Formula used
A₂₁ = (8πhν³ / c³)B₂₁g₁B₁₂ = g₂B₂₁τ = 1 / A₂₁Rabs = B₁₂u(ν), Rstim = B₂₁u(ν), Rspont = A₂₁u(ν,T) = [8πhν³ / c³] / [exp(hν/kT) − 1]This page uses spectral energy density per hertz. Therefore, each B coefficient has SI units of m³ J⁻¹ s⁻². Other textbook conventions can place factors of 2π elsewhere.
How to use this calculator
- Choose which coefficient or transition quantity is already known.
- Enter a wavelength, frequency, photon energy, or wavenumber.
- Provide the lower and upper statistical degeneracies.
- Select a direct radiation field or blackbody temperature.
- Add populations when total upward and downward rates are needed.
- Press calculate, then review coefficients, rates, checks, and steps.
Einstein coefficients and radiative transitions
Einstein coefficients describe how two quantum energy levels exchange radiation. The A coefficient measures spontaneous emission from an upper state. It gives a probability per unit time for decay without an applied field.
The B coefficients describe field-driven processes. B₁₂ controls absorption from the lower state. B₂₁ controls stimulated emission from the upper state. Their ratio depends on the statistical degeneracies of both levels.
A transition frequency links all three coefficients through the radiation mode density. Higher frequencies strongly increase spontaneous emission because the relation contains ν cubed. This dependence explains why many optical transitions decay faster than low-frequency transitions.
The reciprocal of A₂₁ gives the ideal radiative lifetime. Measured lifetimes can differ when nonradiative decay, collisions, quenching, or multiple channels are present. A real upper level may require summing every allowed A coefficient.
Spectral energy density determines induced transition rates. The calculator converts wavelength-domain and angular-frequency-domain inputs into u(ν). This conversion is essential because each spectral convention carries different differential units.
Blackbody mode uses Planck’s law at the selected temperature. It estimates the equilibrium radiation field at the transition frequency. The result helps compare spontaneous emission with thermally stimulated emission.
Population fields extend the calculation beyond one particle. They estimate total upward and downward events per second. A negative net downward result indicates that upward absorption dominates under the entered conditions.
Dipole and oscillator-strength modes provide practical spectroscopy links. These formulas assume an electric-dipole transition and simplified level structure. Magnetic-dipole, quadrupole, cavity, and strongly dispersive systems need specialized models.
The optional medium calculation replaces c with c divided by refractive index. It is a useful approximation for simple comparisons. Accurate condensed-matter work may require local-field corrections and a full photonic environment.
Use consistent degeneracies, transition definitions, and spectral units. Check whether a source quotes B coefficients per hertz or per angular frequency. That distinction prevents hidden factors of 2π.
Example data
| Example | Known input | Transition | g₁ | g₂ | Use |
|---|---|---|---|---|---|
| Hydrogen Lyman-α style example | A₂₁ = 6.265 × 10⁸ s⁻¹ | 121.567 nm | 2 | 4 | Coefficient conversion and lifetime |
| Optical dipole example | Dipole = 2.5 D | 500 nm | 1 | 3 | Estimate A₂₁ from dipole strength |
| Thermal microwave example | B₂₁ = 1 × 10¹⁸ SI | 100 GHz | 1 | 3 | Compare blackbody and spontaneous rates |
Frequently asked questions
What does A₂₁ represent?
A₂₁ is the spontaneous-emission probability per unit time. Its SI unit is s⁻¹.
What is the radiative lifetime?
For one isolated decay channel, the lifetime is 1/A₂₁. Multiple channels require the reciprocal of their summed A values.
Why are B₁₂ and B₂₁ different?
They differ when the two levels have unequal degeneracies. The relation is g₁B₁₂ = g₂B₂₁.
Which radiation-density convention is used?
The calculator reports u(ν) per hertz. Inputs in other domains are converted before rates are calculated.
Can I enter wavelength instead of frequency?
Yes. The calculator converts wavelength, energy, or wavenumber into transition frequency automatically.
What does critical spectral density mean?
It is the u(ν) value where stimulated emission equals spontaneous emission for one upper-state particle.
Does the dipole mode work for every transition?
No. It assumes an electric-dipole transition. Forbidden or higher-multipole transitions need other formulas.
Why can measured lifetime differ?
Collisions, quenching, nonradiative loss, and multiple radiative channels can alter the observed lifetime.
Is the refractive-index option exact?
No. It is a simplified c/n model. Complex media may require local-field and photonic-density corrections.