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Physical Review A

Theoretical treatment of radiation trapping: Steady-state conditions and quenching experiment

Xingxiao Ma, Re Lai

Published Feb 1, 1994• 8 citations
Publisher

Abstract

Theoretical treatment of resonance radiation trapping under steady-state conditions, particularly of quenching experiments, is presented. Applying operator manipulations, Holstein's equation of radiation trapping [Phys. Rev. 72, 1212 (1947)] is solved by a perturbation expansion, which can be physically interpreted in terms of multiple scatterings. The resulting expression shows that in a steady-state quenching experiment radiation trapping invalidates the relation of the Stern-Volmer type except when the amount of the buffer gas is small. In the limit of low buffer-gas pressure, a simple expression is given for the effective lifetime in the Stern-Volmer relation: the average time of the detected resonant photons remaining within the enclosure. While it is true that this suggested approach is perturbative in nature, the asymptotic behavior of the high-order expansion coefficients, however, enables us to reduce the summation over infinitely many alternatives to some leading terms. Such a reduction procedure helps curtail the computation time and improve the precision. Finally, some numerical illustrations of our suggested ideas are presented, which shows this approach is a good tool to treat the steady-state trapping effect.

DOI
10.1103/physreva.49.787

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Theoretical treatment of resonance radiation trapping under steady-state conditions, particularly of quenching experiments, is presented. Applying operator manipulations, Holstein's equation of radiation trapping [Phys. Rev. 72, 1212 (1947)] is solved by a perturbation expansion, which can be physically interpreted in terms of multiple scatterings. The resulting expression shows that in a steady-state quenching experiment radiation trapping invalidates the relation of the Stern-Volmer type except when the amount of the buffer gas is small. In the limit of low buffer-gas pressure, a simple expression is given for the effective lifetime in the Stern-Volmer relation: the average time of the detected resonant photons remaining within the enclosure. While it is true that this suggested approach is perturbative in nature, the asymptotic behavior of the high-order expansion coefficients, however, enables us to reduce the summation over infinitely many alternatives to some leading terms. Such a reduction procedure helps curtail the computation time and improve the precision. Finally, some numerical illustrations of our suggested ideas are presented, which shows this approach is a good tool to treat the steady-state trapping effect.
The purpose of this paper is to propose a theoretical treatment of radiation trapping in a multiple lines coupled system in which coupling includes collisional energy transfer and cross absorption between the excited states. First, to describe the coupled system, the Holstein equation of radiation trapping is extended to an integral differential master equation group in which the vector and matrix form is adopted. Then by use of operator manipulations, the asymptotic solution in the form of a series expansion in terms of operators is presented. Finally, the asymptotic solution is reduced, leading to a result which can not only give the system a complete description, but also has far more advantages than the simple cut off solution. A numeral illustration of a two excited states system is also presented.
A simple stochastic formulation of the multiple scattering representation solution of the classical linear incoherent trapping problem is presented for a broad audience. A clear connection with the alternative Holstein's solution ansatz is emphasized by the (re)interpretation of the fundamental mode as the one associated with a relaxed nonchanging spatial distribution of excitation. Expressions for overall relaxation parameters (ensemble emission yield and lifetime) as well as time-resolved (decay and spatial distribution) and steady-state quantities (spectra and spatial distribution) are given with the fundamental mode contribution singled out. The multiple scattering representation is advocated for final undergraduate and beginning graduate physics instruction based on physical insight and computation feasibility. This will be illustrated in the following instalment of this contribution.
The key to the treatment of the resonance radiation imprisonment lies in the solution to Holstein's equation. A new method to solve Holstein's equation is proposed in this paper, which integrates the equation directly by means of operator calculation and results in a formula for the excited particles in an infinite series in an integral operator. Our study on the behavior of the high-ordercoefficients of this series leads to the success in overcoming the difficulty to obtain the sum of infinite terms, and a formula of finite terms under any given precision is acquired, which applies to the whole time scale.
An experimental study using time-resolved fluorescence techniques together with theoretical simulations has been conducted and used to determine the quenching cross-sections of rubidium–methane and rubidium–ethane. Radiation trapping was significant under many of the experimental conditions (temperatures 40–130 °C and pressures 50–700 Torr) and a detailed analysis of the interplay between radiation trapping and quenching kinetics was carried out. Modifications of the Holstein equation for radiation trapping were implemented to account for the quasi-2 level behaviour of the Rb atom for high buffer gas pressures, the absolute frequency-dependent absorption cross-section for Rb with variable buffer gas pressures which accounts for the hyperfine splitting of 87 Rb and modification of the trapping factors so that radiation trapping and quenching by an additive quenching gas could be treated simultaneously. Experimental results supported by theoretical simulations bound the quenching cross-sections (σ) of methane and ethane at 40 °C to be σ ⩽ 0.019 Å 2 and σ ⩽ 0.033 Å 2 , respectively. These values are nearly two orders of magnitude smaller than previously reported.
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.
We investigate radiation trapping in an atomic vapor which has been excited by a strong short laser pulse. Since the saturation of the vapor by the pulse leads to a decrease in the effective absorption coefficient, the radiation trapping becomes nonlinear. We derive approximate analytical expressions for the excited-state density in the directly excited region, the fluorescence-excited region, and the density averaged over the whole cell. Starting out from fairly simple approximate expressions based on a prescribed distribution of excited atoms, we then develop physically motivated correction factors that drastically improve the accuracy. All these expressions are given for three important cell geometries: the plane-parallel slab, the infinite cylinder, and the sphere. We compare our results to accurate numerical solutions, and find agreement within 5--10%. We then derive the decay time of the emergent radiation, and find that it can be smaller than the natural lifetime of the excited atoms, in agreement with recent experimental results obtained for sodium vapors.