Plasmon Ruler: Nanoparticle Dimer Coupling

Dimer Geometry

Surface-to-surface gap 12.0 nm

Particle diameter D 40 nm

Decay constant τ 0.23

Max fractional shift A 18%

Coupled Resonance

gap / D = 0Δλ/λ₀ = A·e^−s/τgap / D = 2

Normalized gap s = gap/D

0.300

Fractional red-shift

8.1%

Coupled resonance λ

562 nm

Hot-spot enhancement

×3.2

How it works

An isolated gold nanosphere has a localized surface-plasmon resonance (LSPR) near λ₀ ≈ 520 nm — its conduction electrons slosh collectively at the Fröhlich frequency ω = ω_p/√(1+2ε_m). Bring a second nanosphere close by and their near fields overlap: the two dipoles hybridize into a lower-energy "bonding" mode, red-shifting the coupled resonance.

Jain, Huang & El-Sayed (2007) showed this shift follows a near-universal exponential law once the gap is normalized by particle size:

s = gap / D
Δλ/λ₀ = A · e^(−s/τ)
λ_dimer = λ₀ · (1 + Δλ/λ₀)

with τ ≈ 0.2–0.3 and A ≈ 0.15–0.20 for gold nanosphere dimers, almost independent of particle diameter, shape or dielectric medium — which is exactly what makes it useful as a "plasmon ruler": measuring the resonance shift of a dye-free nanoparticle pair tells you their separation directly, and tracking it over time reveals conformational changes in whatever tether (DNA, a protein hinge) holds the pair together.

Gap — surface-to-surface distance between the two spheres; the scene and the dipole field glow update live.
Diameter D — sphere size; the ruler equation is normalized by D, so the same physical gap produces a smaller shift for bigger particles.
τ, A — the two fit constants of the universal curve; drag them to see how sensitive the ruler's calibration is.
Hot-spot enhancement — the same near-field overlap that shifts λ also concentrates the local |E|-field in the gap, the effect exploited in SERS; shown here as a qualitative glow scaled by the same exponential.

Gold nanospheres Near-field hot spot

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