Absorption Cross-Section Calculator

σ = ε · 1000 · ln(10) / N_A — convert the molar absorption (extinction) coefficient to the cross-section per molecule and back, with number density and Napierian absorption coefficient.

Model: Exact

Cross-section σ

1.912e-16 cm²

1.912e-20 m², 1.912 Ų

Number density n

6.022e+15 cm⁻³

n = N_A · c

Napierian coefficient α

1.151 cm⁻¹

α = σ·n; 1/e length 0.8686 cm

Absorbance over 1 cm

0.5000

A = ε·c·l = α·l / ln 10

σ = ln(10) · ε / N_A

σ (cm²) = 3.8235 × 10⁻²¹ × ε (L·mol⁻¹·cm⁻¹)

T = e^(−σ·n·l) = 10^(−ε·c·l)

The 1000 in the lab-unit form is cm³ per litre, and ln 10 turns the decadic ε into the Napierian σ. Typical peak values: fluorescent proteins and dyes, ε ≈ 5 × 10⁴–2.5 × 10⁵, give σ ≈ 2–10 × 10⁻¹⁶ cm² (EGFP, ε = 55 900 at 488 nm: 2.14 × 10⁻¹⁶ cm²); aromatic amino acids in proteins at 280 nm, ε ≈ 1.5 × 10³ (Tyr) to 5.5 × 10³ (Trp), give 6 × 10⁻¹⁸–2 × 10⁻¹⁷ cm² per residue. A free atom on a resonance line is far stronger: a two-level atom at rest peaks at 3λ²/2π, 1.66 × 10⁻⁹ cm² at the 589 nm sodium line, and Doppler broadening in a vapour lowers the peak by roughly two orders of magnitude.

References: IUPAC Gold Book, "absorption cross section"; FPbase (EGFP); Pace et al. (1995), Protein Sci. 4, 2411 (Trp, Tyr at 280 nm); Foot, Atomic Physics (2005) (resonant cross-section).

Model and references

Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers

Exact conversion σ = ln 10 · ε / N_A between the decadic molar coefficient and the Napierian cross-section, with α = σ·N_A·c. The typical values quoted are peak values for single absorbers in dilute solution or vapour.

  1. IUPAC Compendium of Chemical Terminology (Gold Book), entries “absorption cross section” and “molar (decadic) absorption coefficient”.
  2. Pace C. N., Vajdos F., Fee L., Grimsley G., Gray T. (1995). How to measure and predict the molar absorption coefficient of a protein. Protein Sci. 4, 2411–2423. doi:10.1002/pro.5560041120
  3. Lambert T. J. (2019). FPbase: a community-editable fluorescent protein database. Nat. Methods 16, 277–278 (EGFP ε). doi:10.1038/s41592-019-0352-8
  4. Foot C. J. (2005). Atomic Physics. Oxford University Press (resonant cross-section 3λ²/2π).
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