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Formulation of the Classical Problem

Andreas F. Molisch, Bernhard P Oehry

Published Nov 26, 1998• 0 citations
Publisher

Abstract

Abstract There are three principal mathematical approaches to describing trapping: the Holstein equation, the method of combining the equation of radiative transfer with the rate equation for the excited state atoms, and the multiple-scattering representation. Ultimately, these three descriptions are equivalent-as they should be, since they describe the same physical process. Their different formulations stem from different views of the trapping problem. Each method has its specific advantages and disadvantages for the evaluation, hence it is very helpful to fully understand all three formulations.

DOI
10.1093/oso/9780198538660.003.0004

Related papers

Advantages of the multiple scattering representation approach to radiation trapping

Eduardo J. Nunes-Pereira, A. R. Alves-Pereira, José M. G. Martinho +1 more

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.

arXiv:0705.1063•May 8, 2007•1 citation

Radiation Trapping in Atomic Vapours

Andreas F. Molisch, Bernhard P Oehry

Abstract Radiation from spectral lines can be absorbed and re- emitted many times in atomic vapours before it reaches the boundaries of the container encasing the vapour. This effect is known as radiation trapping. It plays an important role practically everywhere where atomic vapours occur, e.g. in spectroscopy, in gas lasers, in atomic line filters, in the determination of atomic lifetimes, in measurements of atomic interaction potentials, and in electric discharge lamps. This book for the first time assembles all the information necessary for a treatment of practical problems, emphasizing both physical insights and mathematical methods. After an introduction that reviews resonance radiation and collisional processes in atomic vapours, physical effects and mathematical methods for various types of problems (e.g. with or without saturation, particle diffusion, reflecting cell walls, etc.) are explained in detail. The last part of the book describes the applications of these methods to a variety of practical problems like cross-section measurements or the design of discharge lamps.Atoms can both absorb and emit electromagnetic radiation like visible light In an assembly of many atoms, e.g., in a vapour or in a gas, radiation emitted by one atom may be reabsorbed by one of its neighbours, just to be reemitted a little later. The game can go on for quite some time—until the photon manages somehow to escape from this prison. At first glance, quite easy to understand. On the other hand, there must be some reason for this voluminous book full of sums, integrals and matrices. Our interest in this subject originally stemmed from investigations on atomic line filters we did for the European Space Agency, ESA. We believed that radiation trapping would have some influence on the performance of these filters and were looking for a simple means to take it into account We started out the same way as probably anyone new to the field would have done when faced with the problem of trapping. We read the classical paper of Holstein, and thought that was all there was to it. We soon had to realize, however, that the assumptions made by Holstein were not fulfilled in our problem, and decided to give the whole matter a closer look. That ‘closer look’ obtained a momentum of its own, and trapping became the centre of our own research in the following years. As our overview of the subject increased. we found that trapping plays a role in a staggering variety of situations. Trapping appears to be important in most experimental setups and in most engineering applications where atomic vapours are involved.

•Nov 26, 1998•191 citations

THEORETICAL TREATMENT OF RADIATION TRAPPING IN A MULTIPLE LINES COUPLED SYSTEM

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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.

Acta Physica Sinica•Jan 1, 1997•0 citations

Polarization

Andreas F. Molisch, Bernhard P Oehry

Abstract In theory, all radiation trapping problems must be treated under consideration of the polarization state of the radiation. Even with an unpolarized excitation source, the geometry of the problem can introduce some degree of polarization to the fluorescence radiation. Of course, when the excitation source itself is polarized, as is usually the case for a laser, polarization can become even more important. One should not forget, however, that incorporation of polarization makes for a very complicated problem. The Holstein equation will become a Fredholm integral equation for a four-component vector instead of the usual scalar function (the excited-state density). Having seen the difficulties in solving the comparatively simple ‘usual’ trapping problem, we see that polarization should only be included when it is not to be avoided.

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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.

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Heisenberg's Quantum Mechanics

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A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities

WORLD SCIENTIFIC eBooks•Jan 1, 2011•16 citations
Abstract There are three principal mathematical approaches to describing trapping: the Holstein equation, the method of combining the equation of radiative transfer with the rate equation for the excited state atoms, and the multiple-scattering representation. Ultimately, these three descriptions are equivalent-as they should be, since they describe the same physical process. Their different formulations stem from different views of the trapping problem. Each method has its specific advantages and disadvantages for the evaluation, hence it is very helpful to fully understand all three formulations.
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.
Abstract Radiation from spectral lines can be absorbed and re- emitted many times in atomic vapours before it reaches the boundaries of the container encasing the vapour. This effect is known as radiation trapping. It plays an important role practically everywhere where atomic vapours occur, e.g. in spectroscopy, in gas lasers, in atomic line filters, in the determination of atomic lifetimes, in measurements of atomic interaction potentials, and in electric discharge lamps. This book for the first time assembles all the information necessary for a treatment of practical problems, emphasizing both physical insights and mathematical methods. After an introduction that reviews resonance radiation and collisional processes in atomic vapours, physical effects and mathematical methods for various types of problems (e.g. with or without saturation, particle diffusion, reflecting cell walls, etc.) are explained in detail. The last part of the book describes the applications of these methods to a variety of practical problems like cross-section measurements or the design of discharge lamps.Atoms can both absorb and emit electromagnetic radiation like visible light In an assembly of many atoms, e.g., in a vapour or in a gas, radiation emitted by one atom may be reabsorbed by one of its neighbours, just to be reemitted a little later. The game can go on for quite some time—until the photon manages somehow to escape from this prison. At first glance, quite easy to understand. On the other hand, there must be some reason for this voluminous book full of sums, integrals and matrices. Our interest in this subject originally stemmed from investigations on atomic line filters we did for the European Space Agency, ESA. We believed that radiation trapping would have some influence on the performance of these filters and were looking for a simple means to take it into account We started out the same way as probably anyone new to the field would have done when faced with the problem of trapping. We read the classical paper of Holstein, and thought that was all there was to it. We soon had to realize, however, that the assumptions made by Holstein were not fulfilled in our problem, and decided to give the whole matter a closer look. That ‘closer look’ obtained a momentum of its own, and trapping became the centre of our own research in the following years. As our overview of the subject increased. we found that trapping plays a role in a staggering variety of situations. Trapping appears to be important in most experimental setups and in most engineering applications where atomic vapours are involved.
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.
Abstract In theory, all radiation trapping problems must be treated under consideration of the polarization state of the radiation. Even with an unpolarized excitation source, the geometry of the problem can introduce some degree of polarization to the fluorescence radiation. Of course, when the excitation source itself is polarized, as is usually the case for a laser, polarization can become even more important. One should not forget, however, that incorporation of polarization makes for a very complicated problem. The Holstein equation will become a Fredholm integral equation for a four-component vector instead of the usual scalar function (the excited-state density). Having seen the difficulties in solving the comparatively simple ‘usual’ trapping problem, we see that polarization should only be included when it is not to be avoided.
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.
A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities