻¹ for many allowed transitions). Export CSV/PDF to document assumptions, then refine with measured lifetimes or calibrated spectroscopy.

FAQs

1) What does the rate A represent?

A is the probability per second that an excited state emits a photon spontaneously. Higher A means faster decay, shorter radiative lifetime, and a larger natural linewidth.

2) Should I enter wavelength and frequency together?

You can, but the calculator uses wavelength when both are provided. Enter one value with the correct unit to reduce input conflicts and simplify validation.

3) Which dipole unit should I choose?

Use Debye for molecular transitions, C·m for SI values, and eÅ for electronic structure outputs. The tool converts each to C·m before computing A.

4) What is a typical oscillator strength range?

Weak transitions can be around 10⁻³ to 10⁻², moderate around 10⁻¹, and strong allowed transitions can approach 1. Values outside this range may indicate a data or unit issue.

5) What does the linewidth Δν mean?

Δν is the lifetime-limited spectral width from spontaneous decay, computed as A/(2π). It is a fundamental lower bound on linewidth in the absence of additional broadening mechanisms.

6) How should I use the refractive index option?

The refractive index scaling is a quick approximation for comparing vacuum and dielectric environments. For accurate device modeling, consider local-field factors, geometry, and cavity or waveguide effects.

7) Can this compute non-radiative decay?

No. This calculator targets radiative spontaneous emission. If you have measured total lifetime, you can compare it to the radiative τ to estimate non-radiative contributions indirectly.