Einstein Coefficients Calculator

Calculator

Solve for Derive A21, B21, and B12 (recommended) A21 from B21 B21 from A21 B12 from B21 B21 from B12

Pick a target, then provide the required known values.

Spectral input

Wavelength (λ) Frequency (ν)

nm µm m

Hz kHz MHz GHz THz

Wavelength and frequency are internally converted using c = 299,792,458 m/s.

Lower degeneracy (g1)

Needed for B12↔B21 conversions and full derivation.

Upper degeneracy (g2)

Uses g1·B12 = g2·B21.

A21 (s⁻¹)

Provide when deriving B21 or the full set.

B21

Used with A21 relation: A21 = (8πhν³/c³)·B21.

B12

Convert with degeneracies: g1·B12 = g2·B21.

Calculate

Formula used

How to use this calculator

1. Select what you want to solve for from the “Solve for” menu.
2. Choose wavelength or frequency, then enter the value and unit.
3. Enter degeneracies g1 and g2 when converting B-values.
4. Provide the known coefficient (A21, B21, or B12) required by your choice.
5. Press Calculate to view results above the form.
6. Use CSV or PDF buttons to export the results table.

Example data table

1) Why Einstein coefficients matter

Einstein coefficients quantify how atoms and molecules exchange energy with radiation. The spontaneous coefficient A₂₁ sets the natural emission rate (s⁻¹), while B₁₂ and B₂₁ describe absorption and stimulated emission driven by an external spectral energy density. These constants connect spectroscopy, laser thresholds, and radiative lifetimes in a single framework.

2) Two-level model and transition notation

This calculator assumes a two-level transition between an upper state (2) and lower state (1). The transition frequency ν (Hz) and wavelength λ (m) are related by ν = c/λ. Degeneracies g₁ and g₂ represent the number of quantum sublevels in each state and directly scale the absorption and stimulated emission strengths.

3) Core relations used in the calculator

The key radiative linkage is A₂₁ = (8πhν³/c³)·B₂₁, which ties spontaneous emission to stimulated emission through fundamental constants. The symmetry relation g₁·B₁₂ = g₂·B₂₁ ensures detailed balance at thermal equilibrium. Together, they allow you to compute any missing coefficient from common experimental inputs.

4) Practical data sources for A21 and B values

A₂₁ is frequently reported as a radiative decay rate or inverse lifetime (τ_rad ≈ 1/A₂₁) in spectroscopy tables. B values are often inferred from measured cross sections, oscillator strengths, or calibrated gain/absorption measurements. When you input either A₂₁ or B₂₁, the calculator derives the consistent set and reports a relation check to confirm degeneracy consistency.

5) Interpreting outputs and magnitudes

Large A₂₁ implies short radiative lifetimes and broader natural linewidths, common for strongly allowed transitions. Smaller A₂₁ corresponds to longer-lived excited states, as in weakly allowed or forbidden lines. B coefficients scale with ν⁻³ through the A–B relation, so infrared transitions can show very different B magnitudes compared to ultraviolet transitions even when lifetimes are comparable.

6) Degeneracy effects and selection rules

Degeneracies enter through g₂/g₁. If g₂ > g₁, absorption can be stronger than stimulated emission for the same B₂₁ baseline because more upper sublevels are accessible. In practice, g values follow the level’s angular momentum structure (for example, 2J+1 for many atomic cases) and the chosen polarization selection rules.

7) Laser relevance and gain intuition

Stimulated emission B₂₁ links directly to optical gain when a population inversion exists. For a fixed transition, larger B₂₁ supports higher stimulated emission rates at a given radiation density, which helps reduce threshold requirements. Combining ν, A₂₁, and degeneracies lets you benchmark candidate laser media and compare transitions on a consistent radiative basis.

8) Quality checks and reporting

Use the calculator’s “Relation check” to verify g₁·B₁₂ equals g₂·B₂₁ within rounding. For publications, record the spectral input (λ or ν), the known coefficient source, and the computed set. The built-in CSV export is convenient for lab notebooks, while the PDF export is suitable for attaching to reports.

FAQs

1) What does A21 physically represent?

A21 is the spontaneous emission rate from level 2 to level 1. It is the probability per second that an excited state emits a photon without external stimulation.

2) When should I enter wavelength instead of frequency?

Use the form you measured most directly. The calculator converts between them using ν = c/λ, so results are identical when values are consistent.

3) Why do I need g1 and g2?

Degeneracies account for how many sublevels exist in each state. They set the balance between absorption and stimulated emission through g1·B12 = g2·B21.

4) Can I compute all coefficients from A21 alone?

Yes, if you also provide the transition wavelength or frequency. The calculator derives B21 from A21 and then uses degeneracies to compute B12.

5) What units do the B coefficients use here?

This tool reports B values in a radiation-density form consistent with A21 = (8πhν³/c³)·B21. Keep units consistent when comparing with other conventions.

6) Why does B depend strongly on frequency?

The A–B relation contains ν³. As frequency changes, the conversion factor between A21 and B21 changes rapidly, so B magnitudes can vary widely across spectra.

7) How can I check if my inputs are consistent?

Compute the full set and review the “Relation check” row. Matching values indicate g1, g2, and the provided B value agree with detailed balance.