🟡 Localized Surface Plasmon Resonance (LSPR)

The Fröhlich Condition

In the quasi-static limit (r ≪ λ), a metal nanoparticle is a polarizable sphere. Resonance occurs when the dipole polarizability denominator is minimised — the Fröhlich condition:

ε_r(ω) = −2 ε_m    where ε_m = n_m²

Au in water (ε_m ≈ 1.77):  ε_r ≈ −3.54 → resonance ~490–520 nm

Extinction cross-section:
  C_ext ∝ ε_i / [(ε_r + 2ε_m)² + ε_i²]

Replacing water (n = 1.333) with a protein layer (n ≈ 1.45) shifts ε_m upward, red-shifting the resonance. This is the LSPR sensing principle. Au's broad peak (~80–100 nm FWHM) reflects large ε_i from d-band interband transitions; Ag's peak is sharper because its interband threshold is in the UV.

Extinction is the sum of absorption and scattering: C_ext = C_abs + C_sca. The quasi-static expression above is the absorption cross-section in the small-particle limit. For r ≲ 25 nm extinction is almost entirely absorption (∝ V); for r ≳ 50 nm scattering (∝ V²) contributes substantially, broadening and red-shifting the peak via radiation damping. The strong scattering of larger nanoparticles is what makes single-particle dark-field LSPR sensing possible.

Dielectric data: uses Johnson & Christy (1972) tabulated n,k values, converted to ε(λ) = (n + ik)² on a 2 nm grid (380–1000 nm) — avoids Drude model double-counting of interband contributions.

⚠ For r < 10 nm: surface electron scattering broadens the linewidth (Kreibig correction: γ_eff = γ_bulk + A·v_F/r) — the simulator shows a warning badge but does not apply the correction.

Shape Effects — Tuning Across the Spectrum

Nanorods have two distinct plasmon modes:

• Transverse (~520 nm for Au, insensitive to aspect ratio)
• Longitudinal — red-shifts strongly with AR: λ_max ≈ 96 × AR + 418 nm (Link, Mohamed & El-Sayed 1999, Gans theory; coefficients shown are for ~10 nm-diameter Au rods in aqueous medium and shift with rod diameter and surrounding ε_m)

AR=2 → ~610 nm   AR=3 → ~706 nm   AR=4 → ~802 nm   AR=5 → ~898 nm

The longitudinal mode also has higher bulk RI sensitivity (m ≈ 300–800 nm/RIU for rods vs ~100 nm/RIU for spheres). Nanoshells (silica core + gold shell) are tunable 600–1500 nm; nanostars have tip-localised hot spots giving SERS enhancement 10³–10⁴×.

LSPR Sensing — Wavelength Shift and Figure of Merit

Δλ_res = m × Δn × (1 − exp(−2d / l_d))

m    bulk RI sensitivity (nm/RIU)
     ~100       40 nm Au sphere
     ~300–600   Au nanorod (longitudinal)
Δn   n_layer − n_medium ≈ 0.117 (protein in water)
d    adlayer thickness (nm)
l_d  evanescent decay length ~ 0.2–0.5 × r (5–15 nm typical;
     shape- and geometry-dependent; Haes & Van Duyne 2004)

FOM = m / FWHM  (RIU⁻¹)
	     Au sphere  ~0.5–1.5    Au rod  ~3–8    Ag sphere  ~4–6

For a protein monolayer (d ≈ 5 nm) on 40 nm Au: Δλ ≈ 7 nm. Detectable limit ~0.3 nm → LOD ≈ few hundred pg/mm². Less sensitive than SPR per unit area, but the short l_d makes LSPR robust in complex biological matrices.