Shape of Collision-Broadened Spectral Lines

Van Vleck and Weisskopf and Fröhlich have derived a microwave line shape by studying the interruption by collisions of the motion of a classical oscillator. They assume that after the instantaneous impact the oscillator variables are distributed according to a Boltzmann distribution appropriate to the value of the applied field at collision. In contrast to the earlier theory of Lorentz, they obtain the correct static polarization. The procedure involves an assumption of very large velocity during collision. This is criticized on the grounds that the duration of collision is short compared to the resonant period and energy exchanges are of the order of $\mathrm{kT}$. We have derived a line-shape formula assuming that the positions are unchanged after impact. Two extreme models are studied. In one, the oscillators have a Maxwellian distribution of velocities after impact; the second is a Brownian motion treatment. The resulting line shape in both cases is that of a friction-damped oscillator. For collision frequency much less than the resonant frequency, the polarization postulated by the above authors is reached as a result of kinematic motion between collisions, and the line shapes agree. However, to obtain equal line widths and peak absorptions, the collision frequency is twice as large for the present theory. For collision frequency comparable to resonant frequency a less distorted line shape results. For testing the theories, experiments on foreign-gas broadening in the microwave region at pressures of the order of an atmosphere are required. Differences between the theories are small for conditions accessible experimentally at present.