Spectral Deconvolution & Band Model
Build a spectrum from overlapping peaks and see how well their areas can be recovered, or model the vibronic shape of an emission band.
Band maximum
520.0
height 1.001
Band FWHM
40.08
around the main maximum
Band centroid
520.7
first moment over the window
Total area
71.85
Σ height × FWHM × shape factor, whole lines
| Peak | Centre | FWHM | Height | Area | Area share | Fitted height | Fitted area |
|---|---|---|---|---|---|---|---|
| 1 | 480.0 | 25.00 | 0.7000 | 18.63 | 25.9 % | 0.7010 ± 0.0031 | 18.65 ± 0.083 |
| 2 | 520.0 | 35.00 | 1.000 | 37.26 | 51.9 % | 1.002 ± 0.0026 | 37.32 ± 0.099 |
| 3 | 570.0 | 30.00 | 0.5000 | 15.97 | 22.2 % | 0.5014 ± 0.0029 | 16.01 ± 0.091 |
| Pair | Separation / mean FWHM | Dip between them | Height correlation |
|---|---|---|---|
| 1 – 2 | 1.33 | 32.0 % (resolved) | -0.0885 |
| 2 – 3 | 1.54 | 45.1 % (resolved) | -0.0382 |
Band model and simulated spectrum
- Simulated data (sum + noise)
- Peak 1
- Peak 2
- Peak 3
- Fitted sum
I(x) = Σᵢ hᵢ · shape((x − xᵢ)/wᵢ)
Gaussian exp(−4 ln2 u²), area √(π/(4 ln2)) h w; Lorentzian 1/(1 + 4u²), area (π/2) h w
pseudo-Voigt η·L + (1 − η)·G with one FWHM; heights: (BᵀB) h = Bᵀy, cov = s²(BᵀB)⁻¹
The spectrum is the sum of the peaks you enter, sampled at 601 points, plus seeded Gaussian noise. The heights are then fitted to that simulated spectrum by linear least squares with every centre and width held at its true value, which is the best case: the standard errors and the negative correlation between neighbours show how overlap alone trades area from one peak to the next. A real deconvolution also fits the centres and widths (nonlinear least squares from starting guesses); then the errors grow, and peaks closer than about one FWHM are often not unique. Gaussian bands describe inhomogeneous broadening (a distribution of environments), Lorentzian bands homogeneous broadening (lifetime, collisions). Two equal Gaussians show a dip only when they are more than 2σ ≈ 0.849 FWHM apart, two equal Lorentzians more than FWHM/√3 ≈ 0.577 FWHM.
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
Sums of height-normalised Gaussian, Lorentzian or pseudo-Voigt peaks; the heights are fitted by linear least squares with the centres and widths held at their true values, so the errors are a best case. The vibronic band has one displaced harmonic mode at T ≈ 0 (Poisson Franck–Condon factors), ν̃ and ν̃³ frequency factors and the λ⁻² Jacobian for emission.
- Huang K., Rhys A. (1950). Theory of light absorption and non-radiative transitions in F-centres. Proc. R. Soc. Lond. A 204, 406–423. doi:10.1098/rspa.1950.0184
- Strickler S. J., Berg R. A. (1962). Relationship between absorption intensity and fluorescence lifetime of molecules. J. Chem. Phys. 37, 814–822. doi:10.1063/1.1733166
- Mooney J., Kambhampati P. (2013). Get the basics right: Jacobian conversion of wavelength and energy scales for quantitative analysis of emission spectra. J. Phys. Chem. Lett. 4, 3316–3318. doi:10.1021/jz401508t
- Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P. (2007). Numerical Recipes, 3rd ed. Cambridge University Press, ch. 15 (general linear least squares).
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