Sum Frequency Generation Spectroscopy
SFG vibrational spectrum of an interface: the SFG wavelength and |χ⁽²⁾|² of Lorentzian CH or OH resonances interfering with a non-resonant background of chosen amplitude and phase.
SFG wavelength range
457.73 – 463.03 nm
IR 2800–3050 cm⁻¹ on 18797.0 cm⁻¹
Strongest SFG peak
2878.7 cm⁻¹
|χ_eff|² = 2.747, λ_SFG 461.35 nm
Apparent peaks in range
4
4 resonances; interference merges, shifts or hides peaks
SFG intensity vs IR wavenumber
- |χ_eff|² (with background)
- |χ_R|² (resonances only)
- |A_NR|² (background only)
Resonances and where their peaks appear
| Mode | ν̃_q (cm⁻¹) | Γ_q (cm⁻¹) | A_q/Γ_q | λ_SFG (nm) | Nearest apparent peak (cm⁻¹) |
|---|---|---|---|---|---|
| CH₂ symmetric (d⁺) | 2850 | 6 | 0.25 | 461.96 | 2849.8 (−0.2) |
| CH₃ symmetric (r⁺) | 2878 | 5 | 1.60 | 461.36 | 2878.7 (+0.7) |
| CH₃ Fermi resonance (r⁺_FR) | 2938 | 7 | 0.71 | 460.09 | 2940.6 (+2.6) |
| CH₃ asymmetric (r⁻) | 2965 | 6 | 0.33 | 459.52 | 2968.4 (+3.4) |
Assignments from Lambert et al. (2005) and Du et al. (1993). Amplitudes and widths are illustrative: real values come from fitting a measured spectrum with this model. The water preset gives the whole H-bonded band one sign, opposite to the free OH, as heterodyne-detected spectra show (Nihonyanagi et al. 2015); an early positive 3200 cm⁻¹ band is still debated (Tang et al. 2020). The nearest peak can belong to a neighbouring mode when lines overlap.
1/λ_SFG = 1/λ_vis + ν̃_IR (ω_SFG = ω_vis + ω_IR)
χ⁽²⁾_eff(ν̃) = A_NR e^(iφ) + Σ_q A_q / (ν̃ − ν̃_q + iΓ_q)
I_SFG ∝ |χ⁽²⁾_eff|² · I_vis · I_IR
χ⁽²⁾ vanishes in centrosymmetric bulk media in the dipole approximation, so the signal comes from the interface, where the symmetry breaks. The terms add as complex amplitudes before squaring: with a real background (φ = 0) a positive resonance moves to higher wavenumber and turns dispersive; at φ = +90° the background cancels part of a positive resonance (a weaker peak or a dip) and at φ = −90° it adds to it. Some papers write the resonance as A_q/(ν̃_q − ν̃ − iΓ_q): the same model with every A_q negated, or φ moved by 180°.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Standard SFG lineshape: Lorentzian resonances plus a constant non-resonant term, added as complex amplitudes before squaring. Fresnel factors, beam geometry and polarisation combination are folded into the amplitudes, which (with the widths) are illustrative; the band assignments are from the literature. The SFG wavelength is exact.
- Shen Y. R. (1989). Surface properties probed by second-harmonic and sum-frequency generation. Nature 337, 519–525. doi:10.1038/337519a0
- Lambert A. G., Davies P. B., Neivandt D. J. (2005). Implementing the Theory of Sum Frequency Generation Vibrational Spectroscopy: A Tutorial Review. Appl. Spectrosc. Rev. 40, 103–145. Lineshape model and CH stretch assignments. doi:10.1081/ASR-200038326
- Du Q., Superfine R., Freysz E., Shen Y. R. (1993). Vibrational spectroscopy of water at the vapor/water interface. Phys. Rev. Lett. 70, 2313–2316. OH stretch bands. doi:10.1103/PhysRevLett.70.2313
- Nihonyanagi S. et al. (2015). Accurate determination of complex χ(2) spectrum of the air/water interface. J. Chem. Phys. 143, 124707. Signs of Im χ(2): free OH positive, H-bonded band negative. doi:10.1063/1.4931485
- Tang F. et al. (2020). Molecular structure and modeling of water–air and ice–air interfaces monitored by sum-frequency generation. Chem. Rev. 120, 3633–3667. The 3200 cm⁻¹ sign debate. doi:10.1021/acs.chemrev.9b00512
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