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.
The description of the self-imprisonment of resonance radiation in gases or solids and of the density distribution of excited species depends upon a transmission coefficient $T(\ensuremath{\rho})$, the probability that a quantum of emitted radiation will traverse a thickness $\ensuremath{\rho}$ of material. This coefficient is shown to be given by the rapidly convergent series $T(\ensuremath{\rho})=\ensuremath{\Sigma}{n}^{}\frac{{(\ensuremath{-}{k}_{0}\ensuremath{\rho})}^{n}}{n!{(n+1)}^{\frac{1}{2}}}$, where
k0 is a constant characteristic of the excited species and of the temperature. The above expression is used to give the first accurate discussion of resonance radiation confined between infinite parallel planes, and of radiation inside a sphere over a wide range of ${k}_{0}\ensuremath{\rho}$ values.
The theory of the imprisonment of resonance radiation developed by Holstein and by Bieberman is applied to the analysis of experimental measurements of excitation cross sections for the resonance (
n1P) states of helium. The transport equation for the density of atoms in the resonance state is solved numerically for the case of a thin sheet of exciting electrons between parallel plane electrodes. The fraction of the resonance atoms producing visible radiation is then calculated for parallel plane geometry and estimated to within five percent for cylindrical geometry with an axial electron beam. Predictions of the theory are compared with the available experimental data for helium. The theory shows that at the helium pressures commonly used (${10}^{\ensuremath{-}3}$ to ${10}^{\ensuremath{-}1}$ mm of Hg) the observed visible radiation may easily be a factor of ten greater than that expected when imprisonment effects are neglected. As a result, the cross sections for the excitation of the
n1P states given in the literature are much too large. For example, our analysis of the available experimental data suggests that the cross section for excitation by electrons to the
31P state of helium at 100 electron volts is 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$
cm2 instead of 4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$
cm2 as given in the literature.
The method of numerical simulation was used to study the effective lifetime of the excited level of the resonance transition in sodium and lithium vapour. Taking into account the partial frequency redistribution and the cylindrical geometry of the medium, the rate equations of the population balance of a two-level atom were solved with the equation of radiation transfer. The Biberman–Holstein escape factor was calculated as a function of the optical thickness of vapour for different geometries of the luminescing gas and different models of the frequency redistribution functions. The dependence of the escape factor on the optical thickness investigated experimentally by Romberg and Kunze (J. Quant. Spectrosc. Radiat. Transfer. 1988 39 99) is discussed.
Following the theory of Holstein, the density,
Nr, and the imprisonment lifetime of resonance radiation,
T, has been investigated for a gaseous discharge between parallel plates. A general solution is given. Two approximate solutions occur according as the number of electronic de-excitations of the resonance state during the imprisonment time of a photon is much less than or much greater than unity. When this number is much less than unity,
Nr can be given by a simple relation which compares well with computations based on the exact solution in the case of Doppler broadening.
T is then essentially equal to the decay time as calculated by Holstein for the decay of resonance radiation following optical excitation. For large numbers of de-excitations
T falls somewhat below the decay value and, as expected,
Nr is given by its thermodynamic equilibrium value at the temperature of the electrons.
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.