Calculator Inputs
Use any one known coefficient or lifetime, then provide one spectral quantity and the state degeneracies.
Formula Used
The calculator applies the standard Einstein relations for two-level radiative transitions. You can start from one known coefficient or the radiative lifetime.
How to Use This Calculator
- Select the known input type: A21, B21, B12, or radiative lifetime.
- Enter the numerical value for that known quantity.
- Choose one spectral input mode: wavelength, frequency, or energy gap.
- Enter the spectral value and select the matching unit.
- Provide lower and upper degeneracies, g1 and g2.
- Optionally add spectral energy density to estimate absorption and stimulated rates.
- Press the calculate button to show derived coefficients above the form.
- Use the CSV or PDF buttons to export your computed results.
Example Data Table
These sample rows illustrate realistic workflows for using the calculator.
| Case | Known Input | Wavelength | g1 | g2 | Derived A21 | Derived B21 | Derived B12 |
|---|---|---|---|---|---|---|---|
| Sodium-like example | A21 = 6.200000e+07 s⁻¹ | 589 nm | 2 | 4 | 6.200000e+07 s⁻¹ | 7.607495e+20 m³·J⁻¹·s⁻² | 1.521499e+21 m³·J⁻¹·s⁻² |
| Rubidium-like example | τ = 2.620000e-08 s | 780 nm | 2 | 2 | 3.816794e+07 s⁻¹ | 1.087644e+21 m³·J⁻¹·s⁻² | 1.087644e+21 m³·J⁻¹·s⁻² |
| Lyman-α style example | A21 = 6.265000e+08 s⁻¹ | 121.6 nm | 2 | 6 | 6.265000e+08 s⁻¹ | 6.764348e+19 m³·J⁻¹·s⁻² | 2.029304e+20 m³·J⁻¹·s⁻² |
Why These Results Matter
Einstein coefficients connect microscopic atomic transition probabilities with measurable optical behavior. They help analyze spontaneous emission, laser gain, radiative lifetimes, detailed balance, and population-dependent light-matter interactions. This calculator is useful for spectroscopy, astrophysics, photonics, atomic physics, and teaching applications.
FAQs
1) What does A21 represent physically?
A21 is the spontaneous emission probability per second from the upper state to the lower state. Its inverse gives the radiative lifetime when that decay channel dominates.
2) Why are B12 and B21 often different?
They differ when the lower and upper degeneracies are unequal. The detailed-balance relation g1B12 = g2B21 sets the correct ratio between absorption and stimulated emission.
3) Can I enter wavelength instead of frequency?
Yes. The calculator accepts wavelength, frequency, or energy gap. It converts the chosen spectral quantity internally using ΔE = hν = hc/λ.
4) What unit does this calculator use for B coefficients?
The displayed unit is m³·J⁻¹·s⁻², consistent with using spectral energy density per unit frequency interval in J·m⁻³·Hz⁻¹.
5) Why is a logarithmic chart used?
Einstein coefficients and transition rates can differ by many orders of magnitude. A log-scale graph lets you compare them visually without hiding smaller values.
6) What happens if I leave spectral energy density blank?
The calculator still computes A21, B12, B21, lifetime, and spectral properties. It simply skips absorption and stimulated-rate calculations that require ρ(ν).
7) When is the critical energy density useful?
It marks the spectral energy density at which stimulated emission equals spontaneous emission. That threshold helps interpret when radiation fields begin dominating downward transitions.
8) Can this tool be used for laser and spectroscopy studies?
Yes. It is helpful for estimating transition strengths, radiative lifetimes, gain-related quantities, and consistency checks in spectroscopy, laser physics, and atomic modeling.