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Profile measurements of the lines HeI 4471 and 4922 in a dense plasma Nelson, Robert Howard 1970

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PROFILE MEASUREMENTS OF THE LINES Hel 4471 AND 4922 IN A DENSE PLASMA by ROBERT HOWARD NELSON B.Sc, UNIVERSITY OF BRITISH COLUMBIA, 1965 M.Sc, UNIVERSITY OF BRITISH COLUMBIA, 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA MAY 1970 In p r e s e n t i n g t h i s t h e s i s in 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 C o l u m b i a , I ag ree tha 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 pu rposes 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 . It i s u n d e r s t o o d tha 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 hou 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 o f B r i t i s h Co lumbia Vancouver 8, Canada Date 2>3 ^JxJL, 1970 ABSTRACT Detailed measurements were made of the p r o f i l e s of the l i n e s Hel 4471 and 4922 A i n a pulsed arc plasma, for electron d e n s i t i e s i n the range 3 x l O 1 ^ to l O 1 ^ cm"3 and at a temperature of 4.5 eV. P h o t o e l e c t r i c detection was employed i n a rapid scan spectrometer. The r e l a t i v e i n t e n s i t i e s agreed w e l l with the computed values, but the component peaks for each l i n e were closer together. i i TABLE OF CONTENTS Page CHAPTER I INTRODUCTION 1 CHAPTER II THEORY 2.1 Introduction 4 2.2 Ion Broadening 5 2.3 E a r l y Treatments of E l e c t r o n Broadening 8 2.31 Lorentz Impact Theory 8 2.32 Types of C o l l i s i o n s 8 2.33 Adiabatic Theory 10 2.4 Modern E l e c t r o n Broadening Theory 11 2.41 Impact Approximation 11 2.42 C l a s s i c a l Path Approximation . . . . . . . 11 2.43 The Line Shape 12 2.44 V a l i d i t y of the Assumptions 18 CHAPTER I I I APPARATUS 3.1 Introduction 21 3.2 Plasma Vessel 21 3.3 Discharge C i r c u i t 23 3.4 Mono chroma tor 23 3.5 Rapid Scan Spectrometer 25 3.^ 51 Construction 25 3.52 Alignment . . 27 3.53 C a l i b r a t i o n 28 3.6 Photomultiplier 35 3.7 E l e c t r o n i c Timing Devices . . . . 37 3.8 Oscilloscopes 40 i i i Page CHAPTER IV EXPERIMENTAL WORK 4.1 I n t r o d u c t i o n 42 4.2 Side-On Observations . 42 4.21 E l e c t r o n Density . 44 4.22 E l e c t r o n Temperature . . . . . . . 45 4.23 I n i t i a l S p a t i a l l y Resolved Measurements. . . 47 4.3 End-On Observations 48 4.31 C a l c u l a t i o n of O p t i c a l Thicknesses 49 4.32 S p a t i a l l y Resolved Density Measurements. . . 51 4.33 F i n a l P r o f i l e Measurements . . . . . . . . . 57 CHAPTER V DISCUSSION OF ERRORS 5.1 I n t r o d u c t i o n 67 5.2 Instrument E r r o r s . . . . . . . . . 68 5.3 Measuring E r r o r s • • • • 68 5.4 Plasma V a r i a t i o n s . . . . . 69 CHAPTER VI CONCLUSIONS 6.1 Observed P r o f i l e s 72 6.2 P u l s e d Arc Plasma .. . . . . . . . ..... 75 6.3 Rapid Scan Spectrometer . . . . . . . . 75 APPENDIX I D e r i v a t i o n of the E f f e c t i v e Sweeping Arm . 78 APPENDIX I I The C o r r e c t i o n f o r Instrument Broadening . . . . . . . 80 APPENDIX I I I E l e c t r o n i c Timing Devices. . 84 APPENDIX IV ADDITIONAL PROFILES . . . . . . . . . . . . . . .86 BIBLIOGRAPHY 88 iv LIST OF FIGURES Number ' • Page 1 Stark E f f e c t I n t e r a c t i o n of the 4'S, 4'P, 4'D and 4'F Levels i n Helium 6 2 Ion F i e l d D i s t r i b u t i o n Function W(F) (Baranger and Mozer, 1959) 7 3 A T r a n s i t i o n (X—>fi with Perturbing Level 0Cf 9 4 The A and B Functions of Griem et a l , 1962 16 5 Types of Broadening C o l l i s i o n s . . . . . . . . . 20 6 Plasma Vessel 22 7 Discharge C i r c u i t and Timing Devices 24 8 Rapid Scan O p t i c a l System 26 . 9 Two P o s s i b l e Orientations of Entrance S l i t S^ and E x i t S l i t S . . . . . 27 " r 10 Width of T o t a l Instrument P r o f i l e . . . . . 29 11 The Constant C( X ) . . • 31 12 Reduced Aperture f o r Non Central Wave Lengths. . . . . . . . . 32 13 Transmission of Spectrometer at X 5460 33 14 Transmission of Spectrometer at X 8942 (Second Order). . . 34 15 External C i r c u i t f o r the Phot o m u l t i p l i e r . . . . . . . . . . 35 16 Non-Linear Response of the Pho t o m u l t i p l i e r . . . . . . . . . 36 17 Timing Sequence 38 18 Sequence at Coincidence I 38 19 Sequence at Coincidence II . . 39 20 T r i g g e r i n g Sequence. 39 21 Time V a r i a t i o n of T y p i c a l Hel and H e l l Lines . . . . . . . . 43 22 T o t a l Wave Length Response of Monochromator and Photo-m u l t i p l i e r 46 v Number • Page 2 3 Hypothetical Radial Variations of 7 t e 4 7 24 Hypothetical P r o f i l e of X 4 4 7 1 or 4 9 2 2 47 25 O p t i c a l Arrangements for S p a t i a l l y Resolved Side-On Measurements 49 26 O p t i c a l Arrangements f or S p a t i a l l y Resolved End-On Measurements 5 2 27 R e p r o d u c i b i l i t y of Hel 4 4 7 1 Line In t e n s i t y 5 3 28 End-On S p a t i a l l y Resolved X 3 8 8 9 P r o f i l e s . . . 5 5 29 End-On S p a t i a l l y Resolved X 3 8 8 9 P r o f i l e s 56 30 O p t i c a l Arrangement f or S p a t i a l l y Resolved Measurements Along The Axis 57 3 1 Comparison of End-On and Side-On X 3 8 8 9 P r o f i l e s 58 3 2 Comparison of End-On and Side-On X3889 P r o f i l e s 59 3 3 T y p i c a l P r o f i l e of Hel 4 4 7 1 with Parameters Labelled . . . . 6 1 34 V a r i a t i o n of F u l l Width, W With T i g 6 3 3 5 V a r i a t i o n of Separation, S With T i g . 64 36 V a r i a t i o n of I n t e n s i t i e s , I f and I With "Vlg . 6 5 37 Observed and Calculated (Barnard et a l , 1 9 6 9 ) P r o f i l e s of Hel 4 4 7 1 and 4 9 2 2 . 66 3 8 A Correction to the Computed P r o f i l e of Hel 4 4 7 1 . 73 39 Improved Designs f o r the Rapid Scan Spectrometer . . . . . . 77 4 0 Image S h i f t When Mirror Mfi Rotates By A0 . 79 4 1 Approximate Correction f o r Instrument Broadening . . . . . . 8 3 . . • • v i ; ACKNOWLEDGEMENTS • I wish to thank Dr. A. J. Barnard f o r h i s kind help and support throughout the experiment and i n the preparation of this t h e s i s . I wish to thank also the other members of the Plasma Physics Group with whom I have had many h e l p f u l and stimulating discussions; i n p a r t i -c ular, Dr. D. E. Roberts, Mr. J. Burnett and Dr. J. Meyer. Thanks are due also to the technical s t a f f of the group, i n p a r t i c u l a r , Mr. J. Dooyewerd and Mr. D. Sieberg who designed and b u i l t the e l e c t r o n i c timing devices used i n this experiment; Mr. J . Lees and Mr. E. Williams who b u i l t and maintained the vacuum system; Mr. T. Knop who b u i l t the r o t a t i n g mirror assembly; and Mr. R. Haines and Mr. D. Brydle who a s s i s t e d i n b u i l d i n g the o p t i c a l mounts for the rapid-scan spectrometer. The f i n a n c i a l assistance of National Research Council and H. R. MacMillan Family Fellowships i s g r a t e f u l l y acknowledged. v i i CHAPTER I INTRODUCTION Inte r e s t i n spectroscopy dates back to S i r Isaac Newton who was the f i r s t person to recognize the composite nature of white l i g h t . Since that time spectroscopy has yielded an enormous amount of information about the atom and has led eventually to the development of quantum mechanics. Spectroscopy i s now extensively used f o r research i n physics, chemistry and astrophysics and has widespread i n d u s t r i a l a p p l i c a t i o n s . I t i s natural that spectroscopy should be used i n the study of laboratory plasmas. There are two main reasons f o r doing so. One i s the quest for f u r t h e r basic knowledge concerning the i n t e r a c t i o n of charged perturbers (ions and electrons) with atoms emitting v i s i b l e l i g h t . Although the basic quantum mechanical theory i s now well understood, the approximations required to derive numerical r e s u l t s ( i n p a r t i c u l a r l i n e shapes for various e l e c t r o n and ion temperatures and densities) are open to some doubt and require v e r i f i c a t i o n by experiment. The second important reason for studying s p e c t r a l l i n e s emitted by plasmas i s diagnostics. For laboratory plasmas, spectroscopic measure-ments have the advantage that they do not perturb the plasma whereas for means astrophysics, spectroscopic measurements are one of the few for studying A s t e l l a r structures. Hence experimentally v e r i f i e d p r o f i l e s are required. Two important l i n e s i n s t e l l a r atmospheres are the neutral helium l i n e s A4-922 ft'p-VD) and X 447/ (2 3 p-4 - 3 D ) for each of which the ZP-fyF forbidden component and to a lesser extent the 2P-4-P forbidden component appear strongly. These components a r i s e from the 2 mixing of the 4*D > 4 -F a n c* *^ P levels of different parity by the JL, intense electric fields in the stellar plasma. Griem (1968) and Barnard, Cooper and Shamey (1969) have calculated profiles of these lines for various electron temperatures and densities. Laboratory measurements for the lines W4471 and 4922 have been carried out by Wu'lff (1958) for an electron density, T l of 3 x 10^ cm while Sadjian (1961) has measured the A. 4922 profile for V*. = c* 16 3 5 x 10 cm . Rather poor agreement is apparent between Sadjian's and Wulff's measurements o f A 4922, and Wulff's measurement of A 4471 shows some discrepancy with the theoretical profiles. Hence i t was decided to measure again the profiles of these lines, this time for a range of electron densities in order to better check the theory. (The profiles may be expected to vary strongly with Y l and only weakly with "~[~ , the electron temperature.) Chapter II outlines the theory used by Griem and Barnard in the calculation of the line profiles. The v a l i d i t y of the assumptions is then examined. Chapter III describes the rapid scan spectrometer which was built to scan each spectral line in one shot. The timing devices are bri e f l y described as well as the construction of the pulsed arc. Chapter IV describes preliminary studies of the plasma, the measurement of Y l p and kTL , and an examination of the spatial homo-geneity as well as the measurement of the profiles of interest. An analysis of errors is presented in the next chapter with the conclusions in Chapter VI. * \4471 only Appendix I presents a d e r i v a t i o n of the e f f e c t i v e sweeping arm of the spectrometer and a c o r r e c t i o n f o r the e f f e c t of instrument broadening i s derived i n Appendix I I . The c i r c u i t diagrams for the timing devices are given i n Appendix I I I . CHAPTER I I THEORY 2.1 Introduction The sources of spectral l i n e broadening for o p t i c a l l y thin plasmas can conveniently be grouped into three main categories: 1) Natural broadening, which i s a r e s u l t of the f i n i t e l i f e t i m e of the emitting states, 2) Doppler broadening, which i s a r e s u l t of the motion of the emitting atoms (or ions), and 3) Pressure broadening, which i s a re s u l t of charged (or neutral) perturbers interacting with, the emitting atom (ion). In the highly ionized plasmas studied i n this experiment pressure broadening i s the o dominant mechanism as a t y p i c a l natural linewidth i s only 0.0002 A and o the Doppler width (for neutral helium lines) i s t y p i c a l l y 0.36 A , com-o pared with observed widths of 15 - 50 A for Hel 4471 and 4922. In par t i c u l a r the dominant perturbers are the ions and electrons since the neutral - neutral interactions (occurring i n Van der Waals and resonance broadening) are much weaker than those with charged perturbers. The ions and electrons perturb the emitting atom by the same mechanism - the interaction between the perturbing charge and the charge d i s t r i b u t i o n of the atom. For most purposes (neutral emitters only) i t i s s u f f i c i e n t to keep only the f i r s t term i n the multipole expansion of the interaction. The inte r a c t i o n potential i s then V= cL • £ where cL i s the dipole moment of the atom and i s the e l e c t r i c f i e l d produced by the perturbers at the center of the atom. Although the interaction p o t e n t i a l s are i d e n t i c a l , t h e i n t e r a c t i o n or c o l l i s i o n times for the ions and electrons are widely d i f f e r e n t and hence the mathematical treatment f o r each i n the c a l c u l a t i o n of l i n e p r o f i l e s w i l l be d i f f e r e n t . One can h a l f width of the l i n e i n frequency units) and compare this time with the c h a r a c t e r i s t i c c o l l i s i o n time f o r the electrons or ions. The c o l l i s i o n distance and \f , the mean ion or el e c t r o n speed. If TJ. , the ion c o l l i s i o n time i s very much greater than the inverse linewidths, then the motion of the ions i s unimportant i n perturbing the emitter and the " q u a s i - s t a t i c approximation" can be used to c a l c u l a t e the broadening e f f e c t due to the ions. On the other hand i f Tg , the ele c t r o n c o l l i s i o n time i s much less than the inverse linewidth then the impact approximation can be used f o r the el e c t r o n broadening. This l a s t approximation i s us u a l l y v a l i d at l e a s t near the center of the l i n e and w i l l be discussed i n the next section. 2.2 Ion Broadening In using the q u a s i - s t a t i c assumption, one i n f e r s that the t y p i c a l ion c o l l i s i o n times are long compared to times of i n t e r e s t i n the per-turbed emitter. The motion of the ions can therefore be neglected i n c a l c u l a t i n g the broadening. The f i r s t step i s to ca l c u l a t e the Stark s p l i t t i n g f o r an a r b i t r a r y e l e c t r i c f i e l d . This i s well known. (See F i g . 1, from Condon and Shortley, 1935.) In hydrogenic atoms one has the l i n e a r Stark e f f e c t ; i n other atoms, i t i s quadratic f o r small f i e l d s and l i n e a r f o r large ones. (For close l y i n g l e v e l s one has to ca l c u l a t e f o r think of a r e l a x a t i o n time for the emitter TJ - — (where A W i s the (semi) time can be estimated by f i e l d combinations of L S states of d i f f e r e n t p a r i t y . This mixing of states allows t r a n s i t i o n s normally forbidden under the e l e c t r i c dipole s e l e c t i o n rules.) too I | 1 1 1 1 I 1 1 1 | -520 -I I L. v\y 0.2 0.+ O.b 0.8 1.0 £ in T O 5 v o l t s / c m FIG. 1.--Stark E f f e c t I n t e r a c t i o n of the 4"*S , 4"'P > ^0 and J^- p Levels i n Heli .ium The next step i s to c a l c u l a t e the ion f i e l d d i s t r i b u t i o n , VV(F) • The f i r s t such c a l c u l a t i o n was made by Holtsmark (1919) for independent ions A c t u a l l y i n f u l l y ionized plasmas the ions are correlated between them-selves and also with the electrons. Later c a l c u l a t i o n s of the ion f i e l d d i s t r i b u t i o n (Ecker and M i i l l e r , 1958) replaced the Coulomb f i e l d used by Holtsmark by the Debye f i e l d (see Baranger, 1962 p. 536). Mozer and Baranger (1959, 1960) and Hooper (1968) have furt h e r r e f i n e d the c a l c u l a l a t i o n s of W ( F ) . Their r e s u l t s are close to those of Ecker and Miiller. Once the e l e c t r i c f i e l d d i s t r i b u t i o n i s known, one can c a l c u l a t e the broadened p r o f i l e due to ions alone by taking the (sharp) Stark com-ponents f o r an a r b i t r a r y e l e c t r i c f i e l d and averaging over the e l e c t r i c f i e l d d i s t r i b u t i o n . This procedure was followed by some e a r l y workers who erroneously assumed that since the e l e c t r o n c o l l i s i o n s were f a s t , t h e i r e f f e c t s averaged to zero. A c t u a l l y the electrons are frequently a more important source of broadening. Since the two e f f e c t s are s t a t i s t i c a l l y independent, the usual procedure i s to c a l c u l a t e the Stark s p l i t t i n g f o r an a r b i t r a r y e l e c t r i c f i e l d , c a l c u l a t e the e l e c t r o n broadening i n t h i s f i e l d and then average the r e s u l t i n g p r o f i l e s over the e l e c t r i c f i e l d d i s t r i b u t i o n . 0.5 0.4 0.3 0.2 0.1 0 W(F) Average I n t e r p a r t i c l e Distance (*/3 TI r / n » i ) f>0 = Debye Length r i e e % 4 ro/P0 = O.O (Holtsmark) — r./p0 = o.8 0 4 P =F7FC FIG. 2.—Ion F i e l d D i s t r i b u t i o n Function W(F) (Baranger and Mozer, 1959) 8 2.3 E a r l y Treatments of Electron Broadening 2.31 Lorentz Impact Theory The f i r s t treatment of c o l l i s i o n broadening was made by Lorentz (1906) who assumed that the (electron) c o l l i s i o n s are instantaneous and occur with a frequency l)^ , each c o l l i s i o n i n t e r r u p t i n g the wave t r a i n completely r e s u l t i n g i n a random phase s h i f t . The l i n e p r o f i l e can e a s i l y be derived c l a s s i c a l l y by taking Fourier transforms (see Margenau A fj-\ - A i^o^ and Lewis (1969), p. 571). A wave t r a i n of amplitude A v t ; ~ P\Q Q emitted f o r time T w i l l have a Fourier transform -r, f A -t~>* r - t (co-tojt Q ~ \ J(<*j) ~ J A ( i ) e <* Je = -tcco-co.)— I f t h i s were the only r a d i a t i v e event the l i n e shape would be given by J J (OL>}"T") | . However the observed r a d i a t i o n w i l l be from many atoms emitting for a Poisson d i s t r i b u t i o n of times, T " . If the mean emission time i s given by T>~^/l)c then the p r o b a b i l i t y a given atom w i l l r adiate for a time T i s X * «3 and the l i n e shape w i l l be -T/z 1 O C This i s the well known Lorentz d i s p e r s i v e p r o f i l e whose f u l l width at h a l f maximum i n t e n s i t y (frequency units) i s . The c o l l i s i o n f r e -quency i s given by l)^ •=• J\J^VJ"^> where N i s the number density of per-turbers, "V t h e i r speed and CT the c o l l i s i o n or " o p t i c a l " cross section. (The average i s performed over a l l perturber speeds.) 2.32 Types of C o l l i s i o n s In c o l l i d i n g with the atom the perturber can a f f e c t the emission process i n two ways: 1) I t can s h i f t , during the c o l l i s i o n time, the energies of the upper and lower states of the t r a n s i t i o n , producing a net phase s h i f t i n the emitted r a d i a t i o n ( e l a s t i c c o l l i s i o n ) , and 2) I t can terminate the emission e n t i r e l y ( i n e l a s t i c c o l l i s i o n ) . The occurrence of the two cases can be found from the quantum mechanical c r i t e r i o n f o r adiabacity. (See Margenau and Lewis (1959), p. 578 f f or Bohm (1951), p. 448.) A E , FIG. 3.--A T r a n s i t i o n OL -»y2 with Perturbing Level QL I f the product of the c o l l i s i o n time T c = f'/'V and the energy d i f f e r e n c e A £ , i s much greater than "ft. , the c o l l i s i o n s w i l l be mostly adiabatic OL oL (and therefore e l a s t i c ) . In the converse case, f o r /iE < "Vl the V OCOL' c o l l i s i o n s w i l l be mostly d i b a t i c (and i n e l a s t i c ) . (See F i g . 3.) A c o l l i s i o n can also be described as strong or weak. A c o l l i s i o n w i l l be weak when the product of the i n t e r a c t i o n energy V -cL'£> a n ^ the c o l l i s i o n time Tc — p/lt i s much smaller than . The dipole moment ci can be written as d — P R. . and the e l e c t r i c f i e l d as & — ~ ~ ~ 7, 10 so that for weak c o l l i s i o n s — <^ C VX or J 5 V » a constant. 2.33 Adiabatic Theory Lenz (1924) and Weisskopf (1932) were able to c a l c u l a t e actual phase s h i f t s f o r the adiabatic case and thereby obtain an expression for the o p t i c a l cross-section. In the absence of perturbations, the phase ^ would be a l i n e a r function of time. If during a c o l l i s i o n the upper and lower energies of the t r a n s i t i o n were s h i f t e d by an amount A Efjg ("fj) and AE-("fc) then the net phase s h i f t would be P A T ° = f (LO-UJJ dt = ~ J [ A Ot) -_\(-£) J dt where the i n t e g r a t i o n i s over the c o l l i s i o n time *Cg . The energy d i f f e r e n c e AE C"t) — AEa (~t) i n the a d i a b a t i c case w i l l be given cc p . . . , 2. . 2 / i 1 * by C"(rv for r — (jD + "U~ "L / , the distance between the perturber and the center of the atom, and the net phase s h i f t A w i l l be a function of j3 and "V . Lenz and Weisskopf c a l c u l a t e d phase s h i f t s f o r various j> and 1 / * and found for a given average * V a cut-off parameter f o r which A - f . Hence the Lorentz o p t i c a l cross section can be w r i t t e n as CT" = 77. J 3 Q . C o l l i s i o n s f o r which p < p^ were regarded as strong and contributed to the broadening whereas the weak c o l l i s i o n s f o r p > po were neglected. This theory and f u r t h e r improvements by Lindholm (1941) and Foley (1946) which predicted a l i n e s h i f t as well as a width met with some degree of success; however, they were v a l i d only for the adiabatic case. (The a d i a b a t i c assumption i s v a l i d for low per-turber energies £ — f< , small compared to E" , .) However, at high temperatures the i n e l a s t i c c o l l i s i o n s w i l l dominate, making the a d i a b a t i c assumption i n v a l i d . Furthermore i n the case of hydrogen the l e v e l s are degenerate and one w i l l have A E , O . A c o l l i -sion can be e l a s t i c and s t i l l change the state of the atom - again making the adiabatic assumption i n v a l i d . A s i m i l a r problem i s met i n c a l c u l a t i n g the ion broadening where one has close l y i n g Stark components. 2.4 Modern E l e c t r o n Broadening Theory 2.41 Impact Approximation An extension of the impact approximation helped overcome these d i f f i c u l t i e s . (See Baranger (1962), p. 498.) E s s e n t i a l l y i t says that the average c o l l i s i o n i s weak. Perturbation theory can then be used and the e f f e c t s of several c o l l i s i o n s , which may be occurring simultaneously, can be added a l g e b r a i c a l l y . This does not mean that strong c o l l i s i o n s cannot occur - t h e i r e f f e c t can be estimated roughly, f o r example, by the Lorentz theory. However the strong c o l l i s i o n s must be well separated i n time, occurring one at a time so that t h e i r e f f e c t s w i l l not be over-estimated. The impact approximation w i l l be v a l i d i f the weak c o l l i s i o n s produce most of the broadening. V a l i d i t y c r i t e r i a w i l l be examined l a t e r . 2.42 C l a s s i c a l Path Approximation A second approximation u s u a l l y made i s the c l a s s i c a l path approxi-mation. I t states that the perturbers can be treated as c l a s s i c a l p a r t i c l e s which follow predetermined t r a j e c t o r i e s ( s t r a i g h t l i n e paths f o r neutral emitters). I t i s furth e r assumed that the i n t e r a c t i o n p o t e n t i a l between the perturber and emitter (atom) i s independent of the state of the emitter. This l a s t condition i s s l i g h t l y i n c o n sistent. The perturber can have an i n e l a s t i c c o l l i s i o n with the atom, thereby r a i s i n g i t s energy, but the atom i s not allowed to transfer this energy back to the motion of the perturbers. The atoms would eventually reach an i n f i n i t e temperature. However, i f the linewidth i s large enough then the t y p i c a l r e l a x a t i o n times w i t h i n the atom w i l l be short enough to prevent this occurrence. In order f o r the c l a s s i c a l path approximation to be v a l i d one must f i r s t be able to use wave packets. A wave packet of s i z e X Q w i l l have a momentum spread A p — which, i n order for the wave packet to hold together for any length of time, must be much smaller than the average momentum p = "^/iJL f ° r ^ the de Broglie wavelength. Then X Q ^ W ~ "fa /~Yr\~\y . But i n order f o r the perturber to be a c l a s s i c a l p a r t i c l e the c o l l i s i o n distance p must be much larger than X 0 . Hence one must have jD » ^h./yri'V or Wvp/I^ -Jl >/> 1 f o r JL , the o r b i t a l angular momentum quantum number. There w i l l always be some perturbers with small JL but the approximation w i l l be good i f they do not contribute much to the broadening. Note that i f the c l a s s i c a l path approximation i s v a l i d f o r electrons i t i s c e r t a i n l y v a l i d for ions since l i = M V p = £ i y « . W v p = y ^ i a (=864 * , f e i ( i « ) . 2.43 The Line Shape The following treatment of e l e c t r o n broadening w i l l not be a rigorous d e r i v a t i o n of the l i n e shape but rather a sketch of the approach followed by Baranger (1962) and gi v i n g r e s u l t s by Griem (1964). P a r t i -cular a t t e n t i o n w i l l be paid to the assumptions made and t h e i r v a l i d i t y i n this case. A general expression f o r the p r o f i l e of a l i n e F Cw) emitted i n the presence of an (electron) perturber i s given by Baranger (1962): F M - 2 5(to-w. f) \<f\d\i>\*p. xvhere the summation i s carried over a l l the i n i t i a l states 1 and f i n a l states £ of the t o t a l system. The energy separation of i n i t i a l and f i n a l states i s given by " K '•> d and p^ are the dipole operator of the system and the density matrix respectively of the t o t a l system. The Fourier transform of f~(<~o) i s then oo r — TUJ 5 —I Z. UJ. ~ S n e Fo>) du> = Z e I f |<f ! ^ U > I P . - 0 0 The time evolution operation T(S) - 6Xp(-fHs/^a) for H , the Hamiltonian i s introduced and 4* (S) can be written as 4* ( 2 ) = TV [d T ( s ) ct Tcs) p ] without reference to i n i t i a l and f i n a l states. The c l a s s i c a l path approximation i s then usually made which allows one to average over perturber states and sum the trace only over the atomic states. Then d , ~X and j O operate only on the bound states. Another approximation usually made i s that one can neglect the lower state broadening. This i s because the interaction Hamiltonian i s proportional to the dipole moment of the atom which i s much greater for the higher p r i n c i p a l quantum numbers. (This i s a good approximation for the neutral helium lines studied.) One then defines the lower state as zero energy and assumes that the density matrix jD^ i s approximately constant over the upper states. Then <p (s) = T^j] D £(s/j ^ where A V refers to an average over impact parameters p and perturber speeds U- and [) =2 ^ I c C j /S><C/6| dM\ Ot'y i s an operator acting oLoL' ftp " ' r r between upper states J Of } and | G£ . i s a lower state. Baranger (1962) argues that, since the impact approximation implies that the average c o l l i s i o n i s weak, i t takes many c o l l i s i o n times to distu r b the atom. (Strong c o l l i s i o n s are assumed to be infrequent -t h e i r e f f e c t i s treated separately.) Only the average e f f e c t of the ele c t r o n c o l l i s i o n s matters. " A l l happens as i f some time independent perturbation Jr£ has been added to the atomic Hamiltonian . The times involved i n the c a l c u l a t i o n of 3*£ are much larger than c o l l i s i o n times so that 36 depends only on the net r e s u l t of a c o l l i s i o n , not on i t s d e t a i l e d development. The l i g h t i s the same as i f i t were emitted by an i s o l a t e d atom with Hamiltonian "f* and, since i s not Hermitian, the energy l e v e l s have an imaginary part and the l i n e s have a width." Hence the evolution operator can be wr i t t e n as The t o t a l l i n e shape due to both ion and el e c t r o n perturbers i s then given by, (Griem, 1964) and the l i n e shape due to ele c t r o n impacts w i l l be states of the ion f i e l d F . The function W(F ) i s the ion f i e l d d i s t r i b u t i o n . The angular frequency i n t e r v a l (-O depends on the 15 f i e l d strength: oo - E (F) ~ E„ The matrix elements of the operator $ are (taking only the second order terms): oo £ a " -00 -OO I J OO  where , = E" CF) - E (F) "~ oca." <x ot" and v / / s _ j e - egf°(£ T- yrt) V ( t ) - - ( p a + T r i t * ) 3 / a The brackets ^ ^ i n d i c a t e an average over perturber impact parameters and v e l o c i t i e s If" . The expression f o r ^^^i c a n ^ e s i m p l i f i e d by taking the angular average f i r s t . Then oo -o3 _ oo where j ' C v ) i s the electron v e l o c i t y d i s t r i b u t i o n . Introduce the var i a b l e s x = -wt/p x ' = -vl-'/p and Z = p c O ^ / v - Z # = f ^ V " / " ^ and the r e a l functions A and B def ined by 16 These functions, which are tabulated by Barnard et a l (1969), are generalizations of the A and B functions of Griem et a l (1962); they reduce to these when Z - % • The functions A(5S) and B(«) are p l o t t e d i n F i g . 4. FIG. 4.--The A and B Functions of Griem et a l (1962) With these d e f i n i t i o n s the matrix elements O . can be written i n the form ' " ^ ' ( t f ? <*> "V""**'14>'> K ¥ [A (%,*) + f Sfsytv] The dipole matrix elements ^cU | c^u | between upper states can be r e a d i l y calculated by the Coulomb approximation (a good model f o r helium). The i n t e g r a l over perturber speeds converges at both l i m i t s and i s r e l a t i v e l y i n s e n s i t i v e to the type of speed d i s t r i b u t i o n $C^~) which i s u s u a l l y assumed to be Maxwellian. However, the i n t e g r a l over p diverges at O and o£> and the 17 usual procedure i s to cut o f f at some l i m i t s 0 . and Q„,_ . Physi-c a l l y an upper cut-off should e x i s t since the charges are screened at large distances and one should truncate at some n 0 the Debye length. (Chappell et a l (1969) set 0 •= 0 .682.P •) At the lower l i m i t J M I X jo the perturbation theory used i n the c a l c u l a t i o n of ^ breaks down and one should truncate at a minimum impact parameter j-^i'rt " Griem (1964) estimates J^f^ by ^ Tl/Wir for "H. = p r i n c i p a l quantum number of the upper state Od . The e f f e c t of the strong c o l l i s i o n s i s estimated by the Lorentz-Weisskopf theory; to the diagonal elements ^<^cC ^S ac^ec* the term — "Yl & J T l P ^ IT £(-v) cLlr which corresponds to the c o l l i s i o n frequency = Nl^n/CT^ mentioned previously. The c a l c u l a t e d l i n e s h i f t i s .somewhat more s e n s i t i v e than the width to the upper cutoff parameter Pyndy a s t ^ i e ^ ° 1 lowing considerations w i l l i n d i c a t e . To evaluate the l i n e shape F(£-°) one has to i n v e r t a matrix of c o e f f i c i e n t s (V , . For i s o l a t e d l i n e s this matrix i s one x (HOC1 by one. Baranger (1962) shows that for t h i s case ( e l e c t r o n broadening only) the l i n e shape i s Lorentzian and given by for C a constant and - w + id — . Then the r e a l part of which i s r e l a t e d to A (Z<) i s equal to the width, and B (%•) i s associated with the s h i f t . For overlapping l i n e s the r e l a t i o n s h i p w i l l not be so simple; however, the diagonal terms a r e u s u a l l y much larger than the o f f diagonal terms and there w i l l s t i l l be a strong r e l a t i o n s h i p between the width and the r e a l part of $ ' , and between the s h i f t and the imaginary part of ^ctoi' ' evaluating the matrix elements ^Q/OC' o n e i n t e S r a t e s over impact parameter and truncates at some j 3 m a ? ( • The r e a l part, which i s dependent upon 18 A (SEjjS ' ) w i l l be r e l a t i v e l y , i n s e n s i t i v e to the upper cutoff j O w a x since A (S^JE'J f a l l s to zero r a p i d l y with increasing 5£ . However Q(%t?L') , which determines the imaginary part of $ Q ^ 0 ^ / » g ° e s t o zero more slowly and the upper cutoff parameter, j 3 m C L > f that one choses i s more important. In th i s way the s h i f t of the calculated l i n e p r o f i l e i s more s e n s i t i v e than the width to the upper cutoff J^m0iX • 2,44 V a l i d i t y of the Line Broadening Assumptions The plasma studied i n this experiment was found to have an electron temperature of 4.5 eV and an el e c t r o n density of 3 x 1 0 ^ to 1 0 ^ cm ^ 16 3 (see section 4.1). Using a mean value of 5 x 10 cm and assuming an i d e n t i c a l value f o r the ion density the following parameters were c a l -culated: Debye Length p^ s ^ ^ l = 5.0 x 10" m. rne^ 13 -1 Plasma Frequency CO - A / T = 1.26 x 10 sec * e 16 A (at A 4922) Holtsmark Normal F i e l d Strength F0 = ^ TiT N i = °- 5 1 x 1 0 v 1 c m Fre f quency Separation (jj . — ^^oCot'/'rX rom Closest Level c C o C' ' / i i n 1 2 - 1 4.3 x 10 sec (23 cm"1) Elec t r o n Thermal Speed 'y-r — 3 k T . 1.54 x 10 m/sec Ion Thermal Speed i3kTt 1.80 x 1(T m/sec 19 t -9 Cutoff Impact Parameter J - ^miK ~ " = 1.24 x 10 Linewidth (Semi-halfwidth) A CO = 0.87 x 1 0 1 3 s e c " 1 (11 A" ) The types of c o l l i s i o n s for t h i s plasma were then depicted on a p-'V p l o t ( F i g . 5) due to Roberts (1968). The l i n e p = 'v/0^*.' indicates the e l a s t i c and i n e l a s t i c c o l l i s i o n s ; since ^-^ot' i s small, most of the c o l l i s i o n s are i n e l a s t i c . The curve p-ys C indicates the weak and strong c o l l i s i o n s . (See section 2.32.) The Maxwellian speed d i s t r i b u t i o n for an el e c t r o n temperature of 4.5 eV i s also p l o t t e d . The v a l i d i t y of the assumptions r e f e r r e d to i n the previous sections can then be checked. 1) Q u a s i - s t a t i c Approximation. This i s v a l i d since the t y p i c a l ion c o l l i s i o n time, given by %± ~ j ^ m a * , / ' ^ = ^ ' ^ x ^ s e c ^ s much greater than the inverse linewidth i/ato = 1.1 x 10 sec. 2) Impact Approximation. This i s v a l i d since the longest electron c o l l i s i o n times, T g = pma.K/'V'e = 2.2 x 10 ^ sec are shorter than the inverse linewidth. The strong c o l l i s i o n s are w e l l separated i n time since the duration of a strong c o l l i s i o n , given by j^min/"We ~ ^"^ X ^ ^ sec i s much smaller than the mean time between strong c o l l i s i o n s , which - i „ , . - 1 2 i s es timated by |/»Vl£ = 2.6 x 1 0 " 1 2 sec. 3) C l a s s i c a l Path Approximation. This i s v a l i d since the minimum angular momentum (strong c o l l i s i o n s ) i s given by Jl0 — '*ne"^e Pvnin __ ^ e ft 4) Neglect of Lower State Broadening. This i s v a l i d since the 1 1 5 s h i f t of the upper states 4 D, 4 F for a f i e l d of 0.5 x 10 v/cm i s i 11 cm 1 (Condon and Shortley, 1935). The s h i f t of the lower l e v e l , 20 1 -1 2 P f o r this t r a n s i t i o n i s only 0.007 cm . The r e l a t i v e e f f e c t on the lower l e v e l by e l e c t r o n impacts w i l l be s i m i l a r l y small. 5) Small Strong C o l l i s i o n Broadening. This i s v a l i d . The width 2. 11 due to strong c o l l i s i o n s i s given by VV^  = 7 7 T L Pm(rt = 3.7 x 10 -1 ° sec . This corresponds to a ( f u l l ) h a l f width of 0 . 9 4 A which i s small compared with the observed widths. If-N E L A S T I C E L A S T I C FIG. 5.--Types of Broadening C o l l i s i o n s M A X W E L L I A N S P E E D D I S T R I B U T I O N CHAPTER I I I APPARATUS 3.1 Introduction This chapter presents a d e s c r i p t i o n of the apparatus used to pro-duce and measure the overlapping helium l i n e s . The plasma source chosen for this endeavor was a w a l l - s t a b i l i s e d pulsed arc driven by a slow capaci-tor bank (see sections 3.2 and 3.3). I t has several advantages over other devices such as d.c. arcs, shock tubes and z-pinches, commonly used for l i n e p r o f i l e measurements. The present device i s much more uniform than a d.c. arc; the tedious and inaccurate s p a t i a l unfolding of p r o f i l e s i s unnecessary (see s e c t i o n 4.22). On the other hand t h i s source i s much less t r a n s i t o r y than z-pinch or a shock tube; a longer time i n t e r v a l can be used f o r recording the l i n e p r o f i l e s . (See Barnard et a l 1968.) Each l i n e p r o f i l e was recorded i n a separate shot by means of a rapid scan spectrometer. The p h o t o - e l e c t r i c detection used i s l i n e a r and i s much more s e n s i t i v e than a photographic technique which might require several shots to record a p r o f i l e ; also a much greater range of i n t e n s i -t i e s f o r each l i n e can be recorded. The tediousness of a (photo-electric) shot-to-shot technique i s avoided along with the errors due to the non-r e p r o d u c i b i l i t y of the l i g h t source. 3.2 Plasma Vessel The discharge tube (see F i g . 6) was f a b r i c a t e d from two standard end pieces of "Kimax" two inch tubing which had been tapered down and joined to the ends of a 10 cm length of 19 mm (O.D.) pyrex tubing. The aluminium electrodes were sealed to the tube with neopreme "0" rings and 21 22 SCALE: HALF-SIZE QUARTZ WINDOW ALUMINUM TUBE 0 - RING ALUMINUM ELECTRODE PYREX TUBE ALUMINUM ELECTRODE OUTLET TO VACUUM SYSTEM FIG. 6.--PLASMA VESSEL standard f i t t i n g s . An aluminum viewing tube with a quartz window was attached to the positive electrode; extending the window reduced corrosion b}' the plasma to a minimum. The shape of the tube was chosen so that the constricted center portion would produce a hot, dense (wall stabilized) plasma whereas the contamination from the larger electrodes would be minimized. 3.3 Discharge Circuit The discharge c i r c u i t (see Fig. 7) consisted of a capacitor bank (30 ^.F), an inductor (43 |XH) and an open a i r spark gap switch placed in series with the discharge tube. The inductor consisted of 24 turns of 1/4" copper tubing, 7 inches in diameter, wound and clamped on a frame of 1/2" lucite. After several thousand firings no damage to the inductor was apparent. The switch consisted of two hemispherical brass electrodes one inch in diameter enclosed in a brass can. The electrode at ground potential had been d r i l l e d out to receive a tungsten trigger pin surrounded by porcelain insulation. Electrode spacing for a bank voltage of 11 kV was 6 mm. To trigger the switch a fast (20 vtsec risetime), high voltage (40 kV) pulse from a thyratron c i r c u i t was applied between the electrode and the trigger pin. When the switch was fired (bank voltage - 11 kV, capacitance -30 jaF) a damped sinusoidal current of period 226 LA. sec and amplitude 9.2 kA resulted. 3.4 Monochromator The rapid scan system ut i l i z e d a 3/4 meter "Spex" Czerny-Turner MONOCHROMATOR FIG. 7.--Discharge C i r c u i t and Timing Devices 25 monochromator having an aperture of S/3 and a r e c i p r o c a l d i s p e r s i o n of 11 A/mm. i n f i r s t order and 5 A/mm. i n second order f o r a l i n e of wave-o length 4500 A . The grating, which has 1200 lines/mm, i s blazed at a 7500 A making measurements i n second order convenient. The instrument has a r e s o l v i n g power of 80,000 i n second order which corresponds to a o O width of the instrument function of 0.05 A at 4500 A . 3.5 Rapid Scan O p t i c a l System 3.51 Construction The rapid scan o p t i c a l system i s shown i n F i g . 8. The external o p t i c a l system makes use of two f r o n t surfaced s p h e r i c a l mirrors, and M ? (3" diameter x 18" f . l . ) , a f a s t (up to 10,000 RPM) f l a t r o t a t i n g mirror M£, and two f l a t (2" x 3") f r o n t surfaced mirrors, M„ and Mr to 6 3 5 image the s p e c t r a l l i n e s from the plane of s l i t S^ (opened wide) onto an external s l i t S^ behind which i s mounted a photomultipler. Mien the mirror i s rotated, the s p e c t r a l l i n e s are swept across the external s l i t S^ i n the d i r e c t i o n of i n c r e a s i n g wavelength. Mirror M^ i s placed at the f o c a l distance, along the o p t i c a l a x i s , from the plane of s l i t . S ^ so that a p a r a l l e l beam i s formed between M^ and M^. The f u l l aperture of the monochromator can then be used by the external o p t i c s . The r o t a t i n g mirror assembly was constructed and balanced separately i n a r i g i d frame. A f r o n t surface mirror was attached to a 2" x 3".rotor mounted by two high speed b a l l bearings. The rotor was driven by a 10,000 RPM motor connected to the rotor shaft by means of a f l e x i b l e coupling with a rubber cushion to damp v i b r a t i o n s . At the other end of the rotor shaft a d i s c was attached which contained a small piece of magnetized i r o n at the periphery. A recording tape head, which was mounted 26 on the frame near the edge of the d i s c , picked up a pulse whenever the magnet passed i t . This pulse was used f o r timing purposes, The p o s i t i o n of the tape head was movable f o r coarse adjustment of the timing. MONOCHROMATOR EXTERNAL ! M A ' V \ IMAGING \ [ \ SYSTEM \ P. M. FIG. 8.--Optical Arrangement of the Rapid-Scan Spectrometer The external s l i t S^ was a Hilger s l i t assembly attached to an adjustable s l i t holder which allowed f i n e control of the s l i t angle and p o s i t i o n along the ax i s . A l l o p t i c a l elements were securely mounted to 3" x 3" x 1 /4" thick brackets which were bolted to a large plate of 1/2" aluminum. Slots i n the brackets and i n the aluminium plate allowed coarse adjustment of the p o s i t i o n of the brackets. Mirrors M^, M^, M^, and M^ were attached to the brackets by means of spring-loaded screws which allowed f i n e adjust-ment of the mirror angle about a v e r t i c a l axis and a h o r i z o n t a l axis. 27 3 .52 Alignment The external optics were i n i t i a l l y aligned using a He-Ne l a s e r . However a l a s e r , despite i t s very narrow linewidth, i s not r e a l l y s u i t -able since i t acts as a coherent point source and cannot uniformly i l l u m i n a t e the entrance s l i t , S^. A better source was a 1 0 0 watt mercury o lamp which despite i t s wide l i n e s ( ~ 1 A ) could be used to produce a narrow " l i n e " at the e x i t s l i t The lamp was f i r s t focussed on s l i t (opened to 1 0 0 ) so that the f i r s t mirror, was uniformly illuminated. Next the wavelength d i a l on the monochromator was adjusted so that the b r i g h t e s t l i n e , Hgl 5461 was centered on the e x i t s l i t which was then closed down to 10 |u» . The r e s u l t i n g " l i n e " could then be used to check the imaging of the external o p t i c s . In the external imaging system well-known aberrations such as s p h e r i c a l aberration, coma and astignatism were present. However, these aberrations were minimized by making the distances —* and —> equal to the f o c a l length and angle , equal to 0^ with both the minimum po s s i b l e . There were also two possible o r i e n t a t i o n s of s p h e r i c a l mirrors and s l i t s (see F i g . 9 ) . FIG. 9.--Two Possible Orientations of Entrance S l i t S^, and E x i t S l i t S„. 28 Sawyer (1963) states that system (b) i s better and tends to cancel some of the aberrations present with one mirror alone - hence that o r i e n t a t i o n was used i n t h i s device. To a l i g n the system the o p t i c a l elements were f i r s t placed roughly as shown i n F i g . 8 with mirror M. at i t s f o c a l distance from S, and angles ® f , 0£ set at minimum. Then the angular p o s i t i o n of each mirror was s u c c e s s i v e l y adjusted so that the r e f l e c t e d beam was centered on the next mirror. L a s t l y the l a t e r a l and a x i a l p o s i t i o n s of S^ were adjusted so that = 0^ and the image of the l i n e was sharpest. (Mirror M^ was l e f t s t a t i o n a r y . ) Distance S^—^ M^ and M^ —T» S^ were then compared and i f necessary the former length was adjusted to the mean of the two distances and the above procedure repeated. When mirror M^ was rotated, a s i g n a l was recorded by the photo-m u l t i p l i e r on each r e v o l u t i o n of the r o t a t i n g mirror. The width of t h i s pulse could be "tuned" to minimum width by successive f i n e adjustment of the mounting screws f o r each mirror. The minimum instrument width thus obtained was 0.28 A i n f i r s t order and 0.13 A i n second order. Curves f o r the instrument width as a f u n c t i o n of s l i t width (S^ = S^) are given i n Fig. 10. 3.53 C a l i b r a t i o n In order to determine observed widths the scanning rate had to be c a l c u l a t e d . The a n a l y s i s i n Appendix I. shows that the scanning rate can be w r i t t e n I-\ where OO - the angular frequency of the r o t a t i n g mirror = the r e c i p r o c a l d i s p e r s i o n of wavelengths at s l i t S^ (or S^) 29 -r—r -j H—-| j — T T o < Z UJ ZD f -.10 1 S T O R D E R W I D T H - 1 S T O R D E R T H E O R Y 2 N D O R D E R W I D T H 2 O R D E R T H E O R Y 1.0 0 . 7 h 0 . 5 0 . 3 0.2 0 . 1 5 I ' t l — L J L 5 0 S L I T W I D T H (fx) 1 0 0 FIG. 10.--The Width of the Instrument P r o f i l e f o r S l i t Widths (Sj = S 3) 30 R. = the e f f e c t ] :ive sweeping arm 2 ( L | + L l - i i ^ i ) here L, = the distance along the o p t i c a l axis from M, to M-, = 1 6 7 56.8 cm ± 0.5% L^ , = the distance along the o p t i c a l axis from to = 45.2 c m ± 0.5% R = the radius of curvatureof the concave mirror = 91.4 cm i 1% The angular frequency, CO of the r o t a t i n g mirror could be preset to an accuracy of 1% by the e l e c t r o n i c timing devices (see the following s e c t i o n ) . The r e c i p r o c a l d i s p e r s i o n d(x) i s a weak function of wave-length. This quantity was tabulated f o r various wavelengths by the manu-factu r e r of the monochromator; i t was calculated using the grating equation and i t was also checked experimentally by noting the change i n wavelength s e t t i n g on the monochromator as a sharp l i n e was passed across the 3 mm wide e x i t s l i t S^. The agreement between the various determina-tions or d(X) was better than 17o. In p r a c t i c e a more convenient way of w r i t i n g the scanning rate i s d\ CM JL-t T c where "T = period of r o t a t i o n of (m sec) and C W = ZURdM X 10 i s p l o t t e d i n F i g . 11. Note that for second order one replace.s C O O by C(x)/Z . 3.54 Transmission Curve of External Optics In order f o r the rapid scan device to be used for trac i n g the pro-f i l e s of Hel 4471 and 4922 a s u f f i c i e n t wavelength range had to be trans-mitted by the monochromator and scanning o p t i c s . The e x i t s l i t was (Scan Rate, ^ | = ^ A / p S e C f o r T, period of r o t a t i o n of mirror, i n msec) H 1 1 G ( X ) ( 1 0 3 A ) » calculated points -f SPEX DATA d U ) Hie X Measured from AX over 3mm s l i t width / ? , \ ( A / m m ) 8 7 L 3 0 0 0 5 0 0 0 JL 7 0 0 0 9 0 0 0 W A V E L E N G T H ( A ) J J . I 1 1 0 0 0 1 3 0 0 0 FIG. 1 1 . — V a r i a t i o n of C with Wavelength, A 6 1 5 0 0 C 32 too r e s t r i c t i v e as i t could only be opened to 3 mm, passing a wavelength o range of 30 A i n f i r s t order. Removing the s l i t assembly gave an open-ing of more than 3 cm; however, then the edges of the mirrors i n the external o p t i c a l system wouH provide a stop f o r wavelengths g r e a t l y d i f f e r e n t from the cen t r a l wavelength (see F i g . 12). The central beam of wavelength X,0 passes unobstructed whereas part of the beam of wave-lena th X„ + A A. s t r i k e s the' edge of mirror M~. PLANE OF SLIT $ 2 FIG. 12.--Reduced Aperture f o r Non-central Wavelengths Another problem was that the imaging of the non-central wavelengths might not be as good as that measured previously, g i v i n g a larger instrument width. In order to check the wavelength range/of the o p t i c a l system, the Hgl 5461 l i n e from a mercury lamp was scanned at a slow speed (0.96 A/^tsec) o with wavelength s e t t i n g A 0 = 5461 A • The traces from several scans 33 ansmission 7o of Peak) A X - W A V E L E N G T H ( A ) I N F I R S T O R D E R FIG. 13.--Transmission of the Scanning Mirror System As A Function of A X X0= 5 4 6 ! A 0 - 1 0 0 - 5 0 0 + 5 0 A X - W A V E L E N G T H ( A ) I N S E C O N D O R D E R FIG. 14.--Transmission of Spectrometer at X 8942 (Second Order) 35 were recorded on a storage o s c i l l o s c o p e and the average peak i n t e n s i t y was taken. The measurement was repeated f o r d i f f e r e n t wavelengths by s e t t i n g the monochromator d i a l to 5461 - A A . The r e s u l t s are plotted i n F i g . 13 as percent transmission ( r e l a t i v e to % 0 ) as a function of the change i n wavelength A "K . For a d i f f e r e n t wavelength s e t t i n g "\ 0 the r e c i p r o c a l d i s p e r s i o n of the monochromator w i l l be d i f f e r e n t , r e s u l t i n g i n a d i f f e r e n t trans-mission curve. An adjusted curve for A-^ = 8942 (Hel 4471 i n second o r d e r ) i s given i n F i g . 14. The v a r i a t i o n of instrument width over the wavelength range was checked by recording the h a l f width of the Hgl 5461 trace f or d i f f e r e n t A A when wide s l i t s S^, were used. (The ha l f o width of Hgl 5461 was about 2.4 A ) . The instrument width was found to change by less than 20%. 3.6 Photomultiplier The photomultiplier was an EMI 9558 B with an S-20 surface (peak o s e n s i t i v i t y 4500 A ). I t has a low dark current and a f a s t risetime (16 n sec). (The external c i r c u i t f o r the dynode chain with the cable termination!is shown i n F i g . 15.) The maximum allowable (d.c.) anode current was 1 mA. I t was found that signals had to be kept below 100 mV to prevent saturation (not withstanding the speed-up capacitors over the l a s t few dynodes). For a constant l i g h t source, the cathode voltage i s plotted versus the s i g n a l strength. For signals greater than 100 mV the logarithm of the response s t a r t s to become non-linear, (see F i g . 16). ANODE CATHODE I I I —NAA-56 K (AA-2 Z K - H V fo -vV -vAA 21K T % -vA/— 2.2. K I l k HhHh Z2K 3 €. OSCILLO-SCOPE rrrr t l SI-FIG . 15.--External C i r c u i t f o r the Photomultiplier 36 C A T H O D E P i t f E N T I A L ( V ) FIG. 16.--Non Linear Response of Photomultiplier 37 3.7 E l e c t r o n i c Timing Devices The timing was accomplished by feeding the pulses from the tape head on the r o t a t i n g mirror assembly into a timing c i r c u i t (see Appendix III) that consisted of a pulse shaper, delay c i r c u i t and coincidence c i r c u i t . The sequence i s depicted i n F i g . 17. The incoming pulse A t r i g g e r s a pulse shaping network and Schmitt t r i g g e r which produce a sharp (40 n sec risetime) pulse B . This pulse i s fed into both a delay c i r c u i t and a coincident c i r c u i t (an "AND" gate). The delay c i r c u i t emits, a f t e r some preset delay time T^ a pulse C which i s then fed into the other terminal of the coincident c i r c u i t . I f the r o t a t i n g mirror D i s s t i l l coming up to speed i t w i l l have a period of r o t a t i o n T* < TL and the coincident c i r c u i t w i l l not react. However at the ins t a n t that T = Tp , the AND gate r e g i s t e r s a coincidence and triggers a one-shot c i r c u i t that emits a large (20 V ) , f a s t (100 ns risetime) pulse which i s then used to i n i t i a t e the f i r i n g sequence. (The one-shot c i r c u i t i s r e s e t t a b l e by a push-button so that only one pulse need be emitted by the timing c i r c u i t for each run.) Over a period of time the timing u n i t would experience a d r i f t of approximately 1% i n the preset time , n e c e s s i t a t i n g small adjust-ments from time to time i n the settings for the delay u n i t s . The j i t t e r ( d r i f t from shot-to-shot) was approximately 0.27«, which was t o l e r a b l e . In order to minimize the j i t t e r the following e f f e c t had to be considered. As the mirror comes up to speed, the subsequent periods of r o t a t i o n T T* T w i l l diminish by d i s c r e t e amounts (see F i g . 18). However, another sequence might r e s u l t (see F i g . 19). In F i g . 19 pulse (m - 1) does not quite coincide with pulse C and i t i s not u n t i l the next pulse (m) that coincidence i s obtained. Hence a r e l a t i v e error of 38 f j / T*D i n the morror speed i s obtained. This error was e a s i l y minimized by bringing the mirror up to speed slowly; then Vj , the d i f f e r -ence between subsequent mirror periods near coincidence was small. In p r a c t i c e most of the error i n timing came from j i t t e r i n the delay c i r c u i t of the timing u n i t . A - PULSE FROM TAPE HEAD. B -INITIAL H 5 -PULSE _ i l C-DELAYED PULSE D-OUTPUT P U L S E AS MIRROR COMES TO SPEED T (PERIOD op ROTATING MIRROR) T0  ( PREOETERMIWEP &ELAy_) J l n 11 i i i i FIG. 17.--The Timing Sequence (Time Base Approximately 500 |A. sec/cm) f 71-Z SUBSEQUENT PULSES B L -n-i PULSE C 1 C O I N C I D E N C e -FIG. 18.--The Sequence at Coincidence I (Time Base Approximately 100 n sec/cm) Hence the timing c i r c u i t emits a pulse when the r o t a t i n g mirror i s at a known frequency. This pulse i s then fed into two delay units (see Appendix III) , one of which t r i g g e r s the thyratron c i r c u i t i n i t i a t i n g the bank discharge and the other t r i g g e r s , an i n t e r v a l l a t e r , the o s c i l l o s c o p e recording the l i n e p r o f i l e . The sequence i s shown i n F i g . 20. 39 f S U B S E Q U E N T P U L S E S B PULSE C - m - 3 < 'vn.-'i. 1 •>n-i L V N ~< Tj 5* r FIG. 19.--The Sequence at Coincidence II (Time Base Approximately 100 n sec/cm) P U L S E D E L A Y E D P U L S E 1 B A N K C U R R E N T D E L A Y E D P U L S E 2 O S C I L L O S C O P E T R A C E FIG. 20.--The Triggering Sequence (Time Base Approximately 20 u. sec/cm) 4 0 In p r a c t i c e the f a s t e s t (10,000 RPM) mirror speeds were used for scanning the l i n e s of i n t e r e s t (Hel 4471 and 4922 A ) . Therefore the p o s i t i o n of the pick-up head on the r o t a t i n g mirror assembly was adjusted i n order to minimize the delay time , required f o r this speed, thereby minimizin the absolute errors i n triggering the recording o s c i l l o s c o p e . Normally a scanning rate of 4.8 A /M. sec was used (time base: 0 i 5 or 1.0 £J- sec/ div.) with a t o t a l j i t t e r i n t r i g g e r i n g the o s c i l l o s c o p e of 0.2 sec, a n e g l i g i b l e amount. However, i t was also necessary to scan the neutral helium l i n e X 3889 for diagnostic purposes (see the next s e c t i o n ) . This l i n e i s o much narrower ( ~ 1 A ) than the above l i n e s ; hence i t was desired to scan i t at a lower speed thaii the former l i n e s (0.74 A / j U s e c ) so that a large number of photons would s t r i k e a photomultiplier producing a smoother p r o f i l e . In this case, since i t was inconvenient to move the pick-up head, the delay time " t ^ became large (~800 |LC sec) - producing a larger (1 |LX sec) j i t t e r i n the t r i g g e r i n g of the o s c i l l o s c o p e . This f i g u r e was t o l e r a b l e although i t necessitated r e j e c t i n g some p r o f i l e s which were of f the screen of the o s c i l l o s c o p e . 3.8 Oscilloscopes The o s c i l l o s c o p e s used were a,Tektronix dual beam type 551 and a Tektronix type 549 storage o s c i l l o s c o p e . The type 551, which has a r i s e -time of 12 "K sec was used i n conjunction with a Dumont camera to record the l i n e p r o f i l e s . The type 549 was used with a slower sweep speed to monitor the timing. An integrated s i g n a l from a small bank pick-up c o i l was added to the s i g n a l from the photomultiplier and displayed on the face of the storage o s c i l l o s c o p e . The time at which the l i n e appeared i n the current cycle of the bank could be r e a d i l y seen. The time bases of each o s c i l l o s c o p e were checked by a Dumont time mark generator immediately before the f i n a l r e s u l t s were taken. The time base of the type 549 was l i n e a r to better than 1% over the center screen and the type 551 time base was l i n e a r to 17. over the center 8 d i v i s i o n s of the screen. The v e r t i c a l l i n e a r i t y f o r each was also better than 17.. CHAPTER. IV EXPERIMENTAL WORK 4.1 Introduction Preliminary observations were made side-on since Barnard et a l (1968) found that for s i m i l a r plasma conditions i n Argon the strongest l i n e s can be appreciably s e l f absorbed. ' An attempt was made (Section 4.23) to determine the r a d i a l temperature v a r i a t i o n of the plasma by the Abel unfolding of r e l a t i v e i n t e n s i t y measurements. However, the considerable s c a t t e r i n g and d i f f u s i n g of l i g h t at the tube walls prevented t h i s . End-on observations were then made. This method had the advantage that a f l a t end window, well removed from the plasma could be used. Other advantages were that the observed l i n e i n t e n s i t i e s would be enhanced and that no r a d i a l deconvolution would be required. Calculations (Sec. 4.31) f o r the o p t i c a l thicknesses of the l i n e s W 4471, 4922 and 3889, made using data from the side-on observations, indicated that s e l f absorption was n e g l i g i b l e . A l l f i n a l measurements were made end-on. 4.2 Side-on Observations I n i t i a l observations were made with the discharge tube h o r i z o n t a l and perpendicular to the s l i t of the monochromator. The time v a r i a t i o n of several l i n e s of i n t e r e s t was observed by using narrow s l i t s S^, S^ and by leaving mirror Mg sta t i o n a r y and placed so that the image of S^ was centered on S^ (opened wide). The neutral helium l i n e s W 3889, 5876, o 4713 and 4471 A and the l i n e He II 4686 were then observed for various conditions of i n i t i a l pressure, bank voltage and capacitance. 42 T I M E (JDLSGC) FIG. 21.--Time V a r i a t i o n of T y p i c a l Hel and H e l l Lines 44 I t was noted that for the higher bank energies a strong background si g n a l appeared over a wide wavelength range, reaching a maximum i n t e n s i t y at a time TJ = 100 LL sec a f t e r the i n i t i a l breakdown. This s i g n a l , which may have been due to recombination r a d i a t i o n at the tube walls, decreased sharply as the bank voltage was lowered and became n e g l i g i b l e for the i n i t i a l conditions of bank voltage, VQ ~ 11 KV and capacitance, Q •=. 30 This i n i t i a l condition was then chosen for a l l subsequent observations. ( I t was desi r a b l e to keep the bank energy high enough that a hot, dense arc would be formed, r e s u l t i n g i n a w a l l - s t a b i l i s e d , and therefore f a i r l y s p a t i a l l y uniform plasma.) Some t y p i c a l traces of the helium l i n e i n t e n s i t i e s are shown i n F i g . 21. The trace for the ionized helium l i n e ( H e l l 4686) peaked at a time "t = 60 LL sec a f t e r breakdown i n d i c a t i n g that a maximum ele c t r o n temperature occurs near there. Hence th i s time was chosen for measure-ments. 4.21 E l e c t r o n Density An i n i t i a l estimate of the ele c t r o n density, T i g was obtained by measuring the p r o f i l e of Hel 3889 i n second order of the monochromator. The widths and s h i f t s for this l i n e for various e l e c t r o n d e n s i t i e s have been calculated by Griem (1964) and Cooper (1969) and have been well checked experimentally by Berg (1962) and Grieg (1968). For i n i t i a l pressure of 10 torr an e l e c t r o n density at "fc =• 60 |-c sec was 5 x 10 -3 cm . Since subsequent r e s u l t s indicated that the plasma was almost f u l l y i onized, t h i s e lectron density would seem too low as the i n i t i a l . number density of helium atoms f or a pressure of 10 torr i s 3.5 x 1 0 ^ -3 cm . However, the plasma pressures were s u f f i c i e n t l y high and the helium ions, s u f f i c i e n t l y mobile f o r appreciable numbers of ions and electrons 45 to move a x i a l l y i n 60 p. sec to cooler regions of the plasma at the ends of the tube where they could recombine. Hence a much lower e l e c t r o n density seems reasonable. 4.22 E l e c t r o n Temperature The e l e c t r o n temperature can be estimated from the r a t i o of i n t e n s i t i e s of the strong l i n e s H e l l 4686 and Hel 5876. Griem (1964) and Mewe (1.967) have ca l c u l a t e d the r a t i o of these i n t e n s i t i e s as a function of e l e c t r o n temperature. This r a t i o of i n t e n s i t i e s v aries strongly with temperature so that the i n t e n s i t i e s need not be measured very accurately. (At k T e = 4.5 eV an error of 207» i n the measured r a t i o of i n t e n s i t i e s would produce an error i n Tg of only YL.) However i t was f e l t necessary to c a l i b r a t e the wavelength response of the monochromator and photomultiplier since the two diagnostic l i n e s o are more than 1000 A apart. To accomplish the c a l i b r a t i o n a tungsten ribbon lamp, whose temperature (2000°K) had been measured by an o p t i c a l pyrometer, was imaged on the entrance s l i t of the monochromator. The i n t e n s i t y of s i g n a l emitted by the photomultiplier was then recorded f o r o a number of wavelengths from 4000 - 8000 A . These i n t e n s i t i e s were divided by the corresponding values for the r e l a t i v e e m i s s i v i t y of a 2000° K grey body to give a t o t a l response c\irve for the photomultiplier and monochromator. This curve, which i s p l o t t e d i n F i g . 22 indicates that the measured r a t i o 1(4686)/ I (5876) should be m u l t i p l i e d by a factor of 1.5. This corresponds to a small c o r r e c t i o n i n the temperature. The measured ( t o t a l ) i n t e n s i t y r a t i o I (4686)/ X (5876) (at 60 u^. sec) and Mewe's c a l c u l a t i o n s indicated an e l e c t r o n temperature of 4.5 eV. 46 4 0 0 0 5 0 0 0 6 0 0 0 W A V E L E N G T H ( A ) 7 0 0 0 FIG. 22.--Total Wavelength Response of Monochrometer and Photomultiplier 47 4.23 I n i t i a l . S p a t i a l l y Resolved Measurements The observed temperature represents an average over the regions of the plasma seen by the o p t i c a l system. I t i s very important to know the d i s t r i b u t i o n of or k T ^ over the various regions of a plasma when making l i n e p r o f i l e measurements. For example, one might have a plasma whose r a d i a l d i s t r i b u t i o n of T i g i s shown i n F i g . 23(a). If this d i s t r i -bution i s then approximated by two " s h e l l s " of constant T i g , then the observed p r o f i l e (of say, Hel 4471) w i l l be the sum of two p r o f i l e s emitted by regions 1 and 2 (see F i g . 23(b)). FIG. 24.--Resultant P r o f i l e of X 4471 or 4922 48 The t o t a l observed p r o f i l e ( F i g . 24) would be d i s t o r t e d and would not correspond to that emitted by any region of intermediate electron density. . Note that the width of the dip of the composite p r o f i l e i s r e l a t i v e l y narrower compared to the other p r o f i l e s . (Although a d i r e c t check was necessary, one would not expect the r a d i a l density d i s t r i b u t i o n of F i g . 23 since previous measurements on a s i m i l a r plasma (Barnard et a l , 1967) indicated a f l a t d i s t r i b u t i o n of T ig which dropped o f f sharply at the tube walls.) A measurement of the r a d i a l temperature d i s t r i b u t i o n at ~b = 60 |U sec was attempted by mounting the discharge tube v e r t i c a l l y and imaging a part of the plasma on the entrance s l i t S^, by means of a lens L and mirror MQ (see F i g . 25). This mirror could be rotated about a v e r t i c a l axis by means of a micrometer screw. The t o t a l l i n e i n t e n s i t i e s of Hel 5876 and H e l l 4686 were then measured at the time of i n t e r e s t f o r various positions across the tube. I t was hoped that the Abel ( r a d i a l ) deconvolu-t i o n could then be performed (Bockasten 1961, Barr 1962) to give the r e l a t i v e e m i s s i v i t y f o r each l i n e and hence the e l e c t r o n temperature as a function of radius. However, the observed i n t e n s i t i e s were strongly af f e c t e d by r e f r a c t i o n and d i f f u s i o n at the tube walls which would become scarred and corroded a f t e r only a few shots. When the mirror M 0 was turned to a p o s i t i o n which should have imaged the edge of plasma on s l i t S^, the i n t e n s i t i e s of the two l i n e s were s t i l l appreciable and did not drop to zero as one would require. Hence the r a d i a l d i s t r i b u t i o n of e l e c t r o n temperature was not measured. 4.3 End-on Observations I t was then decided to look end on. This had the advantage that a f l a t window, well removed from the plasma could be used; hence corrosion 49 would be reduced to a minimum. Furthermore, no r a d i a l deconvolution of observed i n t e n s i t i e s would be necessary. Another advantage was that the observed i n t e n s i t i e s of a l l s p e c t r a l l i n e s would be increased. D I S C H A R G E T U B E FIG. 25.--The O p t i c a l Arrangement for S p a t i a l l y Resolved Side-On Measurements 4.31 C a l c u l a t i o n of O p t i c a l Thicknesses One had to be sure, however, that the l i n e s of i n t e r e s t were not appreciably self-absorbed. Griem (1964, p. 173) gives an expression for the o p t i c a l thickness f(co) of a l i n e observed through a length of plasma, ? M = J Ac^x) dx = i nV0 c jl - e j L M i f • . 50 where k^,X) = absorption c o e f f i c i e n t as a function of frequency and p o s i t i o n Y"o = c l a s s i c a l radius of the ele c t r o n C = speed of l i g h t absorption o s c i l l a t o r strength for the t r a n s i t i o n = number density of atoms i n the lower state of the t r a n s i t i o n "ft CO = energy d i f f e r e n c e between l e v e l s OZ and j$ |_(to) = l i n e i n t e n s i t y normalised to un i t area JL = length of plasma through which the l i n e i s observed. (Use was made of the assumption that the r a t i o of populations of the upper and lower l e v e l s of the t r a n s i t i o n could be replaced by the Boltzmann f a c t o r ) . In order that the widths of the observed p r o f i l e s are not excessively broadened by s e l f absorption, one requires that at the l i n e center, XC^o) (which i s e s s e n t i a l l y the p r o b a b i l i t y that photon of energy * t \CU 0 w i l l escape from the plasma) be small. Cooper (1966) shows that, f o r small optical.thicknesses (X < 0.5) the r a t i o of the observed to true h a l f width can be w r i t t e n as wtrMe - 1 + t For accurate p r o f i l e measurements one would require Y (CL>6) < O.i and preferably less than 0.05. For Lorentzian l i n e shapes the expression f o r the o p t i c a l thickness at the l i n e center becomes: -2/ -2. / --'fta>/k"T\ f o r A- = wavelength of t r a n s i t i o n , /\ o A A- = f u l l h a l f width of the p r o f i l e , A 51 Calculated values f o r the absorption o s c i l l a t o r strengths are given i n Griem (1964) and Wiese (1966). To estimate the density of atoms i n the lower state a computor program was wr i t t e n which made use of the Saha and Boltzmann equations (the assumption of LTE i s not r e a l l y v a l i d here) to ca l c u l a t e the populations i n the plasma. The p a r t i t i o n functions for the neutral and s i n g l y ionized species a (TV - 2J <J^  Q i were calculated by summing over the tabulated atomic of i o n i c l e v e l s and estimating the remainder up to some cutoff l e v e l using the hydrogenic approximation (see Griem, 1964, Chapter 6). Input parameters f o r the program were i n i t i a l pressure and e l e c t r o n temperature. In order to compensate for loss of p a r t i c l e s at the ends of the plasma by j~ = 60 |A sec the input value f o r the i n i t i a l pressure was adjusted so that the calculated value of T i g agreed with the measured value. Mewe's (1967) ca l c u l a t i o n s i n d i c a t e that f o r the non-LTE conditions present i n this plasma the density of helium atoms i s greater than the calculated LTE values by a factor of 10. (At a temperature of 4.0 eV the plasma i s approximately 977. ionized.) Hence the computed values for (neutral l i n e s only) were m u l t i p l i e d by this f a c t o r i n estimated t(tOe) . The o p t i c a l thickness f o r the important helium l i n e s are l i s t e d below. Line Hel 5876 0.27 H e l l 4686 0.13 Hel 3889 0.01 Hel 4471 0.01 Hel 4922 0.01 For t h i s plasma a l l but the strongest l i n e s i n the helium spectrum were o p t i c a l l y t h i n . (Hel 4471 and 4922, although strong l i n e s are. o p t i c a l l y t h i n because of the i r large width.) 4.32 S p a t i a l l y Resolved Measurements The r a d i a l v a r i a t i o n s i n T i g (at ~t = 60 sec) were examined by 52 measuring the width of Hel 3889 when observed along several a x i a l " s l i c e s " (see F i g . 26). F i r s t the center of the plasma column was imaged on the s l i t (shortened to 2 mm height) by means of lens which had been stopped down to £ /30. Next a 1" thick l u c i t e block having a known index of r e f r a c t i o n ( Vt = 1.50 i 0.01) was mounted between the lens and the discharge tube. A curve f o r l a t e r a l d e f l e c t i o n of the beam as a function of inc i d e n t angle 8 was calculated using geo-metric o p t i c s . The angle 0w a. x f o r which the edge of the plasma should have been imaged on S^, was checked by s i g h t i n g along the discharge tube towards when a continuous source was shone backwards through the monochromator. Systematic errors i n the r a d i a l p o s i t i o n V"R were seen to be 5% or smaller. FIG. 26.--The O p t i c a l Arrangement for S p a t i a l l y Resolved End-On Measurements With the small aperture and short s l i t S^ the i n t e n s i t i e s of Hel 3889 were too low for tracing the p r o f i l e by the rapid scan device and i t was necessary to use shot-to-shot measurements. This requires good reproduci-b i l i t y between shots. I t was found that the best r e p r o d u c i b i l i t y was obtained by f i l l i n g the tube once with fresh gas and then f i r i n g the bank at evenly spaced i n t e r v a l s (1 min. apart). On a tes t run i n which the t o t a l i n t e n s i t y of the Hel 4471 l i n e was recorded, the i n t e n s i t i e s 100 FIG. 27.--Reproducibility of Line I n t e n s i t y from Shot-to-Shot. Hel 4471 i s Observed End-On at f/30 (t = 60/x,sec ) with S l i t s S 1 = 5^A, , S 2 = 3mmx2mm High. I n i t i a l Pressure = 10 Torr 500 400 -mean 346 ± 7.5% 300 200 mean 348 mean 344 10 15 20 25 Shot Number 30 35 40 45 50 Co 54 s e t t l e d down a f t e r a dozen shots to a mean value about which they deviated by 7.57o (see F i g . 27). P r o f i l e measurements could then be made a f t e r the tenth or twelfth shot. In t his way p r o f i l e s f o r Hel 3889 were measured at 10 torr f o r various normalised r a d i i , T = 0.0, 0.5, 0.7 and 0.9 using the same gas. As a check the p r o f i l e f o r Y = 0.0 was measured again with the same gas at the end of the sequence. Two measurements were taken f o r each point and averaged. The above procedure was repeated for i n i t i a l pressures of for two pressure* 6 t o r r , 17 torr and 28 t o r r . The r e s u l t s ^ a r e plotted i n F i g s . 28, 29. For a given pressure the f u l l h a l f width was constant to 67>. In addi t i o n no systematic v a r i a t i o n of h a l f width with radius was detected. The a x i a l behaviour of the plasma was examined by observing the plasma side on, using a rotatable mirror, M* set by a spring-loaded micrometer (see F i g . 30). P r o f i l e s of Hel 5876 and H e l l 4686 were taken with the rapid scan device f o r a x i a l positions 5£ = 0 (center), 5£ = -4cm (near the end of the co n s t r i c t e d p o r t i o n of the tube) and "2 = -7.5 cm (at the wide end portion of the tube). Temperatures were ca l c u l a t e d from the t o t a l l i n e i n t e n s i t i e s , ( kTe = 4.55, 4.5, 3.55 eV resp.), i n d i c a t i n g a f a i r l y uniform hot region i n the co n s t r i c t e d part of the tube with cooler regions at the end. A rough estimate of the r a d i a l v a r i a t i o n s of kTg at the end of the tube ( % = -7.5 cm) was made by observing the two l i n e s using d i f f e r e n t s l i t heights (2, 5 and 10 mm - which were magni-f i e d by a factor of 5). A cool, outer region was d e f i n i t e l y indicated there with the temperature d i s t r i b u t i o n i n the hotter c e n t r a l region less c e r t a i n . However, the important f a c t o r was the ele c t r o n density d i s t r i b u t i o n FIG. 28.--End-On S p a t i a l l y Resolved X 3889 P r o f i l e s at 10 Torr 56 — L _ 1 I I 1 I I I - 3 - 2 - 1 0 1 2 3 4 W A V E L E N G T H ( A ) FIG. 29.--End-On Spatially Resolved at 28 Torr X3889 Profiles 1 1 \ I \ ! * \ ! N V \ * * i \ \ \ O -if -7.5 FIG. 30.--The Op t i c a l Arrangement for S p a t i a l l y Resolved Measurements along the Axis which was more d i f f i c u l t to measure. To check t h i s d i s t r i b u t i o n the p r o f i l e of Hel 3889 was measured side-on from the center portion of the tube and compared with the p r o f i l e taken end-on. If there were any large a x i a l v a r i a t i o n s of T i g , the two p r o f i l e s would have d i f f e r e n t widths and shapes. As shown i n F i g s . 31 and 32, the corresponding p r o f i l e s , which were taken i n the same sequence of shots, are c l o s e l y s i m i l a r and the widths agree to 57o. Hence there are no appreciable v a r i a t i o n s i n 'YLg i n the region of i n t e r e s t along the center of the tube (where the f i n a l measurements were taken) which are hot enough to emit strongly Hel 3889 (and Hel 4471, 4922). 4.33 F i n a l P r o f i l e Measurements With this demonstrated s p a t i a l uniformity i t was possible to use the f u l l s l i t height (20 mm) and f u l l aperture (£/8 ) i n scanning the l i n e s of i n t e r e s t , Hel 4471 and 4922 (plus Hel 3889 to check T i g ) . In order f o r the rapid scan device to be used f o r tracing these p r o f i l e s a s u f f i c i e n t wavelength range had to be transmitted by the mono-chromator and scanning o p t i c s . The e x i t s l i t S„, when opened to i t s 58 I n i t i a l Pressure, 10 torr o Width, I n s t r . P r o f i l e , 0.28 A 1 1 I J I I » t - 4 - 3 - 2 - 1 0 1 2 3 W A V E L E N G T H (A) FIG. 31.--Comparison of End-On and Side-On X3889 P r o f i l e s at 10 Torr FIG. 32.--Comparison of End-On and Side-On X3889 P r o f i l e s at 28 Torr 60 o maximum width, transmitted a range of only 30 A ( f i r s t order); con-sequently the whole e x i t s l i t assembly had to be removed. A wavelength o range of 150 A (second order) was then transmitted. (The r e s t r i c t i o n s are caused by the geometry of the external o p t i c a l system and are discussed i n Section 3.54). The p r o f i l e of Hel 4471 was recorded using the rapid scan technique. Because only one p r o f i l e could be obtained at a time, Y l g had to be obtained by scanning Hel 3889 i n another shot. In order to obtain the maximum r e p r o d u c i b i l i t y the following procedure was used. The discharge tube was f i r s t f i l l e d to the desired:pressure and shots f i r e d every 2 minutes using the same gas. For the f i r s t four shots the p r o f i l e of Hel © 3889 was recorded (second order) at an optimum scan speed (0.74 A / sec) with optimum s l i t width (S^ = S^ = 20 | X , s l i t height = 20 mm). In the next two shots the p r o f i l e of A 4471 was recorded (second order at the o . f a s t e s t scan speed (4.8 A / sec) with wider s l i t s (S^ = S^ = 50 L t , s l i t height 20 mm). A t y p i c a l p r o f i l e f o r an i n i t i a l pressure of 10 torr i s shown i n F i g . 33. As a check the p r o f i l e of Hel 3889 was measured on the next shot. Variations of less than 6% were obtained i n the d i f f e r e n t estimates of Tig . Several runs were made at each i n i t i a l pressure (6, 10, 17, 28 torr) and the following parameters were taken from the oscillograms using a p a i r of div i d e r s (see F i g . 33): 1) The combined ( f u l l ) width at h a l f i n t e n s i t y , W 2) The separation (A) °f component peaks, b 3) The r e l a t i v e i n t e n s i t y of the forbidden peak, 1^  4) The r e l a t i v e i n t e n s i t y of the dip. —I • , , 1 • 1 . 1 . 1 , 1 , , , r 4450 60 70 80 90 4-500 10 20 W A V E L E N G T H (A) FIG. 33.--A T y p i c a l P r o f i l e of Hel 4471 at an I n i t i a l Pressure of 10 Torr These parameters are p l o t t e d along w i t h the t h e o r e t i c a l values of Barnard (1969) and the experimental values of Wulff (1958) i n F i g s . 34, 35, and 36 as a f u n c t i o n of T i g . The p r o f i l e of Hel 4922 was then examined. In t h i s case a strong Hp l i n e (which apparently came from hydrogen i m p u r i t i e s i n the g l a s s w a l l s ) appeared a f t e r the f i r s t shot and obscured the helium l i n e . Hence the X4922 p r o f i l e measurements had to be made during the f i r s t shot a f t e r f i l l i n g the tube w i t h f r e s h gas. Timing d i f f i c u l t i e s made i t d i f f i c u l t to determine "VL^ i n the next shot. (The d r i f t i n the timing u n i t produced l a r g e r e r r o r s i n t r i g g e r i n g the o s c i l l o s c o p e at the lower scan speeds used f o r Hel 3889. This problem d i d not occur i n the procedure f o r measuring the X 4471 p r o f i l e s because then Hel 3889 was scanned f i r s t and e r r o r s i n t r i g g e r i n g could be cor r e c t e d i n the f i r s t one or two shots.) I n any case, the presence of hydrogen i n the plasma a f t e r the f i r s t shot cast doubt on the assumption that Y l g would be unchanged from 62 the f i r s t shot. However i t was n o t i c e d t h a t , f o r a given i n i t i a l pressure i n the tube, the e l e c t r o n d e n s i t y obtained f o r the f i r s t shot w i t h f r e s h gas was r e p r o d u c i b l e to 107o. The e l e c t r o n d e n s i t y assigned to a given "X. 4922 p r o f i l e could then be taken as the average val u e obtained from s e v e r a l shots at that f i l l i n g pressure. Values f o r Ylg were obtained f o r the f i l l i n g pressures of 6, 10, 17 and 28 t o r r . P r o f i l e s f o r X 4922 were then taken f o r these f i l l i n g pressures. I n t h i s case i t was found that the e r r o r s i n c u r r e d i n measuring the various parameters on the o s c i l l o g r a m s could be minimized by u s i n g a f a s t e r time base and d i s p l a y i n g on the o s c i l l o g r a m o n l y the center of the l i n e , ( e n c l o s i n g the f u l l width W ). The zero l i n e was obtained by tu r n i n g the wavelength d i a l on the monochromator to a nearby r e g i o n of the spectrum where no helium l i n e s were seen, f i r i n g another shot and super-imposing the tr a c e on the same o s c i l l o g r a m w i t h the \ 4922 p r o f i l e . In t h i s way the continuum, and any spurious s i g n a l picked up i n the cables were a c c u r a t e l y s u b t r a c t e d from the observed l i n e i n t e n s i t y . Again the v a r i o u s parameters were p l o t t e d ( f i g s . 34, 35, 36) as a f u n c t i o n of T i g along w i t h the t h e o r e t i c a l values of Barnard (1969) and the experimental values of Wulff (1958) and Sad j i a n (1961). The observed p r o f i l e and the c a l c u l a t e d p r o f i l e (Barnard e t a l 1969) f o r Hel 4471 f o r an e l e c t r o n d e n s i t y of 4.5 x 10^ are compared i n F i g . 37. 16 3 A s i m i l a r comparison f o r Hel 4922 at an e l e c t r o n d e n s i t y of 4 x 10 cm i s presented i n F i g . 37A. A d d i t i o n a l p r o f i l e s are given i n Appendix IV. X 4471 X 4922 50 40 -30 20 x o This Experiment Best F i t Wulff (1958) , S a d j i a n e t a l (1961) T h e o r e t i c a l Curve (Barnard e t a l , 1969) 10 4 8 10 FIG. 34.--Variation of F u l l Width, W with n t 4 6 n e d 0 1 6 c m - 3 ) X 447! X 4922 FIG. 3 5 . — V a r i a t i o n of Separation, S with T l e 1.0 0.8 X 4 4 7 1 x o This Experiment Best F i t Wulff (1958) Sadjian et a l (1961) T h e o r e t i c a l Curve *~ (Barnard et a l , 1969) FIG. 36.--Variation of Intensities I f and 1^ with T l f X 4 9 2 2 4 71 6 ( 1 0 1 6 c m - 3 ) 444,0 4470 4500 x 4500 CHAPTER V DISCUSSION OF ERRORS 5.1 Introduction This chapter presents a discu s s i o n of the errors involved i n the l i n e p r o f i l e measurements and of the v a l i d i t y of the r e s u l t s obtained. Such considerations are c l e a r l y very important for comparison of t h e o r e t i c a l and experimental p r o f i l e s . The errors can be divided into three types: 1) instrument errors, 2) errors i n taking the p r o f i l e parameters from the Polaroid oscillograms, and 3) errors from v a r i a t i o n s i n the plasma. The magnitudes of these errors f o r each of the parameters measured are l i s t e d below. s=small (~17») , n=neglible,-=not appl i c a b l e n e W S I f I d 1) INSTRUMENT ERRORS a) v a r i a t i o n s i n time base of o s c i l l o s c o p e 1 % 1 % 1 % — •— b) n o n - l i n e a r i t y i n v e r t i c a l a m p l i f i e r -YX •— "KL c) d i s t o r t i o n s i n scope camera/film "Kb 'VL d) n o n - l i n e a r i t y of photo-m u l t i p l i e r on. 'YL — e) uncertainty i n mirror p e r i o d , T 0.5 % 0.5-% — — f ) uncertainty i n constant f C C A . ) 1 % 1 % f % — — g) neglect of transmission curve -ru s s h) u n c e r t a i n t i e s from ins.tr. broadening 0 . 5 - 3 % -n — — s 2) MEASURING ERRORS i ) uncertainty i n baseline on trace — 1 7o i % j) l i m i t of measuring p r e c i s i o n Z-5 % 1-5% 4~/o % 5% 5 % 67 68 s=small (~17o) , n=negligible, -=not appli c a b l e w S I f u 3) PLASMA VARIATIONS k) temporal v a r i a t i o n s i n plasma during scan 2 . - 7 % 2 . -5% 1 - 1 % 1) shot-to-shot v a r i a t i o n s i n the plasma — — — — m) s p a t i a l v a r i a t i o n s i n Tig — s T T . s 5.2 Instrument Errors The instrument e r r o r s , which are mostly small include uncertainty i n the constant C(X) ( i n the r e l a t i o n ) and uncertainty i n the mirror period T (due to d r i f t i n the timing u n i t ) . These un c e r t a i n t i e s were observed to be 0.5 and 1% r e s p e c t i v e l y . The p r o f i l e s ^ of Hel 4471 and 4922 were not corrected for v a r i a t i o n i n the trans-mission of the mirror system with wavelength as the r e s u l t i n g errors were small. The instrument broadening was n e g l i g i b l e for the l i n e s XX. 4471 and 4922; however, f o r "X 3889 the instrument broadening was not n e g l i -g i b l e and a rough c o r r e c t i o n was needed (see Appendix I I ) . This c o r r e c t i o n assumed a Lorentzian p r o f i l e and a trian g u l a r instrument p r o f i l e . Uncertainties due to these assumptions and to the measured instrument widths gave an error i n X 3889 widths (and hence V l e ) estimated at 0.5 to 37o. (The narrowest p r o f i l e s were the l e a s t certain.) 5.3 Measuring Errors The measurement of p r o f i l e parameters from the Polaroid oscillograms induced several e r r o r s . The uncertainty i n the p o s i t i o n of the baseline produced on the average a 27. error i n T\,g and W and a 17, error i n 1^ . and I . (less f o r these because a r a t i o was taken). The usual procedure • 69 f o r recording a l i n e p r o f i l e was to di s p l a y only the center of the l i n e on the oscillogram and estimate the p o s i t i o n of the baseline from a separate shot i n which the wings of the l i n e were displayed. A better procedure, used l a t e r , was to trace on the same frame and during a separate shot, the continuum from a nearby wavelength region i n which there were no helium l i n e s . In this way the continuum and any spurious s i g n a l picked up i n the cables were accurately subtracted from the observed i n t e n s i t i e s . Another error occurred because of the f i n i t e p r e c i s i o n with which one could measure parameters from the oscillograms with a pair of d i v i d e r s . This error, which was estimated by having several observers measure the same l i n e independently, was set at 0.08 div. or 1/3 the width of the trace, whichever was la r g e r . For some parameters the r e l a t i v e error was quite large (as large as 107o f o r S ). However, the worst measurements were l a t e r repeated with a f a s t e r o s c i l l o s c o p e sweep speed (and consequently a smaller measurement e r r o r ) . Good agreement between the determinations was noted. 5.4 Plasma Variations The errors due to v a r i a t i o n s i n the plasma were more important and harder to estimate. One of the assumptions made f o r the rapid scan device was that conditions i n the plasma were e s s e n t i a l l y constant during the scan time. This i s not s t r i c t l y true for t h i s plasma where the t o t a l l i n e i n t e n s i t i e s were observed to vary by as much as 27o/sec at the time of i n t e r e s t . In measuring these p r o f i l e s one i s recording i n t e n s i t i e s I(X,+AX,t,+ At) etc., instead of the i d e a l I (\, t,) I flX, t, ) The discrepancy between I + A < > "t,) and X (\ + "fc, + A"t) was e s t i -70 mated by examining oscillograms for the time varying i n t e n s i t i e s (rotat-ing mirror stationary) at various preset wavelengths, A, X,+^ X . A rough c o r r e c t i o n was then applied to the observed l i n e p r o f i l e s and the e f f e c t on the linewidths examined. Errors i n the X 4471 width, were found to be between 2 and 77, ; errors f or the A 3889 width were approxi-mately h a l f this amount. The errors i n 1^ from the same source were estimated by assuming an i n t e n s i t y v a r i a t i o n of 27,/ kt, sec and multiply-ing by S/r where S = peak separation and the scan rate. Errors i n T. were hal f this amount (1-27,). The shot-to-shot v a r i a t i o n s i n the ci plasma induced another error because the p r o f i l e s f o r X4471 (or A- 4922) and X 3889 (for 71 ) were recorded i n d i f f e r e n t shots. The magnitude of this error (for T i g only) was estimated at 57. by considering the v a r i a t i o n i n A. 3889 widths obtained when this l i n e was scanned many times i n successive shots using the same gas. For this estimate the errors occurring i n taking linewidths from the polaroid oscillograms (discussed above) were taken into account. The unc e r t a i n t i e s i n the AA4471 and 4922 p r o f i l e s due to s p a t i a l v a r i a t i o n s i n the plasma are d i f f i c u l t to estimate. The most important parameter Ylg was seen to vary by less than 67. out to 0.9 times the tube radius when the p r o f i l e of \ 3889 was measured end-on. In add i t i o n no systematic v a r i a t i o n of TL© with radius was observed. The a x i a l v a r i a -tions of Tig , which were estimated by comparing the side-on and end-on p r o f i l e s of X3889, were small. Hence the errors i n the measured values The p r o f i l e s f o r which this error was worst ( i n i t i a l pressure 27 torr) were l a t e r checked by measuring i n f i r s t order where the scan times (and hence errors from temporal v a r i a t i o n s ) were smaller by 507.. Again good agreement was noted. of 'Vlg due to s p a t i a l v a r i a t i o n s i n the plasma were estimated at 67>. In summary, the t o t a l uncertainty i n any parameter due to the aforementioned errors w i l l be less than 107o (see table i n Sec. 5.1). Some of the t h e o r e t i c a l values for these parameters do not f a l l w ithin t h i s error estimate; these are discussed i n the next chapter. CHAPTER VI CONCLUSIONS 6.1 Observed P r o f i l e s With the experimental l i m i t a t i o n s of Chapter V i n mind, i t i s apparent that the observed p r o f i l e s do not agree i n d e t a i l with the t h e o r e t i c a l p r o f i l e s of Griem (1968) and Barnard (1969). While the r e l a t i v e i n t e n s i t i e s of the forbidden components X^  show good agree-ment for each l i n e , the combined widths W are a l i t t l e narrower than c a l c u l a t e d and, the peak separations S and r e l a t i v e i n t e n s i t i e s 1^  show poor agreement with the calculated values. I t would appear that the calculated wavelength s h i f t s of the allowed ( 2.P - ty-D ) and forbidden ( >LP - ty p ) peaks from the zero f i e l d values are i n error as the following considerations would suggest. The theore-16 —3 t i c a l p r o f i l e of % 4471 f o r Y l e = 4.5 x 10 cm was resolved by in s p e c t i o n i n t o two components. These components were s h i f t e d together o by 2.0 A and added. The r e s u l t a n t p r o f i l e then agreed remarkably well with the observed p r o f i l e (see F i g . 38). The adjustment i n the calculated separation S produced good agreement i n W and with 1^. unchanged. The c a l c u l a t e d i n t e n s i t y of the 2P - 4P component i s also i n good agree-ment with the observed i n t e n s i t y which has a larger uncertainty (about 30%). The discrepancy i n the peak separation S i s understandable since i n l i n e p r o f i l e c a l c u l a t i o n s , l i n e s h i f t s are more d i f f i c u l t to determine than l i n e widths. One reason, already mentioned (Section 2.43) i s that the imaginary part of > which can be associated with the l i n e s h i f t , 72 0.02 b 4440 FIG. 38.--An Empirical Correction to the Computed P r o f i l e of Hel 4471 © H"X This Experiment (Three Runs) Best F i t f o r Observed • P o i n t s Reconstructed T h e o r e t i c a l P r o f i l e . 4470 4500 4520 74 i s more s e n s i t i v e than the r e a l part (width) to the upper cutoff para-meter Q that one uses. Another d i f f i c u l t y i s that the imaginary part i s more s e n s i t i v e than the r e a l part to the number of l e v e l s " included i n the summation. The imaginary terms w i l l be e i t h e r p o s i t i v e or negative since 2 — t J ^ ^ p^ir^ and BCS^X') i s an odd function (see Barnard et a l , 1969). In the summation over Oi" the imaginary terms w i l l tend to cancel and the number of l e v e l s one includes w i l l be important. On the other hand A(2,z9 i s an even function making the r e a l terms p o s i t i v e . The contributions from d i s t a n t l e v e l s w i l l then be of diminishing s i z e so the number of l e v e l s included i s less important for the width. I t should be noted that, although the t h e o r e t i c a l assumptions d i s -cussed i n Section 2.44 appear to be v a l i d i n t h i s case, Griem 1s (1968) and Barnard's (1969) c a l c u l a t i o n s were f o r a hydrogen plasma with a small amount of helium present. The presence of doubly ionized helium p a r t i -cles i n this plasma should be taken into account. Chandrasekhar (1943) shows that for the Holtsmark normal f i e l d strength p = a.ti e ( the ion density N^ . can be w r i t t e n as N j = £ ^ 3 A n ^ f o r -nf = n ( M e + ; etc. Assuming for the sake of argument that i n t h i s plasma H2='H< then ' - 2.61 e _3/i _ yL/3 _ 2. 4 75 = 1-13 F e where F , was calculated with M _ - Trt. - ~ ~y\. I e f The new value, p would suggest an increase of 18% i n the calculated s h i f t and broadening due to the ions. However, th i s increase would be o f f s e t to some extent by changes i n the ion f i e l d d i s t r i b u t i o n , \ A / ( F ) due to increased co r r e l a t i o n s between the ions. In any case the agreement with the observed p r o f i l e s would be s l i g h t l y worse than before. I t i s hoped that these r e s u l t s w i l l suggest further ways to improve the c a l c u l a t i o n s f or these l i n e p r o f i l e s . For example, Barnard's ( 1 9 6 9 ) c a l c u l a t i o n s included, f or computational s i m p l i c i t y , only the y\ = 4 perturbing l e v e l s . Perhaps the i n c l u s i o n of the "KL = 3 and V\. = 5 l e v e l s would improve the calculated s h i f t . Another p o s s i b i l i t y i s that the assumptions used i n the c a l c u l a t i o n of the ion f i e l d d i s t r i -bution (not considered) may not be v a l i d i n this case. 6.2 Pulsed Arc Plasma The plasma produced i n the pulsed arc described i n this work was observed to be a good source for l i n e p r o f i l e measurements. E l e c t r o n d e n s i t i e s from *Ylg = 3 x 1 0 ^ to 1 0 ^ cm ^ were obtainable with a very uniform s p a t i a l d i s t r i b u t i o n (at "fc = 60 JJLsec) both i n the r a d i a l and a x i a l d i r e c t i o n s . The e l e c t r o n temperature obtained, 4.5 eV was a l i t t l e higher than optimum for the neutral helium .lines studied (the plasma was approximately 997o ionized at t h i s temperature); however, the e m i s s i v i t y was s u f f i c i e n t l y high that smooth p r o f i l e s were s t i l l obtained. 6.3 Rapid Scan Spectrometer ; • The rapid scan p h o t o e l e c t r i c technique was very s u i t a b l e f or this plasma, representing an improvement over the tediousness and larger errors 76 incurred i n the shot-to-shot p r o f i l e measurements and also providing greater s e n s i t i v i t y than with a photographic technique (Barnard et a l , 1968). There are some improvements possible i n the design of the spectro-meter. A provision for recording several p r o f i l e s simultaneously would have been desirable to eliminate a l l shot-to-shot errors i n the measure-ments ( H i l l et a l 1966). A better quality external imaging system i s also possible. Such a device could make use of parabolic mirrors to eliminate the spherical aberration and a near axis geometry to reduce the astigmatism and coma (see F i g . 39a). A l t e r n a t i v e l y the external imaging system could be dispensed with altogether by rotating (see F i g . 39b). These designs would improve the t o t a l resolving power or a l t e r n a t i v e l y permit the use of wider s l i t s , giving greater i n t e n s i t i e s . The present device however was e n t i r e l y adequate for the lines studied and did not represent a serious l i m i t a t i o n i n the measuring accuracy. I t i s hoped that further studies can be made of overlapping helium l i n e s , for instance Hel 4026 and 4387. I t would also be highly desirable to measure the W 4471 and 4922 p r o f i l e s at a lower electron 15 -3 density (~10 cm ) although for this purpose a d i f f e r e n t plasma source would be required. 77 FIG. 39.—Improved Designs for the Rapid Scan Spectrometer 78 APPENDIX I Derivation of E f f e c t i v e Sweeping Arm The scan rate was wr i t t e n i n Section 3.43 as ol"t f o r A. = wavelength CO = angular frequency of the r o t a t i n g mirror o((X.) = r e c i p r o c a l d i s p e r s i o n of wavelengths at s l i t S^ (or S^ i f the imaging system i s one to one) R. = the e f f e c t i v e sweeping arm. The expression f o r R. w i l l be derived, using geometric o p t i c s , from lengths L ^ L ^ ay\J. R. , the radius of curvature of the s p h e r i c a l mirror M^ , (see F i g . 40). When mirror rotates by an angle of A 0 , the beam i s deflected o would by an angle of 2, &0. If were a plane mirror the cen t r a l ray move from X | to X ^  • Then A X — Z ( L 1 + Lz) • A 0 However, the normal at O i s not T l , p a r a l l e l to T l - i t i s T L which passes through the center of curvature of as does T L . Define S as the angle between T l / and T l " . Then S ~ D / R and D = ZL^AQ. Now the "corrected beam" w i l l be deflected by an angle of 2.(5 from X ^ or AX = 2.(1^ + 1-^)4 0 -2-L-z£ 79 w FIG. 40.—Image S h i f t when Mirror M, Rotates by A 0 80 APPENDIX II The Correction f or Instrument Broadening j When a dispersive instrument such as a monochromator or spectrograph i s used to observe a l i n e source, some of the broadening i n the observed l i n e s w i l l be caused by the f i n i t e r e s o l v i n g power of the instrument. This r e s o l v i n g power w i l l depend on such factors as the d i s p e r s i v e element (grating, prism), the geometry of the o p t i c a l system and the s l i t widths. This "instrument broadening" can be expressed by a transmission curve |—| (f) for wavelength di f f e r e n c e s "% - \ ~ A. about a ce n t r a l wavelength . I f for 7| = X"X0 i s the true l i n e p r o f i l e , (having a f u l l h a l f width normalised to u n i t y here), then the observed i n t e n s i t i e s w i l l be given by the convolution i n t e g r a l : oo - o o (Note that i f one has a source with l i n e s of n e g l i g i b l e width, i f becomes a S function and one measures the instrument p r o f i l e H ^ ^ ) On the other hand i f these l i n e s are extremely wide H ( f ) can be W 81 represented by a (§ function and one measures fC^) d i r e c t l y . ) For convenience H (.^ ) was approximated by a tri a n g u l a r shape i n these c a l c u l a t i o n s , Hw(t) = 1 - T 7 • fof I t M W W O for i l l > W (This r e l a t i o n i s exact for wide s l i t s ; f o r narrower s l i t s , i t i s only a f a i r approximation because the instrument p r o f i l e becomes more rounded. However, these c a l c u l a t i o n s were l a t e r checked with Voigt functions (Van de Hulst and Reesinck, 1947) using reasonable approximations for the instrument p r o f i l e : (1) Gaussian, and (2) Voigt with ^Jjk. = 0.1. For the narrowest l i n e studied the deviations i n the corrected widths used 1.6 and 2.47. from the value calculated below. If the true l i n e p r o f i l e f (T^) i s Lorentzian (which was a good approximation f o r Hel 3889), the observed i n t e n s i t y d i s t r i b u t i o n becomes: ~ w o w ° W f d1 22 f d1 n f d1 -w -w 1 fJ2LL Wj fy,+ w % (n+w) 82 Then _L w U ian U + Vta«f V " 1% t a ^ X + { h r + for u V X z(r/ + w) 1 (>t - w) P r o f i l e s I (T^) were plotted f or various values of Vs/ and the ( f u l l ) h a l f widths, A A . measured. These values are given below. Then obs W/A XOFC)S was p l o t t e d versus W/AX . ( AX i s the true ( f u l l ) h a l f width of the p r o f i l e , normalised to u n i t y here.) This graph ( F i g . 41) could then be used as an approximate co r r e c t i o n f or the instrument broadening, ( i . e . given oys one could f i n d W / A X ). w A X 0bs 0.1 1.01 0.2 1.04 0.3 1.08 0.4 1.14 0.5 1.20 0.6 1.28 FIG. 4 .—Approximate Correction f or Instrument Broadening + 6 V ? 10 K (I0T) I ! — W V — + 5 0 - 6 0 V o s s i p r JOI 1 4 6 J I 1 • 0 2 I30J IN ' I O r03 1 2 O I I C VARIES FROM l20pF to .002 uF DEPENDING ON DESIRED DELAY RANGE OUT D E L A Y - U N I T W I T H H I G H - V O L T A G E , L O W I M P E D A N C E ( 5 0 J 1 ) O U T P U T . C O M P R I S E D O F : D E L A Y - O N E S H O T , O U T P U T P U L S E W I D T H O N E S H O T B O T H W I T H T E R M I N A T I N G N E T W O R K S , A N A M P L I F I E R , A H I G H - V O L T A G E A M P L I F I E R , A N D E M I T T E R - F O L L O W E R O U T P U T W I T H P R O T E C T I V E C I R C U I T R Y . + 20-25 V Q + 5 V 100 -vw MC846P SHAPER 6 SCHMITT TRIGGER 6 7 NOTE: QI,Q2,Q4,Q7 ARE 2N3704 TRANST. Q6 IS A 2N656 TRANSISTOR Q8 IS A 2N697 " INPUT COMES FROM A TAPEREC. HEAD BEING ACTIVATED BY A REVOLVING MAGNET. UNIJUNCTION J Z L I 2 3 12 10 II 14 MC846P AND GATE FLIP FLOP 9 "1PBI 6 + 5 V Q3 IS A T I 5 4 3 TRANSISTOR R l IS A 50KI0TURN POT, IT IS THE VARIABLE CONTROL FOR THE UNIT. R 2 IS A 20K TRIMPOT USED TO SET THE UPPER AND LOWER LIMIT FOR Rl SI SELECTS THE HIGH OR LOW RANGE FOR Q3 IN CONJUNCTION WITH R l . POSITION A IS LOW, POSITION B IS HIGH. PBI RESETS THE FLIP FLOP TURNING ON READY LAMP. WHEN ONE SHOT # 3 FIRES ON COMMAND FROM FLIP FLOP, LAMP GOES OUT. ONLY ONE PULSE OUT AT COINCIDENCE. Q3 AND DIODE ( IN9I4) FROM Q 3 S ' BASE 2 ARE IN SAME HEAT SINK FOR GOOD STABILITY. 2.2K READY 6 V LAMP N.O.C. PRE-SET REPETITION RATE INDICATOR I • I . I . I , I , I 4900 4910 4920 4930 4940 4950 Al'PIWDTX TV - Linear Plolii oC 4922 WAVELENGTH (A) 86 BIBLIOGRAPHY Baranger, M. 1962. Chapter- 13 i n D. R. Bates (ed.) . 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A.,Palumbo, G. and Griem, H. R. 1968. Phys. Rev. 172, 148. Griem, H. R., Baranger, M., Kolb, A. C. and Oertel, G. 1962. Phys. Rev. 125, 177. 1964. Plasma Spectroscopy (McGraw-Hill, New York). . 1968. Astrophys. J. 154, 111. 89 H i l l , R. A. 1965. App. Opt. 4, 1593. and Beckner, E. H. 1964. App. Opt. 3, 929. and F e l l e r h o f f , R. D. 1966. App. Opt. 5, 1105. Holtsmark, J. 1919. Ann. Phys. (Leipzig) 58, 577. Hooper, C. F. 1968. Phys. Rev. .165, 215. Huddlestone, R. H. and Leonard, S. L. '1965. Plasma Diagnostic Techniques (Academic Press, New York). Lenz, W. 1924. Z. Phys. 25, 299. Lindholm, E. 1941. Ark. Mat. Astron. Fys. 28B, 3. Lochte-Holtgreven, W. (ed.) 1968. Plasma Diagnostics (North Holland Publishing Co., Amsterdam). Lorentz, H. A. 1906. Proc. Roy. Acad. S c i . (Amsterdam) 8, 591. Margenau, H. and Lewis, M. 1959. Rev. Mod. Phys. 31, 569. Messiah, A. 1958. Quantum Mechanics (Wiley, New York). Mewe, R. 1967. B r i t . J. App. Phys. 18, 107. Moore, C. E. 1959. A M u l t i p l e t Table of Astrophysical Interest NBS (U.S.) Tech. Note 36. Mozer, B. and Baranger, M. 1960. Phys. Rev. 118, 626. Roberts, D. E. 1968. JQSRT 8, 1241. Sadjian, H. and Wimmel, H. K. and Margenau, H. 1961. JQSRT 1, 46. Sawyer, R. A. 1963. Experimental Spectroscopy (Dover, New York). Van de Hulst, H. C. and Reesinck, J. J. M. 1947. Astrophys. J. JL06, 121. Weisskopf, V. 1932. Z. Phys. 75, 287. Wiese, W. L., Smith, M. W. and Glennon, B. M. 1966. Atomic T r a n s i t i o n  P r o b a b i l i t i e s Vol. I NBS 4 U.S. Dept. of Commerce, Washington. Wulff, H. 1958. Z. Phys. 150, 614. 

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