Experimental investigations have been made of the transfer of mercury 2537-\AA{} radiation through a slab of natural mercury gas of various densities and dimensions, using a sinusoidally modulated ($\ensuremath{\omega}=2\ensuremath{\pi}f$,
f=5to50 kc/sec) radio-frequency-excited (50-Mc/sec) source of 2537-\AA{} radiation. Transmitted radiation has been found to gain in phase relative to the source wave, while that which is scattered at 90\ifmmode^\circ\else\textdegree\fi{} loses phase. From the radiation scattered at 90\ifmmode^\circ\else\textdegree\fi{}, decay time constants of the imprisoned photons have been calculated by using the relation between the time constant
T and the phase shift $\ensuremath{\varphi}$, $\ensuremath{\varphi}=invtan\ensuremath{\omega}T$. The relation $T\ensuremath{\propto}\frac{{C}^{2}}{{K}^{2}}+{C}^{2}$), where
K is a constant, has been found for the dependence on the breadth
C of the resonance tube as well as the well known dependence $\mathrm{ln}T\ensuremath{\propto}\mathrm{ln}n$ on the molecular density
n. On the other hand, the transmitted radiation obeys a relation ${T}_{c}\ensuremath{\propto}{(\mathrm{nA})}^{\frac{1}{2}}$ between the time constant
Tc corresponding to the advanced phase shift, the thickness
A of the resonance tube, and
n. The dependences on other factors have also been examined.
The predictions of the incoherent scattering theory of the transport of resonance radiation developed by Holstein and by Biberman are shown to agree with laboratory measurements of the decay constants for the intensity of the mercury 2537 A resonance radiation following a period of optical excitation. Also, the relative importance of the diffusion of resonance atoms and the escape of resonance radiation as mechanisms for the destruction of mercury atoms in the
3P1 or resonance state are determined from measurements of decay constants as a function of mercury density and cell radius. The experimental results show that diffusion of the resonance atoms is negligible and that the predictions of imprisonment theory are confirmed to within 15%. The experiment sets an upper limit to the diffusion coefficient at unit gas density for atoms in the
3P1 state of 5\ifmmode\times\else\texttimes\fi{}
1017 ${\mathrm{cm}}^{\ensuremath{-}1}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ at 340\ifmmode^\circ\else\textdegree\fi{}K, which is consistent with a value of 4\ifmmode\times\else\texttimes\fi{}
1016 ${\mathrm{cm}}^{\ensuremath{-}1}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ predicted using the frequency of excitation transfer collisions calculated by Holstein. The success of the theories based on completely incoherent scattering of resonance radiation points to the desirability of including this feature in the treatment of those astrophysical problems in which the spectral line shape is determined by Doppler broadening or by collision broadening.
We solve the Holstein equation describing radiation trapping in an atomic vapor by the geometric quantization technique. The treatment is based on studying the integral trapping equation as a wave equation for an associated quasiparticle with a complicated form of its dispersion law. The latter is determined by the spectral properties of the vapor medium. Extending our previous work, which dealt with one-dimensional (1D) geometries, we consider here more realistic two- and three-dimensional (2D and 3D) vapor cell geometries. For this, an explicit representation of the phase factors is derived when the quasiparticle is reflected under arbitrary angles from the surface of the cell confining the vapor. We give closed-form equations to obtain analytically the complete 3D spectrum of escape factors in finite cylinder, sphere, parallelepiped, and finite prismlike geometries for all practically occurring line shapes and opacities. In addition, the relevant semiclassical variational method is applied to develop a perturbation theory for the evaluation of escape factors in more complicated geometries that are sometimes adopted in experiments, including infinite (or finite) elliptical cylinders, prolate and oblate ellipsoids. Comparisons with numerical calculations show that results are accurate within
5 for the ground modes and even better for higher-order modes.
We introduce an analytical method to investigate radiation trapping problems with Doppler frequency redistribution. The problem is formulated within the framework of the Holstein-Biberman-Payne equation. We interpret the basic integro-differential trapping equation as a generalized wave equation for a four-dimensional (4D) classical system (an associated quasiparticle). We then construct its analytical solution by a semiclassical approach, called the geometrical quantization technique (GQT). Within the GQT, it is shown that the spatial and frequency variables can be separated and that the frequency part of the excited atom distribution function obeys a stationary Schr\"odinger equation for a perturbed oscillator. We demonstrate that there is a noticeable deviation of the actual spectral emission profile from the Doppler line in the region of small opacities. The problem of calculating the spatial mode structure and the effective radiation trapping factors is reduced to the evaluation of wave functions and quantized energy values of the quasiparticle confined in the vapor cell. We formulate the quantization rules and derive the phase factors, which allow us to obtain analytically the complete spectrum of the trapping factors in 1D geometries (layer, cylinder, sphere) and other (2D and 3D) geometries when the separation of space variables is possible. Finally, we outline a possible extension of our method to treat radiation trapping effects for more general experimental situations including, for instance, a system of cold atoms.
The Holstein-Biberman integrodifferential equation for the transport of resonance radiation has been solved for a gas contained in a long cylindrical container. The solution involves the following assumptions: (i) The excited atoms are initially distributed uniformly along the axis of the cylinder; (ii) the pressure and temperature of the gas are such that the absorption coefficient has either a pure Doppler-broadened or a pure pressure-broadened profile; (iii) the radius of the cylinder corresponds to many optical depths at the frequency corresponding to the maximum of the absorption coefficient. It is also shown that steady-state solutions corresponding to a line source along the axis of the cylinder can be calculated by the same procedure. The geometry and the initial conditions considered here are of particular interest because the situation can be simulated experimentally. Hurst and Thonnard have pointed out that an approximate line of excited atoms can be produced by sending a well-collimated pulse of protons or electrons down the axis of a long cylinder containing the gas of interest. The present paper also considers the decay of resonance radiation from a steady state built up by a line source. The latter situation is even easier to arrange experimentally. All experimental work to date has concerned itself with situations where the initial distribution of excited atoms is not known, and theory and experiment can only be compared at times which are large enough so that the system has decayed to its lowest eigenmode.