Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 36A / b / iii

Table of contents
- Solution iii

Solution

The Doppler relation is the affine change of variables

v_{x} = c (\frac{ω}{ω _{0}} - 1), \frac{d v _{x}}{d ω} = \frac{c}{ω _{0}} .

Transforming the one-dimensional Maxwell-Boltzmann velocity distribution therefore gives the normalized thermal Doppler broadening profile

I (ω) = \frac{c}{ω _{0}} \frac{m}{2 π k _{B} T} exp [- \frac{m c ^{2}}{2 k _{B} T ω _{0}^{2}} (ω - ω_{0})^{2}] .

Thus

a = \frac{m c ^{2}}{2 k _{B} T ω _{0}^{2}} .

The measured Gaussian line width satisfies

σ_{ω} / ω_{0} = k_{B} T / m / c

, so the temperature follows from

T = \frac{m c ^{2}}{k _{B}} (\frac{σ _{ω}}{ω _{0}})^{2} .

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