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Measurement of some relative transition probabilities in singly ionized argon Campbell, Hugh Daniel 1968

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MEASUREMENT OF SOME RELATIVE TRANSITION PROBABILITIES IN SINGLY IONIZED ARGON by HUGH DANIEL CAMPBELL B.Sc, University of B r i t i s h Columbia, 1962 M.Sc., University of B r i t i s h Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the department of PHYSICS We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n -t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada - i i -ABSTRACT Relative t r a n s i t i o n p r o b a b i l i t i e s were measured for several t r a n s i t i o n s in the v i s i b l e A r i l spectrum. An argon-nitrogen pulsed arc of duration 90 psec, with electron 17 -3 densities of 2.3 x 10 cm and electron temperature of 2.6 ev, was used as a source of radia t i o n . Measurements were performed with photographic diagnostics and a ro t a t i n g -mirror shutter system. The fourteen values of r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s obtained agreed reasonably well with previous measurements. - i i i -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i - LIST OF TABLES i v LIST OF ILLUSTRATIONS V ACKNOWLEDGEMENTS v i Chapter I INTRODUCTION 1 1.1 Introduction to the Subject 1 1.2 Introduction to the Present Experiment 3 1.3 Outline of the Thesis 4 Chapter II THEORETICAL CONSIDERATIONS 5 2.1 Local Thermal Equilibrium 5 2.2 Line Emission 6 2.3 Intensity of the Line Emission 8 2.4 Relative Transition P r o b a b i l i t i e s 9 Chapter III EXPERIMENTAL WORK 11 3.1 Introduction 11 3.2 Generation and Dynamics of the Plasma 14 a) Capacitor Bank 14 b) Discharge Vessel 15 c) The Generated Plasma 20 3.3 Diagnostic Equipment 23 a) Optical Equipment 23 b) Light Shutter and Timing System 25 c) Carbon Arc 27 d) Neutral Density Step F i l t e r 29 3.4 Exposure of the Plates, and Development 30 3.5 Measurements on the Plates; the Comparator 32 Chapter IV ANALYSIS AND RESULTS 34 4.1 Processing the Plate Data 34 4.2 D e t a i l s of the Calculations 36 4.3 Relative T r a n s i t i o n P r o b a b i l i t i e s of A r i l 40 4.4 Discussion of Errors 44 Chapter V CONCLUDING DISCUSSION 48 Appendix I PHOTOGRAPHIC DIAGNOSTICS - 52 Appendix II SPECTROGRAPHIC TEMPERATURE MEASUREMENT 57 Appendix III VOIGT ANALYSIS OF SPECTRAL LINES 60 Appendix IV TEMPERATURE PROFILE ERROR 64 Appendix V REDUCTION OF SELF-ABSORPTION 70 BIBLIOGRAPHY 73 LIST OF TABLES T i t l e RELATIVE TRANSITION PROBABILITIES IN Ar EXPERIMENTAL AND THEORETICAL ESTIMATES OF RELATIVE TRANSITION PROBABILITIES IN THE A r i l SPECTRUM STANDARD VOIGT PROFILES REDUCTION OF OPTICAL OF A r i l 4348 THICKNESS -v-LIST OF FIGURES Number T i t l e Page 1 LAYOUT OF BASIC APPARATUS 13 2 SCHEMATIC OF DISCHARGE CIRCUIT 16 3 CROSS-SECTIONAL VIEW OF ELECTROLYTIC RESISTOR 17 4 BANK CURRENT AND LIGHT INTENSITY WAVEFORMS 18 5 PLASMA VESSEL 19 6 SYSTEM FOR GATHERING AND SHUTTERING PLASMA LIGHT 24 A l TYPICAL RESPONSE OF THE PHOTOGRAPHIC EMULSION 53 A2 TYPICAL CASE OF RECIPROCITY FAILURE AT TWO DIFFERENT WAVELENGTHS 55 A3 VOIGT PROFILES 62 A4 ASSUMED TEMPERATURE PROFILE 64 A5 RECIPROCAL DISPERSION OF THE SPECTROGRAPH 67 A6 APPARATUS FOR STEP FILTER CALIBRATION 68 A7 DISPLAYING NON-NEUTRALITY OF STEP FILTER AT DIFFERENT WAVELENGTHS 69 A8 RELATIVE INTENSITY PROFILES OF A r i l 4348 FOR DIFFERENT MIXTURES 72 - v i -ACKNOWLEDGEMENTS I wish to thank Dr. A.J. Barnard for suggesting and supervising the v/ork described i n t h i s thesis. Thanks are also due to Dr. B. Ahlborn for his he l p f u l suggestions i n the writing of thi s thesis, as well as to Dr. J.H. Williamson for his enthusiastic aid. The assistance of Mr. J.T. Dooyeweerd for main-tenance of the el e c t r o n i c s equipment, and of Mr. J. Lees for glassblowing services i s also acknowledged. F i n a l l y I wish to express my appreciation for the many h e l p f u l and stimulating discussions held with members of the plasma physics group. In p a r t i c u l a r I should mention the following names; Mr. R. Morris, Mr. H.G. James, Mr. S.S. Medley, and Mr. C.S. MacLatchy. - 1 -CHAPTER I  INTRODUCTION 1.1 Introduction to the Subject The quantity A.. i s the t r a n s i t i o n p r o b a b i l i t y J between two energy states of an atom E i and E., and i s defined as the pr o b a b i l i t y per unit time that an atom in state i w i l l spontaneously make a t r a n s i t i o n to the state j . Ratios of t r a n s i t i o n p r o b a b i l i t i e s are known as r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s . This thesis des-cribes an experiment which has been performed to measure r e l a t i v e A ^ j 1 s of singly ionized argon gas (referred to as A r i l henceforth). Although measurements of A^ • 's of neutral atoms have been attempted for the past t h i r t y years, i t has been only i n very recent times that accurate and reproducible r e s u l t s have been achieved. Such i s not the case with the measurements for ionized p a r t i c l e s . The technology developed i n the experiments with neutral atoms has not been overly suitable for applications to measure the ionized species, and consequently, measurements on ionized p a r t i c l e s were not commenced u n t i l the early s i x t i e s . OLSEN (1963) was the f i r s t one to take measurements on the A r i l spectrum. Making use of a high current dc arc, and photoelectric diagnostics, he obtained values for fourteen t r a n s i t i o n s , and quoted accuracies of -2-approximately 20%. In 1965 POPENOE and SHUMAKER completed a s i m i l a r experiment (again employing photoelectric diagnostics with a dc arc) and obtained measurements for two of the strongest l i n e s i n the spectrum. Although POPENOE quoted accuracies of 25%, his values were approxi-mately twice as large as those of OLSEN's. In the same year RICHTER (1965) reported a measurement for one t r a n s i t i o n p r o b a b i l i t y of the A r i l spectrum. Using a source and diagnostic technique s i m i l a r to the ones just mentioned, he obtained a value very close to the corresponding one measured by OLSEN. The most recent work on A r i l by BERG (1967) has added another set of data to consider. His l i g h t source was a transient plasma (conical theta-pinch), and using photoelectric devices he measured for ten t r a n s i t i o n s values which were midway between- the previous measurements. In the experiments mentioned above, the experimenters measured absolute values of A — ' s of the A r i l spectrum. The success of these experiments depends c r i t i c a l l y upon the establishment of equilibrium conditions in the l i g h t sources. According to GRIEJVP s c r i t e r i a (1963) for equilibrium (and the quoted values of electron densities) i t would appear doubtful that such was the case i n these experiments (see Section 2.1). This may have been a factor a f f e c t i n g the accuracy of their measurements. -3-1.2 Introduction to the Present Experiment Since our objective i s the measurement of tran-s i t i o n p r o b a b i l i t i e s , we are very concerned with a suitable choice of a l i g h t source. Generally we are lim i t e d to two sources, namely shock tubes and arcs. Considerable, work has been done with shock tubes i n t h i s laboratory (see SIMPKINSON, 1964) and i t was found, that thermal equilibrium conditions were not achieved in these p a r t i c u l a r devices. Consequently, a high current arc sim i l a r to the one described by DURAND (1963) was chosen. The arc was produced by discharging a capacitor bank through a sui t a b l y constructed vessel. The design of the capacitor bank and discharge vessel was such that a f a i r l y uniform high temperature plasma was generated, which lasted for 90 L t s e c . Photographic diagnostics were employed throughout the experiment for measuring the l i n e shapes and r e l a -t i v e i n t e n s i t i e s of the spectral l i n e s . A carbon arc was used for c a l i b r a t i n g the spectral s e n s i t i v i t y of the photographic emulsion. Although, i n p r i n c i p l e , photoelectric devices could have been used, i t was f e l t that the photographic technique was superior when diag-nosing a pulsed source such as described above. -4-1.3 Outline of the Thesis A short summary of the relevant spectroscopic theory i s given i n Chapter II. P a r t i c u l a r attention i s given to thermodynamic equilibrium conditions, as well as to the concept of l i n e emission from atoms and ions. In Chapter III are presented d e t a i l s concerning the apparatus and laborabory techniques used i n t h i s i n v e s t i -gation. The equipment required for analyzing data obtained from our plasma i s discussed, and parameters characterizing t h i s plasma are given. The contents of Chapter IV deal with the analysis employed to determine the f i n a l values of the r e l a t i v e A..'s of the A r i l spectrum. These values are tabulated i n Table I, and a discussion of errors follows. The f i n a l chapter compares the work described i n t h i s thesis with previous experimental work . Values published by OLSEN and BERG are l i s t e d i n Table II, along with two sets of values t h e o r e t i c a l l y determined. CHAPTER II THEORETICAL CONSIDERATIONS The theory of plasma spectroscopy i s f a i r l y well developed i n the l i t e r a t u r e . For example, an excellent review a r t i c l e was published by COOPER in 1966, which to-gether with GRIEM's textbook (1964) f provide a thorough treatment of the theory. In thi s chapter, the important physical concepts pertaining to our experiment are explained. 2.1 Local Thermodynamic Equilibrium The concept of l o c a l thermodynamic equilibrium (LTE) i s one that helps to simp l i f y the very complex problem of determining the nature of emitted radiation from a hot gas. According to GRIEM (1963) t h i s i s defined as follows: " I f the d i s t r i b u t i o n s over the various bound states and i o n i z a -t i o n stages and the v e l o c i t y d i s t r i b u t i o n s of atoms and. ions are as in a thermodynamic equilibrium system of the same temperature, mass density, and chemical composition, then one speaks of complete LTE". This condition can be r e a l i z e d when the r a d i a t i v e processes become n e g l i g i b l e compared with the c o l l i s i o n a l ones i n determining the population densities of excited states of the atoms or ions. Since most processes are dominated by c o l l i s i o n s with electrons, i t i s the electron temperature (T e) which i s used i n the d i f f e r e n t equilibrium r e l a t i o n s (eg. see Equation 2.11). -6-In general, complete LTE i s very d i f f i c u l t to achieve due to the very high electron density requirements. Conse-quently, one often has to be s a t i s f i e d with " p a r t i a l LTE", i . e . only c e r t a i n l e v e l populations w i l l be equal to those i n a thermodynamic equilibrium system. GRIEM (1963) compared the rate of t r a n s i t i o n s induced by electron c o l l i s i o n s from a l e v e l n ( p r i n c i p a l quantum number) to a l l higher l e v e l s n' with the rate of radiative decay of l e v e l n. Postulating that the former should be at least ten times as large as the l a t t e r , he derived from s e m i - c l a s s i c a l arguments that 2.1 i n order to have l e v e l n in p a r t i a l LTE with a l l higher l e v e l s and the electron continuum, where 3 X\e i s the electron density (number of electrons per cm ) 2. i s the e f f e c t i v e charge acting on the ra d i a t i n g electron (equal to 2 for sing l y ionized p a r t i c l e s ) Te i s the electron temperature (units of ev) En i s the io n i z a t i o n potential of hydrogen (units of ev) S t r i c t l y speaking, Equation 2.1 applies only to l i g h t e r hydrogenic ions. However, i t i s generally f e l t that t h i s i s a reasonable approximation for heavier ions as well, i n which case the factor i s replaced by the appropriate i o n i z a t i o n p o t e n t i a l (equal to 15.7 ev for argon). Taking the 1 7 - 3 values of n o = 2.4 x 10 cm and T„ = 2.6 ev ( t y p i c a l values of our plasma) we f i n d that p a r t i a l LTE should hold down to the n = 3 l e v e l . When measuring r e l a t i v e A^j's equilibrium i s only necessary amongst the upper states from which the tr a n s i t i o n s originate. Hence t h i s degree of p a r t i a l LTE i s s a t i s f a c t o r y for our experiment, as a l l of the tr a n s i t i o n s studied here have p r i n c i p a l quantum number n = 4. However i t i s consider-ations such as these that make one doubt that the experiments mentioned i n Chapter I achieved complete LTE. 2.2 Line Emission Consider a system of atoms/ions i n a high temperature region. Depending upon the temperature, there w i l l be d i f f e r -ent densities of atoms e x i s t i n g i n the possible excited states. For a p a r t i c u l a r t r a n s i t i o n , say from upper state i to lower state j , the t o t a l power radiated £;j (per unit volume per unit s o l i d angle) i s given by = >W-Ayn< 2.2 where the frequency of the emitted radiation, i s given by E\. Ej " ? a n c j f\- the number density of p a r t i c l e s i n state i . The quantities , , refer to the energies above the ground state of the states i and j . Due to the di f f e r e n t broadening mechanisms present the emitted radiation w i l l not be monochromatic, but w i l l instead be di s t r i b u t e d about the central frequency, namely Vo . This spread i n f r e -quencies i s characterized by the normalized l i n e p r o f i l e Lev), where -8-Now the spe c t r a l emission i s given by i n units of power per unit volume per unit s o l i d angle per unit frequency i n t e r v a l . 2.3 Intensity of Line Emission When the plasma i s o p t i c a l l y thin (that i s when a l l emitted r a d i a t i o n escapes out of the radiating plasma), the observed i n t e n s i t y I ( V ) of a pa r t i c u l a r t r a n s i t i o n i s d i r e c t l y proportional to the emission. However, i f there i s appreciable absorption of the radiation, then r a d i a t i o n transfer has to be considered (see CHANDRASEKHAR, 1950, and AMBARTSUMYAN, 1958). For a plasma i n LTE, which i s homogeneous over i t s physical length X , the observed i n t e n s i t y w i l l be given by iyw) - B v ( T ) i - e Bvfr) 2.4 where: Lj(v) i s the observed i n t e n s i t y of the emitted radiati o n of the t r a n s i t i o n i to j , i n units of power per unit area per unit s o l i d angle per unit frequency i n t e r v a l SV(T) i s the Planck Function. The o p t i c a l thickness Ty i s given by C V ~ By(T) * * 2.5 -9-and for an o p t i c a l l y thin emission l i n e we have the following condition xv « 1 Hence Equation 2.4 approximates to Ljtv) ^  &ip)l and the t o t a l integrated i n t e n s i t y i s then given by l i n e The problem of determining the i n t e n s i t y p r o f i l e of an o p t i c a l l y thick l i n e i s very complex indeed, and the reader i s referred to COOPER for a detailed treatment. 2.4 Relative T r a n s i t i o n P r o b a b i l i t i e s We now proceed, to show how the above information can be used to obtain values for the r e l a t i v e A..'s. For convenience we refer to the t r a n s i t i o n from state i to state j as t r a n s i t i o n 1, and any other t r a n s i t i o n as t r a n s i t i o n 2. Writing the spectral i n t e n s i t y of t r a n s i t i o n 1 in the form I,(V) L, R(V) 2.8 where I 0|is the peak in t e n s i t y at the l i n e center Vo , and P ( V ) i s the i n t e n s i t y l i n e shape with P(V©) = 1, then the integrated i n t e n s i t y becomes I, ~ (l ,(vJdV = IafRWaV S E I o l di 2.9 2.6 2.7 -10-where (L = mfvVlv i s referred to as the "peak-normalized" area of the i n t e n s i t y p r o f i l e . Note that except for a d i f -ferent s c a l i n g factor, P(V) and L(V) are i d e n t i c a l . Taking the r a t i o s of integrated i n t e n s i t i e s of two l i n e s , and using Equations 2.9, 2.7, and 2.2 we obtain Al = ,1M a. .J4..IU. A Z I 0 2 Q 2 y, r\» 2 - 1 0 which i s the expression for the r e l a t i v e value of A-^  with respect to k,^. When the plasma i s in LTE, the upper state densities of these t r a n s i t i o n s are related by the Boltzmann r e l a t i o n : where: 9»)9zare ^ e s t a t i s t i c a l weights of these lev e l s are the energies above the ground state of these l e v e l s , and £ E — E, — E 2 Using the r e l a t i o n i n Equation 2.10 then gives /4. I01 a, % 2.12 This i s the basic equation used i n t h i s thesis. The values for ^ijV^C} 1;^ a n d c a n b e f o u n d i n standard spectroscopic data tables (see MOORE, 1959). The experimental techniques employed to determine the i n t e n s i t y p r o f i l e s , and t h e i r r e l a t i v e magnitudes w i l l be discussed i n Chapter IV. -11-CHAPTER III  EXPERIMENTAL WORK 3.1 Introduction We have outlined the t h e o r e t i c a l background r e l a t i n g to our experiment - the end r e s u l t of which i s expressed i n Equation 2.12. Recall that use of t h i s expression demands that the source of rad i a t i o n s a t i s f y the following conditions. -At least p a r t i a l LTE must be established. -The source must be o p t i c a l l y thin for the emitted r a d i a t i o n of the relevant t r a n s i t i o n s . A r i l r a d i a t i o n was produced i n a high-current pulsed discharge s i m i l a r to the arcs used by WULFF (1965) and DURAND (1963). Ionization of the argon was achieved by discharging the stored energy of a cha-rged delay-line capacitor bank through a suitable vessel. Figure 1 shows the general layout of the basic apparatus used i n t h i s experiment. Both time-integrated and time-resolved spectroscopy were employed i n the investigation. The movable mirror Mg could be positioned to send l i g h t into the monochromator-photomultiplier arrangement, which was used, to study the temporal behavior of the A r i l r a d i a t i o n . In order to photograph the emitted spectrum during optimum conditions, a rotating-mirror l i g h t shutter was used to admit l i g h t into the spectrograph only in a selected time i n t e r v a l . The A r i l r a d i a t i o n was recorded on spectrographic plates once the synchronization of the l i g h t shutter was completed. When using photographic diagnostics i t i s necessary to determine both the energy response and the spectral s e n s i t i v i t y of the emulsion (see Appendix I ) . A neutral density f i l t e r and a carbon arc were used to achieve these ends. As mentioned e a r l i e r , i t was necessary for the source to s a t i s f y two conditions to be suitable for our experiment. F i r s t , the existence of LTE conditions were reasonably established when measurements of the Stark-broadening of the spectral lines revealed that the electron 17 -3 density was approximately 2.4 x 10 cm . This value was s u f f i c i e n t l y high to ensure that the plasma was a collision-dominated one. Second, the condition r e l a t i n g to the o p t i c a l thickness was met only after a gas mixing technique was used i n the source. D i l u t i n g a r e l a t i v e l y large quantity of nitrogen with a.small quantity of argon (5-14% by volume) ensured, that the ef f e c t of s e l f -absorption was minimized for the argon radiation. The chapter concludes with a description of the experimental conditions and techniques used to obtain the f i n a l data. -13-PLASMA DISCHARGE VESSEL T '"LIGHT SHUTTER I SYSTEM N" STANDARD SOURCE I I 1 L_ X JACO 82-010 HONOCIIROMATOR CZZ3 HILGER E742 SPECTROGRAPH TO OSCILLOSCOPE IP 28 PHOTOMULTIPLIER' FIGURE 1 LAYOUT OF BASIC APPARATUS -14-^•^ Generation and Dynamics of the Plasma aj Capacitor Bank The capacitor bank used to supply energy to the plasma vessel was i n the form of a lumped parameter delay l i n e . In Figure 2 i s given a schematic diagram of the bank and other components of the discharge c i r c u i t . A discussion of the theory of delay l i n e s can be found in MILLMAN and TAUB ( 1 9 5 6 J . The sixteen LC stages, with C = 5 LIF, and L = 2 LIH, resulted i n a c h a r a c t e r i s t i c impedance of approximately 0.6iT for the bank. Inductor c o i l s were constructed by winding copper tubing around a r i g i d l u c i t e support. As the c o i l s were subjected to high electromagnetic forces in the course of a discharge, considerable precautions had to be taken to ensure s u f f i c i e n t strength in these devices. After charging the bank to the f i r i n g voltage (generally 14 kV) the stored energy was then discharged through a spark gap switch i n series with a terminating r e s i s -tor (R T) and the discharge vessel. This switch was of conventional design consisting of two brass electrodes mounted i n a c y l i n d r i c a l container of brass and l u c i t e . A tungsten trigger pin was inserted through a hole in one of the electrodes and the arc was i n i t i a t e d by applying a high-voltage pulse from a THEOPHANIS ( I 9 6 0 ) trigger unit between this pin and the electrode. The lower potential electrode was held at ground potential by means of a 5 MQ. -15-r e s i s t o r (R Q) connected to ground, so that the entire bank voltage was i n i t i a l l y applied across the two electrodes. One of the main d i f f i c u l t i e s encountered i n the construction of t h i s bank was that of finding a suitable terminating r e s i s t o r . This r e s i s t o r was to match the c h a r a c t e r i s t i c impedance of the bank, and be capable of di s s i p a t i n g most of the energy stored i n the bank (for example, at 14 kV, bank energy = 8 kJ). An e l e c t r o l y t i c r e s i s t o r was chosen (see BISHOP and EDMONDS, 1964), with two large copper plates immersed i n a solution of copper sulphate (see Figure 3). After the r e s i s t o r was connected into the c i r c u i t , the variable inductor c o i l s were then i n d i v i d u a l l y adjusted to produce the square current pulse shown in Figure 4a. b) Discharge Vessel An "egg-timer" discharge tube was designed to produce a high density, high temperature plasma. Large aluminum electrodes were cemented into the glass tube with epoxy. Considerable care was taken to prevent the epoxy seal from coming into contact with the gas inside the tube. Figure 5 gives a diagram of thi s discharge vessel. The discharge tube was connected to a vacuum pump and gas metering system. A chamber was also f i t t e d to allow mixing gases from two seperate bottles and using t h i s mixture i n the vessel. The entire vacuum system could be pumped down to a base pressure of about 0.02 Torr FIGURE 2 SCHEMATIC OF DISCHARGE CIRCUIT 50 CM FIGURE 3 CROSS-SECTIONAL VIEW OF ELECTROLYTIC RESISTOR -18-4a 4b 4c 4d 4e Time i n useconds FIGURE 4 BANK CURRENT AND LIGHT INTENSITY WAVEFORMS 4a- Bank Current Waveform 4b- On Axis 4c- 2 mm off Axis 4d- 4 mm off Axis 4e- 6 mm off Axis Relative I n t e n s i t i e s of A r i l 4806 at Different Radial Positions. 25 CM GLASS EPOXY RESIN FIGURE 5 PLASMA VESSEL -20-(or much lower when cold traps were used), with no noticeable pressure r i s e within t h i r t y seconds of turning off the pump. Performance of the tube was found to be very s a t i s f a c t o r y , although after 50 f i r i n g s , "crazing" became quite evident on the inner walls, and after 200 f i r i n g s , contamination of t h i s wall became so high that the tube had to be replaced. c) The Generated Plasma A l l experimental evidence gathered on the plasma strongly indicated that a uniform plasma had been generated i n the central portion of the tube, and t h i s uniformity continued for a s u f f i c i e n t period of time for observations to be made. Evidence for t h i s statement i s given in several ways which w i l l now be discussed. F i r s t , high speed photographic observations were made on the time development of the arc (see NEUFELD, 1966) with a Barr and Stroud framing camera. These studies showed that at low i n i t i a l presures (less than 0.5 Torr) the plasma column displayed, a highly non-uniform structure. However, at the operating pressure of 10 Torr, and within the timing resolution of the framing camera (less than 1 usee), no well defined s p a t i a l structure could be detected. The resultant current waveform (see Figure 4 a ) was completely reproducible within the accuracy available on -21-the Tektronix 551 oscilloscope (approximately +_ 1%). With an i n i t i a l bank voltage of 14 kV, a current of approximately 12 kA was produced which lasted for 90 psec. By adjusting the variable inductor c o i l s , t h i s current was made constant to within 5% over the duration, of the discharge. Temporal behavior of the A r i l l i n e s was analyzed by using a Jarrel-Ash 0.5 m monochromator and an RCA IP28 photomultiplier (see Figure 1). The A r i l l i n e s 4014, 4348, 4426 and 4806 were studied and a l l displayed the same time v a r i a t i o n shown in Figure 4b. The shot-to-shot v a r i a t i o n of the i n t e n s i t y was less than 10%. It i s evident that the l i g h t i n t e n s i t y decreases at 50 usee after arc i n i t i a t i o n -a s a t i s f a c t o r y explanation for t h i s e f f e c t has not yet been determined. As w i l l be described in Section 3.3b a shutter was used which would admit l i g h t into the spectrograph i n the time i n t e r v a l 15-30 usee after the onset of the discharge. Preliminary spectroscopic studies in t h i s time i n t e r v a l showed that the plasma was o p t i c a l l y thick for several of the A r i l s pectral l i n e s . This d i f f i c u l t y was overcome by using a gas d i l u t i o n technique] Mixing nitrogen gas with the argon in the r a t i o s of 10:1 and 3:1 provided e f f e c t i v e d i l u t i o n factors of 20:1 and 6:1 respectively. With these d i l u t i o n r a t i o s the maximum value for the o p t i c a l thick-ness of any t r a n s i t i o n s studied in t h i s experiment was less than 0.02 (see Appendix V). Measurements of the electron density p r o f i l e of -22-our generated plasma were ca r r i e d out by JAMES (1968). He used photographic techniques i d e n t i c a l to those described i n t h i s thesis to determine the true i n t e n s i t y p r o f i l e s of the spectral l i n e s ( i . e . the p r o f i l e s were corrected for instrumental broadening e f f e c t s ) . From the Stark-broadening theory of GRIEM (1964; a theory which predicts the broadening of spectral l i n e s due to the presence of a high density of charged p a r t i c l e s i n the v i c i n i t y of the emitting p a r t i c l e s ) , and from the experimental r e s u l t s of JALUFKA (1966; who found that GRIEM's th e o r e t i c a l predictions had to be scaled by a factor of 2.6 i n order to agree with experimental measurements of electron density and associated broadening), i t was possible to obtain an estimate of the electron density once the half-widths of the true p r o f i l e s had been obtained. JAMES took measurements at four d i f f e r e n t r a d i a l positions with both gas mixtures, and. found that the electron density was constant out to 6 mm o f f - a x i s , and of 17 — o value n e = (2.4 +_ 0.5} x 10 cm °, i n the i n t e r v a l 15-30 psec. An estimate of the electron temperature was obtained by comparing the r e l a t i v e i n t e n s i t i e s of two spectral l i n e s (see Appendix I I ) . These l i n e s were N i l 3995 and NIII 4097. Using the experimental arrangement shown i n Figure 1 (in p a r t i -cular, the monochromator and photomultiplier section), measurements were taken at two r a d i a l positions and gave a temperature of T g = (2,6 + 0.3) ev. Now that the general properties of the l i g h t source have been discussed, we proceed to examine the more s p e c i f i c -23-diagnostic equipment required for the measurement of r e l a t i v e A..'s i n the next section. 3.3 Diagnostic Equipment a) O p t i c a l Equipment Light from the plasma source has to be focussed onto a spectroscopic plate. The o p t i c a l arrangement designed to focus t h i s r a d i a t i o n onto the spectrograph entrance s l i t i s shown in Figure 6. With the axis of the tube l y i n g in the horizontal plane, th i s system gathers l i g h t from a thin h o rizontal s t r i p of plasma, and the positioning of th i s sampled s t r i p can be e a s i l y varied i n the v e r t i c a l d i r e c t i o n . There are four p r i n c i p a l components of t h i s system. Lens L^ (a combination of 2 lenses) focusses the axis of the discharge vessel onto the shutter s l i t S-^ . As the spectrograph s l i t i s v e r t i c a l , and as the axis of the discharge tube i s horizontal, a Dove-prism (P) was used to rotate the image of S-^  .through 90° , thereby enabling l i g h t from the horizontal s t r i p of plasma to be sent d i r e c t l y into the spectrograph. Lens Lr. of the l i g h t shutter focusses the image of S-, onto S, (operation of t h i s shutter i s described i n the following section). The beam deviator G (a thick piece of plane glass) determined the v e r t i c a l position from which l i g h t was c o l l e c t e d . This was accomplished by rotating the deviator about a horizontal axis p a r a l l e l to the axis of the plasma vessel. The CARBON ARC B D l ^2 G M l M2 M 3 P ST T INCANDESCENT BULB DIODE FOR LOGIC C I R C U I T DIODE FOR MONITORING SYNCHRONIZATION GLASS DEVIATOR LEN S SYSTEM LENS STATIONARY MIRROR ROTATING MIRROR REMOVABLE MIRRORS DOVE PRISM S L I T SPECTROGRAPH ENTRANCE SLIT PLASMA DISCHARGE VESSEL LENS FIGURE 6 SYSTEM FOR GATHERING AND SHUTTERING PLASMA LIGHT -25-change i n o p t i c a l path length r e s u l t i n g from this rotation was n e g l i g i b l e . A Hilger E742 prism spectrograph with glass optics was used to photograph the spectrum of the arc. The collimating lens of t h i s instrument was 7.5 cm i n diameter and of f o c a l length 170cm. The resolving power was approximately 10^, and a graph of the r e c i p r o c a l dispersion i s given in Figure A5. The net e f f e c t of the o p t i c a l system was that of sampling a horizontal s t r i p of the plasma, of dimensions 0.5mm thick x 2 mm long. Below are l i s t e d the actual dimensions used to achieve t h i s sampling: T - L X 20 .cm L l - S l 45 cm S l - Mx 30 cm M l - L2 11 cm L2 - M 2 10 cm M 2 " S2 27 cm The width of S-j_ was always 1 mm, while the width of the spectrograph s l i t S 2 varied from 20 u. (for photographing the plasma radiation] to 750 u (when photographing the carbon arc r a d i a t i o n ) . b) Light Shutter and Timing System As previously mentioned, the i n t e n s i t y of the A r i l r a diation was observed to vary considerably i n the course of the discharge. However, investigation of this v a r i a t i o n -26-showed that i n the i n t e r v a l 15-30 usee after arc i n i t i a t i o n the i n t e n s i t y changed by less than 10% (see Figure 4). This r e s u l t led to the conclusion that plasma conditions were s i m i l a r l y f a i r l y constant in t h i s i n t e r v a l , and consequently, i t was decided to sample the plasma radiati o n in t h i s i n t e r v a l . A l i g h t shutter was constructed which would perform t h i s task - the elec t r o n i c s section of which served to synchronize the bank i n i t i a t i o n with the desired viewing time. A rotating mirror, trigger lamp and photodiode, together with an electronic frequency gate are the p r i n c i p a l components of the shutter. As the mirror rotates, the photodiode receives one pulse of l i g h t from the trigger lamp per revolution of the mirror (see Figure 6). These pulses are sent into an elec t r o n i c frequency gate, and when the rate of pulses being sent into the gate i s equal to the required frequency, a pulse i s sent out of the gate into a delay unit. After the necessary delay, the delay unit then sends out a pulse to trigger the Theophanis unit, which in turn f i r e s the bank. The mirror w i l l have rotated through a small angle during t h i s delay, and w i l l then be properly positioned to admit l i g h t into the spectrograph. The reader i s referred to NEUFELD (1966) for a more detailed description of the shutter. Only one s i g n i f i c a n t change was made to NEUFELD's shutter. A d i f f e r e n t lens was used which would focus s l i t S-i d i r e c t l y onto the spectrogaph s l i t S 0 when the -27-ro t a t i n g mirror was i n position. This modification increased the s p a t i a l resolution of the o p t i c a l system, and allowed much more f l e x i b i l i t y i n the choice of the duration of the sampling i n t e r v a l . Adjustment of the system was accomplished by placing a bright lamp d i r e c t l y i n front of the plasma vessel, and mounting an RCA-IP28 photomultiplier i n the image plane of the spectrograph. This photomultiplier gave di r e c t i n f o r -mation on the temporal behavior of the shutter system. F i r s t the trigger lamp and photodiode (see Figure 6) were properly positioned and aligned. Then the frequency gate and delay unit were adjusted. F i n a l adjustments were made to the system by f i r i n g the bank. When data was being recorded on the spectrographic plates the timing was monitored by the pulse from a second photodiode D 2 mounted d i r e c t l y i n front (and s l i g h t l y below) of the spectrograph s l i t . Under normal operating conditions the j i t t e r i n the viewing pulse was less than 1 usee, which was quite acceptable for our purposes. c) Carbon Arc When comparing the radiant i n t e n s i t i e s of a source at d i f f e r e n t wavelengths i t i s es s e n t i a l to have previously determined the spectral response of the recording medium. In our experiment, c a l i b r a t i o n of the spectral s e n s i t i v i t y of the photographic emulsion was accomplished by using a carbon arc. MacLATCHY (1965) of t h i s laboratory had used -28-such an arc and found i t to be s a t i s f a c t o r y for thi s purpose. The c i r c u i t consisted simply of a dir e c t current power supply (150 v o l t s , 15 amps, Sorensen-Nobraton) i n series with the arc and a high current variable carbon r e s i s t o r . The arc was a standard commercial model made by Leybold, and Ringsdorf spectroscopic carbons RW202 and RW401 were used for the anode and cathode respectively. Operating conditions prescribed by NULL and LOZIER (1962) were c a r e f u l l y adhered to, which ensured optimum accuracy from the source. Molecular band radiati o n was observed i n several portions of the v i s i b l e spectrum (see NULL and LOZIER), and these regions were avoided for c a l i b r a t i o n purposes. Referring to Figure 6, i t i s evident that d i f f e r e n t o p t i c a l systems were used to c o l l e c t l i g h t from the plasma and the carbon arc. This technique i s quite acceptable in our measurement of r e l a t i v e A--'s, as we are not measuring absolute values of the l i n e i n t e n s i t i e s . In measurements of absolute A. ,'s. i d e n t i c a l optics for both sources i s very desirable. The lens used to focus l i g h t from the carbon arc into the spectrograph (see Figure 6) magnified the anode spot of the arc (approximately 4 mm diameter) to a size much larger than the height of the spectrograph s l i t . Consequently, only the central uniform region of the luminous anode was e f f e c t i v e in sending radiation into -29-the spectrograph, and a very uniform image could be photographed. As the i n t e n s i t y of r a d i a t i o n from the carbon arc was much lower than the i n t e n s i t y of the plasma spectral l i n e s , s l i t widths of 300-750 LI were used to produce s i m i l a r densities on the plates ( r e c a l l , that a s l i t width of 20 p. was used when photographing the plasma arc spectrum). A neutral density step f i l t e r placed d i r e c t l y i n front of spectrograph s l i t was used when photographing the carbon arc l i g h t , and i s described i n the following section. d) Neutral Density Step F i l t e r A Hilger F1273 neutral density step f i l t e r was used to determine the i n t e n s i t y p r o f i l e s of the spectral l i n e s , and also to compare the peak i n t e n s i t i e s of these p r o f i l e s . The quartz f i l t e r , having six steps of varying transmissions and a clear portion, provided seven c a l i b r a t i o n points from which the usual energy response curves could be measured (see Appendix I ) . The n e u t r a l i t y of the f i l t e r was determined with the experimental arrangement shown i n Figure A6. The system comprises a photomultiplier and monochromator, l i g h t source, chopping wheel and phase sensitive detector, and chart recorder. F i r s t the l i n e a r i t y of . the photomultiplier was examined. Placing a pin-hole card d i r e c t l y i n front of the lamp, and then increasing the distance from the pin--30-hole to the monochromator entrance s l i t i n measured i n t e r v a l s , the photomultiplier signal was found to decrease with the expected inverse square law dependence. Then the transmission of each step of the f i l t e r was measured at 200 A i n t e r v a l s over the region of i n t e r e s t . The accuracy of these measurements was limited by noise i n the system (photomultiplier and chart recorder) and trans-missions could only be determined to 5% for the highest density step ( a l l lower density step measurements were more accurate). This data, shown in Figure A7, was used in place of the manufacturer's s p e c i f i c a t i o n s in a l l experimental c a l c u l a t i o n s . 3.4 Exposure of Plates, and Development We have described our source of rad i a t i o n and given d e t a i l s of the diagnostic equipment used i n this i n v e s t i g a t i o n . We w i l l now outline the procedure employed to obtain a spectroscopic plate. F i r s t the timing and synchronization of the system had to be completed. Second, the bank had to f i r e d several times to obtain the correct exposures for the d i f f e r e n t spectral l i n e s and the carbon arc pattern had to be photographed. F i n a l l y , the plates had to be c a r e f u l l y developed. The timing adjustments have been described i n Section 3.3b. Photodiode Dg, positioned d i r e c t l y in front of the spectrograph s l i t , had to be aligned prior to commencement of the plate exposure. Its sole purpose -31-was to monitor the r e p r o d u c i b i l i t y of the shutter system when photographing the arc spectrum. After the timing adjustments were made, the bank was f i r e d several times and the plate exposed. The number of shots required for the proper exposure of a l i n e depended on the strength of the t r a n s i t i o n . Occasional misfires of the bank were detected by the photo-diode monitoring system; when thi s happened the plates were discarded. The carbon arc spectrum was then photographed. With the neutral density step f i l t e r and s l i t widths of 300-750 u reasonable exposures were obtained aft e r 0.25 - 0.5 seconds. A l l spectroscopic plates analyzed i n thi s i n v e s t i -gation were KODAK-IF plates taken when we were working with the following conditions* i ) Nitrogen:Argon gas mixture of 10:1 I n i t i a l pressure of 10 Torr I n i t i a l bank voltage of 14 kV i i ) Nitrogen: Argon ga.s mixture of 3:1 I n i t i a l gas pressure of 10 Tori-I n i t i a l bank voltage of 14 kV. After the plates had been successfully exposed, they were then c a r e f u l l y developed. To ensure that the spectral l i n e s were not dist o r t e d by the adjacency e f f e c t , the plates were brushed continuously with a camel-hair brush while undergoing development. The standard procedures prescribed in the KODAK MANUAL (1967) were c l o s e l y adhered to. -32-3.5 Measurements on the Plates; the Comparator Once s a t i s f a c t o r y spectrographic plates had been obtained, i t was then necessary to extract the data from them. A Grant spectrum l i n e measuring comparator was used to accomplish t h i s task. The e s s e n t i a l feature of t h i s instrument i s very simple i n p r i n c i p l e ; i t reads out both the p o s i t i o n and average transmission of any desired small portion of the plate. It i s beyond the scope of t h i s thesis to present a d e t a i l e d description of the Grant comparator, and so only a sketch of i t s main features w i l l be given. The reader i s referred to an a r t i c l e by TOMKINS and FRED (1951] i f more information i s required. The plate i s f i r s t positioned on a carriage which can be driven in the d i r e c t i o n of changing wavelength on the plate. Below the carriage i s mounted a lens which focusses l i g h t onto a small area of the plate. The transmitted l i g h t i s then detected by a photoelectric device situated above the carriage. When scanning a l i n e p r o f i l e , transmission readings were recorded along with, and as a function of, carriage position by a Datex CDS-1 system. The output of t h i s system was then sent into an IBM 526 card punch. The accuracy available on t h i s instrument was very high. Transmission readings (incremental steps of 1 u n i t . i n the range 0-999) were found to be reproducible to better than than + 1 % . And the plate carriage could be driven with -33-an incremental precision of better than + 1 (i. As a l l of the A r i l l i n e p r o f i l e s were approximately 100 u. i n width (i.e.. approximately 1 A, with a r e c i p r o c a l dispersion of 10 A/mm; see Figure A5), t h i s precision was s u f f i c i e n t l y accurate for our purposes. In the following chapter we describe the analysis of the data stored on the IBM punch cards. -34-CHAPTER IV  ANALYSIS AND RESULTS The present chapter describes the a n a l y t i c a l techniques employed, to arr i v e at the f i n a l values for the r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s . The density p r o f i l e s of the spectral l i n e s as found from the Grant comparator were then translated into i n t e n s i t y p r o f i l e s by means of a computer program. D e t a i l s of th i s program, and the inherent approximations of i t , w i l l be discussed i n Section 4.1. We then examine the steps of the c a l c u l a t i o n , with emphasis directed towards the use of the p r o f i l e data obtained from the computer. A table of f i n a l values i s next presented. In the error analysis which follows we consider the d i f f e r e n t approximations which were used and give an estimate for the o v e r a l l accuracy of the experiment. 4•1 Processing the Plate Data Analysis of experimental data was greatly aided by using an IBM 7040 computer. A program was used which would translate the l i n e scan data obtained from the plates into r e l a t i v e i n t e n s i t y p r o f i l e s (see NEUFELD, 1966). The transmissions of each step from the neutral density step f i l t e r were punched onto cards, and the -35-seven points obtained were used to plot the H-D curves ( i . e . energy response curves; see Appendix I ) . As the H-D plots from the plasma source and carbon arc were found to be equivalent, the carbon arc patterns were used both for determining the i n t e n s i t y p r o f i l e s , and also for comparing the peak i n t e n s i t i e s of the spectral l i n e s . Relative i n t e n s i t y p r o f i l e s could then be obtained from the density p r o f i l e s by int e r p o l a t i o n on the H-D curves. These p r o f i l e s were then smoothed out by applying a second degree polynomial least square f i t t i n g to consecutive groups of f i v e points of the p r o f i l e . When the computer had vdetermined the best parabolic f i t to the f i v e points nearest the center of the p r o f i l e , a l l ordinates of the p r o f i l e were scaled to make the peak value equal to unity. The computer then calculated the Voigt widths (see Appendix III) i n the following manner. It found three points on each shoulder of the p r o f i l e nearest 0.8I o ( i . e . those points nearest the 80% ordinate of the p r o f i l e ) . A quadratic i n t e r p o l a t i o n was then used to determine the abscissae of the .8I e on both sides of the shoulder. It should be noted that t h i s operation was a repeated and hence unnecessary step, but as i t had no ef f e c t upon the measure-ments the program was l e f t unchanged. The difference between the two abscissae then gave the .8I 0 width. This procedure was repeated at the . 7 I 0 , . 6 I 0 , .1I 0 ordinates thus giving their respective widths. These widths were then normalized to the .5I C width, so that the -36-standard tables of VAN DE HULST and REESINCK (1947, see Table III) could be r e a d i l y employed. 4.2 D e t a i l s of the Calculations Recall Equation 2.12 for c a l c u l a t i n g the r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s , 2.12 It w i l l be seen that i f the temperature i s known (Section 3.2c) the r e l a t i v e A — ' s can be found from a measurement of the r a t i o of the peak i n t e n s i t i e s and the r a t i o of the areas of normalized i n t e n s i t y p r o f i l e s . F i r s t we discuss the measurement of the r a t i o of the areas of the p r o f i l e s . This would appear to a t r i v i a l problem once the i n t e n s i t y p r o f i l e has been obtained from the computer ca l c u l a t i o n s . However, thi s i s not the case, because the s e n s i t i v i t y of the photographic emulsion decreases markedly for weak signals, i . e . for the wings of the l i n e p r o f i l e s . As the p r o f i l e s are predominately Lorentzian, there i s considerable area i n the wings, and so an accurate measurement of the area by numerical integration i s not s a t i s f a c t o r y . Consider the following. If the i n t e n s i t y p r o f i l e has been accurately determined down to the . 1I 0 ordinate, the area remaining in the wings of a pure Lorentzian l i n e represents 25% of that area under the center of the p r o f i l e . -37-S i m i l a r l y , with an accurate p r o f i l e measurement down to . 2 I 0 , the area of the wings represents 43% of that area under the center of the p r o f i l e . Hence, i t i s evident that an accurate measurement of the areas i s impossible by numerical integration, when the photographic technique only gives accurate information down to the .1I 0, or .2I 0 values. Voigt analysis of the l i n e p r o f i l e s was employed to overcome t h i s d i f f i c u l t y (see Appendix I I I ) . Before using t h i s analysis however, one has to be convinced that the p r o f i l e s have a true Voigt character. There are three important l i n e broadening mechanisms to be considered: 1) Stark broadening - due to the presence of a high density of charged p a r t i c l e s i n the v i c i n i t y of the emitting atoms, 2) Doppler broadening - due to the high thermal v e l o c i t i e s of the emitting atoms, 3) Instrumental broadening - a consequence of the lack of i n f i n i t e resolution i n the o p t i c a l components of the measuring system. While the f i r s t two mechanisms s a t i s f y the requirements (being Lorentzian and Gaussian r e s p e c t i v e l y ) , there i s some doubt as to the nature of the instrumental broadening function. NEUFELD (1966) and JAMES (1968) have considered t h i s problem in d e t a i l , and found that a reasonable Voigt f i t could be determined for the instrumental p r o f i l e of the o p t i c a l equipment used in thi s experiment. This being -3 8-the case, one could expect that the i n t e n s i t y p r o f i l e s could be analyzed with the Voigt technique with reasonable accuracy. The Voigt widths printed out by the computer were used to the determine the appropriate Voigt parameters for each p r o f i l e . The areas (X could then be calculated, by using. Equation A.6 from Appendix III, a A . e where (D i s the Voigt parameter of the i n t e n s i t y p r o f i l e , and K i s the f u l l halfwidth of the p r o f i l e . The remaining unknown in Equation 2.12 i s the r a t i o of the peak i n t e n s i t i e s which we w i l l now consider. As described i n Appendix II, r e c i p r o c i t y f a i l u r e of the emulsion was anticipated because the i n t e n s i t y of the carbon arc radiati o n was so much lower than that of the plasma source. To overcome t h i s d i f f i c u l t y , the experimental re s u l t s of BLITZ and WEBB (1948) were used. These measure-ments showed that i f two sources of d i f f e r e n t i n t e n s i t i e s are exposed for d i f f e r e n t times, and i f the r e s u l t i n g densities at two d i f f e r e n t wavelengths are equal for the two sources, then JLLJL- =r JLi£_ A.4 where I|JJ, 1^, are the i n t e n s i t i e s of the plasma source at wavelengths 1,2, and I ) c , l i t are the corresponding i n t e n s i t i e s of the carbon arc at the same two wavelengths. -3 9-In order to f i n d an equal density point on a plate at the two d i f f e r e n t wavelengths for the two sources, i t was necessary to take measurements on the shoulders of the density p r o f i l e s of the spectral l i n e s . As the carbon arc radi a t i o n had been intercepted, by the neutral density step f i l t e r , a wide range of densities was automatically a v a i l -able for t h i s source. The following r e l a t i o n s were then used when working at a p a r t i c u l a r density D-^ : I|,2{o ~ ^1)2 Io,;i ; 4.1 where J 0 ( ^  i s the peak i n t e n s i t y of the i n t e n s i t y p r o f i l e ) at wavelengths 1,2 Xi ip i s that i n t e n s i t y of p r o f i l e 1,2 required to ' produce the density CK\}i r e l a t e s X<?i,2 a n d X.2_jp» , and has the range .2 = °^./Z = 1 , where X| 2C i s ^ n e f u l l i n t e n s i t y of the carbon arc * r a d i a t i o n at wavelengths 1,2 X\ iC i s that i n t e n s i t y of carbon arc radiatio n * required to produce density i s the transmission of the neutral density ; f i l t e r required to reduce l ^ i c - down to l ' ( ^ c 6 Employing Equation A.4 for the density then gives and using Equations 4.1 and 4.2 we obtain Ie>i ~ ti l,c -40-This r e s u l t reduces further to Several values of densities were chosen for each pair of l i n e s to be measured, and an average value of the quantity was determined. i t was then possible to calculate values of the r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s by using Equation 2.12. 4.3 Relative Transition P r o b a b i l i t i e s i n A r i l b i l i t i e s measured in t h i s experiment are displayed i n Table I. Included i n t h i s table are the measurements of the p r o f i l e areas, and the r a t i o s of the peak amplitudes of each t r a n s i t i o n to the peak amplitude of the t r a n s i t i o n A r i l 4380. The t r a n s i t i o n s are designated according to the multiplet convention used by MOORE (1959). Values given under the column t i t l e d I c a , refer to the i n t e n s i t i e s of the carbon arc at the d i f f e r e n t wavelengths. As only the r a t i o s of i n t e n s i t i e s was required, the values of I a are given in a r b i t r a r y units. As the majority of measured tr a n s i t i o n s could be compared d i r e c t l y to the t r a n s i t i o n A r i l 4380, this l i n e was chosen as the reference l i n e . The density p r o f i l e of t h i s t r a n s i t i o n was of intermediate strength; consequently, for both higher i n t e n s i t y and lower i n t e n s i t y p r o f i l e s the Once the quantities had been measured The f i n a l values of the r e l a t i v e t r a n s i t i o n proba-MULT. NO. WAVE LENGTH ( A ) ENERGY OF UPPER LEVEL (ev) DEGENERACY OF UPPER LEVEL (k ± f (%) (units arb.) (units arb. ) A r e l 7 43 80 19.56 2 1.00 1.67 ± 4% 1.29 _ 1.00 7 4267 19.46 6 0.99 ± 3% 1.16 8 1.17 1 0.39 _+ 15% 7 4331 19. 53 4 1.30 3 1.63 2 1. 24 1 0. 59 12 7 4348 19.41 8 4.02 2 1.45 4 1.26 1 0. 80 13 39 4482 21.50 6 0.44 2 1.90 5 1.41 8.5 0.39 16 17 4579 19. 89 2 0.45 4 1. 59 4 1.51 2 0.60 13 31 4590 21.04 6 0.76 1 1.99 4 1. 52 7 0.61 14 31 ' 4610 21.05 8 1.48 2 1.98 5 1.54 7 0.93 15 15 4658 19.72 2 0.50 4 1.91 3 1.60 1 0.76 12 14 4727 19.68 4 0.58 2 1.56 11.5 1.67 1 0.38 17 6 4736 19.18 4 0.76 2 1.78 5 1.68 2 0.46 13 15 4765 19.78 4 0. 54 4 1.98 4 1.71 1 0.46 13 6 4 806 19.14 6 ' 1.48 1 1.77 5 1.75 2 0.60 13 6 4848 19.22 2 0.60 4 1. 54 6 1.80 2 0.68 14 14 4 880 19.60 6 0. 82 1 2.19 5 1. 83 1 0. 52 13 TABLE I RELATIVE TRANSITION PROBABILITIES IN ARII -42-factor ^ ( ^ g Q ^ could r e a d i l y be measured. For f i v e t r a n s i t i o n s (see Table I) ^ ("^go") could not be d i r e c t l y determined. Instead, a two-step process was required: where \ i r e f e r s to those t r a n s i t i o n s which could not be d i r e c t l y compared to A r i l 4380 "X(t\ ref e r s to some other t r a n s i t i o n which could be compared d i r e c t l y to A r i l 4380. For the higher i n t e n s i t y t r a n s i t i o n s A r i l 4348, A r i l 4610, the factor ^('^^gol could be obtained v i a the factor \ 4b%0' ' *"° r i n t e n s i t y t r a n s i t i o n s A r i l 4658, A r i l 4765, A r i l 4848, the factor jf was determined by at least three independent paths (in p a r t i c u l a r V i a ^4Mo) t ^(tt'$o) ? T (^^O ) )• T h e values obtained from these independent paths were then averaged to give the f i n a l value of T'I'J^I'IQJ ' ^ e expected accuracy of the values of r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s i s discussed i n the following section. 4.4 Discussion of Errors When presenting new experimental data on t r a n s i t i o n p r o b a b i l i t i e s i t i s mandatory to give a ca r e f u l error analysis of the measuring technique. Past experience of s c i e n t i s t s i n t h i s f i e l d of research has shown that too much care cannot be excercised when measuring these quan-t i t i e s . The main sources of error can be categorized by the following questions. -43--How suitable i s the plasma generator for such studies? -How accurate i s the standard source, and data recor-ding technique? -How sens i t i v e or appropriate i s the a n a l y t i c a l technique employed? A discussion attempting to answer these questions follows, a) S u i t a b i l i t y of the Plasma Generator A l l experimental evidence obtained strongly indicates that a f a i r l y uniform plasma has been produced. F i r s t , the electron density p r o f i l e measurements showed that n e was constant over most of the cross-section of the discharge vessel (see Section 3.2c). Also, the time-resolved spectro-scopic studies showed that the i n t e n s i t y of the A r i l r a d i a -t i o n did not vary by more than 10% over the sampling i n t e r v a l 15-30 psec. Time-resolved measurements of the electron temperature indicated that the T e p r o f i l e also was constant over most of the cross-section of the vessel. A treatment of the error introduced as a consequence of c o l l e c t i n g l i g h t from the lower temperature region near the vessel walls i s given i n Appendix IV. An upper l i m i t of 2% was the r e s u l t of t h i s c a l c u l a t i o n . A s i m i l a r error i s expected due to the change in temperature of the plasma over the i n t e r v a l during which l i g h t was gathered. The sampling technique was shown to be highly repro-ducible. This statement i s mainly a comment on the j i t t e r time of the e l e c t r o n i c s equipment. Not only was the i n i t i a t i o n time of the arc reproducible to better than +_ 1 u.sec, but -44-also the e l e c t r o n i c s frequency gate was accurate to better than +_ 1 psec. There are two more features of our plasma generator which should be discussed; namely LTE and o p t i c a l thickness. The LTE assumption for lev e l s above p r i n c i p l e quantum number n = 3 was shown to be v a l i d , at least according to GRIEM's c r i t e r i o n (see Section 2.1). Although i t i s d i f f i -c u l t to assess the error which would arise from departures of the population densities from th e i r equilibrium values, i t i s evident that t h i s departure should contribute only a small error to the measurement of r e l a t i v e A ^ ' s (since the upper l e v e l s of a l l t r a n s i t i o n s studied here are so close to each other, see Table I ) . Varying the gas mixtures of argon and nitrogen (by volume) assured that the plasma was o p t i c a l l y thin for the t r a n s i t i o n s studied i n t h i s experiment The appropriate mixture for each t r a n s i t i o n was determined by ensuring that the l i n e p r o f i l e did not a l t e r when a higher d i l u t i o n r a t i o was used (see Appendix V). The maximum value of the o p t i c a l thickness for any t r a n s i t i o n studied here was calculated to be less than 0.02. Accor-ding to BURGESS(1965) the emitted i n t e n s i t y for a plasma with o p t i c a l thickness r c w i l l be lower by a factor (1 - r 0/4) than that emitted by a completely thin plasma. It then follows that our values for the r e l a t i v e A —'s would be i n error by no more than +_ 1% as a consequence of t h i s e f f e c t . -45-b) Standard Source and Photographic Errors The spectral s e n s i t i v i t y of the photographic emulsion was determined by using a carbon arc as a standard source. Many experimenters have studied t h i s source (in p a r t i c u l a r NULL and LOZIER, 1962) and have found that the emitted ra d i a t i o n appears very close to that t y p i c a l of a black-body of (3800 +.10) °K temperature. As only r e l a t i v e i n t e n s i t i e s were measured, the error introduced into the calc u l a t i o n s due to departures from the black-body depen-dence should be less than 2-4%. Di f f e r e n t o p t i c a l components were employed to gather l i g h t from the two sources (see Figure 6). As most of the components were quartz, and as a l l mirrors were front surfaced, i t can be assumed that the e f f e c t of varying transmissions with respect to wavelength i s n e g l i g i b l e , and that the two o p t i c a l paths are v i r t u a l l y the same. This e f f e c t should introduce an error not larger than 1-2%. C a l i b r a t i o n of the neutral density step f i l t e r has been described i n Section 3.3d. The agreement with the manufacturer's s p e c i f i c a t i o n s was found to be good, except for wavelengths below 4200 A, and also for the highest density ( i . e . lowest transmission) step. Although the success of the o v e r a l l experiment depends c r i t i c a l l y upon the accuracy of t h i s c a l i b r a t i o n , the transmissions of each step could only be measured to an accuracy of 5%. Hence the possible error introduced here w i l l be approximately 5%. -46-The errors a r i s i n g from the inherent short-comings of the photographic emulsion are extremely d i f f i c u l t to assess. Reciprocity f a i l u r e was the source of most concern, but experimental evidence indicated that t h i s e f f e c t did not manifest i t s e l f ( i . e . the H-D curves of both the plasma source and the carbon arc were found to be sim i l a r within the experimental accuracy a v a i l a b l e ) . Measurements by SAUVENIER (1960) corrobrated t h i s r e s u l t . He found that there was no r e c i p r o c i t y f a i l u r e for exposures down to 1 0 seconds, for ce r t a i n emulsions. Also, the technique used to analyze the data (see Section 4.2) was such, that i f r e c i p r o c i t y f a i l u r e were present to some degree, i t s ef f e c t would be minimized. Development of the plates was c a r e f u l l y performed, and use of a camel-hair brush reduced the possible errors which might otherwise have arisen as a consequence of the adjacency e f f e c t . Having taken many precautions i n processing the photographic plates, we f e l t that the possible error due to f a i l i n g s of the emulsion had been reduced to a s a t i s f a c t o r y degree. c) S e n s i t i v i t y to Temperature Measurement, and Voigt Analysis. Although an accurate estimate of the temperature was not obtained in t h i s experiment, i t should be stressed that the measurement of r e l a t i v e A.j's i s f a i r l y i n s e n s i t i v e to the temperature measurement. Using Equation 2.12, i t can e a s i l y be shown that an error AT i n measuring T w i l l manifest i t s e l f as an error A ^'//Q i n the following way: -47-A / A 2 T * T The three t r a n s i t i o n s for which t h i s uncertainty could produce an appreciable error are A r i l 4482, 4590, and 4610. With the temperature estimate of T = (2.6 + 0.3) ev, the uncertainty in the t r a n s i t i o n p r o b a b i l i t y should be 6-8%. For a l l other t r a n s i t i o n s t h i s possible error w i l l be less than 2%. Determination of the i n t e n s i t y p r o f i l e areas by Voigt analysis was found to be s a t i s f a c t o r y with the exception of two t r a n s i t i o n s . Averages over many l i n e p r o f i l e s were taken to determine the Voigt parameter p and the halfwidth h of each t r a n s i t i o n examined. The standard deviation i n the area measurement was found to be less than 6% for a l l but two l i n e s . Transitions A r i l 4727 and 4267 had standard deviations of 11% and 7% respectively, primarily because i t was d i f f i c u l t to f i n d reasonable Voigt parameters for these l i n e s . d) Overall Estimate of the Error In the previous three sections we have examined the main sources of error which a f f e c t the accuracy of the measure-ments obtained i n t h i s experiment. The l i m i t a t i o n s of the Grant comparator have been discussed i n Section 3.5, and at most should contribute no more than 1-2% to the error. The expected o v e r a l l accuracy of the values of r e l a t i v e A-j-j's i s given i n Table I. The uncertainty • . . • Jx 100% J A 7 A 2 was calculated by taking the square root of the sum of squares of a l l the aforementioned possible errors. -48-CHAPTER V  CONCLUDING DISCUSSION Relative t r a n s i t i o n p r o b a b i l i t i e s of several t r a n s i t i o n s i n the spectrum of singly ionized argon have been measured. Although t h i s work does not represent the f i r s t attempt at measuring these quantities, the history of such experiments c l e a r l y shows that several measurement using d i f f e r e n t techniques and sources are required i n order to obtain r e l i a b l e values for these atomic constants The experiment described i n th i s thesis not only employed a d i f f e r e n t diagnostic technique, but also made use of a d i f f e r e n t source of radiati o n than that used by others. POPENOE and SHUMAKER (1965) have made ca r e f u l measurements of the absolute t r a n s i t i o n p r o b a b i l i t i e s of two A r i l l i n e s , and RICHTER (1965) has obtained a measurement of the absolute t r a n s i t i o n p r o b a b i l i t y for one A r i l l i n e . Since we have only measured r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s i t i s not appropriate to compare th e i r values with ours. OLSEN (1963) and BERG (1967) obtained values for several of the A r i l t r a n s i t i o n s , and these are l i s t e d i n Table II. A discussion comparing thei r diagnostic techniques and sources employed with our technique and source w i l l now be given. Both BERG and OLSEN made use of photomultipliers to gather data. While these devices possess highly linear -49-EXPERIMENTAL THEORETICAL VALUES VALUES MULT. WAVE- THIS BERG OLSEN GARSTANG STATZ LENGTH, A ' et a l NO. EXPT. (1967) (1963) (1954) (1965) 7 4380 1.00 1,00 1.00 1.00 1.00 7 4267 0.39 0.63 0.60 7 4331 0.59 0.47 0.53 7 4348 0.80 0.95 0.99 1.14 1.14 39 4482 0.39 17 4579 0.60 0.64 0.009 0.69 31 4590 0.61 0.56 0.56 0.89 31 4610 0.93 1.00 0.86 0.96 15 4658 0.76 0.76 0.60 0.01 0.62 14 4727 0.38 0.27 6 4736 0.46 0.54 0.33 0.52 15 4765 0.46 0.47 0.63 0.59 6 4806 0.60 0.76 0.68 0.70 0.72 6 4848 0.68 0.44 0.08 0.71 0.69 14 4880 0.52 0.59 0.57 0.69 0.73 TABLE II EXPERIMENTAL AND THEORETICAL VALUES OF RELATIVE TRANSITION PROBABILITIES IN THE ARII SPECTRUM - 5 0 -response c h a r a c t e r i s t i c s , they are not as convenient as our technique for measurements of t r a n s i t i o n p r o b a b i l i t i e s . Our technique exploited the photographic emulsion as a recording medium. The panoramic feature of photography i s p a r t i c u l a r l y suitable for studying a pulsed source -information for several t r a n s i t i o n s i s simultaneously recorded. Having to cope with the non-linear character-i s t i c s of the emulsion i s more than compensated when the benefits derived from the ease of gathering experimental data are considered. As a source of radiatio n OLSEN used a high current atmospheric dc arc. The a t t r a c t i v e feature of thi s source i s that f l u c t u a t i o n s of plasma parameters can be reduced v i r t u a l l y to zero. However, the r a d i a l temperature depen-dence i s d i f f i c u l t to deal with, and the assumption of LTE i s open to some doubt. BERG, on the other hand, u t i l i z e d a highly transient plasma source; namely, a conical theta-pinch. It i s possible that the s p a t i a l and temporal variati o n s in the pinched plasma could be d i f f i c u l t to analyze. * Our experiment was performed on a source which was a compromise between the previously.mentioned ones. Although the r a d i a t i o n was generated on a pulsed basis, the duration of the discharge was s u f f i c i e n t l y long to ensure that steady conditions had been achieved. The c o n s t r i c t i n g walls of the discharge vessel served to produce a plasma with high s p a t i a l uniformity, thereby eliminating the need -51-for s p a t i a l unfolding which i s inherently inaccurate. In addition to the experimental investigations described there are two reports i n the l i t e r a t u r e of t h e o r e t i c a l calcu-l a t i o n s . The values of r e l a t i v e t r a n s i t i o n p r o b a b i l i t i e s obtained i n these works - GARSTANG (1951) and STATZ et a l (1965) - have been included i n Table II. The purpose of presenting these values.here i s to enlighten the reader as to the present state of agreement between t h e o r e t i c a l and experimental work. The agreement, except for is o l a t e d cases, i s not overly s a t i s f a c t o r y . It i s hoped that the values obtained i n our experi-ment w i l l be of use, both i n future t h e o r e t i c a l c a l c u l a t i o n s , and also as diagnostic aids for other experimentalists s t u d y i n g various plasma sources. -52-APPENDIX I PHOTOGRAPHIC DIAGNOSTICS Of the many references which could be given on the subject of photography, probably MEES and JAMES (1966) covers the entire range of topics most adequately. This book gives many experimental findings, and also presents a very compre-hensive treatment of the th e o r e t i c a l aspects of the subject. Only those topics which are of intere s t to the present thesis w i l l be discussed here. When comparing the i n t e n s i t i e s of d i f f e r e n t spectral l i n e s by photography, there are e s s e n t i a l l y two non-l i n e a r i t i e s which must be considered. The highly non-l i n e a r response of the emulsion to varying i n t e n s i t i e s must be determined; and also the s e n s i t i v i t y of the emulsion to d i f f e r e n t wavelengths has to be accounted f o r . The energy response of the emulsion i s best displayed on a H-D plot (after HURTER and DRIFFIELD) - a t y p i c a l curve i s shown in Figure A l . The H-D plot i s read i l y obtained by using a neutral density step f i l t e r . This f i l t e r i s placed v e r t i c a l l y i n front of the s l i t of the spectrograph and provides a convenient method of varying the exposure. The exposure of l i g h t transmitted through the i ^ * 1 step of the f i l t e r i s then A . l where i s the transmission of that step (see Figure Al). -53-P log (E) FIGURE A l TYPICAL RESPONSE OF THE PHOTOGRAPHIC EMULSION D i s the density (an approximate measure of the density of developed grains i n the emulsion) and i s mathematically defined by 16 i s the in t e n s i t y of l i g h t transmitted through an unexposed portion of the spectroscopic plate i s the i n t e n s i t y of l i g h t transmitted through the exposed and developed portion of the plate E i s the exposure (a measure of the radiant energy to which the emulsion i s exposed to) and defined Then, by scanning the plate i n the v e r t i c a l d i r e c t i o n (the d i r e c t i o n of increasing or decreasing T^ of the d i f f e r e n t steps of the f i l t e r ) the corresponding density D^ can be measured. The seven points obtained with the f i l t e r (see Section 3.4) can then be used to plot the H-D curve. A.2 -54-When l i g h t from an emission l i n e i s recorded on a photographic plate, the f i r s t step i n determining the inten-s i t y p r o f i l e of t h i s l i n e i s to scan the plate with some instrument (see Section 3.5) to obtain the density p r o f i l e of that l i n e . Then, by means of the H-D plot, the density p r o f i l e can be translated into a r e l a t i v e i n t e n s i t y p r o f i l e of the recorded l i n e . If the i n t e n s i t y of the l i g h t used to determine the H-D plot i s accurately known, then the density p r o f i l e can be translated d i r e c t l y into an absolute . i n t e n s i t y p r o f i l e . As we were concerned with r e l a t i v e measurements i n t h i s experiment, i t was not e s s e n t i a l to determine the i n t e n s i t y of the l i g h t used to c a l i b r a t e the energy response of the emulsion. However, as l i n e s of d i f f e r e n t wavelengths were being compared, i t was necessary to determine the spectral s e n s i t i v i t y of the emulsion. This was accomplished by using a carbon arc, which could be used to calculate the r e l a t i v e i n t e n s i t i e s of radiat i o n in d i f f e r e n t regions of the spectrum. There i s a serious deviation from the H-D plot to which the photographic emulsion i s subject to, known as r e c i p r o c i t y f a i l u r e (referred to as RF henceforth). It can be seen from Figure A l that I and t are r e c i p r o c a l variables; i . e . halving I and doubling t should produce the same density i n an emulsion as an exposure equal to the product of I and t - f a i l u r e to behave i n t h i s manner i s known as RF. -55-As the in t e n s i t y of the carbon arc radiation was much lower than the plasma radiation, t h i s could have been a serious problem to deal with. BLITZ and WEBB (1948), and WEBB (1933) obtained experimental r e s u l t s which could be used to overcome t h i s d i f f i c u l t y . E s s e n t i a l l y , they found that RF i s approximately constant at a l l wavelengths, as shown i n Figure A2. The RF curves for d i f f e r e n t wavelengths were found to be p a r a l l e l to within 1% over a wide range of time scales. -p log (t) FIGURE A2 TYPICAL CASE OF RECIPROCITY FAILURE AT TWO DIFFERENT WAVELENGTHS (the emulsion density i s constant along both curves) This information can be used i n the following manner. Consider t and tQ to be the exposure times of the plasma source and carbon arc respectively. From the paral l e l i s m of the two constant density curves we have -56-where I| C , I 2 C are the i n t e n s i t i e s of radiatio n at two d i f f e r e n t wavelengths of the carbon arc, and I,p , I2^> are the corresponding i n t e n s i f i e s of radiation from the plasma source. It then follows that A . 4 This r e s u l t was used i n the analysis of experimental data to compare the r e l a t i v e values of the peak i n t e n s i t i e s of the sp e c t r a l l i n e s . -57-APPENDIX II SPECTROSCOPIC TEMPERATURE MEASUREMENT To speak of the temperature of a plasma, i s to imply that there i s some degree of thermal equilibrium present. As the time required for a gas of electrons to achieve a Maxwell-Boltzman d i s t r i b u t i o n in the i r energies i s of the order of I O - 1 1 seconds (see SPITZER, 1956), i t i s almost always meaningful to speak of an electron temperature T , which characterizes the d i s t r i b u t i o n . As described in Section 2.1, i t i s a consequence of the high numbers of c o l l i s i o n s between electrons and ions/atoms that the concept of LTE i s v a l i d . When such i s the case, i t i s T e which characterizes the population densities of the excited states of the atoms/ions. The most common method of obtaining a temperature estimate of a plasma i s by spectroscopic techniques. These techniques generally require a measurement of the i n t e n s i t i e s of d i f f e r e n t s p e c t r a l l i n e s . In order to extract a value for T , some knowledge of the population densities of the rad i a t i n g atomic l e v e l s i s necessary - t h i s i s where LTE i s so useful, because then the densities of d i f f e r e n t states are related v i a the Boltzman d i s t r i b u t i o n (Equation 2.11). Consider the r a t i o of integrated i n t e n s i t i e s of two spectral l i n e s as expressed i n Equations 2.9, 2.10: i = _AL AL HJ. The i n t e n s i t i e s I, , I^ are now i n t e n s i t i e s of l i n e s emitted by d i f f e r e n t i o n i c species (eg. subscript 1 may refer to -58-the neutral species, and subscript would then refer to the singl y ionized species). Boltzman's equation may then be used to give where N_, N r e f e r to the t o t a l number densities of p a r t i c l e s i n species 1,2 respectively, and Z i , Z2, are the p a r t i t i o n functions of species 1,2 respectively. Saha's equation i s required to r e l a t e and where E(p i s the i o n i z a t i o n potential of species 1 (that energy required to remove an electron in the ground state of species 1 into the continuum), mg i s the mass of the electron, and h i s Planck's constant. These three r e l a t i o n s can then be used to give the r e s u l t ,43<t{ This equation was used to obtain an estimate of T , by measuring the i n t e n s i t i e s of two nitrogen l i n e s . It was not possible to perform the measurement with the argon l i n e s , because only A r i l l i n e s were present. By measuring the i n t e n s i t i e s of the l i n e s N i l 3995 and NIII 4097 with A.5 -59-a monochromator and photomultiplier arrangement (see Figure 1) the value of T e = (2.6 +_ 0.3) ev was obtained. The value of T g was found to be constant at the d i f f e r e n t r a d i a l positions at which the temperature measurement was performed. It i s i n t e r e s t i n g to examine the s e n s i t i v i t y of t h i s measuring technique. Rewriting Equation A.5 in the form and d i f f e r e n t i a t i n g T e with respect to R, we obtain ATe „ f i + <-5 R/Te + J09^ \ ( 1.5 4 B 4 I 0 3 R / R This r e s u l t then approximates down to AT e ~ J _ A R T e ~ 7 Bs for the conditions i n our plasma, when performing the N i l , NIII r e l a t i v e i n t e n s i t y measurement. The uncertainty i n R i s p r i -marily a consequence of the uncertainties present i n the meas-urement of ^ /iz, and also i n the values of the j ' s used i n th i s c a l c u l a t i o n . However, i t i s evident that even with 100% uncertainty i n the measurement of R, there would only be 14% uncertainty i n the value of T g obtained. It was f e l t that R could be measured with an error less than 100%, and so the value of T e = (2.6 +_ 0.3) ev was the r e s u l t of these measurements. It must also be mentioned here, that use of Saha's equa-tion and Boltzmann's equation (when used down to the ground, state) require that complete LTE be present i n the plasma. This was not expected to the case as discussed i n Section 2.1, however i t was not possible to obtain a better estimate of T than by the technique described above. - 6 0 -APPENDIX III VOIGT ANALYSIS OF SPECTRAL LINES Voigt functions are defined as the convolution i n t e -grals between Lorentzian (or Cauchy) and Gaussian functions. Consider the general convolution i n t e g r a l of two functions Ux) and -F (*) which defines another function f W by If both and P are Gaussian functions, of the form where c and ^ z are constants, then 4- i s also Gaussian, and the parameters ^ of the three functions f/P^ are related by S i m i l a r l y , i f both -p' and -f" are Lorentzian functions where c a n d a r e constants, then -p i s also Lorentzian, and the paramters are related by f . = f! + Referring to the convolution i n t e g r a l and our f i r s t d e f i -n i t i o n , i t follows that the in t e g r a l which defines the Voigt function i s where M i s constant. The functions ^ 6 ^ ) and <•?(*) can be thought of as extreme examples of Voigt functions, and as the function defined by the convolution of two of either of these functions i s -61-i t s e l f a Gaussian of Lorentzian function, i t can be shown that the convolution of two d i f f e r e n t Voigt functions i s also a Voigt function. The parameters of these Voigt functions then s a t i s f y the r e l a t i o n s $ - V + Figure A3 displays three Voigt p r o f i l e s , and serves to define the parameters h and c, which are the f u l l h a l f -width and peak value of the p r o f i l e respectively. The area (\ under the curves of the p r o f i l e s can be obtained by using numerical integration techniques on the i n t e g r a l J_oO Taking t h i s value of Ck , another parameter p of a Voigt p r o f i l e can be defined by the r e l a t i o n (X = f>hc, A.6 VAN DE HULST and REESINCK (1947) have provided tables (see Table III) giving values of p for a l l p r o f i l e s which have Voigt character. To determine t h i s parameter for a p a r t i c u l a r p r o f i l e however, i t i s necessary to perform a Voigt analysis on that p r o f i l e . The analysis proceeds i n the following manner. The widths of the p r o f i l e are measured at the .8, . .1, .05, .02 ordinates ( i . e . at the ordinates .8c, .7c, .. .02c). Then, after normalizing these widths to the .5 ord nate width, Table III can be r e a d i l y used to determine p. The normalized widths are then referred to as the Voigt widths of the p r o f i l e . -62 FIGURE A3 VOIGT PROFILES: G = Gaussian Type L = Lorentzian Type V = Intermediate Type The preceding analysis was applied to the spectral l i n e p r o f i l e s determined in the experiment described i n th i s t h e s i s . As i t was only the area of the l i n e p r o f i l e which was required, i t i s evident that only p and h had to be evaluated to achieve t h i s end ( a l l p r o f i l e s were normalized so that c = 1) Parameters Ordinates in Terms of Central Ordinate P 2 A P 2/h 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0.01 Widths in Terms of Half-width (b,./h) 0.000 0.025 0.050 0.075 0.100 •0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.00 0.04 0.09 0.14 0.19 0.24 0.30 0.36 0.43 0.51 0.59 0.69 0.79 0.92 1.07 1.26 1.50 1.83 2.38 3.54 o O 0.60 0.59 0.57 0.55 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.35 0.33 0.30 0.27 0.23 0.19 0.13 0.00 0.36 0.34 0.32 0.31 0.29 0.27 0.25 0.23 0.21 0.20 0.18 0.16 0.14 0.12 0.11 0.09 0.07 0.05 0.04 0.02 0.00 1.06 1.08 1.11 1.13 1.16 1.18 1.20 1.23 1.25 1.28 1.30 1.33 1.35 1.38 1.40 1.43 1.45 1.48 1.51 1.54 1.57 0.57 0.56 0.56 0.56 0.56 0.56 0.55 0.55 0.55 0.54 -0.54 0.53 0.53 0.53 0.52 0.52 0.52 0.51 0.51 0.51 0.50 0.72 0.72 0.71 0.71 0.71 0.71 0.71 0.70 0.70 0.70 0.70 0.69 0.69 0.68 0.68 0.68 0.67 0.67 0.66 0.66 0.66 0.86 0.86 0.86 0.86 0.86 0.86 0.85 0.85. 0.85 0.85 0.84 0.84 0.84 0.84 0.84 0.83 0.83 0.83 0.82 0.82 0.82 1.00 1. 1. 1. 1. 1. 1. 1. 00 00 00 00 00 00 00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1. 1. 1. 1. 1. 00 00 00 00 00 1.15 1.15 1.15 1.16 1.16 1.17 1.17 1.17 1.18 1.18 1.18 1.19 1.19 1.19 1.20 1.20 1.21 1.21 1.22 1.22 1.22 1.32 1.33 1.33 1.33 1.34 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.44 1.45 1.47 1.48 1.50 1.52 1.53 1.52 1.53 1.54 1.56 1.57 1.59 1.60 1.62 1.64 1.66 1.68 1.71 1.74 1.77 1.81 1.85 1.88 1.92 1.96 1.98 2.00 1.82 1.84 1.87 1.90 1.94 1.98 2.02 2.06 2.10 2.15 2.19 2.24 2.29 2.34 2.40 2.46 2.54 2.64 2.74 2.87 3.00 2.08 2.12 2.19 2.25 2.34 2.42 2.54 2.64 2.-75 2.87 2.98 3.12 3.26 3.39 3.54 3.70 3.85 4.00 4.13 4.25 4.36 2.38 2.49 2.63 2.79 3.00 3.24 3.52 3.80 4.14 4.44 4.73 5.03 5.32 5.57 5.83 6.07 6.30 6.55 6.76 6.92 7.00 2.58 2.82 3.13 3.56 4.08 4.58 5.05 5.50 5.96 6.40 6.78 7.15 7.52 7.86 8.21 8.55 8.86 9.18 9.50 9.77 9.95 TABLE III STANDARD VOIGT PROFILES I to i -64-APPENDIX IV TEMPERATURE PROFILE ERROR Throughout the analysis of data i n thi s experiment we have assumed a constant temperature p r o f i l e out to the walls of the discharge tube. However, there has to be a region near the walls i n which the temperature drops very r a p i d l y to room temperature (or a value close to room temperature). The radiatio n emitted from t h i s lower temperature region w i l l be characterized by that lower temperature, and w i l l contribute to the error i n the measure-ment of the r a t i o s of the i n t e n s i t i e s of the spectral l i n e s (see Equation 2-12). The following analysis shows that t h i s e f f e c t i s n e g l i g i b l e . Assume the temperature p r o f i l e to have the form shown i n Figure A4. For an o p t i c a l l y thin, but non-uniform plasma as shown i n Figure A4, Equation 2.4 reduces to Radial distance from axis FIGURE A4 ASSUMED TEMPERATURE PROFILE -65-Integration over the l i n e p r o f i l e then gives the integrated i n t e n s i t y : Taking the r a t i o of i n t e n s i t i e s of two l i n e s then gives li £ * r z + J* £ a( r)dr Factoring, and expanding th i s expression subject to the condition (r 0 - r, ) <^  r, then gives In order to avoid unnecessary mathematical complications at a l a t e r stage i n the c a l c u l a t i o n , i t i s convenient to i n t r o -duce another approximation at th i s time. The upper l i m i t of the two i n t e g r a l s can now be replaced by the value rz~ ( ri + *v)/2. This i s equivalent to stat i n g that there w i l l be no A r i l r a d i a t i o n emitted from the plasma for temperatures less than T c/2. For our case T c = 2.6 ev, and the approximation i s j u s t i f i e d because the temperature of 1.3 ev i s too low to cause s i g n i f i c a n t population densi-t i e s i n the upper states of the A r i l ions. ~~ We now make use of Equations 2.2, 2.11 to obtain the following expression for r u r n i C r -66-i V, where n 2 c i s the population density of the upper state of t r a n s i t i o n 2, i n the constant temperature range. If we now assume that n 2 ( r ) has the same r a d i a l dependence as the temperature p r o f i l e ( i . e . constant value of n 2c out to r, , and then decreasing l i n e a r l y to zero at the walls), then can be rewritten i n the form: f AT _ 'AX- i j dr' where r' has the range 0 = r' = Ar, and ^ r = r 0 - r, This expression further reduces to Hi = f f {.125+4-04 + } Spectroscopic measurements revealed that the temper-ature p r o f i l e was constant out to the r , = 6 mm radius, and so with r 0 = 7 mm (see Figure 5) -r- = 7. The maximum value ' 1 0 of AE was 1.94 ev (see Table I) for the A r i l 4482 t r a n s i t i o n and so ^ = .02. This represents a 2% correction factor which i s n e g l i g i b l e compared to the accuracy obtained i n t h i experiment. For the remaining t r a n s i t i o n s studied i n our invest i g a t i o n ^ would be approximately 1/10 of the above value. -67-JACO 82-010 MONOCHROMATOR NEUTRAL DENSITY STEP ^FILTER -t-E.M.I. 9558 B PHOTOMULTIPLIER D DIODE LIGHT H CHOPPER PHASE SENSITIVE DETECTOR I CHART RECORDER SLIT G.E. OSCILLOGRAPH TUNGSTEN RIBBON LAMP FIGURE A6 APPARATUS FOR STEP FILTER CALIBRATION - 6 9 -0.6 0.4 •2 0-2 cn CO •H s cn a a u 0.1 0.06 J _ 3800 4000 4200 4400 4600 4800 Wavelength i n Angstroms FIGURE A7 DISPLAYING NON-NEUTRALITY OF STEP FILTER - AT DIFFERENT WAVELENGTHS _ Manufacturer's Specifications K # Experimental Measurements 5000 -70-APPENDIX V REDUCTION OF SELF-ABSORPTION The degree of self-absorption for any p a r t i c u l a r t r a n s i t i o n can be estimated by c a l c u l a t i n g the o p t i c a l thickness (see Equation 2.5). Of pa r t i c u l a r interest i s the value of the o p t i c a l thickness at the center of the l i n e , t0. When working with argon i n the discharge vessel To was calculated, to be approximately (0.8 +_ 50%) for the stron-gest l i n e s ( A r i l 4348 and A r i l 4806). These values were calculated for the plasma conditions T e = 2.6 ev and 17 -3 n e = 2.4 x 10 cm , together with OLSEN's (1963) values for the absolute t r a n s i t i o n p r o b a b i l i t i e s , and JAMES's (1968) estimates for the half-widths of the p r o f i l e s . The observed integrated i n t e n s i t y of such a s e l f -absorbed l i n e would then be decreased by a factor (1 - ) according to BURGESS(1965), and the measured t r a n s i t i o n p r o b a b i l i t y would s i m i l a r l y be reduced by approximately 20%. As stated i n Section 3.2c, a gas d i l u t i o n technique was used to reduce the self-absorption of the emitted radia-t i o n . The e f f e c t i v e number of absorbing p a r t i c l e s - and hence Te- was then reduced according to the pa r t i c u l a r d i l u -t i o n r a t i o being used. Table IV gives approximate values for X> for the the t r a n s i t i o n A r i l 4348 at three d i f f e r e n t mixtures. -71-Mixing Ratio Optical Thickness (N 2:Ar) at Line Center 0:1 0.8 10:1 0.04 20:1 0.02 TABLE IV REDUCTION OF OPTICAL THICKNESS OF A r i l 4348 The e f f e c t of self-absorption on a l i n e p r o f i l e i s shown in Figure A5. It i s evident that the p r o f i l e s taken with the mixtures 20:1 and 10:1 are very s i m i l a r , while that of the mixture 0:1 i s apparently much wider. It should be stated that these p r o f i l e s represent averages over several p r o f i l e s . For the weaker tr a n s i t i o n s studied, i t was only necessary to go to the d i l u t i o n r a t i o 3:1 to ensure that the e f f e c t s of self-absorption were n e g l i g i b l e . -72-RELATIVE INTENSITY PROFILES OF A r i l 4348 FOR DIFFERENT MIXTURES -73-BIBLIOGRAPHY AMBARTSUMYAN, V.A. (1958), "Theoretical Astrophysics", Oxford; Pergamon Press. BERG, H.F. and ERVENS, W. (1967), Z. Physik 206, 184. BISHOP, A.E. and EDMONDS, G.D. (1965), Plasma Physics ( J . Nuc. Energy-C) T_, 423. BLITZ, M. and WEBB, J.H. (1948), J. Opt. Soc. Am. 52, 1156. BURGESS D.D. and COOPER, J. (1965), Proc. Phys. Soc. J35, 1261 CHANDRASEKHAR, S., (1950), "Radiative Transfer", New York; Dover. COOPER, J. (1966), Rep. Prog. Phys. 2£, 35. DURAND, J. (1963), Z. Naturforsch 18a, 281. GARSTANG, R.H. (1954), Monthly Notices Hoy. Soc. 114, 118. GRIEM, H.R. (1964), "Plasma Spectroscopy", McGraw-Hill. GRIEM, H.R. (1963), Phys. Rev. 131, 1170. JALUFKA, N.W., OERTAL, G.K., and OFELT, G.S. (1966), Phys. Rev. Letters li6, 1073. JAMES, H.G. (1968), Ph.D. Thesis, Univ. of B r i t . Col. KODAK (1967), "Kodak Plates and Films for Science and Industr Kodak Publications P-9, Rochester; Eastman Kodak. MacLATCHY, C.S. (1965), M.Sc. Thesis, Univ. of B r i t . Col. MEES, C.E.K. and JAMES, T.H. (1966), "The Theory of the Photographic Process", New York; MacMillan. MILLMAN J. and TAUB, H. (1956), "Pulse and D i g i t a l C i r c u i t s " , McGraw-Hill. MOORE, C.E. (1959), "A Multiplet Table of Astronomical Interest", N.B.S. Tech. Note 36 NEUFELD, C.R. (1966), Ph.D. Thesis, Univ. of B r i t . Col. NULL, M.R. and LOZIER, W.W. (1962), J. Opt. Soc. A.m. 52, 1156 OLSEN, H.N. (1963), J. Quant. Spectr. Rad. Transfer 3, 59. -74-POPENOE, CH. and SHUMAKER, J.B., J r . (1965), J. Res. N.B.S. 69A, 495. RICHTER, J. (1965), Z. Astrophys. 61_, 57. SAUVENIER, H. (1960), Proc. 5th Int. Congress High Speed Photography B-3, 72. SIMPKINSON, W.V.S. (1964), Ph.D. Thesis, Univ. B r i t . Col. SPITZER, L. J r . (1956), "Physics of F u l l y Ionized Gases", Interscience j3. STATZ, H., HORRIGAN, F.A., KOOZEKANANI, S.H., TANG, C.L., and KOSTER, G.F. (1965), J. Appl. Phys. 36, 2278. TOMKINS, F.S. and FRED, M. (1951), J. Opt. Soc. Am. 41, 641. THEOPHANIS, G.A. (1960), Rev. S c i . Instr. 31, 427. VAN DE HULST, H.C. and REESINCK, J.J.M. (1947), Astrophys. J. 106, 121. WEBB, J.H. (1933), J. Opt. Soc. Am. 23, 316. WULFF, H. (1965), Max-Planck Inst ..-P.A.E. , P i . 

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