Scattering, absorption and extinction cross-sections

The scattering cross section (symbol C_sca or σ_s) is an area; its meaning: multiplied by the intensity of the incident light (power per area) it gives the total power the particle scatters in all directions. A particle with C_sca = 1000 nm² scatters as much light as falls on a 1000 nm² window.

The absorption cross section C_abs is the area that, times the incident intensity, gives the power the particle absorbs and turns into heat. It sets how hot a nanoparticle gets under illumination and how strongly a dye or particle absorbs.

Extinction is everything removed from the beam: C_ext = C_sca + C_abs. It is what a UV-vis spectrometer measures: for N particles per volume and path length ℓ the absorbance is A = N C_ext ℓ / ln 10 (Beer–Lambert).

For a sphere, exactly (Mie theory, derived from Maxwell's equations by expanding the fields in multipoles): C_sca = (2π/k²) Σ (2n+1)(|a_n|² + |b_n|²) and C_ext = (2π/k²) Σ (2n+1) Re(a_n + b_n), with k = 2πn_m/λ and the Mie coefficients a_n, b_n.

For particles much smaller than the wavelength (Rayleigh limit): C_sca = (8π/3) k⁴ r⁶ |(m²−1)/(m²+2)|² and C_abs = 4πk r³ Im[(m²−1)/(m²+2)], with m the particle's index relative to the medium.

The SI unit is the square metre (m²); for nanoparticles nm² (1 nm² = 10⁻¹⁸ m²) and for molecules cm². Nuclear and particle physics use the barn, 10⁻²⁸ m², for the same idea.

The cross section of a particle, layer or molecule as an equivalent area that accounts for everything it does to the beam: an effective scattering or absorption cross section of, for example, a nanoparticle in a medium or an averaged ensemble. With the Beer–Lambert law, an absorption cross section σ and N absorbers per volume give the absorbance A = N σ ℓ / ln 10.

The cross section divided by the particle's geometric cross section: Q = C / πr². It is dimensionless and can exceed 1: a plasmonic particle at resonance has Q_ext of several, and for large particles Q_ext tends to 2 (the extinction paradox: the particle removes the light that hits it and, by diffraction, as much again).

ε = N_A C_ext / (1000 ln 10), with C_ext in cm² and ε in M⁻¹ cm⁻¹; numerically ε = 2.615 × 10²⁰ × C_ext[cm²]. A 20 nm gold sphere in water (C_ext ≈ 430 nm² at 521 nm) has ε ≈ 1.1 × 10⁹ M⁻¹ cm⁻¹.

The cross section belongs to one particle (an area). The absorption coefficient belongs to a material or a suspension: α = N C_abs, in 1/length, for N particles per volume. The molar absorption coefficient ε is the same per mole concentration, in M⁻¹ cm⁻¹.

dC_sca/dΩ: the power scattered into a small solid angle around a direction, per unit incident intensity. Integrated over all directions it gives C_sca. Small particles scatter symmetrically forward and backward; large ones mostly forward.

At a resonance the particle interacts with light well beyond its geometric shadow: a 40–80 nm gold sphere at its plasmon has Q_ext of about 3–6, a 40 nm silver sphere about 20. And any large particle reaches Q_ext ≈ 2 because diffraction around its edge removes as much light from the forward beam as its shadow.

Small particles absorb more: absorption grows with the volume (r³), scattering with the volume squared (r⁶). For gold spheres in water the two are equal at about 85 nm (43 % scattering at 80 nm, 55 % at 90 nm); for silver at about 35 nm; aluminum particles over 30 nm mostly scatter.

Use a scattered-field (total-field / scattered-field) setup and integrate the Poynting flux of the scattered field over a closed surface around the particle for C_sca, and the power dissipated inside the particle for C_abs; divide by the incident intensity. For spheres, layered spheres, spheroids and cylinders the exact methods here give the answer in a second, which is a good check of your mesh and boundary settings.

The radar (backscattering) cross section is σ = 4π × dC_sca/dΩ in the backward direction. For a perfectly conducting sphere much larger than the wavelength it tends to the geometric area πr²; near r ≈ λ/2π it oscillates (the Mie or resonance region).