QB_PH_SEC6_022: Molecular Spectroscopy

A Raman spectrometer using an excitation laser of wavelength $\lambda_0 = 500\text{ nm}$ observes a Stokes vibrational line with a Raman shift of $\Delta\bar{\nu} = 1000\text{ cm}^{-1}$. What is the wavelength (in nm) of the corresponding Anti-Stokes line? (Round off to two decimal places)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Laser Wavenumber Computation**: $\bar{\nu}_0 = \frac{1}{\lambda_0} = \frac{1}{500 \times 10^{-9}\text{ m}} = 2.0 \times 10^6\text{ m}^{-1} = 20000\text{ cm}^{-1}$ 2. **Raman Shift Definitions**: - Stokes line: $\bar{\nu}_{S} = \bar{\nu}_0 - \Delta\bar{\nu}$ (longer wavelength, lower energy photon) - Anti-Stokes line: $\bar{\nu}_{\text{AS}} = \bar{\nu}_0 + \Delta\bar{\nu}$ (shorter wavelength, higher energy photon) 3. **Anti-Stokes Frequency Calculation**: $\bar{\nu}_{\text{AS}} = 20000\text{ cm}^{-1} + 1000\text{ cm}^{-1} = 21000\text{ cm}^{-1}$ 4. **Wavelength of Anti-Stokes Line**: $\lambda_{\text{AS}} = \frac{1}{\bar{\nu}_{\text{AS}}} = \frac{1}{21000\text{ cm}^{-1}} = \frac{1}{2.10 \times 10^6\text{ m}^{-1}} = 4.76190 \times 10^{-7}\text{ m} = 476.19\text{ nm}$ 5. **Official Range**: $[475.50, 477.00]$.