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Physical review. B, Solid state

Imprisonment of Resonance Radiation in Solids and Gases

Charles W. Scherr

Published Dec 1, 1971• 14 citations
Publisher

Abstract

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 ${k}_{0}$ 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.

DOI
10.1103/physrevb.4.3885

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Imprisonment of resonant radiation in gases on the basis of the numerical solution of the rate balance equation for the excited atoms and the transfer resonant radiation equation is investigated. Calculations of the escape factor for the slab, cylinder and sphere at Doppler and Lorentz forms of absorption and scattering profiles are executed. Calculation results of the escape factor for the cylinder and slab are compared with Holsten's asymptotical solutions. The numerical data for time dependence of a spectrum, the intensity of resonant radiation and the spatial distribution of the excited atomic concentration in a regime of afterglow are also considered.

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Imprisonment of Resonance Radiation in a Gaseous Discharge

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Following the theory of Holstein, the density, ${N}_{r}$, 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, ${N}_{r}$ 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, ${N}_{r}$ is given by its thermodynamic equilibrium value at the temperature of the electrons.

Physical Review•Jul 15, 1957•45 citations

Effect of the Imprisonment of Resonance Radiation on Excitation Experiments

A. V. Phelps

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 ($n^{1}P$) 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 $n^{1}P$ 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 $3^{1}P$ state of helium at 100 electron volts is 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$ ${\mathrm{cm}}^{2}$ instead of 4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$ ${\mathrm{cm}}^{2}$ as given in the literature.

Physical Review•Jun 15, 1958•117 citations

Experimental Study of Resonance Radiation Transfer

H. Mochizuki, R. G. Fowler

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 ($\ensuremath{\omega}=2\ensuremath{\pi}f$, $f=5 \mathrm{to} 50$ 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 ${T}_{c}$ corresponding to the advanced phase shift, the thickness $A$ of the resonance tube, and $n$. The dependences on other factors have also been examined.

Physical Review•Jan 4, 1965•2 citations

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

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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.

Acta Physica Sinica•Jan 1, 1993•0 citations

The Decay of Resonance Radiation by Spontaneous Emission

Emil J. Hellund

A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.

Physical Review•Mar 1, 1953•45 citations
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(\ensuremathρ)T(\ensuremath{\rho})T(\ensuremathρ), the probability that a quantum of emitted radiation will traverse a thickness \ensuremathρ\ensuremath{\rho}\ensuremathρ of material. This coefficient is shown to be given by the rapidly convergent series T(\ensuremathρ)=\ensuremathΣn(\ensuremath−k0\ensuremathρ)nn!(n+1)12T(\ensuremath{\rho})=\ensuremath{\Sigma}{n}^{}\frac{{(\ensuremath{-}{k}_{0}\ensuremath{\rho})}^{n}}{n!{(n+1)}^{\frac{1}{2}}}T(\ensuremathρ)=\ensuremathΣnn!(n+1)21​(\ensuremath−k0​\ensuremathρ)n​, where k0{k}_{0}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 k0\ensuremathρ{k}_{0}\ensuremath{\rho}k0​\ensuremathρ values.
Imprisonment of resonant radiation in gases on the basis of the numerical solution of the rate balance equation for the excited atoms and the transfer resonant radiation equation is investigated. Calculations of the escape factor for the slab, cylinder and sphere at Doppler and Lorentz forms of absorption and scattering profiles are executed. Calculation results of the escape factor for the cylinder and slab are compared with Holsten's asymptotical solutions. The numerical data for time dependence of a spectrum, the intensity of resonant radiation and the spatial distribution of the excited atomic concentration in a regime of afterglow are also considered.
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.
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.
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=5to50f=5 \mathrm{to} 50f=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 TTT 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 KKK is a constant, has been found for the dependence on the breadth CCC of the resonance tube as well as the well known dependence $\mathrm{ln}T\ensuremath{\propto}\mathrm{ln}n$ on the molecular density nnn. On the other hand, the transmitted radiation obeys a relation ${T}_{c}\ensuremath{\propto}{(\mathrm{nA})}^{\frac{1}{2}}$ between the time constant Tc{T}_{c}Tc​ corresponding to the advanced phase shift, the thickness AAA of the resonance tube, and nnn. The dependences on other factors have also been examined.
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.
A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.