Thermal Doppler Broadening In Rubidium
Atoms which are in motion relative to an observer will have Doppler shifted emission and absorption energies. For example, if an excited atom with energy $E_1$ moving towards an observer with velocity $v \ll c$ makes a transition to a lower energy state $E_2$ by emitting a photon, that photon will be observed to have a higher energy, $E'$, than the energy of the atomic transition $(E_1-E_2)$ as given by
| $E' = (E_1 - E_2) \left(1 + \dfrac{v}{c}\right)$ | (3) |
Similarly if the same observer wishes to excite the atom by hitting it with a photon, the observer would have to use photons with energy slightly lower than the transition to be excited.
Atoms in a gas at a temperature $T$ will have random velocities given by the Boltzmann distribution,
| $P(v)dv = \left(\dfrac{M}{2\pi kT}\right)^{1/2}\exp{\left(-\dfrac{Mv^2}{2kT}\right)}dv$ | (4) |
where $k$ is the Boltzman constant and $M$ is the mass of the atom. This distribution of velocities leads to a corresponding distribution of Doppler-shifted photon energies for any given atomic transition. The resulting spread in photon energies, about the transition energy, is referred to as Doppler broadening. In terms of frequency, $ν$, the full-width at half-maximum of the Doppler-broadened line is given by
| $\Delta v_{1/2} = 2\dfrac{v_0}{c}\left(\dfrac{2kT}{M} \ln 2 \right)^{1/2} = \delta \sqrt{\ln 2}$ | (5) |
where $v_0$ is the un-shifted frequency of the transition and $\delta$ is called the linewidth parameter. Later we will identify Doppler-broadened spectral features in terms of the final energy level of the transition. For example, the notation ${}^{87}Rb(F^\prime= 2)$ refers to the Doppler-broadened feature that contains all allowed transitions from any of the $5^2P_{3/2}(F^\prime = 0, 1, 2 \;\mathrm{or}\; 3)$ states to the $5^2S_{1/2}(F = 2)$ state. In all, there are four Doppler broadened features: ${}^{87}Rb(F = 1)$, ${}^{87}{Rb}(F = 2)$, ${}^{85}Rb(F = 2)$, ${}^{85}Rb(F = 3)$.