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An Experimental investigation of the Stark broadened line shape of Helium I 3965 Pilon, Peter James 1978

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AN EXPERIMENTAL INVESTIGATION OF THE STARK BROADENED LINE SHAPE OF HELIUM I 3965 by PETER JAMES PILON B . S c , U n i v e r s i t y of New Brunswick, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Dept. Of P h y s i c s , U n i v e r s i t y of B r i t i s h Columb We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September,1978 © P e t e r J . P i l o n , 1978 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements f an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree tha the L ibrary sha l l make it f ree l y ava i l ab le for reference and study. I further agree that permission for extensive copying of th i s thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of P H Y S I C S The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date October- 3,19 78 ABSTRACT Stark broadened He I 3965 (4P-2S) has been st u d i e d e x p e r i m e n t a l l y using a plasma j e t as a source. Integrated i n t e n s i t y p r o f i l e s were obtained by scanning the c y l i n d r i c a l l y symmetric plasma across i t s width and using a spectrometer and o p t i c a l m u l t i c h a n n e l analyser coupled to a magnetic tape d r i v e system to record the p r o f i l e s . The r e s u l t i n g data was converted to s p e c t r a l l i n e shapes as a f u n c t i o n of plasma r a d i u s through the use of an Abel u n f o l d i n g procedure c a r r i e d out by a computer program. The r a d i a l dependence of the plasma temperature and d e n s i t y was determined and then used to match e x p e r i m e n t a l l y determined p r o f i l e s w i t h t h e o r e t i c a l p r o f i l e s which take the two forbidden components of the Stark broadened l e v e l s i n t o account. Comparison of the experimental and t h e o r e t i c a l p r o f i l e s y i e l d e d good agreement. i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES v i AC KNOWLEDGEMENTS v i i i CHAPTER I INTRODUCTION 1 CHAPTER I I THEORY OF THE HE I 3965 LINESHAPE 4 A. Development of the P r o f i l e 4 B. Methods of C a l c u l a t i o n 9 CHAPTER I I I EXPERIMENTAL APPARATUS 14 A. The Plasma J e t 14 B. The Experimental Setup 18 CHAPTER IV PLASMA DIAGNOSTICS 25 A. O p t i c a l Depth 25 B. LTE and the Plasma J e t 27 C. Temperature Measurements 30 D. Density Measurements ..32 i i i CHAPTER V DATA ANALYSIS 39 A. Abel I n v e r s i o n 39 B. The Software Package 41 C. Comparison of T h e o r e t i c a l and Experimental P r o f i l e s 44 CHAPTER VI CONCLUSIONS 52 BIBLIOGRAPHY 55 i v LIST OF TABLES TABLE I I I - l Component S p e c i f i c a t i o n 23 TABLE IV-1 Reduced F u l l H a l f Width of H-Beta Versus Temperature 35 v LIST OF FIGURES FIGURE I I - l Stark S p l i t Energy Levels i n He 5 FIGURE II-2 Estimated P r o f i l e of He I 3965 6 FIGURE I I - 3 E l e c t r i c M i c r o f i e l d D i s t r i b u t i o n s 11 FIGURE II-4 T h e o r e t i c a l l y P r e d i c t e d P r o f i l e s f o r Varying Temperatures 12 FIGURE I I - 5 T h e o r e t i c a l l y P r e d i c t e d P r o f i l e s f o r Varying Density 13 FIGURE I I I - l The Plasma J e t 15 FIGURE I I I - 2 The Plasma J e t Power Supply C i r c u i t 17 FIGURE I I I - 3 The Plasma Dimensions 20 FIGURE I I I - 4 The Experimental Setup 21 FIGURE IV-1 E l e c t r o n Density Versus Temperature 33 FIGURE IV-2 Temperature Versus Plasma Radius. 34 FIGURE IV-3 E l e c t r o n Density Versus F u l l H a l f Width of Balmer Beta 36 FIGURE IV-4 E l e c t r o n Density Versus Plasma Radius. ...38 v i FIGURE V - l Abel I n v e r s i o n Geometry 40 FIGURE V-2 T y p i c a l Integrated I n t e n s i t y P r o f i l e s . ...42 FIGURE V-3 Exp e r i m e n t a l l y Determined P r o f i l e s 45 FIGURE V-4 S h i f t s Versus E l e c t r o n Density 46 FIGURE V-5 F u l l Half Width Versus E l e c t r o n Density. .47 FIGURE V-6 T h e o r e t i c a l and Experimental P r o f i l e s . ...48 FIGURE V-7 T h e o r e t i c a l and Experimental P r o f i l e s . ...49 FIGURE V-7 T h e o r e t i c a l and Experimental P r o f i l e s . ...50 v i i ACKNOWLEDGEMENTS I would l i k e to extend my s i n c e r e a p p r e c i a t i o n to the f o l l o w i n g members of the P h y s i c s Department of the U n i v e r s i t y of B r i t i s h Columbia. F i r s t and foremost, I would l i k e to thank Dr.A.J.Barnard f o r showing i n t e r e s t i n me when I f i r s t came to t h i s s c h o o l and, i n the c a p a c i t y of my s u p e r v i s o r , f o r g i v i n g u n s e l f i s h l y of h i s time and experien c e i n the f i e l d of l i n e broadening. Dr. E l i s a b e t h K a l l n e was always a v a i l a b l e f o r d i s c u s s i o n of theory and her advic e on the p r o j e c t i s g r e a t l y valued and a p p r e c i a t e d . A l s o the e n t i r e Plasma P h y s i c s group, f a c u l t y members and f e l l o w students a l i k e , were always w i l l i n g to d i s c u s s problems and lend a h e l p i n g hand where needed. On the t e c h n i c a l s i d e , I would a l s o l i k e to thank Mr.A.Cheuck f o r a s s i s t a n c e with the e l e c t r o n i c s of the p r o j e c t , Mr.D.Pawluk f o r h i s l a r g e c o n t r i b u t i o n to the computer programming and Mr.P.Hass and Mr . O . C h r i s t i a n s e n f o r c o n s t r u c t i o n of the main body of the plasma j e t . F i n a l l y I would l i k e to thank my wife Charlene f o r her a s s i s t a n c e i n the p r e p a r a t i o n of t h i s r e p o r t . v i i i 1 CHAPTER I INTRODUCTION In the f i e l d of a s t r o p h y s i c s there e x i s t s a s u b s t a n t i a l amount of i n t e r e s t i n the study of white dwarfs. Since many white dwarfs c o n s i s t p r i m a r i l y of helium plus some hydrogen and metals, a w e l l e s t a b l i s h e d , coherent theory of the s p e c t r a l character of helium i s fundamental to the determination of the mechanisms ; of the white dwarf. He I 3965 has been observed i n the sp e c t r a of white dwarfs by s e v e r a l authors (Berger and Fringant (1978) and Shipman et a l (1977)). Many helium s p e c t r a l l i n e s have been w e l l s t u d i e d and t h e i r c o l l i s i o n a l l y broadened p r o f i l e s adequately determined. These l i n e s are u s u a l l y r e l a t i v e l y intense and t h e i r p r o f i l e s w e l l d e scribed by e l e c t r o n impact theory. Their p r o f i l e s are L o r e n t z i a n i n shape and t h e i r widths have been used both i n the l a b o r a t o r y and i n i n t e r s t e l l a r plasmas to determine values f o r plasma e l e c t r o n d e n s i t y and temperature. Other helium l i n e s , such as He I 3965 (4P-2S), are not described simply by e l e c t r o n impact theory but are a l s o broadened by v i r t u e of the ion e l e c t r i c f i e l d . U n f o r t u n a t e l y some of these l i n e s have not been stud i e d 2 adequately; t h e r e f o r e the main purpose of t h i s r e p o r t i s to compare a computer c a l c u l a t e d p r o f i l e from a t h e o r e t i c a l model fo r He I 3965, which takes the Stark e f f e c t i n t o account, w i t h experimental p r o f i l e s obtained from a l a b o r a t o r y plasma j e t source. He I 3965 was chosen f o r t h i s work not only because i t i s under s t u d i e d but a l s o because i t has a rather unusual l i n e s h a p e . In s i n g l e t helium the 4P upper l e v e l of the t r a n s i t i o n l i e s only 46 cm-'- higher than the 4D l e v e l and 41 cm"l higher than the 4F l e v e l . Consequently, when these l e v e l s are Stark s p l i t i n the e l e c t r i c f i e l d of the plasma ions and broadened by atomic c o l l i s i o n s w i t h e l e c t r o n s , the 4P s p l i t l e v e l s o v e rlap w i t h the 4D and 4F s p l i t l e v e l s and a c o l l i s i o n a l t r a n s f e r e q u i l i b r i u m occurs between the three s e t s of s t a t e s . This allows r a d i a t i v e t r a n s f e r from the 4D and 4F l e v e l s to the 2S l e v e l to occur independent of the d i p o l e s e l e c t i o n r u l e s . The combination of the two 4P-2S r a d i a t i v e components produces an asymmetric l i n e shape whi l e the 4D-2S and 4F-2S forbidden components produce f u r t h e r c o m p l i c a t i o n s to the p r o f i l e i n the form of bumps to the high wavelength s i d e of the main peak. This unusual s t r u c t u r e i n i t s e l f makes He I 3965 an i n t e r e s t i n g s p e c t r a l l i n e to study independent of i t s a s t r o p h y s i c a l i n t e r e s t . This r e p o r t i s d i v i d e d i n t o the four f o l l o w i n g s e c t i o n s . Chapter I I b r i e f l y o u t l i n e s the development of 3 the theory of the l i n e s h a p e and d i s c u s s e s the approximations i n which the v a r i o u s terms i n the e x p r e s s i o n are e v a l u a t e d . No d e t a i l e d d e r i v a t i o n of the l i n e p r o f i l e e x p r e s s i o n i s g i v e n as i t can be found i n most books on s p e c t r a l l i n e broadening, such as Griem (1964) . Chapter I I I c o n s i s t s of a d e s c r i p t i o n of the plasma j e t and of other equipment used i n the d e t e r m i n a t i o n of the e x p e rimental l i n e p r o f i l e s . In Chapter IV the d i a g n o s t i c s of the plasma are d i s c u s s e d . Methods f o r the d e t e r m i n a t i o n of the r a d i a l e l e c t r o n d e n s i t y and temperature i n the plasma are d i s c u s s e d and p l o t s of d e n s i t y and temperature as a f u n c t i o n of r a d i u s g i v e n . Chapter V d i s c u s s e s the manner i n which the data i s analysed as w e l l as g i v i n g comparisons between the t h e o r e t i c a l and the experimental l i n e p r o f i l e s f o r a range of e l e c t r o n d e n s i t i e s . Chapter VI c o n t a i n s a d i s c u s s i o n of the r e s u l t s . 4 CHAPTER I I THEORY OF THE He I 3965 LINESHAPE A. Development of the P r o f i l e Since the theory f o r the determination of a * s p e c t r a l lineshape i s complicated, i t i s d e s i r a b l e to t r y and understand the p r o f i l e by f i r s t c o n s i d e r i n g what happens i n the plasma e l e c t r i c f i e l d s . The manner i n which the He s t a t e s s p l i t i n a s t a t i c e l e c t r i c f i e l d and the t r a n s i t i o n s that occur to produce the f i n a l lineshape w i l l be considered f i r s t . Since the e l e c t r o n d e n s i t y of the plasma used i n t h i s experiment i s of the order of 2X10^-^ cm-*, the average e l e c t r i c f i e l d can be c a l c u l a t e d and the r e s u l t i n g magnitude of the s p l i t t i n g of the l e v e l s can be determined from theory (See J.S.Foster (1927) f o r example). Figure I I - l shows how the p e r t i n e n t l e v e l s s p l i t i n a 40 kV/cm f i e l d and what t r a n s i t i o n s are allowed due to the c o l l i s i o n a l i n t e r a c t i o n of the l e v e l s . (The 40 kV/cm f i e l d represents the mean f i e l d produced i n a plasma of 2X10l6 Cni 3 e l e c t r o n density.) A l l of these t r a n s i t i o n s are described by e l e c t r o n impact theory and can be shown to have approximately L o r e n t z i a n p r o f i l e s . The components add together to produce the f i n a l p r o f i l e and Figure I I -2 shows how these components merge. The r e l a t i v e strengths FIGURE I I - l Stark s p l i t energy l e v e l s i n helium w i t h allowed and f o r b i d d e n t r a n s i t i o n s shown. 6 COMPOSITION OF UNESHAPE 1.0-E = 40 kV/cm itrary) x> >_ o J» o INTENSITY (s< / \ /.A'P-2'S V V f V ^ s ^ s . ^ / / ^ \ . ^ ^^>\ Hr^ -: I I I I 3 -4 0 4 8 ( 3964.7) X (A) FIGURE II-2 E s t i m a t e d p r o f i l e o f He I 3965. 7 of the t r a n s i t i o n s have been c a l c u l a t e d from F o s t e r ' s c a l c u l a t i o n s . The f u l l h a l f width of the L o r e n t z i a n components are simply estimated as 4 A, which corresponds approximately to the width f o r He I 3965, as quoted by Griem (1964) f o r the r e l e v a n t d e n s i t y . Since the a c t u a l p r o f i l e i s a c o n v o l u t i o n of p r o f i l e s due to d i f f e r e n t p o s s i b l e e l e c t r i c f i e l d s , Figure II-2 i s only a simple f i r s t approximation to t h i s l i n e shape and i s included only to c l a r i f y the r e s u l t s . When c o n s i d e r i n g c o l l i s i o n a l broadening, there e x i s t s an e n t i r e range of c o l l i s i o n types that must be accounted f o r . They range from f a s t impacts w e l l separated i n time to very slow c o l l i s i o n s i n which the p e r t u r b i n g p a r t i c l e s move s l o w l y w i t h respect to the perturbed p a r t i c l e s . As a f i r s t approximation one can assume that the c o l l i s i o n s l i e at one or the other end of t h i s range and and consider the two types as being s t a t i s t i c a l l y independent so that t h e i r e f f e c t s on the p r o f i l e can be c a l c u l a t e d s e p a r a t e l y . E l e c t r o n c o l l i s i o n s occur at the f a s t end of the range due to t h e i r high v e l o c i t y (Ve=Vi (Mi/Me) 1/ 2 =80 X V i f o r He). Ion c o l l i s i o n s , on the other hand, occur s l o w l y so only the Stark e f f e c t due to the s t a t i c ion e l e c t r i c f i e l d needs to be c a l c u l a t e d . These two approximations are named the e l e c t r o n impact approximation and the q u a s i s t a t i c ion approximation r e s p e c t i v e l y . 8 F i r s t l e t us consider the e l e c t r o n impact approximation. In order to use time-dependent p e r t u r b a t i o n theory to determine the e l e c t r o n impact p r o f i l e , the c l a s s i c a l path approximation i s u s u a l l y made. In t h i s the motion of the p e r t u r b i n g p a r t i c l e s i s considered to be along c l a s s i c a l t r a j e c t o r i e s . The c r i t e r i a f o r the v a l i d i t y of t h i s assumption are that broadening c o l l i s i o n r a t e s dominate e x c i t a t i o n and decay c o l l i s i o n a l r a t e s and that the average d i s t a n c e of c l o s e s t approach during c o l l i s i o n s , i s greater than the d e B r o g l i e wavelength of the perturbed p a r t i c l e . Both of these c r i t e r i a are u s u a l l y w e l l obeyed i n l a b o r a t o r y plasmas, t h e r e f o r e the e l e c t r o n impact lineshape can now be c a l c u l a t e d using time-dependent p e r t u r b a t i o n theory. Doing t h i s , the p r o f i l e f o r e l e c t r o n impact broadening i s found to be S ( u) =-Re^ T * d . d 0<"\—, \ ; H O ( 2 _ 1 ) e TT / . «8 <=gx 1 x(co-to „)-n Y 1 x i * «8 O where « and J are the Stark s p l i t quantum s t a t e s of the upper l e v e l i n the f i e l d E and g the lower l e v e l . 3 and d , are the d i p o l e matrix elements, and \p i s an operator given by the p e r t u r b a t i o n theory which c o n t a i n s the v e l o c i t y , time and impact parameter i n f o r m a t i o n . As mentioned before, the ion broadening i s 9 u s u a l l y n e g l i g i b l e with respect to the e l e c t r o n broadening f o r the m a j o r i t y of l i n e s , however f o r He I 3965 t h i s i s not the case and the e l e c t r i c f i e l d dependence of the terms i n Equation 2-2 must be considered. This i s done by i n t e g r a t i n g Equation 2-1 over a l l p o s s i b l e e l e c t r i c f i e l d s as f o l l o w s V " ' " 42/ d E W ( E ) d - -B d -B<- ' l (a,- 1,. B)-n o < , l - , > <2-a / CC oc Where W(E) i s the e l e c t r i c m i c r o f i e l d p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n . The problem now i s to evaluate t h i s equation f o r given plasma parameters and determine a t h e o r e t i c a l p r o f i l e . B. Methods of C a l c u l a t i o n As can be seen by examination of Equation 2-2, the determination of the t o t a l c o l l i s i o n a l l y broadened l i n e shape i s not a simple task. The i n t e g r a t i o n over a l l p o s s i b l e e l e c t r i c m i c r o f i e l d s i s complicated by the f i e l d dependence of and u)ag. Barnard (1969) has c a r r i e d out c a l c u l a t i o n s of Equation 2-2 fo r He I 4471 and has used these methods to c a l c u l a t e the p r o f i l e f o r He I 3965. The e l e c t r i c m i c r o f i e l d d i s t r i b u t i o n f u n c t i o n was obtained from Hooper 10 (1968) who c a l c u l a t e d t h i s d i s t r i b u t i o n i n terms of the Holtsmark f i e l d s t r e n g t h (E ) f o r d i f f e r e n t values of the H parameter « (See Figure I I - 3 ) . The d i p o l e f a c t o r was c a l c u l a t e d i n the Bates-Damgaard approximation i n which the s o l u t i o n s of the r a d i a l wave equation were constructed from s e r i e s expansions of the hydrogenic wave f u n c t i o n s . The p e r t u r b a t i o n theory operator Y was c a l c u l a t e d f o r weak c o l l i s i o n s and then a strong c o l l i s i o n c ross s e c t i o n term was added to i t s diagonal . elements. The base frequencies can be determined from the Stark s p l i t t i n g of the upper l e v e l s . The 4F and 4D l e v e l c o n t r i b u t i o n s were accounted for i n the summation. The above c a l c u l a t i o n s were c a r r i e d out with the a i d of a computer and p l o t s of the t o t a l c o l l i s i o n a l l y broadened p r o f i l e of He I 3965 produced f o r the range of e l e c t r o n d e n s i t y and temperature of i n t e r e s t . Figure II-4 shows how the p r o f i l e v a r i e s w i t h e l e c t r o n temperature wh i l e Figure I I - 5 shows the v a r i a t i o n with e l e c t r o n d e n s i t y . I t can be seen t h a t the important parameter i s the d e n s i t y w h i l e a change of a f a c t o r of two i n the temperature changes the p r o f i l e very l i t t l e . FIGURE II-3 E l e c t r o n m i c r o f i e l d d i s t r i b u t i o n (Hooper (1969)). ^ /] THEORETICALLY DETERMINED rl.O LINESHAPE (N e= 1.7 XIO , 6cm" 3) \ 7 T e = 20000 °K \ / / T e s 10000 °K I 1 r- i i -10 -8 -6 -4 -2 < i l i • • 0 2 4 6 8 10 X(A) FIGURE II-4 T h e o r e t i c a l l y p r e d i c t e d p r o f i l e s f o r v a r i n g e l e c t r o n temperature. THEORETICALLY I 1 1 1 T 1 1 1 1 1 1 -10 -8 -6 -4 -2 0 2 4 6 8 10 X(X) FIGURE I I - 5 T h e o r e t i c a l l y p r e d i c t e d p r o f i l e s f o r v a r i n g e l e c t r o n d e n s i t y . 14 CHAPTER I I I EXPERIMENTAL APPARATUS A. The Plasma J e t A plasma j e t i s used to produce. a time independent helium plasma at atmospheric p r e s s u r e by p a s s i n g c o l d helium through a DC a r c d i s c h a r g e s t r u c k between a p e n c i l shaped cathode and a r i n g shaped anode. The gas i s p a r t i a l l y i o n i z e d w h i l e p a s s i n g through the d i s c h a r g e s u r f a c e , and then blown out through the hole i n the anode r i n g to produce a f r e e s t a n d i n g plasma (See F i g u r e I I I - l ) . The j e t i s s i m i l a r to one used by L.A.Godfrey (1977). The f o l l o w i n g i s a b r i e f d i s c u s s i o n of the d e s i g n and o p e r a t i o n of the helium j e t . The cathode t i p i s c o n s t r u c t e d from a rod of sharpened t h o r i a t e d tungsten. A copper tube i s melted i n t o the t i p , using a n i c k e l i n t e r f a c e , and then f i t t e d to a bras s tube support s t r u c t u r e using hard s i l v e r s o l d e r . The cathode i s c o o l e d by p a s s i n g water through a t h i n tube which runs through the middle of the support tube. The water s t r i k e s the t i p and then flows out between the w a l l s of the support and t h i n tubes. T h i s can be seen i n F i g u r e I I I - l . The anode i s b u i l t of copper and i s i n the shape of a cone which s i t s over the cathode. Copper i s used 15 TUNGSTEN CATHODE WATER OUTLET COPPER ANODE COPPER INTERFACE BRASS TUBE CERAMIC SPACER WATER INLET NYLON SUPPORT i WATER OUTLET WATER INLET FIGURE I I I - l The plasma j e t (Scale 1:1). 16 here because of i t s high thermal and e l e c t r i c a l c o n d u c t i v i t i e s . A small hole at the top of the anode acts as the j e t o r i f i c e . C ooling here i s achieved by f o r c i n g water between the anode surface and an i n t e r i o r w a l l and e j e c t i n g i t between the i n t e r i o r w a l l and the e x t e r i o r w a l l of the anode s t r u c t u r e . This i s best seen i n Figure I I I - l . Research grade gas i s passed through flowmeter r e g u l a t o r s and p l a s t i c tubing i n t o a chamber below the main j e t c a v i t y . I t i s then blown upwards through small holes i n an i n s u l a t i n g p l a t e c i r c l i n g the cathode i n t o the main chamber and out through the j e t o r i f i c e . The j e t i s powered by a DC welder c u r r e n t source which can be adjusted to d e l i v e r any cu r r e n t between 20 and 300 amps, or by a 240 amp-hour, 48 v o l t bank of b a t t e r i e s c o n s i s t i n g of 24 two v o l t b a t t e r i e s i n s e r i e s . The j e t i s s t a r t e d by using a high v o l t a g e spark i n i a t e d c a p a c i t o r discharge to break down the gap and allow the welder to operate. The j e t i s best s t a r t e d on argon, rather than helium, because of i t s r e l a t i v e l y low i o n i z a t i o n p o t e n t i a l . When an experimental run i s to be made, the gas i s switched to helium and a bank of r e l a y s i s used to switch o p e r a t i o n from the. welder to the b a t t e r i e s and to switch a b a l l a s t r e s i s t o r bank c o n s i s t i n g of s e v e r a l one ohm, 1000 watt power r e s i s t o r s to t r i m the cu r r e n t to the de s i r e d value (See Figure I I I - 2 f o r d e t a i l s ) . 17 I/4-I n r - r V ^ JET SHUNT AAA/— I 1/4-1 n '(6 UNITS) 1/4-1A £ T 4H 48 Volts DC nil 500. 20 CHARGING UNIT WELDER CURRENT SUPPLY - W — STARTING CIRCUIT 0 FIGURE III-2 The plasma j e t power supply c i r c u i t . 18 A s t e e l p e d e s t a l i s used as a mount f o r the j e t . I t i s f i n e l y a d j u s t a b l e i n the two d i r e c t i o n s p e r p e n d i c u l a r to the experiment o p t i c a l a x i s and can be adj u s t e d c o a r s e l y along the a x i s . A c o n t i n u o u s l y v a r i a b l e speed motor i s used to d r i v e the j e t a c r o s s the o p t i c a l a x i s i n order to scan the plasma d u r i n g the experiment. These speeds are slow (approximately 2 cm/min.) and do not e f f e c t the s t a b i l i t y o f the plasma. The v o l t a g e drop a c r o s s the j e t i s measured by a d i g i t a l v o l t m e t e r and the c u r r e n t p a s s i n g through the j e t i s monitored by measuring the v o l t a g e drop a c r o s s a low r e s i s t a n c e shunt. T h i s v o l t a g e i s measured through a p o t e n t i a l d i v i d e r i n such a way t h a t the meter i s c a l i b r a t e d to one m i l l i v o l t / a m p e r e . B. Experimental Setup Even though the j e t flame i s produced by p a s s i n g i o n i z e d gas through an o r i f i c e , the r e s u l t i n g plasma i s q u a s i - s t a t i o n a r y i n t h a t i t m a i n t a i n s the same s p e c t r a l c h a r a c t e r a t any p o s i t i o n at any time. At s u f f i c i e n t l y h i g h c u r r e n t s the d i s c h a r g e o c c u r s evenly about the a x i s of the system and the plasma produced i s i n the shape of an i n v e r t e d cone. The s p e c t r a l dependence i s a f u n c t i o n of r a d i u s and h e i g h t such t h a t S=S((iJ , r , z ) . By l i m i t i n g the o p t i c a l acceptance of the system i n the v e r t i c a l 19 d i r e c t i o n ( f i x i n g z=h), a c y l i n d e r i c a l l y symmetric d i s c of plasma can be observed by moving the j e t across the o p t i c a l a x i s (See Figure I I I - 3 ) . Any i n t e n s i t y measurement through t h i s d i s c w i l l g i v e the i n t e g r a t e d i n t e n s i t y across the plasma. In order to o b t a i n the p r o f i l e as a f u n c t i o n of r a d i u s the plasma must be scanned and the r e s u l t i n g data A b e l - i n v e r t e d . Therefore, what i s needed i s a system which w i l l scan across the plasma wh i l e t a k i n g i n t e g r a t e d l i n e p r o f i l e measurements as a f u n c t i o n of d i s t a n c e from the plasma cent e r . This s e c t i o n w i l l e x p l a i n how t h i s i s achieved. F i g u r e I I I - 4 i s a schematic diagram of the system. The j e t can be moved across the o p t i c a l a x i s of the system by means of a motor d r i v e as mentioned e a r l i e r . The f o c u s i n g lens c o l l e c t s the l i g h t from the plasma and images i t on the spectrometer s l i t . The l i g h t i s d i s p e r s e d at the g r a t i n g and then focused on the d e t e c t o r face of an o p t i c a l m u l t i - c h a n n e l analyser (OMA). The spectrometer i s operated i n second order g i v i n g a r e s o l u t i o n of approximately 0.125 A/channel at the OMA y i e l d i n g s p e c t r a 65 A i n width. The OMA i s i n t e r f a c e d to a b u f f e r e d formatter and tape d r i v e . The o p e r a t i n g c h a r a c t e r i s t i c s f o r the above mentioned equipment are given i n Table I I I - l . T h i s system i s capable of recording ten complete s p e c t r a per second. With t h i s the e n t i r e plasma can be scanned i n l e s s than 20 seconds g i v i n g enough data to determine the PLASMA AFTER FLAME DISK OBSERVED zone of optical acceptance DIRECTION OF PLASMA MOTION FIGURE III - 3 The plasma dimensions. (r =2mm and h =20mm). o o 21 SPECTROMETER TAPE RECORDER oo BUFFERED FORMATTER OMA CON SEL O SCOPE DRIVE He-Ne LASER FIGURE III-4 The e x p e r i m e n t a l setup. 22 s p e c t r a l p r o f i l e as a f u n c t i o n of r a d i u s f o r the l i n e measured. A p a r t i a l l y r e f l e c t i n g m i r r o r i s used along the o p t i c a x i s to a l l o w wavelength c a l i b r a t i o n s p e c t r a from both G e i s s l e r tubes and an i r o n a r c to be o b t a i n e d . A He-Ne l a s e r i s used to a l i g n the l e n s system and spectrometer. 23 Table I I I - l - Component S p e c i f i c a t i o n Spectrometer S u p p l i e r Focal Length Apeture G r a t i n g D i s p e r s i o n OMA S u p p l i e r R e s o l u t i o n No. Of channels Maximum count L i n e a r i t y i n i n t e n s i t y I n t e g r a t i o n time D i g i t a l output l e v e l s D i g i t a l data output Spex Co. 3/4 meter, double m i r r o r F6. 8 1200 l i n e s / i n c h blazed at 10,000 A 5 A/mm i n f i r s t order P r i n c t o n A p p l i e d Research 20 channels/mm 500 channels 732 i n any channel ±1% from 10-500 counts 32.768 msec. TTL 5 BCD d i g i t s + s i g n Buffered Formatter S u p p l i e r B u f f e r s i z e Tracks Kennedy Co. 2 by 1024 9 t r a c k s 24 Tape D r i v e S u p p l i e r Tape speed Dens i t y Tracks Kennedy Co. 37.5 inches/second 800 bpi 9 t r a c k s 25 CHAPTER IV PLASMA DIAGNOSTICS A. O p t i c a l Depth In order to make a temperature measurement on a plasma source, p r a c t i c a l l y a l l methods r e q u i r e that the plasma be i n a s t a t e of l o c a l thermodynamic e q u i l i b r i u m (LTE). This means that the motion of both the e l e c t r o n s and the ions can be described by a Maxwellian v e l o c i t y d i s t r i b u t i o n having the same c h a r a c t e r i s t i c temperature (T=Te=Ti). I t has been shown by M o r r i s (1968) and Godfrey (1977) that t h i s c o n d i t i o n i s not upheld i n an atmospheric plasma j e t source. The ion temperature i n t h i s type of plasma i s l e s s than the e l e c t r o n temperature. However, Godfrey (1972) has shown, by l a s e r s c a t t e r i n g , that the e l e c t r o n v e l o c i t y d i s t r i b u t i o n i s Maxwellian so i t does have an a s s o c i a t e d temperature Te. To measure t h i s temperature by conventional means the amount of u n c e r t a i n t y i n the measurement must be determined. To do t h i s , the amount of departure from LTE must be c a l c u l a t e d . Since a l l LTE v a l i d i t y c r i t e r i a r e q u i r e that the plasma be o p t i c a l l y t h i n so that photo-e x c i t a t i o n can be neglected, the o p t i c a l depth of the plasma must f i r s t be c a l c u l a t e d . The o p t i c a l depth, To, i s given by 26 A T = 8.853 x I O 2 1 f x2S(X) / n ( r) dr (4-1) O 1 U -r o where n(r) i s the p o p u l a t i o n d e n s i t y of the lower l e v e l i n i the t r a n s i t i o n of i n t e r e s t , and i s a f u n c t i o n of r a d i u s . dr i s the r a d i a l length increment,f the o s c i l l a t o r i y s t r e n g t h , \ the center wavelength of the t r a n s i t i o n of i n t e r e s t and S(X) the c e n t r a l l i n e i n t e n s i t y , w i t h t o t a l i n t e n s i t y normalized to u n i t y . The worst case i s taken where the o p t i c a l a x i s passes through the center of the plasma. The lower l e v e l p o p u l a t i o n d e n s i t y i s given by the Boltzmann d i s t r i b u t i o n , n = n (g /g ) exp(-E /kT) (4-2) 1 0 1 0 10 Equation 4-1 can be s i m p l i f i e d by assuming that n remains constant across the plasma and that S (x)=l/w, i where w i s the f u l l h a l f width of the t r a n s i t i o n of i n t e r e s t . Combining equations 4-1 and 4-2 gi v e s — 21 n g f T. = 17. 706 x 10 (~2-i 2—UL) exp(-E /kT) (4-3) 9o w *o Using an approximate temperature value f o r the j e t quoted by M o r r i s (1968) of Te=1.3 ev, and e v a l u a t i n g equation 4-3 27 using Hel 3965 as the t r a n s i t i o n , the o p t i c a l depth i s -4 found to be 2.9x10 which i s n e g l i g i b l y , s m a l l . Since the plasma i s c e r t a i n l y o p t i c a l l y t h i n , the v a l i d i t y of LTE can be considered. B. LTE and the Plasma J e t The e l e c t r o n p o p u l a t i o n d i s t r i b u t i o n s i n the v a r i o u s atomic and i o n i c energy l e v e l s i n a steady s t a t e plasma such as the j e t plasma remain constant. This means that the r a t e of p o p u l a t i o n and depopulation of any one l e v e l are equal. Since i t has been shown that the plasma i s o p t i c a l l y t h i n , then the r a t e of p h o t o e x c i t a t i o n i s n e g l i g i b l e and the c o l l i s i o n a l e x c i t a t i o n r a t e i s equal to the sum of the c o l l i s i o n a l and r a d i a t i v e decay r a t e s . I f i t i s assumed that the c o l l i s i o n a l decay ra t e dominates the r a d i a t i v e decay r a t e and that the p o p u l a t i o n d i s t r i b u t i o n s respond i n s t a n t l y to a change i n the plasma parameters then no knowledge of the atomic cross s e c t i o n s i s needed to c a l c u l a t e the p o p u l a t i o n d i s t r i b u t i o n as i t i s given p u r e l y i n terms of s t a t i s t i c a l mechanics. That i s , the l e v e l s are populated according to the Boltzmann and Saha equations by atoms c o l l i d i n g with f r e e e l e c t r o n s having a Maxwellian v e l o c i t y d i s t r i b u t i o n and a c h a r a c t e r i s t i c temperature, Te. These assumptions form the b a s i s of the LTE 28 model. LTE c r i t e r i a have been developed f o r o p t i c a l l y t h i n plasmas by many authors, such as Griem (1964) . This i s done by forming an i n e q u a l i t y between the c o l l i s i o n a l and r a d i a t i v e decay r a t e s and using a f a c t o r of ten as a s a f t e y margin. That i s , C o l l i s i o n a l Rates > 10 x R a d i a t i v e Rates (4-4) I f t h i s i n e q u a l i t y h o l d s , then the po p u l a t i o n d i s t r i b u t i o n can be shown to be w i t h i n ten percent of a Boltzmann d i s t r i b u t i o n . These c r i t e r i a are developed f o r hydrogenic ions to maintain s i m p l i c i t y but, can be used on the helium plasma as the m a j o r i t y of the plasma c o n s t i t u i e n t s are n e u t r a l helium atoms with one e l e c t r o n i n the ground s t a t e over twenty e l e c t r o n v o l t s away from the other e l e c t r o n . Therefore the e x c i t e d e l e c t r o n sees an e f f e c t i v e nucleus wi t h charge of one and the atom i s approximately hydrogenic. These LTE v a l i d i t y c r i t e r i a can now be used to compare the r a d i a t i v e and c o l l i s i o n a l decay r a t e s so that an u n c e r t a i n t y f o r a temperature measurement made by assuming LTE can be estimated. Since the plasma j e t i s an inhomogeneous source, i t must s a t i s f y two c r i t e r i a i n order to be i n LTE. F i r s t , equation 4-4 must hold and second s p a t i a l v a r i a t i o n s i n e l e c t r o n temperature must be small over 29 d i s t a n c e s of the order of the e q u i l i b r a t i o n time of the atoms. For complete LTE to h o l d , Griem (1964) show that i t i s u s u a l l y s u f f i c i e n t to consider the r a t e s of the resonance l i n e l e v e l s . S o l v i n g f o r the e l e c t r o n d e n s i t y i n the c o l l i s i o n a l r a t e of the c r i t e r i a equation 4-4 gives n > 9 x 1 0 1 7 ( E /E j 3 (kT/E } / 2 (4-5) e r H H where E i s the i o n i z a t i o n p o t e n t i a l of hydrogen and E i s H R the energy of the resonance l i n e . Using an approximate value f o r the e l e c t r o n temperature quoted e a r l i e r of 1.3 ev, i t i s found that that the e l e c t r o n d e n s i t y must be 1 8 - 3 greater than 10 cm which i s not s a t i s f i e d i n the j e t . The j e t plasma has a maximum d e n s i t y of approximately 16 — 3 2x10 cm . This means tha t the r a d i a t i v e processes i n the j e t dominate the c o l l i s i o n a l processes by a f a c t o r of f i v e i n the resonance l e v e l s . Although complete LTE i s not s a t i s f i e d , a s t a t e of p a r t i a l LTE e x i s t s f o r s t a t e s with p r i n c i p a l quantum number of three and g r e a t e r . This occurs because the higher l e v e l s are more hydrogenic and much c l o s e r together so that the c r i t e r i o n can be r e l a x e d . This new c r i t e r i o n i s given by Griem (1964) as n >7 x 1 0 1 8 z 4 n _ 1 1 / 2 ( A E / E ) exp( AE/E u) e ti ti (4-6) 30 where AE i s the d i f f e r e n c e between the i o n i z a t i o n energy and the energy of the nth l e v e l . For n=2 and Te=1.3 ev, 17 — 3 Ne>2xlO x / cm-" which i m p l i e s that the c o l l i s i o n a l and r a d i a t i v e decay r a t e s are approximately equal. For n=3 15 —3 and Te=1.3 ev, Ne>1.3xl0 cm which demonstrates that p a r t i a l LTE holds i n the j e t f o r l e v e l s with p r i n c i p a l quantum numbers three and above. M o r r i s (1968) showed that the temperature v a r i a t i o n across the j e t i s approximately 1.3 to 0.9 ev which i s a drop of t h i r t y percent. The e q u i l i b r a t i o n d i s t a n c e of the atoms was c a l c u l a t e d and found to be of the order of the j e t diameter. This means that a t h i r t y percent temperature change occurs over the e q u i l i b r a t i o n d i s t a n c e and that t h i s c r i t e r i o n f o r LTE i s m a r g i n a l l y obeyed. C. Temperature Measurements The s p e c t r a l p r o f i l e of He I 3965 i s the t o p i c of i n t e r e s t i n t h i s r e p o r t . As i t i s very i n s e n s i t i v e to changes i n the e l e c t r o n temperature (15% decrease i n width and l e s s than 0.5 A increase i n s h i f t from 5,000 K to 40,000°K) an accuracy of plus or minus t h i r t y percent i n a temperature measurement would only incur aproximately f i v e percent maximum e r r o r i n the t h e o r e t i c a l p r o f i l e . Even though the plasma i s not i n complete LTE, i f i t i s assumed 31 t h a t the Saha equation i s u n c e r t a i n by a f a c t o r of ten the r e s u l t i n g e r r o r i n the temperature would be o n l y about p l u s or minus ten p e r c e n t . T h e r e f o r e , as a rough temperature measure, LTE c o u l d be assumed and Te determined by the c o n d i t i o n of charge n e u t r a l i t y , D alton's Law and the Saha-Boltzmann eq u a t i o n f o r ground s t a t e atoms. These are r e s p e c t i v e l y , N = n +2n +3n + (4-6) a 0 1 2 P = N kT (4-7) a n n /n = 6 x I O 2 1 (gr / Q ) ( k T ) 3 / 2 exp(-I /kT) (4-8) e 1 0 1 0 0 where N i s the t o t a l d e n s i t y , g and g are the a 0 1 d e g e n e r a c i e s of the n e u t r a l and i o n i c ground s t a t e s of helium and 1^  the i o n i z a t i o n energy of n e u t r a l helium. A l l d e n s i t i e s are i n cm and e n e r g i e s i n ev. Since no H e l l r a d i a t i o n i s observed i n the plasma n =n and 1 e e q u a t i o n 4-6 reduces to N = n + 2n (4-9) a 0 e S u b s t i t u t i n g 4-9 i n t o 4-7 and s o l v i n g f o r n g i v e s 0 n = P/kT - 2n (4-10) 0 e 32 Equation 4-10 can now be s u b s t i t u t e d i n t o 4-8 to gi v e a r e l a t i o n s o l e y i n terms of known or measureable parameters and kT, so that the e l e c t r o n temperature can be measured. Figure IV-1 i s a graph of n versus kT over the e range of i n t e r e s t and Figure IV-2 i s a diagram of the r a d i a l temperature dependence i n the j e t . D. Density Measurements The accurate determination of the r a d i a l e l e c t r o n d e n s i t y p r o f i l e f o r the plasma i s very important as i t w i l l be the main parameter f o r matching the ex p e r i m e n t a l l y obtained emission p r o f i l e s to the t h e o r e t i c a l p r o f i l e s . Therefore, rather than using l i n e widths of L o r e n t z i a n helium p r o f i l e s , which may be temperature s e n s i t i v e or be so narrow as to introduce a p p r e c i a b l e e r r o r due to instrument broadening, i t was decided to add a small percentage of hydrogen to the helium (approx.. 0.5%) and measure the width of Balmer beta. The f u l l width of t h i s l i n e i s p r o p o r t i o n a l to d e n s i t y to the two - t h i r d s power and t h e r e f o r e i s an e f f e c t i v e t o o l f o r measuring the r a d i a l e l e c t r o n d e n s i t y p r o f i l e . The advantages of using H rather than other methods are s e v e r a l f o l d . H i s approximately 11 A wide 6 at the maximum plasma d e n s i t y . This makes i t easy to 33 SOLUTION TO BOLTZMANN-SAHA EQUATION ( P= I.OATMOS.) ELECTRON ELECTRON TEMPERATURE at ONE ATMOSPHERE (EV) 1.0 I.I 1.2 1.3 14 f I l l i FIGURE IV-1 E l e c t r o n d e n s i t y v ersus temperature f o r an atmospheric plasma. 34 ELECTRON TEMPERATURE (°K) U 20000 10000 0 • 0 6 0 0 0 0 PLASMA RADIUS (mm) 1.0 _ J _ 2.0 FIGURE IV-2 E l e c t r o n temperature v e r s u s plasma r a d i u s . 35 measure and a l s o renders any other form of broadening n e g l i g i b l e . For example the instrument width i s only approximately 0.25 A. A l s o t h i s t r a n s i t i o n i s p r a c t i c a l l y temperature independent. The f u l l h a l f width i s given by -10 2/3 w = 12.48 x 10 « n (4-11) where <* i s given i n Table IV-1. ALPHA Te°K .0808 5000 .0840 10000 .0860 20000 .0860 30000 .0861 40000 Table IV-1 I t can be seen that t h i s introduces l e s s than three percent e r r o r from Te over the range of i n t e r e s t (10000-20000 °K). Figure IV-3 i s a graph of Equation 4-11 showing d e n s i t y versus f u l l h a l f width of H^. The main disadvantage to using H^ i s the presence of He I 4921 which has a t a i l o v e rlapping H . This g i v e s a l a r g e r u n c e r t a i n t y i n the h a l f width at the 36 ELECTRON DENSITY FIGURE IV-3 E l e c t r o n d e n s i t y versus f u l l h a l f w idth o f Balmer b e t a . 37 cen t e r of the plasma where 4921 i s more i n t e n s e and the St a r k broadening more pronounced. T h i s u n c e r t a i n t y can be minimized by c a r e f u l l y s u b t r a c t i n g the t a i l by using c a l c u l a t e d wing p r o f i l e s or by measuring the wing p r o f i l e s i n a pure helium j e t plasma with the same c o n d i t i o n s as the helium and hydrogen plasma. When t h i s i s done, the r a d i a l d e n s i t y p r o f i l e can be o b t a i n e d . T h i s i s shown i n F i g u r e IV-4. 38 -5 - 4 ELECTRON DENSITY -3 (cm"3 ) 16 - 1 X 10 - 9 - 8 - 7 - 6 5 - 5 - 4 - 3 PLASMA RADIUS (mm) " o l 1 1.0 1 i 2.0 J L FIGURE IV-4 E l e c t r o n d e n s i t y versus plasma r a d i u s . 39 CHAPTER V DATA ANALYSIS A. Abel I n v e r s i o n As mentioned i n Chapter I I I - B , at any given height the plasma can be considered to be c y l i n d r i c a l l y symmetric. Therefore, any i n t e n s i t y measurement through the plasma y i e l d s the i n t e g r a t e d i n t e n s i t y across the l i n e of s i g h t . I f the plasma i s o p t i c a l l y t h i n then t h i s i n t e g r a t e d i n t e n s i t y can be w r i t t e n i n terms of the r a d i a l emission c o e f f i c i e n t s , e ( r ) , as f o l l o w s The v a r i o u s geometric values f o r t h i s expression are shown p i c t o r a l l y i n Figure V - l . needed f o r t h i s work, t h e r e f o r e equation 5-1 must be i n v e r t e d to g i v e e (r) i n terms of the measured I ( y ) . This i s done by t a k i n g the Abel Transform (Hormann (1935)). r Ky) (5-1) y I t i s the r a d i a l emission c o e f f i c i e n t s that are (5-2) r GEOMETRY FOR ABEL INVERSION 40 I(y ) <v Y A •>• X FIGURE V - l Abel i n v e r s i o n geometery. 41 With t h i s transform and the a i d of a computer the r a d i a l emission c o e f f i c i e n t s and hence the s p e c t r a l p r o f i l e f o r any t r a n s i t i o n present i n the plasma j e t can be obtained. B. The Software Package The 9-track magnetic tape used i n the experiment co n t a i n s s e v e r a l data f i l e s . Any one data f i l e i s produced by re c o r d i n g the i n t e g r a t e d i n t e n s i t y versus OMA channel number (wavelength) at d i f f e r e n t d i s t a n c e s from the plasma center as the experiment scans the j e t from one s i d e of the flame to the other.. Therefore a f i l e c o n s i s t s of a s e r i e s of records, each 500 data values i n l e n g t h . A complete scan produces approximately 100 records. A t y p i c a l subset of these i n t e g r a t e d i n t e n s i t y p r o f i l e s are shown i n Figure V-2. This tape i s analysed on the U n i v e r s i t y of B r i t i s h Columbia IBM 370 computer using a F o r t r a n program. As can be seen i n Figure V-2, the p r o f i l e s are not smooth and they a l l possess a c h a r a c t e r i s t i c background. The roughness i s caused by j i t t e r i n the OMA and i s p a r t i a l l y removed during a n a l y s i s by the use of a second degree polynomial smoothing r o u t i n e . The background i s a f u n c t i o n of the i n t e g r a t i o n time of the OMA and has no wavelength dependence. Therefore i t can be subtracted 1.0 H CHANNEL I FIGURE V-2 T y p i c a l i n t e g r a t e d i n t e n s i t y p r o f i l e s . 43 from each p r o f i l e . As to not throw away any d a t a , the p r o f i l e s from each s i d e of cente r are averaged together and the end p r o f i l e s , produced b e f o r e the plasma reached the o p t i c a x i s , are t r u n c a t e d . What i s produced to t h i s p o i n t i s a two d i m e n s i o n a l a r r a y o f i n t e g r a t e d i n t e n s i t y v a l u e s v a r y i n g i n both wavelength and i n the d i s t a n c e from the plasma c e n t e r . T h i s a r r a y forms the in p u t to the Abel i n v e r s i o n program which i s b u i l t around the r o u t i n e w r i t t e n by F l e u r i e r and C h a p e l l e (1974) . As can be seen i n Equation 5-2, the a r r a y o f i n t e g r a t e d i n t e n s i t y v ersus d i s t a n c e from plasma cen t e r f o r each of the 500 wavelength v a l u e s must be passed to the Abel i n v e r s i o n program. In t u r n , each a r r a y i s smoothed, d i f f e r e n t i a t e d , smoothed again and then i n t e g r a t e d to g i v e the r a d i a l e m i s s i o n p r o f i l e v e rsus r a d i u s f o r t h a t p a r t i c u l a r wavelength. F i n a l l y t h i s g i v e s a group of un f o l d e d s p e c t r a l l i n e shapes as a f u n c t i o n of r a d i u s . 44 C. Comparison of T h e o r e t i c a l and Experimental R e s u l t s . Figure V-3 shows four e x p e r i m e n t a l l y determined He I 3965 p r o f i l e s f o r a range of d e n s i t i e s found i n the plasma j e t . Although the l i n e can be observed at lower d e n s i t i e s , r e s u l t s are not given here as the i n t e n s i t i e s are low and shrouded i n n o i s e . The s h i f t to the blue of the main t r a n s i t i o n with i n c r e a s i n g d e n s i t y i s c o n s i s t e n t with theory. The 4F-2S forbidden component i s c l e a r l y d i s t i n g u i s h a b l e , although i t i s l e s s intense and broader than p r e d i c t e d by theory (Figure I I - 5 ) . The small r i s e i n the v i c i n i t y of +10 A i s due to the 4D-2S t r a n s i t i o n . I t i s a l s o p e r c e p t i b l e i n Figure I I - 5 . Figures V-4 and V-5 compare the experimental s h i f t s and f u l l h a l f widths with other experimental and t h e o r e t i c a l f i n d i n g s . I t can be seen that both the experimental width and e s p e c i a l l y the s h i f t compare favourably w i t h the c a l c u l a t i o n s of Barnard. The t h e o r e t i c a l values of Griem are higher i n both cases and p a r t i c u l a r l y so f o r the case of the s h i f t . The measurements of D i a t t a are c l o s e r to those obtained i n o t h i s work although h i s temperature was c l o s e r to 10000 K. Figures V-6, V-7 and V-8 are d i r e c t comparisons for the d i f f e r e n t d e n s i t i e s of e x p e r i m e n t a l l y obtained p r o f i l e s w i t h those c a l c u l a t e d by Barnard. The main p o i n t to note from these diagrams i s that the 4F-2S forbidden components do not match w e l l . This p o i n t w i l l be N = 1.9 X IO* - -hi \ N = 1.7 XIO 1 - -f II N = 1.5 x io 1 - T T / / N = 1.3 X I O 1 6 / / / / (cm^) / x ^ / / EXPERIMENTALLY DETERMINED " , 0 LINE SHAPES ( T = 16000 °K) I I l b -10 -8 -6 -4 -2 ( • i i i ' D 2 4 6 8 10 X(S) FIGURE V-3 E x p e r i m e n t a l l y determined p r o f i l e s f o r v a r i n g e l e c t r o n d e n s i t y . ' ' ' ' • ' 1 ' ' FIGURE V-4 S h i f t v ersus e l e c t r o n d e n s i t y ( H e I 3965). FIGURE V-5 F u l l h a l f width versus d e n s i t y (He I 3965) . FIGURE V-7 Comparison of t h e o r e t i c a l and experimental p r o f i l e s . I 1 1 1 1 1 1 1 1 I 1 -10 -8 -6 -4 -2 0 2 4 6 8 10 X(A) FIGURE V-8 Comparison o f t h e o r e t i c a l and experimental p r o f i l e s . discussed in the conclusions. 52 CHAPTER VI CONCLUSIONS As can be seen i n F i g u r e s V-6 through V-8 the agreement between the theory and the experimental f i n d i n g s i s q u i t e good. However, the f o r b i d d e n components are broader and l e s s i n t e n s e than p r e d i c t e d by th e o r y . T h i s can be a t t r i b u t e d to n e g l e c t i n g ion motion e f f e c t s ( q u a s i s t a t i c i o n approximation) when t r e a t i n g the f o r b i d d e n component 4F-2S. As argued by Burgess (1970), the f o r b i d d e n components u s u a l l y l i e i n the t a i l of a Stark broadened l i n e where the q u a s i s t a t i c approximation holds t r u e f o r the main t r a n s i t i o n (p /v>>l/<jj) . However when c o n s i d e r i n g these f o r b i d d e n components, the frequency d i f f e r e n c e must be taken from the f o r b i d d e n l i n e c e n t e r , not the cente r of the main t r a n s i t i o n . The q u a s i s t a t i c approximation t h e r e f o r e does not hold f o r the c e n t r a l p o r t i o n of the f o r b i d d e n t r a n s i t i o n s . In f a c t , the q u a s i s t a t i c approximation does not apply to the c e n t r a l p a r t of the main t r a n s i t i o n e i t h e r . However i t can be shown t h a t the i n t e r a c t i o n between the r a d i a t o r s and the moving ions i n the plasma does not envoke any e x t r a t r a n s i t i o n s and the s t a t i c case can be assumed without c a u s i n g problems f o r the main t r a n s i t i o n . 53 This n o n - r a d i a t i v e i n t e r a c t i o n i s c a l l e d the a d i a b a t i c i n t e r a c t i o n (or approximation) and holds under the c o n s t r a i n t s that E x>> h and I V /E I <<1 (VI-1) ac ac ac where T i s a t y p i c a l c o l l i s i o n time, E i s the energy ac d i f f e r e n c e between the r a d i a t i n g Stark s p l i t l e v e l s and V i s the matrix element of the charged p a r t i c l e - r a d i a t o r ac i n t e r a c t i o n . I t i s p r e c i s e l y the E term which d i s a l l o w s the ac use of the a d i a b a t i c approximation f o r the forbidden components of He I 3965. As can be seen from the energy l e v e l diagram (Figure I I - l ) the 4P l e v e l i s s p l i t by s e v e r a l cm~l w h i l e the forbidden t r a n s i t i o n s p l i t upper l e v e l s , 4F and 4D, l i e too c l o s e together to be resolved on the diagram. Thus the c r i t e r i a f o r the a d i a b a t i c approximation i n Equation VI-1 are upheld f o r the 4P-2S t r a n s i t i o n but not f o r the 4D and 4F-2S t r a n s i t i o n s . T h i s e x p l a i n s why the t h e o r e t i c a l p r o f i l e forbidden components are not i n agreement with the e x p e r i m e n t a l l y obtained p r o f i l e s . The blue s h i f t of the experimental forbidden components (with respect to the t h e o r e t i c a l p r o f i l e ) with i n c r e a s i n g d e n s i t y i s simply due to the f a c t that the main l i n e i n t e n s i t y drops o f f f a s t e r on the red side than the forbidden l i n e i n t e n s i t y i n c r e a s e s . This causes an 54 apparent shift of the peak to the blue. A similar occurance does not result in the theoretical profiles of Figure II-5 as there the forbidden component is narrower due to the effect mentioned above and the two profiles f a l l and rise at approximately the same rate. In conclusion the results found in this work have yielded good agreement with present theory with promise of better results when more theoretical work is done to calculate the ion motion contribution to the broadening, and more experimental work done on other sources to give experimental profiles for different and more varied plasma conditions. 55 BIBLIOGRAPHY Barnard,A.J.,Cooper,J. ,and Shamey,L.J., Astronomy and A s t r o p h y s i c s 1 , 28 (1969) . Barnard,A.J.,Cooper,J.,and Smith,E.W., J o u r n a l of Q u a n t i t a t i v e Spectroscopy and R a d i a t i v e Transfer 14 , 1025 (1974) . Barnard,A.J.,and P i l o n , P . J . , To be published.(1978). Berger,J.,and Fringant,A.,Astromomy and A s t r o p h y s i c s 64 , No.1/2, L9, (1978) . Burgess,D.D., J o u r n a l of Physics B 3 , L70 (1970). Diatta,C.S., Ph.D. T h e s i s , U n i v e r s i t e d'Orleans, (1977). F l e u r i e r , C . , a n d C h a p e l l e , J . , Computer Physics Communications 10 ,200 (1974). F o s t e r , J . S . , Proceedings of the Royal S o c i e t y A 177 , 13-7 (1927). Godfrey,L.A., M.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, (1972). Godfrey,L.A., Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, (1977) . Griem,H.R., Plasma Spectroscopy , McGraw-Hill Book Co., New York,(1964). Griem,H.R., S p e c t r a l Line Broadening by Plasmas , Academic Pr e s s , New York, (1974). Hooper,C.F.,Jr., P h y s i c a l Review 165 , N o . l , 215 (1968). Hormann,H., Z. Physik 97 ,539 (1935). Morris,R.N., Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, (1968). Shipman,H.L., Greenstein,J.L.,and Boksenberg,A., Astronomy J o u r n a l 8_2 , No.7,480 (1977). 

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