GATE 2022 PH – Question 32
A Raman line at $300\ \mathrm{cm^{-1}}$ has Stokes intensity four times anti-Stokes intensity. Find the temperature in K, nearest integer. Given $1\ \mathrm{cm^{-1}}=1.44$ K.
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Correct answer: 312
Explanation
In Raman scattering the Stokes line comes from molecules in the lower level (v = 0), and the anti-Stokes line comes from molecules in the upper level (v = 1). The intensity ratio therefore reflects the Boltzmann population ratio:
$$\frac{I_S}{I_{AS}}\approx\frac{N_0}{N_1}=e^{h\nu/k_BT}.$$
**Numbers.** The Raman shift of $300\text{ cm}^{-1}$ corresponds to $h\nu/k_B=300\times1.44=432$ K:
$$4=e^{432/T}\;\Rightarrow\;\frac{432}{T}=\ln4=1.3863\;\Rightarrow\;T=\frac{432}{1.3863}=311.6\approx\mathbf{312\ K}.$$