On Radiation Diffusion and the Rapidity of Escape of Resonance Radiation from a Gas

The radiation diffusion process is considered from the standpoint of the free paths of the diffusing resonance quanta as influenced by the Doppler and other line broadening effects. Abnormally long free paths are found to be of such importance as to enable resonance radiation to escape from a body of gas faster than has usually been supposed. It is assumed that a large concentration of diffusing resonance quanta will, on the basis of Doppler broadening only, give rise to a characteristic excitation of atoms, as dependent on their speeds, which can be represented by a distribution function which will lie between two limiting distribution functions, namely (1) Maxwell's distribution function and (2) a distribution function expressing a lower relative excitation of the high speed atoms than that of Maxwell, based on the excitation of all atoms as if by absorption of the core of the line. On the basis of (1) and (2), limiting expressions are derived for: (a) the fraction of emitted quanta traversing at least a given distance before absorption, (b) the diffusion coefficient, (c) the average square free path, (d) the average free path. A fundamental difference between radiation diffusion and molecular diffusion appears in that whereas (a) decreases exponentially with the distance in the latter case it is found to decrease only linearly (roughly) with the distance in the former case. For this reason very long free paths are found to be of relatively great importance in radiation diffusion. It is found that, for a gas container of infinite size, (b), (c), and (d) are all infinite. For a gas container of finite size, estimates of the order of magnitude of the apparent or effective values of (b), (c), and (d) are made on the basis of special assumptions. It is found that $(\mathrm{b})=\frac{(\frac{1}{6}){p}^{2}}{\ensuremath{\tau}}$ where $\stackrel{-}{{p}^{2}}$ is the average square free path and $\ensuremath{\tau}$ is the mean life of the excited atom. The same formula is found to hold for the coefficient of molecular diffusion if $\stackrel{-}{{p}^{2}}$ is used to denote the average square molecular free path and $\ensuremath{\tau}$ is used to represent the mean time between collisions. It is pointed out that coupling and other line broadening effects will still further increase the importance of extremely long free paths and hence also the rapidity of escape of resonance radiation from a gas. The results of the Hg resonance radiation imprisonment experiments of Zemansky and of Webb and Messenger are discussed and are shown to be in accord with the conclusions arrived at above that resonance radiation escapes from a gas faster than according to classical theories of radiation diffusion, the effect being the greater the larger the gas volume or density. The rapidity of escape of $\ensuremath{\lambda}2537$ at the lower gas densities ($N={10}^{15}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ or less) in these experiments can be largely accounted for by the Doppler effect on the basis of a diffusion coefficient proportional to the effective average square free path as calculated by the methods of the present paper; at the higher gas densities ($N=30\ifmmode\times\else\texttimes\fi{}{10}^{15}$ ${\mathrm{cm}}^{\ensuremath{-}3}$) the escape is more rapid than can be accounted for by these methods and this is attributed to coupling or other pressure broadening.