Progress in Plasmas and Gas Electronics, Volume 1 [Reprint 2021 ed.] 9783112545805, 9783112545799

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Progress in Plasmas and Gas Electronics Volume 1

Progress in Plasmas and Gas Electronics Volume 1

Edited by R . R o m p e a n d M. S t e e n b e c k

With 211 Figures and 80 Tables

AKADEMIE-VERLAG 1975

BERLIN

Translated from the original German „Ergebnisse der Plasmaphysik und der Gaselektronik", Band 1, published by Akademie-Verlag, with revisions and additions.

Erschienen im Akademie-Verlag, 108 Berlin, Leipziger Str. 3 —4 © Akademie-Verlag, Berlin, 1975 Lizenznummer: 202- 100/573/74 Gesamtherstellung: VEB Druckerei „Thomas Müntzer", 582 Bad Langensalza Bestellnummer: 761 745 4 (55 70/1E) • LSV 1195 Printed in GDR EVP 1 1 5 , -

Foreword

Plasma physics has assumed considerable proportions in recent times. This applies both to exploratory research and technical application; new phenomena are discovered almost every day in the field of plasma physics and related fields of gas electronics, theoretically interpreted and, in some cases, applied in practice. While in other fields of contemporary physics one occasionally hears t h a t following a phase of rapid development we are now approaching a calmer era, this does not generally hold true for plasma physics. The extraordinary multiplicity of the field — physics, geophysics, astrophysics, chemistry, electronics are all contributing their share — complicates as a matter of course adequate arrangement of the individual results and justifies the need for a complete, systematic representation. I n view of the rapid pace of development, representations of this kind must be realizable in a relatively short time to make them useful. Contemplating this task it seems that it is hardly possible to give a coherent representation of our present knowledge of plasma physics and its technical application in the conventional form of a text-book or handbook. I t is remarkable t h a t attempts to give comprehensive representations of plasma physics, which were still possible some 25 to 30 years ago, have no longer been made in recent years, apart from literature dealing with special fusion problems, which, despite its up-to-date nature, is still rather one-sided. Plasma physics is the physics of the most frequent aggregate state occurring in nature. The universe is immersed in a sea of matter in plasma state featuring a wide variety of the characteristic parameters. Only some celestial bodies, such as our own planet, where temperatures of several hundred °K are encountered at high densities, have condensed phases — this is but a small fraction. To represent plasma physics only from the point of view of fusion systems would mean to omit broad fields of knowledge. This would also be the disadvantage of trying to elaborate a representation of plasma physics solely on the basis of astronomic phenomena occurring in the cosmos. The representation of plasma physics must be approached differently, by taking into account the latest advances in the field of physics. The courage to attempt comprehensive representations of plasma physics, which many physicists possessed twenty-five to thirty years ago — we may consider ourselves to be among their number — originated in the situation

VI

Foreword

extant in physics of that time characterized by the permeation of the classical gas discharge physics by advances in the fields of quantum physics of electrons, atoms and ions. Its charm lay in the possibility of fundamental simplification making it possible to comprehend the physics of gas discharge. Essentially, it was a matter of viewing all phenomena in plasma from the point of view of the kinetics of its components, a task which, as a whole, could be fulfilled. The predominant approach was based on the individual particles; the binary impacts primarily determined the interpretation of the phenomenon. Since this method of treatment has been verified in a wide range of circumstances, it could be applied successfully to plasmas of astrophysics, of the ionosphere, of sparks and arc discharges, and of gas discharges at low pressure. I t is also suitable for many problems of plasma chemistry, whose importance is increasing nowadays. This approach has resulted in extraordinary methodical progress and clarified the common physical foundation in the great majority of plasmas. Although the work was confined to the classical realm in the field of plasma, this progress has been comparable and took place nearly concurrently with the development in the field of solid state physics, which began after the studies conducted by SOMMERF E L D and B L O C H in the field of the one-electron theory of metals, semiconductors and insulators. If the kinetics of homogeneous isotropic plasmas are approached on the basis of discrete particles, two physically significant problems must necessarily recede into the background: the question of various collective interactions of the plasma components, of typical multi-particle problems and the question of the morphology of plasma, of the outer form assumed by plasma for various reasons. I n special cases one speaks of the "problem of support" — a problem of a very general nature with which we are confronted in the cosmos as well as in the laboratory. But it is precisely these two aspects that give plasma its identity as a special medium, comparable perhaps with liquid being characterized by a free surface and its greater degree of disarrangement than observed in crystalline structures. A great deal has been written about the fundamental significance of these two questions in monographs appearing in the past ten years. But the initiates will not be surprised to hear that the time is not nearly ripe for answering these two questions. Particularly in this area, a great deal of persevering and purposive work must still be done in order to arrive at tangible results. Multi-particle problems have become part of the group of major problems in the theoretical foundations of modern physics. I t will suffice to mention the extraordinarily successful work being done in this connection in the fields of superconductivity, superfluidity, states of excitation of solids (exciton, polaron, plasmon) and of the theory of atomic nuclei. Our German series entitled Ergebnisse der Plasmaphysik und der Gaselektronik appears as a number of self-contained volumes. Specific areas of study t h a t have, in our opinion, attained the necessary degree of maturity will be summarized by researchers who are themselves actively working in the respective

VII

Foreword

fields. The emphasis will be on topical areas of research and technology that have an important bearing on the development of plasma physics in the broadest sense. We believe that although it is not possible to give a systematic overall survey of the field today, we can at least formulate the major problems in research and application. We ask all colleagues interested in promoting the development of plasma physics and gas electronics along these lines to cooperate with us, be it as authors — we do not presume to be able to find all those best suited for this purpose or as readers of the contributions by their constructive criticism. Robert Rompe Max Steenbeck

Contents

Spectroscopic Methods of Plasma Diagnostics by W. Neumann Chapter 1.

Introduction

3

Chapter 2.

Method for Determining t h e Emission Coefficients and Line P r o f i l e s . . .

5

Chapter 3.

Classification of Spectra

87

Chapter 4.

Calculation of t h e Composition of a Thermal Plasma with a Boltzmann Distribution of the Electronic Levels

95

Chapter 5.

Determination of Plasma Parameters via t h e I n t e n s i t y of Spectral Lines, emitted b y Atoms and Ions in an Optically Thin Layer. Plasma in Thermal Equilibrium Determination of Temperatures and Particle Densities of Plasmas Displaying Appreciable Absorption Determination of Plasma Parameters f r o m t h e Band Spectrum of Diatomic Molecules Determination of Particle Densities in Plasmas b y Means of t h e Refractive Index. Laser Diagnostics

Chapter 6. Chapter 7. Chapter 8.

Chapter 9. Determination of Plasma Parameters by Means of Continuous Radiation Chapter 10. Determination of Plasma Parameters f r o m Line Profiles. Fading of Spectral Lines a t t h e Series Limit and t h e Problem of t h e Reduction in t h e Ionization Energy Chapter 11. Validity Limits of t h e Methods of Thermal Spectroscopy discussed in Chapters 5 to 10. Some Special Methods of Nonthermal Spectroscopy . . Bibliography

119 198 237 258 289

324 404 430

Magnetohydrodynamic Waves (Waves in Ideal Media) by K. Baumgartel and G. Wallis Chapter 1. Chapter 2. Chapter 3.

Introduction 513 Propagation of Plane Waves in Homogeneous Media 516 Interaction of MHD-Waves with Surfaces of Discontinuity. Propagation in Inhomogeneous Media 541 Chapter 4. Propagation in Inhomogeneous Media 563 Chapter 5. Appendix: Matrices 582 Bibliography 589

Contents

X

Validity Conditions for Local Thermodynamic Equilibrium by H. W. Drawin Chapter 1.

Introduction

Chapter 2.

P a r t i a l L.T.K, in T i m e - I n d e p e n d e n t a n d H o m o g e n e o u s Optically Thin Plasmas 597

593

Chapter 3.

Complete L . T . E . in T i m e - I n d e p e n d e n t a n d H o m o g e n e o u s Optically Thin Plasmas 606

Chapter 4.

Influence of Resonance A b s o r p t i o n on t h e E s t a b l i s h m e n t of Complete L . T . E . in Homogeneous S t a t i o n a r y P l a s m a s 608

Chapter 5.

Influence of Resonance Absorption on t h e E s t a b l i s h m e n t of P a r t i a l L.T.E. in H o m o g e n e o u s S t a t i o n a r y P l a s m a s 611

Chapter 6.

I n f l u e n c e of H e a v y Particle Collisions on t h e E q u i l i b r i u m P o p u l a t i o n of Highly E x c i t e d Levels 614

Chapter 7.

Validity of P a r t i a l L . T . E . in Quasi-Stationary a n d Quasi-Homogeneous P l a s m a s (optically t h i n a n d thick) 617

C h a p t e r 8.

R e l a x a t i o n Times for t h e E s t a b l i s h m e n t of P a r t i a l a n d Complete L . T . E . in Homogeneous T r a n s i e n t P l a s m a s 631

Chapter 9.

R e l a x a t i o n L e n g t h s for t h e E s t a b l i s h m e n t of P a r t i a l a n d Complete L . T . E . in S t a t i o n a r y N o n - U n i f o r m P l a s m a s 637

C h a p t e r 10. P e r t u r b a t i o n of t h e Electron T e m p e r a t u r e b y E x t e r n a l l y Applied Electric Fields

64

C h a p t e r 11. Calculation of Optical R e d u c t i o n F a c t o r s Aj{ and A t

648

C h a p t e r 12. Concluding R e m a r k s

657

Bibliography

658

Subject Index

661

W. Neumann

Spectroscopic Methods of Plasma Diagnostics

CHAPTEE I

Introduction

The investigation of plasmas (ionized gases) has come during the last two decades into the focus of technological and scientific interest. Of special interest are high- and low-pressure plasmas, produced by steady and pulsed electric discharges, by adiabatic and shock-wave compression, by focusing of strong electromagnetic radiation, by thermal heating, by chemical processes, and by injection of energetic particles into magnetic bottles. Plasmas find application in lighting, welding, cutting, melting, and heat treatment of materials used at high temperatures. Furthermore, plasmas are encountered in gaselectronic devices, in circuit breakers, in high-temperature chemistry, in devices for conversion of heat into electric energy, such as thermionic converters and MHD-generators, and also in research on controlled thermonuclear reactions in high-temperature plasmas, in which nuclear energy is to be converted into other, technologically useful forms of energy. For the description and explanation of phenomena occurring in a plasma, it is necessary to determine as many parameters of the plasma as possible with the help of as many as possible mutually independent methods, as accurately as possible. Among the relevant parameters are the material composition of the plasma, the temperature and density of the different plasma components, the magnetic and electric field strengths, and the flow velocities of the individual components of the plasma. To the numerous methods of plasma diagnostics, which were developed for the purpose of determining the plasma parameters, belong the methods of plasma spectroscopy. Their application has the advantage over other methods in t h a t the plasma remains practically undisturbed. At the present time the useful range of wavelengths for plasma spectroscopy extends from the infrared (maximum wavelength 0.1 cm) to the vacuumultraviolet (wavelengths down to a few A). Shortwave spectroscopy is of special interest in the study of high-temperature plasmas with temperatures of the order of 106 °K and higher. The spectroscopic methods were primarily developed for the determination of the plasma parameters in thermal plasmas, i.e., plasmas in a local thermodynamic equilibrium. Most methods, however, can also be applied in those cases where the equilibrium is disturbed due to the difference in temperature of the different plasma components and when certain conditions are met: each plasma component must have a Maxwellian velocity distribution and the

4

Spectroscopic Methods of Plasma Diagnostics

electron states must be occupied with electrons in accordance with the Boltzmann distribution. In plasmas containing molecules the occupation of the rotational and vibrational terms must also correspond to a Boltzmann distribution. Several spectroscopic methods are known which are valid even though neither the Maxwell distribution nor the Boltzmann distribution occur. To these methods belong those for the determination of density, whereby the refractive index is measured. Besides these, there are special methods which are only applicable to lowpressure plasmas with an extremely disturbed BoLTzmann distribution and in which the line intensities emitted from relatively low levels are measured. In the following discussion we shall first deal with the general measuring and computational methods for the determination of line intensities and line profiles (Chapter 2). In Chapter 3 we shall briefly treat the classification of spectra. Chapter 4 deals with the calculation of the plasma composition in a thermal atomic plasma or in a plasma in which chemical reactions take place. In Chapters 5 —10 are discussed in detail those plasma diagnostic methods, which were developed under the assumption of thermal equilibrium. In these methods the intensities in the line spectrum and in the continuous spectrum of emission and absorption spectra are measured, the refractive index is determined, and the line profiles and line shifts are obtained. In Chapter 11 the limits of application of the above methods are stated and certain specific methods are mentioned in which the plasma parameters in low-pressure plasmas with non-Boltzmann electron distributions can be determined.

CHAPTER

2

Method for Determining the Emission Coefficients and Line Profiles 2.1.

Spectroscopic Instrumentation

Spectroscopic devices are used to provide photographic or electric recording of the spectra and to determine the line intensities and line profiles. With the help of prisms, gratings, and interferometers the light can be analyzed. Where a resolution of 0.1 A is sufficient, prism spectrographs can be used*. When higher resolution is requisite, grating spectrographs or interference devices are necessary which require a preresolution of the rays by means of a monochromator, prism spectrograph, or grating spectrograph. For grating spectrographs the interfering overlapping orders must be eliminated by filters or by a prism. The prism devices are distinguished from the grating spectrograph in that a higher light intensity is obtained in the former. If stray light is unwanted as it may bring serious errors into the experimental results especially when working in the ultraviolet spectral region, then for its reduction those spectral devices must be used which have a high linear dispersion or include two light dispersing systems in series (for example double monochromators). Among the last kind of instruments there are to mention double monochromators enclosing a prism and grating in series, giving by this means a relative high dispersion over a relative broad wavelength region. Within the frame of this section, only basic types of spectral devices are dealt with. For more details see for instance [2.52], About special devices used for high-speed spectroscopical work see section 2.5. 2.1.1.

Prism Spectrographs

With a prism spectrograph a spectral range of 0.11 to 40 um can be covered. The individual spectral ranges for different prism materials are listed in Table 2.1. The optical image is effected by means of glass lenses, in the visual range. In the other spectral ranges lenses made of the above materials can be employed. B y combining several of these substances (e.g., quartz and lithium fluoride) colour-correcting lenses can be produced. Mirror lenses are universally suitable for optical images within the total spectral range covered with prism substances. * Regarding the two main forms of spectral devices — spectrographs and monochromators there is no principal difference in their working principle, therefore we restrict our attention in the following, if not otherwise stated, to spectrographs only.

6

Spectroscopic Methods of Plasma Diagnostics T a b l e 2.1. Spectral ranges for different prism materials Material

Chemical formula LiF CaF2 Si0 2 NaCl KBr KJ

Lithium fluoride Calcium fluoride Quartz Rock salt Potassium bromide Potassium iodide Flint glass Sylvite Thallium bromoiodide J)

KCl TI Br J

Wavelength range 0.11 - 6 0.12 - 9 0.185- 3.5 0.20 - 1 7 0.21 - 2 8 0.25 - 3 1 0.34 - 1.0 0.38 - 2 1 1 ) 0.50 - 4 0

[im um um um um um (im um (im

Absorption bands at 3.2 and 7.1 [Am.

The intensity of the spectrograph depends essentially on the relative aperture l/o of the camera lens (o = focal length:effective diameter). Relative apertures of 1:2 to 1:30 are typical. T o photograph low-intensity light waves or shorttime processes high-power spectrographs are preferable, provided the generally small linear dispersion and the lower resolution of the high-power instruments can be allowed for. ^ The linear dispersion ^ is usually given in A/mm and is dependent on the wave mode. The relationship between the wavelength A and the position on the plane of the photographic plate where this wavelength appears is given by the Hartmann interpolation formula. Excluding the vacuum ultraviolet it has the form ([2.53]* p. 438) =

X

Xq

(2-1)

The constants A0, c and x0 in this case apply only to a small spectral range and can be determined for this range from 3 measured lines with known wavelengths. For the vacuum-ultraviolet the Hartmann dispersion formula ([2.53], page 438) becomes A — A0 = f - i — ) ° ' 6 .

(2.2)

The resolution of the spectrograph is given by ^A theore t. = —:—-—-.— > sin p • Ax

(2.3)

where o denotes as explained above the f/number of the camera lens and ft the angle of inclination of the photographic plate to the ray direction. T o obtain the maximum resolving power it is suitable to choose the width of the entrance slit as follows B = °coii. ' A , * For references see page 436.

(2.4)

7

2. Determining t h e Emission Coefficients and Line Profiles

where ocoll is t h e f / n u m b e r of the collimator. W h e n t h e slit width is too small t h e resolution is worsened due to diffraction a t t h e slit, a n d too large slit widths cause pure geometrical blurring in t h e plane of t h e photographic plate. The resolution can be written after t r a n s f o r m i n g (2.3) as follows AX theorct.

1.7 X

AX

T ~

AS

where b denotes the utilizable diameter of t h e cameralens. t h e size of t h e prism, a n d ~ t h e linear dispersion

(2.5) This depends on

is the angular dispersion, which is connected t o

by AX Ax

=

1 f

AX M

.

sin I 1

(2.6)

If a high linear dispersion is desired for a given prism material, i.e., if ^

is

Ax

to be small, t h e largest possible focal length of t h e camera lens is used a n d several prisms are inserted in series. W h e n high resolution is desired, t h e largest possible prisms are employed and likewise several prisms are inserted in series. I n the last case the angular dispersion is increased as well (i.e., AX becomes smaller). Ad

A f u r t h e r i m p r o v e m e n t of resolving power a n d dispersion is a t t a i n e d , if t h e light after passing t h e prism set is directed back b y means of t h e "autocollimation mirror" and passes t h e same prism set once again in t h e opposite direction. This principle is found within autocollimation spectrographs. Before using this t y p e of spectrograph, it has t o be t a k e n in mind t h a t as a result of u n w a n t e d

F i g u r e 2.1. Optical diagram of t h e 4-prism quartz spectrograph Fuess 110 H (R. Fuess, Berlin, prospectus) 1 — slit, 2 — plane mirror, 3 — collimator objective, 4, 5, 6, 7 — prisms, 8, 9 — plane mirror, 10 — camera objective, 11 — plate 2

Progress in Plasmas

8

Spectroscopic Methods of Plasma Diagnostics

reflection by the prism surfaces there will exist a disturbing spectrum superimposed to the main spectrum. Then the former spectrum must be shifted relative to the last by tilting the autocollimation mirror. A n example of a prism spectroscopic device is given in Figure 2.1, which illustrates the optical diagram of a 4-prism quartz spectrograph. Below we list some commercial spectroscopic apparatus. T a b l e 2.2. Data on spectroscopic apparatus

Spectrograph type Manufacturer

3-prism spectrograph VEB Carl Zeiss Jena with glass prisms 4-prism quartz spectrograph 110 H R. Fuess Berlin Quartz prism spectrograph Q 24 V E B Carl Zeiss Jena Mirror monochromator with prisms (P) and grating (G) Type SPM 2 V E B Carl Zeiss Jena Plane-grating spectrograph PGS 2 V E B Carl Zeiss Jena

y)

Spectral range [A]

/camera

O

[cm]

12 2.3 27 5.5 10000 130 (Autom.) 26 3600

2100

A [Ä]

Al Ax [A/mm]

^-theorct.

[A]

5000. 5000 5000

105 47 4.9

0.2 0.2 0.1

95

ca. 16

3000 5000

5 25

0.04 0.3

50

11

3000 5000

13.5 58

0.08 0.5

3000 5000 3000 5000

7.4 7.4 3.7 3.7

0.1 0.2 0.05 0.1

8,000 2000 5,800 P: 2000

40

8.4

400000 G: 2000 -10000 2000 207.5 -

28000 207.5 2)

28.5 28.5

T h e s e d a t a c o r r e s p o n d t o a g r a t i n g w i t h 650 lines/mm a n d t o t h e f i r s t o r d e r of d i f f r a c t i o n .

' ) T h e d a t a in t h e t w o last lines c o r r e s p o n d t o a r a y p a t h w h e r e the r a d i a t i o n is t w i c e d i f f r a c t e d b y t h e g r a t i n g .

2.1.2.

Grating Spectrographs

The most widely used spectrographs of this type are plane-grating spectrographs with a reflecting grating, in which the design is such that the greater part of the radiation is reflected into a certain angle (Blaze-gratings). The favoured wavelength within a given spectral order is called the "Blaze-wavelength". Figure 2.2a illustrates the optical diagram of such a spectrograph with an Ebert mount, manufactured by V E B Carl Zeiss Jena; Figure 2.2b shows the external view of the instrument.

2. Determining the Emission Coefficients and Line Profiles

9

Rowland gratings, which do not require additional optical components, since they are shaped like concave mirrors, are also used, and t h e y are of high value for vacuum ultraviolet work. A shortcoming is the astigmatic image obtained when usual grating mounts are employed. As a measure of the light intensity one can again use the relative aperture l/o (effective diameter: focal length of the camera object lens). However, it should be stressed t h a t the light intensity decreases toward higher orders. As mentioned above, the overlapping of several orders can be prevented by preresolving the radiation by means of filters or monochromators. The linear dispersion for the case in which t h e rays are incident normal to the plane of the plate, is given b y dA da;

d • cos /3

(2.7)

/ n

where d is the distance between adjacent grating rules (grating constant), fi t h e angle between the normal to the grating and the ray diffracted by the grating, / t h e focal length of the camera objective, and n the order of t h e spectrum. If the incident radiation is normal to the grating, the condition for obtaining diffraction maxima is given by t h e inequality ^ < 1 -

(2-8)

Taking n = 1 we get for the maximal wavelength the relation Amax = d

(for normal incidence) .

For oblique incidence Amax can become twice the above value. Accordingly, a grating with 1000 lines/mm in the most extreme case can be used for a maxim u m wavelength of 20,000 A. Radiation of longer wavelength (infrared radiation) can therefore be diffracted only by gratings with larger intergroove spacings. The theoretical resolving power is given b y ¿l^theoret.

=

(2.9)

n N

where N is the total number of illuminated rulings of the grating. According to equation (2.9) it is possible to increase t h e resolution for a given grating by using a higher order n, b u t this procedure is limited, if n o t in consequence of the decreasing light intensity (which can be countered by using especially adapted Blaze gratings [2.15a]) then in consequence of formula (2.8), giving for the maximal resolving power of a grating the relation —

X

^^theoret.

, T , diV = nN
0.3 ¡xm this contribution is < 3 % , while for A < 0.25 [xm this contribution is in excess of 2 0 % . Because this contribution is sensitive against changes in geometry, the carbon crater ceases to be a radiation standard for wavelengths A œ © Ol © 00 n >o io IO © © © © © ci-H 1-1 ^ F—1

3.00

2.87

2.74

2.64

!

2.34

2.29

2.24

IO oi

2.19

OÎ

2.10

Ol ©

2.06

1.98

1.90

1.84

OS

! 2.46 2.54

m "3

(M ©

t00

ors

T3

1

-d

i-H ©

1.82

00 IN œ >o Ttl Ol T|H Tf IO l> 00 Ol Ol © C © io © co IO © ©O © Ol co io © l> 00 œ IO 00 © i—i Ol CO IN Ol Ol oi Ol Ol Ol Ol Ol 00 œ © i—i Ol Til io l> 00 © Ol CO CO eo co co CO CO CO CO CO CO CO Tf< Tt< >o >o >o I—1 1-1 t—1 l—l i—l 1-1 r—1 ^

>o IO >o © © r - o 1—< I—1 i—i

r- 00 00 00 œ l—l

i—l Ol Ol Ol a> œ © © (M Ol Ol Ol Ol Ol Ol

© © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © i-H

CO

IO "O IO io ^ TH co co co ©

t- © © © Ol Ol © 00 00 CO © © r©- © r- œ © © © © © © © co © © t- tt- tc© © © © © © © © © © © © © © © © © © © © ©

00 ©

© © © © © >o io IO -

(2-74)

where xn and xn+1 denote the absorption coefficients at the points rn and rn+1, respectively, and amn the normalized geometrical path length: = \in + I) 2 - m2]1'2 - [w2 - m2]1'2 .

(2.75)

By using expression (2.74) for the optical depth in each interval, we can obtain for the lateral distribution of the spectral intensity I(x), taken for discrete values x = xm for a source, whose radial distribution of emission coefficient is s(r) and whose radial distribution of the absorption coefficient is x{r), the expression Im = I E n where

+oen+1-Amin

N n-1 2 (1 — e-*«») 1 + e-1™" tm nJ JI ieCk — 2 Tfli 77I e~Tm' . , -r Xn + l l=m l = n +1

m, n —

(2.76) (2.77)

A m: n is a triangular matrix, which can be analytically transformed into the inverse matrix Bn m Bnm = Amn • (2.78)

78

Spectroscopic Methods of Plasma Diagnostics

The inversion equation for the determination of the volume intensity distribution s(r) from the spectral intensity distribution I(x) now becomes (2.79) The prerequisite for applying this equation is t h a t the radial distribution of the absorption coefficient x(r) is known. I n some cases, a direct determination of x(r) is possible (see the foregoing section), while in others it is not (for example, when the plasma is enclosed in a cylindrical quartz tube with nonideally transmission properties of t h e envelope leading to a disturbance of the absorption measurements). Then the inversion problem may be solved by an iterative procedure, beginning with an estimated distribution for x(r) (we may also choose x(r) = 0 initially). Thus, we obtain for e(r) a zeroth approximation. F u r t h e r measures are aimed at determining whether we are dealing with the radiation of a continuum or with t h a t in the range of a spectral line, and whether t h e temperature distribution is already known or should be first determined from the volume intensity to be calculated. W e shall consider the case, in which the temperature distribution is to be determined from the volume intensity sv of the continuum. Let the relationship between sv and T be known. We can now carry out the inversion of I(x) to £(r0,(r) by assuming t h a t the absorption coefficient x = 0. The corresponding temperature distribution will be designated by T (0) (r). Furthermore, we can obtain with the help of Kirchhoff's law, which is valid for local thermal equilibrium, a zeroth approximation for t h e absorptoin coefficient x according to the equation

(B p is the Planck function (see (2.19))). Applying the values x(0>(r) a second inversion can be performed, whereby ev(r), T(r), and x(r) are obtained as first approximations. Should the first approximation not suffice, a second approximation can always be made. Let us consider the case where the temperature distribution is to be determined from t h e integral emission coefficient of a spectral line with known transition probability A „ ^ m . A complication arises from the wavelength dependence of the absorption coefficient across the spectral line profile. If the degree of self absorption is not too great, then a mean absorption coefficient can be introduced. Assuming for the spectral line a dispersion profile with a full half-width 2 /9 (in frequency units) we obtain for the absorption coefficient taken with respect to the radiation of the line integrated over frequency x(T)

gmAmn

c°(l - e-*>/*T) n(T) 16jti»2/J Q(T)

-

U„lkT

(2.81)

79

2. Determining the Emission Coefficients and Line Profiles

TJn is t h e excitation energy of t h e lower level, n t h e n u m b e r of particles per cm 3 , which emit t h e relevant spectral line, Q(T) t h e partition function, a n d 2 ft t h e full half-width of t h e spectral line, which also depends on t e m p e r a t u r e . I n t h e first a p p r o x i m a t i o n t h e full half-width can be considered as constant and can be found f r o m t h e spatially integrated spectral line profile. The inversion follows in t h e same m a n n e r as for continuous radiation: first of all we determine £ (r) f r o m I(x) without allowing for absorption. W e t h e n calculate T w>(r), a n d subsequently from (2.81) W i t h t h e values I(x) a n d x (r), a n d % (r). See Olsen [2.102] for a more precise determination of s(r) a n d x(r) for t h e case when t h e radial t e m p e r a t u r e distribution T(r) is k n o w n and t h e transition probability is the wanted q u a n t i t y . 2.10.2.3. Inversion of measured intensity d a t a into local emission coefficients for absorbing plasmas. I t e r a t i v e inversion a f t e r Elder et al. I n this section we follow a n iterative inversion procedure proposed b y Elder et al. [2.35b] to derive t h e radial distribution of local emission coefficient s(r) f r o m t h e measured cross distribution of intensity I(x) and absorptivity A(x). The convergence of this procedure has been proofed b y Birkeland a n d Oss [2.15b], At first, we write down t h e formula for t h e intensity I(x) a t t h e cross coordinate x

I(x) =

fx* - x 2 r

/

J -1/ip-X*

respectively

I(x) = e

-

-

dy s(r) • e

yj? 2 - x'

! 0

xdv

F r o m this we obtain I(x) TW(x)

¡][r 2-x 2 \ / "(c) d»!

\ »

JiR'-x*

r • / o 0

/,

p 2 = X 2 + rt 2

K

1

y % + v y2 r 2 = yx

/ y

y

\

I f "(.e) di - / x(e) d n l e(r) I e» ' + e » Id y .

R f e(r) cosh cosh G(x, r) • r dr f °( x> r) • r dr J (r 2 - x 2 )l/2 '

in which

R r K(r) r dr 2 2 2 'J (r -x )'l

(2.82)

(2.83)

(2 K • 84i I

?

T(x) = e

1

(2.85)

denotes t h e t r a n s m i t t a n c y of the plasma, which is connected with t h e absorpt i v i t y by t h e relation T(x) = 1 -

while t h e q u a n t i t y G is given b y

A{x) ,

80

Spectroscopic Methods of Plasma Diagnostics

If the transmittancy is greater than 80%, then the term cosh G(x, r) in (2.84) can be substituted by 1, and we have the usual Abel integral equation which can be solved as shown in section 2.10.1. For this region of transmittancy, Freeman and Katz [2.41] obtained the same result. For transmittancies below 80%, a solution of (2.84) can be obtained in form of a series s(r) = em(r) + £

= half 1/e-width (see equation

_Anmgmn(T)e-UnlkT(1

_e-*,/tr)t '

x

When applying equations (2.92) and (2.93) it is assumed that the width of the relevant spectral line (/Jx or fl2) is known and that there exists information on the transition probability. For estimates these quantities need only be approximately known. If a line width due to smallness can not be determined with a spectroscopic device, it is still always possible to establish a lower limit (usually the Doppler width). The half 1/e-width in frequency units is

with ¿2 _ 2 kT

where M is the weight of the ion or atom, which emits the relevant spectral line. Then for an estimate of Av max we simply have the following inequality A

^ Anmgmn(T)

e-UnlicT

{1

_ e-h,ikT)

i

2. Determining the Emission Coefficients and Line Profiles

83

When a continuous background of appreciable intensity is superimposed on the spectral line, the portion of the continuous absorptivity is added to the absorptivity of the spectral line. 2.12.2. Experimental Check ior Low Seli-absorption The basis of our considerations lies in the expression for the emission from a homogeneous layer of length I = (1 -

e—'1) J J , ,

(2.95)

which for an optically thin layer (x„l 1) can be replaced by an expression proportional to I Iv~xvlBv. (2.96) Deviations between (2.95) and (2.96) of 5 % or more begin to occur when x,,>0.1. Equations (2.95) and (2.96) are also valid as an approximation for inhomogeneous plasmas, when T is understood as the temperature in the center of the plasma and for I a suitable value is taken (effective length of plasma). (These interpretations are not allowed in the case of resonance lines). These equations are based on the assumption that the radiation can be spectrally resolved. This is not always the case, such as for example, when the radiation of a narrow spectral line is examined. Now, instead of (2.95) and (2.96), we have I = Bv / (1 - e-*w) dv (2.97) (line)

as well as I as x I Bv

(2.98)

for small optical layer thicknesses }«, max I 1. I here denotes the integral line intensity and x the mean absorption coefficient. Based upon equations (2.95) —(2.98), we shall now examine some criteria for low self-absorption and in connection herewith some practicle examples. Criterion 1. The radiation is optically thin when it can be verified that the spectral intensity I v or the total intensity I of a spectral line is proportional to the plasma length I or increases linearly with I. This criterion was used by Knopp et al. [2.66] to verify that the line A I 4158.6 A from an argon plasma jet is laterally emitted from an optically thin layer. For this purpose the line intensity was first measured in the usual manner, after which the line intensity was measured when the length of the source I was doubled. The doubling of I was effected by placing a mirror behind the plasma ray. The intensity with the mirror was 95 % higher than that without. Assuming a loss due to reflection (1 — /) of a few percent, it follows from (2.70) that the absorptivity of the plasma becomes vanishingly small.

84

Spectroscopic Methods of Plasma Diagnostics

When employing wall-stabilized arcs and observing along the axis it is possible to measurably alter the plasma length I. I n this case it can be very readily confirmed whether /„ or I varies linearly with I. Criterion 2. We consider two sources with source lengths lx and l2> respectively, having similar temperature distributions. Let us put l2 lv The exact values of and l2 are not required. Let us examine the spectrally resolved radiation Iv 1 and Iv 2 for the two frequencies v1 and v2, respectively, whose separation lies within the range where the Planck function Bv can be taken to be constant. Say I„i i — 0 Pe

(4.16)

' M 3 / 2 ¿5/2 & + 1 /p5/2 } Qi

e-(Ui-AUi)lkT

(4.17)

and (for the case n a better value A C7 for the reduction in the ionization energy can be calculated from equation (4.18). With this a better ^¡,-value, can be determined from the above system

4. Calculation of the Composition of a Thermal Plasma

101

of equations (4.15) —(4.17) and (4.19). Further iteration is usually no longer required. Accordingly, the particle concentrations are obtained from the expressions

When estimates are made it is often sufficient to replace the partition function by the statistical weight gf 0 of the deepest lying multipet. The error thus involved is usually small for a small degree of ionization. For this purpose we list in Table 4.1 the statistical weights gf 0 of the deepest multiplet for different elements along with their respective ionization energies XJf. The tabulation procedure corresponds to that of Unsold [4.41], The symbol * * next to the statistical weight signifies that the term immediately following the ground term lies within a distance of 1 eV (corresponding to 8068 cm" 1 ). The symbol * means that the term immediately following the ground term lies within a distance of 2 eV (wave number 16,136 cm - 1 ). The data in the table were taken from [4.2a], [4.11], [4.13c], [4.19], [4.21], [4.23b], [4.25], [4.39], and [4.41], Numerical values of the complete partition function of 39 elements and their ions, given as function of temperature T and lowering of ionization energy A Ui, can be found in [4.10b]. Up to this point in our discussion we have not considered the case in which negative ions occur in the plasma. This happens when the plasma contains electronegative gases, such as hydrogen, oxygen, and chlorine. No essential changes have to be introduced into equations (4.1) — (4.4) for the calculation of the ionization equilibrium. Only additional terms with z = — 1 appear and the quasineutrality condition assumes a slightly different form and in the case of a simple plasma reduces to n+ = ne + n_. For details on the calculation of the reduction in the ionization energy in a plasma containing negative ions, see Ecker and Kroll [4.12]. Saha functions S(T) for a hydrogen plasma containing negative ions are given by Drawin and Felenbok [4.10b] p. 227. Boldt [4.3] studied the case where negative ions are formed by attachment of electrons to excited atoms.

4.2.

Calculation oî the Component Composition oî a Chemically reacting Plasma as a Function of Temperature and Pressure

When calculating the equilibrium concentrations we must consider the chemical reactions taking place between the neutral plasma components. Also, we need only limit ourselves to the case of a simply ionized thermal plasma, since in more highly ionized plasma there are practically no molecules present.

102

Spectroscopic Methods of Plasma Diagnostics oae^woo-^r-osio a t OOOCO®NO«q«OOt;QOWNOO«OOOOW OOOXNWlObhOlN^XCOflìOONMaiflìOO H NCliiliMWW^^HHHHCqiNlNNHHHIN 1—1Hr—t

CO to 00 Ös o CD

N

®OOO«OWM«00^

t>

to

COOìaMWWMMMCOHHrtHHHHHHHHHH

t^Ofl5 HC N HM HM HM H

«S

( N H ( M H C D f l ì ^ f l ì O H N H i O O ì T | i O ) C C ) H OHH(M C O(M i Owi O O « IO CO 00 Ci 05 00 CO F - i l i J W M f O O C O I > ^ M t > 00 ifl O qqHHqrtio^HTijqqHq^qt-4qq ©(NCOhhddo^OMhHOCOHfliCOOiflffiNOOrtCO «lCWiOiCt*CO®ONCOfl5flìOOHH(N0505050000 T^lO"—I^HI—Ii—Ii—Ii—1 OHMH(D

ö >

C I o « H i a i t j o 5 H N i o ® n o ^ t o t - o o o H

to" ta H

«

iH

Nffl«OÌ

3

h

From quasineutrality considerations we also have Pe= ¿pi i=1

,

(4.24)

and the expressions for the total pressure P=Pe

+ S

(Pt + Pt)

•

(4-25)

¿=1 Finally, we still have to write the expression for the material composition of the plasma. Let C; j be a number denoting how often the plasma constituent* Lj ( j = 1 , . . . , r — s) occurs in the plasma component Kt or Kf (i = 1, . . ., r). Then the condition for a constant material composition becomes r

r

21 (Pi + p t ) Cn •• 2 j=1 i=1

r (Pi + pi)

Z ( P i + p t ) Ci,r-S

¿=1

= c1:c2

- • • : cr__s ,

(4.26)

* B y plasma constituents are meant those heavy particles which are not further dissociated, independent of whether t h e y are electrically neutral or charged. See [4.13].

108

Spectroscopic Methods of Plasma Diagnostics

where the relative numbers of plasma constituents Cj, . . ., cr_s are known beforehand. Expression (4.26) contains r — s — 1 mutually independent equations. W i t h equations (4.22) —(4.26) we have 2 r + 1 equations at our disposal, with which we can determine the 2 r + 1 unknown partial pressures. Once these are known we can readily obtain the particle densities M; etc., with the help of the expression pt = nik T for an ideal gas. W e need not go into deviations from the ideal gas law. These only begin to play a role at pressure exceeding 100 atm. Let us briefly outline the procedure for setting up the system of equations for the component composition of a reacting plasma, such as, for example, argon-free air, consisting of 21% oxygen and 79% nitrogen. The plasma is composed of 1. 5 neutral plasma components N , 0 , N 2 , NO, 0 2 , 2. 5 simply ionized plasma components N + , 0 + , N f , N 0 + , O j , 3. electrons. The plasma constituents are N and O. W e can specify 3 dissociative reactions: N + N o,

The quasineutrality condition is Pe = i>N, + 2V +

+

+ Pot *

»lkT .

u

(5.3)

for the emission coefficient. Here gm is the statistical weight of the upper level*, nf the number of particles of type A with charge number z, which emit the spectral line under investigation, Qf the corresponding partition function (for computation see section 4.1.1), and U m the excitation energy of the upper level. In the case that the gas temperature is not equal to the electron temperature, T must be replaced by Te in equations (5.2) and (5.3). Below we shall assume a unique plasma temperature. We shall now formulate problems of plasmaparameter determination which can be solved by the measurement of absolute emission coefficients of spectral lines. * In the following we assume that gm is constant. In reality, this is not the case when the excitation energh Um of level m lies in the neighbourhood of the ionization limit f7j. About the consequences thereof for plasmadiagnostics see section 10.8.2. 9

Progress in Plasmas

120 5.1.1.

Spectroscopic Methods of Plasma Diagnostics

Determination of the Plasma Temperature when Pressure and Material Composition are known

Let the pressure p and the material composition of the plasma be known. The temperature T is determined from a measurement of enm. Firstly, the particle concentration nf of the plasma particles, emitting the spectral line vnm, must be calculated as a function of temperature with the help of the formulas given in section 4.1. Then the emission coefficient of the spectral line is given as a function of temperature by substituting the particle density into (5.3). The plasma temperature T is then found from a plot of enm = f(T) by inserting the measured value for enm.

plasma radius (mm) • F i g u r e 5.1. Determination of the radial temperature distribution (lower figure) in a plasma of a 1.1 atm, 400-A high-current argon arc from the measured absolute radial intensity distribution (upper right) and the calculated temperature dependence of the absolute intensity (upper left) for 2 spectral lines Ar I 7635 Â and Ar II 4806 Â, and for continuous radiation A = 5535 Â (see section 9.5.1) (after Olsen [5.303])

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

121

How the radial temperature distribution T(r) is determined from the calculated curves enm = f(T) and the experimentally determined curves enm = f(r) is demonstrated in Figure 5.1 for the case of the radial temperature distribution in a high-current argon arc (current intensity I = 400 A, arc length = 5 mm) at a distance of 2 mm in front of the cathode. The spectral lines A I 7635 and A I I 4806 A were used. The figure stems from a work of Olsen [5.303]. I t follows from the figure on the upper left t h a t the emission coefficient of each spectral line is maximum for a specific temperature (norm temperature). I n the neighborhood of this temperature the temperature cannot be precisely determined with the spectral line in question. However, if we measure the emission coefficient of several spectral lines with different norm temperatures (spectral lines of the same element but different ionization stages, or spectral lines, which are emitted by different elements), the temperature can be measured in a continuous fashion via the norm temperatures. As a further example we shall discuss the temperature measurement after Jurgens [5.187] in a water-stabilized arc (under conditions of atmospheric pressure and a current intensity of 50 A) by means of the absolute emission T a b l e 5.1. Temperature determination from absolute intensities of spectral lines when pressure and material composition are known Plasma object

arc, plasma jet, Ar

arc, Ar arc, CI arc, SF e arc, Ar + 5% H 2 magnetically 1 confined arc, H 2 / waterstabilized \ arc / arc, Kr + H 2 arc, Hg

p [atm]

1

0.01 - 1

0.03 1

0.36-1 0.13-1 1

arc, N 2 arc, Ne arc, 0 2

1

2)

9*

/

J

1 1

T-range [10 3 °KJ

Ari

8

-20

Aril

9

-20

Ar I, Ar II CI I F I 6240 A H«> H/s

1.1 1 1 1

arc, plasma jet, 1

n2

Emitter

Hp, H y H„ — H y % Hg I 5790 He I NI Nil Nell 01

10 - 2 0 7 - 8 9 -14 12 - 1 4 8

-12

12.6 10.8-11.4 5 - 6.3 9

-19

10 21 8

-13 -23 -13

References are given in sh ort form, for e sample, 3d stand for [5.3d]. Demixing processes are allowed for.

Reference 1 ) f 3d, 131a, 131c, 144e, 1 144f, 184, 254, 254 a, | 289b, 303, 399, 415b, ( 455 d, 470 a. l b , 122,184a,254, 254a, 303, 455 d, 455 e, 473. 303 290 c 290 b 2 ) 183 80 a 187 111a 26 b 84 b 58, 184, 399, 400, 415b, 470a. 58 111a 84b, 296a, 400.

122

Spectroscopic Methods of Plasma Diagnostics

coefficients of the lines H a , Hg, and H y . The measurement gave an axial temperature of T — 12,615 ± 280 °K. The smallness of the optical layer is checked by comparing the line profiles for end-on and side-on observations (see section 2.12.2). A survey of temperature measurements by means of measured absolute emission coefficients of spectral lines is given in Table 5.1. Day and Griem [5.72 a] considered the case of temperature determination in a plasma from absolute line emission coefficients when instead of the pressure p the electron density ne is known. The object of investigation was a plasma formed in a T-shaped shock tube containing the elements N and He with known concentration ratio. The electron temperature was derived from the sum of the absolute emission coefficients of the spectral lines N I I 5676/79/86 A .

T

no3 w-

F i g u r e 5.2a. Emission coefficient e of the spectral line Ar I 7635 À (a) and Ar I I 4806 À (b) in an argon plasma as a function of temperature and pressure

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

123

5.1.2. Determination of the Plasma Temperature and the Plasma Pressure from the Absolute Intensity of Two Spectral Lines when the Material Composition is known The emission coefficients of the two spectral lines are denoted by and e2, respectively. Both are functions of temperature and pressure and can be determined according to well-known methods: ei=A(T,p),

e2=f2(T,p).

Figure 5.2b

(5.4)

124

Spectroscopic Methods of Plasma Diagnostics

Let gj and £2 assume the values a and b in a certain volume element of the plasma. Then, for the determination of T and p we obtain the two equations f1{T,p)

= a,

MT,p)

= b,

(5.5)

from which the temperature as well as the pressure can be found. Here it must be assumed, however, t h a t the temperature dependence of the intensity is different for both lines. Let us consider an example. Say the emission coefficient of the A I-line 7635 A in a volume element of an argon plasma is found to be 8.0 W c m - 3 sr _ 1 and that the emission coefficient of the A I I line 4807 A in the same volume element is 1.4 • 10~2 W c m - 3 sr _ 1 . I n Figure 5.2 a and b the emission coefficients of these lines are plotted as a function of p and T. In Figure 5.2a a straight line is drawn at e = 8.0 W c m - 3 sr - 1 , and in Figure 5.2b at e = 1.4- 10~2 W c m - 3 sr _ 1 . The points at which these lines intersect the p = const curves are given below: A I line

A II line

p [atm]

T [°K]

p [atm]

T [°K]

1.74 2.29 3.16 5.62 10.0

13230 12660 12180 11480 10910

1.0 1.74 2.29 3.16 5.62 10.00

12610 12530 12470 12350 12100 11930

piatì F i g u r e 5.3. T vs. p diagram (see text) for the determination of T and p from the measured line intensities e = 8.0 W cm - 3 sr _1 (Ar I line 7635 À) and e = 1.4 • 10~2 W cm - 3 sr - 1 (Ar II line 4806 À) for an argon plasma. The curves intersect at the point T = 12,400 °K and p = 2.7 atm

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

125

From the pairs of values we obtain two p(T)-curves, which are shown in Figure 5.3. The point, at which these two curves intersect, gives the sought values T = 12,400 °K , p = 2.7 atm . In an analogous manner, Thornton and Cambel [5.421a] determined the electron temperature and particle number densities in an argon plasma generated in a shock tube via the emission coefficients of the spectral lines A I 4158 and 4259 A, and A II 4379 and 4579 A. They obtained typical values of T = (13 - 17) • 103 °K, n = (8 - 9) • 10 le cm" 3 . See also Lincke and Griem [5.251 d] who determined the electron temperature in a He —H-plasma with known concentration ratio of the substituted elements from the intensity ratio /(He I 5876 A) : I ( H J . e

5.1.3.

e

Determination of the Material Composition of a Complex Plasma from Absolute Emission Coefficients of Suitable Selected Spectral Lines with known Transition Probabilities

The material composition of a complex plasma can be determined by measuring the absolute intensities of spectral lines, emitted by the different Plasma components. According to equation (5.3) the emission coefficient of a spectral line, emitted by particles of element A and charge z and when disregarding the pressure dependence of the partition function Qf, is only a function of the temperature T and the concentration nf of the particle characterized by A and z. nf in turn (see section 4.1) is a function of the temperature T, the pressure p, and the material composition of the plasma, which, for a «-component plasma, is defined to within a constant factor by the unknown parameters cx, . . ., cs (element concentrations). The concentrations are normalized as follows: C

1 +

c

2 H

+

Cs =

1 .

(5.6)

Thus, we obtain the following expressions ei

= f i ( P , T, c 1; . . , , c s ) ,

e

=

M P >

=

f s i P t

2

T , c

l

t

Cj,

. . . ,

. . .,

c

s

)

,

c

s

)

,

s £

c

n

=

i

.

n = 1

Here, e v . . ., e s denote the emission coefficients of the spectral lines, which are respectively emitted by the elements 1, If these are known from measurement, then (5.7) represents a system of equations with which, for known pressure p, the temprature T as well as the concentrations cv . . ., cs can be determined.

126

Spectroscopic Methods of Plasma Diagnostics

Foster [5.107] applied this method to a plasma at 1 atm pressure, which was created in a vortex-fluid stabilized arc chamber, when a swirling water benzene mixture was injected into the chamber. The chemical composition of the medium consists of C 6 H 6 -f x H 2 0 with unknown mixture ratio x, which was to be determined along with the temperature. To do this the intensities of the lines H^ and O I, A — 3947 A, were calculated as a function of temperature and the mixture ratio x. The isophots were plotted in an a: — T diagram (Figure 5.4). Introducing the two measured emission coefficients into the diagram, we obtain the sought quantities x and T from the points where the two isophots intersect.

F i g u r e 5.4. Isophot diagram in the x — T coordinate system for the determination of the chemical composition and the temperature of a C 6 H 6 + x H 2 0 plasma by means of the absolute intensities of the lines H^ 4860 A and 0 I 3947 A (after Foster [5.107]) The steeper isophots correspond to the H p line, the flatter ones to the O I 3947 A line. The n u m b e r s beside the i s o p h o t s indicate the absolute intensity in the following u n i t s : [10 8 erg s " 1 c m " 2 s r - 1 ] for H p , [10 s erg s - 1 c m - 2 s r - 1 ] for O I 3947 A

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

127

However, if the pressure is not known, the emission coefficient of an additional spectral line, which is emitted by one of the s elements, must be determined. Among these s + 1 spectral lines there have to be at least 2 lines, whose emission coefficients have a quite different dependence on the temperature. I t is favorable when the 5 + 1 spectral lines are not all emitted by particles in the same ionization stage. At this point it should be mentioned that Melchenko [5.267] in an analogous manner determined the temperature T, the effective degree of ionization 1 / ' x = 2 J x i U t 2J rii, and the particle densities of the components of a plasma ¿=0 / ¿=o at 1 atm, consisting of air, Mg, and Zn, whereby the intensities of the Mg I, Mg II, and Zn I lines had to be measured. The plasma constituents were at the most singly ionized. Richter [5.367 a] has determined the temperature and plasma composition in a cascade arc burning in the binary gas mixtures He—N, H—A, N—A, H—N from absolute emission coefficients of the spectral lines A I 4044 A, H I 4861 A (H^), He I 5876 A, N I 4935 A and N I I 4613 A. A temperature in the range of 11,200-13,700 °K was found. Neumann and Serick [5.296b] determined the temperature and the concentration of Cu and Na in a rotating argon arc containing the elements Cu and Na as additives from absolute intensities of the spectral lines (A in A) Cu I 5105, 5218, 5220 and Na I 4983. The temperature range was 4000 — 5600 °K. The pressure was in every case 1 atm. 5.2.

Determination of the Plasma Parameters via Relative Intensities of Spectral Lines

5.2.1. Determination of the Particle Density in a Complex Plasma with known Yalues for the Pressure and Electron Density from Relative Line Intensities (Boldt [5.45]) We shall limit ourselves to the case where the chemical reactions between the plasma constituents can be neglected. Let the plasma be made up of X different constituents K and of J7 different components L. (For example, in an A—C0 2 plasma, which is completely dissociated, we have the constituents A, C, and 0 ; the components, for temperatures not appreciably exceeding 16,000 °K, are: A I, A l l , C I , C I I , 0 1 , O i l , and e). For the calculation of the individual component densities nL. (i = 1, . . ., Y) and the temperature T we have at our disposal Y — 1 — X Saha equations 1 quasineutrality condition (5-8) 1

Y

Dalton's law p = H pi i= 1

giving a total of

Y + 1 — X equations.

128

Spectroscopic Methods of Plasma Diagnostic

The number of unknowns is Y + 1, so t h a t we still require R = X remaining equations to determine the densities nLi and the temperature T. Assuming t h a t an independent determination of the electron density, for example, via the line broadening, is possible, the number of unknowns reduces by 1 to Y, so t h a t now only R = X — 1 equations are missing. Further assuming t h a t the relative concentrations of the plasma constituents are constant and can be found from the composition of the gas, which is injected into the arc, the X — 1 missing equations are given by the system nKl-nK;.

• • • :nKx = c i : c 2 . . .:cx

(5.9)

where the values nEi, i = 1, . . ., X give the concentrations of the constituents. When diffusion processes lead to demixing (see [5.111]), equation (5.9) no longer holds. Then for the determination of the wanted quantities one may start from relative intensity measurements of properly selected spectral lines and may proceed as follows: Next we can add to the Y + 1 — X equations N further equations, which connect the relative emission coefficients Sj of the spectral lines j ( j = 1, ..., N), emitted by N different plasma components, with the particle densities nlt . . ., nK of the N different plasma components. The N equations are e, = K.nJ^T)

,

j = l,...,N.

(5.10)

K j is a constant which depends only on the spectral line, and is essentially the relative transition probability. fj(T) in essence contains the Boltzmann factor as well as the partition function. If we know the measured values for the emission coefficient Sj, we have with (5.10) N further equations at our disposal. However, these equations contain N new unknowns Kj. To find a solution the measurement of the emission coefficients is not carried out on one plasma but on M different plasmas, which are distinguished by different constituent compositions, whereby always the same constituents are used, but with varying concentrations. If the same optical arrangement is used for the determination of the emission coefficients of the different plasmas, then the 7T r values are equal for all plasmas, while the values of nL{ and T are different for all plasmas. We obtain a system of equations with i l f ( r - Z + l + iV) equations in M(Y) + N

unknowns.

The above system can be solved when the number of equations is larger or equal to the number of unknowns, i.e., when M (N + 1 - X) > N .

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

129

This can always be achieved with a suitable choice of M and N, for example, by setting M = N = X = number of constituents. At this stage we shall not go into any more detail on this subject and refer the reader to section 5.3.2 (b) and [5.45]. 5.2.2.

Temperature Determination from the Ratio of the Emission Coefficients of Two Spectral Lines, Emitted by Particles of the Same Element A and Same Charge z (Two-line Method)

According to (5.3) the ratio of the emission coefficients of lines 1 and 2 is given by _fi = _£i. d i . H . e - i U t - u j n m _ £2 ^2

(5.11)

I t can be seen that this ratio no longer depends on the particle density nf but only on temperature. Determination of the temperature from (5.11) requires that the excitation energies of the lines, their statistical weights, and relative transition probabilities be known. Further the lines must be selected in such a way that the excitation energies of both lines are quite distinct. I t is also desirable that the two spectral lines are not so far displaced from one another, then we can reckon under certain conditions with a constant spectral sensitivity of the radiation detector used during measurement. In the latter case it is not necessary to use a standard source, whose relative radiation density usually has to be known with sufficient accuracy for both lines to be measured. If a plasma does not emit suitable spectral lines which can be used to find the temperature by the above method, this can be effected by admixture of a suitable substance, so that the desired spectral lines are now emitted. The added element is then called a "thermometric element". Hill [5.171] used this method to determine the temperature in an argon plasma jet from the intensity ratio of the copper lines X = 5153 and 5700 A, shown in Figure 5.5 as a function of temperature. The intensities were measured from the side. Temperatures between 5000 and 6000 °K have been obtained. Hill did not analyze the radial distribution of the radiation. He did not measure the ratio of the emission coefficients of these lines but rather that of the intensities, i.e., of the emission coefficients integrated along the line of sight. The largest contribution to the intensity is made by the region having the highest emission coefficient. This lies either in the center of the plasma or outside the center when the norm temperature is exceeded. If the plasma temperature remains below the norm temperature (for the copper lines this lies between 8400 and 9700 °K), then the temperature determined from the intensity ratio is characteristic for the region in the center of the plasma. If the norm temperature is reached or exceeded then formally a value is given for the temperature in the line of sight which hardly differs from the norm temperature. Thus, temperature determination by this method no longer has any sense. Therefore, either the radiation must be spatially resolved or suitable

130

Spectroscopic Methods of Plasma Diagnostic

spectral lines (e.g., ionic lines) must be used to determine the temperature. We shall further discuss temperature measurements of an air plasma jet with the help of the lines C u i 5106 and 5153 À. (Grechikhin and Shimanovich [5.139]). The measured temperatures (between 6800 and 8300 °K) lay below the norm temperature.

F i g u r e 5.5. Intensity ratio of the copper lines 5153 and 5700 A as a function of temperature (after Hill [5.171])

When the partial pressure of the thermometer element becomes too high, some lines of this element will be affected by self absorption. Then these lines are not suitable for the purpose of temperature determination. This case is partially realized in an iron arc. As in all types of free burning metal arcs with eroding electrodes, the partial pressure of the metal and consequently the degree of self absorption grows with growing current intensity, so that especially for high current intensities the thermometer lines must be carefully selected. Hirschfeld and Weinschenk [5.173] determined the temperature of an iron arc in the current range I = 30 — 250 A on the basis of the intensity ratios of 12 Fe I lines in the spectral range from 2577 to 2648 A with term series a5 F — xb G and a5 D — y3 D and of 16 Fe I I lines in the spectral range from 2563 to 2631 A with term series a 4 D — z* P and a6 D - z6 D. From these lines different line pairs were taken from which the temperature was found (6000 °K at 150 A) in accordance with equation (5.11). The relative transition probabilities were determined by the above authors from intensity measurements at a 30 A iron arc the temperature of which has been measured by Krysmanski [5.227]. The relative transition probabilities determined by Sobolev [5.404] for the Fe I and Fe I I lines in the UV-range could not be used, since these led to discrepancies in the calculations. Accurate data on the degree of self-absorption of the lines employed are unfortunately not given in the work.

131

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

5.2.3.

Temperature Determination from the Relative Emission Coefficients of Several Spectral Lines of the Same Element and Same Charge z

W i t h this method a more accurate temperature determination is possible t h a n by the method described above in section 5.2.2. From equation (5.11) we obtain after taking logarithms lg ( E J - M - 1 » " 1 ) = const - ^

Um

(Um in eV, T in °K) .

(5.12)

If data exist on t h e statistical weights g, transition probabilities A (these only have to be known in relative measure), and term values Um, we can calculate the temperature from the measured relative values for the emission coefficients e. For this purpose lg (e g'1 A_1 is plotted as a function of the energy V of the upper level. W h e n self-absorption is negligible the measured values lie along a straight line with slope 5040/T, from which the desired T-value can be obtained. Morozova and Startsev [5.288] employed this method in determining the temperature from the relative intensities of Fe I lines in an arc burning between two electrodes in air for current intensities between 1 and 6 A. The 16 spectral lines in the range from 3175 to 3355 A and t h e 17 lines in t h e range from 4120 t o 4268 A, which were used, are listed in Table 5.2. The t e m p e r a t u r e (about 4500 °K) was taken from the graphs of lg / ( ^ o ) 3 = - ~

Um + const = / ( U m ) ,

which are plotted in Figure 5.6 (see equation (5.44) for t h e connection between / and A). Note, t h a t I is the spatially unresolved integral line intensity. Figure 5.7 shows t h a t in the center of the arc the given temperatures are too low by 500 °K when t h e radiation is not spatially resolved. Toward the arc edges these differences become smaller. (According to Krysmanski [5.227] the maximum temperature for a 100-A iron arc is only 200 °K higher t h a n t h a t formally determined from the relative line intensities without spatial resolution.) As mentioned above, in this method it should be ensured t h a t the selected lines are not affected by self-absorption. This was t h e case for the lines employed b y Morozova and Startsev. On the other hand, however, p a r t of the lines used by Hefferlin [5.155] for finding the temperature of a n iron arc of t h e same type, displayed self-absorption. Hefferlin showed t h a t in this case it is also possible to determine the temperature b y applying t h e theory of self-absorption of spectral lines [5.155], [5.157]. We shall not discuss this any further, since the method is too formal and because in most cases it is possible to select spectral lines free of self-absorption or by slight addition of another element (e.g., copper in the case of an iron arc) other suitable lines can be obtained which do not exhibit self-absorption. Accordingly, Krysmanski [5.227] found t h a t even for current intensities above 100 A, where most of the Fe lines display self-absorption, there are still a few iron lines with known B»-pD a5D — z3D° alg — w'-F" z7D° — e!F zW°-fD b3H - x 3 / 0 a5D —z3D° as1 J) -z3F° cfiD - z 3 D ° a5p _ yspa a 5 p _x3p0 b3H -x3P lap _ yspo b3H — u3H° b3H —u3H°

155 156 8 547 155 156 620 8 7 8 90 95 620 379 617 617

123 94 117 21 208 227 119 65 113 60 55 29 81 57 50 44

0.00 0.00 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.04 0.04 0.06 0.06 0.07

3.63 3.52 1.06 3.43 3.94 3.95 4.47 0.80 1.04 0.83 3.00 2.69 4.35 3.62 4.03 3.99

6.28 6.27 3.87 6.89 6.32 6.30 7.04 3.87 3.86 3.91 5.96 5.99 7.05 6.53 6.96 6.97

I

31gf 0

ig (gf)

U m [eV]

27 29 50 71 87 31 28 83 136 89 52 20 128 60 40 104 29

0.00 0.00 0.00 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.05

2.97 2.83 3.70 1.80 3.89 0.77 0.71 3.27 3.10 0.27 3.70 2.86 3.05 3.75 2.95 3.92 3.02

5.97 5.81 6.38 4.45 6.35 3.87 3.86 5.77 5.40 2.93 6.34 5.98 5.36 6.32 5.75 6.26 5.99

Lines in visible spectrum No.

A [A]

Transition

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

4120.21 4121.81 4137.00 4147.67 4153.91 4174.92 4177.60 4184.89 4210.35 4216.19 4217.55 4220.35 4222.22 4225.46 4245.26 4247,43 4267.83

b3G - z 1 ^ 0 b3P -3?F° a i p -¿/ID» a3 F —z3G° zsF° -pF abF —z3D° a5 F —z3F° b3p ~i/po z>D°-e>D abD -z7Pa z5F°-ebG c3P

zW-e'D

-x3po

b3P -z38° z$F° — e5£? C3P

-X3P°

Number of multipet 423 356 726 . 42 695 19 18 355 152 3 693 482 152 693 352 693 482

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

133

U(eVJ F i g u r e 5.6. Temperature determination from the relative intensities I of 16 F e l lines in the range from 3175 to 3355 A (curve 1) and 17 F e l lines in the range from 4120 to 4267 A (curve 2) (after Morozova and Startsev [5.288])

mm mm F i g u r e 5.7. Radial temperature distribution of an arc between iron electrodes, determined from relative Fe I line intensities (Morozova and Startsev [5.288]) T(r) — precise temperature, T(x) — formally obtained temperature without spatial resolution of the radiation. Above the relative intensities I(x) and c(r) of the lines used for the temperature determination are given for excitation energies of 3.9 and 6.4 eV, respectively

4227, 4247, 4375, and 4427 A), which are suitable for temperature determination by the above described method. Table 5.3 lists a number of references in which the temperature determination from relative intensities is described.

134

Spectroscopic Methods of Plasma Diagnostics

T a b l e 5.3. Determination of temperature from relative intensities of spectral lines emitted from atoms or ions of the same ionization stage when relative transition probabilities are known. Abbreviations in column4: HF. = high frequency discharge, H.P. = high pressure plasma, M.Arc = magnetically confined arc, MD. = medium pressure discharge, PL = plasma jet, Pu. = pulse discharge, Sh. = shock wave Thermometer line A in [Ä]

Excitation energy [eV]

T e -range [103 °K]

Agi Al I I I 4480-5722

5 . 3 - 6.7 1 7 . 7 - 23.5

4.6 17 -- 3 9

Al VII 2 4 0 - 3 5 6 Ari 4198-5606

35 - 59 1 4 . 4 - 15.3

200 4.1--5.0

Ar I I Ar I I As I Aul

19 - 23 19 - 23 6 . 3 - 8.4 4 . 6 - 7.7

11.6-- 1 3 20 - 8 4 6.5-- 9.5 5.0-- 8.4

3 . 1 - 4.2 1 . 6 - 4.5 2 . 5 - 5.2

3 4.3-- 5.0 7 20 - 4 0

Bal Bal Ba I I CIII Cal Cd I Col Cri Csl

4000-5010 3545-5009 4792-6278 4726-6110 4900-4934 4647, 5696

Plasma object

Arc, Cu-Air, 1 atm Pu., metal electrodes, p > 1 atm vacuum spark afterglowing PL, Ar + He, 2 0 - 1 4 0 TonPL, Ar, p < 1 atm M. Arc, Ar Arc, Ar + metal Arc, C + Ar; N 2 1 atm Ar + BaCl Arc, Cu + Air, 1 atm HF., Air Pu., He + Air, p < 1 atm Arc, Cu + Air, l a t m Arc, Cu + Air, 1 atm Arc, Cu + Air, 1 atm Arc, Cu + Air, 1 atm MD., including Cs

6.1 8.0 5.7 6.2 3.8

4.2-4.8-5.4-4.3 1.0 -

4.9 5.1 5.5

3611-6723

1.96.32.93.01.8-

Cul

5105-5782

3 . 8 - 6.6

3.5 -

8.3 high pressure plasma including Cu

Cul Cu I I Gai Fe 1

2303-3279 2276-2473

3.88.23.12.4-

Fe I I HI Hei Hg I Hg 1 Ini

2562-2631 4101-6562 3188-3889

2576-5455

4339-5769

2.5

7.0 13.6 4.3 7.0

6.6 - 1 2 . 6 6.6 - 1 2 . 6 5.8 2.4 - 1 1 . 1

PL, Ar PL, Ar Arc, Cu + Air, 1 atm HP.

4 . 8 - 5.8 12.1--13.2 23.0--23.7 7 . 7 - 8.5 8 . 8 - 9.5 3 . 0 - 4.1

5 - 8 6.0 - 1 3 . 3 20 6.2 5.8 4.5

Arc Arc, H 2 0 Sh., He Arc, Cu + Air, 1 atm MD. Arc, Cu — Air, 1 atm

Reference 1 )

67 a 74a, 74b. 246 d 302 c 225 b, 255 f. 421b 225 a 40 f 17a 67 a 214d 18f 67 a 67 a 67 a 67 a 59 a, 70 c, 139b, 287b. 18a, 23, 84a, 86b, 139, 168b, 171,214d,277a, 262 a, 348 a, 349 g, 454. 418a 418a 67 a 67 a, 76, 88a, 112b, 115c, 131 d, 155, 157, 173, 214d, 227, 227 a, 288, 295 a, 336. 173, 436a. 84a 2 ), 187. 114b 67 a 295 b 67 a

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

135

T a b l e 5.3. (Continuation) Excitation energy [eV]

Tc-range [10 3 °K]

7774-8446 4590-9676

1 . 6 - 3.4 5 . 1 - 7.2 4 . 4 - 8.9 2 . 2 - 4.5 21.1-26.2 21 - 3 0 3 . 5 - 4.6 10.7-14.1 25.6-28.4

5.4 3 - 5.8 10.0 - 1 3 . 4 3.9 19 - 5 0 20 - 4 0 4.7 19 - 3 4 15 - 2 1 . 7

3739-4661 4254+4642

25.6-29.6 25.6-34.2

30 30

Thermometerline I in [A] K I Mg I Mg I I Mn I Nil Nil Nil 0 1 Oil Oil Oil OIV Sol SnI Sri Til Til VI Zn I

3829-5183 2791-2803 3995-5005

554-

790

2850-3264 4301-5192 2771-3346

15

-22

2.02.51.92.43.22.23.8-

4.1 5.4 5.0 5.0 5.3 5.2 8.6

-44

1 0 0 - 300 5.1 4 4.4 2.6 6.4 5.3 4.8 -

9.4 5.3 6.1 6.6

Plasma object

Arc, Cu — Air, 1 atm HP. Arc, Air, 1 atm Arc, Cu — Air, 1 atm HP., p < 1 atm MD., He + Air Arc, Cu — Air, 1 atm Pu., Air, 1 atm HP., 0 + He + H, 1 atm Air, < 1 atm He + C0 2 , ne « 1017 cm" 3 Stellarator, H, ne = 1013 c m - 3 Arc, Cu — Air, 1 atm Arc Arc, Cu + Air, 1 atm HP., p > 1 atm Arc, Cu + Air, 1 atm Arc, Cu + Air, 1 atm HP., 1 atm

Reference 1 )

67 a 67a, 418d 421 67 a 26a, 311a 18f 67 a 311a 84b 26a, 298a 369 a 172b 67 a 4b, 131d 67 a 67a, 121m, 336 67 a 67 a 49a S. 108 3 ), 67a, 80b, 131b, 133b, 349g

References a r e given in s h o r t symbols. F o r e x a m p l e , 67a stands for [ 5 . 6 7 a ] . 2

) I n s t e a d of the integral line intensity, the center intensity of the s p e c t r a l line is used as indicator for the t e m -

3

) Given a r e calculated diagrams

perature.

5.2.4.

— f(T).

Temperature Determination irom Relative Intensities of Spectral Lines by the Comparison Method

To find the temperature by this method the relative intensities of at least two spectral lines, emitted by regions with two different temperatures, must be known as well as data on the excitation energies of these lines. On the other hand, however, we do not have to know the transition probabilities of the spectral lines. 0 Let us consider the ratio -p of the emission coefficients enm and f}m of a spectral em line with excitation energy Um for the temperatures T0 and Tx, respectively. This is given in accordance with equation (5.3), assuming weak ionization ¡ O f « const, nf ~ i - j , by

10

Progress in P l a s m a s

136

Spectroscopic Methods of Plasma Diagnostics

or, taking logarithms

To find T This is done urements on The plotted

equation (5.13) can be solved either numerically or graphically. by logarithmically plotting the e^/e^-values, obtained from measdifferent spectral lines, as a function of the excitation energy Um. points must lie along a straight line, whose slope is given by

5040 ( — — — j . The value of T^jT^ is obtained from the intercept of the or\T j T0J dinateaxis (Um = 0). Now T0 and T1 can be found separately (see Figure 5.8).

F i g u r e 5.8. Determination of two temperatures T0 and T1 from relative emission coefficients cJJj and for several spectral lines with distinct excitation energy Um (schematic)

Using the above method Elenbaas [5.90] determined the radial temperature distribution in a wall-stabilized high-pressure mercury arc, where p = 0.88 atm, power = 35 W/cm, d{ = 4.1 cm, mercury filling 12 mg/cm, 0 . 1 % Cd admixture. I n the measurement the lines Hg I 4916 and 5770/91 À as well as the lines Cd I 6438, 5086, 4800, and 4678 À were used. The line X = 5461 A could not be employed, since it exhibited self-absorption. Temperatures between 5900 ° K (r = 0) and 4600 ° K (r = 1 cm) were obtained. 5.2.5.

Determination of Temperature and Electron Density îrom the Relative Intensities of Spectral Lines, emitted by Differently Charged Ions of the Same Element

5.2.5.1. Determination of temperature The application of this method presupposes that the relative transition probabilities or oscillator strengths of the relevant spectral lines are known. Further, we must know the plasma composition (in the case of complex plasmas) and the pressure or electron density.

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

137

Then, from the expression „(2)

_ nz9n

(z)

f(z) Jmn

a

-U (ml(kT)

n

(the index z refers to the charge of the ion; G is a factor which comprises physical constants), or taking logarithms gfn

z

= const -

504(

*; T

Um

we obtain a linear relationship between lg

( V m in eV, T in ° K ) E 0 )? gfnz

(5.14)

and U m (in (5.14) some of the

indices have been left out for the sake of simplicity). In this scheme the experimentally determined points must lie along a straight line, from whose slope the temperature can be found. If the particle number densities n z entering into formula (5.14) are not known beforehand, then, knowing the the electron density ne, we can use the Saha equation in the form (4.1) (nz = f(ne, T)) to calculate the particle densities nz. Since nz depends on T the temperature determination can be effected iteratively. First an approximate value for the temperature T ( 0 ) is put into the Saha equation. With the particle densities n

where nf is the number of particles of element A with ionic charge z (z = 0 refers to neutral atoms). Writing (see Olsen [5.304])

*

£ m

we get

nfhVnm e 71

z

~UmlkT =

f { T )

-

(5 19)

(5.20) m

If the emission coefficient s n m of the spectral line is directly measured, according to (5.20) the transition probability appears as the quotient of a measured value and a value determined b y the temperature. In plasmas with rotational symmetry the transition probability can be very accurately found when the norm temperature T * of the spectral line along the plasma axis is exceeded (Olsen [5.304]). The point where T = T* can be well localized. The normalized emission coefficient enm(T*) is then fixed with greater accuracy, and Anm can be determined with greater precision, as well: a

_ enm{T*)

/snii

)• We shall now consider the case where the standard temperature of the spectral line is not attained. The transition probability Anm can again be found from equation (5.3) respectively (5.20), when the temperature T at the point, for which the emission coefficient of the spectral line studied will be determined, is known. The temperature can, for example, be determined by means of the absolute intensity of a second spectral line (see section 5.1.1), when the absolute transition probability of the latter is known. If this second spectral line is emitted by particles of the same element and at the same ionization stage, the accuracy can be raised even more by employing a graphical method suitably to be applied on axialsymmetric plasmas (Olsen [5.304]). Let us write equation (5.3) in the simplified form =

nm

— r* tr*\ 1 ^nmK-

I

A

l i Z

. e ~ W .

(5.22)

150

Spectroscopic Methods of Plasma Diagnostics

The indices 1 and 2 refer to the spectral lines with known transition probability A1 and unknown transition probability A2, respectively, n is the number density of particles which emit the spectral lines 1 and 2, Q is the partition function, t/j and TJ2 are the respective excitation energies of the spectral lines. The constants K1 and K2 are given by hcgi, 2 (5.23) Kl, 2 = JI/l, 2

From (5.3) follows the expression !g £2 = ^

lg e i

lg

K2 A2 n/Q (Kt A1 n/Q)V,H\

(5.24)

If one plots lg e2 as a function of lg ex after the radial emission coefficients e^r) and e2(r) are determined, we can draw through the measured points a slightly bent curve, having a slope d i g e2 dig £l

kT2 kT2

An

AT An AT

+ nU2

+

(5.25) nU1

When the temperatures belonging to the measured values lie below the norm temperature, the curvature is very low ( i t vanishes completely when U2 = f/j), and a straight line with slope U2IU1 can be drawn through the points. Figure 5.15 gives an example of a lg e2, vs lg £j plot for A I I lines. 100

mVArl'lO' 1

UM Ari U933firI'W1 À WS Ari ÀiëOSArïï

I ìc 10 Ci ts

±10%

U76>tkrl U657Arïï

I

4806Ârl emission coeffizient Watt Hem3sr)P i g u r e 5.15. Emission coefficient of different Ar I I lines plotted logarithmically as a function of the emission coefficient of the Ar I I line 4806 Â, also represented on a logarithmic scale. The slopes of the straight lines correspond to the ratios of the excitation energies (after Olsen [5.304])

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

151

A2 can now be found from the intersection of the measured curve with the ordinate. For this we have the expression „ _ K2 A2 n/Q gj = 1 W cm" 1 sr~ (5.26) A1 ?i/Q)u*lui ' njQ depends on T and is known, since T is given by the absolute emission coefficient of line 1, whose transition probability is known. Since njQ is only slightly dependent on temperature for T T*, the accuracy with which A2 is determined from (5.26) is nearly as high as that with which A1 is known, neglecting errors inherent in the intensity measurements. The accuracy becomes higher the smaller the difference between the excitation energies U2 and U1. In the limiting case U2 = Ut, the evaluation of A2 no longer depends on the temperature. Suitable plasma sources for the determination of transition probabilities are arc systems, in which the working substance is injected in the form of a gas or liquid (see Figures 5.16 — 5.19), and shock tubes. With these systems we can determine the transition probabilities of spectral lines of those elements, which at room temperature or somewhat higher temperature exist in the form

F i g u r e 5.16. Gas-flow stabilized arc (after Neumann [5.296]) K — tlioriated tungsten cathode,

— copper disc anode, Z — ignition rod for It /•'-ignition, F — quartz window,

Jt — arc. Gas inlet via supply pipes (IE and screen holes .S'i; gas outlet via screen holes S2 11

Progress in Plasmas

152

Spectroscopic Methods of Plasma Diagnostics

T3 O

s

.1

f l l "

CO

^

V —j

b V Ms

1

I5

X! I O I

V m —

ci t 60' O

,

1« V k

kl

H

ISSp c3 V

cS •n

ce c3

OD

-

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

153

F i g u r e 5.18. Vortex stabilized arc with tangential gas supply. Water-cooled guide plates mounted in front of the electrodes serve to center the arc (after Pfender and Bez [5.349]) fralitule profile section^

J^f

channel.

Jurbine'

mterskin carbon electrode

fangenfia! wafer Inflow

F i g u r e 5.19. Vortex stabilized arc with tangential fluid supply (after Maecker [5.260]) il»

154

Spectroscopic Methods of Plasma Diagnostics

of a gas or liquid, or can be converted into substances of this type by chemical treatment. To find the absolute transition probabilities in complex plasmas, it is recommended, when possible, to employ vortex stabilized arcs as light sources [5.106], Here the plasma is rendered more turbulent t h a n in wall-stabilized arcs, and gas separation effects, which in relatively calm plasmas alter the stoichiometric composition of the plasma in a way which is difficult to control (see Frie [5.110]), are here kept to a minimum. I n this connection it may be mentioned t h a t even in pulsed arc plasmas gas separation effects due to ordinary diffusion and thermo diffusion can occur, as has been shown by Hermel [5.164b] for a He-H-plasma, who found for these processes a relaxation time of the order 20 ¡j.s. I n shock tubes used as radiation sources, diffusion processes are not so i m p o r t a n t t h a n in arc sources, because the shock w a v e plasma has over some distance a high degree of homogeneity. Shock tubes are therefore used in a growing number of cases as radiation source for the determination of transition probabilities of spectral lines emitted by metal atoms which are usually introduced into the plasma device in form of a volatile (for example metalorganic) compound, together with a buffer gas, in most cases Argon. See, for example, [5.54e], [5.302d], Due to the high degree of plasma homogeneity in the region behind the shock front, at least with respect to a direction perpendicular to the direction of f r o n t propagation, the evaluation of transition probability datas from shock t u b e measurements is relative simple. I t requires side-on measurements of spectral line intensities and the knowledge of the optical p a t h length within t h e radiating volume, the temperature and the number density of radiating atoms. While the latter two quantities can be derived from the initial experimental parameters and the measured shock front velocity by means of the continuity conditions at the shock front (see, for example, Oertel [5.302d], p. 239 — 242), t h e temperat u r e can also be found on a pure experimental basis, for example via the line reversal method (see section 6.1). 5.3.2.

Determination of Transition Probabilities from the Emission of Spectral Lines in an Optically Thin Layer when the Concentration of the Emitting Element is not known beforehand

The transition probability of the spectral line to be studied is in principle f o u n d , as above, by means of equations (5.19) and (5.20). However, t h e initially u n k n o w n quantities J 1 (temperature) and n z (concentration of particles, emitting the relevant spectral line) must be determined beforehand by other means. W e shall consider four typical cases, for which the determination of the transition probability is effected in this fashion. Case a. For a mixture of two substances, T and nz can be found from the absolute intensity of two spectral lines when the pressure is known. The details

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

155

of this method were already given in section 5.1.3. I n this connection we mention a work of Foster [5.107], who employed a turbulent-fluid stabilized arc with water and benzene as working substances t o obtain d a t a on the t e m p e r a t u r e , plasma composition, and t h e not yet known C I transition probabilities, f r o m t h e line intensities. The mixture ratio x of t h e plasma with composition C6H6 x H20 and the plasma t e m p e r a t u r e were determined f r o m the absolute intensity of t h e spectral lines H^ and 0 1 3947 A. Case b. The method of Boldt [5.45] for the determination of transition probabilities in complex plasmas with u n k n o w n mixture ratio is based in principle on the determination of t h e plasma composition a n d t e m p e r a t u r e f r o m measurements of t h e relative intensity of spectral lines (see section 5.2.1). After this t h e absolute transition probabilities are found with t h e help of (5.19) and (5.20). W i t h this method Boldt determined t h e transition probabilities of 24 lines and line groups of the plasma component CI in an A-C-0 plasma in the spectral range f r o m 1156 t o 1765 A. The plasma is produced b y a cascade arc located between bored carbon electrodes (Figure 5.20). The arc b u r n s essentially in argon, which is supplied via inlets 1 —5. Through the gas inlet Z a second gas (C0 2 or 0 2 ) can be injected. This gas only fills t h e space between t h e nozzle (where argon is injected via inlet 2) and the anode. I n this region 3 different plasmas are produced: 1. a pure argon plasma; 2. an argon plasma with an a d m i x t u r e of 0 2 , having plasma constituents A and O; 3. an argon plasma with a n a d m i x t u r e of C0 2 . The plasma constituents in this case are A, 0 and C. F u r t h e r m o r e , a slight a d m i x t u r e of hydrogen is used, which, however, is of n o significance for t h e composition of the plasma proper. For each of t h e three plasmas the following was determined b y side-on observation via window J : 1. t h e relative emission coefficient p A I of an A L line; 2. the relative emission coefficient £ 0 I of an 0 1 line; 3. the relative emission coefficient e C I of a C I line. F u r t h e r , for each of t h e three plasmas the line width AX of t h e H^ line is determined b y end-on observation t h r o u g h a hole in t h e anode. Accordingly, we can find t h e electron density f r o m t h e expression (see Griem, Kolb, and Shen [5.140]) n, =-- /¿(AX)3'2 ,

/? = 3.40 • 1014 cm" 3 A " 3 / 2 .

(5.27)

All 4 quantities are measured with respect to t h e plasma axis (r = 0). We can now calculate t h e plasma composition for each of t h e three plasma f r o m t h e obtained d a t a . F o r this we h a v e a t our disposal t h e following equations, when we include t h e electron density ne in the system of equations: 1. For t h e Ar-plasma: 9ld) - „, A " m = f M \ T {1) ) («aha equation) , ,l Al + »All (5.28) + n f n + nW = -JL^ , p = 1 atm , nV

= n^u

,

AX»

= ft{n™)

,

= K,

• f3(T)

.

156

Spectroscopic Methods of Plasma Diagnostics

F i g u r e 5.20. Cascade arc for t h e production of a plasma f r o m composite gases and a vacuum lock for connecting the arc chamber to a vacuum spectrograph (after Boldt[5.45]) 1 — 5 — argon in, 'A — inlets for second gas, Kr — krypton in, J — for side-on observation of the arc

5. D e t e r m i n a t i o n of P l a s m a P a r a m e t e r s , P l a s m a in T h e r m a l E q u i l i b r i u m

157

We thus have 5 equations in the 5 unknowns n[[\, w*1', T m , and K l t of which the first 4 unknowns refer to the A-plasina only, while K1 (relative transition probability of the A I line) depends only on the measuring arrangement and has identical significance for all 3 plasmas. The system can be solved for all five unknowns. 2. Equations for finding the composition of the A-0 2 -plasma: "A II = Un?, »AI + »A II no II (2)

( 2)

» o i • «0 II

= /«(n vn

,„ e

"

.

(5.33)

Assuming that only the element A is ionized (ne = uf), the Saha equation becomes (we shall now leave out the symbol A for the emitting element) »2

2Q, (2 TT M)3/2

=

is e _ Vi!kr

_

^

W„

Combining (5.33) and (5.34), we can eliminate n0 and obtain an expression, linking the electron density, temperature, and intensity of the spectral line in question: n2

=

4 M r )

Q1F(T)

ffm ^rii/i

eU„llkT

;

F ( T }

=

2

hvnm

(

2

e

-

f

.

W

( 5

.

3 5 )

h3

A further connection between T and ne is provided by the current transport equation, which is given by Eicke in the form ( j = current density) j =

l '- T ' i «' 2 ; -" A ' 3

2 V'2 m

(5.36)

kT

where a = 0.75. Together with the expression for the total current it

I = 2 n J j r dr o

we finally have i/g ji I!Anm

=

c ?

E

r

^

I

(5.37)

R

( u

T(r)-]/2

( / f t J-(5T)

e,lm(r)

\

exp ^ - j r dr

(5.38)

o

where C =

= 7.56 • 10" 9 [Amp. (Volt)" 1 • cm • (°K) 1 / 2 ] .

e2 a

V 2 me

(5.39)

h

If we have no further information, we assume

with A0 (¡as = gas kinetic mean free path for p = 1 atm and T ----- 273 °K. When the temperature is determined, for example, by the two-line method (see section 5.2.2), the absolute transition probability can now be calculated from (5.38). To obtain reliable values for the transition probability, the mean free path of the electrons Xe must be known with adequate precision.

160

Spectroscopic Methods of Plasma Diagnostics

In the cases investigated by Eberhagen and Eicke the arc burned essentially in free air, with a slight admixture of the readily ionizable element to be studied, so that only collisions between electrons and air molecules were decisive. 6 Eberhagen gave for this case a value of Gas = 8.0 • 10~ cm, independent of the element added. When the transition probabilities of ionic lines of the clement studied are desired (this only applies to singly charged ions), we start with the equation for the emission coefficient enm(r) of the ionic line: enm(r)

=

t1- 7?

4 71

A , n

& vnrn

e - ™

(5.41)

and the current transport equations (5.36) and (5.37). Combining, we get R Anm

= i

10 t h e absorption profile is practically only determined by the dispersion profile [5.321]. T h e n we have for the damping constant y n«fm n l Y = ¡^ï

l

A

•

(5-74)

F r o m (5.70) and (5.74) we finally obtain 71 A m

n

A \Mmn/

W e thus see t h a t the damping constant now only involves the total absorption AX and t h e inter-hook distance A m .„. B o t h quantities can be measured one after t h e other or simultaneously. I n t h e latter case y can be measured with an a c c u r a c y of 3 % [5.325]. T o determine AMN and A À simultaneously, t h e interferometric arrangement (Figure 5.24) is supplemented b y a monochromator with a double exit slit. One slit is centered on the line, the other on t h e undisturbed continuum in t h e vicinity of t h e line. T h e radiation passing through one slit is alternately screened b y means of a rotating sector and is recorded via a photo-electric radiation detector. W e have AA =

7

JLpi M ,

(5.76)

where 7 0 and I respectively denote the photocurrents when the undisturbed continuous radiation and t h a t attenuated b y absorption fall upon the radiation detector. T h e wavelength interval AX limited b y t h e slit must be so large t h a t it contains the full absorption line. T h e oscillator strengths can be determined as follows from the damping c o n s t a n t y. T h e damping constant is found in accordance with y = -

+ -

+ y. t

(5-77)

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium

171

from the natural lifetimes of the upper (m) and lower (n) levels as well as from the collision damping constant y s t . If the transition is studied in the ground state or in a metastable state, we have in the case of no collisions y = — = £An,m Tm n'

(5.78)

since l / r n = 0 and y s t = 0. Here, the summation is over all «/-values, for which there exist transitions connected with the radiation from the m-state. If there exists only one transition from level m to level n' = n, then we directly have y = Anm, w h e n c e f m n from equation (5.44). When more transitions exist, then in addition to y the relative transition probabilities of the lines studied must be known or measured. This can be done, for example, by means of the simple hook method with the help of equations (5.70) and (5.44). When applying the above method the following conditions must be fulfilled: 1. the quantity nnfmnl must be sufficiently large, so that the condition AJAXu > 10 is satisfied. 2. the total concentration n 0 of the metal atoms and the pressure of the residual gas must be sufficiently small, so that the term yst in (5.77) can be neglected. In [5.325] it was shown t h a t for an absorption tube length from I = 50 cm to 60 cm conditions 1 and 2 can be satisfied for the case where resonance lines of alkaline earth and alkali atoms are studied. This has been verified by determinations of the /-numbers of the resonance lines of Mg I , Ca I , Sr I , B a I, and Na I by the above described method. 5.3.8.

Oscillator Strengths from the Broadening of the Resonance Level

For a sufficiently high atom density the width of the resonance level is appreciable and can exceed the natural linewidth as well as the Doppler width. The resonance broadening depends in accordance with ^«Res =

TO CO0

(5-79)

(see Fursov, Oganov, and Striganov [5.115]) on the oscillator strength / of the resonance line and the concentration n0 of the atoms in the ground state of that atomic species, which emits the corresponding resonance line, i.e., foreign atoms are not counted. The lines corresponding to transitions to a resonance level, in general have the same width (when expressed in frequency units) as the corresponding resonance line, when the pressure is not too low and the degree of ionization is not to high, since the higher level width of the former lines can be neglected compared too the width of the resonance level. Fursov, Oganov, and Striganov [5.115] applied this method for the determination of the /-number of the argon resonance line (] * H f dr = extremum , / xp * y> dr = 1 (5.96) it is most suitable to carry out the computation via the dipole velocity [5.425]. Accurate eigenfunctions can be obtained for hydrogen and comparable ions as well as for helium (see Hylleraas [5.179]) and helium-like ions, including the negative ion H " (see Chandrasekhar [5.63]). For systems with more than two electrons one-electron approximations are used to compute the wave functions. One of the most important methods for calculating the eigenfunctions is the self-consistent field method (see Hartree [5.149]). The method lies herein, that each electron (i) is considered independently in the field of the nucleus and the remaining electrons, whereby the latter field is averaged over the spherical shells in such a way that it becomes spherically symmetric. In order to compute the wave function of the ith electron, approximate functions are used for the other electrons. Carrying out the computation for all electrons, we obtain first approximations for all wavefunctions. These yield a first approximation for the self-consistent field, with which the wave functions can be calculated to a second approximation. The method has in the meantime been improved. Fock [5.104] took into account the exchange effects between electrons having the same spin. Trefftz, Schliiter, Dettmar, and Jorgens [5.427] introduced further correlations between the electrons into the calculation. Biermann and Trefftz [5.38] allowed for the polarization of the other electrons due to the excited electron. In [5.422] the configuration interaction is included.

178

Spectroscopic Methods of Plasma Diagnostics

The error in the transition probabilities for lighter atoms calculated by the self-consistent field method seems, as comparison with experiment has shown, to be about + 2 0 % . As an example we give in Table 5.9 a comparison between measured and computed transition probabilities of Ar I I lines (see Table 5 of Olsen [5.28]). The line intensity can be calculated in a relatively simple way when the field acting on the emission electron is approximated as a Coulomb field (Bates and Damgaard [5.21]). Reliable results on line strengths can be obtained by this method only for transitions within the same multiplet system. Further the upper and lower level must possess the same parent configuration. Assuming Russel-Saunders coupling, the line strength S can be split into three factors (5.97) S = S(M) • S(L) - a 2 . The first factor S(M) gives the relative strength of the multiplet, the second factor S(L) the relative strength of the line within the multiplet. S(M) depends on 1. the number k of equivalent eelctrons, 2. the multiplicity, 3. the angular momentum quantum numbers of the individual electrons, and 4. the orbital angular momentum quantum numbers L of the initial and final T a b l e 5.9. Transition probabilities of some A l l lines. An „, (,Xp.: experimentally obtained averaged transition probability (Olsen [5.304]; An m thcoret.: transition probability computed by Garstang [5.118]) A [A]

A n m cxp. [10 7 s" 1 ]

4348.11 4379.74 4579.39 4589.93 4609.60 4637.25 4657.94 4764.89 4806.90 4847.90 4879.90 4933.24 5009.35 5062.07

11.5 11.6 7.44 6.40 9.06 0.64 6.95 5.40 7.86 0.95 6.59 1.60 1.70 2.20

12.8

11.2

0.10*

10.0

10.7

0.81 0.11* 7.09 7.88 0.81 7.71 2.41 1.39 2.81

X o t e . T h e v a l u e s designated by * were i n d i c a t e d b y G a r s t a n g as especially sensitive t o t h e c h o i c e of t h e p a r a m e t e r .

level. For systems of up to 5 equivalent electrons we list S(M) in the following Table 5.10 (after Allen [5.8]). Further data can be found in [5.127] and [5.128] (Goldberg).

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium b l e 5.10.

179

Relative multiplet strengths for systems with 1 to 5 equivalent electrons (after Allen [5.8])

1 electron V

d

2P

*D SO 1

2S 6 I

2D

ip 7sLLI

3D ip 90 ' D >P I JO I

3F 3D 375 IF Ml

A

3f 3q 3p 3p 3D 726 22 -5 1-5 ~3p\ 67-5 22S 30 IF W 42 7-5 *îp\22-5 7S I

\dd P d \ 3Q 3F 3D 3P 162 42 6 1 3[)\84 52S spUisjos

2G

lF ~m\

2F

\ ¡P 3P3S 3p 15 9 3 ip ip 1S 1P 5 3 1\

99Ò]

\ P sp\

2

3P 3P 18 w is 1p 10 21

s a T \ 3F 3p 3p 3P 21 15 9 •'F ID 1P W 7 5 31

3D 3p 135 45 7 F ID ip 1D 84 15 1 1S\20 7P OS 75 TO

\pf pd\

3p 18 1G 1F W 54 14 2 | IE1 W\28 17 S 4S ip \ 10-5 13S

S

3G 3 F 3D 1 3F f05 35 3p\2S0 36 3P 189 7 6 iF W JF 135 11-7 0-3 11-7 ip]93-3 63 vd 2 3F J £ 3 F ~84\3p 3D .168 27 3P\&1 1G ID IF 108 4 ~1D\35 TP zi 12]

3 electrons

ip

tp 4P 20 12 4 zp 2p 2S ZP 10 6 2 zp *p zs IP 10 6 Z\

S ' S l i \ *F "D "P *D 28 20 12 2F 2D 2P 14 10 6 2 F 2D ZP 2D 14 10

2F 2P 2P *D «P sp.pX, 28 20 121 "P 168 30 2 *P\90 30 *S\40 2F 2Q 2P 84 15 1 1P\45 15 2F ZD zp 2P 2P 2S 2P 2P 10 18 2 | 4 IS 1 zp\45 15

•¿G

180

Spectroscopic Methods of Plasma Diagnostics

T a b l e 5.10 (cont'd). Relative multiplet strengths •3

electrons

*s

*p

12 2D W 15 2

P

15

zp 5 9

«Li

18-7 3-3 4p I 18

*6 *F 540 47 1-3 "D \373 47 4 P 1252

4P *s

120 Z^F22D 210 75

VP

751

18 6 2F ZF 18-7 9-3

2 D 9-3 1-7

5. Determination of Plasma Parameters, Plasma in Thermal Equilibrium T a b l e 5.10 (cont'd). Relative multiplet strengths •3 electrons

2

H

2 I 2 H 2G 156 S3 11\2F TG\7$ 69 32\lD

V | pà2

"F 4H *G «F *G 264 81 15\*D «71 135 105 m «P *D\ 48 80~7Ï\2H 26 2f 2 8^D G 132 40 2f\68 S2 2Ö\ip 2 D\24 4036

\ 4F *G 36 4F 252 4P 2*D 216 105 4P 48 «S 64 2H *G 2 9 H 79 22G 317 115

4

s*P2\

électrons

3D 3£ -tt 3P 15 9 12 W 1P 1 û 15 5 I

3

\ 2P 2 D 10 2P 18 2 S 2

3Q 3F3Q F 54 41 30 3q 3p 3 3p 30 18 6

1H

1Q iF

2 2 2 H 52 44 36 H G F 2 2 2 2 6 44 36 28 G F D 2F 36 28 20 22F 22D 2*P 32D Ï2P 2S 22D 56 40 24 2D 2p ? S 2 tp 190 90 20 I P 20 12 4 1 2

sp*

ïP

182

Spectroscopic Methods of Plasma Diagnostics

The relative line strength S(L) within a multiplet, which is tabulated among others, also by Allen [5.8], depends on the orbital angular momentum quantum numbers L of the initial and final state, on the multiplicity, and on the quantum numbers J for the total angular momentum of the initial and final state. We list S(L) in Table 5.11, where we have restricted ourselves to a formula representation (see Sobelman [5.403a], p. 372). T a b l e 5.11. Relative line strength S(L) within a multiplet. The formulas are still valid when the direction of the arrows f o r the transitions is reversed Transition

i

Relative line strength

{L (L + 1) + J (.7 - f 1) - S (S + 1 ) } 2 (2 S + 1) 4 L (L + 1) (2 L + 1) J (J + 1) (2 S + 1) (S + L + J + 1) (L + J - S) (S + L - J) (S + J - L + 1) (2 ./+ 1 L — L - l J —J 4 ./(./ : l i (2 L • 1 ) (2 /. - 1) /. {2 S - 1) (S + L + J + 1) (L + J - 8) (8 + J - L)\ (L + S + 1 - J) L-+ L J - J - 1 4 1.(1. • 1) (2 /, : 1) •/ (2 .S' : 1} (L + S + ,/) (L + S + J + 1) [L + J - S - 1) (L + J - S) L -*L — 1 J ->-J — 1 ' 1 (2 L 1)7.(2 7. • I )./ (2 .S' • Ii (S + J - L - 1) (S + J - L) (S + L - J + 1) (8 + L - J + 2) 1 J ->J 4 (2 L • I ! (7. 1) (2 L + 3) J (2 S + 1) L

L

J -* J

The data on S(L) enable us to calculate the intensity ratios of lines inside a multiplet and in certain cases to compare these with measured intensity ratios. This is, among others, of significance in connection with checking a plasma for low self-absorption (see section 2.12.2). The last factor in (5.97) is defined by

ff2

=

(

4/-D (//'/'

rdr

)2'

(5'98)

where l i and R ' are the radial components of the wave function of the initial and final state, respectively. a can be represented in the form " = frequence at the line center,

r = optical depth = xl . The condition t'i < 0.02 < t'j

(0.6)

must be fulfilled, i.e., for the line edge the plasma must display a weak absorption in the direction of the larger plasma length, i.e., I [ must be given with better accuracy than 1 % by the approximate expression t\ B'„. If we further assume that in the second observation direction the optical depths r'z and T'2 are both smaller than 0.02, we have a) for observation in direction 2 I'

= T s B ' , ,

a

1",

b) in direction I

Since

[

=

r

[

B

v

=

T.:

B ' ;

(B'

v

»

B "

=

B

r

)

,

( 6 . 7 )

1 /',' = (1 -e-r'i)Br

,

.

(6.8)

= y,' llt t'., = y,' l2, r'[ = x" llt and r'.l = x" l2, we get 4 1

T

= -Ti

(6.9)

T2

and it follows from (6.7) and (6.8) that T'J' is given 1

T

r'

• r\:.

/j i2

by (6.io)

If a measurement of the absolute intensity I 1 , in the line center in the direction 1 is made, then BV(T) is given in accordance with (6.8) by BV{T)

=

1 -

h. c ~

T

' i

'

(6.11)

with BV(T) we now also know the plasma temperature T. Bauer considered in more detail the error in determining T in high-pressure xenon lamps due to the temperature drop at the arc edge. This error lies within the permissible limits when the optical depth in the axial direction is smaller than 5. The most accurate measurements are possible for 3 Tj 5. 6.1.6.

The Two-path Method

This method is quite suitable for high-speed recording of plasma temperatures, especially in the case of shock waves (see [6.17], [6.29]), [6.30]). The principle of the method can be explained on the basis of Figure 6.5.

208

Spectroscopic Methods of Plasma Diagnostics

T h e radiation from the continuous source reaches P.M. 1 via t h e path Llt 3I1, P, A, L2, and the interference filter IF. T h e radiation, which reaches the multiplier P . M . 2 along the second light path via the plasma, must pass in addition through a neutral filter NF. A.s a consequence t h e black body temper-

F i g u r e 6.5. Optical layout for high-speed determination of the line-reversal temperature by the two-path method (after Gaydon and Hurle [6.29]) •S — source of c o n t i n u o u s r a d i a t i o n ; NF

— n e u t r a l f i k e r ; P — p l a s m a ; A — a p e r t u r e s t o p ; IF

— interference

f i l t e r ; V. i l l . — s e c o n d a r y e l e c t r o n m u l t i p l i e r

ature T1 of t h e source is reduced to a lower temperature T2. T h e sensitivity of t h e multipliers is adjusted in such a manner t h a t t h e anode currents become equal when t h e neutral filter is removed. I f t h e plasma temperature lies between T1 and T2, t h e anode current of P.M. 1 is reduced b y a value a compared t o t h e current when t h e plasma is not present, w hile t h e anode current of P . M . 2 increases b y a value e. W h e n t h e difference between Tx and T2 is not too large, t h e plasma t e m p e r a t u r e is obtained by interpolation: T = T, + "M^zlA. a + e

.

(6.12)

Gaydon and Hurle [6.30] indicated an accuracy of ¿ 3 0 ° K for t h e case when T , = 2 5 0 0 ° K and T2 = 2 3 5 0 ° K . B y recording t h e photocurrents e and a on a cathode-ray oscilloscope a t i m e resolution of t h e order of 1 fxs can be achieved without a n y difficulty. W h e n i ' j and T 2 differ strongly, or when t h e plasma temperature T is larger t h a n t h e b l a c k body temperature T1 of t h e u n a t t e n u a t e d continuous radiation, we must refer b a c k to the system of equations (6.3) and (6.5) t o determine T more accurately.

ft. Determination of Temperatures and Particle Densities of Plasmas

209

Let us consider the latter case where the plasma temperature T is larger t h a n the black body temperature Tx of the continuous source. The experimental layout (Figure 6.5) is altered in such a way t h a t instead of the neutral filter a nontransparent disk is employed. The sensitivities of the two multipliers are again adjusted so t h a t both multipliers have the same anode current when the opaque disk is removed. During measurement only radiation from the plasma reaches P.M. 2, while multiplier P.M. 1 receives radiation from the plasma as well as the continuous background radiation attenuated by the plasma. Denoting the increase in the anode current of P.M. 1 by ilt which results when the plasma moves through the measuring range or when in the measuring range the plasma is produced at a point where it did not exist beforehand, and denoting the resulting increase in the anode current of P.M. 2 due to t h e same circumstances b y i2, we have the following expression for these two quantities (see also equation (6.5)) i, = K BV(T) f Av D{v) dv + K B j p j J Ap D(v) dv , h = K B„(T) / A, D(v) dv .

(6.13)

Hence it follows B,(T) = BjTJ

lkT

For the case e~'"' law:

.-J*-^-. (6.14) t2 — Av

(6.25)

.

We introduce the expression / xr{r)

dv = a(r)

(6.26)

(integral absorption) and obtain + >/(z)

A{x)=i;

J a{r)dy • -y(x)

(6-27)

The solution of this integral equation is of the Abel type (see section 2.10.2) and is given by

Av=

da:

—— f

7i J y^ri"^ r

(6 28)

We now divide (6.25) by (6.20), allowing for (6.26), and obtain

I (X)

+y(x) j ^ B(T(r)) dy Av «/

= zm

:

(6.29)

6. Determination of Temperatures and Particle Densities of Plasmas

217

w h o s e solution is

r

f r o m which T(r) can be d e t e r m i n e d . 6.2.2.2. S t r o n g self-absorption Simon [6.51] calculated t h e line-reversal t e m p e r a t u r e s of s t r o n g l y a b s o r b i n g p l a s m a s h a v i n g r o t a t i o n a l s y m m e t r y in t h e following w a y . S t a r t i n g w i t h e q u a t i o n s (6.20)—(6.22) for t h e m e a s u r e d intensities Ix(x), / 2 (.r), a n d I3(x), consider t h e case in which for each x t h e black b o d y t e m p e r a t u r e Ts of t h e comparison source is controlled in such a w a y t h a t line-reversal occurs. T h e condition for t h i s is = / 2 . F r o m this follows t h e expression

Bv{T.)

1 _

-

/

f

x,.(r) (1 y

r

- f Mr)df *„(»•) Bv(T{r)) e

d" =

e

dy dv . (6.31)

F o r spectral lines w i t h a dispersion profile t h e following r e l a t i o n s h i p holds a p p r o x i m a t e l y for t h e a b s o r p t i o n coefficients (see also e q u a t i o n (5.68))

* (r) = "

e2n

n(r)îmnV(r) m, c

1 _ (v — v0)2

g2)

H e r e y(r) is t h e d a m p i n g c o n s t a n t or t h e full h a l f - w i d t h of t h e dispersion c u r v e (in u n i t s of circular f r e q u e n c y ) . nn is t h e n u m b e r of occupied levels in t h e s t a t e n. Thus, w i t h (6.31) we o b t a i n Bv(Ts) = Bv(T(r v For 0 < £

h

ay . • ^'M Ay

= x))

'

~y{x)

b (x)

1/2 . •n^t y^t . 4 • y(x) ~ifi\ .,Ax) / BJj) y.v(r) ) of the curves Y(r0, p) as a function of (after Bartels [6.3])

p

Since the peak intensity I v P ( x ) can be measured, we can also find Bv (r = x) via (6.55) and thus the temperature T (r = x). Applying equation (6.55) at different distances x of the line of sight from the symmetry axis, we get the radial temperature distribution T(r), though not entirely up to the plasma edge where the optical depth is no longer sufficient to produce self-reversal.

224

Spectroscopic Methods of Plasma Diagnostics

The theory of self-reversal for the determination of the temperature of a discharge cylinder is applied in a somewhat different form to exploding wires (Bartels [6.5]). When the wire is sufficiently thin a cylindrically symmetric plasma is obtained. Since the spectrum is essentially continuous and the optical depth r practically does not vary with wave length, it is now possible to reproduce the function Y(T) in the sense that the radiation is not only observed at right angles to the cylinder axis but also at different angles cp to this normal direction. Due to these rotations the initially unknown optical depth r 0 is increased by a factor ljcosq>. Under these conditions p and M, which initially are also unknown, remain unchanged. Then the relative changes in the intensity corresponds exclusively to the change in the function Y(r0) due to a change in T0. Now the measured values should be fit into the F(T0, p) — plot (see Figure 6.11)

F i g u r e 6.11. Determination of plasma temperature in exploded wires (after Bartels [6.5])

in such a way that a curve p = const is described with the closest possible approximation. In the example shown p = 0.5; in addition, r 0 {

o

fi! S3 L£J ce ce ^ -p o> ® J co -p a Ut Jt J 3 ce o o 02 m >

>

— ,,

O >o CJ CO cS I—1 00 hjl 0> ¿6 pi c6 " o M f>

Diagnostics

CS a 3 J3 o m

( l

io 1CD 1

®

s

®

» f, > .2 03 C ®

«5

n

®

I

.2

ff

£ ° r-1 §

•2 ®

C

xs

1

,0 C3

oj

© n —I I 1 . H 1 0

&

a

a

c

T3

^

-D

T. e « .2

«5 „ I I in

2

H

2 2 .

00

O

I CM

I

H

a,.:

CO ao,—,

1 ..£

« . „ „ r-H o I I I—I I 1 . "O H ® t -

o 00 CO US s o CO

(7.2)

or written in short ev"

j",v'j'

—

—

nv'

4 71

r

h

v

-Av»

j'

j »

(7.3)

where j< is the number of molecules per cm3 in the quantum state m v' J'. The transition probability A can be replaced by the oscillator strength / or the line strength 8 : _ 8 ?r2 ea v» gn (2 J" + 1) " "

J

"

^

A •"-v" j " , v' j'

rnc*gm(2J' 64 n l "3 _ — ^ t t 3— 5 — j , 3 hc gm{2 J

+

\)

.—rr + 1)

'

J v

'

r

amv'J'A' » » J"

v

"

A

J

"

'

'

{

f 7

' '

e , '

238

Spectroscopic Methods of Plasma Diagnostics

where gm and gn are respectively the statistical weight of the upper and lower electronic level. Expressing the emission coefficient in terms of S, we obtain 16 ev"

J",

v' J'

3 c3

n3

vl

(2 J'

gm

+

1)

•m v' J' 81 v" J"

n,,',, v' J'

A' A"

(7.6)

(compare Nicholls and Stewart [7.24]). The line strength 8 can be split into two factors: qmv'J'A"nv" J"

A"

qj-fJ' A' °J" A"

—

Pv"

(7.7)

v' •

The first of these, also called the Hônl-London factor, can be represented as follows (see [7.11], p. 208): For transitions with A A = 0 e J' A' " J ' - l , A'

(J'

+

A')

=

(J'

-

A')

or

A'

(iî-branch) sj;a'' +

(J' 1,

(2 J' + 1) /L'2 ./'(./' • 1)

_

J'

+

(Q-branch)

1 +

A'

A') (J'

+

(7.8)

1

(J'

- A ' )

1)

(P-branch) For transitions with A A = + 1 (A' — A" = 1): „J- Ak j ' - U A ' - i

_

(J' +

A')

(J' - ¿ y

1

+

A') ,

(R- branch) aJ' A" & J', A'-l

(J'

+

A')

(J'

+

1 -

(2

A')

4J'(J'

+

J'

+

1)

1)

(7.9)

(Q-branch) (J'

qJ' A' °J' +1,

A'-l

A') (J ' + 2 4(J' + 1) (P-branch) +

1 -

A')

For transitions with A A = — 1 (yl' = y l " — 1): («/'

aJ' A• " J ' - l ,

aJ' °J'

A'~l

A')

(J'

-

1 -

A')

1./' (ii-braneh) (J'

A'

-

-

A')

A'+l

(J'

+

4 J'

' 1 + ( J'

A')

(2 J'

+

1)

(J'

+

1 +

A')

(J'

+

2+

4(J' + 1) (P-branch)

1)

(7.10)

(Q-branch) aJ' A' V j ' + l, A' + l

+

A')

7. Determination of Plasma Parameters from the Band Spectrum

239

The factors have a definite meaning in the determination of the rotational temperature, which we shall return to later on. The following sum rule holds for the ^-factors: E S';,.^,, = ( 2 J ' + 1 ) .

(7.11)

Thus, the sum intensity of all rotational lines belonging to a band v" , then, according to (7.28), from their relative intensities the vibrational temperature Tv can be derived. Considering the emission coefficients e v " „' of several bands, which correspond to the same electron transition, and assuming the rotational partition functions Qr(m v') to be independent of the vibrational quantum number v', we obtain the following expression for the relative emission coefficient of the band ~ Ptr'v'' e ~ U v , * l T .

(7.29)

(See section 3.2.2 for the calculation of the vibrational contribution U'v to the internal energy). From (7.13) we again have pf

(7.30)

v*

or, taking logarithms l g 27 ^ ^ = const .„

& t

vl

2.3026 kT

.

(7.31)

v

'

The vibrational temperature can be found with the help of (7.31) by taking the expressions on the left-hand side of (7.31) from the measurements and plotting them as a function of Uv> (see Figure 7.2: the intensity was taken instead of the emission coefficient). The temperature results from the slope of the straight line through the measured points.

7. Determination of Plasma Parameters from the Band Spectrum

247

F i g u r e 7.2. Plot for determining the vibrational temperature in a carbon are in air from the relative band intensities, lg E I/v4- is linearly dependent on v"

U'v, the vibrational contribution to the internal energy of the upper level. The temperature follows from the slope of the straight line (after Ornstein and Brinkman [7.25]) F i g u r e 7.3. //(v*) (I = band intensity) as a fune" tion of the vibrational energy U*, for different band series, whereby Av = v' — v" = const (after Ornstein and Brinkman [7.25])

Specifically (see [7.25]), t e m p e r a t u r e m e a s u r e m e n t s in a c a r b o n arc in air were dealt with on t h e basis of t h e relative b a n d intensities of t h e B1 E — X2 E b a n d system of CN, a n d a t e m p e r a t u r e of 5500 ° K was t h u s o b t a i n e d . Sometimes it can h a p p e n t h a t t h e intensity c a n n o t be measured for all b a n d s ; overlapping, for example, can h a v e a d i s t u r b i n g influence on t h e m e a s u r e m e n t . I t was f o u n d t h a t b a n d s with v' — v" = c o n s t a n t are described b y L ~

e"a r*ltT ,

«=/(v'-v").

(7.32)

E q u a t i o n (7.32) can be regarded as a n i n t e r p o l a t i o n f o r m u l a t o find t h e missing intensities f r o m incomplete b a n d i n t e n s i t y m e a s u r e m e n t s . 17 1'rogress in Plasmas

248

Spectroscopic Methods of Plasma Diagnostics

Figure 7.3 serves to illustrate the functional dependence of lg I j v 4 on from which the relationships become evident. The measured points are plotted as circles and crosses, while the extrapolated values are indicated by squares

7.5.

Temperature Determination from Molecular Spectra for Nonresolved Rotational Structure

7.5.1.

Rotational Temperature for Nonresolved Rotational Structure

I t can happen t h a t the rotational lines within a molecular band can no longer be observed separately due to blurring and overlapping. This is, for example, the case when a prism spectrograph is used, which in general possesses a fairly low dispersion. Furthermore, the possibility exists t h a t the rotational lines are so strongly broadened due to physical causes t h a t they can no longer be observed separately. We shall a t t e m p t to find out whether it is possible to determine the rotational temperature from the spectral energy distribution within the band. For this it is necessary to calculate the spectral intensity distribution for several temperatures. The method becomes more effective the stronger t h e profiles for different temperatures differ from one another. To calculate the blurred intensity we start with the equation for the spectrally nonblurred energy distribution within a band, whereby the rotational lines can be approximated by ¿-functions

s(y) = Z sj.. j . d ( v -

'

Uj

~

U j

"} .

(7.33)

Sj" j ' is given by (7.24) and depends on temperature. To obtain the blurred distribution ep, equ. (7.33) is joined either with the true slit function (if necessary, the line profile function), with a rectangular function or with a triangular function R(v) (half-width b and height 1): = ¡ j ? ej" J' , 1 , 1 . 1 . 10 20 30 40

50

,

/ MOO

i . i . i 60 70 80 v/clcm'V

30

F i g u r e 7.5. Calculated intensity profiles of the 0 - 0 and 1—1 CX-bands at 3883 and 3871 A, respectively, as a function of the temperature T for a constant slit width 6 = 8 c m - 1 (after Smit-Miessen and Spier [7.36])

Figure 7.5 illustrates the blurred intensity distributions of the 0—0 and 1 — 1 CN-bands (3883 and 3871 A, respectively), calculated by Smit-Miessen and Spier [7.36] for a spectrographic slit width b of 8 c m - 1 and for different temperatures between 4000 and 7000 °K. The temperature can now be determined: a) from the ratio of the areas under the profile curves when the bands do not overlap ([7.37] and [7.38]) and from the ratio of portions under the profile curves when the bands partly overlap (see Figure 7 in [7.33]), b) from the ratio of the maximum intensities at the band heads ([7.37] and [7.38]). For this the ratio of the areas or that of the partial areas (case a), or the maximum intensity ratio (case b) must be calculated as a function of temperature. Figure 7.6 shows the relevant plots for the ratio of the areas under the profile curves of the 0 - 0 and 1 - 1 CN-bands (3883 and 3871 A) for different slit widths b.

7. Determination of Plasma Parameters from the Band Spectrum

251

The vibrational temperature is finally obtained from the measured values and the graphs. The situation becomes more simple when the rotational profile depends only slightly on temperature. Then the maximum spectral band intensity measured at the band head is proportional to the intensity integrated over the band.

0.80

•-I

0.75

0.70

0.65

0.60

WOO

5000

6000

rm-

7000

F i g u r e 7.6. Determination of the vibrational temperature from the intensity ratio of the nonresolved 0 - 0 and 1 - 1 CN-bands (3883 and 3871 A) (after Spier and Smit-Miessen [7.38]) C u r v e s / : r a t i o of t h e i n t e g r a t e d i n t e n s i t i e s under t h e profiles of t h e 1 — 1 a n d 0 — 0 b a n d s ; Curves I I : r a t i o of t h e m a x i m u m s p e c t r a l i n t e n s i t i e s of t h e ] —1 a n d 0 — 0 profiles. T h e s p e c t r o g r a p h » : slit width b is given in i-ni-* as t h e ourve p a r a m e t e r

Thus, the possibility arises (see Ortenberg andNesterko [7.29]), to determine the vibrational temperature for known values of the band strengths .

The experimental determination of the transition probability, oscillator strength, and band strengths iV'»'> is usually effected by means of intensity measurements on one or several molecular bands, or also by means of absorption measurements, whereby in both cases the plasma temperature and composition must be known or else determined in some other way. The desired quantities then result from equations which are given in sections 7.1 and 7.2.1, so that we need not repeat them here. I t should only be mentioned, that when, applying methods which presuppose a small optical depth, the plasma must be checked for slight self-absorption (see [7.16]). For details on the determination of oscillator strengths from lifetimes, see Bennett and Dalby [7.2],

7. Determination of Plasma Parameters from the Band Spectrum

255

W h e n absolute quantities are measured, t h e y are usually specified in t h e form of oscillator strength /,/,.»; relative values arc usually given in t e r m s of t h e b a n d strength pv,,< (rel.). The band strength can also be theoretically calculated. I t is given by the expression p,. , = Z W i df| a (7.42) j (see Nicholls and S t e w a r t [7.24]). The m a t r i x elements are s u m m e d over t h e degenerate upper level i and lower level j. Expression (7.42) can be broken u p into two factors, one of which essentially depends only on t h e electron transition and the other on t h e vibrational transition: i W =q,-v z m ^ • (7.43) i,) qrV'

Franck-Condon-factor is given b y [7.24] ?,»,• = [ / r " dr]* ,

(7.44)

where Wv is t h e solution to t h e wave equation +

2

F

{ u

' ~~ U [ r ) } V« =

0

•

(7 45)

-

fj, is the reduced mass of t h e two nuclei of t h e molecule. U(r) can bo described by U(r) = D e { l - e-P0--re)}2 (7.46) (Morse-potential) where De is the energy of dissociation and re is the equilibrium internuclear distance. T h e q v " r ', of course, determine first of all the distribution of t h e radiated energy over the different vibrational transitions v' v" occurring in one electron transition. However, for a given electron transition t h e q u a n t i t y Z |R;,-| 8 is dependent on t h e internuclear distance as well as on the vibrational j state. Nevertheless, this dependence is weaker t h a n t h a t of r> on v" and v' and can be expressed as Z |«fj|2 = , (7.47) ¡j in which the electronic transition moment R" is an analytically expressible function of the "7?-centroid" defined b y (7.48) J %ih'' y>e" diLike q„'r", the q u a n t i t y rv-,,» can be evaluated immediately from t h e potential curve U(r) [7.23c], F o r more details on t h e calculation of molecular transition probabilities a n d equivalent quantities, we refer t h e reader t o Nicholls and Stewart [7.24], Ortenberg and Antropov [7.28a], Spindler [7.38a] a n d Nicholls [7.23q]. On t h e determination of oscillator strengths f r o m lifetimes, see B e n n e t t a n d D a l b y [7.2], —

256

Spectroscopic Methods of Plasma Diagnostics

I n c o n c l u s i o n , w e g i v e a s u r v e y of t h e r e l e v a n t l i t e r a t u r e o n t r a n s i t i o n p r o b a b i l i t i e s , o s c i l l a t o r s t r e n g t h s , a n d b a n d s t r e n g t h s of m o l e c u l a r b a n d s (see T a b l e 7.3) a n d of F r a n c k - C o n d o n f a c t o r s ( T a b l e 7.4).

T a b l e 7.3. Bibliographic references on transition probabilities, oscillator strengths and band strengths ]>,•••,:' of molecular bands. The symbols left hand of the reference numbers characterize the methods of determination of the respective d a t a and have t h e following meanings: A — absorption measurements, Calc. — calculated data, Comm. — comments, E — emission measurements, L — life time measurements, Oth. — other methods, rel — relative values. References are given in short form (for example, 10a instead of [7.10a]) Molecule AIO B„ BH BO BaO BeO C2 CF: CH CN CO: CO + : CaO Cl2 d2 F, H2 H;l Hi HeH+ Li„ LiH MgH MgO N2 NÎ NH NO Na, 02" 0+ OH PN PbO SiN SiO TiO VO

Reference E: 38i. E, rel: 10a, 39a. Calc.. rel: 23 u Calc., rel: 23u Calc., rel: 23u. E: 35a. E: 27 b Calc., rel: 381. E: 6 i . 27b, 38k. A: 38d, 38e. Calc.; 6a, 35d. Calc., rel: l l g , 23u. E: I d . 7b, 9r, 11 x, 201, 23, 38 f. E: 9 r L: lm A: 9h, 18a. A, rel + Comm.: 18c. Calc.: 35d. Calc., rel: 23u. Comm.: 9g. E: 6k, 7a, 23, 33h, 34c. L: 21a, 42a. Oth.: 29f, 33c. A: 34d. E: 20m, 23b, 35, 381, 38m. E, rel: 35. ¿ : 2 1 a . Calc.x: 11 h, 35d. E: 12, 19a, 23f, 35a, L: 11, 12, 19a, 35e. Oth.: l i d . E: 2 7 b Calc., 21 f Calc.: 11 j Calc.: 21 f. E: 20d Correction 20f. Calc.: l l j , 29h. Calc.: l h , l i , l j , 20a. Calc.: 91 Calc.: l e , I f , 6 d . Calc. : 11 e Calc.: 12 c E: 20 d h Corr. 20 f. E: 20d Correction 20f. 21 n. 27 b. A: 5f. Calc.: 11m, 12d, 35d. Calc., rel: l l i , 111. Comm.: (if. E: 5c, 6 j , 6 p . 12f, 18b, 34b, 39e, 42c. E, rel: 5 d . L: 6b, 7g, 23i, 24b. Calc.: 12d, 35d. E: 1, 6 p , 6 q . 12f, 14a, 18b, 34b, 35], 39e, 42d. L: 7g, 23i, 24b. E: 9s. L: l m , 7 g. A: l a , l b , 29c, 44d. A, E: 20h. Calc.: 12d, 12f, 35d. Calc., rel: 23u. Comm.: l i a . E: 42b. E, rel.: 35. Calc.: 6e A: 7c. A, E: 20g. Calc.: 12d. Calc., rel.: 23a. E: 12e, 12f. E, rel. : 35. L: l l w . A: 9i. Calc.: tig. Calc., rel: 23u. Comm.: Un. E: 2, 9b, 23. L: 2a. E, rel: 35 c Calc., rel: 9q. E, rel: 38 c A: 20e E: 27 b E: 27 b

7. Determination of Plasma Parameters from the Band Spectrum

257

T a b l e 7.4. Bibliographic references on calculated Franck-Condon-factors. References are given in short symbols (for example 23 v is used instead of [7.23 v]) Molecule A10 B2 BH BN BO BaO BeO C2 CH CH+ CN CO €0+ CO ^ C0+ cs CaH CaO CrO h2 H„ H„ He,, HgCl I2 LaO Li 2 MgH MgHH MgO N, -NJ n ; .NO N0+ NO - • X 0 + Na 2 02 O.t o i - O.r OD OH PN PbO RbH SD SH SO ScO SiN SiO SrH SrO TiO VO YO

Reference 23h, 35g, 38h, 38i. 23 u 23 u 23 u 23u, 23 v. 27b, 28b, 39b. 23v, 27b. 3 8 k . 381. 9c, 9e, 11 g, 23s, 23u, 28, 3 8 a . 5e 23 v 9a, 9c, 11 q, 23k, 23u, 33e, 38a, 44b. | 9 a , 9o, l i p , l i s , 23g, 23v, 28, 33b, 33e. 33g. 35m, 35n, 39d. i 9d, 11 h. l i s , 23f, 23g, 33e. 21b, 21 d, 23t, : 7f, 33 f. 27b 28b, 35i, 4 4 h . 44a. Comments: 21k. 9p ; 9o, 9p, 21c, 23n, 29g. ! 231 35q 38 p 23j, 38b, 44e, 44f. 21 h, 28b. 21 e 27 b 33 a 2 1 m , 21 n, 23h, 27b, 28b, 33k, 331. : 5e, 9 b , 9d, 9 n , 9o, 11m, l l r , l i s , l i t , 23d, 23g, 23p, 35k, 44g. 1 9c, 9j, 9 k . 9 n , 11 r, 23d, 23e, 23g, 35k. i 23g 1 5 e , 7 h , 7 i , 9 d , 9 n , 11 x, l i s , 14b, 14c.23h,231.23u.23 v . 2 8 , 2 9 a , 2 9 c , 2 9 d , 39d. ; 231 1 231 9o, 21b, 21d, 44e. l c , 9c, 9d, 9 m . 9 n , 9o, 9p, l l f , l l n , l l o , l i s , l l u , 12f, 21b, 23a. 23e, i 23g, 23r, 23u, 23v, 29b, 29c. 29e, 39d, 44c. 9 n , l l r . l i s , 23c, 2 3 v , 38o. I 231 : 7d 2, 7d, 7e. l i s , 2 3 m . 23u, 38n. 35c 23 u ! 33d, 44e. 1 23 v 23 v 35 p 28 b 38 c 20c, 23h. 35 o ! 23 h 9c, 21i, 27b, 33j. 211, 27b, 33i, 33j. 21 j, 27b, 28b.

CHAPTER 8

Determination of Particle Densities in Plasmas by Means of t h e Refractive Index. Laser Diagnostics 8.1.

Relationship between the Optical Refractivity and Particle Number Density

Inferences on t h e particle densities in plasmas can be drawn on the basis of t h e optical refractivity. The following relationship exists (see [8.5]): (11 — 1 )plasma = ( n

l)atoms + ( n

l)ions + ( 1 l ~

Selections •

(&-1)

For a given frequency (n — 1 ) p i asma is a linear function of the particle densities (in the case of a simple plasma with at the most singly-ionized particles, (n — 1) is a linear function of N0, Nj, and Ne, when these respectively denote t h e n u m b e r of neutral particles, ions, and electrons per cm 3 *. The refractive index of the neutral gas components is given by the expression (see [8.3]) ("

l)a„„,

2.

(8.2)

where the s u m m a t i o n is to extend over all resonance lines with cyclic frequency o>k and oscillator strength fk. If t h e resonance lines fall into t h e ultraviolet region (this applies for example in the case of Argon), t h e n it is allowed to use the formula of Cauchy (see [8.3]) (n ~ v

2 l) a t „ms = v(A +1 BIX 1 )

— ' N0(0°C,

N

-°, 760Torr)

V(8.3)

'

where N0 (0 CC, 760 Torr) = 2.69 • 1019 cm" 3 is the particle density under normal conditions. Values for A and B, t a k e n f r o m [8.1b|, [8.3], and [8.26], are listed in Table 8.1 for several gases. As an illustration t h e tables also lists the index of refraction for the relevant gas under normal conditions for a wavelength in the range of the N a D-lines at 589 nm. F u r t h e r d a t a about refractive indices are available for the species Ar [8.9 b], [8.151]: B+, Be, B e + + , C++ [8.15h]; CH 4 , C 2 H 6 , C 0 2 ( [ 8 . 2 0 d ] p. 390): Cu [8.12a]; F - [8.15h]; H [8.8g]; He [8.15h]; K r [8.151]: Li+ [8.15h]; N [8.2 a]: X+++,. Na+, Ne [8.15h]; 0 [8.2a]; Xe [8.151], I n addition, the refractive index in a t h e r m a l Argon plasma can be found in [8.14o] under the c o n d i t i o n ^ = 1.1 a t m , T = 5 0 0 0 - 2 0 , 0 0 0 °K, X = 6328 A, or in [8.14p], for the same pressure, for T = 1 0 , 0 0 0 - 2 2 , 0 0 0 °K and I = 6328 A respectively I = 1.15 (xm ( H e - X e Laser wavelength). * To avoid confusion, we h a v e designated t h e particle densities b y upper-case letters.

8. D e t e r m i n a t i o n of Particle Densities in P l a s m a s b y Means of t h e R e f r a c t i v e I n d e x

259

T a b l e 8.1. Constants A and B for determining the refractive index of gases in accordance with (8.3) Atom or molecule

B

A

He Ne Ar Kr X Hg Air H2 02 N2 CO NO OH

3.486.66 • 27.92 • 41.89 • 68.23 • 87.8 • 28.71 • 13.58 • 26.63 • 29.06 • 32.7 • 28.9 •

N -

[10" 14 cm 2 ] 10"5 10"5 10"5 10~ 5 10"5 10"5 10"5 10"5 10"5 10"5 10"5 10"5

0.08 0.16 1.56 9.92 6.92 19.9 1.63 1.02 1.35 2.24 2.65 2.02

1 (589 nm)

3.50 6.71 28.37 42.73 70.2 93.5 29.18 13.84 27.2 29.7 33.4 29.7 25.0 ±

10- 5 10"5 10"5 10" 5 10"5 10" 5 10" 5 10" 5 10" 5 10" 5 10"5 10"3 3 • H)"5

For the refractive index of t h e ion components we assume, insofar no other information is given, t h e same constants A and B as for the n e u t r a l gas components: (N -

1 )I011S « ( A

\

+

•

A2/

^

2.69 • 1019 cm

3

(8.4)

If a higher precision is needed, (n — l) i o n s m u s t be calculated according to a formula of t y p e (8.2) in which N0 is substituted b y NI. The index of refraction of t h e electron component is given b y (n

l)electrons

=

^ •

(8-5)

TO CO

Formula (8.5) needs to be corrected when the electronic collision f r e q u e n c y comes into the order of the frequency of light, as m a y be t h e case for a f a r infrared test beam [8.9], [8.14n]. Combining we obtain t h e expression /

-

,, A + -B/(A2) 1 pla-m« = , y Q - i m 19 2.69 • 10 cm

,,T J

2 JIE2 N4E ,- . m co2

. ,, , + NI)

„. (8.6)

To gain some idea on t h e order of magnitude of t h e influence of the n e u t r a l a n d electron gas on the index of refraction, we give t h e following formulative relationship in t h e case of argon for the wavelength 546 n m : l)ArI =

+

1.06 • 1 0 ~ 2 3 NAT

lJelectron, =

~

13-33 • 1 0 " « NE .

(N (N -

L

,

F r o m this it is evident t h a t a t this wavelength t h e electrons h a v e a 10-fold greater influence on the refractive index t h a n t h e n e u t r a l particles. F u r t h e r more, the sign is reversed, i.e., the electrons t e n d to decrease t h e index of refraction.

260

Spectroscopic Methods of Plasma Diagnostics

Since from (8.6) it follows that for a given frequency (n — l) p i aslna is a linear function of N0 as well as of Ne( = N{), the quantities N0 and Ne cannot beseparately found on the basis of one measurement of n — 1. If the degree of ionization is small (smaller than 1%), a choice of sufficiently short wavelengths and a monochromatic measurement of n will enable the determination of the neutral gas particle density N0 alone, since in this case the influence of the electrons on the refractive index is practically negligible. Both N0 and Ne can be calculated with the help of (8.6) when the index of refraction n is measured for at least 2 different wavelengths, since the respective influences of the neutral gas and the electrons are dependent on wavelength to different degrees (the refractive index of the atoms varies in contrast to the refractive index of the electrons only slightly with wavelength). In a gas mixture composed of two components with number densities Nt and N 2 , the quantity (n — 1) is a function of both number densities. In consequence of this, it is not possible to determine the quantities and N2 from a measurement of (n — 1) alone. To get N1 and N2, besides of the quantity (n — 1), further quantities respectively sets of quantities with different behavior with respect to a variation of iVj and N2 have to be measured (for example, the mass density, or the temperature together with the pressure). On this basis, Carv [8.9a] has determined the dissociation degree and the dissociation rate in shock heated N2. The neutral components in this experiment are N and N2. Vice versa, starting from the resultant refractivity (n — 1) of a partially dissociated two-atomic molecular gas in the equilibrium zone of a shock wave with known degree of dissociation, it is possible to obtain values for the atomic contribution to the refractivity index A (n — l)/(/1 AT.,tom), as has been shown by Alpher and White [8.2 a] for 0 and N and by Brinkschulte and Muntenbruch [8.8g] for H. There are several techniques for determining the refractive index. Essentially these are: 1. the interferometric method, 2. the Schlieren technique, and 3. the shadowgraph technique. These will be treated below. 8.2.

Interferometric Determination of the Refractive Index n

8.2.1.

Interference Fringes

We shall consider a Mach-Zehnder interferometer system as shown in Figure 8.1 The light source S, which we assume to be monochromatic, is focussed on an aperture or slit A. The lens B renders the light bundle parallel, after which the latter is split by C1 into two mutually interfering bundles. These two bundles are reunited behind the beam splitter C2. A phase shift between the bundles occurs when a measurement object, whose refractive index differs from that of the surrounding medium, is placed at G1. The object G1 is imaged via the lens E on the film F. When the plates C1 and C2 and the mirrors Dl

8. Determination of Particle Densities in Plasmas b y Means of the Refractive Index

261

and Z)2 are all parallel to one another, no interference fringes are observed when the object is not located in the light path or when the measurement object is homogeneous and possesses a constant length along the light path.

F i g u r e 8.1.

Mach-Zehnder interferometric system for producing interference fringes (after Weinberg [8.25], p. 207)

S — l M i t source, .1

a p e r t u r e , C\ a n d C 2 — b e a m s p i t t e r s , D l a n d J) 2 — m i r r o r s , iV, — m e a s u r e m e n t o b j e c t , E — c a m e r a lens for focussing G'j on tlie film F

The more or less uniformly illuminated image created in F is dark or more bright depending on the extent of the phase shift. Figure 8.2a shows an example of this in the case of an interference photograph of a carbon arc, which is not operating. The photograph shows nothing besides the carbon electrodes. When the carbon arc is actuated, the object in the ray path becomes optically inhomogeneous and interference fringes appear on the photograph (Figure 8.2 b). Interference fringes can also be produced when the object is not located in the light path by turning C\ with respect to C2 by an angle and I) l with respect to D2 by an angle a 2 . Then the wavefronts which interfere are at an angle of 2 (,oc, + a 2 ) to one another. The separation q between two neighboring fringes, relative to the plane G1 (which is photographed on F) where the interference takes place, is given for the case when the angles a x and a , are small by the expression q1 =

2 (ax + a2)

.

(8.7)

If a homogeneous object of length L is placed lengthwise along the light path a shift in the interference results which can be expressed in units of the fringe interval as follows A{n - 1)• L s = - -V- „ -, (8.8) where A (n — 1) is the difference between the refractive index of the object and that of the surrounding medium. For more details on interferometric techniques see Ladenburg et al. [8.17] and Kinder [8.15k]. For constructional details see, for example, [8.15f].

262

Spectroscopic Methods of Plasma Diagnostics

As an example we give an interference photograph obtained by Ascoli-Bartoli and R a s e t t i [8.7] for a longitudinally homogeneous object (Figure 8.3). The left portion of the photograph refers to t h e case in which a 32 cm long t u b e filled with argon (0.5 Torr) is placed in t h e light p a t h . The right portion of the photograph displays a shift in t h e interference fringes, which results when a n HF-discharge is triggered in the tube. I n the above case a shift of s = — 0.06 was obtained. The change in t h e refractive index A (n — 1) is f o u n d f r o m s with t h e help of equation (8.8). If A (n — 1) is measured for 2 wavelengths, t h e neutral particle density and the electron density can be determined in accordance with the considerations in section 8.1. To get two interference pictures of the same object and for t h e same time a t two different wavelengths, the interferometric system shown in Figure 8.1 can be completed by addition of a second section E' F' placed below t h e half t r a n s p a r e n t mirror C 2 , and by addition of two suitably chosen monochromatic filters placed between C2 and E respectively C2 and E' [8.2b]. While in the optical spectral region t h e interference p a t t e r n is recorded mainly photographically, in t h e I R region it is obligatory to use photoelectric recording [8.2a], [8.17b] (Only for very high intensive infrared radiation sources, photographically recording can be effected by means of liquid crystal screens (see section 2.3.1.). 8.2.2.

Monochromatic Intcrferometric Technique for Axially Symmetric Inhomogeneous Plasmas

W e shall t r e a t an example of the photographic and computational procedure, and shall base our arguments on a s t u d y performed b y Schmitz [8.5]. Figure 8.4b illustrates a corresponding interferometer photograph of a vertically burning carbon arc, which is located in the interferometer light p a t h at 01 (see Figure 1) and is irradiated at right angles to the axis of t h e arc. For comparison Figure 8.4a gives an interferometer photograph of the arc arrangement when t h e arc is not burning. The computational procedure is as follows: for a given plane normal to t h e axis of the arc (indicated by an arrow in Figure 8.4b) the shift s of t h e fringes is measured as a function of the transverse coordinate x of the arc (x = 0 corresponds t o the center of the arc), s is taken, as shown already above, with respect to the fringe interval. Figure 8.5 gives a corresponding s(x)diagram. I n s t e a d of (8.8) we now have the relationship (8.9) where A (n — 1) is the difference between t h e refractive index of t h e object a n d t h a t of the surrounding m e d i u m : y is the coordinate along t h e line of

Plate 5

F i g u r e 8.3. I n t e r f e r e n c e p h o t o g r a p h for a 32 cm long t u b e , filled w i t h a r g o n ; initial pressure p 0 = 0.5 Torr. left: w i t h o u t , r i g h t : with triggered HF-diseharge. W a v e l e n g t h = 546 n m , (after Ascoli-Bartoli and R a s e t t i ([8.61 a n d [8.7J), L a b o r a t o r i o Gas-Ionizzati, E U R A T O M - C X E X , F r a s c a t i . Rome) F i g u r e 8.2. I n t e r f e r o m e t e r p h o t o g r a p h s for t h e case when t h e plates a n d mirrors are parallel (after Schmitz [8.21]) a) when arc is not burning 1>) Imrnhifi homogeneous carbon are

Plate 6

F i g u r e 8.4. I n t e r f e r o m e t e r p h o t o g r a p h of a vertically b u r n i n g c a r b o n a r c (b). c o m p a r e d t o t h a t for t h e arc w h e n it w a s not b u r n i n g (a) ( a f t e r S c h m i t z [8.21])

a

b F i g u r e 8.8. I n t e r f e r o g r a m of a h o m o g e n e o u s gas column. W a v e l e n g t h varies a l o n g t h e h o r i z o n t a l d i r e c t i o n , ( a f t e r Ascoli-Bartoli [8.3]. L a b o r a t o r i o G a s Ionizzati, E U R A T O M C X E X , F r a s c a t i , Home) a ) w i t h c o m p e n s a t o r ( s t r a i g h t i n t e r f e r e n c e linos) li) "without c o m p e n s a t o r ( s l a n t e d i n t e r f e r e n c e lines)

8. Determination of Particle Densities in Plasma by Means of the Refractive Index

F i g u r e 8.5.

263

Fringe shift as a function of the transverse coordinate x of the arc (after Schmitz [8.21])

R K

distance from arc ax is (cm) F i g u r e 8.6. Radial temperature distribution in a homogeneous carbon arc, derived from the fringe shift (solid curve). The dashed curve refers to the case in which the molecular dissociation is not allowed for in the interpretation of the measurements (after Schmitz [8.21])

IS

P r o g r e s s in P l a s m a s

264

Spectroscopic Methods of Plasma Diagnostics

sight, a n d —y 0 a n d -\-y 0 are t h e coordinates along this same direction w h i c h f o r m t h e o u t e r limits of t h e arc. Owing t o t h e axial s y m m e t r y t h e radial d i s t r i b u t i o n A (n — 1) (r) ( c o m p a r e section 2.10.1) follows f r o m t h e t r a n s v e r s e d i s t r i b u t i o n s(x) in accordance w i t h R

(8.10)

F r o m A (n — 1) (r) we finally o b t a i n t h e q u a n t i t y o(r) — o (295 °K) (assuming t h e t e m p e r a t u r e of t h e s u r r o u n d i n g m e d i u m is 295 °K) (via t h e r e l a t i o n s h i p gl(n — 1) = const. W i t h t h e k n o w n value for q (295 °K) we t h e n a r r i v e a t t h e d e n s i t y q as a f u n c t i o n of t h e r a d i u s r. F r o m t h i s we o b t a i n t h e r a d i a l t e m p e r a t u r e d i s t r i b u t i o n T(r) (Figure 8.6) via t h e relationship o = f(T), p = 1 a t m , w h e r e b y it should be t a k e n i n t o a c c o u n t t h a t t h e arc p l a s m a cont a i n s C 0 2 a n d (CN) 2 , a n d t h a t a d d i t i o n a l dissociation p r o d u c t s are p r e s e n t .

8.2.3.

Application of the Interferometric Technique to Density Determination in Homogeneous Plasmas by Utilizing a Larger Spectral Range

I n c o n t r a s t t o i n t e r f e r o m e t r i c p h o t o g r a p h s of i n h o m o g e n e o u s p l a s m a s (which are s u r r o u n d e d b y t h e free a t m o s p h e r e ) , f r o m which t h e positions of t h e und i s t u r b e d interference lines can be f o u n d , in p h o t o g r a p h s of h o m o g e n e o u s p l a s m a s t h e u n d i s t u r b e d lines c a n n o t be o b t a i n e d on t h e s a m e p h o t o g r a p h . T h u s , in order t o d e t e r m i n e t h e fringe shift a t least t w o p h o t o g r a p h s are required. T h e choice of a concrete m e a s u r i n g m e t h o d d e p e n d s on t h e s t r e n g t h of t h i s shift. 8.2.3.1. Shift of t h e interference fringes b y a f r a c t i o n of t h e fringe i n t e r v a l Ascoli-Bartoli et al. [8.3] considered t h i s case. T h e p l a s m a was p r o d u c e d b y a n H F - d i s c h a r g e in an a p p r o x i m a t e l y 30 cm long t u b e with plane-parallel e n d p l a t e s filled w i t h a r g o n or x e n o n u n d e r a slight pressure of several Torr (see r a y p a t h 1 in F i g u r e 8.7). A n o p e n t u b e filled w i t h air is placed in r a y p a t h 2. This serves t o eliminate streaks which otherwise would occur in t h e vicinity of t h e discharge t h u s d i s t u r b i n g t h e m e a s u r e m e n t s . T h e light source employed was a m a x i m u m - p r e s s u r e x e n o n l a m p X B O 162, which e m i t s intense c o n t i n u o u s r a d i a t i o n . T h e object t o be investigated is i m a g e d on t h e s p e c t r o g r a p h slit in such a w a y t h a t t h e interference fringes are directed along t h e dispersion direction (i.e., horizontally) a n d come t o lie one a b o v e t h e other. To o b t a i n reference fringes on t h e p h o t o g r a p h i c p l a t e in order t o f a c i l i t a t e t h e m e a s u r e m e n t of t h e fringe shift, horizontally s t r e t c h e d 0.01 m m t h i c k t u n g s t e n f i l a m e n t s were placed in f r o n t of t h e s p e c t r o g r a p h slit, (see F i g u r e 8.8) which a p p e a r as s h a r p d a r k lines on t h e i n t e r f e r o g r a m s .

8. Determination of Particle Densities in Plasmas by Means of the Refractive Index

265

In order to determine fringe shifts it is necessary to obtain two interferograms in direct juxtaposition, i.e., one for the case when the discharge is triggered and one when the gas in tube 1 is not excited. For this purpose a screen with 18 consecutive rectangular openings was placed before the photographic plate. X B0162

F i g u r e 8.7. Layout employed by Ascoli-Bartoli et al. [8.3] for determining the particle densities in a tube filled with argon, triggered by an HP-discharge. (Laboratorio Gas Ionizzati, EURATOM-CNEN, Frasoati, Rome) 1 — discharge t u b e with optical windows, 2 — open, air-filled t u b e , 3 — diaphragms, 4 — water-filled compens a t o r , 5 — i n t e r f e r o m e t e r plates, 6 — screen w i t h 18 windows. 7 — p h o t o g r a p h i c p l a t e . The circuit m a r k e d C1)E

serves to trigger the

II^'-discharge

First an interferogram was taken before the discharge was triggered (see Figure 8.8). The fact that certain wavelength ranges are blocked by the screen has no further disturbing effect. Then the photographic plate was horicontally shifted. A second interferogram was then taken after the discharge was triggered. In this fashion we obtain in each case 18 interferogram sections in direct juxtaposition which refer to the state with discharge and that without. Figure 8.3 shows a section of such an interferogram. In this particular case the fringe shift, expressed in units of the fringe interval, amounted to s = 0.0G. The sensitivity of this system is so large, that a fringe shift of about 0.01 can still be observed. 18»

266

Spectroscopic Methods of Plasma Diagnostics

When the fringe shift is measured for two suitably separated wavelength ranges, the neutral gas density N0 as well as the electron density can be determined in accordance with the relationships given in section 8.1. The electron density in this case was found to be 7.6 • 1014 cm - 3 . The sensitivity of the device is sufficient to measure 1014 electrons per cm - 3 or about 1015 argon atoms/cm3 for a plasma length of 30 cm. In conclusion it should be mentioned that due to the differing dependence of th e refractive index on wavelength along the interferometer pathes 1 and 2 (Figure 8.7), the interference fringes are somewhat slanted (Figure 8.8b). This effect can be practically eliminated by placing a water-filled compensator in ray path 1 (4 in Figure 8.7). The refractive index of water and that of air depend on wavelength to nearly the same degree. Figure 8.8a illustrates that the interference fringes are in fact straightened out again due to the compensator. 8.2.3.2. Fringe shift over several fringe intervals Shukhtin [8.22] developed a modified technique for the interferometry of plasmas enclosed in tubes. Here, the undisturbed interference fringe system and that disturbed by the discharge in the tube are directly superposed. We shall consider the case in which the interference pattern shifts over several fringe intervals as well as the case in which the shift amounts to only a fraction of the fringe interval. We shall first treat the former case. The interference pattern obtained when there is no discharge burning in the tube will be designated as the unshifted pattern, while that resulting as a consequence of the discharge will be referred to as the shifted pattern. WThen the unshifted and shifted patterns are photographed one on top of the other, spectro-interferograms, such as shown in Figure 8.9, are obtained. Assuming that the electrons do not contribute to the refractive index n (weak ionization), n and the gas density o are then connected by N - 1= A Q.

(8.11)

The change in the optical path length is LA (N — 1) or LA AQ, where L denotes the path length through the plasma. If AQ is sufficiently large, then for certain wavelengths the change in the optical path length is a whole multiple of the light wavelength A. At these points the interference fringes become more intense. At other wavelengths, for which the change in the optical path length is only an odd multiple of half the wavelength, the shifted fringes fall into the spaces of the unshifted pattern. In this case, either places displaying blurring or places with twice as many fringes can be seen on the spectro-interferogram.

8. Determination of Particle Densities in Plasmas by Means of the Refractive Index

267

The positions where blurring occurs are given by the expressions LAAe

=

2kl

LAAg=

2

+ l?.1 ^ A

2

(8.12)

k2 — whereby the three unknowns kt, k2, and Ao can be calculated. In (8.12) kt and k2 are whole numbers, while i is the number of places where a sharp pattern occurs between two blurred portions. When the refractive index is determined by the electrons, we have the following relationship (see (8.5)) N e2 znmc2 The interpretation of the corresponding interferogram follows as above, except that instead of (8.12) we have to solve the system of equations L N, e : „ 2 hi k, + 1 5 2 7i m c2Al = L N„ e2 2 h, 1 (8.14) K = 2 n m c2 ' 2 22

—*

—

^ >

where kx and k2 are whole numbers and i has the same meaning. When the electrons and the neutral gas simultaneously play an important role in determining t h e refractive index, the measuring technique remains the same. Here again a n unshifted and a shifted interference pattern are superimposed on one another. In this case 3 places are observed displaying blurring of the interference pattern. The positions of these places are given by the equations LN, LAAg + X\ 'lki ' '/,1 2 jttoc 2 1 2 2 1 LNee 2 LAAq + 2 2 n m c2 (8.15) L Ne e2 2*3 + 1 LAAq + 2 71TOC 2

k2 —

k3 — k1 - f - ¿2

where ij denotes the number of sharp regions between the blurred portions at wavelengths A1 and 12 and i2 denotes the number of sharp spots beween the wavelengths ?.3 and klt k2, k3, Ao, and N can be obtained from the above system of equations. The signs of ix and i2 are as yet unknown, i.e., the direction in which the interference pattern shifts cannot be immediately deduced.

268

Spectroscopic Methods of P l a s m a Diagnostics

To determine t h e direction in which the p a t t e r n shifts a compensator which can be filled to different pressures, is inserted into the second r a y p a t h of the interferometer system. First the interference p a t t e r n is photographed when the gas in t h e t u b e and in t h e compensator is not excited. Then the gas density in the compensator is changed by a given a m o u n t , t h e discharge is triggered in t h e tube, and the second interference p a t t e r n is superposed on the first. W h e n the sign of the change in the gas density inside the compensator agrees with t h a t in the test volume, t h e a m o u n t b y which t h e p a t t e r n shifts equals t h e difference between the shifts separately produced by t h e change in t h e compensator gas density and by t h a t in t h e test volume. W h e n the signs are opposite to one another the shift becomes equal to t h e sum of t h e individual shifts. 8.2.3.3. Shift in the interference p a t t e r n by a small fraction of a fringe width [8.22] H e r e t h e method described in section 8.2.3.2 cannot be applied in its original f o r m , since t h e variations in intensity of the superposed and t h e nonsuperposed p a t t e r n cannot be seen in t h e visible spectral range. Blurred fringes are obtained in the following way: A. An interferogram is t a k e n when the gas in t h e compensator is m a i n t a i n e d at a certain pressure and when t h e gas in the discharge t u b e is not excited. This interferogram is t h e n superposed by another, which is t a k e n when t h e gas pressure in t h e compensator is lowered by an a m o u n t Zlo0. The position of t h e k-til order blurring is given b y £o

2* + 1 = (8.16) ¿0 k 2 B. An interferogram is t a k e n f o r a given initial pressure in the compensator when the gas in t h e discharge t u b e is not excited. Then a second photograph is t a k e n on t h e same film when t h e density in the compensator is decreased b y a measurable (via the pressure) a m o u n t Aq0 and the test volume is excited by t h e discharge. The position of t h e blurring is now shifted, and is given b y 2

k

+

1

) "

r,

A

An

T.

N

2

el

k

+

1

2

(8.17)

where k is the order of t h e blurring and ?.k corresponds to the position where t h e blurring occurs, k is obtained f r o m t h e calibration interferogram. W e are now left with two unknowns AqX and NE in (8.17), which refer to the excited gas. To determine these we must write two equations, which correspond to t h e blurrings of order kx and k2: 2

k ,

+

1 ,.

,

.

T

A

A

I

L

N

e

e -

,

(8.18) 2

J l . ± l

2

ü

!

-

x

k

)

*

=

L

A

A o

x

+ 2

J i m

.32

c

This system of equations can be solved for NE and AO-

h

'

•

L'Iato 7

F i g u r e 8.9.

S j i e c t r o - i n t e r f c r o g r a m s obtained w h e n two interference p a t t e r n s are

super-

posed (after .Shukhtin [8.22J) Frolli

tu]) t o 1 Mittniii: a ) s u p e r p o s i t i o n of t w o m i s h i t t e d

portion

of tile p a t t e r n

the shift

amounts

to

d) t h e p a t t e r n

half

patterns;

a fringe

are shifted

width:

h) for w a v e l e n g t h s e o r r e s p o n d i n ^ e) the

hy more than

shift

oeeilrs over several

to

the middle

triune

widths:

1 IH) f r i n « e s

F i g u r e 8 . 1 0 . P h o t o g r a p h s of t h e i n t e r f e r e n c e p a t t e r n of a t h e t a - p i n c h a t t i m e i n t e r v a l s of 4(10 ns w i t h e x p o s u r e t i m e s of 10 ns. t a k e n b y m e a n s of a n i m a g e c o n v e r t e r . The s p a t i a l e l e c t r o n d e n s i t y d i s t r i b u t i o n c a n b e d e t e r m i n e d f o r d i f f e r e n t i n s t a n c e s of t i m e f r o m t h e i n t e r f e r e n c e p a t t e r n s ( a f t e r K i i p p e r |8.1(>])

F i g u r e 8 . 1 2 . T i m e - r e s o l v e d i n t e r f e r o m e t e r p h o t o g r a p h a l o n g t h e a x i s of a n e l e c t r o d e l e s s d e u t e r i u m d i s c h a r g e in a c y l i n d r i c a l c h a m b e r , i n i t i a l p r e s s u r e 0 . 5 T o r r . S l i t a l o n g t h e r a d i u s . T i m e a x i s f r o m left t o r i g h t ( a f t e r M e d f o r d et a l . [8.20|)

8. Determination of Particle Densities in Plasmas b y Means of the Refractive Index

269

By this method fringe shifts as small as 1/500 of a fringe width can be still observed. For a t u b e length of 1 m an electron density of 10 13 cm" 3 or a neutral particle density of 1014 c m - 3 can be measured. This differs somewhat according to the particular gas.

8.2.4.

High-speed Interferometry

This is applied in those cases where neutral gas or electron densities are t o be measured in plasmas which change rapidly with time (spark discharges, shock waves). If one is satisfied with a single photograph of short exposure time, the usual interferometer arrangement can be used when a synchronizable short-time pulse discharge [8.8] or a (^-switched r u b y laser [8.1a], [8.15c] is employed as t h e light source. Several high-speed photographs of this type, t a k e n at short time intervals f r o m a test object rapidly changing with time, can be obtained by means of a n image converter (see Kiipper [8.16]). I n this case t h e interference p a t t e r n is projected u p o n t h e photocathode of t h e image converter. B y means of a suitable deflecting system it is ensured t h a t consecutive images a p p e a r a t different locations on the screen and can be photographed. The illumination t i m e of the pulse l a m p used t o illuminate t h e system m u s t be correspondingly great. Figure 8.10 illustrates 3 interferograms of a theta-pinch t a k e n a t t i m e intervals of 400 ns with a 10 ns exposure time. F r o m these photographs t h e spatial distribution of the electron density can be determined for different instances of time. Monochromatic interferometer photographs with continuous time resolution can be t a k e n by using a combination of an interferometer arrangement and a rotating-mirror camera. Such a system was employed b y Medford et al. [8.20], [8.26c] to determine the electron density in a n electrodeless discharge in a cylindrical chamber (see Figure 8.11). The discharge was irradiated in t h e axial direction. A pulse l a m p served as t h e light source. An example of a time-resolved interferogram is shown in Figure 8.12. The electron density can be derived f r o m t h e deflection of t h e interference fringes; deflection in the direction of t h e arrow indicates a n increase in t h e electron density. On the left-hand portion of t h e p h o t o g r a p h t h e effect of t h e background luminosity of t h e discharge itself, which has a shock-wave structure, can be recognized. W h e n t h e photograph was t a k e n t h e initial pressure of the deuterium gas was 0.5 Torr. The sensitivity of this system is sufficient to record a shift in t h e fringe p a t t e r n by 1/5 the interfringe separation, i.e., for optical p a t h lengths of 1 cm t h e method is suitable for measuring electron densities of t h e order of 1017 c m - 3 . W h e n the shift of the interference lines a m o u n t s to more t h a n one order, t h e n it is practical to detect this shift photoelectrically [8.9f], Using a double

270

Spectroscopic Methods of Plasma Diagnostics

slit detection system, together with the amount of the interference line shift also its direction can be recorded [8.91], As light sources for time-resolved interferometry not only pulse discharge tubes (for constructional details see, for example, [8.12b]) can be used but also continuously working He-Ne-Laser [8.9f], [8.26d]. I n [8.9f], a time resolution of 10 ns was obtained.

interferometer F i g u r e 8.11.

camera

Interferometer layout with rotating-mirror camera for photographing time-resolved interferograms (after Medford et. al. [8.20])

S — source, F — filter, transmission range 4670 + 35 A, C — compensating plates, E — electrodeless discharge J — iris diaphragm, SI — Camera Slit, It — rotating mirror

8.2.5.

Other Types of Two-beam Interferometers

Brown, Bekefi, Llewellyn-Jones and Whitney [8.9], [8.17 b] and also Weinberg ([8.25] p. 213) have described an interferometer, in which a diffraction grating was used as the light dispersion element. For details we refer the reader to the relevant literature. A double slit interferometer illuminated by a He-Ne-Laser was used by German and Schreiber [8.14 c] to evaluate the radial electron density profile of a high current argon arc. I n the Twyman-Greene-type interferometer used among others by Besse and Kelly [8.8d], the test beam crosses the plasma two times. Thus the sensitivity is doubled. I n the common path interferometer used by Dyson et al. [8.11], [8.11a], [8.12 a] as well as in the differential interferometer used among others by Zimmermann [8.26h], t h e light splitting needed to get interferences is effected by means of crystal optical elements. I n the differential interferometer, both the test beam and the reference beam cross the plasma with a slight lateral shift between these beams, thus the shift of interference fringes is proportional to L • dnjdr, where L denotes the effective plasma length and dn/dr the gradient of the refractive index perpendicular to the direction of the test beam.

8. Determination of Particle Densities in Plasmas by Means of the Refractive Index

271

For optical probing of a laser produced plasma, David et al. [8.9f] have used a Mach-Zehnder interferometer, adjusted to give a concentric interference pattern. The electron density was obtained from the phase shift of intensity within the central part of the Airy disk, which is detected photoelectrically. —

—

M

TB

P ,STM

M PB

F i g u r e 8.13. Twyman-Greene-type interferometer used in [8.8d] (schematic)

M

— mirror,

STM —

semitransparent mirror,

diation source, F

8.2.6.

TB —

test b e a m ,

Itli —

reference beam,

P —

plasma,

— ja-

— p h o t o g r a p h i c f i l m or p h o t o e l e c t r i c d e t e c t o r a s s e m b l y

Laser Interferometer

The He-Ne-laser interferometer in the form used by Ashby, Jephcott et al. [8.7 a] is schematically shown in Figure 8.14. photo multiplier

laser

gas discharge plasma

laser mirrors

-200cm-500cm -

external mirror

electrode

F i g u r e 8.14. He-Ne-laser interferometer (after Jephcott et al. [8.7a])

The discharge plasma is optically coupled to the laser resonator by means of an external mirror. Changes of the refractive index of the plasma are indicated by the modulation of the laser beam intensity which is recorded by means of a photomultiplier on the left hand side from the laser. The number of intensity modulation periods corresponds to the number of fringe changes 2 An • L, where L is the effective length of the plasma.

272

Spectroscopic of Methods Plasma Diagnostics

T h e p r o b i n g w a v e l e n g t h , either 0.63 (im or 3.4 |im, can be isolated b y m e a n s of a filter. D u e t o t h e coupling of t h e laser o u t p u t i n t e n s i t y a t b o t h working wavelengths, t h e s h i f t of interference fringes can be recorded either via t h e i n t e n s i t y a l t e r a t i o n of t h e visible r a d i a t i o n or t h e i n f r a r e d r a d i a t i o n , i n d e p e n d e n t f r o m t h e probing wavelength. The i n t e r f e r o m e t e r in t h i s f o r m is useful for t h e m e a s u r e m e n t of p l a s m a densities which produces a shift of more t h a n one interference fringes. W i t h t h e i n f r a r e d t e s t b e a m , t h e m i n i m a l d e t e c t a b l e electron density is (3.10 1 6 /¿)cm~ 2 . W i t h a n indicator w a v e l e n g t h of A = 0.63 fim, t h e t i m e resolution is relative low (1 — 10 ¡as), d u e to t h e inertia of laser m e c h a n i s m . This l i m i t a t i o n can be overcome by a direct recording of t h e t e s t r a d i a t i o n a t 3.4 [i.m which is monitored f r o m a hole in t h e e x t e r n a l m i r r o r [8.14a], [8.15d], T h e n a t i m e resolution of 20 —30 n s is obtainable. T h e sensitivity of t h e laser i n t e r f e r o m e t e r can be raised b y a t least t h r e e methods: 1. S u b s t i t u t i o n of t h e plane e x t e r n a l mirror b y a spherical mirror of s u i t a b l y chosen focal length [8.15dj. 2. As i n d i c a t o r for t h e change in p l a s m a r e f r a c t i v i t y t h e v a r i a t i o n of laser oscillation f r e q u e n c y is used [8.14b], [8.15e]. 3. Using as i n d i c a t o r for t h e change in p l a s m a r e f r a c t i v i t y An t h e c h a n g e of t h e laser o u t p u t respectively t h e change of t h e corresponding p h o t o c u r r e n t AI, t h e n t h e sensitivity AljAn can be raised b y i n s t a l l a t i o n of a n optical f e e d b a c k amplifier in t h e r e s o n a t o r v o l u m e [8.9k]. W i t h a p l a s m a l e n g t h of 50 cm, t h e m i n i m a l d e t e c t a b l e electron d e n s i t y was f o u n d t o be 10 12 c m - 3 . T h e lower limit for t h e particle n u m b e r d e n s i t y t o be d e t e c t e d is d e t e r m i n e d by mechanical vibrations. A laser i n t e r f e r o m e t e r s y s t e m including a v i b r a t i n g e x t e r n a l mirror h a s b e e n described b y F r e u n d [8.13 a]. W i t h t h i s system, densities as low as 10 15 c m - 3 with a t i m e resolution of 0.5 ¡AS can be d e t e c t e d . As probing w a v e l e n g t h of t h e Hc-Ne-laser i n t e r f e r o m e t e r , besides A = 0.63 a n d 3.4(xm also t h e w a v e l e n g t h X = 1.15 [xm can be used [8.14b]. R e l a t i v e long p r o b i n g w a v e l e n g t h s (for e x a m p l e 10.6 [xm) are o b t a i n a b l e b y m e a n s of t h e C0 2 -laser i n t e r f e r o m e t e r [8.17e], [8.20c], D u e t o t h e small divergence angle of laser r a d i a t i o n , a relative high s p a t i a l resolution (0.3 m m ) is o b t a i n a b l e [8.9k]. F o r more detail a b o u t m o d e r n interf e r o m e t e r s see [8.20q]. 8.2.7.

Multiple Beam Interferometer

T h e e x p e r i m e n t a l l a y o u t is such t h a t t h e parallel m o n o c h r o m a t i c light of t h e laser l a m p t r a v e r s e s t h e F a b r y - P e r o t i n t e r f e r o m e t e r (gaseous discharge t u b e w i t h 2 end plates, which serve as b e a m splitters, see section 2.1.3) a n d is

8. Determination of Particle Densities in Plasmas b y Means of the Refractive Index

273

focussed by a converging lens in a plane, on which the interference fringes appear. If the plasma excitation mechanism is alternately switched on a n d off in the tube, shifts in t h e interference p a t t e r n will occur, which constitute a measure for the change An in t h e refractive index in t h e tube. The actual relative shift of t h e fringes ( = absolute shift of t h e interference p a t t e r n : separation between two fringes) amounts to „

A

(8.19)

The absolute effectiveness of t h e method depends on t h e m i n i m u m value of s t h a t can be determined. This is determined b y t h e relative fringe width As, and t h u s depends on 1. t h e line width of the laser radiation, a n d 2. t h e resolution (resonance width) of t h e F a b r y - P e r o t interferometer. Assuming t h a t t h e resolving power of t h e interferometer limits t h e sensitivity, we obtain for t h e relative fringe width (again expressed in units of t h e interfringe separation) As=

i^-', n \r

(8.20)

where r is the reflectivity of the interferometer plates. Dougal [8.10] estimated t h e sensitivity of t h e technique when measuring electron densities, while assuming for the plates the very high reflectivity 0.98, and obtained a sensitivity of Ane = 10 13 c m - 3 for t h e wavelength 6328 A.

8.2.8.

Holographic Interferometry

Three dimensional interferometric charts f r o m a plasma or a hot gas volume can be obtained using t h e holographic single or double exposure t e c h n i q u e (see, for example, [8.14j]). Referring t o the latter, a t one of these exposures t h e plasma is switched on, while at the other exposure t h e plasma is n o t excited. Both exposure are superposed on t h e same photographic p l a t e (by each of these exposures, t h e photographic plate is illuminated a t t h e same t i m e interval a) by direct laser light, b) by laser light passing t h r o u g h t h e test volume). W h e n the developed photographic plate (interference hologram) is illuminated b y coherent light (for example, b y light f r o m a He-Ne-laser), regular interference pictures of t h e test volume can be photographed f r o m this plate. W h e n these photographs are t a k e n for different visiting directions, a n u m b e r of interference pictures (interferograms) are obtained. These interferograms correspond to different test beam directions. F r o m these interferograms, t h e spatial distribution of particle n u m b e r density in t h e plasma can be evaluated. One of t h e advantages of holographic double exposure technique for obtaining interferograms consists in the elimination of disturbances due t o p l a s m a enclosing walls of low optical quality.

Spectroscopic Methods of Plasma Diagnostics

274 CO 1-3 3 SO S © "5 S3 S 3 SO © ti '—1 •5 m 60,0 2 o a

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o . 43 _ rfS , s ® t," h ¿p tT S cs X . l9 e3 m3/2 (kT)1!2 v ' ' '

v

(9.U)

'

If only low electron velocities are considered satisfying the relation v K*T)

(1

_

e -A,/

„


(e»„/(tr)

Vg ,

_

1} >

(9

.18)

V < Vn •

A comparison between (9.15)—(9.18) shows t h a t in t h e high f r e q u e n c y region (defined b y

h vj(kT)

1 f o r v < vg r e s p e c t i v e l y v > vg f o r h vj(kT)

1)

the

free-free radiation is relative u n i m p o r t a n t while in t h e low frequency region (defined by h vj{kT) Vn

''

£ n

'~

128 ji 4 e10 3f3(2^)3/2

wi/2c3

ZlGNeNl n^kT)3/2

_v)RkT)

h(v 6

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( y

'ly)

in which n means t h e principal q u a n t u m n u m b e r . The " G a u n t " factor G which in most cases differs from 1 b y no more t h a n 2 0 % can be t a k e n f r o m t a b u l a t i o n s [9.9] —[9.11], [9.36c], [9.38b], [9.60d], A characteristic feature of the isolated recombination continua is t h e sudden onset of intensity a t frequency v = vn followed b y a n exponentially decrease of intensity in t h e direction of increasing frequency with a steepness inversely proportional to t h e t e m p e r a t u r e T. * I n the original t e x t [9.45], the term (Z + .s)4 stands instead of the term Z* 6.

20*

298

Spectroscopic Methods of Plasma Diagnostics

The continua corresponding to recombination into the discrete lower electronic energy levels n are superimposed one to eachother, thus the resulting continuum becomes saw tooth shaped as indicated in figure 9.5. For completion if must be noticed that the spectral emission coefficient ev H of an individual recombination continuum is connected with the photoionization cross section n by the formulas v, n =

e

K,n ' Bv{T) = Nncrv

n

- BV(T)

in which B„(T) is the Planck function (see formula (2.19a)), while Nn is the number density of atoms in level n. Difficulties arise when the formulas (9.15) and (9.16) are applied to real plasmas. The reasons for this are: 1) It is difficult to define g in the case of recombination radiation. 2) The electric field at place of a bound electron is no longer a pure Coulomb field given by the ionic charge number Z. To overcome these difficulties, the factor g Z2 is substituted by a new factor (Z + s)2 [9.63] which depends (like g) from frequency and temperature. The factor (Z + s)2 can be derived by calibrating experiments as demonstrated by Schirmer [9.63] for the continuum emitted by a high pressure xenon lamp. For shorter wavelengths for which the transitions to lower energy states belonging to the zone of close spaced levels give the main contribution to the continuous intensity, a value of 2.5 was found for (Z + s)2, while in the direction of longer wavelengths (for frequencies v vg) a decrease of the factor (Z -j- s)2 was observed which approaches to the value 1. On the other hand, the factor (Z s)2 can be found theoretically by comparing the quantummechanically derived continuous emission coefficient with the expressions (9.17) and (9.18). In general, the exact calculation of continuous intensity is very laborious (see [9.47 b] chapter 3). A fairly good approximation can be obtained by means of the quantum defect method (QDM) to which reference is made in the next section. 9.2.3.

Free-bound and Total Continuous Emission Coefficient in the Nonhydrogenic Case, evaluated by Means of the Quantum Defect Method (QDM)

For nonhydrogenic atoms (ions), the contribution of free-bound continuum £„ f h to the total continuous emission coefficient ev is found by summation over the discrete recombination continua: evJb

= Zertnl, n, I

U t

~

V n

*l V T h c coefficients aiv,biv and civ are given in [9.13b], [9.13c], n* is t h e effective q u a n t u m n u m b e r corresponding to t h e threshold frequency vg; fii and are t h e q u a n t u m defects corresponding t o the angular q u a n t u m numbers I a n d V. s = h v/Z2 scales in R y d b e r g units ( = 13.6 eV). Xii'(n*> £ ) c a n calculated b y means of a formula given in [9.13b]. Analogous formulas for £{v, T) in the case when fi{s) varies linearely with e are presented in [9.5c], |(j>, T) varies relatively slowly with t e m p e r a t u r e T. Graphical representations of t h e factor £(v) respectively £(v, T) for a n u m b e r of a t o m s respectively ions can be found for example in [9.5 c], Based on experiments with an argon plasma containing atoms, electrons and single charged ions, Giindel a n d N e u m a n n [9.27 b], [9.27 c] have found t h e following simple half analytical expression for t h e continuous emission coefficient useful for plasmadiagnostical applications: £„ = A{v) • N0 kT

(9.28)

9. Determination of Plasma Parameters by Means of Continuous Radiation

301

Na is t h e number density of neutral argon atoms and Ua is interpreted as "continuous excitation e n e r g y " . T h e advantage of formula (9.28) over (9.26) for plasmadiagnostical application in the special case of argon is t h a t in t h e whole visible-near ultraviolet spectral region t h e nonanalytical coefficients A and Ua in (9.28) are only functions of a single argument (j>) while t h e nonanalytical q u a n t i t y £ in (9.26) and (9.27) would become a function of two variables (v and T). T h e values of A(v) and Ua(v) in t h e case of argon are t a b u lated in [9.27 c] for frequencies between 4.4 • 10 1 4 s _ 1 and 1.0 • 10 1 5 s - 1 . 9.2.4.

Effect of Lowering of Ionization Energy on the Spectral Distribution of Continuous Emission Coefficient. Comparison between calculated and measured Spectral Distributions

Only in a high diluted plasma is t h e ionization limit, up t o which discrete electronic levels corresponding to single-electron excitation exist, a well defined q u a n t i t y which can be found in corresponding tables. Due t o t h e Coulomb interaction of charged particles in real plasmas, t h e ionization energy is suppressed b y a q u a n t i t y A Ui which depends from plasma temperature and plasma density as shown in section 4.1.1. and figure 4.1. D u e to this suppression, t h e formulas for t h e continuous emission coefficient are changed in t h e following way: F o r m u l a (9.15) is formally retained, but its range of validity is shifted along t h e frequency scale b y t h e q u a n t i t y AUilh, t h a t means formula (9.15) holds now for frequencies v < vQ — A Uijh. F o r m u l a (9.16) has t o be interchanged with k(vg —v) — A Ui v > vt -

A Uijh:

Ev = 6.36 • 1 0 " « Z 2

e

™

g .

(9.16a)

F o r t h e isolated photoionization continua described originally b y formula (9.19), the positions of t h e photoionization edges at which continuous emission sets in ar e also shifted b y t h e q u a n t i t y Av = A Uijh. T h e n we get 1 9ftn -7Te* PZ1« O 128 3

ft N

N e

i e +

h{yn-y)-AVi .

(9.19a)

(2 tc)3/2 A2 mi/2 c 3 n3 (kT)W

A formula for A Ui has been derived b y Armstrong [9.1 o] under t h e assumption t h a t the apparent shift of ionization edge is primarely caused b y t h e effect of line merging at t h e series limit due t o electron collision broadening of spectrum lines. F o r a simple plasma (N e = Ni) he obtains , „ , ATT I AUi = (AUn) =

4.707 • 10- 6 N2'7 { k T ) i p

r

[eV].

F o r comments on t h e problem of edge shift see also [ 9 . 7 5 c ] and sections 10.8.1 and 10.8.2.

302

Spectroscopic Methods of P l a s m a Diagnostics

F o r a model plasma, the resulting distribution of continuous intensity as function of frequency and the shift of the photoionization edges A T.J¡¡h are schematically shown in figure 9.5.

1 i

^—

main series limit

v

free -free radiation

^^

AUjkT

—1 0

hUrUi/kT

whv/kTF i g u r e 9.5. Schematic representation of t h e continuous spectrum e m i t t e d b y a t h e r m a l plasma. One m a y notice in t h e figure t h e diverse kinds of continuous r a d i a t i o n (free-free r a d i a t i o n a n d r a d i a t i o n due t o recombination of electrons into t h e group of close spaced levels a n d into isolated levels) a n d t h e shift of t h e emission edges b y t h e a m o u n t A Ui(/kT) •where AUt is t h e lowering of ionization energy. (After Finkelnburg a n d P e t e r s [9.21])

F i g u r e 9.6. Continuous emission coefficient £„ of a n argon plasma a t p = 1 a t m a n d T = 12,000 ° K as function of frequency v. (After Giindel a n d N e u m a n n [9.27 c])

This idealized intensity distribution will be in some cases a good a p p r o x i m a t i o n of t h e real one. This has been shown b y Maecker and Peters [9.45] for a plasma containing t h e elements C, N and 0 (see also figure 9.9). I n contrast t o this idealized intensity distribution we show in figure 9.6 t h e intensity distribution of t h e continuum emitted f r o m an argon plasma for a pressure of 1 a t m a t T = 12,000 °K, as observed by Giindel and N e u m a n n . The d a t a are t a k e n from [9.27c], The plasma was produced in a cascade arc.

9. Determination of Plasma Parameters by Means of Continuous Radiation

303

The spectral intensity distribution shows principal deviations from that of the Maecker-Peters model: At the expected limiting frequency vg = 4.6 • 10 14 s _ 1 (corresponding to the position of level 3 d6 down to which the electronic energy levels lie fairly close one to another, see figure 9.7), no marked edge is observed from the beginning

13

F

s2 s3siss

p1p2p3pi.pspeprpâp3p,0s'1 s;s;s; d,d;d2d3didld5d6 ionization Unit

z^^Lßl^-zz

— z^Zzz

£ - - - - — '

g

11 10

Jy

F i g u r e 9.7. Level diagram of Argon I. (After H. N. Olsen [9.60])

of which the continuous emission coefficient in the model plasma will show an exponential decrease according to the law s r ~ e ~ h r ^ k T \ The observed decay begins at v = 6 • 10 14 s _ 1 and is much weaker than expected. Furthermore, there are expected some other edges in the continuous intensity distribution in the visible-near ultraviolet spectral region, among them an edge at v = 1.01 • 10 15 s - 1 corresponding to the position of the level 1 s5. In reality, no edge is found in the neighborhood of this frequency (see also figure 5 in the work of Wende [9.74d).* As Giindel and Neumann [9.27b] have shown, the nonvisibility of photoionization-photorecombination edges is a consequence of the statistical character of lowering of ionization potential, which causes not only a shift but also a smearing out of ionization edges along the frequency scale. It must be noted that these conclusions are based on observations in the visible-near ultraviolet spectral region in which the photoionization edges of Ar are relative close spaced and they must not necessarily be valid in the vacuum ultraviolet region. * The edges noticed in an older work of Olsen [9.60] are only fictitious, and they can be identified with Ar I spectral lines, as shown in [9.27 b].

304 9.3.

Spectroscopic Methods of Plasma Diagnostics

Electron-atom Continuum

The theoretical description of the continuous radiation originating from collisions between electrons and neutral atoms (e —a continuum) is more complex than that of the electron-ion continuum arising from a pure Coulomb interaction, where the spectral intensity distribution can be fairly accurately estimated from classical calculations. The e —a continuum has been observed with considerable intensity in those plasmas containing atoms which have a high elastic cross section Q relative to electron collisions [9.55], The existence of a correlation between intensity of e — a continuum and cross section Q has been confirmed indirectly by Taylor and Kivel [9.68] showing that the quantum mechanical formulas for the collision cross section and for the continuous radiation intensity include analogous terms. As a further argument for this correlation the authors refer to measurements of the continuous radiation intensity I„{T) in air and nitrogen in the parameter range T = 6000 — 9000 °K and A = 2 — 8(xm performed by Taylor [9.66], [9.67] using as radiation source a reflected shock wave. In the wavelength region mentioned the continuous radiation stems almost exclusively from freefree transitions. Considering only that part of radiation stemming from e —a collisions, the emission coefficient due to free-free radiation can be described formally by an expression analogous to (9.13) ev = 6.36 • 1 0 " «

A7 P-hvl(kT) e

,

(kT)ll2

,,„

„

„

E Z\ N,

(9.29)

in which the symbol I stands for the diverse kinds of neutral particles (N, N 2 , O, 0 2 ). The quantities Zf which have in first instance only formally meaning have been obtained on a pure experimental basis by applying formula (9.29) to emission coefficients obtained for variable plasma conditions (nitrogen plasma, air plasma, variable temperature). The quantities Z\ have irrespective of their dimensionless character the meaning of inelastic cross sections relative to continuous emission. In the following table (taken from Taylor and Kivel) the values for Z\ are compared with the corresponding values for the elastic cross section Q relative to electron collisions. T a b l e 9.2 Element O N

N2

ZÌ (0.2 ± 0.3 )10- 2 (0.9 ± 0.4) 10" 2 (2.2 ± 0.3) 10~ 2

Q [10" 1 5 cm 2 ] 0.2 (0.5) 2

The data for Q are taken from Lin [9.41], [9.42] and from Neynaber et al. [9.56], Irrespective of the fact that the contributions of the diverse kinds of

9. Determination of Plasma Parameters by Means of Continuous Radiation

305

neutral particles to the resulting free-free continuum has been derived with only low precision, a marked conformity between Z\ and Q can be noticed. While the atoms of mercury and hydrogen have evidently great cross sections relative to elastic electron collisions (Qng = 1-2 • 10~14 cm2 after Brode [9.13], Qh = 2.24 • 10 - 1 4 cm2 after Drawin [9.19]), the corresponding cross sections of the rare gases are relatively low (of the order 10~16 cm 2 ). Accordingly, the e — a continua can be generally well observed in plasmas containing the elements Hg and H in a considerably amount (see [9.7], [9.43] and [9.55]), while in heavy rare gas plasmas these continua are of importance only for low degrees of ionization (such continua have been observed in low pressure discharges [9.61a], [9.61b], [9.61c], [9.61 o]). Intensity formula for the e—a bremsstrahlung continuum derived on a quantum mechanical basis applicable both to atoms and to molecules for which the elastic collision cross section Q relative to electrons is only a slowly varying function of electron energy can be found in a number of papers (see for example [9.1b], [9.If], [9.1 r], [9.3a], [9.5d], [9.21a], [9.24f], [9.36d]). They can be brought to the form of equation (9.29), in which the effective charge number squared is asymptotically given by the expressions (see [9.5d]) o T/o Z 2 = t i t Q(T) (kT)2

if

7i2 e

Z2

=

h

v/(2 kT)
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>j H

O® o e a < tHS

s s

0) g T3 73 'o "o S s a 2 CO W £ ce eg be fi =3 FT PH S

'Ph'C^S

s I

K K o ffl

fi JS a

S

®

a S

fi"

-c o fi

s ®

10. Determination of Plasma Parameters from Line Profiles

©

t-

«

'—1

c3 IO cq CN IM

©

t> p—I C3 IO I-

I © o

M o

co ©

te ©

E-i

o 2 ® £o .2 ¡> 02

^

"i

ft § ft h ö h

05 d

oo d

CO © I «5 S © CO © O ö

•s b sß

2 g 'S) o °

^ . _ I00 M ® H I g ¡*j X! xj ¡xj ^

p
m _ Cim Aa>n, i, m AUn> h rl

an(j

¿j — _!_2 for the Stark constant the expression r * CA

A(0n, —

I,

E2

i

(10.72)

Now Ci results by taking a suitable mean over the different m-components. In the general (non hydrogenic) case, the Stark effect constant is given by 0 4 — C4 i — C t j ,

10. Determination of Plasma Parameters from Line Profiles

359

w h e n i and / respectively denotes the initial and final spectral energy level. 04

i

is g i v e n b y ([10.166a] p. 116)

2

ft

3 ft2 j

a>ij

3 ft2 gt

a>ij

when the sum is t o be taken over all levels j in the neighbourhood of level i f o r which optical transitions exist. I n terms of oscillator strength this formula changes into (see equation (5.86)) Citi

= 7.8- 10-i Z f t j f y , j

(10.74)

(Xij in c m ; n e g a t i v e j for emission, compare ([10.166a] p. 116)). A n analogous expression holds for C 4 j . H e r e it should be mentioned that in the presence of large electric field strengths the quadratic Stark effect can go over into the linear Stark effect. This phenomenon was observed, f o r example, in helium (see Unsold [10.168] and [10.170], Ch. 11). Data on Stark constants can be obtained, f r o m the works of Joos and Saur [10.72] and K o p f e r m a n n [10.90] as well as f r o m the references listed in the appendix at the end of section 10.5. 10.5.3.3. Resonance broadening (p =

3)

This broadening takes place upon the interaction of like excited and nonexcited atoms. T h e broadening involves resonance levels and thus those spectral lines originating f r o m transitions between the resonance levels and the ground level as well as f r o m transitions f r o m higher levels t o a resonance level. W h e n the perturbations take place the light quanta are transferred f r o m one a t o m to another. T h e resonance broadening was calculated, quantum mechanically among others, b y Fursov and Vlasov [10.39]. Their results are in a p p r o x i m a t e agreement with those of the statistical theory (see [10.166], p. 231). I n each case the Stark constant was found to be = where fnm

(10.75)

2 m a>0

is the oscillator strength of

the resonance line, originating f r o m

a transition f r o m l e v e l m with the unperturbed circular frequency co0. T h e line profile is g i v e n in accordance with [10.39] as

( to — a>o\ 0,394 d)

Uto)

= 9=1 --T^ 2 n*

S- •

(co — co0)J

6 is the full half-width of the profile, and is g i v e n b y = 12 C3N. 2N 0 „ 6m,feco 24

Progress in Plasmas

(10-76)

(10.77)

360

Spectroscopic Methods of Plasma Diagnostics

Other authors have found other numerical factors, which all together do not deviate more t h a n 6 0 % f r o m one another. The formula for t h e line width derived b y Ali and Griem [10.1 h] contains an additional factor (galge)112'¿AH, Griem = 3.84 (galge)ll- N m eo . 0 I n this formula, ga and ge denote the statistical weights of t h e ground ("absorbing") level and t h e resonance ("emitting") level. The case when t h e resonance broadening and the broadening b y dispersion forces are of t h e same order has been t r e a t e d in [10.97a]. A survey of different calculations can be f o u n d in [10.20], p. 231 (Breene). I n addition t o its importance in particle density measurements, resonance broadening is also significant in determining oscillator strengths. This was discussed previously in section 5.3.8. W e accentuate t h a t t h e validity of the theory of resonance broadening as quoted in this section has been verified for a greater n u m b e r of elements including H g [10.43], Cd [10.128], Ar [10.115], He [10.180d] and I n [10.134d]. An unusual high broadening leading to a p p a r e n t line widths much in excess of those following from formula (10.77) was observed for the resonance lines of t h e elements Ca, Sr, Mg [10,31 f], Na, K, Rb, Cs [10.138k], [10.151], Li [10.180d]. I n t h e case of alkali resonance lines, Ya'akobi [10.180d] and Cayless [10.21 f] were able t o show t h a t the excess broadening is merely a result of t h e simultaneous effect of customary resonance broadening and self absorption. The degree of self absorption in t h e center of t h e resonance lines in cases where t h e resonance broadening is dominant, is given b y the formula (see (10.77) a n d (5.68) with the substitution y = 8 441, 93a 1 ), 138t. 4 0 - 900 1.4 - 1 4 1 0.07 - 0 . 0 9 115, 116. 138t 235 0.24 7 0 - 1700 0.04 - 55 8 3 ), 93d, 128 3 ). 0.4 4 93 d1) 7 0 - 200 100 2 7 123 c 2 ) 0 . 0 3 - 10 0.2 - 24 1, 25b, 31k, 44c, 44u, 45c, 108a 2 ), 162. 3 - 38 0.07 - 1.4 5c, 119a 4 ). 1 . 2 - 3.8 2 4 56 a 5) 7 4 - 120 134b 12, 13, 14c, 23d, 0 . 0 1 3 - 4300 0.2 - 4 3 0 31, 41a, 52, 58, 60d, 79, 101b, 105, 106, 106a, 119b, 123d, 132, 134c, 148, 172, 177, 178, 180g. 22 1 3 - 30 - 40 60 d 0.013- 5 0.3 - 12 13, 57, 149, 171a, 31 o. 15 20 - 40 58 1.2 5.3 171b, 171c. 0.013 0 . 2 - 200 1 - 70 l c , l i , 2, 3a, 20e, 44b, 69, 78, 79, 81, 82, 130 d, 134, 135a, 156, 174a, 174b. 138t 235 16 700 1 0 0 - 400 1 0 0 - 600 5 - 400

3 0.3 0.4 0.1

-

41) 2 6 15

382

Spectroscopic Methods of Plasma Diagnostics T a b l e 10.8. (Continuation)

Emitter

z

Main elements in plasma H, 0

H«-«

H, H, H, H,

He Kr N Ne

H, L y a He I

1 2

H, Ti H, Ar C, H , He, N Na, 0

Hell

2

He

Hgl

80

K IX KrI Kr II Li I

19 36 36 3

Mg I

12

Mg I I NI NU

12 7 7

N He, N

NUI Na I

7 11

He, N He, N, Na

Nel Nell Ol

10 10 8

Ne Ne H, 0

Oil 0 VI PI SI SII Sil Si I I I

8 8 15 16 16 14 14

N, 0

Hg, Xe Kr

H, Ne, 0 , S Ne

Parameter range 3

T [10 °K]

AT e [10 15 cm- 3 ] -200

Reference

5c, 17, 24, 25, 45d, 73, 89, 156, 178a. 0 . 2 - 1 0 0 0 0 0.7 - 2 5 0 > 1 0 - >23 31 j, 78, 79, 82,98c. Or 10 95 9 81 12.1 0.82 40 156 10a, 95, 104, 131, 0.015-290 0.1-- 1 0 0 6 - 20 134. 4 25 c 26 20 5) 84 12 17 d I d , 16b,23g, 23n 7 ), 0.1 - 2 8 0 1.86)— > 4 9 0 . 0 2 - 7 0 0 0 31 j, 50, 59c 8 ), 70a, 97 c, 98 c, 112 a, 114c, 118, 1381, 139b 5 ), 145, 155, 171a, 172, 178d, 179. 2a 5 ), 11a, 12, 13, 0.08-4000 0.2 - 4 0 4 3 - 200 23g, 36b, 59d 1 ), 9 ), 92,109,138 e, 138h, 178d, 179. 6 - 30 3 - 1 2 0 0 0.08--38 3 ) 8, 43, 85, 135, 138 c 10 ), 140, 142. 980000 0.26 23k. 68-259 2 -7 158 a 9 . 5 - 10.6 3 0 - 40 100 0.4 - 1 . 8 110, 112a, 118. 5 - 30 0.7-800 0.2 - > 1 6 0 44y, 45, 60a, 101 d, 180b, 180 e 11 ) 100-1200 45, 91, 96, 145, 8 - 30 0.1 - 1 . 3 180a. 50-152 101a 1.5 —122) 8 - 15 10-160 130c 9 ), 130e, 159. 1 2 - 14 0.001 - 1 4 . 3 15-3000 12a, 12b, 23n, 2 0 - 60 0.3 - 1 0 112b, 138o, 165c. 4 0 - 45 110-190 0.2 - 0 . 4 31 e 0.5-70 44r, 77a, 80, 91, 5 - 20 0.01 - 1 . 5 132h, 158. 10-100 44 f 2 ), 131 6 - 12 0.08 - 5 0 100 3 0 - 40 0.1 - 1 . 9 110, 112a, 118. 0.002 - 7 1 74, 123c 2 ), 130c 9 ), 1 2 - 14 10-129 130 e 9 ), 176, 178a. 3 4 - 36 50-156 0.15 - 0 . 7 54a, 131n. 980000 0.5 - 1 . 1 23 k 12 123 c Or 12 100 1 1 - 12 5-100 1.5 - 3 2 20d, 20f, 123c. 1 1 - 12 0.4 - 0 . 7 20 d, 20 f. 63-75 12 123 c 100 6- 1 4 0.2 - 2 . 6 2 0 - 40 100-600 110, 112a, 119. 7 - 16

20

AX [A] 5

-

80

10. Determination of Plasma Parameters from Line Profiles

383

T a b l e 10.8. (Continuation) Emitter

Si IV Sn IV Til Ti II TI I Xe I Znl Zn II *) ®) 8) 4) s)

Z

14 50 22 22 81 54 30 30

Main elements in plasma

Parameter range T [103 °K] ^ e [ 1 0 1 5 c m - 3 ] 30 20

Xe C, H, Zn C, H, Zn

-40 -40

6 7 . 3 - 9.0 8 -40 10 - 1 2

1 0 0 - 600 1 0 0 - 600 3060 30 - 60 0.6 3 7 - 196 50-2000 1 0 0 - 700

Al [Â] 0.4 0.3 -

1.4 2.8

0 . 0 5 - 0.1 -12 4 0.04-18 0.6 - 1 8

Reference

112a, 119. 110, 112a, 119. 119c 119c 80 138 d, 158a. 8, 93c, 165c. 93 c 1 )

Values for ylNe are given. Line shift data are given. Line width is influenced by resonance broadening . Line width is influenced b y Doppler-broadening. Line wing investigation.

•) This temperature value corresponds to electron energy distribution in the low energy range. ' ) Comment. 8 ) Data for the separation of the two line maxima are given. 9 ) Vacuum UV lines. 1 0 ) Corrected for van der Waals broadening. J 1 ) Corrected for self absorption.

W h e n determining t h e plasma parameters from t h e line shift, simple relationships can be obtained only under t h e assumption t h a t t h e a d i a b a t i c t h e o r y of Lindholm is applicable: p = 4 (quadratic S t a r k effect):

A =

y ,

(10.128)

y ,

(10.129)

where y follows from (10.122): p = 6 (van der W a a l s broadening):

A =

2.75

where y is found from (10.123). I n other cases, appropriate formulas must be derived q u a n t u m mechanically. F o r interpolation, t h e formula ^r=B-T

b

(10.130)

can be used, where t h e coefficient B must be derived from calibrating experiments or published data, i f a n y . 10.5.10. Table Appendix I n concluding section 10.5 we have listed a n u m b e r of references on experimental and theoretical investigations of t h e line broadening produced b y particle interaction, whereby we have also cited those works, in which plasma parameter are determined from t h e line broadening. T a b l e 10.8 contains experimental d a t a on t h e line broadening, produced b y t h e linear and quadratic S t a r k effect and their application t o plasma p a r a m e t e r

384

Spectroscopic Methods of Plasma Diagnostics T a b l e 10.9. Calculated data for profiles of spectral lines broadened by Stark effect a) Tabulation of statistical line profiles Spectral lines

Reference Schmaljohann [10.150] Verwej [10.171] Underhill and Waddell [10.167]

Ha — Hv H a — Hy H I, Lyman-Balmer-Paschen- and Brackettseries lines up to n = 18

b) Tabulated profiles of hydrogen and helium lines, taking into account electron collision broadening

Reference

Griem et al. [10.47 a], [10.50]

Griem et al. [10.47a], [10.52] Griem et al. [10.47 a], [10.52] Griem et al. [10.48]

Spectral line A in [A]

T-range [10 3 ° K ]

Ne-range [10 15 cm" 3 ]

Ha H^, Hy Ha Ly. Ly/s He I I 4686 He I I 3203 He I 3965 He I 4471 He I 5 2 2 - 4 6 9 5 0

10-40 10-40 10-40 10-40 10-40 5-80 10-80 20 20 5-80

101000 1100 0.110 1 0 0 - 10000 1000 101 - 10000 11000 300 300 1 0 - 100 000

c) Other line broadening data. Literature is given in short symbols (see head of table 10.8)

Emitter

Z

Parameter range Te [10 3 ° K ]

All Al I I Ari Ar I I

13 13 18 18

2.5-80 2.5-80 2.5-80 2.5-200

B II Bal Ba I I Bel Be I I CI CII

5 56 56 4 4 6 6

2.5-80 2.5-80 2.5-80 2.5-80 2.5-80

Ne [10 15 cm" 3 ]

1.3 - 7 0 0 0 100

Reference

23o, 47a. 47 a 47 a I f , l g , 18a, 18b, 23e, 231, 47a, 47 c 47 a 44 q.1) 23 e 47 a 47 a 47a, 178b. 18b, 18d, 23e, 47a, 47c, 178b.

10. Determination of Plasma Parameters from Line Profiles

385

T a b l e 10.9. (Continuation)

Emitter Cal Ca I I Cd I Cil

Z

Parameter range Te [IO3 °K]

20 20 48 17 17 55 29 9 9 1 1 1

2.5-80 2.5-80

He I

2

1.8-80

K I Li I Li I I Mg I Mg I I NI

Nil

19 3 3 12 12 7 7

2.5-80 2.5-80 2.5-80 2.5-80 2.5-80 2.5-80 2.5-80

NaI Na I I Nel Neil 01 0 II PI PII SI SII SUI Sii Si I I Si I I I Znl

11 11 10 10 8 8 15 15 16 16 16 14 14 14 30

2.5-80 2.5-80 2.5-80 2.5-80 2.5-80 2.5-80 2.5-80 2.5-80 2.5-80 2.5-80 10-20 2.5-80 2.5-80 10-20

cui

Cs I Cui PI PII H a — Hó H6-H14 Ly*

) ) *) 4) б)

1 а

2.5-80 2.5-80 5 2.5-80 2.5-80 5-50 1.8 10-20

Reference

N( [IO15 cm" 3 ]

0.01-10000 0.0132) 84-1000

47 a 18b, 18c, 23e, 47a, 170e. 44 q1) 47 a, 138 q. 47 a 44p, 44q 1 ), 44u, 47a. 44 p 47 a 47 a 5a, 5b, 75c. 136a, 136b. 47b 2 ), 131 h, 156c, 156d, 156e 5 ),

5.0 - 9 4 0

8b 3 ), 8c 3 ), 18c, 23f, 44a, 47a, 47d, 47e 3 ), 112a, 132f, 136a 2 ), 136c2), 136 d2), 138p. 44p, 47a. 44r, 44z, 47a. 47 a 44q 1 ), 47a. 18 d, 47 a. 47a, 178b. 18b, 23e, 23k, 23n, 47a, 47c, 178b. 18c, 44r, 47a. 47 a 47 a 47 a 47a, 178b. 47a, 178b. 47 a 47 a 47 a 18 d, 23 e, 23 j, 47 a, 47 c, 138 s. 18d 47 a 18d, 47a. 18d 44 q1)

9.3 - 6 7 0 0

0.5 - 2 0 0

mg4).

The effect of broadening in inhomogeneous electric microfield is considered. Line wing investigation. Forbidden components are included in calculations. Calculated with many body formalism. Incomplete presentation.

386

Spectroscopic Methods of Plasma Diagnostics

T a b l e 10.10. Reference list about van der Waals type broadening. References are given in short symbols, see head of table 10.8 a) Broadening by rare gases Emitter or Absorber Agi Ari Ba I Ba I I Cai Cd I Cs I Cu I Fe I I Ga I HI He I Hgl

Ini K I Kr I Li I Mg I 2 ) Na I

Ne I Ni I Rb I Si I Sri2) Ti I Ti I I TI I Xe I

Perturbing gas Ar 23 b 63b, c, d, 91e, 94a, 158a, 158b. 134 e 54g 63a, b, 134e, 138 j. 8a 1 ), 138 j, 166 c. 9a, 22, 211, 31 h, 36d, 138z. 94 60 134 d

He

Kr

Ne

Xe

23 b

134e. 54g 61, 62, 63, 63a, b, 134e, 138h. 8a 1 ), 138]. 22, 31 h, 36 d, 40 a. 94

134d 123 63b 63 b 3b, 37a, 54d, 311, 44 h, 441, 93 e, 121, 121a, 120, 138 v, 138z, 165e. 138w, z, 144, 165 d. 134 d 134 d 37 b, 63 e, 65 a, 36e, 37b. 138u, 138z. 128b 128b 63 b, 94.

63 b, 94.

9, 10, 12 c, 37c, 63b, 84a, 84b, 114a, 138 z, 139.

10

54 f, 109 a, 123a, 175b. 96 a 96 a 21 j, 54b, 21 j, m, 44j, 1, 54b, 84a, 132 b, c, 132 b, c, 138x, z. 138 x, z. 23m, 31n. 10, 134e. 10, 134e. 23h, 93b. 93b. 31i 31 i

134 e

134 e

63a, b, 134e, 138 j.

63a, b, 134e. 138 j.

138j

8a 1 ), 138 j.

8a 1 ), 138j.

8a 1 ), 138 j.

21 n

9a, 36c, 36d, 40b.

31 h, 97 d.

134 d 63b 54 d

63 b 8f, 63 b, 158a, 170e. 63b.

311, 121, 121a.

54 d, 138c.

134d 37 b 128b 63 b.

63b, 91 d, 91 e, 123a. 44i, 4413), 54b.

44k, 441, 132b, c.

134e.

134e

158a

10. Determination of Plasma Parameters from Line Profiles

387

T a b l e 10.10 (Continuation) b) Broadening by D 2 , H 2 and N 2 Emitter or Absorber Gal Cs I Fel Hgl K I Nal Rbl Sri Til

Perturbing gas D2

N2

H2 93 36 d 93 21b, 21c 4 ), 37a, 441, 138w, z. 37 b 37 c, 114a. 21 j, 21c, 44g, 441.

21b, c, 441. 44g, 441.

31 i

114c 36 c, 36 d, 138 z. 76 b 37a, 138 w, z, 144. 37b, 65a, 173b. 9, 10, 37c, 77a, 114a. 21c, 138 x, 138z. 10 31i, 77a.

c) Broadening by other gases Emitter or Absorber Ca I Cu I Fe I HGL K I Li I Na I Sri Ti I Ti I I Zn I l

) ) ) •') s

3

Perturbing gas Air

co2

CS

H

H2O

Hydrocarbonflame

0

O2

Rb

158 5c 93 80

37 a 144

5c 37 a

158

21b, 37 a, 144. 21k

80

9 10 10

158 65 158

37 119c 119c

63 h

Broadening cross section is determined b y Hanle- and modulated light double magnetic resonance technique. Reduced broadening parameters scaling per unit polarizability of perturbing gas are given in [10.173], Satellite band broadening is included in the investigation. Also considered as broadener is B I ) and T 2 .

determination, while Table 10.9 refers to calculated Stark profiles and Stark widths, respectively. Table 10.10 lists a number of characteristic studies, in which the van der Waals broadening and shift, respectively, is investigated, while table 10.11 is concerned with resonance broadening of spectral lines. Finally, in Table 10.12 references are listed in which Stark constants are given. A survey of their determination can be found, for example, in [10.17 e]. Not included in table 10.8 are a great number of such investigations concerned with the determination of charged particle number density under the condition

388

Spectroscopic Methods of Plasma Diagnostics T a b l e 10.11. Bibliographic references on resonance broadening Emitter or Absorber Agi Ar I Ba I Cal Cs I Cd I He I Hgl In I K I Nal Nel Rb I Sri

Z

Reference

47 18 56 20 55 48 2 80 49 19 11 10 37 38

14 a 63d, 94a, 136e, 158b. 134f 134f 45a, 180h. 14 a 23i, 36a, 92b, 97a, 136e, 170d. 43 134d 97b, 101c, 165 h, 180h. 165g, 173a, 180h. 91 d, 92a. 21 i, 165i, 180 h. 134 f

Ne = Ni in plasmas containing hydrogen as additive element, from the half width AX of the spectral lines H a and H^, if such communications contain no new information about the dependence of AX from Ne and Te which in these works have been assumed to follow the well approved theory of Griem, Kolb and Shen. 10.6.

Limits of the Ranges in which the Line Broadening is Predominantly Determined from the Doppler Effect and Charge Carrier Interaction

In many cases it is desired to obtain quick insight into the parameter ranges for which a predominant Doppler effect or predominant charge carrier broadening is expected. Margenau [10.113] took this task upon himself and compiles his results in the form of three representations, which we give in Figure 10.21a — c. In the figures N denotes the density of perturbing particles and T the temperature. Figure 10.21 a refers to the linear Stark effect (for a hydrogen plasma with Stark constant C2 = 10 this corresponds approximately to the case in which the H^ line is broadened). Figure 10.21b applies to the quadratic Starkeffect for light, emitting elements, and is based on an atomic weight of 1 and a Stark constant of 0 4 = 10~12. Figure 10.21c is connected with the quadratic Stark effect in the case of heavy emitting particles, whereby the atomic weight of the latter is taken to be 50. Below the dashed line the line broadening due to the Doppler effect is predominant, while above the line broadening caused by particle interaction predominates.

10. Determination of Plasma Parameters from Line Profiles

389

T a b l e 10.12. Bibliographie references on Stark-constants 1 ) System Agl A1 I Ar I Ar I I Au I Ba I Ba I I Bi I Br I CI CII Ca I Call Cd I CII Cr I Cs I Cu I F I Fe I HI He I He I I Hg I 11 K I KrI Kr II Li I Mgl Mg I I N I Nil Na I Nel Nell 0 I 0 II Rb I Si I Sri Xe I Zn I

Z

Reference

47 13 18 18 79 56 56 83 35 6 6 20 20 48 17 24 55 29 9 26 1 2 2 80 53 19 36 36 3 12 12 7 7 11 10 10 8 8 37 14 38 54 30

38 23 o, 67. 72, 126, 131c. 72, 108, 114c, 118, 125, 131b, 131c. 38 44 q 2 ), 90, 124, 137, 138. 4, 72. 59 3 67, 68 a. 441 41, 72, 90. 131a l b , 6a, 7, 8, 37d, 44q 2 ), 75e, 139a. 3 90 8e, 17e, 44p, 44q 2 ), 44u, 55a, 72, 75d, e, 114b, 131a, 161a, 180f. 5 c, 38, 44p. 98 72, 76b. 72, 131 f. 34, 72, 114c, 118, 127, 136c, 145, 163a. 72, 160, 161. 6a, 7, 8, 15, 15a, 15b, 68, 72, 75e, 139a. 3 17e, 44p, 54c, 72, 131a, 156f, 175. 72, 158a. 118 31b, 44r, 44z, 66d, 72, 131a, 145. 6a, 7, 8, 28, 44q 2 ), 72, 135a. 28, 72. 152c, d. 59, 152, 165 c. 39a, 44r, 44w, x, 72, 77, 77a, 89a, 90, 131a. 72 72, 118. 76, 83, 134g, 134h, 165. 134g 8e, 72, 75e, 114b, 131a. 173 72, 90, 178 c. 72, 158a. 6a, 7, 8, 37d, 44q 2 ), 93c, 130g, 138m, 139a, 164, 165c.

*) A survey of the determination of Stark-constants is given in [10.17e]. ) The effect of energy shift due to electric field inhomogeneities is included (the Stark constant is Ca).

8

390

Spectroscopic Methods of Plasma Diagnostics

In addition the diagrams include the division of the N — T region into the regions I, II, and III, in which the broadening due to particle interaction must be described by different theories (see also section 10.5.8.2): Region I: collision theory for ions and electrons, Region II: statistical theory for ions and collision theory for electrons, Region III: statistical theory for ions and electrons.

20 19

F i g u r e 10.21a —c. N — T diagrams with boundary lines (N = perturber density). Below the dashed line the Doppler effect predominates, while above this line the line broadening due to particle interaction is predominant. The solid lines effect a separation of the regions I, I I , and III (see text), in which the broadening produced by electrons and ions, respectively, must be described by different theories (after Margenau [10.113])

>

' 18 17 16 15

Doppler region

%

1

2

3

if

5 IgT

c)

6

T h e individual figures respectively refer to a) linear S t a r k effect, C t = 10, a t . w t . 1 ; b) q u a d r a t i c Stark effect, C, = 1 0 - 1 2 , a t . wt. 1 ; c) q u a d r a t i c Stark effect, C, = 1 0 " u , a t . wt. 50

10. Determination of Plasma Parameters from Line Profiles

10.7.

391

Determination of Electron Density from the Emission Coefficient of Forbidden Lines

The electric microfield in a plasma not only causes a broadening of allowed spectral lines but also gives rise to the appearance of forbidden spectral lines in the neighbourhood of allowed lines. Thus, forbidden/allowed line pairs are formed. The forbidden and allowed component of such a line pair in general have a common lower level. Then the upper levels are neighbored, and in a great number of cases they are coupled one to another by allowed optical transitions. The forbidden transitions can be classified into two groups: Group a. Transitions between odd levels or between even levels, following the selection rules AJ = 0, ± 1 , ± 2 , AM = 0, ± 1 , ±2 . The transition probability is a function of the homogeneous part of the electric microfield (A ~ E 2 if the allowed line component displays quadratic Stark effect) and can be derived from perturbation calculations (see, for example, [10.44n], [10.44o], [10.123b], [10.123d]). Group b. Transitions between odd and even levels and vice versa, following the selection rules AJ = 0, ±1, ±2, ± 3 (excepted J = 0

J = 0) ,

AM = 0, ± 1 , ±2, ± 3 . The transition probability in this case is a function of the inhomogeneous part of the electric microfield and can be found, for example, by perturbation calculations [10.44m], [10.123b], For plasmadiagnostical purpose, practicable formulas for the intensity ratio of forbidden/allowed line components of a line pair taking into account level perturbation not only by the ion field component but also by electron collisions, are given for some elements by Grechikhin and coworkers. They are reproduced by part in the following table 10.13. Further data can be found in [10.44n], [10.180c] (for Li), in [10.44x] (for Na), in [10.44s] for Cs and in [10.171a] (for He). I t must be noticed that in the case of highly turbulent plasmas the forbidden lines can be surrounded by satellite lines stemming from non thermal plasma oscillations, as has been shown by Kunze et al. [10.92 c] in experiments with a He plasma produced in a Theta pinch discharge. 26

Progress in Plasmas

392

Spectroscopic M e t h o d s of P l a s m a Diagnostics

P3

p, Ml Ml

o o io 11 o o Ml

o o CO 11 o io

o o o o o o© f] N N

H t - O O « , 111 x (N r—I i—I 2.57 • 107 T3 • (10.146)

(A Ui in eV) . From (10.146) it follows that for small electron densities the reduction in the ionization energy is dominated by the "polarization interaction", which finds expression in the term x e2. For high densities the reduction due to "electrostatic interaction" predominates. This is expressed by the term C e2/r0, where r 0 is the mean distance between the charge carriers. Recently, Ecker and Kroll [10.30a] have shown that in the range of low electron densities satisfying the relation N, < NetCf, the lowering of ionization

400

Spectroscopic Methods of Plasma Diagnostics

energy A Ui is greater by a factor 1.4 than indicated in the upper part of formula (10.145) and (10.146), respectively, when the actual structure of the microfield is taken into account. Of the same order of magnitude is the correction factor

(1 + £(AUvl(kT)yi* given by Giindel [10.54e], where £ varies between 0.6 and 0.7, and A Up stands 2 for x e . Besides these statistical theories, there exist further theories on the reduction of Ui. Some of these are mentioned in [10.30]. Unsold [10.169] considered the potential energy of an electron in the field of two

14:71

nuclei, each having charge Z, which are separated by a distance rm = I —-iV^ j

\ — 1/3

(corresponding to the mean distance between an atom and the nearest ion). The electron together with one of the nuclei is taken to represent an atom, whose ionization energy is the subject under discussion. The potential curve displays a maximum in the middle which lies below the zero axis by the amount A Ui unsold = £2/3 • 0.7 (J^/IO")1/» eV .

(10.147)

Unsold identified this energy difference with the lowering of ionization energy. According to Ecker and Kroll this derivation is valid only in the case of a quasistationary interaction between the perturbing particles and the atom, whose ionization energy is studied. Quasistationarity in the sense described by Ecker and Kroll occurs when the duration of the microfield influence is larger than the period of an electron around an orbit corresponding to the last existing level. For the validity of the Unsold derivation this leads to the relationship ^>1.11-10

(10.148)

where M* is the molecular weight of the ion. This condition can be satisfied in a whole series of plasmas. For heavier plasmas the condition is more readily satisfied than in lighter ones. For example, taking M * = 10, the condition becomes for T = 104 ° K : 2Ve > 1.11 • 1014 cm" 3 . Now it must be established that in a relative wide parameter range, the reduction in the ionization energy given by Unsold is greater than that found by Ecker and Kroll [10.30], so that the question concerning which theory is actually correct bears some significance. W e still have to discuss the theory developed by Rother [10.143], which is based on the following considerations. A level ceases to exist when the ionization frequency Zi of this level becomes larger than the rotational frequency fu of an electron, which is located at this level. For the last existing level nm it is assumed that Zi and /„ are equal. This

401

10. Determination of Plasma Parameters from Line Profiles

leads to the equation 2 lcT

m

Ne e~UnJ S » (j" o o

P

- ®rr1 -e> Hm

Ei W XI °

o o io o >o

o o Tf

o o II-N H O ®

^ «

o o >o

ft;

®

o » « s o o

A

"5

•3 5 « > H

w> £

®

C a) o o

o CG 60 "B Pi fi2 CO s s

« Js "t ® -H — i1 a ° ?a -b « ^ PH"S O

T. O

to m © o

gp-—0} »fi c .-s H is

cS W stf EH

§ § «

®

-eS ?s °

g K ®

1

Ei • S ^ 1 atm in general fulfilled for p > 1 a t m ; for pure, heavy inert gases either p > 10 atm or p > 1 atm and njn0 ]> 0.1 must hold

f) Tt =

T.

11.2.

Special Techniques of Nonthermal Spectroscopy

Whereas in thermal spectroscopy the emission coefficient of a spectral line is uniquely determined by temperature and pressure as well as by the gas composition, in the case of nonthermal, plasma data are required on the electron temperature (provided a Maxwell distribution exists) or on the electron velocity distribution in order to determine these quantities. Further, the number of different particle species per cm 3 or the partial pressures of the different particle species and their temperatures must be known. In those cases where stepwise excitations from resonance levels play a role, the re-absorption of this resonance emission must be taken into account; then the emission coefficient of a spectral line still depends on the geometry of the emitting plasma (this applies not only to spectral lines corresponding to transitions from higher excited levels but also to resonance lines; see [11.32a] for the emission of the B a I I resonance lines emitted by a quiescent Barium plasma). To evaluate the radiative properties of a plasma which is not in thermal equilibrium one requires data on the effective cross sections for inelastic collisions of the first and second kind as well as information on the transition probabilities for the transitions connected with the radiation. An estimate of 28 Progress in Plasmas

424

Spectroscopic Methods of Plasma Diagnostics

the degree of self-absorption of the plasma requires further data on the line width, which essentially consists of a Doppler broadening, a Stark broadening, and a resonance broadening (insofar as broadening of the resonance levels is concerned). The emission coefficient enm of a spectral line is determined as in a thermal plasma by means of the expression (see equation (5.1)) e

nm

~

n

1 4 n

m - A

n m

h v

n m

(11.27)

.

The occupation number nm for a stationary nonthermal optically thin plasma follows from a system of equations (see Biberman and Ulyanov [11.2]), in which the occupation numbers of all others levels also appear: Ve K j e » +

-

Ê ( k= l

(n°e)2 («ne

£

A

kn +

+

Ve

y . n°ejne)

n°ejkn)

=

-

E yknl k =1

(A

n k

0 .

+

y,

n°ejnk)

(11.28)

Equation (11.28) is a balance for the depletion and refilling of the level n due to radiation and electron collision processes, whereby transitions between different discrete levels as well as transitions to continuous states (free electrons) are considered. Inelastic collisions by heavy particles are neglected. The individual variables signify: y„ = ~

ye =

n.

relative concentration of the excited atoms in level n, referred to the equilibrium concentration in the case of Boltzmann equilibrium, 'l± e

relative electron concentration, referred to the equilibrium concentration,

je „

collision integral for ionizing collisions,

Akn

transition probability for n n

k transitions; Akn — 0 for

k,

11. Validity Limits of the Methods of Thermal Spectroscopy

429

Here is the number density of helium-like ions in the ground state. For Q a value of 0.2 n a\ (a0 is the first Bohr radius), estimated by Seaton, was assumed for both ions. n e was calculated, first assuming a totally ionized deuterium plasma, and secondly for an impurity admixture of 10% with a mean ionic charge of 5. Further, a central plasma concentration was assumed, whose value was found on the basis of the toroidal magnetic field intensification, which was determined by magnetic probes. In determining the contributions of the elements B and C it was assumed t h a t during the discharge these are just as great as before the discharge. Further it was assumed t h a t the major portion of carbon in the center of the discharge tube was in the ground state of the ion C V and the major portion of boron in the ground state of the ion B IV. This was confirmed by the fact the spatially resolved intensity of the B I V line in the center of the discharge tube far exceeded the intensity of all lines emitted by other boron ions (see, e.g., Figure 11.5).

F i g u r e 11.5. Radially resolved intensity of the lines B IV 2822 A (curve A) and B III 2066 A (curve B) (after Williams and Kaufman [11.45])

From the thus obtained data on n e and n v the effective cross sections, and the line emission coefficients s we finally obtain the electron temperature T„ which is listed in the table below. The electron density in (11.35) can be eliminated by first applying it to the emission of the B IV ion and then to the emission of the C V ion. The electron density then cancels out and we are left with the equation ([11.26], [11.45]) U -Um> s' n[ Q' v 1 + UJ(hTe) ^ mkTg (1136) e % Qv 1 + UJ(kTe) The accented quantities correspond to the B IV ion, and the remaining ones correspond to the C V ion. Via equation (11.36) we also obtain an electron temperature, which is listed in the following table

430 Ion CV B IV

Spectroscopic Methods of Plasma Diagnostics

13

4

2

1 • 10 6 • 1013

1.1 • 10" 2.1 • 10"4

304 203

j e [erg cm 3 s ] ]

n1 [cm"3]

p [Torr]

Um [eV]

9 • 10 1 • 105

Te [°K] 2,2 • 10s 2,0 • 105 3,0 • 105

Te from the relative B 1V/(CV) p in t h e tables denotes the p a r t i a l pressure of the admixture (methane or diborane)

The deviations in the measured temperatures are connected with inaccuracies in the data employed or with possible deviations of the electron velocity from the Maxwell distribution. 11.2.2.2. Determination of electron temperature in rarefied plasmas from relative emission coefficients of spectral lines belonging to the same level system Assuming t h a t the upper levels of the spectral lines in question are filled in chief by direct electron excitation from the ground state, then the absolute emission coefficient of a spectral line belonging to the transition m —> n is given by the formula (see [11.34b], p. 44) =

(11.37) 2J

-Ale m

k=1 where denotes the ground state density and jm t the collision integral defined in section 11.2.1. A like relation holds for the emission coefficient of a second spectral line belonging to the transition m' n'. Taking the ratio of both relations, the quantities ne and % cancel, thus we get m' — 1 £

nm e»' m'

v

Z

A

k

• , V \ A ' Jm e) -Anm k—1 V jm' i(Te) An> m> ™ = 1 2- Ah m k=1

= fnm(Te).

(11.38)

Relation (11.38) can be used for the determination of electron temperature from the ratio of measured emission coefficients of such spectral lines which have collision integrals with different temperature dependence. That means in practice, the excitation energies of the two spectral lines must be quite different. It is not profitable for the purpose of temperature determination to use spectral lines with different temperature dependence of the excitation functions when the excitation energies of these lines are nearly the same. Then for not too low particle number densities, the transition rate between the two near resonant upper levels due to inelastic particle collisions becomes so high t h a t the intensity ratio of the two spectral lines becomes a function of particle number density, while the temperature dependence of the line intensity ratio is weakened. The significance of this effect has been demonstrated by Drawin

11. Validity Limits of the Methods of Thermal Spectroscopy

431

cS

aCO

C3 "ft

cS Ph

O ft >>

ft -¡3 J3eS 3 g© £> CO Ph § * a II T3• > «23 0«g "3-

A>>

e X

>> O Q

s ^ 2 üS s

.2® O 1I ä > J o

» 0 2 s 6> D J> ^ A V et*a»2«4)fl Ä Ä «i SO « g2 P, gl 6 ^o S3 flO a ° 11 » Qto-3

432

Spectroscopic Methods of Plasma Diagnostics

and Henning [11.8d] for the triplet-singulet line pair He I 4713 A (4 3S — 2 aP) and 4921 A (4 1D — 2 1P) for particle densities in excess of 1011 cm - 3 . Relation (11.38) has been applied for the purpose of temperature determination to a number of plasmas; a few examples are given in table 11.4. The table contains also, if known, values for the maximal electron density above which step excitation of spectral levels cannot be ignored. 11.2.3.

Determination of the Ionization Temperature for Low Particle Densities

I n a highly rarefied plasma, as in the case of a plasma in thermal equilibrium the electron temperature can be determined, provided the electrons possess a Maxwell velocity distribution, from the intensity ratio of two spectral lines emitted by differently charged ions of the same element. This was the subject of discussion in section 5.2.5.1 for a thermal plasma. The advantage of this method is t h a t the geometrical dimensions of the test object need not be accurately known in the direction of observation. Thus, such a method is well suited for the determination of electron temperatures of astrophysical objects. Let us assume t h a t the spectral lines are excited by direct electron collisions. Then we obtain an expression similar to (11.36) for the ratio of the spectral intensities/' a n d / * , associated with the differently charged ions. To obtain the electron temperature on the basis of the measured value for / ' / / via (11.36), an additional relationship between the ratio of the respective numbers n[ and Wj of the differently charged ions and the electron temperature is required. Such a relationship is designated as an ionization equation. I n the complete ionization (or charge carrier balance) equation, the ionizing collisions and photoionization of all excited states must be allowed for. Further, recombination by electron collisions and by three-body collisions must also be taken into account along with all inelastic collisions and excitation processes, since the latter produce an alteration in the occupation of the discrete levels leading to a change in the ionization and recombination conditions of the plasma. Thus we arrive at the system of equations (11.28). According to Elwert ([11.11], [11.12]) a simplification can be introduced in the case of highly rarefied plasmas, whereby collisions of the excitation type or of the second type are not taken into account at all, photoionizations are neglected due to the low radiation density, while recombination by three-body collision can be neglected with respect to photo recombination because of the small particle density. The balance equation for ionization and recombination processes for the i i + 1 ionization equilibrium is given by j rti ne S12 = ni+1 ne Q21 , (11.39) rti n„ S12 is the number of ionization events per cm 3 and s, n n i+1 e Q21 i® the number of photorecombinations per cm 3 and s. * In the homogeneous plasma approximation used in astrophysical applications, the ratio of emission coefficients E'IE appearing in formula (11.36) can be substituted by the intensity ratio I'jI.

11. Validity Limits of the Methods of Thermal Spectroscopy

433

S12 and Q21 can be regarded as collision integrals, which are connected with the ionization cross section and with the recombination cross section. When calculating S12 and Q21 it should be remembered t h a t not only ionizations originating from the ground state but also from excited states can play a role. The significance of the excited levels in a rarefied plasma, however, is regarded as minor. Further, it should be remembered t h a t photo-recombinations can occur not only in the lowest state n but also in higher states. For a rarefied plasma Elwert obtained the following ionization formula, also called the corona formula: = 8.3 • 10® ^ (E^IL) 2 g n \ V t ) U^kT,)

rat

.

V(11.40)

'

Here two uncertainty factors of the order of 1, which were stated by Elwert, have been left out. Ui H is the ionization energy of hydrogen ( = 13.6 eV), Ui the ionization energy of the ion, n the principal quantum number of the ground state of the ion i, and >~{n) the number of electrons in the ground level of the ion of species i. g is a factor between 1.4 and 4, which allows for the recombination in higher levels. Elwert [11.11] plotted this factor as a function of the ionization energy U¿, the electron temperature T{, and the principal quantum number n of the lowest level. The ratio of the ion concentrations for 2 different ionization stages with ionic charges differing by more t h a n 1 from one another is given as a product of several factors of the type (11.40). The above relationship were used to determine the electron temperature via the intensity ratio of the spectral lines Fe X 6374 A and Fe X I V 5303 A emitted by the solar corona. Neglecting again the uncertainty factors the ratio of the ion densities is given by (see Elwert [11.11]) " > ™ »FeX

=

(

A ^ ) 4 where * = . (11.41) fi * ) (kTe)e\is given by Edlen [11.10] for the relevant electron

e 2 0

\

The intensity ratio / 5303 // 6374 temperature range as

= 2.4 • I (1965) 3526. GERMAN, J . D. a n d P . W . SCHREIBER, Proc. 9. internat, Conf. P h e n o m . Ionized Gases, B u c h a r e s t 1969, p. 640. G E R R Y , E . T . a n d R . M . P A T R I C K , P h y s . Fluids 8 ( 1 9 6 5 ) 2 0 8 . GRASSMANN, P. a n d H . WULFF, Proc. 6. i n t e r n a t . Conf. Ion. P h e n o m . Gases, P a r i s 1963, Vol. 4, p. 113. G R I B B L E , R . F., E . M . L I T T L E , R . L . M O R R E a n d W . E . Q U I N N , P h y s . Fluids 1 1 DOUGAL,

DUSHIN,

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[8.14h] [8.14i] [8.14j] [8.14k] [8.141] [8.14m]

1221.

K . R . , H . E D E L S a n d G . H O W A R T H , Proc. 6 . i n t e r n a t , Conf. Ion. P h e n o m . Gases, P a r i s 1963, Vol. 2, p. 253. H E A R N E , K . R. a n d N. K O N J E V I C , Z . Physik 204 (1967) 443. H E A R N E , K . R. a n d N. K O N J E V I C , Z . Physik 208 (1968) 65. H E F F L I N G E R , L. O., R. F . W U E R K E R a n d R. E. BROOKS, J . appi. P h y s . 37 (1966) 642. H E ARNE,

HEISS, A., Interferometrische Messungen a n Isar I , Rep. I n s t . P l a s m a p h y s . , Garching near Munich, I P P 1/64, J u n e 1967. HOOPER, E. B. a n d G. BEKEFI, Proc. 7. i n t e r n a t . Conf. P h e n o m . Ionized Gases, Belgrade 1965, Vol. 3, p. 209. HOOPER, E. B. a n d G. BEKEFI, J . appi. Phys. 37 (1966) 4083; E r r a t u m : J . appi. P h y s . 38 (1967) 1998.

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483

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[8.15a]

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JAHODA, F . C., R .

[8.15c]

JAHODA,

[8.15d] [8.15e]

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A.

JEFFRIES

E . M. LITTLE,

and

G. SAWYER,

W . E . QUINN,

YAMANAKA,

J . P h y s . Soc.

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F. L. RIBE

and

G. A. SAWYER,

J . appl. P h y s . 35 (1964) 2351. J A N N I T T I , E . a n d G . T O N D E L L O , Proc. 8. i n t e r n a t . Conf. P h e n o m . Ionized Gases, Vienna 1967, p. 510. J O H N S O N , W . B . , A . B . L A R S E N a n d T . P . S O S N O W S K I , Proc. 7. i n t e r n a t . Conf. P h e n o m . Ionized Gases, Belgrade 1965, Vol. 3, p. 220.

[8.15f]

JOHNSTONE, R . K . M. a n d W . SMITH, J . sei. I n s t r u m . 4 2 (1965) 231.

[8.15g] [8.15h] [8.15i]

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[8.15j] [8.15k] [8.151] [8.15m]

KIEFER, J . H . a n d R . W. LUTZ, P h y s . F l u i d s 8 (1965) 1393. KINDER, W., Optik 1 (1946) 413. KINGSTON, A. E., J . Opt. Soc. Amer. 54 (1964) 1145. KLEIN, A. F., P h y s . Fluids 6 (1963) 310.

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A. a n d R. L . T R I M P I , J . Aeronaut. Sei. 1 7 (1950) 311. V. G„ K . T . C H U N G a n d R . P . H U R S T , P h y s . Rev. 1 7 2 ( 1 9 6 8 ) 35. KEGEL, W. H . , Rep. I n s t . P l a s m a p h y s . Garching near Munich, I P P 6/9, Oktober 1963. KANTROWITZ,

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38

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KOMISAROVA, I. I., G . V. OSTROVSKAYA, L. L. S H A P I R O a n d A. N. Z A I D E L , Proc. 9. i n t e r n a t . Conf. P h e n o m . Ionized Gases, B u c h a r e s t 1969, p. 641. K O R O B K I N , V . V . a n d A . A . M A L U T I N , Zh. t e k h n . Fiz. 3 8 (1968) 1095. KUNZE, H . - J . , Z. N a t u r f o r s c h . 2 0 a (1965) 801. K U N Z E , H . - J . , The laser as a tool for plasma diagnostics, in: P l a s m a Diagnostics, N o r t h Holland P u b l . Co., A m s t e r d a m 1968, p. 550. K U N Z E , H . - J . , A. E B E R H A G E N a n d E . F Ü N F E R , P h y s . L e t t e r s 13 (1964) 38. K U N Z E , H . - J . , E . F Ü N F E R , B . K R O N A S T a n d W . H . K E G E L , P h y s . Letters 1 1 (1964) 42. KUNZE, H . - J . ,

E.

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K U N Z E , H . - J . , A. H . GABRIEL

P h y s . L e t t e r s 1!) (1965) 11. R . G R I E M , P h y s . Rev. 185 (1968) 267. Proc. 9. i n t e r n a t . Conf. P h e n o m . Ionized

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K U N Z E , H . - J . , a n d W. D. J O H N S T O N , Gases, B u c h a r e s t 1969, p. 28. KÜPPER, F . P . , Z. N a t u r f o r s c h . 1 8 a (1963) 895. L A D E N B U R G , R . , D A N I E L a n d D . B E R S H A D E R , Physical Measurements in Gasdynamics a n d Combustion, P a r t I, Section 1, 3, Oxford Univ. Press, Oxford 1955.

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c] [8.17d] [8.17e] [8.17

484 [8.18] [8.19] [8.20]

[8.20a] [8.20B]

Bibliography MANDELSHTAM, S. L., Proc. Coll. spectrosc. i n t e r n a t . VI., P e r g a m o n Press L o n d o n 1957, p. 245. M A N D E L S H T A M , S. L., Spectrochim. Acta 15 ( 1 9 5 9 ) 2 5 5 . MEDFORD,

R. D.,

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and

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Proc.

5.

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10

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433.

[8.20c] [8.20D]

[8.20e] [8.20f] [8.20g] [8.20h] [8.20i] [8.20j] [8.20k] [8.201]

[8.20m] [8.20n] [8.20o] [8.20p] [8.20q] [8.20r] [8.20s] [8.20t] [8.20u]

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Bibliography SEATON, M. J . , in: Atomic and Molecular Processes, Academic Press, New York and London 1962, p. 374. SEMENOVA, 0 . P. and Y. Y. KOKHANENKO, IZV. Akad. Nauk SSSR, Ser. Fiz. 14 (1950) 727.

[11.42a] [11.42b] [11.42c] [11.43] [11.44]

SUCKEWEK, S . , P h y s . L e t t e r s 2 5 A ( 1 9 6 7 ) 2 8 4 . STICKEWER, S., P h y s . L e t t e r s 2 5 A ( 1 9 6 7 ) 4 0 5 . VILENSKAYA, T . V . , I z v . V U Z F i z i k a , N o . 6 ( 1 9 6 2 ) 111. VOLKOV, Y A . F . , V . G . DYATKOV a n d I . N . MITINA, T e p l o f i z i k a 3 ( 1 9 6 5 ) 6 5 9 . WALDMEIER, M . , D i e S o n n e n k o r o n a , B i r k h ä u s e r V e r l a g B a s e l 1 9 5 1 .

[11.45] [11.45a]

WILLIAMS, R. V. and S. KAUFMAN, Proc. Phys. Soc. Lond. 75 (1960) 329. WIMEL-PECKER, C., JILA-Report No. 90, April 1967, Joint Inst. Laboratory Astrophysics, Boulder, Colorado. WITTE, H., Z. Physik 88 (1934) 415.

[11.46] [11.46a]

ZAIDEL, A . N . , G . M . MALYSHEV a n d Y E . A . PTITSYNA, Z h . t e k h n . F i z .

(1963) 200.

33

K. Baumgàrtel and G. Wallis

Magnetohydrodynamic Waves (Waves in Ideal Media)

This article is translated from t h e German original without additions.

CHAPTER 1

Introduction

I n t h e first section we shall s t u d y t h e p r o p a g a t i o n p h e n o m e n a of small d i s t u r b a n c e s in a n ideal m a g n e t o h y d r o d y n a m i c m e d i u m . " I d e a l " will refer t o a m e d i u m which is a classical, compressible, a n d dissipationless c o n t i n u u m of infinite electrical c o n d u c t i v i t y . I n t h e presence of a m a g n e t i c field t h e reaction of a c o n t i n u u m of t h i s t y p e t o small d i s t u r b a n c e s is more complicated t h a n in h y d r o d y n a m i c s . A n o r d i n a r y compressible m e d i u m t r a n s m i t s only one t y p e of d i s t u r b a n c e , n a m e l y s o u n d waves which are compression waves. These p r o p a g a t e (locally) isentropically. D i s t u r b a n c e s , which a r e n o t related t o pressure changes, are a t rest r e l a t i v e t o t h e fluid. I n a m a g n e t o - h y d r o d y n a m i c m e d i u m however, n o n c o m p r e s s i v e d i s t u r b a n c e s can also p r o p a g a t e (Alfven-waves). These are b r o u g h t a b o u t b y t h e m a g n e t i c stresses a n d p r o d u c e t r a n s v e r s e oscillations of t h e m a g n e t i c field lines. Compressions of t h e m e d i u m are generally linked w i t h d i s t u r b a n c e s of t h e m a g n e t i c pressure. B o t h can produce similar effects (fast m a g n e t o a c o u s t i c waves o r / - w a v e s ) , or p a r t i a l l y c o m p e n s a t e one a n o t h e r (slow m a g n e t o a c o u s t i c waves or s-waves). T h e a b o v e t h r e e w a v e m o d e s are characterized b y different p h a s e velocities a n d polarizations. T h e m a g n e t i c field produces a s t r o n g a n i s o t r o p y : t h e p h a s e velocities d e p e n d o n t h e direction of p r o p a g a t i o n r e l a t i v e t o t h e m a g n e t i c field direction. Only t h e / - w a v e s can p r o p a g a t e n o r m a l t o t h e direction of t h e m a g n e t i c field. T h e e q u a t i o n s describing these d i s t u r b a n c e s are linear, or m o r e precisely, t h e y comprise a s y s t e m of hyperbolic linear p a r t i a l d i f f e r e n t i a l e q u a t i o n s of t h e first order. This s y s t e m is o b t a i n e d b y linearization f r o m t h e basic m a g n e t o h y d r o d y n a m i c (MHD) e q u a t i o n s (section 2.2). W h e n linearizing it is a s s u m e d t h a t a solution t o t h e basic e q u a t i o n s is k n o w n , i.e., for t h e " u n d i s t u r b e d " s t a t e . T h e p r o p a g a t i o n properties of t h e d i s t u r b a n c e s are essentially d e p e n d e n t on t h e s t r u c t u r e of t h i s s t a t e . T h e simplest relationships are o b t a i n e d w h e n initially considering a u n i f o r m , s t e a d y , infinitely long flow. I n t h i s case we can a s s u m e plane periodic waves, a n d t h e a b o v e t h r e e wave m o d e s will p r o p a g a t e i n d e p e n d e n t l y of one a n o t h e r . P h a s e velocities polarization, a n d g r o u p velocities can be s t u d i e d in a s t r a i g h t f o r w a r d m a n n e r (sections 2 . 3 - 2 . 5 ) . W e shall t h e n discuss t h e reflection a n d r e f r a c t i o n of t h e s e w a v e s a t a p l a n e d i s c o n t i n u i t y f r o n t (section 3.1). T h e d i f f e r e n t w a v e m o d e s t r a n s f o r m i n t o

514

Magnetohydrodynamic Waves

one another: an incident wave in general produces all other modes as reflected and refracted waves. We shall derive the analog of Snell's Law and present a geometrical method for the construction of the angles of reflection and refraction (section 3.2). Upon calculating the amplitudes of the reflected and refracted waves we can reach certain conclusions concerning the stability of MHD-discontinuity fronts (section 3.3). If the disturbed medium is nonuniform, the different wave modes will be linked to one another. (A discontinuity can in this sense be considered as a special type of inhomogeneity). The precise description of the disturbed state is thus given by the linearized equations. Solving these equations is inconvenient and in most cases unnecessary. I t is more convenient to limit ourselves to an approximate description of the propagation process, as for example, is done in geometrical optics or in "geometrical acoustics". Instead of the system of linearized MHD-equations we solve only the relevant characteristic equation, which is determined by the basic solution and no longer depends on the disturbances. We are dealing with a firstorder partial differential equation of the Hamilton-Jacobi type, which describes spatial surfaces changing with time (initial conditions are previously given) (section 4.1). These characteristic surfaces can be physically interpreted (besides being boundaries between disturbed and undisturbed regions) in the following way: 1. They are identical to the so-called weak shock waves or weak discontinuities (surfaces at which the state parameters can undergo small jumps or along which short-time weak pulse disturbances can propagate). 2. They represent the wave surfaces of periodic, local plane waves in the limiting case co -> oo (A —> 0) (the linearized MHD-equations are related to the corresponding characteristic equation in the same way as Maxwell's equations are related to the eikonal equation of geometrical optics). We can construct the wavefronts, which emanate from a surface or point excitation (periodic or pulsed). In general, there are seven such surfaces fronts which propagate independently of one another with different velocities. The characteristic equation itself possesses characteristic properties (the socalled bicharacteristics of the initial system). These determine the spatial curves ("rays") along which the characteristic surface elements move and by means of which we can construct the characteristic surfaces (planes). Their direction at every point corresponds to the direction of energy propagation at that point. As examples the propagation of Alfven disturbances in a dipole field and the propagation of /-waves in the magnetosphere of the earth are discussed. Furthermore, one can obtain additional information on the variation of the excitation intensity (i.e., the amplitudes) of disturbances along the rays.

1. Introduction

515

Among the references listed (p. 589 ff.) the following are of particular importance: J . Bazer and J . Hurley [8], J . Bazer and O. Fleischmann [9], M. Kantorovich [13], as well as the book by A. Jeffrey and T. Taniuti, Nonlinear Wave Propagation with Application to Physics and Magnetohydrodynamics [1], I n several sections we shall make frequent reference to these studies. Further details and comprehensive literature can also be found in them.

CHAPTEE 2

Propagation of Plane Waves in Homogeneous Media

Basic MHD-Equations for Ideal Media

2.1.

T h e basic magnetohydrodynamie equations for ideal media are the following (in rationalized mks-units) rot E

=

div E

=

en

e» Maxwell's equations*

rot B = p i div B = E SQ

+ div

et iëv

(Q V)

{ v grad) 8s

=

0 , -

j =

(2.2)

(2.3)

,

(2.4) (v x B)

,

Ohm's law

(2.5) (2.6)

= 0,

f (».' grad 5) =

it

(2.1)

UT

— grad p + oe E + ( j X B)

hydrodynamic equations

0 ,

(2.7) (2.8)

p = p(g, s)

equation of state

(2.9)

where E, B, j , Qe respectively denote the electronic field strength, magnetic induction, current density, and the electric charge density; p, v, o, s

are respectively the pressure, current velocity, mass density, and the entropy per unit mass;

p, €Q

are the permeability

and the dielectric constant in a vacuum,

respectively. T h e a b o v e system links Maxwell's equations with those of mechanics for the corresponding material continuum. term ( i x B )

T h e connection is essentially due t o the

appearing in the equation of motion (2.7) and Ohm's law (2.5).

T h e latter states that an electric field cannot exist in a coordinate system, which moves with the medium (due to the infinite c o n d u c t i v i t y ) . Equation (2.4) on account of (2.1) is only useful as an initial condition. f r o m (2.6), we are dealing with 16 equations in 16 scalar variables. The displacement current has been left out.

Apart

2. Propagation of Plane Waves in Homogeneous Media

517

The following energy equation is equivalent to (2.8)

where e = e(g, s) is the integral energy per unit mass. E and ; can be eliminated from all the equations so t h a t we are left only with relationships between B and the hydrodynamic and thermodynamic quantities r, o, p, and e, divif = 0 ,

(2.11)

rot ( v x B ) = 0 ,

(2.12)

§ + d i v (e » ) = < > , ct

(2.13)

pii st

o ( J J - \8t

grad) »1 = — grad p + --• (rot B X B), J /,

g + (cgrad«) = 0 , ct

(2.14) (2.15)

p = p(e>«) >

(2-i6)

q. = - e 0 d i v ( » x B ) •

(2.17)

In the equation of motion (2.14) we neglected the term e0(vxB) div (vxB), which results from the term ge E. This is permissible when v2 c2, in which case we have |e0 (rxB)

div

(vxB)\

-1- (rot B x B)

/( \ c

+1

In a system which moves locally with the medium div E — 0, since E = 0. However, the (formal) space charge Qe = — E0 div (« X B) in general does not vanish. The energy equation, equivalent to (2.15), is given by

iff + s e + i) = - divM^+ge)[ rp+i (/ix(^x/i))}• (2-i8) Equations (2.11) —(2.16) represent a quasilinear system of differential equations. We shall write this system in another way and subsequently transform it (except for (2.11)), into the so-called conservation form, which can be applied after determination of the boundary conditions at the discontinuity planes (see section 3.1). dt

+ div Fi = 0 ,

i = 1, . . . , 8 ,

(2.19)

518

Magnetohydrodynamic Waves

where F{ = Fn e1 + Fi2 e2 + Fis e 3 ; F{j = Fi} (aly . . . , a 8 ). (Equations (2.13) and (2.18) are already in this form.) From F{ Q

qv,

Q Vi

(p

+QViVi

+ ±-

(2.20)

6{j -

Bi J 5 } j j e , ,

E (B, Vj - B, Vi) e, ,

(2.21) (2.22) 'S=^

follow the continuity equation, the three components of the equation of motion, the three components of the "induction equation" (2.12), and the energy equation. e1 , e2 , and e3 are unit vectors directed along the axes of a cartesian coordinate system. With the vector

a = («!, . . . , a8 ) = (q, q i>!, q v2 , q v3 , Bl7 B2 , Ba , W)

(2.24)

we can arrive at the following condensed form (2.25)

± + Z A » = 0 . of

where

j= i

/dFij 8c' A, = | : 8Fgj 8a1 '

dXj

ej\j ' 8as : |, SFgj ' 8a_

j = 1, 2,3 .

(2.26)

The "standard form" is similarly obtained. For

u = (g, vlt v2 , v3 , Bv B2 , B3 , S)

(2.27)

we get

The matrices A } and IJ\ are given in the appendix, pp. 582—584. (2.25) and (2.28) are complemented by the expression

Systems

div B = 0 . The quasilinearity lies herein that the coefficients of matrices Aj and Vj are still functions of the a; and «;, respectively. The relationship between the two vectorial representations is given by aj =

. . . , m8) ,

i = 1, . . . , 8 .

(2.29)

Consequently, we have — = M—, 8t 8t

8xj

=

8x]

(2.30)

2. Propagation of Plane Waves in Homogeneous Media

where

519

tda_J

I

8(a"

8 u

:

i '

:

(2.31)

Xa«!'

and

?7} = J/"1 ^ M . * The matrix M is also given in the appendix.

2.2.

Linearization and the Equation for Plane Waves

Let us assume that we know a solution u(x, t) = (g(x, t), . . . , s(x, t)) of the basic equations and that it describes the "undisturbed" state. We shall now investigate the influence of superimposing small disturbances du(x,

t) = ( d g ( x , t), . . . , ds(x, n3) = — , k'

~ C

=

to k'

Lu =

1

iij Uj

we can rewrite the above system in the following way (n SB) = 0 , Ludu

(2.50)

= C du .

(2.51)

\\LU — G I\\ = 0

(2.52)

The secular equation of Lu gives the dispersion relation; the eigenvalues C define the phase velocities of the allowed plane waves. The eigenvectors represent the relevant amplitudes of the disturbed parameters. Calculations of the secular equation and the eigenvectors will be made on the basis of the coordinate system specified by n and B (Figure 2.1): e1 = n ,

ft

nxB \nxB\

t = v(ry,n)=ll~(nli)n '

|B — (n B) n\

.

F i g u r e 2.1. Orientation of the eigenvectors of the nrt-coordinate system 34 *

(2.53) K '

522

Magnetohydrodynamic Waves

In addition, we shall consider the variables in (2.45) in the following order dw = (— dp, dvn, dvt, dbt, — pt ds, dvT, dbr, is composed of longitudinal as well as transverse components. However, with respect to dE, the MA-waves are transverse, i.e., (ndE) = - n(dvxB)

= 0.

since n, dv, and B lie in the same plane.

Figure 2.3 illustrates the orientation of the vectors dv, dB, n, and B for the case (n B) > 0. In accordance with (2.87), we can show that (dvfdvs)=0

and

when

(n B) ^

'

It is possible to derive a relationship between the angles