All Problems Optics of Moving Sources
Problem 5.238
A gas consists of atoms of mass \(m\) being in thermodynamic equilibrium at temperature \(T .\) Suppose \(\omega_{0}\) is the natural frequency of light emitted by the atoms. (a) Demonstrate that the spectral distribution of the emitted light is defined by the formula \[ I_{\omega}=I_{0} \mathrm{e}^{-a\left(1-\omega / \omega_{0}\right)^{2}} \] \(\left(I_{0}\right.\) is the spectral intensity corresponding to the frequency \(\omega_{0}\), \(\left.a=m c^{2} / 2 k T^{\top}\right)\) (b) Find the relative width \(\Delta \omega / \omega_{0}\) of a given spectral line, i.e. the width of the line between the frequencies at which \(I_{\omega}=I_{0} / 2\).
5.238. (a) Let \(v_{x}\) be the projection of the velocity vector of the radiating atom on the observation direction. The number of atoms with projections falling within the interval \(v_{x}, v_{x}+d v_{x}\) is \[ n\left(v_{x}\right) d v_{x} \sim \exp \left(-m v_{x}^{2} / 2 k T\right) \cdot d v_{x} \] The frequency of light emitted by the atoms moving with velocity \(v_{x}\) is \(\omega=\omega_{0}\left(1+v_{x} / c\right) .\) From the expression the frequency distribution of atoms can be found: \(n(\omega) d \omega=n\left(v_{x}\right) d v_{x}\). And finally it should be taken into account that the spectral radiation intensity \(I_{\omega} \sim n(\omega)\) \[ \text { (b) } \Delta \omega / \omega_{0}=2 \sqrt{(2 \ln 2) k T / m c^{2}} \]