## 4. Let's look at an FTIR instrument ### a Mirror Can Move 5 from ZPDIf the FT has a mirror can move up **5 cm from the Zero Path Difference (ZPD)**, the **maximum optical path difference (OPD)** is **10 cm** (since the light goes to the mirror and back). The **spectral resolution (Δν)** is given by: \[ \Delta \nu = \frac{1}{\text{OPD}_{\text{max}}} \] So, \[ \Delta \nu = \frac{1}{10\ \text{cm}} = .1\ \text{cm}^{-1} \] - **You could collect spectra with a resolution of .1 cm⁻¹.** - **The minimum resolvable difference in wavenumber is .1 cm⁻¹.** - **The spectral range depends on the source, detector, and beamsplitter, not just the mirror travel.** --- ### b. Uses 632.8 nm Laser for Mirror Position - The **laser wavelength (632.8 nm)** is used for precise position calibration, not as a limit on measurement. - The laser provides an **accurate reference for path difference**, allowing high-precision interferogram sampling. - **The spectrum collected is not limited to the laser wavelength—** the system can still measure a broad spectrum, as determined by the source and detector. The **laser only improves accuracy and reproducibility**. --- ### c. Wavenumber at 155 nm Wavenumber (\(\tilde{\nu}\)) is: \[ \tilde{\nu} = \frac{1}{\lambda} \] But with \(\lambda\) in cm: \[ 155\ \text{nm} = 1.55 \times 10^{-4}\ \text{cm} \] \[ \tilde{\nu} = \frac{1}{1.55 \times 10^{-4}\ \text{cm}} \approx 6451.6\ \text{cm}^{-1} \] --- ### d. Wavelength Resolution at 155 nm Wavenumber resolution found above: \(\Delta \nu = .1\ \text{cm}^{-1}\) Wavelength resolution (\(\Delta \lambda\)) at a given wavelength: \[ \Delta \lambda = \lambda^2 \Delta \tilde{\nu} \] Substitute \(\lambda = 1.55 \times 10^{-4}\) cm and \(\Delta \tilde{\nu} = .1\ \text{cm}^{-1}\): \[ \Delta \lambda = (1.55 \times 10^{-4})^2 \times .1 = 2.4025 \times 10^{-8}\ \text{cm} \] Convert to nm: \[ 2.4025 \times 10^{-8}\ \text{cm} \times 10^7\ \text{nm/cm} = .240\ \text{nm} \] **So, the wavelength resolution at 155 nm is approximately .24 nm.** --- ## **Summary Table** | Part | Answer | |------|--------| | a | Spectral resolution is .1 cm⁻¹ with 5 cm mirror travel. | | b | Laser is for calibration; spectrum range is not limited by 632.8 nm. | | c | Wavenumber at 155 nm is ~6452 cm⁻¹. | | d | Wavelength resolution at 155 nm is ~.24 nm. |