Spectral Line Broadening by Plasmas

418 Pages·1974·6.453 MB·English

by  HANS R. GRIEM (Eds.)

Spectral Line Broadening
by Plasmas
HANS R. GRIEM
Department of Physics and Astronomy
University of Maryland
College Park, Maryland
ACADEMIC PRESS New York and London 1974
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1974, BY ACADEMIC PRESS, INC.
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Library of Congress Cataloging in Publication Data
Griem, Hans R
Spectral line broadening by plasmas.
(Pure and applied physics, v. )
Includes bibliographical references.
1. Plasma spectroscopy. I. Title. II. Series.
QC718.5.S6G74 543'.085 73-5300
ISBN 0-12-302850-7
PRINTED IN THE UNITED STATES OF AMERICA
Preface
Many problems have been solved by the very active experimental and
theoretical research on Stark broadening that began in the 1950's, although
without question a number of the more difficult problems have not yet
yielded even to rather high-powered approaches. Hopefully, the nature of
these problems will become clear to the reader of the appropriate sections
of this book in spite of the fact that a "minimum theory" approach was
generally preferred. However, not only the formal aspects of theoretical
work had to be somewhat abbreviated, but also the routine functions of
experiments designed principally to measure certain Stark broadening
parameters had to be neglected to a large extent in order to gain space for
the discussion of critical experiments. Such experiments have contributed
more than their share to our present understanding of the subject and will
probably continue to offer serious challenges to the theoreticians.
References (close to five hundred) are numbered throughout the text to
avoid repetition. If several papers are listed under one number, they are
usually distinguished by a), b), c), etc., in the text, unless an entire group
of papers is referred to. No value-judgement should be attached to the
ordering or the multiplicity of papers under one reference number. As a
matter of fact, a large number in such a group may well mean that the
subject of these papers is particularly interesting and the research on it
unusually active.
Future work in this area will be much facilitated by the establishment
of a data center for spectral line shapes and shifts at the United States
National Bureau of Standards. The two main objectives of the center are:
(1) the collection and cataloging of all literature relevant to the broadening
ix
X PREFACE
and shift of atomic spectral lines; and (2) the preparation and publishing
of bibliographies and critical reviews of various topics in atomic line
broadening. Its first publication is a "Bibliography on Atomic Line Shapes
and Shifts" by J. R. Fuhr, W. L. Wiese, and L. J. Roszman (NBS Spec.
Pubi. 366), U.S. Government Printing Office, Washington, D. C, 1972. A
supplement to this publication was issued in January 1974.
Acknowledgments
This monograph was begun while the author was a Guggenheim fellow
at the Culham Laboratory in England during 1968-1969, and the manu-
script was completed in the course of a one year's stay at the European
Space Research Institute at Frascati, Italy. The hospitality of spec-
troscopists, plasma-, and astrophysicists in both laboratories has made
much of the tedious work possible that would have stretched out over an
even longer period otherwise. However, the major part of the manuscript
was written in the two intervening years 1969-1971 at the University of
Maryland, a preliminary version serving as the basis for a special lecture
course in the spring of 1970. The students of this class and other colleagues
and students of the University of Maryland have given so many critical
comments and suggestions that individual acknowledgments are out of
the question.
Equally valuable have been numerous discussions with scientific col-
leagues all over the world, who offered their criticisms of the 1970 Maryland
lecture notes, answered questions regarding their own work, or contributed
unpublished results to this first attempt at a comprehensive review of the
Stark broadening of atomic and ionic spectral lines. Again I have to plead
for the understanding of these readers if I do not acknowledge their con-
tributions individually.
Almost all of the draft manuscripts and the entire final manuscript were
typed patiently and critically by Mrs. Mary Ann Ferg. For this I thank her
with all the unnamed scientific colleagues who contributed so much to
this work.
List of Symbols
a (Integral) width function Fo Holtsmark (normal) field
do Bohr radius strength
A (Differential) width function, $ Bates and Damgaard factor
Transition probability, Fourier g Gaunt factor, Statistical
transform of field strength dis- weight, Two-particle correla-
tribution function, Ion broad- tion function
ening parameter G Green's function
AM Asymmetry Giß) Chandrasekhar function
b Time derivative of reduced h Profile parameter
field, (Integral) shift function fi Planck's constant divided by
B Magnetic field strength, (Dif- 2?r
ferential) shift function, H Holtsmark function, Hamil-
Parameter for dynamical tonian
corrections Ha Balmer a line, etc.
c Velocity of light 3C Effective impact broadening
C Stark effect coefficient, Phase Hamiltonian
shift parameter i Initial state (subscript)
C(s) Autocorrelation function I Intensity
'3 > CA Interaction constants /(« Chandrasekhar function
Ci, Stark coefficient h Bessel function
d Dipole (subscript), Stark shift Im Imaginary part
D Dipole operator 3 Angular momentum quantum
e Electron charge number
En Ionization energy of hydrogen j(x) Reduced line shape
Ei Atomic energy levels J,S Total angular momentum
E„ Ionization energy or series limit quantum number
f Scattering amplitude, Velocity k Wave number, Momentum,
distribution function, Final Boltzmann constant, Trans-
state (subscript), (Collision) formed field variable
frequency, Oscillator strength K Wave number
F Electric field strength KM Modified Bessel functions of
F,F>? l,ft Relaxation theory functions the second kind
xu
LIST OF SYMBOLS Xlll
I Reduced line shape, (Orbital) S Spin quantum number
angular momentum quantum t(s, 0) Schrödinger evolution operator
number, Thickness t Time
L,£ Orbital angular momentum T Transition matrix, Kinetic
quantum number temperature
««) Line shape Tr Trace
L« Lyman a line, etc. u(s, 0) Heisenberg evolution operator
£(ω) Relaxation operator U Interaction Hamiltonian
m Magnetic quantum number, u Electric field energy density of
Q
Electron mass plasma waves
Wir Radiator mass V Velocity
mp Perturber mass Ve Electron velocity
m' Reduced mass Vi Ion velocity
M Ion mass, Magnetic quantum V Volume
number w Stark (half) half-width
max Maximum (of) W Field strength distribution
min Minimum (of) function
9fïl Total magnetic quantum X Reduced wavelength, Carte-
number sian coordinate, Correction
n, n», ri/ Principal quantum number function, Dimensionless vari-
n Integer, Total number of per- able
turbers Xa Coordinate (operator)
Π\ , 7l2 Parabolic quantum numbers y Coordinate
N- Electron density Y Spherical harmonic, Dimen-
N Perturber density sionless variable
P
P Power, Probability, Projection z Dimensionless variable, Co-
operator ordinate
P, Paschen a line, etc. z Nuclear (or core) charge of
Pn Configurational partition func- radiator
tion z» Perturber charge
q Quadrupole (subscript)
Q Perturber coordinates a Scattering angle, Fine struc-
Q(r) Configuration space distribu- ture constant, Index for 1, 2,
tion function and 3, (Holtsmark) reduced
r Distance, Position wavelength
ri Position vector of perturbing ß Reduced field Strength
ion y Damping constant, Euler's
r Mean ion-ion radius (separa- constant
P
tion) r Gamma function
R Reactance matrix, Debye δ Reduced frequency separation,
shielding parameter Kronecker symbol, Dirac's
Re Real part (of) delta function, New variable
rms Root mean square for hyperbolic path functions
s Time variable Δ Difference
S Spectral density, Spin quan- A Dipole operator in line space
d
tum number, S matrix, Line Δ(/3) Correction to Holtsmark func-
strength tion
£+.- Satellite intensities Δω Frequency separation from un-
θ(« Kogan function perturbed line
XIV LIST OF SYMBOLS
e Dielectric constant, Kinetic φ Bates and Damgaard correc-
energy, Dimensionless param- tion function
eter, Eccentricity Φ Phase shift, Polar angle, Im-
η Coulomb parameter, Coulomb pact broadening operator
phase, Decrement, Imaginary
χ Ionization energy
part of phase shift
χ' Screening function
Θ Polar angle, Broadening opera-
φ' Two-particle correlation func-
tor in "line" space
tion
λ Wavelength, Azimuth angle
ψ( y) Generalized phase shift correla-
Ä de Broglie wavelength (divided
tion function
by27r)
μ Summation index ω Frequency
v Summation index cos Mean Stark splitting
£ Parameter in hyperbolic classi- coF Field fluctuation frequency
cal path theory ω Doppi er width
0
P Charge density, Impact param- ω»/ Unperturbed frequency of
eter spectral line
PD Debye radius
coo Unperturbed frequency of
pi Statistical (density) operator
spectral line
σ Cross section, Transition in-
ω (Electron) plasma frequency
tegral ρ
ωα, Separation of unperturbed
©(<£) Relative line strength
energy levels
©(2fTZ) Multiplet strength
T Duration of collision, Relaxa- Ω Frequency of plasma waves,
tion time, Dimensionless time Angle, Solid angle, Potential
variable energy
CHAPTER I
Introduction
Effects of electric fields from electrons and ions (both acting as point
charges) on spectral line shapes can be important over a wide range of
plasma parameters, especially of charged particle densities. At one extreme
of the density range are so-called H II regions (N « 103 cm-3) emitting
radio-frequency radiation due to transitions between highly excited states
(principal quantum numbers n « 102) of atomic hydrogen; at the other are
stellar interiors (N « 1026 cm-3) in which some radiative energy transport
may be provided by Stark-broadened resonance lines of highly ionized
atoms (such as 25-times ionized iron, Fe XXVI) in the X-ray region of
the electromagnetic spectrum. In between are laboratory plasmas with
densities of N « 1013 cm-3 (rf discharges) toiV ~ 1019 cm-3 (laser-produced
plasmas, etc.) at the extremes, and with spectral lines from those of neutral
atoms mostly in the visible part of the spectrum to those of multiply ionized
light or medium atoms in the vacuum ultraviolet region.
The temperature range of both astronomical objects and laboratory
plasmas is smaller in comparison—say, from T « 2 · 103 K in some dis-
charge sources and, perhaps, certain H II regions to T « 2 · 107 K in
both stellar interiors and very high temperature laboratory sources. Other
plasma parameters, namely those describing the spectrum of plasma waves
and details of the electron and ion velocity distribution functions, tend to
be of minor influence in regard to line shapes, unless deviations from
thermodynamic equilibrium are large. These additional parameters, besides
1
2 I. INTRODUCTION
electron and ion number densities and kinetic temperatures, are of course
superfluous in case of complete thermodynamic equilibrium.
The very wide range of densities, temperatures, wavelengths, and ionic
charges would seem to discourage hopes for a unified and practical, as
opposed to formal, theoretical treatment of the subject. However, for any
particular spectral line, the relevant range of plasma conditions for Stark
broadening to be important is fortunately much smaller. The density range
of interest may typically be estimated from the inequality
COD < w < | ω,·/ — co»'/' \j
which involves Doppler width COD (almost always much larger than the
natural width), Stark width w (approximately proportional to N p with p
}
ranging from f to 1 in actual cases), and unperturbed frequencies ω,/ and
co»'/' of the line in question and a neighboring line. Even for widely spaced
lines, the ratio of maximum and minimum densities is therefore only of
order (c/v)1,2p, c being the velocity of light and v the mean velocity of the
emitting or absorbing systems relative to the observer. For all astronomical
and laboratory conditions mentioned above, this ratio stays below
iVmax/iVmin « 104 according to this consideration and is much smaller than
that in most cases—say, N */N i « 102 to 103 for any given line. For all
ma mn
plasmas but those showing extreme deviations from equilibrium, the tem-
perature range is restricted by
10-2χ <kT < χ,
X being the ionization energy of the radiating atom or ion, whose relative
abundance would be vanishingly small at other temperatures.
Also, the frequency separation Δω relative to the unperturbed line fre-
quency coo tends to vary by no more than a factor about 10 2 (corresponding
to variations in relative intensities by factors of 104-105) over directly
observable or otherwise important (e.g., for radiative transfer) portions of
the line profile. At relatively high densities, this factor is still smaller be-
cause of the overlap with neighboring lines, and it is thus fair to say that
for a particular line, the three main variables (N, T, Δω) usually vary only
by about a factor of IO2. A description of the line profile in terms of one or
two rather extreme approximations to a more general, but less practical,
theory thus becomes a much more likely proposition.
Whether or not the same approximations will be useful for a large class of
lines depends on the relative magnitudes of, say, all the characteristic
frequencies entering the general problem. Of these frequencies, the angular
frequency corresponding to the wavelength λ of the line, namely co = 2rc/\,
and that corresponding to a quantum of kinetic perturber energy, namely
cok ~ kT/h, tend to be comparable and much larger than most other char-