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Measurement of small shifts of wide lines Potter, Michael Urwin 1967

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MEASUREMENT OF SMALL SHIFTS OF WIDE LINES by  MICHAEL U. POTTER  B.Sc. Royal M i l i t a r y College, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the department of PHYSICS  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER, 1967  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and Study.  | further agree that permission for extensive copying of this  thesis for scholarly purposes may be granted by the Head of my Department or by h.iJs representatives.  It is understood that copying  or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  y$/cS  The University of British Columbia Vancouver 8, Canada Date  /  Oc  -iiABSTRACT A new technique has been d e v i s e d to o b t a i n timer e s o l v e d measurements o f s m a l l s p e c t r a l l i n e s h i f t s . l i n e a r n e u t r a l d e n s i t y wedges w i t h o p p o s i t e  Two  transmission  g r a d i e n t s were p l a c e d over the top and bottom o f a monochromator e x i t s l i t  i n such a way t h a t a wavelength  shift  of a s p e c t r a l l i n e r e s u l t e d i n a change i n l i g h t i n t e n s i t y transmitted  through t h e wedges, which was monitored by two  photomultipliers.  To t e s t t h i s technique,  an N I I l i n e ,  f o r which Grlem has c a l c u l a t e d the Stark s h i f t as a f u n c t i o n of e l e c t r o n d e n s i t y , was observed i n the emission  spectrum o  of a s m a l l t h e t a - p i n c h u s i n g a monochromator o f 10 A/mm inverse dispersion.  Space and time r e s o l v e d l i n e  shift  measurements were made and were a s s o c i a t e d w i t h the a x i a l v e l o c i t y and the e l e c t r o n d e n s i t y f l u c t u a t i o n s o f t h e plasma. o  o  An u n c e r t a i n t y o f 0.015 A was estimated,  which, f o r t h e 1.5 A  l i n e used, corresponds t o a s h i f t t o w i d t h r a t i o o f 10 b e l i e v e d t o be lower than any p r e v i o u s nique.  ,  s h i f t measuring t e c h -  Time o f f l i g h t v e l o c i t y measurements and c o n s e r v a t i o n  o f mass c o n s i d e r a t i o n s have confirmed t h e r e s u l t s as both reasonable  and s e l f - c o n s i s t e n t .  -lii-  TABLE OF CONTENTS CHAPTER I  1  INTRODUCTION  (a) General I n t r o d u c t i o n  1  (b) Discussion of E x i s t i n g Methods  2  (c) Double Wedge Technique  5  CHAPTER I I  THEORY  8  (a) Fundamental Equation  8  (b) S e n s i t i v i t y of the Measurement  15  (c) A p p l i c a t i o n to Experiment  17  CHAPTER I I I  APPARATUS  (  21  (a) General D e s c r i p t i o n of the Apparatus  21  (b) N e u t r a l Density Linear Wedges  25  CHAPTER IV  EXPERIMENT AND RESULTS  30  (a) Preparations f o r Line S h i f t Measurements  30  (b) Measurements of Small Line S h i f t s  35  (c) Doppler S h i f t Measurements  37  (d) Stark S h i f t Measurements  ^2  (e) Systematic E r r o r s i n the Measurement  ^5  CHAPTER V  CONCLUSIONS  ^8  (a) General Conclusions  ^8  (b) Future Research  ^9  APPENDIX I APPENDIX I I  THE PHOTOGRAPHIC PROCESS AND THE LINEAR WEDGE TIME OF FLIGHT VELOCITY MEASUREMENTS  50  53  -ivACKN OWLEDGEM EN TS I wish to thank my s u p e r v i s o r , Dr. B Ahlbom, f o r h i s patience and guidance during the experimental i n v e s t i g a t i o n s and the p r e p a r a t i o n of t h i s t h e s i s . Thanks are due a l s o to the members of the t e c h n i c a l s t a f f f o r t h e i r help and advice. In p a r t i c u l a r , I wish to acknowledge the i n v a l u a b l e a s s i s t a n c e of Mr. P. Haas, Mr. J . Dooyeweerd, and Mr. E. Williams i n the c o n s t r u c t i o n and maintenance of the apparatus. I am a l s o g r a t e f u l to Mr. J . Strachan f o r the work done i n photographing and i d e n t i f y i n g the spectrum which was used i n t h i s work. • The f i n a n c i a l a s s i s t a n c e of an NRC Studentship i s g r a t e f u l l y acknowledged.  -1CHAPTER I INTRODUCTION (a) General Introduction Plasma spectroscopy has become an extremely  effective  technique f o r measuring both atomic parameters and some macroscopic properties of the plasma.  A great deal of information  may be obtained by measuring the wavelength line.  s h i f t of a spectral  Such s h i f t s can be produced by mass motion  (Doppler  s h i f t s ) , magnetic f i e l d s (Zeeman e f f e c t ) , and internal f i e l d s (Stark e f f e c t ) . Stark effect  electric  In many t y p i c a l laboratory plasmas the  i s dominated by the impact of electrons with the  emitting atoms or Ions.  Hence, v e l o c i t i e s , magnetic f i e l d s and  electron densities can, i n p r i n c i p l e , be determined from l i n e s h i f t measurements.  The d i f f i c u l t y of these measurments i s  due to the very small s h i f t s compared with the l i n e width itself.  For example, i f a plasma has a p a r t i c l e v e l o c i t y of  5 km/sec, then the Doppler s h i f t of a spectral l i n e of S000 A wavelength  i s \ = X..v/c = 8 x 1 0 " A . 2  Stark s h i f t s which, to  a very good approximation, are l i n e a r l y proportional to the electron number density, may by t y p i c a l l y of the order of 10"  1  A f o r a plasma of n  « 1 0 ? cm~^ and T « 10^ °K. 1  For a  o  t y p i c a l l i n e width of 1 A i t i s required to measure s h i f t to width r a t i o s of 1 0 ~ to obtain 10% accuracy i n l i n e s h i f t 2  measurements of the above examples.  The d i f f i c u l t y of such a  measurement i s enhanced by the fact that frequently only high inverse dispersion monochromators are a v a i l a b l e .  For an  o  inverse dispersion of 10 A/mm _?  of the order of 10  mm.  the s h i f t of these examples i s  -2The purpose of this thesis i s to describe a new technique for measuring such small l i n e s h i f t s of wide l i n e s and to report some results obtained from a theta-pinch experiment applying t h i s method to Stark and Doppler shifted l i n e s . important  The  features of the technique are (1) no assumptions  regarding the wavelength p r o f i l e of the spectral l i n e are necessary,  (2) high time r e s o l u t i o n i s possible, and (3) high Q  inverse dispersion (10 A/mm) monochromators may be used.  (b) Discussion of E x i s t i n g Methods Several methods of measuring small l i n e s h i f t s have been previously reported.  Photographic  techniques are the most  common and perhaps the easiest to apply.  A spectrum i s taken  with a low inverse dispersion spectrograph of the plasma of interest,  and, on the same p l a t e , i s placed a wavelength  c a l i b r a t i o n provided by the spectrum of a low-pressure d i s charge, f o r which the l i n e s h i f t s are n e g l i g i b l y small.  The  main disadvantage of t h i s technique Is that, without further complication of the apparatus (rotating mirrors, drum cameras, e t c ) , a time integrated spectrum i s taken and no time resolution i s possible. In 1962 Drobowshevsky (ref. 1) reported that he had measured Doppler s h i f t s of a  few tenths of an Angstrom r e -  s u l t i n g from the azimuthal motion of a plasma i n a homopolar device.  His method consisted of observing a single l i n e from  a monochromator and s p l i t t i n g the wavelength p r o f i l e with the i n t e r s e c t i o n l i n e of two prism faces perpendicular to each other, so that two photomultipliers were exposed to the l i g h t from the two halves of the l i n e p r o f i l e .  He then rotated the  -3prism assembly about an axis p a r a l l e l to the d i r e c t i o n of the l i g h t beam, so that i t was f i g . 1.  set at some angle a, as shown i n  It i s easily seen that a s h i f t i n wavelength of the  Fig. 1  Drobowshevsky»s l i n e s h i f t measurement  l i n e w i l l r e s u l t i n an increased l i g h t i n t e n s i t y f a l l i n g on one photomultiplier and a decreased i n t e n s i t y on the other, and from the v a r i a t i o n i n their voltage signals the s h i f t be calculated.  can  The f i r s t d i f f i c u l t y associated with t h i s tech-  nique i s that i t requires a knowledge of the l i n e p r o f i l e and the i n t e n s i t y d i s t r i b u t i o n along the s l i t . may  These functions  be d i f f i c u l t to measure accurately and could very well  vary during an experiment.  The second d i f f i c u l t y a r i s e s i f  the l i n e i s narrow or the inverse dispersion of the monochromator i s high, i n which case the width of the prism i n t e r s e c t i o n edge may  no longer be small compared to the l i n e width at the  -4exit plane of the monochromator, so that i r r e g u l a r i t i e s i n edge w i l l greatly influence the  result.  Another l i n e s h i f t technique was et a l ( r e f . 2 ) .  applied by  Keilhacker  They used a Fabry-Perot interferometer  monitored the l i n e p r o f i l e photoelectrically eight f i b r e bundles arranged i n a l i n e . .  covered a spectral range of 2.5^  the  and  with an array of  These eight  fibres  o  A.  By monitoring the  line  a p r o f i l e i n 0.32  A steps i t seems unlikely that s h i f t s of  less  o  than 0.05 was  A could be accurately measured.  This s e n s i t i v i t y  s u f f i c i e n t to obtain useful information regarding  the  r o t a t i o n of a plasma i n a theta-pinch, since Doppler s h i f t s several tenths of an Angstrom were found. of interest  i n laboratory plasmas are not  However, l i n e  of  shifts  always so large.  For  example, i t would be d i f f i c u l t to apply such a method to measure p a r t i c l e v e l o c i t i e s i n a membrane shock tube where s h i f t s i n -2 -1 0  the range 10  to 10  A are to be  expected.  Small l i n e s h i f t s have also been measured (ref. 3»^»5) by  selecting two  narrow segments on the wings of the l i n e ( f i g . 2)  two narrow wavelength bands monitored photoelectrically  Fig. 2  Line s h i f t measurement by s e l e c t i n g two segments of the spectral l i n e .  narrow  and recording the signal from each segment.  As the l i n e s h i f t s ,  the r e l a t i v e magnitude of the signals w i l l vary and the size of the s h i f t can be calculated i f the l i n e p r o f i l e i s known. Generally, a small inverse dispersion instrument for  i s required  t h i s technique, except i n Hirschberg»s apparatus (ref. 3),  which, by clever use of polarized l i g h t , separates two d i f f e r e n t wavelengths passing through a single exit s l i t . d i f f i c u l t y associated with t h i s technique  The major  i s that i t i s r e -  quired to know the l i n e p r o f i l e before the s h i f t s can be c a l culated.  In addition, only a f r a c t i o n of the t o t a l l i g h t inten-  s i t y i s used and an acceptable s i g n a l to noise r a t i o may be more d i f f i c u l t to obtain than i f the entire l i n e p r o f i l e were used.  (c) Double Wedge Technique The double wedge technique of measuring l i n e s h i f t s i s designed to permit time resolved measurements of s h i f t s -2 °  down to 10  A.  The basic p r i n c i p l e of t h i s method, which  has already been described (ref. 6), can be best understood by means of f i g . 3.  The monochromator i s used to observe a  single i s o l a t e d l i n e i n the plasma r a d i a t i o n spectrum. At the exit s l i t are mounted two neutral density f i l t e r s , W^ and Wg.  The transmission of these f i l t e r s varies l i n e a r l y i n  the x - d i r e c t i o n and the transmission gradients are equal and opposite f o r the two f i l t e r s .  These f i l t e r s are also aligned  so they have equal transmission at the centre of the s l i t . All  the l i g h t from the upper h a l f of the exit s l i t passes  through the f i l t e r W  1  and i s monitored by photomultiplier P  1  Fig. 3  Double wedge technique  and a l l the l i g h t from the lower half of the s l i t passes through W  2  and onto Pg.  A s h i f t i n the wavelength of the spectral l i n e  w i l l increase the Intensity i n one photomultlplier and decrease the intensity i n the other.  I f the slope of the transmission  gradients, the transmission at the centre of the s l i t , and the dispersion of the monochromator are a l l known, then the wavelength s h i f t of the l i n e can be calculated from measurement of the photomultlplier signals. As w i l l be proved i n Chapter I I , no assumptions need be made regarding the l i n e p r o f i l e or even the constancy of the line profile. l i n e centroid.  This technique always measures the s h i f t of the By recording the photomultlplier signals on an  oscilloscope a temporal history of the l i n e s h i f t can be obtained.  Very good time resolution i s possible and i s limited  only by the response of the photo-electronics which can be —8  made as short as 3 x 10~ sec. A plasma l i g h t source (theta-pinch) was set up to experimentally demonstrate the double wedge technique, and an N I I l i n e was observed f o r which the Stark s h i f t has been c a l culated by Griem (ref. 7),  It was expected that possible ex-2 °  perimental error inherent i n the apparatus was about 10 A and the s t a t i s t i c a l uncertainty, due mainly to the non-reproduc-2 °  i b l e nature of the plasma, was about 3 x 10 p e t i t i v e measurments were taken.  A i f nine r e -  Checks have been made to  show that the Doppler s h i f t and Stark s h i f t r e s u l t s obtained were both reasonable and self-consistent. A major d i f f i c u l t y i n the experimental procedure of t h i s method i s to balance two photomultipliers accurately f o r f a s t r i s i n g signals.  Although response time i s naturally  l i m i t e d by the r i s e time of the photo-electric system, a procedure was devised to eliminate the e f f e c t of small d i f f e r ences i n response c h a r a c t e r i s t i c s of the twin photomultiplier system.  This i s discussed i n Chapter IV.  -8-  CHAPTER II THEORY (a) Fundamental Equation In  t h i s section the fundamental equation w i l l be  derived whereby l i n e s h i f t s can be calculated from the double wedge technique. Consider the intensity p r o f i l e of an unshifted spectral l i n e at the exit plane of a monochromator to be I(x,y) = £(x)a(y).  The function *(x) represents the l i n e  shape at the exit plane and i s due to the natural l i n e shape, the macroscopic properties of the plasma (such as r o t a t i o n a l motion), the instrument p r o f i l e and the s l i t width.  The func-  t i o n a(y) represents the intensity d i s t r i b u t i o n over the length of  the exit s l i t .  This i s dependent upon the c h a r a c t e r i s t i c s  of  the monochromator and upon the optics which focus the l i g h t  from the plasma onto the entrance s l i t .  The a(y) w i l l also  depend upon any non-uniformity of the plasma observed.  During  the experiment cc(y) i s considered to change only by a constant factor and the design of the apparatus (in p a r t i c u l a r , the entrance optics) must ensure that this assumption i s v a l i d . The assumed form of I(x,y) i s j u s t i f i e d i f the monochromator s l i t s are p a r a l l e l and i f the intensity of l i g h t at each point on the exit s l i t has equal contributions from a l l parts of the plasma observed so that there i s no correlation between the p o s i t i o n y and the plasma configuration.  As des-  cribed more f u l l y i n Chapter III both t h i s assumption and the constancy of a(y) require astigmatic focussing of the plasma r a d i a t i o n on the entrance s l i t .  -9The p o i n t , x = 0, y = 0, i s defined to be a t the centre of the e x i t s l i t .  The s p e c t r a l l i n e to be observed i s p o s i t i o n e d  so that i t s c e n t r o i d w i t h respect to the x-coordinate x = 0.  l i e s at  Assume that the width w of the s l i t i s very much  greater than the width L of the l i n e a t the e x i t plane.  Under  these conditions the t o t a l l i g h t i n t e n s i t y passing through the upper h a l f of the e x i t s l i t i s given by d/2 w/2 = /a(y)dy /*(x)dx 0 -w/2 where d i s the length of the s l i t .  (la)  The l i g h t i n t e n s i t y  passing through the lower h a l f of the s l i t i s  G  2  0 = /a(y)dy -d/2  w/2 /f(x)dx -w/2  Two p h o t o m u l t i p l i e r s monitor i n t e n s i t i e s signal  recorded by each p h o t o m u l t i p l i e r  (lb)  and Gg.  The  i s p r o p o r t i o n a l to  the l i g h t i n t e n s i t y i t r e c e i v e s , where the constant of proport i o n a l i t y may be v a r i e d by the p h o t o m u l t i p l i e r power supply adjustment.  These voltage s i g n a l s can be w r i t t e n 1  5  2  5  =  =  k  k  l l G  i Z a  2 2  ^  (2b)  G  The constants k^ and kg are v a r i e d by a d j u s t i n g the supply voltages to make  = S  2#  In f r o n t of the upper h a l f of the e x i t s l i t i s placed a n e u t r a l density wedge whose transmission i s given by t,(x) = t  9  + bx  (3a)  where t  e  and b are constants.  In f r o n t of the lower h a l f o f  the s l i t i s placed a wedge w i t h transmission t (x)  = t  2  4a.  This i s shown g r a p h i c a l l y i n f i g . of  (3b)  - bx  Q  The l i g h t i n t e n s i t y  the upper h a l f of the s l i t i s now given by d/2 w/2 = /a(y)dy / t ( x ) * ( x ) d x 0 -w/2  S  ±  1  d/2 w/2 = t G + b/a(y)dy / x f ( x ) d x 0 -w/2 0  1  (4 ) a  and the s i g n a l recorded by the upper p h o t o m u l t l p l i e r i s s  l = l«l k  = k t G 1  e  1  d/2 w/2 + k b/a(y)dy /x*(x)dx 0 -w/2 1  (5a)  S i m i l a r l y , f o r the lower h a l f of the e x i t s l i t the photomultiplier signal i s  s  2  = k t G 2  e  2  0 w/2 - k b/a(y)dy /x*(x)dx -d/2 -w/2 2  (5b)  Since, f o r any value of y, the c e n t r o i d of the i n t e n s i t y p r o f i l e w i t h respect to the x-coordinate l i e s at x = 0, then w/2 JxS(x)dx = 0 -w/2  (6)  F i g . 4b  S h i f t e d l i n e at the  -11-  e x i t plane  -12and  from equations (5a)  s = k t G  1  (7a)  s =k t G  2  (7b)  1  1  2  Therefore,  (5b)  and  s  1  0  2  = s  e  (8)  2  Assume that the l i n e centroid Ax  and may  i s s h i f t e d by an amount  change i t s shape i n any manner (see f i g . ^b). It  i s assumed that the o p t i c a l setup does not change i n any  way  so that the intensity d i s t r i b u t i o n a(y) along the exit s l i t w i l l change at most by a constant factor, C. One may  write the i n -  tensity d i s t r i b u t i o n of the s h i f t e d l i n e as I*(x,y) = Ca(y)I*(x - Ax) No relationship Now  (9)  i s assumed between the functions I and  I*.  the signal recorded by the upper photomultiplier i s  d/2 w/2 a(y)dy J t ^^xj) s* = k C/a(y)dy 0 -w/2 P  lJ  Let x* = x  (10)  I*(x - A x ) d x  -Ax  d/2 w/2 +Ax s* = k C/a(y)dy J t ^ x ' + Ax)I*(x» )dx» 0 -w/2 +Ax  (11)  1  Since small s h i f t s , such that x « the assuption, w »  w are considered,  and  L, implies I * ( x ) — y 0 as x — > w / 2 , then  the l i m i t s of integration over x* can be changed to ±w/2 negligible  error.  with  -13d/2 w/2 sf = k Ct /a(y)dy/I*(x«)dx» 0 -w/2 1  e  d/2 w/2 + kjCb/afyJdy /(x' +Ax)I*(x' )dx» 0 -w/2 d/2 w/2 = ^Cto/aCyJdy /I*(x» )dx» 0 -w/2 d/2 w/2 +k Cb/a(y)dy ;i*(x»)x»dx» 0 -w/2 1  d/2 w/2 xk Cb/a(y)dy /I*(x»)dx« 0 -w/2  +  1  (12)  Since the centrold of the shifted l i n e l i e s at x =Ax ( i e . x* = 0) then the second term of equation (12) vanishes. (13a)  s£ = Ct S£ + AxCbS* 0  Where S£ i s the signal from the upper photomultlplier when observing the s h i f t e d l i n e with no wedges. f o r the lower half of the exit  slit (13b)  s* = C t S | -AxCbS* 0  From  = 2 S  d/2  0  w/2  w/2  k /a(y)dy Jl(x)dx = k /a(y)dy /I(x)dx 0 -w/2 -d/2 -w/2 1  and i t follows that  Similarily  2  -14d/2  w/2  0  w/2  C3k /a(y)dy /I*(x» )dx» = Ck /a(y)dy 1  0  -w/2  -d/2  Therefore, Equations  /l*(x» )dx«  2  (14)  -w/2 (15)  S* = S* = S* (13a) and (13b) now become s f = CS*(t  G  + bAx)  (16a)  s | = CS*(t  0  - bAx)  (16b)  Solve f o r A x from equations (16a) and (16b) _ s* - s | ^ A  " s* + s | b  X  If the inverse dispersion, dX/dx, of the monochromator i s known then one can write s  ^  "  t  ~ * s  +  s  s  2  t  o  ^  A  s  2 "b" dx ~  t  o  ~  (18)  d ? l  dx  This i s the fundamental equation used to calculate l i n e s h i f t s from the double wedge technique, where AX i s the wavelength change of the shifted l i n e measured r e l a t i v e to the p o s i t i o n where both wedges have the same transmission, t . 0  Notice that equation (18) does not indicate an absolute l i n e p o s i t i o n measurement.  The double wedge tech-  nique i s a sensitive l i n e s h i f t measuring device only insofar as r e l a t i v e l i n e s h i f t s are concerned.  In t h i s respect this  technique has properties similar to a Paby-Perot i n t e r f e r ometer, with which r e l a t i v e measurements have high resolution but absolute wavelengths are known only to the p r e c i s i o n of the pre-selecting spectrograph.  It w i l l , however, become  -15evldent i n section (c) of this chapter how r e l a t i v e  line  s h i f t measurements can y i e l d useful information about a plasma l i g h t source.  (b) S e n s i t i v i t y  of the Measurement  The c r i t i c a l measurement f o r the determination of a small l i n e s h i f t i s the voltage difference,As.  For a given  l i n e s h i f t A.X this quantity w i l l be maximum when the slope b of the wedges i s greatest.  The value of b i s maximum when the  wedges vary from 0% to 100% transmission over the f u l l width L of the spectral l i n e at the exit plane (and t h i s i s only possible f o r a symmetrical  line) y.  t  ie. when transmission t  e  b  =  1 d\ L dx  w i l l be 5®%*  In this case equation (18) becomes AV=f§§  (19)  However, f o r a given experiment the quantity g ^ j i s a g  constant and the s e n s i t i v i t y of the voltage difference A s to a given l i n e s h i f t AX i s maximized when g ^  s  imized.  (which i s  This occurs when the l i g h t intensity  proportional to £s) i s greatest.  j i s min-  However the intensity  must  be maximized by adjustment of the entrance optics and increase of the plasma l i g h t intensity  Itself.  If Es i s increased  by opening the entrance s l i t then L i s also increased by a proportional amount and the s e n s i t i v i t y of measurement w i l l not be improved.  In essence, equation (19) indicates that  -16a bright, narrow l i n e at the exit plane i s required f o r best r e s u l t s of the l i n e s h i f t measurement. However there are other considerations whereby one can optimize the entrance s l i t width f o r a given spectral l i n e . It i s a normally accepted r e s u l t (ref. 8) that f o r a given number of photons incident on a photomultiplier the number of primary electrons emitted by the photo-cathode w i l l be Poisson d i s t r i b u t e d about the mean.  A unique feature of this  d i s t r i b u t i o n i s that the variance i s equal to the mean. By taking the variance of the s h i f t Ax, as given by equation (17) while holding t , b, and 2s constant one has 0  var(Ax) = [var(s») + v a r ( s » ) ] [ but var(s£) =  (mean of s£),  Therefore,  var(Ax) = [ £ ° ] ^  b(s  ^« » ] s  2  }  and var(s^) = s?  2  (mean of s|) (20)  2  since (sj + s ) 2  w  Is  However, Is i s proportional to the entrance s l i t width 1, the intensity of the plasma M, and the centroid transmission t  Q  of the wedges.  equation  Therefore one can write  Is = M l t , and e  (20) becomes var(Ax) =  b^Ml  (21)  The t o t a l l i n e width L at the exit plane i s approximately 1 + 1 , where l 9  c  i s the actual l i n e width as determined by  the natural l i n e shape and the properties of the plasma. It i s required that the wedge transmission vary from 100%  -17to  0% over the width L = 1 + 1  (assuming the expected s h i f t  0  i s small compared with L ) .  Under these conditions the slope b  i s given by  = 17+T  b  which implies that 1 = £ - l  (22)  e  so equation (21) becomes var(Ax) =  (23)  9  b M(± - 1.) 2  To f i n d b f o r the minimum var(Ax), equate ^ [ v a r ( A x ) ] to zero. £*(b - b l ) ~ ( l 2  2  e  i e . 21  Q  - 2bl ) = 0 Q  = i  Prom equations (22) and (24) 1 = l  (24) e  . This r e s u l t indicates  that one should choose an entrance s l i t width equal to the actual l i n e width.  However, i t should be kept i n mind that  t h i s i s required i f one wishes to minimize the variance of the d i s t r i b u t i o n of the voltage signal; but the possible errors introduced by other factors such as electromagnetic pick-up and noise may be of f i r s t consideration, especially when a bank discharge i s involved.  (c) Application to Experiment This method can be d i r e c t l y applied to measure s h i f t s due to Doppler and Stark effects i n a plasma with  with r o t a t i o n a l symmetry and a x i a l motion (such as a thetapinch). To measure the Doppler s h i f t due to the a x i a l motion i t i s necessary to observe the plasma from two d i f f e r e n t directions with respect to the z-axis and , i n e f f e c t , to compare the p o s i t i o n of the l i n e i n each case. However, to measure the Stark s h i f t one need only observe the plasma from one p o s i t i o n , but the absolute s h i f t can be evaluated only i f a precisely known reference l i n e can be observed. Unshifted standard l i n e s may be obtained from low density discharges, such as that described by Minnhagen (ref. 9). However, f o r the experimental work reported i n t h i s thesis such a discharge was not a v a i l a b l e .  Hence the Stark s h i f t s  were measured r e l a t i v e to an arbitrary wavelength  ( the wave-  length where the transmission of both wedges i s t ) , which 0  y i e l d s electron density fluctuations; i e . changes i n the electron density as a function of time. The fundamental nature of the measurement described above can be better understood as follows.  If one observes  the plasma from some angle a from the axis of r o t a t i o n a l symmetry then the l i n e p o s i t i o n observed can be expressed as A\ (t) ±  where:  =AX (t) +AX (t) + X g  d  0  (24)  A X „s i s the absolute Stark s h i f t AX  d  i s the Doppler s h i f t due to mass motion at v e l o c i t y V i n the a x i a l d i r e c t i o n ; i e . AA„ = a  X  Q  X ^cp a  i s the p o s i t i o n of the unshifted l i n e , which i s only known to the accuracy of the monochromator setting  -19Th e plasma i s then observed from some other angle a' without changing the position of transmission t  0  ( see f i g . 5).  / ^  Z  mirrors  Fig. 5  Doppler s h i f t measurement f o r a r o t a t i o n a l l y symmetric plasma  For t h i s second case the l i n e i s observed at the wavelength - X ( t ) = ^ X ( t ) +A\'(t) + X 2  The values o f A X  s  (25)  e  andAX» w i l l be equal i f one can assume s s  that e s s e n t i a l l y the same plasma configuration i s observed from both d i r e c t i o n a and a*.  If a + a'= 180® and one observes  a r o t a t i o n a l l y symmetric, o p t i c a l l y thin plasma with i d e a l optics t h i s assumption i s v a l i d ,  Under these c o n d i t i o n s A X  andAX^ are equal i n magnitude and opposite i n sign.  d  By  subtracting equation (25) from (24) the Doppler s h i f t i s given by AX (t) = d  X,(t) - X„(t) 1  2  2  (26)  To obtain Stark s h i f t data one observes the plasma from a d i r e c t i o n perpendicular to the d i r e c t i o n of mass motion,  -20i e . a = 90°, as shown i n f i g . 6.  Fig. 6  Stark s h i f t measurement of a r o t a t i o n a l l y symmetric plasm  Since AX. = X Q  V  o  o  s  a  =0  f o r a = 90°  then equation (24)  C  becomes X, (t) = A X ( t ) + X 1 s  e  (27)  and i t follows that | ^ A t = ^ ( A X ) At s  (28)  and the fluctuations i n time of the Stark s h i f t can be calculated. To apply and t h i s theory i n practice a plasma l i g h t source was required which was r o t a t i o n a l l y symmetric and whose spectral l i n e s were expected to show Stark and Doppler  shifts.  It was expected that a small theta-pinch device with a conveniently long ringing period, not c r i t i c a l l y damped, should permit measurement of both s h i f t s .  Such a source was set  up and i s described i n the following chapter.  -21CHAPTER I I I APPARATUS (a) General Description of Apparatus A schematic diagram of the experimental setup i s found i n f i g . 7. device.  The plasma observed was created by a theta- pinch  This consisted of a glass tube of 2.5 cm i . d . sur-  rounded by a single turn copper c o i l 2 cm wide and of diameter 3.2 cm.  A 15.6/^P capacitor bank charged to 15 kv was d i s -  charged through the c o i l .  Breakdown of the gas i n the tube  was ensured by a p r e - i o n i z i n g glow discharge.  For these ex-  periments the tube was f i l l e d with a i r at 35<V<Hg pressure. The l i g h t emitted from a narrow cross-section (AZ « 0.5 cm) of the plasma i n the tube was focussed a s t i g matically on the entrance s l i t of a Spex 1700 II monochromator by means of a system of f i e l d stops and c y l i n d r i c a l lenses. The axes of these lenses were p a r a l l e l to the monochromator s l i t s since, with no focussing i n t h i s d i r e c t i o n , the intens i t y d i s t r i b u t i o n ct(y) along the length of the s l i t s was not s e n s i t i v e to the o p t i c a l alignment and could be expected to remain constant during the experiment.  In addition, t h i s  ensured that a p o s i t i o n on the entrance s l i t did not correspond to a p a r t i c u l a r region of the plasma cross-section and the wavelength p r o f i l e of the l i n e was expected to be the same (except f o r a constant factor) at a l l points on the s l i t . Thus, two of the important assumptions of Chapter II are validated.  The o p t i c a l system was arranged so that the l i g h t  from the plasma could be observed from three d i f f e r e n t directions; perpendicular to the tube axis and at an angle of 45®  theophanis triggering circuit  c= i5.§#f » 15 kv pre-ionlzer 1000 v DC 0.3 ma  f i e l d stop mirrors  Spex 1700 II  position f o r neutral f i l t e r  monochromator e 10.6 A/mm f: 9.2  twin ejsltter follow er circuits  l i g h t tight box 6 « containing wedges twin prisms and photo- photomultiplier power supply multipliers  Fig. 7  Schematic diagram of apparatus -22-  © Tetronix 551 dual beam CRO  -2 3-  from the axis i n both directions (see f i g . 7 ) . A suitable spectral l i n e was centred at the exit  slit.  Immediately i n front of the s l i t were mounted the two neutral density l i n e a r wedges.  The l i g h t passing through each wedge  was monitored by an RCA IP 21 photomultiplier and the voltage signal from each photomultiplier was passed through an emitter follower and displayed on a Tetronlx 551 dual beam oscilloscope. The d e t a i l s of the wedge mount are shown i n f i g .  8.  This part of the apparatus had to be of p r e c i s i o n construction since, f o r a s h i f t of 10"  1  A, shot to shot s t a b i l i t y of 10"^ mm  i n the wedge p o s i t i o n was required i f an accuracy of 10% was to be expected.  The l i n e a r wedges, prisms and photomultipliers  were mounted i n a l i g h t tight box.  The wedges were movable  In the x-direction and were attached to a p a i r of micrometer drums i n such a way that they could be adjusted either i n dependently or together. Photographs showing the important parts of the apparatus are shown i n f i g 9.  Pig.  9  exit 'slit a) Main s l i d e , moving both l i n e a r wedges r e l a t i v e to the monochromator exit s l i t b) Micrometer drum moving main s l i d e (a) c) Neutral density l i n e a r wedges d) Secondary s l i d e moving lower wedge e) D i f f e r e n t i a l screw moving secondary s l i d e  i  ro  o  ct  o & H ct  f ) Shield separating l i g h t from v upper and lower halves of exit s i l t g) Mirrors d i r e c t i n g l i g h t to upper and lower photomultipliers h) Photomultipliers j ) Light box containing photomultlplier c i r c u i t s and emitter followers v  X  N  -25(b) Neutral Density Linear Wedges To obtain a suitable neutral density l i n e a r wedge i s of fundamental importance i n applying t h i s technique to measure small l i n e s h i f t s .  For a t o t a l l i n e width i n the exit plane o  o  of 1 A and a monochromator inverse dispersion of 10 A/mm one requires a wedge to be l i n e a r over a distance of about 1 0 " mm. 1  A photographic technique f o r making such a wedge has been developed.  It involves exposing a f i n e grained photographic  plate to the h a l f shadow region of a straight edge, as shown i n f i g . 10. A diffused l i g h t source (square with dimensions a x a) was placed at a height d above the p l a t e .  F i g . 10  A razor  Photographic technique of making the neutral density l i n e a r wedges  -26blade was held i n a h o r i z o n t a l p o s i t i o n at a height h with i t s edge d i r e c t l y beneath the centre of the source and p a r a l l e l to one of i t s sides.  D i f f r a c t i o n effects were n e g l i g i b l e f o r  the small distances between the razor edge and the plate and, therefore, the l i g h t intensity f a l l i n g on the plate i n the shadow region was a l i n e a r function of the distance along the plate and the width of the shadow was  found from similar  tri-  angles. As well as the simple dependence of the wedges on the geometry of the apparatus described above, t h e i r c h a r a c t e r i s t i c s also depended on the H & D curve the photographic p l a t e .  (density vs. log exposure) of  This e f f e c t i s dealt with i n Appendix I.  In order to understand the correspondence between the dimensions i n fig.10 and the wedge c h a r a c t e r i s t i c s , several wedges were made while varying a, d, and the exposure time. The transmission-distance r e l a t i o n of each was measured on a Kipp and Zonen microdensitometer  with a Heath chart recorder  at several points along the wedge.  These r e s u l t s are summarized  i n table 1. As seen i n Chapter II the slope of the wedges i s an important parameter i n c a l c u l a t i n g l i n e s h i f t s . microdensitometer,  Although the  once i t s r e s o l u t i o n was know, gave a good  estimate of t h i s quantity, a control l i n e s h i f t technique  was  developed giving a f a r more accurate average value f o r the slope. The Na 5890 l i n e , emitted from a Gates sodium lamp was  centred  at the exit s l i t of the monochromator and the wedges placed i n the wedge mount such that, f o r each, t This was  0  lay at the l i n e centrold.  done by moving each wedge i n the x-direction u n t i l the  -27Table 1 50 c i , a = 3 cm, h = 0,64 cm exposure Ax A t 20 sec 0.264 mm 40$ 25 «• 0.224 " 48" 30 " 0.218 » 60" 35 0.106 " 50" 40 " 0.112 " 55" 45 » 0.092 " 56" 50 " 0.099 " 60" n  slope b 157^/mm 214 " 275 " 473 " 491 " 609 " 607 "  60 cm, a = 3 cm, h = 0.64 cm exposure 20 sec 25 " 30 " 35 " 40 • " 45 " 50 " 55 "  Ax 0.186 mm 0.207 " 0.165 " O.172 «' O.152 " 0.185 " 0.106 " 0.125 "  At 28$ ^7" 53" 55" 55" 57" 56" 56"  slope b 150^/mm 227 " 322 " 320 " 355 " 309 " 529 " 448 "  A t 33$ 50" 45" 62" 58" 52" 41" 50" 48" 40" 47" 50" 47"  slope b 2l8#/mm 303 » 247 " 375 " 387 " 351 " 283 " 345 " 364 " 357 " 443 « 420 » 399 "  70 cm, a = 2 cm, h = 0.64 cm exposure 30 sec 35 " 40 « 45 « 50 " 55 60 " 65 " 70 " 75 " 80 " 85 " .9.0 " M  Ax 0 . 1 5 2 mm O.I75 " 0.172 " 0.151 " 0.150 " 0.148 " 0.145 " 0.145 " 0.132 " 0.112 " 0.106 " 0.119 " 0.098 "  In the above table A x and A t are the distance and the change i n transmission of the l i n e a r part of the wedge.  -28l i n e i n t e n s i t y as seen by each p h o t o m u l t l p l i e r was reduced by the f a c t o r t . 0  The l i n e was then s h i f t e d across the e x i t plane  by means of a very small r o t a t i o n of a glass p l a t e suspended i n the l i g h t path i n s i d e the monochromator.  By means of care-  f u l measurement of the index of r e f r a c t i o n and thickness of the g l a s s , and the angle of r o t a t i o n , one can c a l c u l a t e the magn i t u d e of the s h i f t of the l i n e at the e x i t plane.  I f , at  the same time, the voltage s i g n a l s of the two p h o t o m u l t i p l i e r s are recorded then the equation (17)  can be a p p l i e d where the  wedge slope b i s the only unknown. Such a measurment of the wedge slope was a valuable e r r o r check.  There were three ways that the wedges could have  introduced e r r o r s i n t o the experiment:  (1)  they may not have  been a l i g n e d p a r a l l e l to the s p e c t r a l l i n e , (2) there were l i k e l y to be imperfections i n the wedge due to non-uniformity and g r a i n i n e s s of the photographic p l a t e , and  irregularities  i n the l i g h t source and r a z o r blade used i n making the wedge, (3) the s h i f t e d , or u n s h i f t e d , l i n e may have spread out to the n o n - l i n e a r r e g i o n of the wedge. In case (1),  i f the wedges were p a r a l l e l to each other,  then the slope b* as i t appears to the s p e c t r a l l i n e would be constant (provided the l i n e l a y w i t h i n the l i n e a r r e g i o n of the wedge at a l l p o s i t i o n s along the e x i t s l i t ) and would be r e l a t e d to the true slope of the wedge by t * ss b cos a, where a i s the angle between the wedges and the l i n e .  This e f f e c t i v e  slope b' would be d i r e c t l y measured by the c o n t r o l l i n e s h i f t technique and was,  i n f a c t , the value to be used to c a l c u l a t e  the l i n e s h i f t s of the plasma s p e c t r a l l i n e .  I f , however, the  -2Q-  upper and lower wedges were not p a r a l l e l to each other then they would have d i f f e r e n t e f f e c t i v e slopes.  This would be  evident from the control l i n e s h i f t experiment  since, under  these circumstances, the sum of the voltage signals from the upper and lower photomultupliers would not remain constant. In cases ( 2 ) and ( 3 ) the slope b would not be constant i n the region of the wedges where the l i g h t passes  through.  However, these.sources of error would also be evident from the control l i n e s h i f t experiment  since the voltage signal  difference A s would vary non-linearly with the l i n e s h i f t . It can be concluded that t h i s technique gives an accurate value f o r the wedge slope and also represents a complete check on the wedges and t h e i r positioning i n the wedge mount.  -30CHAPTER IV EXPERIMENT AND RESULTS (a) Preparations f o r Line S h i f t Measurements To perform the experiment select a strong isolated l i n e .  i t was f i r s t necessary to  In order to survey the a v a i l -  able l i n e s of the theta-pinch at 350 AHg pressure of a i r the /  axis of the tube was imaged onto the entrance s l i t of a Hllger E 1 spectrograph and a time integrated spectrum (superposition of 10 shots) was taken.  F i g . 11 shows a  section of t h i s spectrum and indicates the  correspondence  between the p o s i t i o n on each spectral l i n e with the p o s i t i o n on the tube axis. The N II 399^ l i n e was selected f o r observation since i t f u l f i l l e d a l l three of the important requirements f o r making the l i n e s h i f t measurement: (1) l i n e s of the spectrum,  i t was one of the strongest  (2) i t was isolated, having no ob-  servable l i n e s within a few Angstroms, (3) i t s wavelength was close to that of the peak of the photomultlplier response curve.  In addition, Stark effect calculations have been  made by Griem (ref. 7) f o r t h i s l i n e so that l i n e s h i f t s can be related to electron densities. The l i n e was scanned by the monochromator with the s l i t s at a very narrow setting and the natural l i n e width c  plus instrument p r o f i l e was thereby estimated as 0.5 A.  A  few tests showed that i n order f o r the l i g h t intensity to record a strong signal on the photomultipliers f o r a wide range of positions along the theta-pinch tube i t was necessary to open the entrance s l i t up to 0.1 mm.  The width of the l i n e  theta-pinch c o i l Pig. 1 1 Spectrum of theta-pinch  -32at  the exit plane i s approximately the sum of the l i n e width  estimated above, and the entrance s l i t width; i . e .  about  o  o  1.5 A, since the monochromator inverse dispersion i s 10 A/mm. This indicates that a wedge i s required which i s l i n e a r over the range 0,15 mm plus the expected s h i f t . Prom the investigation of the wedge properties i n Chapter I I I (see table 1) i t was seen that a wedge made with h = 70 cm, a = 0.6 mm and exposure time 4-5 sec would be s u i t able f o r the l i n e to be observed.  A microdensitometer trace  of  The slope of the wedge was  t h i s wedge i s shown i n fig.12.  measured to be 375 #/mm  and i t s range of l i n e a r i t y $» 0.15mm.  Since the resolution of the instrument was 0.015 mm (estimated by scanning a razor blade edge) the actual rangeof l i n e a r i t y was l i k e l y to be about i n the l i n e a r region.  O.165 mm.This trace shows fluctuations These were due to graininess, i r r e g u l -  a r i t i e s i n the f i l m and dust, which were a l l random e f f e c t s . They would be evident when a narrow s t r i p of the wedge i s scanned, as i n the microdensitometer, but not when the i n tegrated e f f e c t of a l l positions along the wedge i s considered, as i n the l i n e s h i f t apparatus. This wedge was cut i n two and each half placed i n the wedge mount such that t h e i r transmission gradient lay i n opposite d i r e c t i o n s .  They were aligned under a t r a v e l l i n g  microscope before being f i x e d to the exit s l i t assembly.  The  control l i n e s h i f t experiment was then used to accurately measure the slope and test the l i n e a r i t y of the wedges. The r e s u l t s of t h i s check are discussed more f u l l y i n a l a t e r sect i o n of t h i s chapter.  -33-  transmission  Exposure time: 45 sec Source h e i g h t : 70 cm Source s i z e : 2 cm sq Razor edge h e i g h t : 0.6  lope b = H £ #  .1  .2 distance  F i g . 12  1  = 375  .3  (mm)  Microdensitometer t r a c e o f l i n e a r wedge  cm  #/mm  -35(b) Measurement of Small Line Shifts A time resolved picture of the N II 3994 l i n e i n tensity emitted by the plasma at a p a r t i c u l a r p o s i t i o n i n the theta-pinch tube i s shown i n f i g .  Fig.  13  13.  N I I 399 * l i n e intensity (upper trace) and di/dt i n the drive loop 1  The upper trace shows the photomultiplier signal and the lower trace i s the voltage signal from a sensor c o i l near to the drive loop, and represents di/dt i n the loop.  The entrance  optics focussed the cross-section of the discharge tube at Z = 0 , 5 cm observed from a d i r e c t i o n perpendicular to the Z-axis onto the monochromator entrance s l i t .  As expected the  l i g h t intensity fluctuates as a series of decaying pulses with a frequency twice the bank r i n g i n g frequency.  I t was decided  to measure the l i n e s h i f t s i n the f i r s t four main pulses of l i g h t that appear i n f i g ,  13,  As indicated i n Chapter I I , the f i r s t step i n making the measurement was to balance the voltage signals from the two photomultipliers with no wedges i n front of the e x i t Fig.  slit.  14 shows the sum (upper trace) and difference (lower trace)  of these signals f o r the best photomultiplier balance.  -36-  Ls As  2v/div 0.5v/dlv  >•  2 /a sec/div Pig. 14  Signal sum (upper trace) and s i g n a l difference (lower trace) f o r best photomultlplier balance  It was found that, In general, the completely balanced cond i t i o n (zero difference signal f o r a l l time) could not be obtained.  This was due to several e f f e c t s of both a systematic  and a random nature, which are l i s t e d below. A. Systematic Effects 1. D i s t o r t i o n of the signal by non-ideal e l e c t r o n i c apparatus (emitter followers, transmission  lines,  oscilloscope). 2. Different frequency c h a r a c t e r i s t i c s of the two emitter followers. 3. Electromagnetic  pick-up due to the bank discharge.  4. Systematic v a r i a t i o n of the d i s t r i b u t i o n function a(y) with time. B. Random Effects 1. Electronic and photoelectric noise of a l l kinds. 2. Random v a r i a t i o n of a(y) due to non-reproducibility of the plasma. The possible errors a r i s i n g from random effects oould be reduced by taking several i d e n t i c a l measurements and c a l culating the mean.  However, i n the case of the systematic  -37-  e f f e c t s i t was necessary to ensure that they were held constant during the entire measurement so that the deviation of the A s trace from zero without the wedges was i d e n t i c a l to the AS signal obtained with wedges, i n the absence of any s h i f t . Since these possible systematic errors may depend non-linearly on the magnitude of the photomultiplier signals, i t was r e quired that, the signals without wedges be reduced to approximately those with the wedges.  This was done by inserting a  homogeneous neutral density f i l t e r i n front of the entrance s l i t , with transmission t  « t . e  In t h i s way the intensity  with and without the wedges was kept approximately equal and any systematic e f f e c t s could be expected to remain equal. Under these conditions, the signal difference A s , recorded f  with the f i l t e r i n p o s i t i o n , m u l t i p l i e d by the factor t / t e  n  represents the amount of the signal difference A s , recorded w  with the wedges i n p o s i t i o n , which i s due to systematic, reproducible effects other than l i n e s h i f t s .  Equation (18)  i s now modified as  =  AX.  (c)  2s  (29)  .' b dx  TT  Doppler S h i f t Measurement A series of l i n e s h i f t measurements were taken by  observing the N II 399^ l i n e from two directions, a = 45°, 135  e  and from equations (26) and (29) the Doppler s h i f t was given by f t  c  As  dX  b dx  4 5 0  w  - |* As f t dX n b dx ^w e  135°  -38-  A t y p i c a l s e t of t r a c e s , from which the c a l c u l a t i o n s of Doppler s h i f t s were made, i s shown i n f i g . 15.  Ss 2v/div As 0.5v/dlv  (a) With 0.3 ND f i l t e r  F i g . 15  (b) With l i n e a r wedges  2JA sec/div  Es and AS traces from which Doppler s h i f t s are c a l c u l a t e d  The two observed c r o s s - s e c t i o n s o f the theta-pinch i n t e r s e c t e d a t Z = 2.5 cm.  Three separate runs under i d e n t i c a l c o n d i t i o n s  were taken to o b t a i n an accurate time r e s o l v e d a x i a l flow v e l o c i t y a t t h i s p o s i t i o n , and to check the r e p r o d u c i b i l i t y of the f i n a l r e s u l t s .  For each run a l l the measurements r e q u i r e d  were taken independently nine times, and f o r each p o i n t i n time the mean values of the voltage s i g n a l s were c a l c u l a t e d .  By  means of such r e p e t i t i v e measurement i t was p o s s i b l e to reduce the u n c e r t a i n t y due to random f l u c t u a t i o n s i n the s i g n a l . F i g . 16 shows the a x i a l v e l o c i t y as a f u n c t i o n of time f o r each of the three runs.  The e r r o r bars represent the standard  e r r o r of the mean, c a l c u l a t e d from the nine measurements. F i g . 17 shows the mean of the three runs p l o t t e d along w i t h the i n t e n s i t y of the l i g h t s i g n a l and d i / d t as measured by the sensor c o i l .  I t can be seen that there i s a c o r r e l a t i o n  -39v (km/sec)  -run  #3-  0.2  A  ill  0.1  _L_  10  L  5  5  8  J  10  1  1  12  L  14  16  J  I  .18  1  L  20  -5  22 d-5  -0.1 NB. Pig.  16  time t = 0 i s arbitrary but i s the same f o r a l l measurements  Doppler s h i f t s due to a x i a l motion as a function of time (Z = 2.5)  -40-  AX  (A)  0.3 0.2 0.1  1  0  J  I  L  10  •0.1  intensity (arb. u n i t s )  di/dt (arb. u n i t s )  4  1  Fig.  '/  17  '  8  1  fo  1'2\  1'4 '  1'6  '  .y  ' 20  '  2'2  Mean of Doppler s h i f t measurements, N I I 399^ l i n e i n t e n s i t y , and d i / d t of d r i v e loop  '  -41between the fluctuations of a l l three curves, the l i g h t i n tensity and v e l o c i t y fluctuate with double the bank frequency. The intensity peak i s delayed about 1.5  sec from the peak  di/dt, which i s the order of magnitude of the time expected for the p a r t i c l e s to t r a v e l from the place of generation to the region being observed. As a v e r i f i c a t i o n of these r e s u l t s a series of time of f l i g h t measurements was made.  The absolute intensity of  the N II 3994 l i n e was observed from d i r e c t i o n a = 90° f o r several Z-positions down the theta-pinch tube.  These traces  were a l l similar to the :sum (2s) curve of fig.15 except that the intensity pulses gradually change t h e i r shape while proceeding down the tube.  However, certain features could be de-  tected i n a l l the oscillograms and i t was assumed that these corresponded to luminous fronts t r a v e l l i n g down the tube. As expected these fronts were recorded at a l a t e r time f o r positions further away from the drive loop.  The complete set  of oscillograms i s reproduced i n Appendix I I .  The displacement  curve of the f i r s t intensity pulse i s plotted In fig.18. The straight l i n e i n t h i s figure represents the slope of the displacement curve at Z = 2.5 cm.  This indicates a front v e l -  ocity at t h i s p o s i t i o n of 6 ± 1 km/sec averaged over the time t = 6yusec to 8/csec.  This front i s l i k e l y to be a shock front  and a Mach number of 20 would be expected.  The p a r t i c l e v e l -  ocity should then be s l i g h t l y smaller than the front speed, and, indeed, the Doppler s h i f t measurments (see fig.17) y i e l d 4.8 km/sec f o r t h i s time i n t e r v a l .  It i s interesting to note  that Simkinson (ref. 10), i n h i s studies of a similar theta-pinch  -42-  /  t (//sec) F i g . 18  Time of f l i g h t measurements  device, observed shock fronts and measured a front v e l o c i t y of 6 km/sec.  (d) Stark S h i f t Measurements Time resolved Stark s h i f t measurements were made by observing the N I I 399^  l i n e perpendicular to the theta-pinch  tube axis at three p o s i t i o n s , Z = 1, 2,  3 cm.  the Stark s h i f t were calculated from equation  Fluctuations i n (18)  but since  no zero s h i f t reference source was a v a i l a b l e absolute s h i f t s were not measured.  F i g . 19 shows the r e s u l t s .  Note that the  three runs were performed so that they a l l had the same a r b i t rary zero s h i f t p o s i t i o n . The equations of conservation of mass and the ideal gas law permit an order of magnitude check on the Stark s h i f t results.  One can compare the t o t a l number of p a r t i c l e s inside  the theta-pinch volume with an estimate of the number of charged  Fig.  19  Stark s h i f t measurements (Z = 1,2,3 cm) as f u n c t i o n o f time; p l o t t e d w i t h l i n e i n t e n s i t y  -44-  p a r t l c l e s observed to t r a v e l through a cross-section at Z = 2.5  cm.  From application of the ideal gas law one finds  17 that p r i o r to discharge the theta-pinch tube contains 1.7 x 10 ' molecules/cm-^. Assume a mixture of N  2  and 0  2  molecules so that upon  d i s s o c i a t i o n one has 3.4 x 10 ? atoms/cm-^, and i f one has com1  plete (single) ionization then the electron density, n electrons/cm^.  = 3.4 x 10*?  In the oscillograms taken i n the time of f l i g h t  v e l o c i t y measurement (see Appendix II) i t appears that the section of the theta-pinch tube up to Z = 2 cm breaks down almost simultaneously.  If one assumes that t h i s includes the  entire region of the tube from the mid-plane of the loop to 2 em i n front of the loop then, since the loop i s 2 cm wide, the  2 t o t a l breakdown region has a volume of nr r = 1.25  x 3 cm, where  cm, the inside radius of the tube.  completely  If the gas i s  (singly) ionized then a maximum t o t a l number of 17  electrons of 3.4 x 10  f  2  x3xTTxr  18  = 5 i 10  1  electrons  would be produced. This figure may  be compared to the minimum number of  electrons observed experimentally to cross a given cross-section of the tube at Z > 2 cm.  A minimum value for the Stark s h i f t  can be estimated as equal to the magnitude of the fluctuations e  observed.  From f i g . 19, Z = 2 and 3 cm, t h i s was about 0.1  A.  This i s equivalent to assuming that the zero Stark s h i f t i s represented by the smallestAX_ observed s  (at Z = 2 cm, t = 9.0  which puts a lower l i m i t on a l l measured s h i f t since, for t h i s p a r t i c u l a r l i n e the Stark s h i f t i s always toward the red.  The  sec),  -45graph i n f i g . 18 represents breaks down.  the 16yu^sec period a f t e r the  gas  Prom f i g . 16 i t i s observed that the mean velocity-  over t h i s time range at Z = 2.5  cm was  5.3 km/sec.  The electron  density calculated by Griem (ref.7) which corresponds to a Stark s h i f t of 0.1 A i s 2 x lO^cm""^.  From t h i s data, i f one  assumes that the electron density i s equal to the ion density, then one can calculate the t o t a l number of ions crossing the 2.5  cross-section at Z =  cm.  18 n  i  A  v  i  t  «  2  x  1°  2  where A =* 1 cm  i s the estimated area on the axis through which  the electrons and ions escape the magnetic confinement i n the theta-pinch.  This estimated number of 2 x 10  18 ions can be  U8  accounted f o r by the i n i t i a l available t o t a l number of 5 x 10 Therefore,  .  t h i s check can v e r i f y that the Stark and Doppler s h i f t  r e s u l t s as reasonable and s e l f - c o n s i s t e n t . (e) Systematic Errors i n the Measurement As indicated i n section (b) of t h i s chapter a good check on the possible errors introduced by the wedges i s the control l i n e s h i f t experiment.  The r e s u l t s of t h i s t e s t ,  using the Na 5890 l i n e and s h i f t i n g the image by r o t a t i n g a piece of glass 12.6 mm i s shown i n f i g . 20. step i s represented  thick of index of r e f r a c t i o n 1.5177, The glass was  rotated i n steps and each  by a dot on the oscillogram.  to ensure that the t o t a l s h i f t represented  Care was  taken  by this oscillogram  exceeded the maximum width of the wedges that would be used to measure the N II l i n e s h i f t . t h i s experiment was  0.05 mm  The entrance s l i t width f o r  so that the l i n e at the exit plane  -46-  thickness o f glass,  d = 1.26 cm  index o f r e f r a c t i o n , n = 1.5177 angle o f r o t a t i o n , a = 0.0353 r a d .  x - d a ( l - J) - O.1517  Fig.  20 R e s u l t s from c o n t r o l  line shift  experiment  -*7-  had a t l e a s t that width.  I t was c a l c u l a t e d that the t o t a l l i n e  s h i f t seen i n fig.20 was 0.1517 mm, which i s approximately the width of the wedges expected to be used i n the l i n e s h i f t measurement.  I t can be seen that i n t h i s range the s i g n a l  d i f f e r e n c e ' s departs from the l i n e a r by a maximum of 0.06v and the s i g n a l sum, Is i s constant to w i t h i n 0.04v. AS/IS  has a p o s s i b l e e r r o r of  +  6  )  Therefore,  = ±7.5$ ±6.0% = ±13.5/*  However, t h i s i s an unduly pessemistic upper l i m i t f o r the p o s s i b l e e r r o r s i n c e , a l l o w i n g f o r the l i n e width used (0.05 mm), f i g . 20 represented a t o t a l wedge range of 0.2 mm.  I t was assumed  that as long as the N I I l i n e was p r o p e r l y centred on the -1  wedge and the s h i f t s were small (of the order of 10  0  A ) then  the e r r o r due to n o n - l i n e a r i t y and p o s i t i o n i n g of the wedges was < 10$.  This e r r o r could, i n p r i n c i p l e , be f u r t h e r reduced  by u s i n g a s m a l l s e c t i o n of a wider wedge. Prom f i g . 20 the c e n t r a l l i n e s h i f t experiment gave a value f o r the wedge slope b of ^00%/mm,  This i s s l i g h t l y  more than the wedge slope measured by the microdensitometer, but the value given by the c o n t r o l l i n e s h i f t technique i s expected to be more r e l i a b l e .  -48CHAPTER V CONCLUSIONS (a) General  Conclusions  I t was concluded that the double wedge technique i s an accurate method by which l i n e s h i f t s of a theta-pinch 0  plasma can be measured w i t h i n an u n c e r t a i n t y of about 0.015 A o f o r a 1.5 A wide l i n e .  From t h i s r e s u l t i t i s b e l i e v e d that  t h i s method i s more accurate than any time r e s o l v e d l i n e s h i f t measurement that has been reported i n the l i t e r a t u r e .  This  experiment showed t h a t , w i t h the N I I 399^ l i n e observed, Doppler s h i f t s and Stark s h i f t s were measured w i t h an uncerta i n t y corresponding  to v e l o c i t i e s of ± 1 km/sec and e l e c t r o n  density f l u c t u a t i o n s of ± 3x10  cm" . 0  With such r e s o l u t i o n  one could e a s i l y measure the r a t i o of Stark s h i f t s f o r d i f f e r e n t l i n e s to compare the p r e d i c t i o n s of the theory.  With improved  r e p r o d u c i b i l i t y of the source or w i t h phase s e n s i t i v e d e t e c t i o n techniques  f o r slowly v a r y i n g sources, s h i f t to width r a t i o s  of 1 0 " s h o u l d be measurable.  The Doppler and Stark s h i f t  measurements were i n good agreement w i t h time of f l i g h t measurements and d i d not v i o l a t e the conservation of mass.  They  were a l s o i n good agreement w i t h Simpkinson's r e s u l t s ( r e f . 10) and Indicated that a shock f r o n t was generated by the discharge. The value of t h i s technique i s obvious.  I t i s pos-  s i b l e to make time and space r e s o l v e d measurements of both mass v e l o c i t y and e l e c t r o n density without d i s t u r b i n g the plasma.  The knowledge of these parameters alone y i e l d impor-  tant information regarding the behaviour of any laboratory plasma.  -49-  (b) Future  Research  In f u t u r e work an accurate check of the double wedge technique should be made by observation of a shock tube plasma, f o r which the v e l o c i t y can be c a l c u l a t e d . A f u r t h e r Improvement of t h i s technique would be to construct a standard low density source of s u f f i c i e n t l y strong, w e l l - d e f i n e d l i n e s which have n e g l i g i b l y small Stark s h i f t . The low pressure AC discharge described by Minnhagen ( r e f . 9) would be appropriate.  Accurate absolute Stark s h i f t measurements  could then be made f o r any l i n e whose wavelength i s c l o s e to a s u i t a b l e l i n e i n the standard source (such that they both l a y w i t h i n the l i n e a r region of the wedges).  This would mean that  absolute e l e c t r o n d e n s i t i e s could be c a l c u l a t e d . A f u r t h e r a p p l i c a t i o n of t h i s technique could be Zeeman s h i f t measurements.  A s l i g h t m o d i f i c a t i o n of the appar-  atus would be necessary so"only one of the components would be observed.  -50-  APPENDIX I THE PHOTOGRAPHIC PROCESS AND THE LINEAR WEDGE In order to make a l i n e a r n e u t r a l density wedge i n the manner described i n Chapter I I I one must show that i f a photographic p l a t e i s exposed to a l i n e a r shadow then the developed p l a t e w i l l have a l i n e a r transmission g r a d i e n t . The most important c h a r a c t e r i s t i c of a photographic p l a t e or f i l m i s the density v s . l o g exposure r e l a t i o n , u s u a l l y known as the "H & D" curve ( r e f . 1 1 ) .  A t y p i c a l curve i s shown  i n f i g . 21. The l i n e a r p o r t i o n o f the H & D curve, which has  log F i g . 21  exposure  T y p i c a l H & D curve  constant slope y» covers most of the u s e f u l range o f exposures f o r ordinary f i l m s .  The value o f y and, indeed, the shape of  the curve i n general, depends on the development, i . e . type of developer and length of development. In the region of constant y the H & D curve can be expressed by the equation  -51(31)  D = k + y log E  (32)  * log (E^kj) where:  k = log k^ —  the density axis intercept of the extended l i n e a r portion of the H„& D curve.  E i s the exposure —  E = I t , where I i s the l i g h t i n tensity and t i s the length of time f i l m i s exposed.  D i s the density of the exposed f i l m . Now,  the transmission of the exposed f i l m i s given by (33)  D = -log T  (34)  therefore, - l o g T = log ( E k ) Y  1  and i t follows from equations  (33) and  T = (E^, )  (34) that (35)  _ 1  Since f o r a l i n e a r shadow E (X x i t i s seen that the transmission of the wedge can never be l i n e a r , but w i l l always vary as For c e r t a i n films, however, the transmission-distance w i l l be l i n e a r to a very close  E~ . Y  curve  approximation.  In order to make the wedges used i n this experiment I l f o r d Special Lantern Contrasty plates were used.  This plate  i s described by I l f o r d as a "fast, non-colour-sensitive, f i n e grain plate.;"  Its H & D curve i s shown i n f l g . 2 2 , ( r e f .  From t h i s f i g u r e equation T = (1.8  12).  (35) becomes x 10 )E" * 6  2  5  This equation i s plotted i n fig.23 f o r transmissions from 10% to 90%.  It can be seen that t h i s p l o t i s l i n e a r from about 25$  to 80% transmission.  This i s i n good agreement with the range  -52of l i n e a r i t y measured i n f i g . 12 f o r the wedge that was used i n the apparatus.  _j  1  1  f  ^  1 T 0 2 . 0 3 7 o log  exposure  /  F i g . 22  H & D curve f o r I l f o r d Special Lantern Contrasty  transmission A  exposure F i g . 23  (arb. units)  Calculated transmission-exposure curve f o r I l f o r d Special Lantern Contrasty  -53APPENDIX II TIME OF FLIGHT VELOCITY MEASUREMENTS As a check on the v e l o c i t i e s recorded by the l i n e s h i f t measurements a series of time of f l i g h t measurements were made and the v e l o c i t y of the luminous front was calculated as a function of i t s p o s i t i o n i n the theta-pinch tube.  To do t h i s  the tube was observed from a d i r e c t i o n perpendicular to i t s axis and the oscillogram was recorded representing the time resolved N II 399^ l i n e intensity at various distances, Z, down the axis from the drive loop.  Care was taken to s t a b i l i z e  the oscilloscope trigger and horizontal s e t t i n g so that d i f f e r e n t oscillograms had the same time zero. taken.  F i g . 24 shows the traces  Since the second peak i s a c l e a r l y marked feature of  a l l these oscillograms, i t s p o s i t i o n i n time as a function of Z was measured and p l o t t e d i n f i g . 25.  The slope of the curve  Z(cm)  Fig. 25  Time of f l i g h t measurements  represents the v e l o c i t y of the luminous front which i s represented by the second intensity pulse.  The slope of the dotted l i n e i s  ''  .— •  MSP  T Z • 0.5cm  Z • 2.5cm  Z - 1.0cm  Z = 3.0cm  «••••••••i n  Z = 1.5cm  l l l l l l i ^ l i n g  Z = 2.0cm  Z = 3.5cm  Z = 4.0cm  2 /<sec/dlv Fig.  24  Line Intensity traces f o r time of f l i g h t measurements  -55about 40 km/sec, which i s an unexpectedly f o r a theta-pinch.  high a x i a l v e l o c i t y  It i s quite l i k e l y , however, that the  breakdown of the gas occurs almost simultaneously f o r regions of the tube very close to the drive loop (Z < 2cm.).  Of  s p e c i a l interest i s the v e l o c i t y at the point Z = 2.5 cm since t h i s i s where the l i n e s h i f t v e l o c i t y measurements were taken. The s o l i d l i n e represents t h i s v e l o c i t y , averaged over 2 - Z - 3 and 6 - t - 8 and shows a slope of 6±1 km/sec.  -56-  REFERENCES 1  Drobowshevsky, T7T3 6, 145, (I963)  2  Keilhacker, et a l .  3  Hirschberg,  Plasma Physics and Controlled Nuclear Fusion Research 1_, 315, (I966)  Project Matterhorn Rept., Matt-Q-21, 322 (1964)  4  Burgess and Cooper,  5  Hubner,  6  Ahlborn and Barnard,  7  Griem,  8  Prescott,  9  Minnhagen,  J . S c i . Instrum. 42, 829, (1965)  Z. Naturforsch, l^A, 1111, (1964) AIAA J . II36,  (-966)  Plasma Spectroscopy, McGraw-Hill (1964) Nuc. Instrum. and Methods, 22, 173,  (1966)  J . of Research (National Bureau of Standards) 68C, 237,  (1964)  10  Simpkinson, PhD Thesis, UBC Plasma Group, (1964)  11  Sawyer,  12  I l f o r d Ltd. Data Sheet (Special Lantern Contrasty)  Experimental Spectroscopy, Dover ( 1 9 ^ )  

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