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Acta Physica Sinica

THEORETICAL TREATMENT OF RESONANCE RADIATION IMPRISONMENT SOLUTION TO HOLSTEIN'S EQUATION

LAI RE, Shilin Liu, MA XING-XIAO, 中国科学技术大学近代化学系,合肥230026

Published Jan 1, 1993• 0 citations
Publisher

Abstract

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.

DOI
10.7498/aps.42.1216

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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.
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.
Following the theory of Holstein, the density, Nr{N}_{r}Nr​, and the imprisonment lifetime of resonance radiation, TTT, 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{N}_{r}Nr​ can be given by a simple relation which compares well with computations based on the exact solution in the case of Doppler broadening. TTT 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 TTT falls somewhat below the decay value and, as expected, Nr{N}_{r}Nr​ is given by its thermodynamic equilibrium value at the temperature of the electrons.
Abstract A differential integral equation system of Holstein type is used to describe the kinetic process in V E energy transfer between molecules and atoms with radiation trapping.A reduced solution of a finite series has been presented,which allows to describe the whole temporal behavior of fluorescence.The further derivative is an analytical expression of the fluorescence decay rate as a function of rate constants and the maximum eigenvalue of the integral operator appearing in the rate equations.This will be especially useful and convenient for experimental determinations.Results are schematically illustrated and mutually compared by means of numerical calculations.
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.
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 (n1Pn^{1}Pn1P) 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 n1Pn^{1}Pn1P 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 31P3^{1}P31P state of helium at 100 electron volts is 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$ cm2{\mathrm{cm}}^{2}cm2 instead of 4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$ cm2{\mathrm{cm}}^{2}cm2 as given in the literature.