The Hook
Your FT-IR spectrometer doesn't measure absorbance vs frequency directly.
It measures an interferogram — intensity vs mirror position. A wiggly, complicated signal.
Then it computes the Fourier transform. Out comes the spectrum.
How does mathematics convert position information into frequency information?
The Core Idea: Any periodic function can be written as a sum of sines and cosines. Sines and cosines form a complete basis for periodic functions.
1. Building Waveforms from Harmonics
A square wave is a sum of odd harmonics with decreasing amplitudes.
Square wave: \(f(x) = \frac{4}{\pi}\sum_{n=1,3,5,...} \frac{1}{n}\sin(nx)\)
Notice the Gibbs phenomenon — the overshoot near discontinuities!
2. The Uncertainty Principle
A Gaussian transforms to a Gaussian. Narrow in time → wide in frequency.
This is why ultrashort laser pulses have broad spectra!
3. Spectral Lineshapes
Different decay mechanisms produce different lineshapes.
| Lineshape | Origin | Time Domain |
|---|---|---|
| Lorentzian | Lifetime/natural | Exponential decay |
| Gaussian | Doppler/inhomogeneous | Gaussian distribution |
| Voigt | Both mechanisms | Convolution |
4. FT-IR: Interferogram to Spectrum
The Fourier transform converts mirror position data to frequency spectrum.
FT-IR Advantages:
• Fellgett: All frequencies measured simultaneously → better S/N
• Jacquinot: No slits needed → higher throughput
• Connes: Internal laser calibration → precise frequency scale
5. Sampling and Aliasing
Undersampling causes high frequencies to appear as low frequencies.
Summary
Transform Pairs
| Time Domain | Frequency Domain |
|---|---|
| Narrow | Wide |
| Wide | Narrow |
| Convolution | Multiplication |
| Exp. decay | Lorentzian |
| Gaussian | Gaussian |
| Delta | Constant |
Chemistry Applications
| Technique | FT converts |
|---|---|
| FT-IR | Interferogram → Spectrum |
| FT-NMR | FID → Spectrum |
| Ultrafast | E(t) → E(ω) |
| QM | ψ(x) ↔ φ(p) |
The Core Message: Fourier analysis decomposes signals into frequencies. Time and frequency are conjugate domains — narrow in one means wide in the other.
Exercises
- A laser pulse has Δt = 50 fs. Estimate its spectral width in nm (around 800 nm).
- An excited state has lifetime τ = 10 ns. What is the natural linewidth (FWHM)?
- What sampling rate is needed to measure up to 4000 cm⁻¹ in FT-IR?
- The convolution of two Gaussians with widths σ₁ and σ₂ gives width = ?