Light & Matter | Complex Refractive Index (ñ = n + ik) & Dispersion Model

Wavelength λ 632.8 nm

Photon Energy E 1.959 eV

Refraction n (calc / exp) 0.191 (exp 0.180)

Extinction k (calc / exp) 3.478 (exp 3.433)

Permittivity ε₁ / ε₂ -12.06 / 1.33

Reflectance R 94.3%

Skin Depth δ 14.5 nm

λ (nm)	Band	Energy E (eV)	n (calc)	n (exp)	k (calc)	k (exp)	ε₁ (Re)	ε₂ (Im)	R (Reflectance %)	α (cm⁻¹)	Skin Depth δ (nm)

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⚛ Outer Orbitals & Physics

Au

79 Au

Gold (Au)

Noble Metal

Full Electron Configuration

[Xe] 4f¹⁴ 5d¹⁰ 6s¹

Material Density (ρ)

19.30 g/cm³

Atomic Concentration (Nat)

5.90 × 10²⁸ m⁻³

Plasma Energy (Ep)

9.06 eV

Plasma Wavelength (λp)

136.8 nm

Measurement Benchmark Johnson & Christy (1972)

At red laser line (He-Ne 632.8 nm): calc n = 0.191, k = 3.478 vs exp n = 0.180, k = 3.433 (Δn = 0.011, Δk = 0.045).

Interband transitions of electrons from 5d to 6s orbitals begin at 2.4 eV (wavelength ~515 nm).

Outer Valence Shells & Dispersion Role

📐 Physical Theory: How outer atomic orbitals determine n and k ▼

The optical dispersion of matter in optical, UV, and IR frequencies is dictated by the response of outer valence electrons to the oscillating electric field of light waves:

1. Complex Permittivity ε̃ and Refractive Index ñ = n + ik

ε̃(ω) = ε₁ + iε₂ = (n + ik)² = (n² - k²) + i(2nk)

Inverting the relationship yields exact analytical equations for n and k:
n = √[ (|ε̃| + ε₁) / 2 ]
k = √[ (|ε̃| - ε₁) / 2 ], where |ε̃| = √(ε₁² + ε₂²).

2. Free Electrons and Core Polarization (Drude Model)

ε_Drude(ω) = ε_∞ - ω_p² / [ ω(ω + iγ₀) ]

Delocalized outer electrons (e.g. 6s¹ in Au, 5s¹ in Ag) form a plasma with frequency ω_p. Screening from filled d-shells provides high background permittivity ε_∞ ≈ 5.9 (Au) and ε_∞ ≈ 3.7 (Ag).

3. Bound Electrons & Interband Transitions (Lorentz Oscillators)

ε_Lorentz(E) = ∑ [ S_j ] / [ (E₀,j² - E²) - i·E·Γ_j ]

Describes dipole transitions between atomic orbitals (e.g. 5d → 6s at the L-point near 2.72 eV and 5d → 6p at the X-point near 3.65 eV in gold).

4. Optical Reflectance and Attenuation

R = [ (n-1)² + k² ] / [ (n+1)² + k² ]

Bouguer absorption coefficient: α(λ) = 4πk / λ (cm⁻¹).
Optical penetration / skin depth: δ = λ / (4πk).