Engineering & Science
Raman Shift Wavenumber (cm^-1) & Anti-Stokes / Stokes Boltzmann Ratio Formula
Convert laser excitation and scattered wavelengths (nm) into Raman vibrational shift Delta_nu = (10^7 / lambda_0) - (10^7 / lambda_s) in cm^-1 and compute the vibrational Boltzmann population ratio EXP((-1.4387769 * |Delta_nu|) / T).
Fingerprint silicon phonons (520 cm^-1) and measure non-contact temperature!
When a green 532.0 nm laser illuminates a silicon wafer, graphene sheet, or pharmaceutical tablet, ~1 in 10^7 photons exchanges a vibrational quantum with the sample (C. V. Raman, 1930 Nobel Prize)! Because vibrational energy is independent of which laser color you use, spectroscopists convert wavelength (in nanometers, 1 cm = 10^7 nm) into Raman Shift Wavenumber (cm^-1): Delta_nu = (10,000,000 / lambda_laser_nm) - (10,000,000 / lambda_scattered_nm)! Red-shifted Stokes scattering (lambda_s > lambda_0, creating a phonon) gives positive Delta_nu; blue-shifted Anti-Stokes scattering (destroying a thermally excited phonon) gives negative Delta_nu!
Even better, Anti-Stokes scattering requires a molecule ALREADY sitting in its excited vibrational level, governed by the Boltzmann factor EXP(-(h * c * Delta_nu) / (kB * T)) = EXP((-1.4387769 * ABS(Delta_nu_cm1)) / T_K) (using CODATA second radiation constant c2 = h*c/kB = 1.4387769 cm*K)! For silicon's optical phonon scattered at 547.1 nm under a 532.0 nm laser (Delta_nu = 518.8 cm^-1) at room temperature T = 298.15 K, the thermal Anti-Stokes-to-Stokes population ratio is 0.0818 (8.18%)—dropping 10-fold to 0.0079 for a higher-frequency 1,003.4 cm^-1 ring-breathing mode (562.0 nm)!
| Scattered Peak (532.0 nm Green Laser, 298.15 K) | Raman Shift Delta_nu (cm^-1) | Boltzmann Ratio EXP(-c2*nu/T) | Vibrational Assignment |
|---|---|---|---|
| lambda_s = 547.1 nm (Stokes) | +518.8 cm^-1 | 0.0818 (8.18% excited) | Crystalline Silicon optical phonon |
| lambda_s = 562.0 nm (Stokes) | +1,003.4 cm^-1 | 0.0079 (0.79% excited) | Aromatic ring breathing mode |
| lambda_s = 532.0 nm (Rayleigh) | 0.0 cm^-1 | 1.0000 | Elastic Rayleigh laser line |
Raman Vibrational Wavenumber Shift Delta_nu in Reciprocal Centimeters (cm^-1)
Evaluates (10^7 / lambda_laser_nm) - (10^7 / lambda_scattered_nm).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Wavelengths", ROUND((10000000 / B2) - (10000000 / B3), 1))Thermal Vibrational Boltzmann Anti-Stokes / Stokes Factor EXP(-c2 * |Delta_nu| / T)
Uses exact CODATA second radiation constant c2 = 1.4387769 cm*K at absolute sample temperature B4 (Kelvin).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Raman Inputs", ROUND(EXP((-1.4387769 * ABS((10000000 / B2) - (10000000 / B3))) / B4), 4))Try the example
Use the original values or change them to test your understanding.