Radiation Trapping in Atomic Vapours

by Molisch, Andreas F.; Oehry, Bernhard P.

• ISBN13: 9780198538660
• ISBN10: 0198538669
• Format: Hardcover
• Copyright: 1999-02-25
• Publisher: Clarendon Press

Summary

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. inspectroscopy, 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, emphasizingboth 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.

Table of Contents

List of Symbols	ixx
I BACKGROUND
Introduction	3	(12)
The physical process of radiation trapping	3	(3)
Historical overview	6	(3)
Applications of radiation trapping	9	(3)
Outline of the book	12	(3)
Atomic Lineshapes	15	(18)
The Einstein theory of radiation	15	(5)
Broadening mechanisms	20	(10)
Natural broadening	20	(1)
Doppler broadening	20	(2)
Pressure broadening	22	(2)
Voigt lineshapes	24	(6)
Strength rules for fine and hyperfine splitting	30	(3)
Collisions, quenching, and particle diffusion	33	(16)
Collisional cross-sections	33	(2)
Collisions between atoms of the same kind	35	(1)
Collisions between atoms of different kinds	35	(1)
Collisions with electrons; ionization and recombination	36	(1)
Particle diffusion	37	(4)
Hyperfine-structure intermixing	41	(1)
Fine-structure intermixing	42	(7)
II THE CLASSICAL RADIATION TRAPPING PROBLEM
Formulation of the classical problem	49	(30)
The Milne equation	50	(1)
The Holstein equation	51	(13)
Derivation of the Holstein equation	51	(5)
Idealized geometries	56	(3)
Simplifying assumptions in the derivation of the Holstein equation	59	(5)
The multiple-scattering representation	64	(3)
The equation of radiative transfer	67	(12)
The escape factor	69	(10)
Mathematical methods for the Holstein equation	79	(15)
The variational technique	79	(7)
Exact solutions in the high-opacity case	86	(3)
Van Trigt's solution	86	(1)
The geometrical quantization technique	87	(1)
The Fourier transform technique	88	(1)
The piecewise-constant approximation	89	(2)
The propagator function method	91	(1)
Other computation methods	92	(2)
Methods for the multiple-scattering representation	94	(9)
Monte Carlo simulations	94	(6)
Analytical solutions	100	(3)
Fitting equations and physical interpretation	103	(25)
Doppler and Lorentz lines	103	(8)
Voigt lines	111	(2)
Hyperfine split lines	113	(5)
Higher-order modes	118	(2)
Steady-state solutions of the Holstein equation	120	(4)
The emergent spectrum	124	(4)
The Milne and Eddington approximations	128	(12)
The original Milne theory	128	(3)
The angle approximation	131	(3)
The generalized Eddington approximation	134	(2)
The frequency approximation	136	(4)
Mathematical methods for the transfer equation	140	(25)
Discrete ordinate solution	140	(2)
The Feautrier technique	142	(7)
The basic Feautrier technique	143	(5)
Modified finite-differencing equations	148	(1)
Choice of the boundary conditions	149	(1)
The variable Eddington factor technique	149	(5)
Modified Feautrier approaches	154	(11)
The Rybicki reorganization	154	(2)
The core saturation method	156	(1)
The implicit integral method	157	(1)
Splitting algorithms	157	(1)
Quadrature perturbation	158	(1)
The discontinuous finite-element (DFE) method	158	(7)
III GENERALIZED TRAPPING PROBLEMS
Simple generalizations	165	(39)
Branching and quenching	165	(2)
Three-level atoms	167	(4)
Reflecting walls	171	(8)
Diffusely reflecting walls	171	(2)
Specularly reflecting walls	173	(6)
Particle diffusion	179	(14)
Formulation and direct solution	179	(2)
The modal combination technique	181	(5)
The discrete-ordinate technique	186	(6)
Results	192	(1)
Inhomogeneous distribution of absorbers	193	(4)
Two- and three-dimensional geometries	197	(7)
The finite cylinder	197	(4)
The parallelepiped	201	(1)
The torus and the hollow cylinder	201	(3)
Partial frequency redistribution	204	(43)
The physical picture of PFR	204	(19)
Redistribution functions in the atomic rest frame	205	(3)
Redistribution functions in the laboratory rest frame	208	(2)
Angle-averaged redistribution functions	210	(1)
Pure Doppler broadening	211	(3)
Doppler plus natural broadening	214	(4)
Doppler plus collisional broadening	218	(1)
Doppler, natural, plus collisional broadening	219	(1)
Branching transitions	220	(1)
The Holstein equation with PFR	221	(2)
Variational solution	223	(7)
The velocity distribution of excited atoms	230	(2)
The propagator function method and the PCA method	232	(4)
Monte Carlo simulations	236	(2)
Transfer equation formulations	238	(2)
Frequency diffusion	240	(2)
Large-scale particle flow	242	(5)
Polarization	247	(15)
Introduction	247	(1)
Formal solution of the vector transfer equation	248	(2)
Trapping problems with polarization	250	(4)
The polarization-free approximation	250	(1)
The Holstein equation without magnetic field	251	(2)
The Holstein equation with magnetic field	253	(1)
Monte Carlo simulations	254	(4)
Physical effects	258	(4)
Non-linear radiation trapping	262	(51)
When do non-linearities occur?	263	(4)
Interaction of strong laser radiation with atoms	267	(4)
Selection of velocity groups	268	(2)
Burn-through and one-dimensionality	270	(1)
Steady-state solutions	271	(17)
Complete linearization	272	(4)
Operator perturbation	276	(9)
Direct iteration	285	(2)
Approximate techniques	287	(1)
Monte Carlo simulations	287	(1)
Transient problems	288	(11)
Numerical solution of the non-linear Holstein equation	288	(3)
Analytical approximations	291	(2)
Numerical solution of the transfer equation	293	(1)
Physical effects	294	(4)
Monte Carlo simulations	298	(1)
Multilevel systems	299	(14)
Complete linearization in the multilevel case	299	(1)
Operator perturbation techniques for multilevel systems	300	(1)
The equivalent two-level atom	301	(2)
Steady-state in three-level atoms	303	(4)
Time-dependent depletion	307	(6)
Combination of techniques	313	(13)
The four basic questions	313	(1)
Linear trapping with complete frequency redistribution	314	(5)
Solution methods for the classical Holstein equation	314	(3)
Inclusion of other physical effects	317	(2)
Nonlinear trapping with CFR	319	(2)
Taking the non-linearity into account	320	(1)
Multilevel atoms	320	(1)
Required accuracy	321	(1)
Trapping with partial frequency redistribution	321	(5)
IV APPLICATIONS
Measurements in chemical physics	326	(20)
Atomic lifetimes	327	(1)
Collision cross-sections	328	(3)
Ionization cross-sections	331	(4)
Quenching and intermixing cross-sections	335	(3)
Atomic beams	338	(1)
Atomic densities	339	(1)
Radiation trapping in cold atoms	340	(4)
Other measurements	344	(2)
Simulation of optically pumped gas lasers	346	(15)
Trapping in gas lasers	346	(1)
Low-opacity formulation	347	(7)
Basic formulation	347	(3)
Approximate solution for single-mode operation	350	(2)
Conclusions	352	(2)
The mercury-nitrogen laser	354	(7)
The operating principle	354	(1)
Setup of the rate equations	355	(3)
Conclusions	358	(3)
Atomic line filters	361	(7)
Discharge lamps and plasmas	368	(19)
Introduction	368	(1)
The theory of discharge lamps	369	(3)
Radiation trapping in discharge lamps	372	(4)
Decreasing radiation trapping in lamps	376	(3)
The optogalvanic effect	379	(2)
Plasmas	381	(2)
Concluding remarks	383	(4)
V APPENDICES
Atomic structure	387	(16)
Models of the atomic structure	387	(1)
The Bohr-Rutherford model	387	(1)
The Schruodinger equation	388	(1)
The hydrogen atom	388	(3)
The periodic table of elements	391	(1)
Alkali atoms	392	(5)
Lifting of the l-degeneracy in alkali atoms	392	(2)
Fine structure of alkali atoms	394	(3)
The helium atom	397	(2)
The alkaline earth elements	399	(1)
Trivalent elements	399	(1)
Hyperfine structure and isotope splitting	399	(2)
Hyperfine structure	399	(2)
Isotope splitting	401	(1)
Effects of external magnetic and electric fields	401	(2)
The Zeeman effect	401	(1)
The Stark effect	402	(1)
Values of the Ak.m matrix elements for the numerical solution of the Holstein equation	403	(10)
The slab	403	(1)
The cylinder---method 1	404	(2)
The cylinder---method 2	406	(3)
The sphere	409	(4)
Publicly available software for the computation of radiation trapping	413	(9)
Rad-Trap	413	(2)
McTrap	415	(3)
Slab3	418	(1)
Altair	419	(1)
Tlusty	420	(1)
Other programs	421	(1)
Fitting equations for the eigenvalues and eigenfunctions of the Holstein equation	422	(5)
The slab	422	(1)
The cylinder	422	(1)
The sphere	423	(4)
Finite difference solution of the inhomogeneous equation of radiative transfer in a finite cylinder	427	(6)
The density matrix	433	(7)
The density matrix for atomic states	433	(1)
The density matrix for photons	434	(2)
Interaction of atoms and radiation	436	(1)
State multipoles	437	(3)
Definition of tensor operators	437	(1)
Definition of state multipoles	438	(2)
High-field effects	440	(6)
Absorption and emission coefficients of homogeneously broadened lines	440	(1)
AC-Stark splitting and Mollow triplets	441	(2)
Effects of AC-Stark splitting on radiation trapping	443	(1)
The validity of the transfer equation	444	(2)
References	446	(55)
Index	501